Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods [1 ed.] 9789811569890, 9789811569906

Table of contents :
Preface
Contents
1 Introduction
1.1 Basic Concepts of Rendezvous and Docking
1.2 Phases of Rendezvous and Docking
1.2.1 Before-Launch Phase
1.2.2 Launch Phase
1.2.3 Far-Range Rendezvous Phase
1.2.4 Near-Range Autonomous Control Phase
1.2.5 Docking Phase
1.2.6 Complex Operation Phase
1.2.7 Departure Phase
1.2.8 Reentry Phase
1.3 Overview of Rendezvous and Docking
1.3.1 Introduction
1.3.2 Classification of Rendezvous and Docking Flight Missions
1.3.3 Typical Rendezvous and Docking Flight Mission
1.3.4 Future Development Trend of Rendezvous and Docking Technology
1.4 Guidance Navigation and Control for RVD
1.4.1 Sensors
1.4.2 Actuators
1.4.3 Controllers
References
2 Rendezvous Kinematics and Dynamics
2.1 Reference Frames
2.2 Orbit Dynamics
2.2.1 Two-Body Problem
2.2.2 Orbital Elements
2.2.3 Orbital Perturbation Equations
2.2.4 Perturbation Acceleration
2.3 Attitude Kinematics and Dynamics
2.3.1 Attitude Kinematics
2.3.2 Attitude Dynamics
2.3.3 Spacial Environmental Torques
2.4 Relative Motion
2.4.1 Relative Motion in Circular Orbital Frame
2.4.2 Relative Motion in Elliptical Orbital Frame
2.4.3 Relative Motion in the Line of Sight Frame
2.4.4 Relative Motion in Cylindrical Frame
2.5 Relative Attitude
2.5.1 Relative Attitude Dynamics for Stable Target
2.5.2 Relative Attitude Dynamics for Rotating Target
2.6 Selection of Reference Frames and Dynamics Equation
References
3 Navigation Method and Scheme Design for Rendezvous and Docking
3.1 Introduction
3.1.1 Relative Measurement System
3.1.2 Relative Navigation Algorithms
3.2 Sensors
3.2.1 Inertial Measurement Sensors
3.2.2 Satellite Navigation Equipments
3.2.3 Microwave Radar
3.2.4 Laser Radar
3.2.5 Camera-Type Rendezvous and Docking Sensor
3.3 Filtering Method for Navigation
3.3.1 Theory of Bayesian Estimation
3.3.2 Linear Minimum Variance Estimation
3.3.3 Kalman Filter
3.3.4 Deterministic Sampling Filter
3.3.5 Particle Filter
3.4 Scheme Design of Navigation
3.4.1 Introduction
3.4.2 Orbital Parameters Estimation
3.4.3 Relative Position Parameters Estimation
3.4.4 Relative Attitude Parameter Estimation
3.4.5 Relative Motion and Attitude United Estimation
3.5 Key Points of Navigation System Design
References
4 Guidance Method and Schematic Design for Rendezvous and Docking
4.1 Introduction
4.2 Rendezvous by Orbital Maneuver
4.2.1 Hohmann Rendezvous
4.2.2 Lambert Rendezvous
4.2.3 Orbit Elements Based Rendezvous
4.3 Multiple-impulse Optimal Rendezvous
4.3.1 Solutions of Optimal Impulsive Rendezvous Between Two Near Circular Orbits
4.3.2 Solutions of Optimal Impulsive Rendezvous of Ellipse-to-Circle Orbits
4.3.3 Guidance Law from Multiple-impulse Optimal Rendezvous
4.4 CW Guidance
4.4.1 Two-Impulse CW Guidance Law
4.4.2 Rendezvous Time and Fuel Consumption
4.4.3 Selection of Rendezvous Time
4.5 Light of Slight Guidance
4.5.1 Guidance Law
4.5.2 Stability Analysis
4.6 Scheme Design for Far Range Rendezvous Phase
4.6.1 2–3 Day Guidance Scheme
4.6.2 Short Rendezvous Guidance Scheme
4.7 Scheme Design for Homing Phase
4.7.1 Determination of Initial and Final States, and Rendezvous Time
4.7.2 Multiple-impulse Optimal Guidance Law
4.7.3 Analysis of Guidance Accuracy
4.7.4 Simulations
4.8 Scheme Design for Closing Phase
4.8.1 CW Guidance Law
4.8.2 Light of Sight Guidance Law
4.8.3 United Guidance LAW
4.8.4 Simulations
4.9 Key Points of Guidance System Design
References
5 Automatic Control Method and Scheme Design for Rendezvous and Docking
5.1 Introduction
5.2 Characteristic Modeling for the Plant to be Controlled
5.2.1 Important Lemma
5.2.2 Characteristic Model
5.2.3 Simulation Verification
5.3 Intelligent Adaptive Control Method Based on Characteristic Model
5.3.1 Golden Section Adaptive Control
5.3.2 Logical Differential Control
5.3.3 Golden Section Phase Plane Control
5.3.4 Logical Differential Phase Plane Control
5.4 Control Scheme Design for Rendezvous and Docking
5.4.1 Attitude Stable Control in Orbit Maneuver
5.4.2 Accurate Six-Degree-of-Freedom Relative Position and Attitude Control
5.4.3 Relative Position Control at the Hold Points for Long Period
5.5 Key Points of Control System Design
References
6 Manual Control Method and Scheme Design for Rendezvous and Docking
6.1 Introduction
6.2 Measurement Principle and Scheme Design
6.2.1 Sensors for Manual Rendezvous and Docking
6.2.2 Relative Motion State Determination
6.2.3 Display of Image and Data
6.3 Control Method and Scheme Design
6.3.1 Control Handles
6.3.2 Control Method of Relative Motion
6.4 Operation Method
6.4.1 Approaching Velocity Control
6.4.2 Lateral Position and Attitude Control
6.4.3 Simulations
References
7 Theory and Design of Thruster Configuration and Control Allocation
7.1 Introduction
7.1.1 Thruster Configuration for Typical RVD Missions
7.1.2 Control Allocation
7.1.3 Thruster Configuration Design
7.2 Problem Description
7.2.1 Thruster Configuration Matrix
7.2.2 Mathematical Model of Thruster Control Allocation
7.3 Theory and Design of Thruster Control Allocation
7.3.1 Thruster Control Allocation Method
7.3.2 Performance Analysis
7.3.3 Synthetical Design of Thruster Control Allocation Algorithm
7.4 Theory and Design of Thruster Configuration
7.4.1 Performance Requirements
7.4.2 Performance Analysis
7.4.3 Design Method
References
8 Method and Scheme Design of Safety for Rendezvous and Docking
8.1 Introduction
8.2 Safe Trajectory Design with Rectangular Keep-Out-Zone
8.2.1 Characteristics of the Free-Flying Trajectories
8.2.2 Characteristic Points for Safe Trajectory Design
8.2.3 Safe Trajectory Design for Homing Phase and Closing Phase
8.2.4 Safe Trajectory Design for Fly-Around Phase
8.2.5 Simulations
8.3 Trajectory Safety Judgement with Rectangular Keep-Out-Zone
8.3.1 Extremune Characteristic Points
8.3.2 Intersection Characteristic Points
8.3.3 Judgment Method of Trajectory Safety
8.3.4 Simulations
8.4 Design Method of Trajectory Safety
8.4.1 Passive Trajectory Protection
8.4.2 Active Trajectory Protection
8.5 Safe Trajectory Estimation and Collision with Safety Belt for Rendezvous and Docking
8.5.1 Safety Belts of Trajectory
8.5.2 Impulse Based Safety Belts and Guidance Law
8.5.3 Simulations
8.6 Safety Design for Rendezvous Missions
8.6.1 System Design
8.6.2 Flight Scheme Design
References
9 Simulation Verification of Rendezvous and Docking
9.1 Introduction
9.1.1 Classification of Simulation Systems
9.1.2 Overview of Simulation Technology
9.2 Mathematical Simulation
9.2.1 System Composition
9.2.2 Errors Sources
9.2.3 Simulation Requirements
9.2.4 Simulation Conditions
9.2.5 Simulators
9.3 Hardware-in-the-Loop Simulation
9.3.1 9-Degree Hardware-in-the-Loop Simulation System
9.3.2 Automatic Control Verification Through Hardware-in-the-Loop Simulation System
9.3.3 Manual Control Verification Through Hardware-in-the-Loop Simulation System
References
10 RVD Verification in Orbit Flight
10.1 Introduction
10.2 Automatic Control System of Shenzhou Spacecraft
10.2.1 Composition and Flight Phases
10.2.2 Scheme Design of GNC
10.2.3 Accuracy Analysis of GNC
10.2.4 Flight Verification of Shenzhou Spacecraft
10.2.5 Summary
10.3 Manual Control System of Shenzhou Spacecraft
10.3.1 Manual Control System Composition
10.3.2 Determination of Relative Movement States
10.3.3 Control Strategy
10.3.4 Operation Strategy
10.3.5 Flight Verification of Shenzhou Spacecraft
10.3.6 Summary
10.4 Prospects
References

Citation preview

Yongchun Xie Changqing Chen Tao Liu Min Wang

Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods

Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods

Yongchun Xie Changqing Chen Tao Liu Min Wang •

•

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Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods

123

Yongchun Xie Beijing Institute of Control Engineering Beijing, China

Changqing Chen Beijing Institute of Control Engineering Beijing, China

Tao Liu Beijing Institute of Control Engineering Beijing, China

Min Wang Beijing Institute of Control Engineering Beijing, China

The book is funded by B & R Book Program. ISBN 978-981-15-6989-0 ISBN 978-981-15-6990-6 https://doi.org/10.1007/978-981-15-6990-6

(eBook)

Jointly published with National Defense Industry Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: National Defense Industry Press. © National Defense Industry Press 2021 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The rendezvous and docking of the spacecraft is a key step in the completion of complex space missions such as space station assembly, on-orbit service of the spacecraft, sampling and return of celestial objects, and manned space exploration. The guidance, navigation, and control technology is one of the core technologies for rendezvous and docking mission. The guidance of the rendezvous and docking is to plan the reasonable trajectory of the rendezvous with consideration of the fuel consumption and the time of the rendezvous, and to give the velocity increment for the maneuver. The navigation of the rendezvous and docking is to estimate the translational parameters and rotational parameters required for the guidance and control process, through using the measurement information of sensors, combining the motion model of the spacecraft, and designing filtering algorithm. The control of the rendezvous and docking is to execute the control force and torque on the spacecraft through the engine and other actuators, which are calculated by the guidance and attitude control law. The final goal of the rendezvous guidance, navigation, and control is to meet the initial conditions for docking, which include relative position, velocity, attitude angle, and angle rate of the chaser with respective to the target. The authors of the book have engaged in the research of rendezvous guidance, navigation, and control in the last 20 years, and have participated in the development of China’s manned space project and lunar exploration. This book is the result of the authors’ achievements in the theory research and engineering design in this field. Based on these results, the rendezvous guidance, navigation, and control scheme has been successfully applied to the respective rendezvous mission of “Shenzhou-8”, “Shenzhou-9”, “Shenzhou-10” spacecraft with “Tiangong” space laboratory. These results also provide great reference to China’s future space exploration missions such as manned lunar mission, deep space exploration, and on-orbit service flight. There are totally 10 chapters in the book: Chapter 1 summarizes the basic concepts and flight scheme of rendezvous and docking. Chapter 2 introduces the kinematics and dynamics of rendezvous and docking. Chapter 3 introduces the navigation method and scheme design for rendezvous and docking. Chapter 4 introduces the guidance method and scheme design for rendezvous and v

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docking. Chapter 5 introduces the automatic control method and scheme design for rendezvous and docking. Chapter 6 introduces the manned control method and scheme design for rendezvous and docking. Chapter 7 introduces the theory and design of thruster configuration and control allocation. Chapter 8 introduces the safety design for rendezvous and docking. Chapter 9 introduces the ground simulation verification for rendezvous and docking. Chapter 10 introduces the on-orbit flight verification for rendezvous and docking. The book is planned by Yongchun Xie, Changqing Chen, Tao Liu, and Min Wang. Yongchun Xie is responsible for the writing of Chaps. 5–6 and 10, and participates in the writing of Chaps. 1–4 and 9; Changqing Chen is in charge of Chaps. 2, 4 and 8. Tao Liu is responsible for the writing of Chaps. 1 and 3; Min Wang is in charge of Chaps. 7 and 9. In addition, Haixia Hu, Hao Zhang, Yong Wang, Xinxin Yu, Yong Hu, Jingjing Liang, Lei Shi, Ning Tang, Youmei Xin,Ying Wang, etc. participated in the research work of relevant content of the book and gave strong support to the writing of the book. During the writing of the book, we received strong support and help from the relevant leaders, experts, and colleagues from China Academy of Space Technology, especially the strong support of the “Shenzhou” manned spacecraft development team. We would like to thank academician Hongxin Wu for his long-time encouragement and advice. We are indebted to academicians including Shancheng Tu, Jianping Zhou, Faren Qi and professors including Baohua Yang, Bainan Zhang, Zhi Shang, Liangdong Liu, Chengqi Sun, Nanhua Wang, Zugui Chen, Jun Hu, Guqing Luo for their help and support to authors in many years of research work; We would like to thank Prof. Duzhou Zhang and Prof. Li Yuan for their valuable comments on this book. After the completion of the draft, Prof. Guojian Tang from National University of Defense Technology made a detailed review of the draft and gave many valuable comments, which we sincerely appreciate. The study of the book has been supported by the National Key Basic Research and Development Program of China (grant number: 2013CB733100). The publication of the book is funded by the National Defense Science and Technology Book Publishing Fund, and Tianjin Key Laboratory of Microgravity and Hypogravity Environment Simulation Technology. National Defence Industry Press of China gives strong support to the publication of the book. Here, the authors are deeply grateful. Due to the limited level of the authors, the book may unavoidably have inappropriate and mistake contents. Readers are welcomed to address us to make corrections. Beijing, China

Yongchun Xie Changqing Chen Tao Liu Min Wang

Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Concepts of Rendezvous and Docking . . . . . . . . . . 1.2 Phases of Rendezvous and Docking . . . . . . . . . . . . . . . . . 1.2.1 Before-Launch Phase . . . . . . . . . . . . . . . . . . . . . 1.2.2 Launch Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Far-Range Rendezvous Phase . . . . . . . . . . . . . . . 1.2.4 Near-Range Autonomous Control Phase . . . . . . . 1.2.5 Docking Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Complex Operation Phase . . . . . . . . . . . . . . . . . . 1.2.7 Departure Phase . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Reentry Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of Rendezvous and Docking . . . . . . . . . . . . . . . 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Classification of Rendezvous and Docking Flight Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Typical Rendezvous and Docking Flight Mission . 1.3.4 Future Development Trend of Rendezvous and Docking Technology . . . . . . . . . . . . . . . . . . 1.4 Guidance Navigation and Control for RVD . . . . . . . . . . . 1.4.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rendezvous Kinematics and Dynamics 2.1 Reference Frames . . . . . . . . . . . . 2.2 Orbit Dynamics . . . . . . . . . . . . . 2.2.1 Two-Body Problem . . . . 2.2.2 Orbital Elements . . . . . .

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2.2.3 Orbital Perturbation Equations . . . . . . . . . . . . . . 2.2.4 Perturbation Acceleration . . . . . . . . . . . . . . . . . 2.3 Attitude Kinematics and Dynamics . . . . . . . . . . . . . . . . 2.3.1 Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . 2.3.2 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Spacial Environmental Torques . . . . . . . . . . . . . 2.4 Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Relative Motion in Circular Orbital Frame . . . . . 2.4.2 Relative Motion in Elliptical Orbital Frame . . . . 2.4.3 Relative Motion in the Line of Sight Frame . . . . 2.4.4 Relative Motion in Cylindrical Frame . . . . . . . . 2.5 Relative Attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Relative Attitude Dynamics for Stable Target . . . 2.5.2 Relative Attitude Dynamics for Rotating Target . 2.6 Selection of Reference Frames and Dynamics Equation . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Navigation Method and Scheme Design for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Relative Measurement System . . . . . . . . . . . . . . 3.1.2 Relative Navigation Algorithms . . . . . . . . . . . . . 3.2 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Inertial Measurement Sensors . . . . . . . . . . . . . . 3.2.2 Satellite Navigation Equipments . . . . . . . . . . . . 3.2.3 Microwave Radar . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Laser Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Camera-Type Rendezvous and Docking Sensor . 3.3 Filtering Method for Navigation . . . . . . . . . . . . . . . . . . 3.3.1 Theory of Bayesian Estimation . . . . . . . . . . . . . 3.3.2 Linear Minimum Variance Estimation . . . . . . . . 3.3.3 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Deterministic Sampling Filter . . . . . . . . . . . . . . 3.3.5 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Scheme Design of Navigation . . . . . . . . . . . . . . . . . . . . 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Orbital Parameters Estimation . . . . . . . . . . . . . . 3.4.3 Relative Position Parameters Estimation . . . . . . . 3.4.4 Relative Attitude Parameter Estimation . . . . . . . 3.4.5 Relative Motion and Attitude United Estimation 3.5 Key Points of Navigation System Design . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Guidance Method and Schematic Design for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Rendezvous by Orbital Maneuver . . . . . . . . . . . . . . . . 4.2.1 Hohmann Rendezvous . . . . . . . . . . . . . . . . . . 4.2.2 Lambert Rendezvous . . . . . . . . . . . . . . . . . . . 4.2.3 Orbit Elements Based Rendezvous . . . . . . . . . 4.3 Multiple-impulse Optimal Rendezvous . . . . . . . . . . . . . 4.3.1 Solutions of Optimal Impulsive Rendezvous Between Two Near Circular Orbits . . . . . . . . . 4.3.2 Solutions of Optimal Impulsive Rendezvous of Ellipse-to-Circle Orbits . . . . . . . . . . . . . . . . 4.3.3 Guidance Law from Multiple-impulse Optimal Rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 CW Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Two-Impulse CW Guidance Law . . . . . . . . . . 4.4.2 Rendezvous Time and Fuel Consumption . . . . 4.4.3 Selection of Rendezvous Time . . . . . . . . . . . . 4.5 Light of Slight Guidance . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Guidance Law . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 4.6 Scheme Design for Far Range Rendezvous Phase . . . . . 4.6.1 2–3 Day Guidance Scheme . . . . . . . . . . . . . . . 4.6.2 Short Rendezvous Guidance Scheme . . . . . . . . 4.7 Scheme Design for Homing Phase . . . . . . . . . . . . . . . . 4.7.1 Determination of Initial and Final States, and Rendezvous Time . . . . . . . . . . . . . . . . . . 4.7.2 Multiple-impulse Optimal Guidance Law . . . . . 4.7.3 Analysis of Guidance Accuracy . . . . . . . . . . . 4.7.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Scheme Design for Closing Phase . . . . . . . . . . . . . . . . 4.8.1 CW Guidance Law . . . . . . . . . . . . . . . . . . . . . 4.8.2 Light of Sight Guidance Law . . . . . . . . . . . . . 4.8.3 United Guidance LAW . . . . . . . . . . . . . . . . . . 4.8.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Key Points of Guidance System Design . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Automatic Control Method and Scheme Design for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5.2 Characteristic Modeling for the Plant to be Controlled . . . . . . . 220

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Contents

5.2.1 Important Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Characteristic Model . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Simulation Verification . . . . . . . . . . . . . . . . . . . . . . 5.3 Intelligent Adaptive Control Method Based on Characteristic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Golden Section Adaptive Control . . . . . . . . . . . . . . 5.3.2 Logical Differential Control . . . . . . . . . . . . . . . . . . 5.3.3 Golden Section Phase Plane Control . . . . . . . . . . . . 5.3.4 Logical Differential Phase Plane Control . . . . . . . . . 5.4 Control Scheme Design for Rendezvous and Docking . . . . . . 5.4.1 Attitude Stable Control in Orbit Maneuver . . . . . . . . 5.4.2 Accurate Six-Degree-of-Freedom Relative Position and Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Relative Position Control at the Hold Points for Long Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Key Points of Control System Design . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Manual Control Method and Scheme Design for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Measurement Principle and Scheme Design . . . . . . . . . . . 6.2.1 Sensors for Manual Rendezvous and Docking . . . 6.2.2 Relative Motion State Determination . . . . . . . . . . 6.2.3 Display of Image and Data . . . . . . . . . . . . . . . . . 6.3 Control Method and Scheme Design . . . . . . . . . . . . . . . . 6.3.1 Control Handles . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Control Method of Relative Motion . . . . . . . . . . . 6.4 Operation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Approaching Velocity Control . . . . . . . . . . . . . . . 6.4.2 Lateral Position and Attitude Control . . . . . . . . . . 6.4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory and Design of Thruster Configuration and Control Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Thruster Configuration for Typical RVD Missions 7.1.2 Control Allocation . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Thruster Configuration Design . . . . . . . . . . . . . . 7.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Thruster Configuration Matrix . . . . . . . . . . . . . . . 7.2.2 Mathematical Model of Thruster Control Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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and Design of Thruster Control Allocation . . . . . . Thruster Control Allocation Method . . . . . . . . . . Performance Analysis . . . . . . . . . . . . . . . . . . . . . Synthetical Design of Thruster Control Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Theory and Design of Thruster Configuration . . . . . . . . . . 7.4.1 Performance Requirements . . . . . . . . . . . . . . . . . 7.4.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . 7.4.3 Design Method . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

9

Theory 7.3.1 7.3.2 7.3.3

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Method and Scheme Design of Safety for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Safe Trajectory Design with Rectangular Keep-Out-Zone . 8.2.1 Characteristics of the Free-Flying Trajectories . . . 8.2.2 Characteristic Points for Safe Trajectory Design . . 8.2.3 Safe Trajectory Design for Homing Phase and Closing Phase . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Safe Trajectory Design for Fly-Around Phase . . . 8.2.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Trajectory Safety Judgement with Rectangular Keep-Out-Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Extremune Characteristic Points . . . . . . . . . . . . . 8.3.2 Intersection Characteristic Points . . . . . . . . . . . . . 8.3.3 Judgment Method of Trajectory Safety . . . . . . . . 8.3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Design Method of Trajectory Safety . . . . . . . . . . . . . . . . 8.4.1 Passive Trajectory Protection . . . . . . . . . . . . . . . 8.4.2 Active Trajectory Protection . . . . . . . . . . . . . . . . 8.5 Safe Trajectory Estimation and Collision with Safety Belt for Rendezvous and Docking . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Safety Belts of Trajectory . . . . . . . . . . . . . . . . . . 8.5.2 Impulse Based Safety Belts and Guidance Law . . 8.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Safety Design for Rendezvous Missions . . . . . . . . . . . . . . 8.6.1 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Flight Scheme Design . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Simulation Verification of Rendezvous and Docking . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Classification of Simulation Systems . . . 9.1.2 Overview of Simulation Technology . . .

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9.2

Mathematical Simulation . . . . . . . . . . . . . . . . . . . . 9.2.1 System Composition . . . . . . . . . . . . . . . . . 9.2.2 Errors Sources . . . . . . . . . . . . . . . . . . . . . 9.2.3 Simulation Requirements . . . . . . . . . . . . . 9.2.4 Simulation Conditions . . . . . . . . . . . . . . . 9.2.5 Simulators . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Hardware-in-the-Loop Simulation . . . . . . . . . . . . . 9.3.1 9-Degree Hardware-in-the-Loop Simulation System . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Automatic Control Verification Through Hardware-in-the-Loop Simulation System . 9.3.3 Manual Control Verification Through Hardware-in-the-Loop Simulation System . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 RVD Verification in Orbit Flight . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Automatic Control System of Shenzhou Spacecraft . . 10.2.1 Composition and Flight Phases . . . . . . . . . . 10.2.2 Scheme Design of GNC . . . . . . . . . . . . . . . 10.2.3 Accuracy Analysis of GNC . . . . . . . . . . . . . 10.2.4 Flight Verification of Shenzhou Spacecraft . 10.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Manual Control System of Shenzhou Spacecraft . . . . 10.3.1 Manual Control System Composition . . . . . 10.3.2 Determination of Relative Movement States . 10.3.3 Control Strategy . . . . . . . . . . . . . . . . . . . . . 10.3.4 Operation Strategy . . . . . . . . . . . . . . . . . . . 10.3.5 Flight Verification of Shenzhou Spacecraft . 10.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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479 479 479 480 483 487 488 489 489 490 491 491 492 493 494 495 495

Chapter 1

Introduction

1.1 Basic Concepts of Rendezvous and Docking Spacecraft rendezvous and docking, a.k.a. space rendezvous and docking, refers to the motion of two spacecraft in orbit approaching each other with the desired position, speed, and time (i.e., rendezvous) and the structural connection of the two spacecraft after the attitude alignment and final approach (i.e., docking) [1]. The two spacecraft in rendezvous and docking are called the chaser spacecraft (i.e., the chaser) and the target spacecraft (i.e. the target), respectively. To realize rendezvous and docking, the chaser adjusts its attitude and position with respect to the target throughout the process. Space rendezvous and docking are highly complex technologies and can be extensively applied in various aerospace tasks, such as space station construction, astronaut rotation in space stations, space transportation, in-orbit service, celestial sample and return, manned lunar landings, and manned missions to Mars. Space rendezvous and docking technologies integrate a variety of aerospace technologies and involve a number of different technical fields. Guidance, navigation, and control (GNC) of rendezvous and docking is a key technology among these. Guidance refers to the algorithm that is designed to plan the trajectories and velocity commands during the orbital transfer by considering the fuel consumption, rendezvous time, and/or other mission requirements. The navigation utilizes a filter to estimate the translational and rotational motion parameters (which are required by the guidance and control) based on measurement information from the sensors and spacecraft kinematics or dynamics model. The control is the algorithm that generates force and control torque to manipulate the spacecraft (generally the chaser) based on the guidance and attitude control requirements. The ultimate goal of the GNC in rendezvous and docking is to satisfy the initial conditions required for docking mechanisms, including the relative position, velocity, attitude angle and angular rate of the chaser with respective to the target.

© National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_1

1

2

1 Introduction

1.2 Phases of Rendezvous and Docking For a complete rendezvous and docking mission, the flight process generally consists of a series of phases: the launch phase, the far-range rendezvous phase, the nearrange autonomous control phase, the docking phase, the assembly operation phase, the departure phase, and the re-entry phase. The rendezvous and docking between China’s Shenzhou manned spacecraft and the target vehicle in low-earth orbit is a representative space rendezvous and docking mission. The rendezvous and docking flight stages are shown in Fig. 1.1 [2]. The tasks and requirements of the GNC system of the rendezvous and docking flight are given as follows.

1.2.1 Before-Launch Phase The before-launch phase refers to the stage in which the GNC system of the chaser is powered on before the rocket is launched. The primary task of the GNC system in this phase is to adjust the strapdown inertial navigation system (SINS) and provide the initial position and attitude parameters for navigation in the launch phase.

1.2.2 Launch Phase

Range

10km

Contact

Docking

Departure

The launch phase begins at the time when the launch vehicle takes off and ends when the chaser enters the desired orbit. The mission of the launch phase is to put the chaser into a predetermined orbit, with the orbit parameters (i.e., apogee, perigee, phase angle difference with the target, orbital inclination, and right ascension of the ascending intersection point) meeting certain constraints. The primary task of the GNC system in this phase is to perform strapdown inertial navigation and provide the position, velocity, and attitude of the chaser with respect to the inertial frame in real-time.

Assembly Operation Phase

P4

P3

P2

P1

A0

30m

140m

400m

5km

52km

Closing Phase Reentry Phase

Approach Phase Near-Range Autonomous Control Phase

Departure Phase

Homing Phase

Insertion

Launch

Far-Range Rendezvous Launch Phase Phase

Dock ing Phase

Fig. 1.1 Rendezvous and docking flight stages of China’s Shenzhou spacecraft

Before-Launch Phase

1.2 Phases of Rendezvous and Docking

3

1.2.3 Far-Range Rendezvous Phase In the far-range rendezvous phase, the chaser is transferred from the initial orbit into the predetermined orbit in which the near-range autonomous control will be performed, and the orbit should meet certain relative motion constraints. The main task of the far-range rendezvous phase includes raising the chaser’s orbit height, adjusting the differences between the phase angles of the two spacecraft, reducing the relative distance, and eliminating the orbit plane deviation. In the farrange rendezvous phase, the orbit transfer of the chaser can be realized either by the telemetering and telecontrol system (in which the determination of the orbit and orbit control are performed by the ground station), or autonomously, by the onboard computer (in which the determination of the orbit and control strategies are fulfilled by the onboard computer).

1.2.4 Near-Range Autonomous Control Phase In the near-range autonomous control phase, the chaser performs relative navigation and then autonomously controls its attitude and orbit such that it can approach and dock with the target. Unlike the other phases, the characteristics of relative motion between the chaser and target should be taken into full consideration by the GNC in this phase, and the performance requirement is much stricter than the others. Therefore, GNC is a vital technology for the research and engineering design of space rendezvous and docking. The near-range autonomous control phase can be further subdivided into a homing phase, approach phase, closing phase, and fly-around phase based on different task functions and performance requirements. 1. Homing Phase In the homing phase, the chaser begins to perform the relative navigation and control at the end of the far-range phase until it reaches the homing terminal aiming point. During the initial flight in the near-range autonomous control stage, the chaser must eliminate the track error of the far-range rendezvous stage and execute orbit guidance and control in order to approach the target and reduce its approach speed. The initial position in the homing phase is the transition position between the farrange rendezvous phase and the near-range autonomous control phase, and it is forms one of the key issues in the design of the flight motion. The selection of this position primarily depends on the measuring range of the relative measurement sensor, the precision of the far-range rendezvous control, and the position and velocity requirements of the near-range autonomous control phase. In the far-range rendezvous phase, the chaser should be controlled such that the target is in the measuring range of the relative measurement sensor, ensuring that the relative orbit configuration can meet the relative measurement sensor to capture the target within the control error range. At the same time, the initial orbit of the homing phase should improve the initial

4

1 Introduction

relative motion conditions for the near autonomous control phase as far as possible. In addition, the design of the initial position of the homing phase is also related to the rendezvous control capability of the chaser and the homing phase relative guidance strategy, which will be analyzed in Chap. 4. Essentially, the GNC functions of the homing phase can be summarized as follows: autonomous attitude estimation, relative navigation, guidance and control law calculation. After a number of orbital controls, the chaser finally enters a holding point, which is on the same orbit but is behind the target by a certain distance. Relative position holding control should be performed for status checks as necessary. 2. Approach Phase The approach phase begins at the aiming point of the homing phase and extends until the chaser captures the end position of the approach phase. The task in the approach phase is to render the chaser to approach the target and capture the docking corridor of the docking port. The docking corridor is usually defined as a cone whose generatrix is the target’s docking axis, and its half angle is about 8°. The relative distance throughout the approach flight process generally ranges from several kilometers to several hundred meters. Holding positions are generally designed to meet the requirements of the measuring range and improve the safety of the flight trajectory. The chaser captures the docking corridor through the transfer between holding points, and as shown in Fig. 1.1, point P2 selected as a transition holding point for the approach process. At the end point of the approach phase (which is also the starting point of the closing phase), the chaser should guarantee that the relative position and attitude meet the working conditions of the relative measurement sensors (usually the camera-type rendezvous and docking sensors), in order to establish a stable relative navigation and simultaneously provide good initial conditions for the closing phase. As shown in Fig. 1.1, holding point P3 is selected as the terminal aiming point for the approach phase. 3. Flying-Around Phase During rendezvous flight, if the chaser can capture the docking corridor of the target at the end of the approach phase, the final rendezvous and docking can be completed by the closing phase. However, if the chaser cannot directly capture the docking interface in the approach phase, or if it transfers between different interfaces, the flying-around phase is necessitated. In this phase, the chaser flies around the target until it captures the target docking corridor. Because the docking interface may be located at any position and orientation with respect to the target, several flying-around strategies exist, among which the most commonly employed ones are the flight between the forward and backward docking interfaces in the orbital plane and the flight between the forward docking interfaces and the radial docking interfaces. The design of the flight path depends on the specified tasks. The starting and ending points of the flight path are determined

1.2 Phases of Rendezvous and Docking

5

by the installation of the relative measurement sensor, the configuration of the target and chaser, the safety of the trajectory, and the guidance strategy. 4. Closing Phase The closing phase begins at the end of the approach phase or flying-around phase and ends when contact is made between the docking mechanisms of the two spacecraft. The task of the GNC system in this phase is to control the chaser to approach the target along the docking corridor and to meet the initial docking conditions for contact between the docking mechanisms of the two spacecraft. The initial conditions for docking are generally described by relative position, velocity, attitude, and angular velocity. The closed-loop control law is adopted to control the relative position and attitude. In this phase, the two spacecraft are very close to each other, and the attitude and position of the chaser with respect to the target must be adjusted frequently. The GNC system performance should be guaranteed to ensure control accuracy, trajectory safety, and system reliability, which are of the highest priority in the design of the entire rendezvous and docking mission.

1.2.5 Docking Phase The docking phase refers to the stage in which the contact and collision of the docking mechanisms of the chaser and the target are achieved; the impact energy is dissipated through the buffering process and the initial deviation of the docking is eliminated, the docking mechanism is tightened and rigidly connected, and the electrical, gas, and liquid paths between the assemblies are connected and conducted. For the rendezvous and docking tasks using the robotic arm, the chaser stops at close range to the target, following which the robotic arm mounted on the target captures the chaser to achieve the goal of the docking of two spacecraft. The success of the docking task requires the relative state to meet certain conditions, i.e., the approximate speed, transverse position, transverse velocity, relative attitude, and attitude angular velocity are required to be within a certain range. Those constraints are determined by the position of the docking interface (such as forward, backward, or radial), the performance of the docking mechanism, control capability, navigation accuracy, spacecraft mass and inertia, and so on.

1.2.6 Complex Operation Phase The complex operation phase begins when docking by the docking mechanism is complete and ends when the chaser and target are separated.

6

1 Introduction

1.2.7 Departure Phase The departure phase refers to the flight motion in which the chaser is evacuated from the safe area of the target after it is separated from the target. The departure flight phase is categorized into normal departure and emergency departure. Normal departure refers to the gradual evacuation of the chaser from the target to the safety zone under the condition that both the chaser and the target control system work normally. When the chaser or target control system is abnormal, emergency departure is performed; the chaser implements an exception handling strategy to quickly evacuate to the target outside the safe zone and into normal operation mode.

1.2.8 Reentry Phase For manned or recycle spacecraft, after the chaser completes its departure, it can return to Earth via re-entry control.

1.3 Overview of Rendezvous and Docking 1.3.1 Introduction By 2014, more than 300 space rendezvous and docking missions had been conducted by the USA, the erstwhile Soviet Union/Russia, Japan, European Space Agency (ESA),and China. Table 1.1 summarizes the overview of the major rendezvous and docking tasks implemented in accordance with the space organization and time [2–9].

1.3.2 Classification of Rendezvous and Docking Flight Missions 1. Classification according to the Rendezvous Orbit According to the orbiting central celestial bodies, the rendezvous and docking missions can be divided into two categories: earth orbit and lunar orbit. (1) Earth Orbit Rendezvous and Docking During the 1960s, i.e., the exploration stage of rendezvous and docking technology, the USA and the erstwhile Soviet Union made a breakthrough in theoretical issues

Dates

1965–1966

1969–1972

1973–1974

1975

Flight program

Gemini

Apollo

Apollo-Skylab

Apollo-Soyuz

CSM

CSM

CSM/LM

Gemini

Chaser

Table 1.1 Overview of rendezvous and docking mission

Soyuz

Skylab

CSM/LM

Titan II 2nd stage REP Gemini Agena ATDA

Target

Probe (CSM) APAS-75 (Soyuz)

Probe (CSM) Drogue (Skylab)

Probe (CSM) Drogue (LM)

Cone & Latches (Agena, ATDA) Latch receptacles (Gemini)

Capture hardware Co-elliptic rendezvous Manual control within 60 m

Guidance

VHF COAS Sextant

VHF COAS Sextant

Co-elliptic rendezvous

CSM: very high Co-elliptic frequency: VHF, rendezvous Sextant, COAS Manual docking LM: Rendezvous Radar, COAS

Rendezvous Radar Crew optical alignment sight: COAS Sextant

Relative sensors

(continued)

Manual

Manual

Manual

Manual

Control mode

1.3 Overview of Rendezvous and Docking 7

Dates

1981–2011

Flight program

Space Shuttle

Table 1.1 (continued)

Space Shuttle

Chaser SPAS IRT Solar Max Palapa-B2 Westar-VI SYNCOM IV-3 SPARTAN PDP radar reflector LDEF GRO IBSS-SPAS II INTELSAT VI EURECA ORFEUS-SPAS Hubble CRISTA-SPAS WSF SFU OAST-Flyer PAMS-STU Mir ISS

Target RMS none TPAD, RMS Stinger Stinger RMS, Grapple Bar E RMS, Hands F RMS none RMS RMS (deploy) RMS Capture Bar, Hands G RMS none APAS-95, APAS-99, RMS K

Capture hardware Radar Star Tracker: COAS; TCS Lidar HHL Lidar Subtended Angle Range Ruler CCTV GPS

Relative sensors (1) 1973–1983, 1983–1997: 1997–2011: R-Bar Rendezvous strategy (4) Perform Lambert guidance law

Guidance

(continued)

Auto/manual

Control mode

8 1 Introduction

Dates

2003

2005

2007

Flight program

XSS-11

DART

Orbital express

Table 1.1 (continued)

ASTRO

DART

XSS-11

Chaser

NextSat

MUBLCOM

Minotaur I upper Stage and other vehicles

Target

Mechanical claw

–

–

Capture hardware

Autonomous rendezvous and capture sensor system

GPS AVGS

LIDAR VCS

Relative sensors Auto

(1) C-W guidance (2) Fored circle fly-around

Auto

(continued)

Control mode

(1) DART vehicle Auto perform a CW transfer to place it 3 km behind the target vehicle (2) DART then perform another CW transfer to place it 1 km behind the target (3) Within 1 km range, DART perform -V-Bar,+ R-Bar and other type relative flight mission

Co-elliptic rendezvous After the rendezvous radar Capture the target,relative guidance is executed

Guidance

1.3 Overview of Rendezvous and Docking 9

1979–1981 1982–1986 1986

Soyuz T (7 K-ST)

T-T4 T5-T15 (11) T15 (2)

Soyuz Ferry 10, 11 14, 15 17–20 (4) 21, 23, 24 25–40 (16)

1971 1974 1975 1976–1977 1977–1981

Soyuz Ferry (7 K-T)

Chaser

Cosmos 188 Cosmos 212 Soyuz 3 Soyuz 4 Soyuz 8

Dates

Soyuz (7 K-OK) 1967 1968 1968 1969 1969

Flight program

Table 1.1 (continued)

Salyut 6 Salyut 7 Mir

Salyut 1 Salyut 3 Salyut 4 Salyut 5 Salyut 6

Cosmos 186 Cosmos 213 Soyuz 2 Soyuz 5 Soyuz 7

Target

Probe–drogue docking system

Probe–drogue docking system

Probe–drogue docking system

Capture hardware

Igla

Igla

Igla rendezvous system

Relative sensors

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) fly-around

Guidance

(continued)

Auto/manual

Auto/manual

Control mode

10 1 Introduction

Dates

1986–2000 2000–2002

2002

2011

Flight program

Soyuz TM

Soyuz TMA

Soyuz TMA-M

Table 1.1 (continued)

TMA-1 to 10

TM-TM30 (30) TM31-TM34 (4)

Chaser

ISS

ISS

Mir ISS

Target

Probe–drogue docking system

Probe–drogue docking system

Probe–drogue docking system

Capture hardware

Kurs

Kurs

Kurs rendezvous system

Relative sensors

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

Guidance

(continued)

Auto/manual

Auto/manual

Auto/manual

Control mode

1.3 Overview of Rendezvous and Docking 11

Dates

1978–1981 1982–1985 1986–1990

1989–2001 2001

2000–2001 2000–

Flight program

Progress (7 K-TG)

Progress M

Progress M1

Table 1.1 (continued) Target

Progress M1

Progress M

Mir ISS

Mir ISS

Progress 1–12 (12) Salyut 6 13–24, Cosmos 1669 Salyut 7 (13) Mir 25–42 (18)

Chaser

Probe–drogue docking system

Probe–drogue docking system

Probe–drogue docking system

Capture hardware

Kurs

Kurs

Igla

Relative sensors

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

(1) Compositive rendezvous maneuver strategy with C-W guidance amd line of sight guidance (2) Fly-around

Guidance

Auto

Auto

Auto

(continued)

Control mode

12 1 Introduction

Dates

2008

1997–1999

2009

2011

Flight program

ATV

ETS-VII

HTV

SZ-X

Table 1.1 (continued)

SZ

HTV

Hikoboshi

ATV

Chaser

TG-1

ISS

Orihime

ISS

Target

Androgynous docking mechanisms

Manipulator

Mechanical claw

Probe–drogue docking system

Capture hardware

RGPS Wave Radar Laser Radar CRDS

GPS RVR

GPS RVR PXS

RGPS Vediometer

Relative sensors

Control mode

(1) C-W guidance Auto/manual (2) LOS guidance (3) Phase plane close loop control of relative six DOF

(1) C-W guidance Auto (2) PID close loop control of relative six DOF

(1) C-W guidance Auto (2) PID close loop control of relative six DOF

(1) Homann Auto rendezvous (2) C-W guidance (3) PID close loop control of relative six DOF

Guidance

1.3 Overview of Rendezvous and Docking 13

14

1 Introduction

on its dynamics, guidance, and control and solved the technical problems in its engineering implementation via projects such as the Gemini program (USA), the Apollo plan (USA), the Vostok spacecraft (Soviet Union), and the Soyuz spacecraft (Soviet Union). Both the USA and the Soviet Union had developed their own rendezvous and docking schemes [10]. (1) The rendezvous and docking scheme of the USA: the USA developed a manned rendezvous scheme, i.e., the co-elliptical rendezvous scheme, by considering the superiority of astronaut intelligence. In the far-range guidance phase, the chaser is inserted in the orbit plane of the target with the line of apsides of the two orbits being aligned and the height of the chaser’s orbit being slightly lower than that of the target. The height difference was determined by the illumination condition of the closing phase as well as the controllability. Following the free-floating phase, the chaser implemented terminal initial impulse (which is designed based on the two-impulse guidance strategy) when the triggering angle of sight reached 27.2°. When the chaser arrived near the target and the brake impulse was realized, the astronauts controlled the chaser to dock with the target manually. Advantages of co-ellipse rendezvous: the process of rendezvous is clear, it is convenient to train the astronauts and plan the rendezvous processing in advance, and it reduces the difficulty of flight parameters monitoring. Disadvantages of co-ellipse rendezvous: it is incapable of controlling the specific arrival time and extending autonomous control owing to the reliance on human control at close range, and the issues related to trajectory safety are not sufficiently considered during the trajectory design. (2) Rendezvous and docking scheme of the Soviet Union: The erstwhile Soviet Union developed automatic rendezvous and docking technology, which is more complex than manned rendezvous and docking technology. In comparison with the simple relative measurement sensor used in the USA’s scheme, the ‘Igla’ measurement system is far more complicated. The system can provide complete relative navigation information, including relative distance, relative angles of sight and the rate, as well as relative attitude measurement at closing range, which is required for automatic rendezvous and docking. The primary characteristics in terms of guidance strategy are Hohmann double-impulse rendezvous and line-of-sight guidance within 5 km. Advantages of the Soviet Union rendezvous scheme: automatic rendezvous reduces the requirement of astronauts, and the strategy is more robust. Disadvantages of the Soviet Union’s rendezvous scheme: the weight and power consumption of the relative measurement system are very large. Additionally, the target is required to execute attitude maneuvering to match the relative measurement. The 1970s to the 1990s saw the mature stage of rendezvous and docking technology. The USA and the Soviet Union/Russia began to develop space laboratories and space stations. The USA developed sky-lab and led the development of the International Space Station (ISS), while the Soviet Union/Russia developed the Salyut space laboratory and the Mir space station and participated in the development of

1.3 Overview of Rendezvous and Docking

15

the ISS. In terms of RVD technology, both the countries have basically followed the mature technology. However, with the development of technology, a variety of new relative measurement sensors have been developed. Since the mid-1990s, autonomous, automatic, and multi-mission rendezvous and docking technologies have gradually become the mainstream of development. In addition to the USA and Russia, ESA and Japan have also carried out relevant research. Owing to the lack of human assistance in autonomous and automatic rendezvous and docking missions, the GNC system design must be robust. While the relative navigation system is required to provide complete relative navigation information and constitute redundancy, the guidance scheme is required to consider trajectory safety, sensor field of view, emergency treatment, and preparation for re-rendezvous as far as possible. There are several types of relative measurement sensors developed by different countries for autonomous and automatic rendezvous and docking: satellite navigation equipment, microwave radar, LIDAR, visible light camera/infrared camera/star tracker, laser range finder, and imaging rendezvous and docking sensor. According to the in-orbit rendezvous and docking tests carried out by various countries, the employed autonomous and automatic RVD technologies can be summarized as follows. The chaser enters the near-circle orbit that is coplanar with the target through far-range guidance, with a slightly lower orbit. During the homing phase, it adopts a multi-impulse trajectory to maneuver to the stable parking point behind the target. The guidance strategy is generally designed based on the time and fuel optimization criteria where the Hohmann rendezvous and Lambert guidance law can be adopted. During the approach phase, the chaser completes the transition between parking points by the CW (Clohessy-Wiltshire, CW) guidance law and captures the docking axis by fly-around if necessary. In the last few hundred meters, the docking is completed by the closed-loop control approach. The above scheme is applicable to rendezvous and docking tasks with nearly circular target orbits. Its principle is based on the chaser gradually approaching the target through the transfer between the parking points, which facilitates the design of a safe trajectory and the handling of emergencies and allows for a more robust rendezvous process. However, there are certain shortcomings, e.g., the guidance strategy includes strict conditions, such as the initial orbital height difference, relatively long rendezvous time, and high requirements for the relative navigation system. (2) Lunar Orbit Rendezvous and Docking Besides the Earth orbit rendezvous and docking, the USA has also successfully carried out six manned lunar orbital rendezvous and docking projects in the 1960s and 1970s as part of the Apollo program. Because manned missions to the moon require a life support system, the rendezvous time should not be excessively long. The entire rendezvous and docking process lasts for several hours. The Apollo program adopted two rendezvous and docking programs: the co-elliptic rendezvous program and the direct rendezvous

16

1 Introduction

program. The Apollo 11/12 basically followed the Gemini co-elliptic rendezvous program. The entire rendezvous process lasted for about four hours, and the astronauts finally took control and carried out the docking manually. Apollo 14/15/16/17 adopted the short rendezvous scheme, in which the chaser directly implemented the terminal rendezvous initialization impulse at the first apogee after insertion, such that the entire rendezvous process was completed within two hours. 2. Classification according to the RVD Time According to the duration of the rendezvous and docking process, the rendezvous and docking tasks can be divided into three categories: early quick rendezvous and docking with small phase angle, two to three day rendezvous and docking, and new quick rendezvous and docking [11]. (1) Early Quick Rendezvous and Docking with Small Phase Angle In the early days of manned spaceflight, limited by the capability of life support systems, the USA and the erstwhile Soviet Union adopted the quick rendezvous program with a small phase angle, and the chaser could complete the rendezvous mission after 1–5 revolutions around Earth. By coordinating the launch time of the target and chaser, a specific initial relative distance or phase angle condition between the two spacecraft can be formed to achieve quick rendezvous. The rendezvous scheme is closely related to the location of the launch site. The early small phase angle quick rendezvous program of the Soviet Union adopted the following procedure: (1) The target was launched into a 1-day regression orbit with an orbital altitude of 200–230 km. (2) Usually a day later, the chaser was launched into the same orbit. (3) When the chaser inserted the orbit, it was already in the adjacent position of the target, and the position was determined according to the maximum measurement range of navigation sensors. Because of the geographical advantage of the Canaveral launching site, USA, the Gemini and the third stage of Agena can execute the co-plane launch twice, with different azimuth angles, within one day, with a small phase angle (about 15°) which is required by the quick rendezvous. On the basis of the successful rendezvous experience in Gemini, the quick rendezvous scheme with a small phase angle was applied to three rendezvous missions of the USA’s Skylab program. In these tasks, the phase modulation requires five orbital periods, and the rendezvous process requires about 8.5 h from the Apollo spacecraft launch to dock with the Skylab when the initial phase angle is 40°–50°. (3) Two to Three Day Rendezvous and Docking When a space mission transitions from a manned space exploration stage to a space application stage that requires an intensive launch to the space station, it becomes

1.3 Overview of Rendezvous and Docking

17

very difficult to provide specific phase angle conditions because of the uncertainty of the space station position. Therefore, it takes a long time to complete the rendezvous when the initial phase angle is within a certain range. Thus, the current rendezvous and docking missions generally take two to three days. At present, this rendezvous and docking program is widely used in several countries’ space missions; the rendezvous mission between the Soyuz-TMA spacecraft and the ISS is taken as an example to briefly introduce it. In the rendezvous mission between Soyuz-TMA and ISS, the initial phase angle range was 170°–320°. In general, the Soyuz-TMA enters the orbit from the Baikonur launch site; the perigee height is 200 km, the apogee height is 242 km, the orbit period is 88.64 min, the orbital inclination is 51.6°, and the precision of orbit control is ±22 s. In the second circle, the GNC system was checked. The Moscow flight control center used the data from the ground measurement stations to estimate the orbit and calculate the amount of the first two phase-modulated impulses, then ordered the chaser to perform these two maneuvers, which transferred the Soyuz-TMA into the phase-modulated orbit. Until the next maneuver, Soyuz-TMA maintained the sunorienteering attitude and rotate around the body coordinate axis at the speed of 2° s−1 , with the axis being perpendicular to the solar panel plane and pointing toward the sun. The spacecraft flew from the 6th to the 11th circle on a “deaf” orbit, in which ground control stations cannot contact the spacecraft. In order to eliminate the accumulated flight error, the third phase modulation impulse was calculated, and the amplitude was about 2 m s−1 from the 12th to 16th circle, which was executed at the beginning of the 17th circle. Following this, Soyuz-TMA maintained the sun-orienteering pose until the next impulse. At the 32nd circle, the status variables were uploaded to the SoyuzTMA’s onboard computer. Based on the uploaded data, the GNC could calculate the quantity of the last impulse, which was executed autonomously to ensure that the spacecraft docked completely at the 34th circle [15]. (4) New Quick Rendezvous and Docking In recent years, with the increasing complexity and diversity of space missions, some missions have put forth new requirements to ensure the rapidity of rendezvous. For example, in missions such as space rescue and in-orbit service, rendezvous needs to be completed as quickly as possible to gain valuable time. In manned flights, quick rendezvous can significantly reduce astronaut fatigue, which preserves energy to complete the space mission and also reduces the consumption of the life support system. Therefore, Russia and other space organizations refocused on the development of novel quick rendezvous and docking programs; Russia has already adopted one such program in the rendezvous and docking missions of the Progress cargo and Soyuz manned spacecraft, both with the ISS. The new quick rendezvous and docking strategy was first tested through the Progress M-16M cargo’s mission. In addition to the Progress M-21M cargo for testing Kurs-NA without performing quick rendezvous and docking, all the remaining cargo execute the quick rendezvous and docking strategy. The first quick rendezvous mission of the manned spacecraft

18

1 Introduction

was performed by the Soyuz TMA-08M, and the subsequent Soyuz spacecraft all carried out this program. 3. Classification based on the Types of Target Spacecraft In the majority of rendezvous and docking missions, the target spacecraft is a cooperative target. Rendezvous missions with cooperative targets typically have the following characteristics: (1) The target spacecraft has active attitude and track controllability, and it shall be in the target orbit and maintain the required attitude direction during the rendezvous and docking stage. (2) To adapt to the relative measurement sensor installed on the chaser, the target is usually equipped with cooperative targets, such as GPS receivers, radar transponders, and angle reflectors. (3) A reliable communication link between the target and the chaser can be guaranteed. (4) The target device is equipped with a docking mechanism that is compatible with the chaser, such that the final docking and combination through rigid connection can be achieved. However, rendezvous and acquisition tasks such as failed satellite maintenance and space debris clearing no longer possess the above characteristics. Such tasks are called non-cooperative target rendezvous and docking. 4. Classification by Contact Mode The ultimate goal of a rendezvous and docking mission is to realize the structural connection between the chaser and the target and form a combined spacecraft. After the completion of the close intersection, the final structural connection operation can be realized in two ways: docking mode and docking capture mode [12, 13]. (1) Docking Mode In docking mode, to satisfy the initial condition for the connection of the docking mechanism, the chaser’s GNC system must control the chaser’s states to ensure that the relative states (usually including position, velocity, attitude angle, and angular rate) meet the conditions of docking between the interfaces of the chaser and the target after closing phase rendezvous. Following this, the docking mechanism closes, and lock action is implemented to realize the agencies connection between the two spacecraft. Table 1.2 shows the primary technical indicators to be met during the docking of typical space rendezvous and docking tasks. The docking mechanism used in the development of rendezvous and docking mainly include: (1) the probe–drogue docking mechanism, adopted by the Soviet Soyuz spacecraft; (2) the androgynous peripheral docking mechanism, the docking process used for the first time in Apollo and Soyuz spacecraft;

1.3 Overview of Rendezvous and Docking

19

Table 1.2 Typical criteria of space rendezvous and docking Program

Approach mode

Docking criteria

Apollo

+R-bar docking V-bar docking

Close-in speed: 0.03–0.3 m s−1 Lateral speed: ≤0.15 m s−1 Lateral position: ≤0.3 m Angular rate: ≤1° s−1 Angular misalignment: ≤10°

Space Shuttle

Manual R-bar docking

Close-in speed: 0.01 m s−1 ≤Vx ≤0.10 m s−1 Lateral speed: ≤0.045 m s−1 Lateral position: ≤0.11 m Angular rate: ≤0.2° s−1 Angular misalignment: ≤4°

Soyuz/progress

V-bar docking

Close-in speed: 0.1 m s−1 ≤Vx ≤0.35 m s−1 Lateral speed: ≤0.10 m s−1 Lateral position: ≤0.34 m Angular rate: ≤ 1° s−1 Angular misalignment: ≤7°

ATV

V-bar docking

Close-in speed: 0.05 m s−1 ≤Vx ≤0.1 m s−1 Lateral position: ≤0.1 m Angular misalignment: ≤5°

ETS-VII

V-bar docking

Close-in speed: 0.01 m s−1 ≤Vx ≤0.05 m s−1 Lateral position: ≤0.1 m Angular misalignment: ≤2°

(3) the weak percussive docking mechanism, used for the first time in the Japanese engineering test satellite-VII. (2) Docking Capture Mode Docking capture mode is defined as follows. The chaser’s GNC system eliminates the relative linear and angular velocity and sets up the chaser holding somewhere close to the target. Then, the mechanical arm on the chaser or target captures the capturing device on the other spacecraft in order to dock the spacecraft in the final position and completely connect the structures of the two spacecraft through the appropriate machinery. Table 1.3 shows the main technical indicators to be met to dock typical space rendezvous and docking tasks. The shuttle teleoperation manipulator (Shuttle Remote Manipulator System, SRMS), which is designed and manufactured by the Canadian MDRobotics company, is the world’s first practical space manipulator Table 1.3 The technical indicators of docking capture mode [6, 9]

Program

Technical indicators

HTV

Translational speed ≤0.024 m s−1 Angular rate ≤0.08° s−1

Dragon

Translational speed ≤0.024 m s−1 Angular ≤0.012° Angular rate ≤0.02° s−1

20

1 Introduction

which serves the shuttle. Its primary function is to capture and deliver satellites, and it can also be used as the astronauts’ extravehicular auxiliary equipment [4]. The company has also designed a mobile service system (MSS) for the ISS, which will dock the Japanese HTV.

1.3.3 Typical Rendezvous and Docking Flight Mission

1. Earth Orbit Manned Rendezvous and Docking—the Rendezvous and Docking between the Shenzhou Spacecraft and Tiangong Space Laboratory [7] In November 2011, China’s first automatic space rendezvous and docking (Fig. 1.2) was carried out by the SZ-8 manned spacecraft and the TG-1 space laboratory. In June 2012, China’s first manned space rendezvous and docking was carried out by the SZ-9 manned spacecraft and the TG-1, and the first manual rendezvous and docking was carried out by astronauts. The SZ-10 spacecraft, which was launched in June 2013 with a full load of three astronauts, completed three rendezvous, two dockings, and one fly-around flight with the TG-1, which once again successfully verified the rendezvous and docking program. This shows that China has fully mastered space rendezvous and docking technology. According to the requirements of the mission, orbit design, configuration of the ground observation and control station, and performance of the spacecraft rendezvous Docking phase

Near-range autonomous control phase Chosing phase

Approach phase

Six DOF control P4 D3 30m

X

C-W and LOS guidance

P3 140m

P2 400m

Launch phase

Far-range rendezvous phase

Before-launch

Phasing phase C-W guidance with time and fuel waste constraint P1 5km

~50km The 5th maneuver for comprehensive The 2th manuever orbit correction for elimination of D 1 M5 M4 plane error

~20km

To autonomous control

dock

393 km The 1th manuever for raising the perigee height, near the apogee ~380km M1

M2

M3

The 4th manuever for rounding the orbit, near the apogee 100km

After docking

The 3th manuever for adjusting the apogee height, near the perigee

Wave radar

Insertion D0 ~200km

20km Laser radar 150m Z

CRDS TV USB GNSS

Doking

Fig. 1.2 The rendezvous and docking process between SZ-8 and TG-1

launch

1.3 Overview of Rendezvous and Docking

21

and docking measurement sensors, the rendezvous and docking process of the spacecraft can be divided into the following four phases: rendezvous phase, docking phase, combined operation phase, and departure phase. The rendezvous phase is divided into a far-range rendezvous phase and an autonomous control phase. The orbit control in the far-range rendezvous phase is carried out by the ground command, while the rendezvous and docking tasks of the autonomous control phase rely on the onboard GNC sub-system for their independent completion. The autonomous control phase is divided into a homing phase, an approach phase, and a closing rendezvous phase. Four holding points at 5 km, 400, 140, and 30 m distances are designed in the autonomous control phase for the rendezvous and docking sensor switching and system status checking. The main task of the far-range rendezvous phase is to measure the orbit of the chaser and the target data by the ground station. Then, under the control command from the ground, the chaser executes five orbit maneuvers to complete the semimajor axis adjustment, the relative phase error correction, the orbital inclination, and the ascending node error elimination. In the end, the chaser is inserted into an orbit which is a coplanar with the target’s orbit, with the height being slightly lower than the target. The primary task of the homing phase is to complete the relative navigation, guidance law, and control quantity calculations based on the data from the relative measurement sensors and then complete the orbit control autonomously. After several orbital controls, the spacecraft enters the same orbit as the target but stays 5 km behind it. The homing phase uses differential satellite navigation equipment, a microwave radar, and a laser radar for relative navigation. The multi-impulse optimal guidance strategy based on the CW equation is used to implement orbit control. The objective of the approach phase is to capture the docking corridors, which is the target organ docking axis as the center of a cone with a half-cone angle of 8°. Two holding points at 400 and 140 m are arranged in the approach phase, respectively. CW guidance law and line of sight guidance law are jointly used to design the orbital control strategy to capture the holding point. The task of the closing phase is to provide initial conditions for docking. A holding point at 30 m is arranged in the phase, and a camera-type rendezvous and docking sensor is used for relative position measurement and estimation. To conquer the vibration of flexible panels, the jet plume, the time delay of actuator, and the serious coupling influence between attitude and orbit control, a novel phase plane adaptive control scheme is designed, based on the characteristic model of the intelligent adaptive control theory. The flight test results show that the proposed scheme can achieve high precision in the relative position and relative attitude control for docking. 2. Unmanned Rendezvous and Docking in Earth Orbit—the Rendezvous and Docking between an Automatic Transfer Vehicle (ATV) Spacecraft and the ISS The ATV is an unmanned spacecraft developed by the ESA, which is employed to provide cargo transportation, propellant refilling, orbit lifting, attitude control, and

22

1 Introduction

garbage delivery services to the ISS. In March 2008, the first ATV was launched and successfully docked with the ISS. The ATV control system is equipped with the following measuring implements [14]. (1) (2) (3) (4) (5) (6) (7)

four 2-axis GYRA components; two star chasers (STR); three 2-axis accelerometers (ACCA); two Global Positioning System (GPS) receivers (the ISS has two GPS receivers); two videometers (VDM); two remote direction finders (TGM); two solar sensor units (SSU).

The rendezvous phases of ATV with ISS are described as follows [14] (1) Phase Adjustment Phase In this phase, the ATV performs orbital maneuvers, gradually raising the orbital altitude to approach the ISS. The terminal point is at the last phase-adjusted maneuver point S-1/2 , which is 39 km behind the ISS and 5 km below. The last phase adjustment maneuver is an orbital circularization maneuver, which makes the ATV drift in a straight line relative to ISS. At S-1/2 , the Proximity Link between the ATV and the ISS begins to work. (2) Rendezvous Phase This phase starts from S-1/2 , the starting point of the drift orbit, and extends until the docking of the ATV and ISS. Aiming points S-1/2 , S0, S1, and holding points S2, S3, S4, and S41 are present on the V-bar along the rendezvous orbit. The drift phase from S-1/2 to S0 is the transition phase from absolute GPS (AGPS) to relative GPS (RGPS). In S0 , RGPS has achieved the desired accuracy and can be put into use. The rendezvous phase can be divided into a far rendezvous phase and close rendezvous phase. The far rendezvous uses RGPS, while the close rendezvous uses optical sensor VDM. Far rendezvous involves pre-homing flight (S-1/2 –S1 ), homing flight (S1 –S2 ) and closing flight (S2 –S3 ). Close rendezvous begins from S3 to docking. The hold and controlled retreat maneuver can be performed at any point during this phase. The final approach consists of final approach 1 (S3 –S4 ) and final approach 2 (S4 to docking). Final approach 2 can be further divided into two parts, i.e., final approach 21 (S4 –S41 ) and final approach 22 (S41 to docking) [14]. 3. Quick Rendezvous and Docking in Earth Orbit—Quick Rendezvous and Docking between the Soyuz/Progress Spacecraft and the ISS The implementation of the new quick rendezvous and docking began in August 2012 between the Progress M-16M and the ISS. Following three flight tests with the

1.3 Overview of Rendezvous and Docking

23

Table 1.4 The quick rendezvous and docking mission of Russian [17] Event

Time (Moscow time)

Flight orbit

Launch

00:43:20

0

Separation

00:52:08

1

Orbit correction 1

01:25:32

1

Orbit correction 2

02:05:35

2

Orbit correction 3

02:46:44

2

Orbit correction 4

03:16:41

2

Starting autonomous rendezvous

04:26:29

4

Starting fly-by and closing rendezvous with ISS

06:11:24

4

Docking

06:32:13

4

Total time

5 h, 48 min, 53 s

–

Progress M-16M, M-17M, and M-18M, on March 28, 2013, the Russian Soyuz TMA08M manned spacecraft arrived at the ISS and the entire rendezvous and docking process required less than 6 h. This shows that the quick rendezvous and docking scheme has matured into the practical using [15–17]. The specific flight processes are shown in Table 1.4. In order to minimize fuel consumption when using quick rendezvous and docking, the rendezvous trajectory must meet the following two constraints: (1) Rocket should carry the chaser into the stable orbit which is coplane with the target; (2) the chaser and the target of the initial phase angle must conform to the phase adjustment strategy. In general, owing to the limitations of Earth’s rotation and direction sector, there is only one chance per day that the first condition can be met. To meet the second condition, it is necessary to select a reasonable launch time or conduct orbital height maneuver of the space station for a given launch time. In a phasing strategy, a poor set of initial phase angles can cause the chaser to brake, resulting in additional fuel consumption. Considering a two to three day rendezvous and docking scheme, a reasonable range for the initial phase angle will be very large (170°–320°). However, to achieve a four-circle rendezvous and docking scheme, the initial phase angle range must be small (20°–40°). Such a short rendezvous time requires a computer onboard the chaser to calculate all rendezvous ignition times. Considering the above rendezvous constraints, the four-circle quick rendezvous shall execute the following schemes successively: (1) Send the Soyuz-TMA spacecraft into the space station orbit plane in the first circle, and obtain the longitude of the ascending node for 20° W, lagging behind the farther east longitude of the ascending node. In this manner, during the second orbital period, as it passes through the next Russian ground station area, the orbital parameters will be uploaded to the chaser’s computer, which will be used to calculate the first two ignition times of the four-lap for phase adjustment. During the third and fourth circles of the phase control, the chaser is located outside the range of the Russian ground station. The third and fourth impulses

24

1 Introduction

are fixed, passing one orbit and half an orbit before the chaser reaches a specific target point. The specific target point mentioned here refers to 8–13 min before the fifth circle (the period during which the chaser passes the equator from south to north). (2) In the third circle, when the chaser passes through the measurement and control range of the Russian ground station, it will send the amount of phase-modulated impulses executed to the ground station through the telemetry signal and compare it with the result of the ground calculation. If the third and fourth impulses match the Soyuz-TMA spacecraft program, a decision is made to execute the final phase of the rendezvous. (3) In the fifth circle, the docking will be carried out within the measurement and control range of the Russian ground station. Using the four-circle quick rendezvous scheme, the time from orbit to docking is about 6 h. 4. Lunar Orbit Rendezvous and Docking—Apollo Lunar Orbit Quick Rendezvous and Docking The Apollo program implemented by the USA in the 1960s and 1970s successfully carried out six manned lunar landing flights, with the command service module (CSM) as the target and the lunar module (LM) as the chaser [2]. In the process of the development of the Apollo program, three rendezvous and docking strategies were demonstrated: direct-ascent rendezvous, co-elliptic rendezvous, and short rendezvous. The latter two strategies have been adopted in the orbital flight [18, 19]. (1) Direct-Ascent Rendezvous The first rendezvous strategy to be proposed was the “direct ascent rendezvous strategy”; research began on it in the early 1960s, before the in-depth study of hardware characteristics and engineering errors. Figure 1.3 and Table 1.5, respectively, show the schematic and maneuver data of the direct-ascent rendezvous. Fig. 1.3 Representative direct-ascent technique [19]

CSM is in an 80-nautical-mile circle orbit Braking

Insertion

1.3 Overview of Rendezvous and Docking

25

Table 1.5 The characteristics of direct ascent rendezvous strategy [19] Nominal rendezvous maneuver

Time from previous maneuver (min)

Insertion

7

Braking

45

V /(m/s)

Aiming orbit Apogee/Perigee (km)

–

153.716/16.668

45.72

148.16/148.16

The major advantage of direct-ascent rendezvous lies in the fact that it takes less time to transition from power ascent to the completion of rendezvous and docking. However, it also involves significant flaws: (1) the approach velocity increment of terminal phase is beyond the capacity scope of the reaction control system, (2) errors in the ascent phase increases the complexity of the astronaut’s working procedures and techniques in the final approach phase. (2) Co-Elliptic Rendezvous In order to overcome the adverse effects caused by various errors, a set of feasible rendezvous methods, i.e., the co-elliptic rendezvous strategy, was finally established at the end of 1964 after several stages of development, and was successfully applied on Apollo 11 and Apollo 12. The co-elliptic rendezvous strategy completes the far-range rendezvous mission through three impulses, corrects the in-orbit and out-of-orbit errors, and provides the standard initial conditions for the final approximation. After about 7 min of ascent, the LM is launched from the lunar surface and enters into a 17 km × 83 km elliptical orbit. An impulse (co-elliptic sequence initiation, CSI) initializes the co-elliptic sequence after 60 min of the launch, such that the LM enters the circular orbit with a height of 83 km. After 90 min from the launch, the orbital plane can be modified as necessary. The constant differential height (CDH) maneuver is carried out within 120 min of the launch, through which LM maintains a constant height difference from the CSM. At 160 min following the launch, the terminal initialization maneuver (terminal phase initialization, TPI) begins, and several midcourse corrections can be made targeting the aim position. Within 205 min of the launch, terminal braking (terminal phase finalization, TPF) is performed. After the braking, the two spacecraft are in the same orbit with a relative distance of several kilometres, and finally, the docking is manually completed by the astronaut. The Apollo orbital maneuver data are presented in Table 1.6, and the flight process of co-elliptic rendezvous is shown in Fig. 1.4. (3) Short Rendezvous The short rendezvous strategy not only has the quick rendezvous characteristic of the direct ascending rendezvous, but it also retains the advantage of the high probability of the co-elliptic rendezvous to reach the final approximation phase. After the first apolune, LM initiates the final approximation process to significantly shorten the rendezvous time. Because the Apollo 11 and Apollo 12 completed their rendezvous tasks excellently and accumulated certain experience, the short rendezvous strategy

26

1 Introduction

Table 1.6 The characteristics of co-elliptic sequence technique [19] Rendezvous maneuver Time from previous maneuver (min) V (m/s) Aiming orbit Apogee/Pergee (km) Insertion

–

83.34/16.668

CSI

50

7

15.24

83.34/83.34

CDH

59

1.524

83.34/83.34

TPI

40

7.62

112.972/83.34

Terminal Braking

43

13.716

111.12/111.12

was applied to the Apollo 14. Owing to the lack of contingency flexibility, the coelliptic rendezvous is used as a backup rendezvous strategy in flight missions to provide a standard final approach in case of contingencies. The flight process of short rendezvous is shown in Fig. 1.5. The LM is launched from the lunar surface, about 7 min after the power ascent, into a 17 km × 83 km elliptical orbit. At 40 min after the launch, the standard terminal phase conducts TPI. If necessary, a midcourse correction of the final approach phase can be made 55 min after the launch. TPF was carried out 85 min after the launch. After the braking, the two spacecraft were in the same orbit with a relative distance of several hundred meters, and the final docking was manually completed by the astronauts. The short rendezvous orbital maneuver data are provided in Table 1.7.

1.3.4 Future Development Trend of Rendezvous and Docking Technology Rendezvous and docking is primarily used to maintain the normal operation of manned space stations/space laboratories. In recent years, with the expansion of human space exploration activities, researchers have begun to focus on the development of rendezvous and docking technologies for new tasks such as in-orbit space services and space security. Beyond Earth and lunar orbits, humans have also explored deep space, and Mars exploration has entered the engineering practice stage. In this type of task, rendezvous and docking is a key technique necessary for sampling returns. From the perspective of mission objectives, rendezvous and docking technologies are faced with multiple types of orbits, including elliptical/circular orbits and Earth/deep space object orbits, as well as different types of target vehicles including cooperative/non-cooperative ones. From the point of view of time and space, rendezvous and docking technology should be capable of completing tasks in all directions. In terms of system safety, rendezvous and docking missions, particularly manned rendezvous and docking missions, have put forth high requirements for the trajectory safety owing to their close relative movements. For the docking

1.3 Overview of Rendezvous and Docking

27

CDH

CSI

Launch

Flight trajectory Insertion trajectory Phasing trajectory with 83 km height Target’s trajectory

PC

TPI

Middle correction CDH

Launch

Middle correction Flight trajectory Phasing trajectory with 83 km height Target’s trajectory

TPF, Docking

Fig. 1.4 Flight process of co-elliptical rendezvous [18]

method, the light and miniaturized weak impact docking mechanism and manipulator are widely used. Space rendezvous and docking is a comprehensive and interdisciplinary technology that is complex and involves a certain level of risk. Achieving all-day, alldirectional, multi-form, and safe rendezvous and docking with different types of targets on various orbits is a challenge in terms of the guidance, navigation, and control theory and method, computer technology, measurement technology, as well as other disciplines. The research on rendezvous and docking technology is of great scientific significance and engineering application value.

28

1 Introduction TPI

Middle correction

Launch

Middle correction Flight trajectory Insertion trajectory Target’s trajectory

TPF, Docking

Fig. 1.5 Representative short rendezvous technique [2]

Table 1.7 The characteristics of short rendezvous [19] Rendezvous maneuver Insertion

Time from previous maneuver (min) 7

V /(m/s)

Actuator

Aiming orbit Apogee/Pergee (km)

–

APS

85.192/16.668

TPI

45

22.2504

APS

112.972/83.34

Terminal Braking

43

13.716

RCS

111.12/111.12

1.4 Guidance Navigation and Control for RVD The rendezvous and docking GNC system has the following functions: (1) the objective of guidance is to generate the increment of orbital control velocity required to make the chaser reach the predetermined intersection point; (2) the objective of navigation is to estimate the position and attitude parameters required for guidance and control functions by using sensor measurement information and combining position/attitude dynamics and kinematics models; (3) the objective of control is to calculate the control instructions using the on-board computer and to drive the executive mechanism, such as the engine and angular momentum exchange device, in order to apply command force and torque on the spacecraft. In order to realize these functions, it is necessary to configure the corresponding measurement sensors, on-board computers, control actuators, and other hardware equipment and develop guidance, navigation and control strategies and algorithm

1.4 Guidance Navigation and Control for RVD

29

software. This section introduces the general rendezvous and docking GNC system from the perspective of system composition. Figure 1.6 shows the composition of the rendezvous and docking GNC system.

Cooperative equipments for relative measurement

Satellite link

Manual control system TV Attitude control handle

Satellite navigation receiver

Transponder for Wave radar

Angle reflectors for laser radar

Reflectors for CRDS

Relative measurement system Satellite navigation receiver

Wave radar

Laser radar

CRDS

Targets for manual control

Target spacecraft Chaser spacecraft

Translation control handle Display instrument and lighting equipment Keyboard and buttons Manual control circuit Attitude measurement system

GNC computer

Flight management

Guidance navigation and control

Gyroscope Infrared earth sensor Solar sensor

Star sensor

Fig. 1.6 The rendezvous and docking GNC system

Propulsion system Angular momentum exchange device

30

1 Introduction

1.4.1 Sensors

1. Attitude and Inertial Measurement Sensors (1) Infrared Earth Sensor The infrared earth sensor functions based on the high gradient of the infrared radiation field of Earth near the infrared horizon (the junction of the Earth and space) to determine the orientation of the geocentric vector of the satellite in the system, thus determining the rolling angle and pitching angle of the satellite relative to the orbit frame. (2) Solar Sensor A solar sensor is an instrument that can obtain satellite attitude information through sensitive sunlight [20]. Based on their output signal types, solar sensors can be divided into two categories: analog and digital solar sensors. Between them, the output signal of the analog solar sensor is a continuous function of the change of the solar angle, while the output signal of the digital solar sensor is a discrete coded digital signal. (3) Star Sensor The star sensor determines the satellite’s triaxial attitude to the celestial coordinate system (inertial system) by detecting the radiation from the star. Because the angle of the star is very small, it can be considered as the point light source; additionally, its high position stability allows very high measurement accuracy of the star sensor. However, because the light from the star is weak, it is easily disturbed by interfering light. Therefore, the star sensor has a small field of view and strict suppression requirements for sunlight and reflected light interference. (4) Gyroscope The gyroscope is an important part of an inertial navigation unit. Based on the different measurement systems, it can be classified into mechanical rotor, optical (fiber optic and laser), electrostatic, and flexible gyroscopes, among others. While different types of gyroscopes have different measuring principles, they can all be used to measure the angular velocity of a spacecraft relative to inertial space. (5) Accelerometer The accelerometer is an important part of the inertial measurement unit. The nongravitational force acting on a unit mass is called a specific force. The accelerometer obtains the acceleration by measuring the inertial force that it generates. Essentially,

1.4 Guidance Navigation and Control for RVD

31

the accelerometer is a dynamometer that can measure the acceleration generated by the specific force in addition to the gravitational acceleration. 2. Relative measurement sensor (1) Satellite Navigation Receiving Equipment Satellite navigation systems receive navigation and positioning signals sent by navigation satellites, using navigation satellites as dynamic known points to measure the position and speed of moving carriers in real time in order to navigate. These systems can provide all-weather, all-day, high-precision navigation and time services for all types of users on Earth’s surface and near-Earth space. At present, there are four main types of satellite navigation systems in the world, including the USA’s GPS, Russia’s GLONSS, ESA’s Galileo satellite navigation system, and China’s Beidou satellite navigation system. The satellite navigation receiving equipment is the equipment with which the motion carrier receives the satellite navigation message and performs signal processing. (2) Microwave Radar and Target Transponder A microwave radar uses electromagnetic waves to detect the distance and azimuth of the target relative to the radar. The radar antenna of the transmitter sends electromagnetic signals to the target, and the feedback signal reflected by the target or fired by its transponder is received by the antenna once more. Signal processing equipment is used to extract the relative distance, relative distance change rate, and azimuth information. (3) Laser Radar and Cooperative Angle Reflector A laser radar is a radar using laser as the carrier, because the laser provides optical wave band electromagnetic radiation, whose wavelength is much shorter than microwave and millimeter wave. Therefore, compared with a microwave radar, the laser radar has a smaller size, mass, and superior angular and ranging resolution. Unlike a microwave radar, which requires transponders on the target, a laser radar requires cooperative target angle reflectors on the target to reflect the laser back to the laser radar. (4) Camera type Rendezvous and Docking Sensor (CRDS) and Cooperative Targets A CRDS is used to measure relative motion parameters between the chaser and target within hundreds of meters. More than three characteristic light points are installed on the target for CRDS. The CRDS camera captures these characteristic points and carries out image processing using an information processor to complete

32

1 Introduction

the extraction and recognition of light points and obtain the image points in the camera coordinate system. Finally, the relative parameters between the target and camera coordinate systems is solved by the measurement algorithm. The relative position and attitude parameters can be obtained according to the sensor installation. (5) Television (TV) and Feature Targets TV and feature targets are the main optical measuring equipment for manned rendezvous and docking. A TV is generally installed on the chaser, and its supporting feature target is installed on the target. The TV provides images of feature targets, and the astronaut estimates the relative motion of the target relative to the chaser by looking at the feature targets and/or feature target images displayed on the monitor.

1.4.2 Actuators

1. Propulsion System The propulsion system is a mass ejection type control actuator. It can be divided into four categories. (1) Air conditioning propulsion system: This system uses compressed gas as a working medium, with low specific impulse, and it is mainly used for satellites with short service life. (2) Single element propulsion system: This system mainly uses anhydrous hydrazine as the propellant. (3) Binary propulsion system: Nitrogen tetroxide and methyl hydrazine are used as propellants in this system. Owing to the high specific impulse, it is suitable for large spacecraft. (4) Electric propulsion system: The specific impulse of electric propulsion is much higher than that of chemical propulsion, but its thrust is small and the technology is complex. In the rendezvous and docking mission, the dual-element and singleelement propulsion systems are primarily used for attitude and orbit control. 2. Angular Momentum Exchange Device An angular momentum exchange device is a type of commonly used spacecraft attitude control actuator, whose primary features include a fixed momentum wheel, control torque gyroscope, frame momentum wheel, and three other types of inertia actuators. When the angular momentum value and direction of the angular momentum exchange device installed by the satellite change in accordance with certain laws, the reaction torque required will be generated so as to implement continuous control of the satellite attitude. The angular momentum exchange device

1.4 Guidance Navigation and Control for RVD

33

is primarily used for attitude stability control in the non-orbit control stage of rendezvous and docking. 3. Translation Control Handle The translational control handle is the main tool used by astronauts to directly control the translation of the spacecraft. It has an independent three-degree-of-freedom output and is used to control the forward, backward, vertical, and horizontal motion of the spacecraft’s center of mass. 4. Attitude Control Handle The attitude control handle is the main tool used by astronauts to directly control the spaceship attitude. It has an independent three-degree-of-freedom output, and under certain external force action, it may separately or simultaneously revolve two or three orthogonal axes;. 5. Keyboard and Buttons The relatively independent dedicated GNC control panel area for astronaut manual control is set on the instrument panel, including a matrix button, two-position switch, instrument coding command board, and handheld control unit, which are used to issue startup command, control mode selection, data input, display selection, fault detection and initialization, etc.

1.4.3 Controllers

1. GNC Computer The main functions of the GNC computer (GNCC) contain these: (1) The computer collects and process the information of each sensor. According to the processing results, the control signals are sent to the energy and propulsion subsystem to complete the normal and fault modes GNC mission; (2) The computer corrects spacecraft orbit data and system parameters and disseminate other information; (3) The computer receives the commands of the data management subsystem, remote control, and astronauts, according to which the working mode of the controller is determined and the corresponding information is provided. The computer sends GNC subsystem status and data information to data management subsystem and telemetry.

34

1 Introduction

2. Manual Control Circuit The manual control circuit is the key equipment for astronauts to carry out human control. It can perform attitude information processing, manual control information processing, and control law generation during manual rendezvous and docking. The manual control circuit receives the output signal of the control handle, calculates the control instruction according to the control requirements, sends the engine control instruction to the drive circuit of the propulsion subsystem, and controls the jet actuator.

References 1. Zhou, J.P.: Space Rendezvous and Docking Technology. National Defense Industry Press, Beijing (2013) 2. Bainan, Zhang: Spacecraft Docking Mission Analysis and Design. Science Press, Beijing (2011) 3. Goodman, J.L., Brazzel, J.P., Chart, D.A.: Challenges of orion rendezvous development. In: AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, South Carolina (2007) 4. Goodman, J.L.: History of space shuttle rendezvous. JSC 63400 (2011) 5. Zhu, R., et al.: Sue/Russian rendezvous and docking technology research. Spacecraft Eng. 20(6), 16–31 (2011) 6. Zhu, R., Wang, H., Xu, Y., Wei, Y.: Study on rendezvous techniques of American spacecraft. Spacecraft Eng. 20(5), 11–36 (2011) 7. Hu, J., et al.: Shenzhou-8 spacecraft guidance navigation and control system and flight result evaluation for rendezvous and docking. Aerosp. Control Appl. 37(6), 1–5 (2011) 8. Cavrois, B., Reynaud, S., Personne, G., Chavy, S.: ATV GNC and safety functions synthesis: overall design, main performances and operations. In: AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii (2008) 9. Ueda, S., Kasai, T., Uematsu, H.: HTV Rendezvous technique and GN&C design evaluation based on 1st flight on-orbit operation result. In: AIAA/AAS Astrodynamics Specialist Conference, Toronto, Ontario Canada (2010) 10. Tang, G.-J., Luo, Y.-Z., Jin, Z.: Space Rendezvous and Docking Mission Planning. National Defense Industry Press, Beijing (2008) 11. Xie, Y., Hu, Y.: Rendezvous strategy review and autonomous rapid rendezvous scheme research. Aerosp. Control Appl. 40(4), 1–8 (2014) 12. Fehse, W.: Automated Rendezvous and Docking of Spacecraft. Cambridge University Press, London (2003) 13. Zhu, R.: Rendezvous and Docking Techniques of Spacecraft. National Defense Industry Press, Beijing (2007) 14. de Pasquale1, E., Bonnet, M., Caluwaerts, D.: In-flight demonstration of safety critical operations of ATV Jules Verne GNC. In: AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii (2008) 15. Murtazin, R.F., Budylov, S.G.: Short rendezvous missions for advanced russian human spacecraft. Acta Astronaut. 67, 900–909 (2010) 16. Murtazin, R., Petrov, N.: Short profile for the human spacecraft soyuz-TMA rendezvous mission to the ISS. Acta Astronaut. 77, 77–82 (2012) 17. Anatoly Zak.: Soyuz TMA-08M [EB/OL] (2013-9-23). http://www.russianspacewebcom/iss_ soyuz_tma08n.html

References

35

18. Young, K.A., Alexander, J.D.: Apollo lunar rendezvous. J. Spacecr. 7(9), 1083–1086 (1970) 19. Alexunder, J.D., Becker, R.W.: Apollo experience report evolution of the rendezvous-maneuver plan for the lunar-landing missions. NASA TN D-7388 (1973) 20. Tu, S.: Satellite Attitude Dynamics and Control. Beijing Astronautics Press (2003)

Chapter 2

Rendezvous Kinematics and Dynamics

The kinematics and dynamics of rendezvous and docking include orbital equations, attitude equations, and relative motion equations that including the relative position equations and relative attitude equations. This chapter mainly introduces the dynamics equations necessary for the following chapters, including the definition of the reference frame, orbital dynamics equations, attitude kinematics and dynamics, relative position dynamics equations, and relative attitude motion equations.

2.1 Reference Frames Definition of a quadrant: from the rear end of the spacecraft along its longitudinal axis, in the clockwise direction, the quadrant is follows the sequence I, II, III, and IV. The quadrant lines are spaced 90° apart, and the I quadrant line points toward the ground during normal on-orbit flight. The reference frames used in this book are for the dynamics of orbit, attitude, and relative motion. Specifically, the following reference frames are used: (1) Earth-Centered Equatorial Inertial Frame OE X I YI Z I The origin OE is the center of mass (CoG) of Earth, the OE X I axis points to the 2000.0 mean vernal equinox, and the OE Z I axis is normal to the 2000.0 mean equatorial plane and is consistent with the direction of Earth’s rotation angular velocity. The OE YI , OE X I , and OE Z I satisfy the right hand rule. (2) Cylindrical Frame O E r θ z Geocentric cylindrical frame O E r θ z: the origin is located at the center of Earth O E , and the O E r axis is radial from the spacecraft CoM to the center of the Earth. The © National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_2

37

38

2 Rendezvous Kinematics and Dynamics

O E θ axis is the counterclockwise direction in the reference plane. The O E z, O E r , and O E θ satisfy the right hand rule. Target cylindrical frame OT r θ z: the origin is located in the target’s CoG OT . The OT r axis is radial from the spacecraft CoM to the center of the Earth. The OT θ axis is the counterclockwise direction in the reference plane. The OT z, OT r , and OT θ satisfy the right hand rule. (3) Spacecraft Orbital Frame Oo X o Yo Z o The origin Oo is the CoG of the spacecraft, the Oo Zo axis points to the CoG of Earth. The Oo Yo axis is normal to Oo Zo and is directed toward the negative direction of the angular velocity of the orbit. The Oo Xo , Oo Zo , and Oo Yo satisfy the right hand rule. (4) WGS-84 Frame OW84 X W84 YW84 Z W84 The origin OW84 is at the center of Earth, the OW84 Z W84 axis points to the BIH1984.0 Conventional Terrestrial Pole (CTP), and the OW84 X W84 axis points to the intersection point of the zero meridional plane of BIH1984.0 and the conventional equator. The OW84 YW84 , OW84 X W84 , and OW84 Z W84 satisfy the right hand rule. (5) Rendezvous and Docking Frame (RVD Frame) OT X r Yr Z r The origin OT is the CoG of the target, and the OT Zr axis points to the CoG of Earth. The OT Yr axis is normal to the OT Zr axis pointing to the angular velocity direction of the orbit. The OT X r , OT Zr , and OT Yr satisfy the right hand rule. (6) TH Frame OT X o Yo Z o The origin OT is the CoG of the target. The OT Xo axis points from the centroid of Earth to the spacecraft. The OT Yo axis is the spacecraft’s flight direction. The OT Xo , OT Yo , and OT Zo satisfy the right hand rule. (7) Line of sight Frame OT ξ ηζ The origin OT is at the CoG of the target, and the direction of OT ξ is the direction of the line of sight from the target to the chaser. OT η is the direction of angular momentum in space along the line of sight, and the OT ζ axis and the OT ξ axis form the rendezvous plane. The OT ζ , OT ξ , and OT η satisfy the right hand rule. (8) Spacecraft Body Frame Oo X b Yb Z b The origin Oo is the spacecraft’s CoG, and Oo X b points to the spacecraft head along the longitudinal axis. Oo Z b is normal to Oo X b and points toward the I quadrant line of the spacecraft. The Oo Yb , Oo Z b , and Oo X b satisfy the right hand rule.

2.1 Reference Frames

39

(9) Target Docking Frame OTd X d Yd Z d The origin OTd is at the center of the target docking mechanism’s docking frame. The OTd X d axis coincides with the longitudinal axis of the target and points to the rear end of the target. The OTd Yd axis is along the axial direction of the target and is normal to the longitudinal axis, pointing to the II quadrant line. The OTd Z d , OTd X d , and OTd Yd satisfy the right hand rule.

2.2 Orbit Dynamics During the far range rendezvous phase of the rendezvous and docking mission, the distance between the two spacecraft is relatively long. Orbital control of the chaser is generally regarded as independent or as being controlled to a predetermined target position.

2.2.1 Two-Body Problem In spacecraft orbital analysis, it is generally assumed that the spacecraft moves in the gravitational field of Earth, ignoring other perturbation forces, to analyze the orbital motion of the spacecraft as a two-body problem. Spacecraft motion is described in the earth-centered equatorial inertial frame OE X I YI Z I . Earth’s mass is given as M. The mass of the spacecraft is m. By Newton’s law, we have G Mm  r  r2 r  G Mm r 

m r¨ m = − M r¨ M =

r2

r

(2.1) (2.2)

Subtracting Eq. (2.1) by Eq. (2.2), we get r¨ = −

G(M + m) r r3

(2.3)

 3 m where G is the universal gravitational constant G = 6.672 × 10−11 kgs 2 . When describing the two-body motion of g the spacecraft and Earth, the mass of the spacecraft is much smaller than the that  of3 Earth. therefore, we have G(M +m) ≈ ; then, the two-body motion equation G M. We denote μ = G M = 398600.5 km s2 is given as

40

2 Rendezvous Kinematics and Dynamics

Fig. 2.1 Schematic description for orbital elements

P

f OE Vernal equinox

A

r¨ +

μ r=0 r3

ω Ω Equator

i

Orbit

(2.4)

2.2.2 Orbital Elements Generally, we use Keplerian orbital elements to describe the motion of the spacecraft. The six orbital elements are the semimajor axis a, eccentricity e, inclination i, longitude of the ascending node Ω, argument of perigee ω, and true anomaly f . In the earth-centered equatorial inertial frame OE X I YI Z I , the ascending node Ω and inclination i determine the orbit orientation in inertial space. In the orbital plane, the orientation of the arch line on the orbital plane is described by argument of perigee ω. a and e depict the elliptical shape, and f reflects the movement of the spacecraft in the orbit with time (See Fig. 2.1).

2.2.3 Orbital Perturbation Equations The forces of the actual spacecraft’s flight are complex. We define the force except for the gravitational force as the perturbation force. Under the perturbation forces, the orbital elements of the spacecraft gradually change. The perturbation equations describe the relationship between the perturbation forces and the orbital elements. The components of perturbation acceleration in the spacecraft orbital frame Oo X o Yo Z o are expressed as the radial component ar , the lateral component au , and the negative normal component ah . The perturbation equations of the six orbital elements are

2.2 Orbit Dynamics

41

2 da = √ [ar e sin f + au (1 + e cos f )] dt ωo 1 − e2 √ de 2 1 − e2 = [ar sin f + au (cos f + cos E)] dt ωo a

(2.5)

(2.6)

r sin(ω + f ) dΩ = ah √ dt ωo a 2 1 − e2 sin i

(2.7)

r cos(ω + f ) di = ah (2.8) √ dt ωo a 2 1 − e2 √    r sin(ω + f ) 1 − e2 dω r = −ar cos f + au sin f 1 + − cos i ah √ dt ωo ae p ωo a 2 1 − e2 sin i (2.9)    √ μp cos f p df r p sin f = 2 + ar − 1+ au (2.10) dt r e μ e p μ where ωo =

μ/a 3 , p = a(1 − e2 ), r =

a(1−e2 ) 1+e cos f

, and cos E =

e+cos f 1+e cos f

.

2.2.4 Perturbation Acceleration The perturbation forces of the spacecraft include Earth’s gravitational field perturbation, air resistance perturbation, Earth-Moon gravity perturbation, and solar pressure perturbation. For near-Earth orbit rendezvous, the gravity of Earth and the air resistance perturbation are the two main perturbation forces. 1. Orbital Perturbation Caused by Non-Centrality of Earth’s Gravitational Field Consider the influence of the gravitational potential of Earth involving a band-shaped harmonic function. The corresponding expression of the gravitational potential is as follows: U = −μ

∞

Jn R nE r −(n+1) Pn (sin φ)

(2.11)

n=2

where Pn (sin φ) is n-th order Legendre polynomial, φ is the geocentric latitude,and R E is Earth’s equatorial radius. Pn (x) =

1 2n n!

dn 2 (x − 1)n dxn

(2.12)

42

2 Rendezvous Kinematics and Dynamics

The radial component of the perturbation gravitational acceleration is obtained as follows (positive to upward): gr =

∞

∂U =μ Jn R nE (n + 1)r −(n+2) Pn (sin φ) ∂r n=2

(2.13)

The meridional component is (positive to the north) gm =

∞

1 ∂U = −μ Jn R nE r −(n+2) Pn (sin φ) cos φ r ∂φ n=2

(2.14)

The latitudinal component is (positive to the east) g p =

∂U 1 =0 r cos(φ) ∂λ

Transforming gr , gm , and g p into perturbation components ar , au , and ah , and letting σ be the angle between the local meridional plane and orbital plane, we get ar = gr = μ

∞

Jn R nE (n + 1)r −(n+2) Pn (sin φ)

(2.15)

n=2

au = gm cos(σ ) + g p sin(σ ) = −μ

∞

Jn R nE r −(n+2) Pn (sin φ) cos φ cos(σ )

(2.16)

n=2

ah = gm sin(σ ) − g p cos(σ ) = −μ

∞

Jn R nE r −(n+2) Pn (sin φ) cos φ sin(σ )

(2.17)

n=2

2. Orbital Perturbation Caused by Air Resistance The air resistance to a spacecraft moving in the atmosphere is 1 D = C D ρϑa2 S 2

(2.18)

where C D is the resistance coefficient, ρ is the atmospheric density, S the windward area, and ϑa the relative velocity of the spacecraft with respect to air.

2.2 Orbit Dynamics

43

Assume that the atmosphere does not rotate with Earth and the windward area of the spacecraft is constant, the acceleration introduced by the air resistance is as follows: at = −C D

1 ρv 2 S 2m

(2.19)

2.3 Attitude Kinematics and Dynamics The spacecraft body frame Oo X b Yb Z b determines its attitude relative to the direction of the reference system. The descriptive forms include direction cosine matrix (DCM), Euler angle, and quaternion. Among them, the Euler angle and quaternion are the two most commonly used forms for engineering design and analysis.

2.3.1 Attitude Kinematics

1. Euler Angle Euler angle is a simple, intuitive, and geometrically significant attitude parameter. According to the Euler principle, the angular displacement of a rigid body around a fixed point is regarded as a composite of several limited rotations about that point. In Euler rotation, the reference system is rotated thrice to obtain the body frame. In the three rotations, the rotation angle of each time is the Euler angle. Therefore, the attitude matrix determined by the Euler angle is the cubic product of the transformation matrix. These matrices have the following form: ⎡

⎡ ⎤ ⎤ 1 0 0 cos θ 0 − sin θ Cx (ϕ) = ⎣ 0 cos ϕ sin ϕ ⎦, Cy (θ ) = ⎣ 0 1 0 ⎦, 0 − sin ϕ cos ϕ sin θ 0 cos θ ⎡ ⎤ cos ψ sin ψ 0 Cz (ψ) = ⎣ − sin ψ cos ψ 0 ⎦ 0 0 1

(2.20)

For the most commonly used 3-1-2 rotation sequence, the angles are denoted by ψ,φ, and θ . They describe the relationship between the spacecraft’s body frame and the reference frame. The DCM of the attitude determined by the Euler angle is the cubic product of the transformation matrix:

44

2 Rendezvous Kinematics and Dynamics C312 (ψ, ϕ, θ ) = Cy (θ )Cx (ϕ)Cz (ψ) ⎡ ⎤ cos ψ cos θ − sin ψ sin ϕ sin θ sin ψ cos θ + cos ψ sin ϕ sin θ − cos ϕ sin θ ⎢ ⎥ =⎣ − sin ψ cos ϕ cos ψ cos ϕ sin ϕ ⎦ cos ψ sin θ + sin ψ sin ϕ cos θ sin ψ sin θ − cos ψ sin ϕ cos θ cos ϕ cos θ

(2.21) The relationship between the 3-1-2 Euler angle and the elements of DCM is given by  C21 C22 ϕ = arcsin(C23 )   C13 θ = − arctan C33 

ψ = − arctan

(2.22)

where Ci j (i, j = 1, 2, 3) is the i-th row and the j-th column element of the attitude matrix C312 . Such Euler’s rotation has a singularity at ϕ = 90° because ψ and θ rotate in the same plane and cannot be uniquely determined. If the reference frame is the spacecraft orbital frame Oo X o Yo Z o , the Euler angles ϕ, θ , and ψ are the roll, pitch, and yaw angles, respectively. The attitude angular velocity ω of the spacecraft relative to the reference system expressed in the spacecraft body frame is T  ω = ω x ω y ωz

(2.23)

This velocity can be regarded as a combination of three Euler angle rotations. For the 3-1-2 Euler rotation, we have ⎡

ω312

⎤ ⎡ ⎤ ⎡ ⎤ 0 ϕ˙ 0 = Cy (θ )Cx (ϕ)Cz (ψ)⎣ 0 ⎦ + Cy (θ )Cx (ϕ)⎣ 0 ⎦ + Cy (θ )⎣ θ˙ ⎦ ψ˙ 0 0

(2.24)

The kinematics based on the Euler angle can be obtained as follows: ⎤ ⎤ ⎡ φ˙ (ωx cos θ + ωz sin θ ) cos φ ⎣ θ˙ ⎦ = 1 ⎣ ω y cos φ + (ωx sin θ − ωz cos θ ) sin φ ⎦ cos φ ψ˙ ωz cos θ − ωx sin θ ⎡

(2.25)

The above equation is a nonlinear equation, and φ = 90◦ is a singular point. In large angle maneuvers, using Euler angles to described kinematics of attitude may lead to singularities.

2.3 Attitude Kinematics and Dynamics

45

2. Quaternion A quaternion is a hypercomplex with four elements that can describe a reference frame or a rotation of vector relative to a certain reference frame. It is defined as   q q¯ = (2.26) q4 where q is the vector part of the quaternion and the real number q4 is the scalar part of the quaternion. Besides, ⎤   q1 φ ⎦ ⎣ q = q2 = e sin 2 q3   φ q4 = cos 2 ⎡

(2.27)

(2.28)

where e is the unit vector in the direction of the rotation axis and φ is the rotation angle. The relationship between quaternion and attitude DCM is ⎡ ⎤ C − C32 1 ⎣ 23 q= C31 − C13 ⎦ 4q4 C12 − C21 1 1 q4 = ± (tr C312 + 1) 2 2

(2.29)

The kinematics expressed by the quaternion is ⎤⎡ ⎤ ⎡ ⎤ q1 0 ωz −ω y ωx q˙1 ⎢ q˙2 ⎥ 1 ⎢ −ωz 0 ωx ω y ⎥⎢ q2 ⎥ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎣ q˙3 ⎦ 2 ⎣ ω y −ωx 0 ωz ⎦⎣ q3 ⎦ q˙4 −ωx −ω y −ωz 0 q4 ⎡

(2.30)

or in a compact form 1 1 ¯ q˙¯ = (ω)q¯ = (q)ω 2 2

(2.31)

    ×   −ω ω q 4 I3 + q × ¯ = (ω) = , (q) −ωT 0 −qT

(2.32)

where

46

2 Rendezvous Kinematics and Dynamics

⎡

⎡ ⎤ ⎤ 0 −ωz ω y 0 −q3 q2 ω× = ⎣ ωz 0 −ωx ⎦, q× = ⎣ q3 0 −q1 ⎦ −ω y ωx 0 −q2 q1 0

(2.33)

In the quaternion kinematics, there is no trigonometric function and no singularity problem. It is therefore widely used in attitude control systems.

2.3.2 Attitude Dynamics Attitude dynamics for spacecraft include rigid body attitude dynamics and attitude dynamics with flexible solar panels. 1. Attitude Dynamics for Rigid Body The rigid body attitude dynamics is derived from the momentum moment theorem of rigid bodies: for a rigid body, the change rate of the angular momentum to a fixed point in inertial space is equal to the sum of the moments of all the external forces acting on that point. In a body frame, the theorem can be expressed as ˙ H+ω ×H=T

(2.34)

where H is the angular momentum of the spacecraft, ω is the angular velocity of the spacecraft relative to the earth-centered equatorial inertial frame, and T is the external torque exerted on the spacecraft, including control torque, disturbance torque, and space environmental torque. The angular momentum vector H satisfies H = Is ω. If the inertia matrix is a ⎡ ⎤ Ix 0 0 diagonal matrix Is = ⎣ 0 I y 0 ⎦, Eq. (2.34) can be expressed by a scalar form: 0 0 Iz ⎧ ⎨ Ix ω˙ x + (Iz − I y )ω y ωz = Tx I ω˙ + (Ix − Iz )ωx ωz = Ty ⎩ y y Iz ω˙ z + (I y − Ix )ω y ωx = Tz 2. Attitude Dynamics with Flexible Solar Panels Spacecraft generally have symmetrical two-wing solar panels. When external torques are exerted on the spacecraft, the solar panels may vibrate under no active vibration control. Generally, we can establish the dynamics for the central rigid body and flexible panels, which includes the central rigid body attitude dynamics and the panel vibration equation, which are following:

2.3 Attitude Kinematics and Dynamics

M X¨ + Ftr s η¨ r s + Ftls η¨ ls = Ps Is ω˙ s + ω˜ s Is ωs + Fsls η¨ ls + Fsr s η¨ r s + Rasls ω˙ als + Rasr s ω˙ ar s = Ts

47

(2.35) (2.36)

T Ials ω˙ als + Fals η¨ ls + Rasls ω˙ s = Tals

(2.37)

T ˙ s = Tar s Iar s ω˙ ar s + Far s η¨ r s + Rasr sω

(2.38)

2 T ¨ T T η¨ ls + 2ξls Ωals η˙ ls + Ωals X + Fsls ω˙ s + Fals ω˙ als = 0 ηls + Ftls

(2.39)

2 T ¨ T T ˙ s + Far ˙ ar s = 0 η¨ r s + 2ξr s Ωar s η˙ r s + Ωar s ηr s + Ftr s X + Fsr s ω sω

(2.40)

where Eq. (2.35) is the system centroid translation Eq. (2.36) is the equation of rotational motion of the system around the centroid. Eqs. (2.37) and (2.38) are the solar panel control Eqs. (2.39) and (2.40) are the solar panel vibration Eqs. (2.37) and (2.38) are based on the attitude control reference frame, whereas Eqs. (2.39) and (2.40) are based on the local reference frame of the solar panel. More details on variables are as follows: ωs ω˜ s M Is Ps Ts Tals , Tars Ials , Iars ωals , ωars Ωals , Ωar s ηls , ηr s ζls , ζr s Ftls , Ftr s Fsls , Fsr s Fals , Fars Rasls , Rasrs

angular velocity vector of the central rigid body, ∈ 3 × 1; skew-symmetric matrix of ωs , ∈ 3 × 3; mass matrix of the central rigid body, ∈ 3 × 3; inertia matrix of the central rigid body relative to CoG, ∈ 3 × 3; external force vector exerted on the central rigid body, ∈ 3 × 1; external torque vector exerted on the central rigid body, ∈ 3 × 1; the control torque vector exerted on the left and right solar panels, ∈ 3 × 1; inertia matrix of the left and right solar panels relative to their respective connection points, ∈ 3 × 3; rotational angular velocity vector of the left and right solar panels, ∈ 3 × 1; modal frequency diagonal matrix of the left and right solar panels, ∈ m × m; modal co-ordinate vector of the left and right solar panels, ∈ m × 1; modal damping coefficient of the left and right solar panels; flexible coupling coefficient matrix of the left and right solar panel vibration and translation of the central rigid body, ∈ 3 × m; flexible coupling coefficient matrix of the left and right solar panel vibration and rotation of the central rigid body, ∈ 3 × m; flexible coupling coefficient matrix of the left and right solar panel vibration and rotation of solar panel itself, ∈ 3 × m; rigid coupling coefficient matrix of the left and right solar panel rotation and rotation of spacecraft, ∈ 3 × 3;

48

m

2 Rendezvous Kinematics and Dynamics

modal order.

2.3.3 Spacial Environmental Torques The environmental torque exerted on the spacecraft is a factor that affects its attitude control accuracy and frequency. Environmental disturbance torques include gravity gradient torque, aerodynamic torque, remanence torque, and solar pressure torque. The gravity gradient torque and aerodynamic torque are the two main factors in near-Earth orbit rendezvous and docking. 1. Gravity Gradient Torque Gravity gradient torque depends not only on spacecraft mass distribution and the mass distribution of Earth but also on the position and attitude of the spacecraft. The general expression for the gravity gradient torque used in engineering is Tg =

3μ EIs E r3

where μ is the gravitational constant of Earth, r is the distance from the center of Earth to the centroid of the spacecraft, E is a unit vector pointing from the centroid of the spacecraft to the center of Earth, and E = − rr . In the three-axis stability spacecraft flight, if the inertia matrix is diagonal Is = ⎡ ⎤ Ix 0 0 ⎣ 0 I y 0 ⎦, the gravity gradient torques expressed in the spacecraft body frame are 0 0 Iz ⎧ g 3μ ⎨ Tx = r 3 (Iz − I y ) cos θ cos ϕ sin ϕ g T = 3μ3 (Ix − Iz ) sin θ cos θ cos2 ϕ ⎩ y g r3μ Tz = r 3 (I y − Ix ) sin θ sin ϕ cos ϕ

(2.41)

2. Aerodynamic Torque Aerodynamic torque depends on the characteristics of the local orbital atmosphere, atmospheric motion generated by the rotation of Earth, the spacecraft’s aerodynamic and mass characteristics, interactions between the spacecraft and atmosphere, and relative velocity of the atmosphere. The general calculation in engineering is as follows: Ta =

ρVR2 CD ri × |Si · vr |vr 2 i

(2.42)

2.3 Attitude Kinematics and Dynamics

49

where ρ is the atmospheric density; C D is the resistance coefficient; Si is the area of the i-th plate of the spacecraft and it is along the normal direction of the surface; ri is the distance from the centroid of the spacecraft to the center of the i-th plate; vr is the unit vector of the incoming flow direction; and VR is the velocity of the atmosphere relative to the spacecraft.

2.4 Relative Motion 2.4.1 Relative Motion in Circular Orbital Frame 1. CW Equation The CW equation was developed in the 1960s by W. H. Clohessy and R. S. Wiltshire. It is a linearized equation of the relative motion in the close range of the spacecraft during rendezvous and docking. The premise of its establishment is the relative distance between the two spacecraft is much less than the target’s orbital radius, the target moves in a circular orbit and with a first-order approximation of the gravitational field [1]. The relative motion of the two spacecraft can be approximately expressed as in the earth-centered equatorial inertial frame as ρ¨ +

RT · ρ μ (ρ − 3 2 RT ) = a 3 RT RT

(2.43)

where ρ is the relative distance vector of the two spacecraft; RT is the distance from the target to the center of Earth; RT is the vector from the target to the center of Earth; a is the acceleration generated by the joint force by the external forces except gravity T  on the chaser; and RT and a are expressed in the RVD frame as RT = 0 0 −RT T  and a = ax a y az . Taking the derivative of ρ in the RVD frame ρ¨ I = ρ¨ T + 2ω × ρ˙ T + ω˙ × ρT + ω × (ω × ρT )

(2.44)

where ρ¨ T , ρ˙ T and ρT are the second-order, first-order, and zero-order derivatives of the relative distance vector ρ in the RVD frame, respectively; ρ¨ I and ρ˙ I are the second-order and first-order derivatives of ρ in the earth-centered equatorial inertial frame; ω is the angular velocity of the RVD frame with respect to the earth-centered equatorial inertial frame, expressed in the RVD frame; and ωoT is the orbital angular velocity of the target. Then, we get T  ω = 0 ωoT 0

(2.45)

50

2 Rendezvous Kinematics and Dynamics

Let x, y, and z be the components of the relative position of the two spacecraft in the RVD frame:  T ρT = x y z

(2.46)

Then, we can write the CW equations as ⎧ ⎪ ⎨ x¨ + 2ωoT z˙ = ax 2 y¨ + ωoT y = ay ⎪ ⎩ 2 z¨ − 2ωoT x˙ − 3ωoT z = az

(2.47)

When the acceleration on the right side of the Eq. (2.47) is zero, that is, a = 0, and the initial relative states x0 , y0 , z 0 , x˙0 , y˙0 and z˙ 0 are known, the following analytical solutions can be obtained:     ⎧ 4x˙0 2˙z 0 2˙z 0 ⎪ ⎪ + 6z t) + cos(ω t) − (6ω z + 3 x ˙ )t + x − sin(ω x(t) = 0 oT oT oT 0 0 0 ⎪ ⎪ ωoT ωoT ωoT ⎪ ⎪ ⎪ ⎪ ⎪ y ˙ 0 ⎪ ⎪ y(t) = sin(ωoT t) + y0 cos(ωoT t) ⎪ ⎪ ⎪ ωoT ⎪ ⎨     2 x˙0 z˙ 0 2 x˙0 + 3z t) + sin(ω t) + 4z + z(t) = − cos(ω 0 oT oT 0 ⎪ ⎪ ωoT ωoT ωoT ⎪ ⎪ ⎪ ⎪ ⎪ z − 3 x ˙ + (6ω z − 4 x ˙ ) cos(ω t) − 2˙ z sin(ω x(t) ˙ = −6ω ⎪ oT 0 0 oT 0 0 oT 0 oT t) ⎪ ⎪ ⎪ ⎪ y ˙ (t) = y ˙ cos(ω t) − y ω sin(ω t) ⎪ 0 oT 0 oT oT ⎪ ⎪ ⎩ z˙ (t) = (3ωoT z 0 + 2 x˙0 ) sin(ωoT t) + z˙ 0 cos(ωoT t)

(2.48) From the Eqs. (2.47) and (2.48), we can see that described by the CW equations, the relative motions inside and outside the orbital plane are decoupled. Therefore, we can analyze them separately. The state equation inside the orbital plane is ⎡ ⎤ x˙ ⎢ z˙ ⎥ ⎢ ⎥= ⎣ x¨ ⎦ z¨

⎤⎡ ⎤ ⎡ ⎤ x 00 0 0 1 0   ⎢ ⎥ ⎢ ⎢0 0 ⎥ 0 1 ⎥ ⎥ ⎢ z ⎥ + ⎢ 0 0 ⎥ ax ⎢ ⎣ 1 0 ⎦ az ⎣0 0 0 −2ωoT ⎦ ⎣ x˙ ⎦ 2 0 3ωoT 2ωoT 0 z˙ 01 ⎡

(2.49)

Its state transfer matrix is ⎡ ⎤ 1 −6ωoT t + 6 sin ωoT t −3t + 4 sinωoTωoT t − ω2oT (1 − cos ωoT t) ⎢ ⎥ 2 1 (1 − cos ωoT t) sin ωoT t 4 − 3 cos ωoT t ⎢0 ⎥ ωoT ωoT (t) = ⎢ ⎥ ⎣ 0 6ωoT cos ωoT t − 6ωoT −3 + 4 cos ωoT t ⎦ −2 sin ωoT t 2 sin ωoT t cos ωoT t 0 3ωoT sin ωoT t (2.50)

2.4 Relative Motion

51

The state equation outside the orbital plane is        y˙ 0 1 y 0 = + ay 2 y¨ −ωoT 0 y˙ 1

(2.51)

Its state transfer matrix is   y (t) =

cos ωoT t −ωoT sin ωoT t

sin ωoT t cos ωoT t

1 ωoT

 (2.52)

2. Relative Motion Analysis For the relative motion inside the orbital plane, time t can be eliminated from the Eq. (2.48) to obtain the following general expression of relative motion: (z − 2a2 )2 (x − a1 + 3a2 ωoT t)2 + =1 (2a3 )2 (a3 )2 where a1 = x0 −

2˙z 0 ωoT

, a2 = 2z 0 +

x˙0 ωoT

, a3 =



2 x˙0 ωoT

+ 3z 0

2

+



z˙ 0 ωoT

2 .

The above equation is an elliptical trajectory equation; its trajectory is determined by three parameters: a1 , a2 , and a3 . As t increases, the ellipse center (a1 − 3a2 ωoT t, 2a2 ) moves parallel to the x-axis; a1 and a2 determine the initial position of ellipse center when t = 0; a2 determines the direction and speed of the moving of ellipse center; and a3 determines the size of the ellipse. Different combinations of parameters a1 , a2 , and a3 will form different relative motion trajectories, which can be categorized into two cases: closed elliptical motion trajectory and non-closed rolling elliptical motion trajectory. Table 2.1 lists the corresponding relative motion trajectories under different initial conditions and summarizes the motion characteristics [2]:

2.4.2 Relative Motion in Elliptical Orbital Frame In 1965, Tschauner and Hempel deduced the TH equation in the study of the elliptical orbit rendezvous and obtained the analytical solution related to the eccentricity and the true anomaly. Next, the TH equation was widely used in studies on rendezvous and docking and formation flying.

52

2 Rendezvous Kinematics and Dynamics

Table 2.1 Free motion characteristics in orbital plane Type

Initial condition

Closed elliptical motion trajectory

a2 = 0

Trajectory figure

a1 > 2a3

Characteristics a2 = 0, i.e. x˙0 = −2ωoT z 0 if and only if the relative motion trajectory is a closed ellipse Satisfying a2 = 0 and certain relationship between a1 and 2a3 , the chaser is behind (a1 > 2a3 ), ahead of (a1 < 2a3 ), or surrounding (|a1 | < 2a3 ) the target and moves in a closed ellipse In particular, when a2 = 0 and a1 = 0, the center of the closed ellipse is the target’s centroid

C T

a1 < −2a3

C T

|a1 | < 2a3 a1 = 0 T C T

x˙0 = 0

z˙ 0 > 0

z0 = 0

C

x

z

z˙ 0 < 0 x

X0 z

C

z˙ 0 = 0

z

X0 C

x

This is a special case of a2 = 0; the motion trajectory is a closed ellipse The initial chaser is located at x(0) = x0 , and without initial velocity in the x-axis, initial velocity along the z-axis determines three motion trajectories. Because    0  a3 =  ωz˙oT , the ellipse size only depends on z˙ 0 ; the direction of the surrounding depends on the positive or negative values of z˙ 0 When z˙ 0 = 0, the chaser will ideally stabilize at the holding point x = x0 on the x-axis, which happens if and only if free motion meets the passive holding point (continued)

2.4 Relative Motion

53

Table 2.1 (continued) Type

Initial condition

Trajectory figure

Non-closed a2 = 0  rolling x˙0 = −1.5ωoT z 0 elliptical z˙ 0 = 0 motion trajectory

z

Characteristics

x

Now a2 = 0, i.e.x˙0 = −2ωoT z 0 ; the relative motion trajectory is a non-closed rolling ellipse. In particular, when  x˙0 = −1.5ωoT z 0 , z˙ 0 = 0 the relative motion trajectory will change to a line  x = x0 − 1.5ωoT z 0 t z = z0

In the TH frame, it is assumed that the target is running in an elliptical orbit, and the distance between the chaser and the target is much less than the target’s orbital radius. The definitions of μ, ρ, RT , a, ω and RT are the same as in Sect. 2.4.1. Similarly, we have   μ RT · ρ ρ¨ + 3 ρ − 3 2 RT = a (2.53) RT RT Let kx , ky and kz be unit vectors along each axis of the reference frame; then, the vectors can be can be expressed in components as ρ = xkx + yky + zkz

(2.54)

  a 1 − e2 kx RT = RT kx = 1 + e cos θ

(2.55)

ω = θ˙ kz =

ωoT (1 + e cos θ)2 kz  3 1 − e2 2

(2.56)

Similar to the derivation of CW equations, we have ρ¨ I = ρ¨ T + 2ω × ρ˙ T + ω˙ × ρT + ω × (ω × ρT ) Then, it follows  T ρ˙ I = ρ˙ T + ω × ρT = x˙ − θ˙ y y˙ − θ˙ x z˙

(2.57)

54

2 Rendezvous Kinematics and Dynamics

 T ρ¨ I = x¨ − 2θ˙ y˙ − θ¨ y − θ˙ 2 x y¨ + 2θ˙ x˙ + θ¨ x − θ˙ 2 y z¨

(2.58)

⎛⎡ ⎤ ⎡ ⎤⎞     x R RT · ρ μ 1 + e cos θ 3 ⎝⎣ ⎦ μ x ⎣ T ⎦⎠ ρ − 3 2 RT = 3 0 y −3 a 1 − e2 RT RT3 RT 0 z ⎡ ⎤   −2x 1 + e cos θ 3 ⎣ 2 = ωoT (2.59) y ⎦ 1 − e2 z Sorting the above equations, we have  ⎧    1 + e cos θ 3 ⎪ 2 2 ⎪ ˙ ˙ ⎪ x¨ = 2θ y˙ + θ + 2ωoT x + θ¨ y + ax ⎪ 2 ⎪ 1 − e ⎪ ⎪ ⎪  ⎪    ⎨ 1 + e cos θ 3 2 2 y¨ = −2θ˙ x˙ − θ¨ x + θ˙ − ωoT y + ay ⎪ 1 − e2 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 + e cos θ 3 ⎪ 2 ⎩ z¨ = −ωoT z + az 1 − e2

(2.60)

Transforming Eq. (2.60) into a matrix form, ⎡ ⎤ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ 2 x˙ x 0 −θ˙ 0 x˙ θ˙ θ¨ 0 d ⎣ ⎦ y˙ = −2⎣ θ˙ 0 0 ⎦ ⎣ y˙ ⎦ + ⎣ −θ¨ θ˙ 2 0 ⎦ ⎣ y ⎦ dt 0 0 0 z˙ 0 0 0 z z˙ ⎡ ⎤ ⎡ ⎤   2x ax 1 + e cos θ ⎣ 2 ⎦ + ⎣ ay ⎦ + ωoT −y 1 − e2 az −z

(2.61)

Equations (2.60) and (2.61) are time-domain representations of the TH equation under the ideal two-body assumption. They can be used to describe the motion of the chaser relative to the target. When e = 0, it degenerates into the CW equation shown in Eq. (2.47). The right side of the Eq. (2.60) is the Coriolis acceleration, the centripetal acceleration related to the rotation of the coordinates, acceleration associated with changes in rotational co-ordinate speed, virtual gravity gradient terms, and acceleration incorporating perturbations and controls. From Eq. (2.60), the motion of the orbital plane is decoupled from the motion outside the orbital plane. In the general study, it is possible to study two problems separately.

2.4 Relative Motion

55

2.4.3 Relative Motion in the Line of Sight Frame In the following derivation, we make preliminary assumptions that both spacecraft are rigid; motion with respect to Earth can be described by an approximate twobody motion; the distance between the two spacecraft is much less than the distance between them and the Earth’s center; and the target moves in a circular orbit [3]. For the line of sight relative motion equation derivation, the main parameters involved are ρ, α and β, where ρ is the relative distance between the two spacecraft; α is the angle between the line of sight and its projection on the plane of the target’s orbit, which we call the deviation angle; and β is the angle between the projection and the target’s flight direction, called the line of sight angle. The relative motion of the two spacecraft is approximated in the earth-centered equatorial inertial frame and is expressed as ρ¨ +

  RT · ρ μ R ρ − 3 = a T RT3 RT2

(2.62)

where ρ is the relative distance vector of the two spacecraft; RT is the distance from the target to the center of the Earth; RT is the distance vector from the center of the target to the center of the Earth; a is the acceleration generated by the joint force by the external forces except the gravitational force on the chaser, and a is expressed as T  a = aξ aη aζ in the line of sight frame. Taking the derivative of ρ in the earth-centered equatorial inertial frame, ρ¨ I = ρ¨ T + 2ω × ρ˙ T + ω˙ × ρT + ω × (ω × ρT )

(2.63)

where ω is the angular velocity of the line of sight frame relative to the earth-centered equatorial inertial frame expressed in the line of sight frame, and T  ω = ωξ ωη ωζ

(2.64)

The components of the relative positions in the line of sight frame are  T ρT = ρ 0 0

(2.65)

Then, ⎡ ⎤ ⎡ ⎤ ⎤⎡ ⎤ ⎡ ρ˙ ρ 0 −ωζ ωη ρ˙ ρ˙ I = ρ˙ T + ω × ρ T = ⎣ 0 ⎦ + ⎣ ωζ 0 −ωξ ⎦⎣ 0 ⎦ = ⎣ ρωζ ⎦ −ωη ωξ 0 −ρωη 0 0 ⎡ ⎤ ρ¨ − ρωζ2 − ρωη2 ρ¨ I = ⎣ 2ρω ˙ ζ + ρ ω˙ ζ + ρωξ ωη ⎦ −2ρω ˙ η − ρ ω˙ η + ρωξ ωζ

(2.66)

(2.67)

56

2 Rendezvous Kinematics and Dynamics

The DCM (3 β—2 α) from the target spacecraft orbital frame O X Y Z to the line of sight frame Oξ ζ η can be obtained as ⎡

⎤ cos α cos β cos α sin β − sin α C(α, β) = ⎣ − sin β cos β 0 ⎦ sin α cos β sin α sin β cos α

(2.68)

The angular velocity ω in the line of sight frame is ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ −β˙ sin α −(β˙ + ωoT ) sin α 0 ωξ ⎦ ⎦ + C(α, β)⎣ 0 ⎦ = ⎣ ⎣ ωη ⎦ = ⎣ α˙ α˙ ωζ (β˙ + ωoT ) cos α β˙ cos α ωoT ⎡

(2.69)

The vector RT in the line of sight frame is ⎡

⎤ ⎡ ⎤ ⎤⎡ cos α cos β cos α sin β − sin α RT cos α sin β 0 RT = ⎣ − sin β cos β 0 ⎦ ⎣ RT ⎦ = ⎣ RT cos β ⎦ 0 RT sin α sin β sin α cos β sin α sin β cos α

(2.70)

Further, we obtain ⎡ ⎤   1 − 3 cos2 αsin2 β RT · ρ μ 2 ρ − 3 2 RT = ωoT ρ ⎣ −3cosα sin β cos β ⎦ RT3 RT −3 sin α cos α sin2 β

(2.71)

Therefore, the relative motion equations in line of sight frame are ⎧ 2 ρ¨ − ρ(ωζ2 + ωη2 ) + ωoT ρ(1 − 3 cos2 αsin2 β) = aξ ⎪ ⎨ 2 2ρω ˙ ζ + ρ ω˙ ζ + ρωξ ωη − 3ωoT ρcosα sin β cos β = aη ⎪ ⎩ 2 − 2ρω ˙ η − ρ ω˙ η + ρωξ ωζ − 3ωoT ρ sin α cos α sin2 β = aζ

(2.72)

In the near range rendezvous, the two spacecraft can be approximately regarded to be in the same orbital plane, i.e. α = 0; then, ⎡

⎤ cos β sin β 0 C(α, β) = ⎣ − sin β cos β 0 ⎦ (2.73) 0 0 1 ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ −β˙ sin α −(β˙ + ωoT ) sin α 0 ωξ ⎦ ⎦ + C(α, β)⎣ 0 ⎦ = ⎣ ⎣ ωη ⎦ = ⎣ α˙ α˙ ˙ ˙ ωζ (β + ωoT ) cos α β cos α ωoT

2.4 Relative Motion

57

⎡

⎤ 0 ⎦ =⎣ 0 ˙ β + ωoT

(2.74)

Equation (2.72) can be simplified to 

2 ρ(1 − 3sin2 β) = aξ ρ¨ − ρωζ2 + ωoT 2 2ρω ˙ ζ + ρ ω˙ ζ − 3ωoT ρ sin β cos β = aη

(2.75)

Consider ωoT to be minimal; then, Eq. (2.75) can be approximated as 

ρ¨ − ρωζ2 = aξ 2ρω ˙ ζ + ρ ω˙ ζ = aη

(2.76)

Substituting Eq. (2.74) into (2.76), we have 

ρ¨ − ρ(β˙ + ωoT )2 = aξ ρ β¨ + 2ρ( ˙ β˙ + ωoT ) = aη

(2.77)

2.4.4 Relative Motion in Cylindrical Frame

1. Motion Description of the Spacecraft in the Geocentric Cylindrical Frame The motion of the spacecraft is described in the earth-centered equatorial inertial frame as [4, 5]. ⎧ μ ⎪ x¨s + 3 xs = 0 ⎪ ⎪ ⎪ Rs ⎪ ⎪ ⎨ μ y¨s + 3 ys = 0 (2.78) R ⎪ s ⎪ ⎪ ⎪ μ ⎪ ⎪ ⎩ z¨ s + 3 z s = 0 Rs where xs , ys and z s are the three components of the geocentric vector of the space craft in the inertial system; Rs = xs2 + ys2 + z s2 . The relationships between xs , ys and z s and the co-ordinate components in the geocentric cylindrical frame are xs = rs cos θs , ys = rs sin θs , z s = z s and Rs = rs2 + z s2 . Then, Eq. (2.78) can be written as

58

2 Rendezvous Kinematics and Dynamics

⎧ μrs ⎪ r¨s − rs θ˙s2 + 3 = 0 ⎪ ⎪ Rs ⎪ ⎨ rs θ¨s + 2˙rs θ˙s = 0 ⎪ ⎪ μ ⎪ ⎪ ⎩ z¨ s + 3 z s = 0 Rs

(2.79)

When the reference plane is the orbital plane of the spacecraft, i.e. z s = 0 and rs = Rs is satisfied, Eq. (2.79) become ⎧ μ ⎨ r¨s − rs θ˙s2 + =0 rs2 ⎩ ¨ rs θs + 2˙rs θ˙s = 0

(2.80)

2. Motion Description of the Spacecraft Relative to the Circular Reference Orbit Considering the motion of the spacecraft relative to the circular reference orbit, the orbital radius of the reference orbit is ar and true anomaly is θr ; then, a˙ r = 0, θ˙r = ωr = μ/ar3 and θ¨r = 0. Taking the origin of the reference system as (ar , θr , 0), and considering the three co-ordinate axes of the reference frame are parallel to the three co-ordinate axes of the geocentric cylindrical frame, then the motion parameters can be expressed as rs = ar + r, θs = θr + θ and z s = z, where r, θ and z are the actual deviation of spacecraft relative reference frame origin in the directions of the three axes in the cylindrical frame. As such, r˙s = ˙r , θ˙s = θ˙r +θ˙ , z˙ s = ˙z ,¨rs = ¨r ,θ¨s = θ¨ and z¨ s = ¨z . When r/ar ,z/ar are minimal, Taylor expansion is performed on the left side of Eq. (2.79) and the linear items are kept, as follows: ⎧ 2 ˙ ⎪ ⎨ ¨r − (2ar ωr θ + 3ωr r ) = 0 ar θ¨ + 2ωr ˙r = 0 ⎪ ⎩ ¨z + ωr2 z = 0

(2.81)

In the relative motion equation described by Eq. (2.81), all three equations contain the angular velocity of the reference orbit ωr . After normalizing Eq. (2.81), the new dynamic equations do not contain ωr and the form will be simplified. = ωr , and we Let r = ar δr, θ = δθ, z = ar δz and τ = ωr (t − t0 ), then dτ dt have: dr d(ar δr ) dτ d(δr ) d(δr ) = = ar = ar ωr dt dt dt dτ dτ 2 d2 r d (δr ) = ar ωr2 dt 2 dτ

2.4 Relative Motion

59

d(δθ ) dτ d(δθ ) dθ d(δθ ) = = = ωr dt dt dt dτ dτ 2 d2 θ d (δθ ) = ωr2 dt 2 dτ dz d(δz) d(ar δz) dτ d(ar δz) = = = ar ωr dt dt dt dτ dτ 2 d2 z d (a δz) r = ar ωr2 dt 2 dτ Furthermore, considering the influence of other accelerations except gravitational acceleration, Eq. (2.81) can be written as ⎧ ˙ ⎪ ⎨ δr¨ =3δr + 2δ θ + a yr δ θ¨ = − 2δr˙ + a yθ ⎪ ⎩ δ z¨ = − δz + a yz

(2.82)

where a yr , a yθ and a yz are the three acceleration components (except gravity acceleration) along the coordinates of the circular orbit reference system. The derivative term in the equation is derived from dimensionless time τ . The Eqs. (2.82) and (2.47) are identical in form except for the state variables. The relative motion described by Eq. (2.82) can also be decoupled into the motion inside and outside the orbital plane.  T Defining state variable x3 = δr δθ δz δr˙ δ θ˙ δ z˙ and reformulating Eq. (2.82) as a state equation, we get ⎤ ⎡ 0 δr˙ ⎢ δ θ˙ ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ δ z˙ ⎥ ⎢ 0 ⎢ ⎥=⎢ ⎢ δr¨ ⎥ ⎢ 3 ⎢ ⎥ ⎢ ⎣ δ θ¨ ⎦ ⎣ 0 0 δ z¨ ⎡

0 0 0 0 0 0

0 0 0 0 0 −1

1 0 0 0 −2 0

0 1 0 2 0 0

⎡ ⎤⎡ ⎤ ⎤ 0 δr 0 ⎢ 0 ⎥ ⎢ ⎥ 0⎥ ⎢ ⎥ ⎢ δθ ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎥ 1 ⎥ ⎢ δz ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ + ⎢ ⎥ ⎢ a yr ⎥ 0 ⎥ ⎢ δr˙ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎣ a yθ ⎦ 0 ⎦ ⎣ δ θ˙ ⎦ δ z˙ 0 a yz

(2.83)

The state transfer matrix for Eq. (2.83) is ⎡

4 − 3 cos τ ⎢ −6τ + 6 sin τ ⎢ ⎢ 0 ⎢ 3 (τ ) = ⎢ ⎢ 3 sin τ ⎢ ⎣ 6 cos τ − 6 0

⎤ 0 0 sin τ 2(1 − cos τ ) 0 1 0 −2(1 − cos τ ) 4 sin τ − 3τ 0 ⎥ ⎥ ⎥ 0 cos τ 0 0 sin τ ⎥ ⎥ (2.84) 0 0 cos τ 2 sin τ 0 ⎥ ⎥ 0 0 −2 sin τ −3 + 4 cos τ 0 ⎦ 0 − sin τ 0 0 cos τ

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3. Dynamics for Rendezvous in Elliptical Orbit with Small Eccentricity This section gives the dynamics for optimal impulse rendezvous in an elliptical orbit with a small eccentricity, which will be discussed in Chap. 4. Defining a cylindrical frame OT r θ z at the centroid of the target, we use the dimensionless dynamics Eq. (2.82) to describe relative motion and provide a method of solving the relative motion state. (1) Dynamics in Cylindrical Frame The target’s orbital radius is ar , and its true anomaly is θr . The radius of the chaser is r = ar + r , and its true anomaly is θ = θr + θ + βr , where βr is the initial phase angle difference between the spacecraft and the reference frame; and θ is accumulated phase angle difference between the spacecraft and the reference frame during motion, and it satisfies θ = 0 at the initial time. The distance from the orbital plane of the reference orbit to the spacecraft is z = z. The relative motions can be described using Eqs. (2.82) and (2.83). (2) Relative Motion As shown in Fig. 2.2, O E is the Earth’s center. O E A denotes the intersection line between the chaser’s orbital plane O E AS and the target orbital plane O E A R; and the plane angle is θ. γ is the angle between the connection to the center of Earth and the plane of the reference orbit; then we have sin γ = sin u sin ϑ

(2.85)

When the spacecraft’s ascending node coincides with that of the reference system, we can see from Fig. 2.2 that u is the latitude angle: u =w+ f

(2.86)

When the deviation of the outer surface of the two spacecraft is not large, the ascending nodes of the two spacecraft can be deemed to be approximately the same. Then, the following relations hold: Fig. 2.2 Description for relative motions

S OE u

γ

O1

θ

B

θ A

H

R

2.4 Relative Motion

61

z = r sin γ = r sin u sin ϑ

(2.87)

˙z = r˙ sin u sin ϑ + r f˙ cos u sin ϑ

(2.88)

Let δa = (a −ar )/ar . Because the orbital radius of the two spacecraft are not very different, the orbital out-of-plane distances are not too large, and the chaser orbit is a small eccentricity ellipse, we could regard e, sinθ, and δa as minimal. Ignoring the high-ordered tiny amount, the components of the relative state variables outside the plane of the orbit are δz = z/ar = (1 + δa)(1 − e2 )(1 + e cos f )−1 sin u sin ϑ ≈ sin u sin ϑ r˙ sin u sin ϑ r f˙ cos u sin ϑ ˙z = + ≈ cos u sin ϑ δ z˙ = ar ω ar ω ar ω Similarly, the relative state inside the plane can also be obtained by ignoring the high order tiny amount and linearization: r = (1 + δa)(1 − e2 )(1 + e cos f )−1 (1 − sin2 u sin2 ϑ)−1/2 − 1 ar ≈ δa − e cos f

δr =

δθ = (θr + θ + βr ) − θr = βr + θ 1 ˙r = (1 − δa)(1 + e)e sin f (1 − sin2 u sin2 ϑ)−1/2 ≈ e sin f ar ω 2  3 3 3 μ 3 ˙ ˙ δ θ = f / 3 − 1 ≈ − δa + 2e cos f + e − e = − δa + 2e cos f ar 2 2 2 2 δr˙ =

(3) Relative Motion for Rendezvous in an Elliptical Orbit with Small Eccentricity The initial relative motion state can be obtained as T  x30 = δa − e cos f βr sin u sin ϑ e sin f −1.5δa + 2e cos f cos u sin ϑ (2.89) From the state transfer matrix 3 (τ ) in Eq. (2.84), the relative motion state at time τ can be obtained as

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2 Rendezvous Kinematics and Dynamics

⎡

x3 (τ ) = 3 (τ )x30

⎤ δa − e cos(τ + f ) ⎢ β − 1.5τ δa + 2e sin(τ + f ) − 2e sin f ⎥ ⎢ r ⎥ ⎢ ⎥ sin(τ + u) sin ϑ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥ e sin(τ + f ) ⎢ ⎥ ⎣ ⎦ −1.5δa + 2e cos(τ + f ) cos(τ + u) sin ϑ

(2.90)

When the orbital plane of the spacecraft motion is in the same plane as the reference orbit, that is, θ = 0, the out-of-plane component is zero, only the components inside the plane are considered, and the corresponding relative states can be obtained: T  x 0 = δa − e cos f βr e sin f −1.5δa + 2e cos f

(2.91)

After time τ, ⎡

⎤ δa − e cos(τ + f ) ⎢ βr − 1.5τ δa + 2e sin(τ + f ) − 2e sin f ⎥ ⎥ x(τ ) = ⎢ ⎣ ⎦ e sin(τ + f ) −1.5δa + 2e cos(τ + f )

(2.92)

2.5 Relative Attitude Here, we consider the relative attitude of the two spacecraft at a very close distance. The target docking frame is set as reference system for relative attitude.

2.5.1 Relative Attitude Dynamics for Stable Target When the target is stably moving at a small angle, the target can be approximated as a stationary target. 1. Relative Attitude Ideally, the attitude equation of the target with respect to the inertial reference is θ˙T = −ωoT = const ψ˙ T = 0 φ˙ T = 0

(2.93)

2.5 Relative Attitude

63

where ψ˙ T , φ˙ T and θ˙T are the angular velocity of the yaw, roll, and pitch of the target with respect to the inertia reference, respectively; and ωoT is the orbital angular velocity of the target. Euler dynamics equation of the chaser is IxC ω˙ xC + (IzC − I yC )ω yC ωzC = TxC I yC ω˙ yC + (IxC − IzC )ωzC ωxC = TyC IzC ω˙ zC + (I yC − IxC )ωxC ω yC = TzC

(2.94)

where IxC , I yC and IzC are the three principal moments of inertia of the chaser; TxC , TyC and TzC are control torques provided for the chaser attitude thruster; attitude angular velocities ωxC , ω yC and ωzC are the components of chaser’s angular velocity with respect to the inertial reference in the chaser body frame. According to the relative rotation relationship, the angular velocity of the chaser relative to the target is ωr = ωC −ωT , where ωC and ωT are the angular velocities of the chaser and the target relative to the inertial space, respectively. Besides, ωr = ψ˙ r + ϕ˙ r + θ˙ r , where ωr is the rotational angular velocity of the chaser relative to the target body frame; and ϕ˙ r , ψ˙ r and θ˙ r are the Euler angular velocities for relative rolling, yaw, and pitch, respectively. Then, ⎡

⎤ ⎡ ⎤ ωxC 0 ωr = ωC −ωT = ⎣ ω yC ⎦ − C312 CboT ⎣ −ωoT ⎦ ωzC 0 And we have [6] ⎡

⎤ ⎡ ⎤ ⎤ ⎡ ωxC 0 ϕ˙r cos θr − ψ˙ r sin θr cos ϕr ⎣ ω yC ⎦ − C312 CboT ⎣ −ωoT ⎦ = ⎣ ⎦ θ˙r + ψ˙ r sin ϕr ˙ ωzC 0 ϕ˙r sin θr + ψr cos θr cos ϕr ⎡

Let C312

⎤ C11 C12 C13 = ⎣ C21 C22 C23 ⎦, we can obtain C31 C32 C33

⎧ ϕ˙ = ωzC sinθr + cosθr ωxC +ωoT (C22 sinψr +C12 cosψr ) ⎪ ⎨ r ! θ˙r = [ωyC cosϕr − sinφr cosθr ωzC + sinφr sinθr ωcx +ωoT (C22 cosψr − C12 sinψr )] cosϕr ⎪ ! ⎩ ψ˙ r = [cosθr ωzC − ωxC sinθr +ωoT (−C22 sinϕr cosψr + C32 cosϕr )] cosϕr

(2.95) In the state estimation mode determined by the relative attitude, the state equation is preferred to use the attitude kinematics equations, which only includes two types of parameters: one is the attitude to be estimated and the other is the attitude angular velocity which can be accurately measured by the gyros.

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2. Coupling of Relative Attitude and Relative Position Because the relative position control force is generally not based on the chaser body frame, it is necessary to convert the coordinates to the body of the chaser. Considering the target is in passive three-axis stability attitude, the thrust acceleration applied to the chaser’s body given by the control law is [7]. aCb = CCbd C312 (θˆr , ϕˆr , ψˆ r )aCd

(2.96)

where θˆr , φˆr and ψˆ r are the instantaneous relative attitude angle derived from the measurement sensor; and CCbd is the DCM of the chaser docking frame to the body frame. The torques caused by the thrusts acting on the chaser body is given by ⎤ ar ⎣ au ⎦ = CCob aCb ah ⎡

(2.97)

where CCob is the attitude matrix of the chaser body to the spacecraft orbital frame. Because of the change of the attitude angle of the chaser body during one control period and the existence of the relative attitude angle measurement error, attitude control will affect relative position control. Similarly, because the position control actuators may not be symmetrically installed and meanwhile the spacecraft’s centroid changes, the position control will generate torques and affect the relative attitude control, leading to the coupled relative position and relative attitude.

2.5.2 Relative Attitude Dynamics for Rotating Target Suppose the target is an uncontrolled spacecraft, the space debris or asteroid, whose motions are complicated may possibly be accompanied by attitude rotation or even maneuvering. In these cases, the target’s dynamic parameters and geometric shape are uncertain, leading to a high control difficulty of carrying out safe rendezvous. Quaternion is more suitable to depict dynamics herein because the attitude angles are not near zero. The attitude dynamics of the target is 1 (ωT )qT 2 IT ω˙ T + ωT × IT ωT = TT q˙ T =

(2.98)

2.5 Relative Attitude

65

The attitude dynamics of the chaser is 1 (ωC )qC 2 IC ω˙ C + ωC × IC ωC = TC q˙ C =

(2.99)

where qT and qC are the attitude quaternions of the target and chaser, respectively; ωT and ωC are the attitude angular velocity of the target and the chaser, respectively; IT and IC are the moment of inertia of the target and chaser, respectively; T   T and TC are −[ω×] ω . the control torque of the target and chaser, respectively; and (ω) = −ωT 0 Defining the relative attitude quaternions of the target and chaser as q = q−1 T ⊗qC , the relative attitude angular velocity is ωr = ωC − CC T (q)ωT , where CC T (q) is the transfer matrix from the target body frame to the chaser body frame. Then, the relative attitude dynamics is 1 (ωr )q 2 ω˙ r = −IC−1 (ωr + CC T (q)ωT ) × IC (ωr + CC T (q)ωT ) q˙ =

−1 + ωr × CC T (q)ωT + CC T (q)I−1 T (ωT × IT ωT − TT ) + IC TC

(2.100)

The rendezvous and docking of the rotating target needs to synchronize the attitude movement of the target and the chaser, that is, its relative attitude quaternion q tends to a constant value.

2.6 Selection of Reference Frames and Dynamics Equation This chapter introduces the reference frames and dynamic equations commonly used in engineering. The choice of reference frames and dynamics forms depends on the distance between the two spacecraft, as well as the flight phase. In different flight phases, measurement sensors are different and the accuracy requirements for position and attitude control are also different. In the far range rendezvous phase of rendezvous and docking, the two spacecraft are generally treated as independent. Position control is generally described by the orbital elements, and the attitude is generally chosen as the attitude of the body frame relative to the spacecraft orbital frame. In the middle-distance or close-distance phases of rendezvous and docking, relative navigation information is needed for position control, and attitude control of the two spacecraft can be independently processed. On the basis of the distance, we can choose the relative motion equation based on the cylindrical frame, the line of sight equation, or the CW equation to design guidance policy.

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Table 2.2 Suggested choices for the reference frame and dynamics during different flight phases Flight phase

Attitude control

Position control

Reference frame

Dynamics

Reference frame

Dynamics

Far range rendezvous

Body frame, orbital frame

Attitude dynamics with flexible panels

Earth-centered equatorial inertial frame

Orbital elements, perturbation equations

Homing

Body frame, orbital frame

Attitude dynamics with flexible panels

Earth-centered equatorial inertial frame, RVD frame

Orbital elements, CW equations

Closing

Body frame, orbital frame

Attitude dynamics with flexible panels

RVD frame, line of sight frame

CW equations, sight equations

Final approaching

Target docking frame

Relative attitude dynamics

Target docking frame

CW equations

In the final approaching phase of rendezvous and docking, it is necessary to consider the relative position and attitude of the two spacecraft simultaneously. Because the two spacecraft finally need to realize the docking of the docking mechanism, the reference system is generally built on the docking surface to accurately describe the final approximation and docking process. Relative position control can be described by the CW equation based on the target docking frame. The relative attitude can also be established in the target docking e frame. The suggested choices for the reference frame and dynamics equations during different flight phases are summarized in Table 2.2.

References 1. Laixing, Lin: Space Rendezvous and Docking Techniques. National Defense Industry Press, Beijing (1995). (In Chinese) 2. Xin, Y.: Research on Control Methods of Flying Around in Rendezvous and Docking. Beijing Institute of Control Engineering (2005) (In Chinese) 3. Hu, H.: Research on Control Methods of Intelligent Autonomous Rendezvous and Docking. Beijing Institute of Control Engineering (2002) (In Chinese) 4. Prussing, J.E.: Optimal multiple-impulse orbital rendezvous. Sc.D. thesis, Dept. of Aeronautics and Astronautics Massachusetts Institute of Technology 5. Chen, C.: Trajectory Safety and Optimization of Rendezvous and Docking. Beijing Institute of Control Engineering (2008) (In Chinese) 6. Zhang, H.: Research on Autonomous Navigation of Rendezvous and Docking Based on CCD Optical Sensor. Beijing Institute of Control Engineering (2007) (In Chinese) 7. Liu, Z.: Research on the Autonomous Docking Control Method and the Simulation Experiment on Ground. Beijing Institute of Control Engineering (2001) (In Chinese)

Chapter 3

Navigation Method and Scheme Design for Rendezvous and Docking

3.1 Introduction In order to achieve successful rendezvous and docking, the relative translation and rotation states, which are required by the guidance and control systems, should be accurately determined by the relative navigation system. The relative navigation system consists of the relative measurement system and the relative navigation algorithm [1].

3.1.1 Relative Measurement System The relative measurement sensor is the hardware system to obtain relative motion information. 1. Relative Measurement System Employed by the Soviet Union/Russia The relative navigation system employed by the Soviet Union/Russia is an automatic system which is developed based on the microwave radar. The radio navigation system includes a microwave radar on the chaser and antennas mounted on two spacecraft, which are used to capture the target and measure the relative attitude between the chaser and the target. The chaser uses the time delay of the return signal from the transponder on the target to determine the relative distance and uses the doppler frequency shift of the returned signal to determine the approach rate; it estimates the relative attitude based on the signal from the docking antenna. This navigation system works reliably and has been successfully applied in the automatic and manual rendezvous and docking missions conducted by the Soviet Union and Russia. However, this relative navigation system is complex, and it is limited by the issues of large mass, large volume, and high power consumption. The early Soviet Soyuz manned spacecraft and the Progress cargo used the “Igla” rendezvous and docking navigation system. Starting from the 1980s, the Soviet Union © National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_3

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replaced the Igla system with the new “Kurs” system for its rendezvous and docking tasks. In contrast to the Igla system, the Kurs system has two advantages: (1) the Igla system’s working distance is only a few tens of kilometers, while the Kurs system can capture the target in the range of several hundred kilometers; (2) the Igla system can be used in the closing phase after the alignment (that is, when the two spacecraft docking surfaces maintain line of sight aim). In order to achieve alignment, the target must implement attitude maneuver, or the chaser must track the interface of the target. If the target is small (such as a satellite), the attitude maneuver is not difficult to achieve. If the target is a space station, the attitude maneuver is not easy to achieve. In this case, the chaser has to fly around the docking interface of the target. In contrast, the Kurs system can compensate for the unexpected deviation of the target, support the rendezvous and docking operation with the non-maneuvering target, and consume less energy [2]. In addition, the Soviet/Russian rendezvous missions were equipped with video cameras (on the chaser) and target devices (on the target) that could be used to assist the astronauts to implement manual rendezvous, and the equipment could also be used by the astronauts in space station to remotely operate the spacecraft for docking. 2. Relative Measurement System used in the USA From the rendezvous and docking of the Gemini spacecraft in the early 1960s to the Apollo lunar exploration program and the space shuttle rendezvous mission in the 1980s, the microwave rendezvous radar has been the main measurement device for relative navigation in the USA (Table 3.1). With the deepening of the research on the space shuttle and ISS, the USA is committed to developing autonomous rendezvous and docking technology, as well as more superior measurement sensors, such as laser range finders, GPS, laser radars, and camera-type rendezvous and docking sensors, which are gradually being applied to rendezvous and docking [3–5]. 3. Relative Measurement System used by Japan Japan’s autonomous rendezvous and docking navigation system has been successfully used in ETS-VII, which is equipped with three types of relative sensors. The navigation system automatically selects the relative sensor based on the relative distance. Outside 500 m, relative GPS is adopted as the main navigation sensor, and the laser radar is used as the main navigation sensor in the range of 2–500 m, as it can measure the relative distance and line of sight. The optical sensor PXS, which can measure relative position and attitude, is used as the main navigation sensor within 2 m. In addition, Japan’s newly developed cargo spacecraft for supporting the ISS, named HTV, uses RGPS and a laser radar system (RVS) as the main sensors in the near-rendezvous phase and closing phase, respectively [6]. 4. Relative Measurement System used by ESA To provide service for the ISS, ESA has developed the automatic transport vehicle (ATV), which successfully docked with the ISS after its first launch in March 2008.

3.1 Introduction

69

Table 3.1 Relative measurement system used in the USA Mission

Time

Gemini

1965–1966 (1) Microwave rendezvous radar (2) Crew optical alignment sight: COAS

Sensor

Apollo-LEM

1969–1971 (1) Microwave rendezvous radar

Apollo-CSM

1969–1975 (1) Very high frequency: VHF (2) Sextant (3) COAS

Space shuttle

1981–2011 (1) (2) (3) (4) (5) (6)

XSS-11

2005

(1) Visible camera system, VCS (2) Light detection and ranging, LIDAR

DART)

2005

(1) GPS (2) Advanced video guidance sensor, AVGS

Ku-radar Star tracker COAS Trajectory control sensor, TCS Hand-Held Lidar, HHL Closed circuit television, CCTV

Orbital express 2007

(1) Autonomous rendezvous and capture sensor system, ARCSS (2) AVGS

Dragon

2012

(1) Star tracker (2) RGPS (3) LIDAR

Cygnus

2013

(1) GPS/INS(SIG1 (2) LIDAR

Orion

(1) (2) (3) (4)

GPS Star tracker S-band radio link VNS

The relative navigation measurement system consists of relative GPS and an optical sensor, called the videometer. The videometer and relative GPS are used as the primary navigation sensors within 0–250 m and 250 m–30 km, respectively. The videometer can provide both the relative position and relative attitude measurements within 30 m and the relative position measurement beyond 30 m [7]. 5. Relative Measurement System Adopted by China In the second phase of China’s manned space program, a relative measurement system was developed by integrating differential satellite navigation equipment, a microwave radar, a laser radar, and CRDS. At the range larger than 140 m in the autonomous control phase, the differential satellite navigation equipment was used as the primary sensor, and the microwave radar and laser radar were used as the backup. Within 140 m, the CRDS is used as the primary navigation sensor to measure the relative position and attitude [8].

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3 Navigation Method and Scheme Design for Rendezvous and Docking

6. Summary At present, the relative navigation sensors commonly used in rendezvous and docking missions mainly include differential satellite navigation equipment (such as GPS), laser radars, microwave radars, optical sensors (such as CRDS), etc. Differential satellite navigation equipment, microwave radars, and laser radars constitute the main measurement devices during the near-range rendezvous phase, and they can provide relative position information, optical sensors can also get relative attitude data. Differential satellite navigation equipment is characterized by the all-day and all-weather application sensor; its measurement accuracy is high, and the measurement error generally does not cause system deviation. Microwave and laser radars, which are autonomous measurement devices, are characterized by their long operating distances. However, their measurement errors usually result in system deviation, and the measurement noise takes the form of thick-tail distribution and other nonGaussian distributions. Optical sensors is the main measurement mean for the closing phase. It is characterized by high precision and abundant information; however, its main disadvantages are its short working range and high technical difficulty. There are two primary factors that cause measurement issues: (1) searching, identifying, and tracking the target, and (2) measurement accuracy.

3.1.2 Relative Navigation Algorithms The relative navigation algorithm estimates the relative motion states, including the relative position, velocity, attitude and attitude rate between the two spacecraft, based on the measurement of relative measurement sensors and the kinematics/dynamics model. The combination of the measurement information and physical model information to estimate the state requires the use of an appropriate state estimation method, usually known as a navigation filter. In engineering practice, the Kalman filter, extended Kalman filter, and even observers are widely used to design the relative navigation algorithm. However, in order to meet the requirements of new tasks, such as in-orbit services and rendezvous and docking of non-cooperative targets, the following issues must be addressed: (1) A model error inevitably exists, as only simplified dynamics/kinematics models can be used in the design of the navigation algorithm. Moreover, the model error is usually related to the relative distance and presents the nature of non-stationary noise. (2) The relative measurement sensors usually possess systematic deviation, and significant measurement model errors will occur owing to the error source in the relative attitude determination when relative attitude transformation is involved. (3) The measurement noise of the sensor generally follows a non-Gaussian distribution.

3.2 Sensors

71

3.2 Sensors 3.2.1 Inertial Measurement Sensors Inertial navigation is a system based on inertial technology, which can provide the carrier with attitude information relative to the reference system and realize the autonomous attitude estimation and position navigation. Inertial navigation sensors primarily include gyroscopes and accelerometers. 1. Gyroscope Generally, any instrument that can measure angular velocity or angular displacement can be called a gyroscope. According to the ranges of angular velocity or angular displacement they measure, gyroscopes can be classified into single-degree-offreedom and two-degree-of-freedom instruments. Figure 3.1 is the principle diagram of the single-degree-of-freedom floating rate integral gyro [9]. The O-xyz frame in Fig. 3.1 is the measuring coordinate, and the O-x G yG zG frame is the gyro component coordinate. x(I)

ωx

y(O)

xG α

S

ωy

Ω O zG(s) H

α T

z

ωz Iy(α+ωy)

Fig. 3.1 The principle diagram of the single-degree-of-freedom floating rate integral Gyro

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3 Navigation Method and Scheme Design for Rendezvous and Docking

The Ox and Oy axis correspond to the input and output axis of the gyroscope, respectively, and the OzG axis corresponds to the rotation axis of the rotor. When the carrier rotates around the Ox axis, the gyro component is subjected to the torque in the direction of the Ox axis, which makes the gyro rotor rotate around the Oy axis with a rotation angle α. The angular momentum equation can be easily obtained as follows:   −H (ωx − αωz ) + Io ω˙ y + α¨ = M OG

(3.1)

where M OG is the external torque along the gyro output axis, M OG = −kα − D α˙ when elastic constraint and damping exist. Omitting the influence of angular velocity in the direction of the non-sensitive axis gives Io α¨ + D α˙ + kα = H ωx

(3.2)

Disregarding the elastic constraint, i.e., k = 0, we get Io α¨ + D α˙ = H ωx

(3.3)

 When the gyroscope reaches steady state, α¨ = 0, α˙ = HD ωx , α = HD ωx dt. Clearly, the rotation angle α is proportional to the integral of angle rate, and this type of single-degree-of-freedom gyroscope is called a rate integral gyro. It is the most commonly used inertial angular velocity measurement sensor on spacecraft. 2. Accelerometer An accelerometer can measure the acceleration of the moving carrier. The force exerted on the carrier can be classified into two categories: celestial gravitational and non-gravitational force. The non-gravitational force acting on the unit mass is called the specific force. The gravitational acceleration of celestial bodies cannot be directly measured, but the specific force can be obtained by using the accelerometer. The principle of the flexible pendulum accelerometer is briefly introduced below [9]. Figure 3.2 is the schematic of the flexible pendulum. The direction of the gravity is perpendicular to the paper and faces inward, which is also the output axis of the accelerometer. The I A axis is the input axis to the accelerometer. When the acceleration of the carrier along axis I A is a I , the pendulum mass will generate angular displacement owing to inertia. Because of the action of the elastic moment of the flexible rod and the damping moment of the floating liquid, the angular displacement tends to be stable, i.e., the acceleration of the mass is the same as that of the carrier, and the angular displacement can reflect the magnitude of a I . Analyzing the dynamics, we have (Fig. 3.3). i ss =

m L Ks Ka K R aI C + Ks Ka K R K T

(3.4)

3.2 Sensors

73 IA aI

(OA,ao) PA

m

aP

θo m

Fig. 3.2 The schematic of the flexible pendulum [9] aP(s)θO(s) aI(s)

−

mL

Md (s)

−

θO(s) 1 Ks IOS2+DS+C

Ka

KR

i (s)

RO

u (s)

KT

Fig. 3.3 Dynamics model of the flexible pendulum

where i ss is the current caused by force feedback in steady state, which satisfies i = K s K a K R θo , i ss reflects the quantity of a I , and in practical application, the voltage u with respect to the current is measured.

3.2.2 Satellite Navigation Equipments Satellite navigation is a system which receives the positioning signals sent by the navigation satellites, using the navigation satellite as the known dynamic reference to measure the position and speed of the moving carrier in real time in order to complete the navigation [10]. Satellite navigation systems can provide all-weather, all-day, high-precision navigation and time service for all types of users on Earth’s surface and near-Earth space. At present, there are four main types of satellite navigation systems in the world, including the USA’s GPS, Russia’s GLONSS, the EU’s GALILEO navigation system, and China’s Beidou navigation system. The satellite navigation receiving equipment is employed by the motion carrier to receive and process the navigation satellite signal. Here, we consider the GPS system as an example to introduce the basic principles of satellite navigation. The GPS is the second generation of satellite navigation systems developed by USA. The navigation constellation of the GPS consists of 24 satellites distributed

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3 Navigation Method and Scheme Design for Rendezvous and Docking

on six nearly round orbits at a height of 20184 km and an orbital period of 11 h 58 min. Because it was built and implemented in 1994, GPS has been increasingly applied in transportation, voyages, geodesy observation, and the field of aerospace. In the application of spacecraft rendezvous and docking, relative differential GPS technology benefits from its short acquisition time, all-weather service, and long working range, as well as being non-sensitive to disturbance and unconstrained by the approaching direction. Therefore, the GPS receiver is often used as the primary navigation sensor in the far-range phase of rendezvous and docking. GPS measurement information primarily includes pseudo-range measurement and carrier phase measurement. 1) Pseudo-Range Measurement Pseudo-range measurement determines the distance between the user receiver and the navigation satellite by calculating the propagation time of the GPS signal from the navigation satellite to the user receiver, which is used as the positioning measurement information. There are three types of time systems employed in GPS navigation systems [11]: (1) time standard of each GPS satellite; (2) time standard of each GPS signal receiver; (3) the ‘GPS time system’ that unifies the above two times standards. In the correlation receiver, the GPS signal receiver generates a group of oscillating P codes (or CA codes) that are identical to the signals transmitted by GPS satellite, which are correlated with the received satellite signals. When these two signals coincide with each other, the time delay (i.e., the propagation time of satellite signals) of the signal can be measured. The clock difference of the k-th receiver is defined as δtk = tk − Tk

(3.5)

where tk is clock read of the receiver and Tk is the ideal receiving time with respect to GPS. The clock difference of the j-th navigation satellite is defined as δt j = t j − T j

(3.6)

where t j is the clock read of the j-th navigation satellite and T j is the ideal receiving time w.r.t. GPS. The ideal propagation time is τ = Tk −T j . When the speed of light, C, is obtained, the pseudo-range is given as   j ρ˜k = C tk − t j   j = ρk + C δtk − δt j

3.2 Sensors

75 j

j

= ρk + Cδtk

(3.7)

j

where δtk = δtk − δt j . Because of the influence of the atmosphere and the thermosphere on the codes, the actual propagation time is longer than the ideal case. Thus, taking this time delay as the distance error, the pseudo-range model can be corrected by j

j

j

ρ˜k (ti ) = ρk (ti ) + Cδtk (ti ) + δρion (ti ) + δρtr op (ti )

(3.8)

where (ti ) is the measurement of the epoch ti . 2) Carrier Phase Measurement The carrier phase measurement method determines the propagation distance by detecting the phase change of the carrier signal transmitted by the GPS satellite from the navigation star to the GPS receiver propagation path, i.e. ρ = λϕ, where λ = Cf , f is the light wave frequency. Set the first k-th receiver in the ti time benchmark carrier phase of φk (ti ), and the first j-th navigation satellite at launch time t j (correj j sponding to receiving time ti ) phase of φk t ; the two signals of the initial frequency and phase are identical. The phase of the signal φk (ti ) emitted from GPS navigaj tion satellite is constant, and thus, the transmission distance ρ = λ.ϕk (ti ) (where j j j ϕk (ti ) = φk (ti ) − φk t ) can be determined by comparing the phase difference j  between the received signal φk (ti ) and the reference signal φk t j . Direct comparison of the phases of a signal usually only yields a phase difference j of less than one period between the two signals Fk (ti ). There still exists ambiguity ∗j resolution, denoting by Nk (ti ), which cannot be determined. Therefore, the actual phase difference should be expressed as ∗j

∗j

j

ϕk (ti ) = Nk (ti ) + Fk (ti )

(3.9)

∗j

where the actual value of Nk (ti ) is difficult to deduce. The phase measurements at different times are given as j

∗j

j

j

ϕk (ti ) = Nk (ti ) + Nk (t1 ) + Fk (ti ) ∗j

j

= ϕk (ti ) + Nk (t1 ) j

(3.10)

where Nk (t1 ) is the ambiguity resolution at t1 . If the connection between the receiver and the j-th navigation satellite is ensured after t1 , the ambiguity resolution of the receiver for the phase measurement of the j-th satellite will be a constant. Because of the bias between the time systems and the carrier delay in the ionosphere, the phase difference measurement is also biased. The distance measurement information obtained from the phase measurement is

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3 Navigation Method and Scheme Design for Rendezvous and Docking

  j j j j j λϕk (ti ) = ρk ti , ti − τk + λNk (t1 ) + Cδtk + Cδτtr op (ti ) + Cδτion (ti )

(3.11)

j

where δtk = δtk − δt j , Cδτtr op (ti ) is the distance error caused by tropospheric delay; Cδτion (ti ) is the electrical distance error due to ionosphere delay. Compared with pseudo-range measurement, the phase difference measurement add ambiguity j resolution Nk (t1 ). 3) Positioning Method Pseudo-range and phase difference measurements essentially provide information about the distance between the receiver and the navigation satellite. There are two main methods used in the practical positioning calculation. (1) The measurement equation related to the state to be estimated is constructed based on the distance measurement information. When there are no less than four observable navigation satellites, linearization is carried out at the initial position. The position correction can be obtained by using the least square estimation, and the position estimation can be obtained by combining with the initial position. By using pseudo-range rate measurement, the velocity estimate can also be obtained by the same method. (2) The measurement equation related to the state to be estimated is constructed based on the distance measurement information. When there are no less than four observable navigation satellites, the dynamic model of the moving body is used as the state equation, and the filter is designed based on the optimal filtering method to estimate the position and velocity of the carrier. Owing to the pseudo-range and phase difference measurement obtained by the measurement containing a variety of errors, such as clock error, ionospheric delay, tropospheric delay, etc., in order to improve the accuracy of the measurement information, the measurement information can be differentially processed to eliminate a part of the measurement error.

3.2.3 Microwave Radar The microwave radar uses electromagnetic waves to detect the range and orientation of the target relative to the radar. The radar antenna of the transmitter should send the electromagnetic wave which will be reflected or transmitted back by the transponder on the target, and the antenna will receive these waves, then the processing equipment could extract relative distance, relative range rate and orientation from these back signal. The microwave radar uses the electromagnetic wave wavelength range from the decimeter wave to the millimeter wave. Among them, the decimeter wave segment includes the L and S bands, and the centimeter band includes the X, Ku, K, and Ka bands [11, 12]. The microwave radar is suitable for relative measurement of long distance in rendezvous and docking. It played an important role in the early rendezvous and

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77

docking missions of the USA. The Gemini used microwave radar with L-band and interferometer angle measurement; the Apollo used an X-band monopulse amplitude continuous radar with side-sound ranging, carrier doppler velocity measurement, and monopulse angle measurement. The Ku-band pulse doppler rendezvous radar has been adopted on space shuttles, as have pulse doppler velocity measurement, pulse ranging, and monopulse specific amplitude measurement angles [13]. 1. Composition of Microwave Radar The composition and functions of a microwave radar are as follows. (1) Transmitter: the function of the transmitter is to generate the waveform of a required power level. (2) Receiver: The function of the receiver is to receive the reflected electromagnetic wave, emitted by the radar, from the target; the weak echo is processed from the accompanying noise and interference, which is filtered and amplified to a fixed level as appropriate for use by a signal processor. (3) Antenna: When transmitting, the antenna concentrates and converges the electromagnetic wave energy in a certain beam range and points the beam in a predetermined direction; when receiving, it forms a beam in a specific direction and collects the beam, and the energy of the reflected electromagnetic waves is then transmitted to the receiver over the transmission line. The orientation of target with respect to the radar is primarily determined by the shape and motion of the antenna lobe. (4) Digital signal processor: The radar signal processor mainly performs the following tasks: digital pulse compression; echo signal processing; constant false alarm detection; digital filtering; closed-loop tracking; range, rate, angle, and angular rate extraction; and system calibration, fault detection, and display. 2. Principle of Distance Measurement The principles of distance measurement include the following. 1) Measurement by Frequency Modulation Method The principle of the continuous wave measurement method of frequency modulation is that the continuous wave of equal amplitude and frequency modulation is generated by a voltage oscillator, where the relative distance is expressed as R=

fb C T 4 f

(3.12)

where f b is the difference frequency between the transmitting signal and the receiving signal, C is the speed of light,  f is the frequency modulation bandwidth, and T is the period of the continuous wave.

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2) Measurement by Pulse Method The pulse measurement methods may be incoherent or coherent, as described below. (1) Incoherent pulse method: Pulse measurement is the most basic method of the pulse radar. An electromagnetic wave propagates in space at the speed of light C = 2.997924562 × 108 m/s, and some energy is returned to the radar after encountering the target, which is amplified and detected by the radar receiver. By measuring the delay between the transmitting pulse and the receiving echo, the relative distance can be obtained, i.e., C τ 2

R=

(3.13)

While the incoherent method is simple, it does not achieve high precision. (2) Coherent pulse method: Compared with the incoherent pulse method, the phase of the transmitted signal and received echo of the coherent pulse is known, because the radar system uses the standard oscillation source, and the phase of various radar signals is determined regardless of the manner in which they change. For a fixed target, the phase of the echo only increases by a fixed amount, i.e., φ = 2π f τ = 2π f R=

φC 4π f

2R C

(3.14) (3.15)

3. Measurement Principle of Range Rate (1) Doppler Based Method When the target moves with respect to the radar, the carrier frequency of the received echo signal will shift with respect to the carrier frequency of the emitted signal, which is known as the Doppler frequency shift. If the target moves toward the radar, the frequency of the echo signal will decrease; otherwise, the frequency will increase. The radial velocity with respect to the radar can be determined once the Doppler frequency shift is measured. Denoting the Doppler frequency by f d , the relative velocity can be obtained as V = where λ is the wavelength of the signal.

λ fd 2

(3.16)

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79

(2) Differentiation-Based Method The measurement of relative velocity can also be obtained by differentiation, i.e., V =

R2 − R1 T

(3.17)

where T is the repeating period of the pulse, R are the distances between the consecutive echo signals. This method is much simpler, but its accuracy is very low. 4. Angle Measurement Angle measurement is classified into phase and amplitude methods. 1) Phase Method Angle measurement by the phase method, also known as interferometer angle measurement, uses the echo phase received by two or more antennas. If the interval between two antennas is d, and the angle between the target and the radar antenna normal is θ , the antennas receive the target echo signal wave path difference for R = d sin θ . The wave path differential is transformed to the phase difference ϕ, which can be measured by the phase comparator as follows: ϕ = 2π R/λ = 2π d sin θ/λ

(3.18)

where λ is the working wavelength of the radar. After using the phase comparator to measure ϕ, θ can be yielded according to (3.18). 2) Amplitude Method Because the antenna beam has a certain direction, during the dwell time of the radar antenna beam irradiating the target, N target echoes can be received, and the corresponding place with the maximum signal amplitude can be obtained from these echo signals; subsequently, according to the center of the radar antenna beam, the angles can be determined.

3.2.4 Laser Radar Laser radar is based on modern optical remote sensing technology, which is a combination of radar technology and laser technology. For the carrier, it uses laser, which has high brightness, and high coherence. Owing to these characteristics, the laser radar has a series of unique advantages: high angular resolution, high range resolution, high speed resolution, wide range of velocity measurement, can obtain a variety

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of target images, with strong anti-interference power. At the same time, the volume and mass of the laser radar are lower than those of the microwave radar. The laser radar used for space rendezvous and docking can measure the relative distance and range rate, as well as the orientation angles and their rates. The measurement range can be from tens of kilometers to less than 1 m. The measurement principle of the laser radar is introduced below [11]: 1) Ranging Principle Laser radar ranging has two modes: coherent detection mode and incoherent detection mode (also known as direct detection mode). While the principle of the coherent detection mode of a laser radar is the same as that of a microwave radar, the high frequency band of the laser significantly improves the ranging accuracy. According to the detected waves, the direct measurement mode can be divided into phase and pulse measurement modes. (1) Phase Measurement Mode The phase measurement is a continuously modulated laser beam emitted to measure the phase difference between the return signal and the local vibration signal. The relative distance is obtained according to the relationship between the phase and distance differences, L=

Cφ 4π f

(3.19)

where φ is the phase difference between the local oscillator and the echo signal, f is the modulation frequency of the laser, and L is the distance from the target to the radar. (2) Direct Pulse Mode Direct pulse measurement is conducted through the laser pulse signal, irradiated to the target and back, L=

CT 2

(3.20)

where T is the time interval between the laser transmitting pulse and the received echo pulse. 2) Angle Measurement Several methods can be used to measure angles, including the four quadrant tracking method, scanning imaging method, single pulse method, and cone scanning method. The former two methods are commonly used. The four quadrant tracking method simultaneously tracks the target by mounting the servo motor that drives the radar

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81

rotation, while on the angle encoder, the line-of-sight azimuth angle and pitch angle are measured. Scanning imaging angle measurement is performed by scanning the laser; the position of the target is obtained via image processing, and the angle measurement is realized. 3) Range Rate Measurement Laser heterodyne velocity measurement and audio frequency measurement based on the Doppler principle are used to directly measure the range rate. The relative range differential is used in the indirect measurement of the velocity, and the accuracy of the former is higher than that of the latter. 4) Angle Rate Measurement In the four-quadrant tracking angle measurement method, the change of the line of sight angle can be measured by the servo platform speed motor rate, whereas in other methods, the angle rate is obtained indirectly by differentiating the angle. 5) Scanning System Because the laser beam is narrow, in order to achieve a wide range of acquisition, beam scanning techniques are used, including double wedge scanning, double galvanometer scanning, rotary drum and galvanometer scanning, and computer grating scanning.

3.2.5 Camera-Type Rendezvous and Docking Sensor The CRDS is universally adopted as the primary measuring sensor in ranges of several hundred meters for docking. It can obtain high accuracy relative position and attitude measurement [11, 14, 15]. 1. Measurement Principles Installing more than three feature reflectors on the target spacecraft, the CRDS camera captures an image of these reflectors, and the corresponding image points are a, b, c, . . .. The image is processed by an information processor. Finally, the coordinate value of the image point in the camera coordinate system is obtained through the measurement. The algorithm solves the relative position between OT X T YT Z T of the reflector coordinate system and OC X C YC Z C of the camera coordinate system (measurement principle is shown in Fig. 3.4). Because the positions of the reflectors and camera in the target and chaser are known, the relative states between the two spacecraft can be calculated based on the above measurements. Let the position of the i-th reflector in the reflector coordinate system be T pi = [X T i , YT i , Z T i ]{T } and that in the machine coordinate system be ri = T [X Ci , YCi , Z Ci ]{C} . Let the position of the corresponding image point in the camera

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3 Navigation Method and Scheme Design for Rendezvous and Docking

A b

c ZT ZC

a

XC

XT B

YC

YT C

Focus plane

Camera lens

Reflector

Fig. 3.4 The measurement principle of CRDS

 T coordinates be Ii = xit , yit , − f {C} ; C CaT ar is the rotation matrix of the camera coordinate system with respect to the reflector coordinate system, and the corresponding attitude angle is being reorganized as θ1 , θ2 , θ3 ; M = [x M , y M , z M ]T is the origin position of the reflector coordinate in the camera coordinate system. Then, ⎤ ⎤ ⎡ X Ci XTi ⎣ YCi ⎦ = C CaT ar (θ1 , θ2 , θ3 )⎣ YT i ⎦ + M (i = 1∼N ). Z Ci ZTi ⎡

(3.21)

According to the image principle, we can get xit = −X Ci f /Z Ci yit = −YCi f /Z Ci

(3.22)

2. Monocular Measurement Algorithm Based on the above measuring principles, the following nonlinear equations can be obtained for the n points fi1 (θ1 , θ2 , θ3 , x M , y M , z M ) = xit + X Ci f /Z Ci = 0; (i = 1 . . . n)

(3.23)

fi2 (θ1 , θ2 , θ3 , x M , y M , z M ) = yit + YCi f /Z Ci = 0; (i = 1 . . . n)

(3.24)

where f is focal length and θ1 , θ2 , θ3 , x M , y M , z M are the parameters to be estimated, which can be obtained by solving the above nonlinear equations. The least-square method is usually utilized to solve nonlinear equations by using the generalized inverse [16]. The above equations can be rewritten as

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83

⎧ ⎨ f 1 (x1 , x2 , . . . , xn ) = 0 ························ ⎩ f m (x1 , x2 , . . . , xn ) = 0

m≥n

(3.25)

where f i (i = 1, 2, defined in an open set D of ⎡ ⎤ real⎡function ⎡ . . . , m)⎤is the nonlinear ⎤ 0 x1 f 1 (x) ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ R n . Let F(x) = ⎣ ... ⎦, x = ⎣ ... ⎦, 0 = ⎣ ... ⎦; f m (x) xn 0 Equation (3.26) can be rewritten as F(x) = 0 ⎡

∂ f1 ∂ x1 ∂ f2 ∂ x1

∂ f1 ∂ x2 ∂ f2 ∂ x2

··· ···

∂ f1 ∂ xn ∂ f2 ∂ xn

⎤

⎥ ⎥ .. .. ⎥ ⎥. . ··· . ⎦ ∂ fm ∂ fm · · · ∂∂ xfmn ∂ x1 ∂ x2 Then, the equation can be solved in iterative manner:

⎢ ⎢ with the Jacobian matrix of F(x) being F (x) = ⎢ ⎢ .. ⎣ . 

xk+1 = xk + αk∗ pk

(3.26)

 + where pk is the least-square solution of F  k pk = −F k , i.e., pk = − F  k F k  + with F  k being the pesudo-inverse of the Jacobian matrix at xk in the kth step, F k being the function of the left-hand of the equation, i.e., F k = [ f 1 (xk ), f 2 (xk ), . . . , f m (xk )]T , and αk∗ being the function of αk which renders (3.26) to be minimum. g(αk ) =

m 

  f i2 xk + αk pk

(3.27)

i=1

3. Binocular Measurement Algorithm Using the binocular measurement algorithm, the analytical least-square solution of the point position can be obtained with respect to the camera frame. The relative attitude between the two spacecraft can be calculated using the two-vector attitude determination algorithm, and thus, the relative position can also be calculated [16]. The binocular measurement algorithm is used to determine the relative position and attitude between two spacecraft using stereo vision, which has been applied in various fields. The basic principle is to use the imaging parallax of the same point in different cameras to calculate the spatial coordinates of the point. Figure 3.5 shows the principle of binocular measurement. In Fig. 3.5, O1 X 1 Y1 Z 1 is the main camera coordinate system, i.e., the left camera system coordinate system, and the origin O1 is the projection center; O1 Z 1 coincides with the main optical axis, O f 1 U1 V1 is the image plane coordinate system, O1 O f 1 = f , and the image plane of the camera is located at the focal length of the camera. O2 X 2 Y2 Z 2 is the reference frame of the right camera, and the origin O2 is in the

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3 Navigation Method and Scheme Design for Rendezvous and Docking P

y1

v1

z1

u2

x2

z2 y2

x1

o2

o1

p2

p1

u1

v2

of 1

of 2

Fig. 3.5 The principle of binocular measurement

projection center; O2 Z 2 coincides with the main optical axis, O2 O f 2 = f , and the image plane of the camera is located at the focal length of the camera. The relative position between the two cameras is denoted by R, and the relative rotation matrix is denoted by CR, i.e., ⎤ ⎤ ⎡ x r11 r12 r13 R = ⎣ y ⎦, C R = ⎣ r21 r22 r23 ⎦ r31 r32 r33 z ⎡

The position of the i-th point with respect to the frame of the first camera (the main T camera) is denoted by ri1 = [X C1i , YC1i , Z C1i ]{C1} ; then, its position with respect to T 2 the second camera ri = [X C2i , YC2i , Z C2i ]{C2} can be obtained as ⎛⎡ ⎞ ⎤ ⎤ X C2i X C1i ⎜ ⎟ ⎣ YC2i ⎦ = C {C2}{C1} ⎝⎣ YC1i ⎦ − M C2{C1} ⎠ Z C2i {C2} Z C1i {C1} ⎡

(3.28)

  where C {C2}{C1}  ci j is the rotation matrix between the frames of the two cameras, and M C2{C1}  [xc2 , yc2 , z c2 ]T is the position of the second camera with respect to the first one. C {C2}{C1} , M C2{C1} are determined by the installation of the cameras. The individual positions of the i-th points with respect to the two frames of the two cameras are given as

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85

xit1 = −X C1i f /Z C1i yit1 = −YC1i f /Z C1i

(3.29)

xit2 = −X C2i f /Z C2i yit2 = −YC2i f /Z C2i

(3.30)

    where xit1 , yit1 , xit2 , yit2 can be obtained from the images. Substituting (3.28) in (3.29) and (3.30) gives ⎧ ⎪ x t1 Z + X C1i f = 0 ⎪ ⎪ it1 C1i ⎪ ⎪ ⎪ ⎪ yit2 Z C1i + YC1i f = 0 ⎨ xi [c31 (X C1i − xc2 ) + c32 (YC1i − yc2 ) + c33 (Z C1i − z c2 )]+ ⎪ [c 11 (X C1i − x c2 ) + c12 (YC1i − yc2 ) + c13 (Z C1i − z c2 )] f = 0 ⎪ ⎪ ⎪ t2 ⎪ y ⎪ i [c31 (X C1i − xc2 ) + c32 (YC1i − yc2 ) + c33 (Z C1i − z c2 )]+ ⎪ ⎩ [c21 (X C1i − xc2 ) + c22 (YC1i − yc2 ) + c23 (Z C1i − z c2 )] f = 0

(3.31)

Let ⎤ f 0 xit1 ⎥ ⎢ 0 f yit1 ⎥ A=⎢ ⎣ c31 x t2 + c11 f c32 x t2 + c12 f c33 x t2 + c13 f ⎦ i i i c31 yit2 + c21 f c32 yit2 + c22 f c33 yit2 + c23 f ⎡

⎡

⎤ 0 ⎢ ⎥ 0 ⎥ B=⎢ ⎣ x t2 (c31 xc2 + c32 yc2 + c33 z c2 ) + f (c11 xc2 + c12 yc2 + c13 z c2 ) ⎦ i yit2 (c31 xc2 + c32 yc2 + c33 z c2 ) + f (c21 xc2 + c22 yc2 + c33 z c2 ) Then, (3.31) can be rewritten as Ari1 = B. If A is full column-rank, we get  −1 r 1i = AT A AT B

(3.32)

The position of the i-th point with respect to the first camera can be obtained according the above equation. When the positions of the N points with respect to the main camera are obtained, the relative attitude can be estimated using a multi-vector determination algorithm, because the positions of the points with respect to the target are already known. The estimation method is detailed as follows. For all the N points, eliminating M T {C} yields   (r i − r 1 ) = C {C}{T } pi − p1 (i = 2∼N )

(3.33)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

  Defining the reference vector  1as V 1 = 1 p2 1− p1 , p13 − p11,. . . , p N − p1 and the observation vector as U = r 2 − r 1 , r 3 − r 1 , . . . , r N − r 1 , U = C {C}{T } V or V = C {T }{C} U can be obtained . The solution with minimal norm is given as follows: C {T }{C} = V U + .

(3.34)

The rank of the observation vector U equals to 2 or 3, as all the points are not collinear.  −1 , (1) If the rank of U equals to 3, letting B = I, C = U, then U + = U T U U T and we get  −1 . C {T }{C} = V U T U U T

(3.35)

In general, the relative attitude matrix C {T }{C} obtained from the above equation is not a orthogonal matrix, and C {T }{C} will not satisfy the observation equation after direct orthogonalization. Therefore, the observation error vector U is utilized as the optimum u criteria, and C {T }{C} is orthogonalized in the optimal case. The optimal solution is given as C ∗{T }{C} =

  1 T C {T }{C} 3I − C {T }{C} C {T }{C} , 2

(3.36)

where C {T }{C} is obtained from (3.35). Once C ∗{T }{C} is determined, the relative position can be calculated by M T {C} =

N  1  1 r i − C ∗{T }{C} pi N i=1

(3.37)

(2) If the rank of U equals to 2, all the points are coplanar, and a normal can be introduced perpendicular to the plane to derive the measurement equation based on the dual-vector method, i.e., adding a column vector to V, 

   p2 − p1 × p3 − p1     ,  p2 − p1 × p3 − p1 

(3.38)

and a column vector to U,  1    r 2 − r 11 × r 13 − r 11  1    .  r − r1 × r1 − r1  2 1 3 1

(3.39)

Then, the new matrix U will be of full rank, and the relative position and attitude can be determined using (3.36) and (3.39).

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87

4. Reflector Arrangement Problem The design for the reflector numbers, reflector layouts, and measurement schemes should consider the measurement algorithm, measurement accuracy, spot recognition, redundancy ability, weight, power consumption, and installation conditions. Additionally, the validity of solving measurement equations must also be considered when a monocular iterative measurement algorithm is used [14]. 1) Determination of the Number of Reflectors In order to obtain the relative position and attitude of the two spacecraft, at least three characteristic points are needed. For the most basic three-point layout, the point coordinates Ii can be obtained with extraction error σ Ii = (σ Ii x , σ Ii y ), and the relative position and attitude extraction errors are σ = (σx M , σ yM , σz M , σα1 , σα2 , σα3 ). The transfer relation between σ Ii and σ can be obtained through the measurement precision analysis algorithm. The transfer coefficients are determined from the relative position, attitude, image point, and reflector position. When the number of reflectors is greater than three, the reflectors can be divided into a number of three-point subsets, and the relative position and attitude of each three-point subsets is solved individually. Finally, the arithmetic mean value of these solutions is taken to obtain the required relative position and attitude. The precision of the estimated position and attitude can be obtained by the above process. If each three-point subset measurement process is independent, and the standard deviation of the relative position and the attitude σi of the solution is the same, i.e., σ . Then, the standard deviation σ¯ is given as σ¯ =



   n  1  D(x) ¯ = D xi n i=1

  n 1  1  σ2 = √ σ = n 2 i=1 i n

(3.40)

For a four-point layout, the number of three-point subsets is n = C43 = 4 For a five-point layout, the number of three-point subsets is n = C53 = 10 For a six-point layout, the number of three-point subsets is n = C63 = 20 For a seven-point layout, the number of three-point subsets is n = C73 = 35 When the number of reflectors increase and the value of N becomes larger, the standard deviation of the arithmetic mean will diminish, the precision can be improved. Figure 3.6 shows the relation between σ¯ /σ and the reflector number N. In the figure, when the number of reflectors increases by N, σ¯ /σ gradually decreases, and the measuring precision will improve. However, when the number reaches a certain value, the number of points is to be increased, accuracy is very low. Therefore, to improve the relative position and attitude solution accuracy, simply increasing the number of reflectors is inadequate. Power consumption and fault redundancy must also be considered comprehensively. Finally, the optimal selection range of the number of points of light on the target is 4–6.

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3 Navigation Method and Scheme Design for Rendezvous and Docking 1 0.9 0.8 0.7 σ/σ

0.6 0.5 0.4 0.3 0.2 0.1 0

4

5

6 7 the number of reflectors

8

Fig. 3.6 The relation between σ¯ /σ and the reflector number N

2) Effectiveness Analysis of Four-Coplanar-Point Pattern The four-spot layout, particularly the square-light spot layout, is a common spot layout type. In general, for the case where only positive solutions are considered, a unique solution of relative position and attitude can be obtained by using the four-spot coplanar layout. However, for the measurement scheme using an iterative measurement algorithm, the iterative algorithm is a single-value algorithm, and if the initial value is not appropriately selected, it will continue to produce a pseudosolution. Theorem 3.1 Zhang et al. [14] The number of reflectors equals to four and the arrangement of the points and the image of these points satisfy the following conditions: (1) the four points are on the same plane, which is parallel to the x-y plane of the target’s frame, and form a quadrangle, (2) the four points are not collinear and do not coincide with each other. There are two solutions C1{C}{T } , M1T {C} and C2{C}{T } , M2T {C} when using four points, and one solution is invalid. The relation between these two solutions can be described by ⎡

C1{C}{T } = C2{C}{T }

⎤ −1 0 0  ⎣ 0 −1 0 ⎦, M1T {C} = −M2T {C} 0 0 1

(3.41)

3) Analysis of the Solution using a Five-Point Pattern The five-point arrangement adds a prominent point which is not in the plane containing the other four points. The analysis of the solution using a five-point arrangement is given in Theorem 3.2.

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89

Theorem 3.2 Zhang et al. [14]. The arrangement of the five points and their images satisfy the following conditions: (1) four points are coplanar and form a quadrangle which is parallel to the x–y plane of the marker’s frame, and the fifth point is outside of the plane; (2) the five image points do not coincide and are not collinear. For the five-point arrangement, the necessary and sufficient condition for obtaining a unique relative position and attitude is that the relative pose of the camera with respect to the marker satisfies Eqs. (3.42) or (3.43): (c11 X T 5 + c12 YT 5 + c13 Z T 5 + x M )(−c31 X T 5 − c32 YT 5 + c33 Z T 5 − z M ) = (c31 X T 5 + c32 YT 5 + c33 Z T 5 + z M )(−c11 X T 5 − c12 YT 5 + c13 Z T 5 − x M ) (3.42) (c21 X T 5 + c22 YT 5 + c23 Z T 5 + y M )(−c31 X T 5 − c32 YT 5 + c33 Z T 5 − z M ) = (c31 X T 5 + c32 YT 5 + c33 Z T 5 + z M )(−c21 X T 5 − c22 YT 5 + c23 Z T 5 − y M ) (3.43)   where C{C}{T }  ci j , i = 1∼3, j = 1∼3. 4) Analysis of the Solution using a Six-Point Pattern The analysis of the solution using a six-point arrangement is given in Theorem 3.3. Theorem 3.3 Zhang et al. [14]: The arrangement of the six points and their images satisfy the following conditions: (1) four points are coplanar and form a quadrangle which is parallel to the x-y plane of the marker’s frame, and the remaining two points are outside of the plane; (2) the six image points do not coincide and are not collinear. For the six-point arrangement, the unique relative position and attitude could be solved by the measurement function. 5. Spot Recognition Problem In the final approach phase of rendezvous and docking, the relative attitude and position between the two spacecraft are constrained in a certain range, because the chaser is gradually approaching in the rendezvous corridor and the relative measurement sensor is required to capture the target at all times. Therefore, if the target recognition algorithm can successfully recognize the patterns, the requirements of the rendezvous and docking mission will be satisfied. At present, there are three types of light spot identification methods [15]: (1) Scintillation Method: Different points of light are flashed at different frequencies, and the scintillation frequency is set to be lower than the measurement frequency rate. However, this method will increase the complexity of hardware design and reduce the reliability of spot. (2) Geometric Method: A specific layout of points of light is set, and they are paired with points of light according to the special geometric relationships identified between them. It is difficult to implement this method when considering the case of spot faults.

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(3) Feature Matching Method: Geometric shape information, constraint conditions, and projection variation of feature points are used. The recognition and matching of feature points are realized by the constraint conditions in the transformation.

3.3 Filtering Method for Navigation The measurement information related to the estimated state can be obtained by using the measurement sensor, but the measurement error is inevitable; In addition, analyzing the motion law of the system, the motion of the system can be established, which can also provide information about the state to be estimated. For state estimation with the measurement and motion information, an appropriate information extraction method, also known as the navigation filtering method, is required.

3.3.1 Theory of Bayesian Estimation The discrete stochastic system is given as X k+1 = f (X k ) + wk

(3.44)

Z k+1 = h(X k+1 ) + vk+1

(3.45)

where X ∈ Rn is the system state, Z ∈ Rm is the measurement of sensor, and wk , vk are the process noise and measurement noise, respectively,   with E[wk ] = 0, Cov(wk , wk ) = Q k , E[vk ] = 0, Cov(vk , vk ) = Rk , Cov wi , v j = 0. In 1964, Ho and Lee illustrated the general statistical filtering problem based on Bayesian theory. A theoretical closed-loop solution is derived, thus forming the Bayesian estimation theory [17]. According to Bayesian estimation theory, the posterior probability density of states contains all statistical information about states and can be regarded as the statistical description of the probability of states. If the state sequence is set as a first-order Markov random sequence, then the posterior probability density of the state can be recursively obtained by Bayesian estimation in the following form: p(X 0:k+1 |Z 1:k+1 ) = p(X 0:k |Z 1:k )

p(Z k+1 |X k+1 ) p(X k+1 |X k ) p(Z k+1 |Z 1:k )

(3.46)

where ! p(Z k+1 |Z 1:k ) =

p(Z k+1 |X k+1 ) p(X k+1 |X k ) p(X k |Z 1:k )d X k:k+1

(3.47)

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91

The above derivation utilizes the time-series of the state X 0:k+1 ; considering only the estimation of the state at k + 1, the marginal distribution is obtained as ! p(X k+1 |Z 1:k+1 ) = =

p(X 0:k+1 |Z 1:k+1 )d X 0:k p(Z k+1 |X k+1 ) p(X k+1 |Z 1:k ) p(Z k+1 |Z 1:k )

(3.48)

p(X k+1 |X k ) p(X k |Z 1:k )d X k

(3.49)

where ! p(X k+1 |Z 1:k ) =

In practical applications, the state is required to be estimated independent of the probability density, and thus, a certain criteria should be designed to determine whether or not the information is used for state estimation. Commonly used methods include the minimum variance, maximum likelihood, maximum a posteriori, and so on. In the Bayesian estimation, high-dimensional integrals must be performed, as shown in (3.47) and (3.49). It is difficult to obtain exact analytic solutions for general nonlinear systems, and only an approximate solution can be obtained. Obtaining the approximate solution from the Bayesian estimation is the most important research interest in the field of statistical filtering of nonlinear systems. The estimation methods include linear minimum variance estimation, particle filtering, Gaussian sum filtering, interpolation filtering [18]. The starting point of these methods is to simplify the description of probability density and approximate the solution of the high-dimensional product to obtain the approximate solution of the system’s Bayesian estimation in terms of a high-dimensional integral.

3.3.2 Linear Minimum Variance Estimation The characteristic of linear minimum variance estimation is: (1) the structure of estimation algorithm is linear, (2) in linear structure filtering, the estimated variance is the smallest, (3) the algorithm only involves the mean value, variance, and covariance of the state/measurement. For the nonlinear system described in (3.44) and (3.45), the recursive form of the linear minimum variance method is given as [19]

where

Xˆ k+1 = Xˆ k+1|k + K k+1 (Z k+1 − Zˆ k+1|k )

(3.50)

P k+1 = P k+1|k − K k+1 P zz,k+1|k K Tk+1

(3.51)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

  Xˆ k+1|k = E X k+1 |Z 1:k P k+1|k = E

"

P x z,k+1|k = E

X k+1 − Xˆ k+1|k

"



X k+1 − Xˆ k+1|k

X k+1 − Xˆ k+1|k



(3.52) T

Z k+1 − Zˆ k+1|k

# |Z 1:k

T

(3.53) #

|Z 1:k

  Zˆ k+1|k = E Z k+1 |Z 1:k P zz,k+1|k = E

"

Z k+1 − Zˆ k+1|k



Z k+1 − Zˆ k+1|k

 −1 K k+1 = P x z,k+1|k P zz,k+1|k .

(3.54) (3.55)

T

# |Z 1:k

(3.56) (3.57)

To solve the equations in (3.52)–(3.56), we should first estimate the predicted possibility density p(X k+1 |Z 1:k ) 和 p(Z k+1 |Z 1:k ). The main factors to be considered in the design of the linear minimum variance method lie in the method of estimating p(X k |Z 1:k ) 和 p(X k+1 |Z 1:k ) and the method of calculating the highdimensional integral of the mean value, variance, and the covariance matrix. Different filtering methods, such as the Kalman filter and deterministic sampling filtering, can be derived based on the manner in which these two issues are handled.

3.3.3 Kalman Filter If the joint distribution of the state and the measurement follow Gaussian distribution, the linear minimum variance estimation is equivalent to the Bayesian estimation. For a linear/Gaussian system, the linear minimum variance estimate is known as Kalman filtering. For general nonlinear systems, it is usually difficult to obtain an accurate analysis of statistical parameters such as the mean, variance, and covariance of state/measurement using linear minimum variance estimation. Extended Kalman filtering (EKF) was proposed in the 1960s. It employs the first-order Taylor expansion to linearize the nonlinear function of the state and uses the approximate linear system to obtain the mean, variance and covariance matrix, thus forming the linear minimum variance estimation of the approximate form. For 50 years, EKF has been widely used in nonlinear system filter design. However, the inherent shortcomings of EKF have long been recognized: (1) owing to the need to apply Taylor expansion for linear approximation, EKF cannot be applied to nondifferentiable nonlinear systems; (2) linearization of nonlinear functions, particularly high-dimensional nonlinear functions, is highly complicated; (3) because it only uses the nonlinear function of the first order approximation, the strong nonlinear system estimation effect of EKF is often not ideal.

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Ever since it was proposed, the improvement of the EKF method has been an area of focus in the field of statistical filtering, resulting in approaches such as second order EKF (SEKF) and iterative EKF (IEKF). However, SEKF needs to solve the Hessians matrix of the nonlinear function, while IEKF requires the system state to be substantially observable. In addition to these harsh application conditions, in many cases, these methods often offer limited improvement in the estimation accuracy, while requiring much more computation than EKF.

3.3.4 Deterministic Sampling Filter The deterministic sampling filter is a type of approximate linear minimum variance estimation, utilizing statistical information to select the appropriate sampling point (Sigma point), and estimate the mean, variance, and covariance matrix of the nonlinear function of the state by weighting the nonlinear mapping of the sampling points. The sampling points are selected in a deterministic manner rather than a random one, which is adopted by particle filtering. 1. Typical Deterministic Sampling Filtering Methods Typical deterministic sampling filtering methods include unscented Kalman filtering, divided difference filtering, Gauss-Hermit filtering, cubature Kalman filtering, etc. (1) Unscented Kalman Filtering Unscented Kalman Filtering (UKF) was proposed by Simon Julier and Jeffrey Uhlmann in the mid-1990s [20]. The basic principle behind UKF is based on the fact that it is easier to approximate the probability density than the nonlinear function. Using unscented transformation (UT), selecting sigma points and the corresponding weights according to the statistical information of the state (usually mean and variance), and estimating the statistical parameters based on the nonlinear mapping of the sigma points will yield the linear minimum variance estimation method. The advantages of UKF are as follows: firstly, UKF does not need to linearize the nonlinear function; thus, theoretically, for any nonlinear system, all systems apply. In addition, the estimation of the statistical parameters of the state nonlinear function obtained by UKF is more accurate than that of EKF, and more accurate state estimation results can be obtained. (2) Divided Difference Filtering The divided difference filter (DDF) was developed by Danish scholar Magnus Nørgaard [21] in the 1990s. It uses Sterling polynomial interpolation formulas to obtain the approximation of the nonlinear function and estimates the mean, variance and covariance matrix of the nonlinear function by combining the statistical information of the states. Following this, the linear minimum variance estimation is

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obtained. Based on the accuracy of the nonlinear function approximation, DDF can be divided into two types: first-order DDF (DDF1) and second-order DDF (DDF2). The nonlinear function approximation method adopted in DDF does not require the differential of the nonlinear function, and it can thus be applied to any nonlinear function. In addition, the nonlinear function approximation method adopted by DDF2 is more accurate than first-order Taylor expansion. The statistical parameter estimation of the state nonlinear function is more accurate than EKF. (3) Gauss-Hermite Filtering If the state is always assumed to be Gaussian, then the mean, variance and covariance matrix involved in linear minimum variance estimation can be solved by the Gaussian integral of the nonlinear function, which can be achieved using numeric integration. K. Itoti employed the Gauss-Hermite numerical integral method, in which the sampling points and their weights are carefully selected, to approximately solve the Gauss integration [22]. The estimations of statistical parameters, such as the mean of the state nonlinear function, are obtained using the nonlinear mapping of the sampling points, and this linear minimum variance estimation is called the Gauss-Hermite filtering (GHF). (4) Cubature Kalman Filtering Arasaranam introduced the cubature integral law into the design of the filtering algorithm, and proposed a method based on Gaussian numeric integration, i.e., cubature Kalman filter (CKF) [23]. The gaussian integral of the nonlinear function is decomposed into a spherical integral and radius integral; the former is done by using the cubature integral law, selecting the integral point and the corresponding weight, while the latter is approximated by the first order Gauss-Laguerre integral law. The corresponding integral points and weights of the two respective parts are multiplied in order to obtain the final integral point and the corresponding weight. Using the corresponding integration points and weights to estimate the mean, variance, and covariance matrix, the linear minimum variance estimation of the approximate form is constructed. 2. Unified Framework for Deterministic Sampling Filtering The deterministic sampling filter has the following characteristics: (1) The design of the algorithm is based on the linear minimum variance principle and uses the known stochastic state X to approximate the mean, variance, and covariance matrix. (2) Appropriate sampling points are selected based on the statistical information of X, and the mean, variance and covariance matrix of the nonlinear function of the state are estimated by weighting the nonlinear mapping of the sampling points. (3) The Jacobian matrix is not involved in the filtering algorithm; therefore, the system function is not required to be differentiable.

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95

(4) Using only the mean and variance of X, the order of estimated means from the above four algorithms can be taken as two, which is higher than that of EKF, while the accuracy of the estimated covariance is equivalent to EKF. When the distribution of X is Gaussian, much higher accuracy can be obtained. It should be noted that the above four algorithms involve the selection of the sampling points, which is similar to the selection of the particles in particle filtering. The difference lies in that the particles are selected using random sampling, while the above algorithms sample the points deterministically. Thus, the above algorithms are called deterministic sampling filters, and they can be summarised using the following unified framework. (1) Generation of Sampling Points The sampling points χ i,k and the weights wi , i = 1 . . . N (where N is the number of sampling points) are generated based on the estimated state and covariance matrix at time k, i.e., Xˆ k , Pˆ k . (2) Time Update Xˆ k+1|k =

N 

wi f (χ i,k )

(3.58)

i=1

Pˆ k+1|k =

N 

  T wi Xˆ k+1|k − f (χ i,k ) Xˆ k+1|k − f (χ i,k ) + Q k

(3.59)

i=1

(3) Re-Generation of Sampling Points χ i,k+1|k , i = 1 . . . N is re-generated based on Xˆ k+1|k , Pˆ k+1|k in the same manner as in (1). (4) Predicting the Mean, Variance, and Covariance Matrix of Measurement Zˆ k+1|k =

N 

wi h(χ i,k+1|k )

(3.60)

i=1

Pˆ zz,k+1|k =

N 

  T wi h(χ i,k+1|k ) − Zˆ k+1|k h(χ i,k+1|k ) − Zˆ k+1|k + Rk+1

i=1

(3.61) Pˆ x z,k+1|k =

N  i=1

  T wi Xˆ k+1|k − χ i,k+1|k Zˆ k+1|k − h(χ i,k+1|k )

(3.62)

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(5) Measurement Update   Xˆ k+1 = Xˆ k+1|k + K k+1 Z k+1 − Zˆ k+1|k

(3.63)

−1 K k+1 = Pˆ x z,k+1|k Pˆ zz,k+1|k

(3.64)

Pˆ k+1 = Pˆ k+1|k − K k+1 Pˆ zz,k+1|k K Tk+1

(3.65)

3. Stability Analysis of Deterministic Sampling Filtering Stability is a basic requirement for filter design. Generally, stability means that the estimates are not affected by the initial state when the time increases. The uniformly asymptotical stability of the linear/Gaussian Kalman filter can be ensured if the system is stochastically observable. However, the proof for the nonlinear case is generally difficult. Taking UKF for a system with a linear state equation and nonlinear observation equation as an example, we analyze the stability of the deterministic sampling filter. A sufficient condition for the stability of UKF is obtained using the boundedness theorem and pseudo-linearization method. The adopted analysis method can also be extended to other deterministic sampling filters. (1) Filtering Algorithm for a System with a Linear State Equation The discrete nonlinear system is given as X k+1 = F k X k + wk

(3.66)

Z k+1 = h(X k+1 ) + vk+1

(3.67)

where X k ∈ R n is the system state, Zk+1 ∈ R m is the measurement, F k is the state matrix, and h(·) is assumed to be continous and differentiable at X k ; wk , vk are the process noise and measurement noise, respectively, which satisfy Cov(wk , wk ) =   Q k , Cov(vk , vk ) = Rk , Cov wi , v j = 0. The UKF of the system in (3.66) and (3.37) is given as 1) Time Update using State Equation

Xˆ k+1|k = F k Xˆ k

(3.68)

ˆk Pˆ k+1|k = F k Pˆ k F Tk + Q

(3.69)

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2) Selection of Sampling Points based on Xˆ k+1|k and Pˆ k+1|k ⎧ ˆ ⎪ ⎪ χ i,k+1|k = X k+1|k $% & ⎪ ⎪ ⎨ χ i,k+1|k = Xˆ k+1|k + (n + κ) Pˆ k+1|k &i $% ⎪ ⎪ ⎪ ⎪ ⎩ χ i,k+1|k = Xˆ k+1|k − (n + κ) Pˆ k+1|k

wi = κ/(n + κ)

i =0

wi = 1/(2(n + κ)) i = 1 . . . n

i−n

(3.70)

wi = 1/(2(n + κ)) i = n + 1, . . . , 2n

3) Predicting the Mean, Variance, and Covariance of Measurement Zi,k+1|k = h(χ i,k+1|k ) Zˆ k+1|k =

2n 

wi Zi,k+1|k

(3.71)

(3.72)

i=0

Pˆ zz,k+1|k =

2n 

ˆ k+1 wi (Zi,k+1|k − Zˆ k+1|k )(Zi,k+1|k − Zˆ k+1|k )T + R

(3.73)

i=0

Pˆ x z,k+1|k =

2n 

wi (χ i,k+1|k − Xˆ k+1|k )(Zi,k+1|k − Zˆ k+1|k )T

(3.74)

i=0

4) Measurement Update Xˆ k+1 = Xˆ k+1|k + K k+1 (Z k+1 − Zˆ k+1|k )

(3.75)

−1 T Pˆ k+1 = Pˆ k+1|k − Pˆ x z,k+1|k Pˆ zz,k+1|k Pˆ x z,k+1|k

(3.76)

−1 K k+1 = Pˆ x z,k+1|k Pˆ zz,k+1|k

(3.77)

ˆ k and R ˆ k+1 are the covariance matrices of the process and measurement where Q nosies, respectively. (2) The Simplification of UKF The pseudo-linearization method is employed here to rewrite the UKF in a simpler form. Define the estimation error as X˜ k+1|k = X k+1 − Xˆ k+1|k , Z˜ k+1|k = Z k+1 − Zˆ k+1|k . Taking the Taylor expansion of Z k+1 at Xˆ k+1|k gives Z k+1 = h( Xˆ k+1|k ) + H k+1 X˜ k+1|k + · · · + vk+1

(3.78)

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where H k+1 =

'

∂h ' . ∂ X X= Xˆ k+1|k

Then, taking the Taylor expansion of Zi,k+1|k = h(χ i,k+1|k ) at Xˆ k+1|k yields 1 Zˆ k+1|k = h( Xˆ k+1|k ) + ∇xT Pˆ k+1|k ∇x h( Xˆ k+1|k ) + · · · 2

(3.79)

T  where ∇x = ∂ ∂X 1 . . . ∂ ∂X n . Subtracting (3.78) from (3.79), we have Z˜ k+1|k = β k+1 H k+1 X˜ k+1|k + vk+1

(3.80)

where β k+1 is a diagonal matrix with dimension m, which is a function of X˜ k+1|k , and thus, β k+1 is a random matrix. Then we have   X˜ k+1|k = F k − F k K k β k H k X˜ k|k−1 + wk − F k K k vk .

(3.81)

The equation for the state update is given as ) ( T P k+1|k = E X˜ k+1|k X˜ k+1|k |Z 1:k    T = F k − F k K k β k H k Pˆ k|k−1 F k − F k K k β k H k + Q k + F k K k Rk K Tk F Tk + Δ P k+1|k

(3.82)

where ˆ k − P k+1|k . δ P k+1|k = F k Pˆ k F Tk + Q

(3.83)

From (3.82) to (3.83), we have   T  Pˆ k+1|k = F k − F k K k β k H k Pˆ k|k−1 F k − F k K k β k H k + Q ∗k

(3.84)

where Q ∗k = Q k + Δ P k+1|k + δ P k+1|k + F k K k Rk K Tk F Tk .

(3.85)

Similarly, the covariance matrix P x z,k+1|k is written as  T P x z,k+1|k = Pˆ k+1|k β k+1 H k+1 + Δ P x z,k+1|k

(3.86)

where Δ P x z,k+1|k is deviation. Let δ P x z,k+1|k = Pˆ x z,k+1|k − P x z,k+1|k ; from (3.86), we can obtain Pˆ x z,k+1|k = Pˆ k+1|k H Tk+1 β k+1 + Δ P x z,k+1|k + δ P x z,k+1|k

(3.87)

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99

For simplicity, an n dimensional time-varying matrix ϕ k+1 is introduced. Pˆ x z,k+1|k = Pˆ k+1|k ϕ k+1 H Tk+1 β k+1

(3.88)

Generally, the system satisfies the condition n ≥ m; thus, we have ( ) −1 ϕ k+1 H Tk+1 β k+1 = Pˆ k+1|k Pˆ k+1|k H Tk+1 β k+1 + Δ P x z,k+1|k + δ P x z,k+1|k . (3.89) Then, P zz,k+1|k can be rewritten as   T  P zz,k+1|k = β k+1 H k+1 Pˆ k+1|k β k+1 H k+1 + Rk+1 + Δ P zz,k+1|k

(3.90)

where Δ P zz,k+1|k is the deviation and δ P zz,k+1|k =

2n 

  T ˆ k+1 − P zz,k+1|k . wi Zi,k+1|k − Zˆ k+1|k Zi,k+1|k − Zˆ k+1|k + R

i=0

(3.91) Then we have   T  Pˆ zz,k+1|k = β k+1 H k+1 Pˆ k+1|k β k+1 H k+1 + R∗k+1

(3.92)

R∗k+1 = Rk+1 + Δ P k+1|k + δ P zz,k+1|k .

(3.93)

with

Substituting (3.88) and (3.92) into (3.76) and (3.77) yields ( )−1   T K k+1 = Pˆ k+1|k ϕ k+1 H Tk+1 β k+1 β k+1 H k+1 Pˆ k+1|k β k+1 H k+1 + R∗k+1 (3.94)



( )−1   T Pˆ k+1 = Pˆ k+1|k − Pˆ k+1|k ϕ k+1 H Tk+1 β k+1 β k+1 H k+1 Pˆ k+1|k β k+1 H k+1 + R∗k+1  T (3.95) Pˆ k+1|k ϕ k+1 H Tk+1 β k+1

(3) Boundedness of Stochastic Process Definition 3.1 Liu and Zie [24]: If there exists η, υ > 0 and 0 < ϑ < 1 such that a stochastic process ζ k satisfies (  )  2 2 E ζ k  ≤ ηζ 0  ϑ k + υ, ∀k > 0, then ζ k is said to be exponentially bounded in terms of the mean-square.

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(4) Sufficient Condition for the Stability of UKF A sufficient condition for the UKF of the nonlinear stochastic system in (3.66) and (3.67) can be formed using Definition 3.1. Condition 3.1 For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), the following four conditions are given: ¯ β, ¯ φ¯ > 0 that satisfy the following inequations 1) There exist positive constants f¯, h, when k ≥ 0:     ¯ β k  ≤ β, ¯ ϕ k+1  ≤ φ. ¯ F k  ≤ f¯, H k  ≤ h,

(3.96)

2) There exist positive constants f , h, β > 0 φ¯ > 0 that satisfy the following inequations when k ≥ 0: f 2 I ≤ F k F Tk , h 2 I ≤ H k H Tk , β 2 I ≤ β k β Tk .

(3.97)

¯ˆ q ∗ , r¯ , r ∗ , p, p¯ > 0 that satisfy the following 3) There exist positive constants q, q, ¯ q, inequations when k ≥ 0: ¯ˆ q ∗ I ≤ Q∗ , R ≤ r¯ I, R ˆ k ≤ qI, ˆ k > 0, r ∗ I ¯ 0≤Q q I ≤ Qk ≤ qI, k k − −

−

≤

∗ Rk+1 ,

p I ≤ Pˆ k ≤ pI. ¯ −

(3.98)

4) F k − F k K k β k H k is uniformly invertible, and there exists f¯∗ > 0 such that  −1   F  ≤ f¯∗ . k

(3.99)

Five lemmas are introduced as follows. Lemma 3.1 Liu and Xie [24]: For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), if Condition 3.1 is satisfied, we get   p f 2 I ≤ Pˆ k+1|k ≤ p¯ f¯2 + q¯ˆ I ∀k ≥ 0

(3.100)

Lemma 3.2 Liu and Xie [24]: For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), if Condition 3.1 is satisfied and 0 < λ < 1, we get 

Fk − Fk K k β k H k

T

 −1  −1 Pˆ k+1|k F k − F k K k β k H k ≤ (1 − λ) Pˆ k|k−1

(3.101)

Lemma 3.3 Liu and Xie [24]: For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), if Condition 3.1 is satisfied and μ > 0, we have

3.3 Filtering Method for Navigation

101

' ) ( −1 −1 ' E wTk Pˆ k+1|k wk + vTk K Tk F Tk Pˆ k+1|k F k K k vk ' X˜ k|k−1 ≤ μ.

(3.102)

Lemma 3.4 Liu and Xie [24]: If wk is n dimensional Gaussian white noise with 0 mean and the covariance matrix Q k , and q I ≤ Q k ≤ q¯ I, then wk is exponentially bounded in terms of the mean-square, i.e., k−1    q¯   E wk 2 ≤ E w0 2 (1 − λw )k + n q¯ (1 − λw )i . q i=1

(3.103)

Lemma 3.5 Liu and Xie [24]: For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), if Condition 3.1 is satisfied, then the predicted state error X˜ k+1|k is exponentially bounded with probability 1 using the UKF, i.e., " " k−1 2 #  #  p¯ f¯2 + q¯ˆ    ˜ 2 k 2 ¯ ¯ X (1 − λ) E  X˜ k+1|k  ≤ E + μ( p ¯ f + q) ˆ (1 − λ)i   1|0 pf2 i=1 (3.104) Theorem 3.4 For the system in (3.66) and (3.67) and the filtering algorithm in (3.68)–(3.77), if Condition 3.1 is satisfied and, then the estimation error of state X˜ k+1|k is exponentially bounded using the UKF. Proof From (3.104), we have " $ & "  # 2 #   p¯ f¯2 + q¯ˆ ˜   ˜ 2 2 2 ¯ E Xk+1|k  ≤ 2 f E X0  + 2E w0  pf2 − −

¯ˆ (1 − λ)k + μ( p¯ f¯2 + q)

k−1 

(1 − λ)i

(3.105)

i=1

and from the state equation, we have   X˜ k+1|k − wk . X˜ k = F −1 k

(3.106)

$ & 2  2   ˜   X k  ≤ 2 f¯∗2  X˜ k+1|k  + wk 2 .

(3.107)

Using (3.106) yields

Taking the mean of each side gives $ " "  # & 2 #   ˜  ˜ 2  ∗2 2 ¯ E  X k+1|k  + E wk  . E Xk ≤ 2 f

(3.108)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

Substituting (3.103) and (3.105) into (3.108), and using the properties of geometric series yields "  # "  #  2  ˜ 2 E  X k  ≤ ηE  X˜ 0  ϑ k + υ

(3.109)

where η = 4 f¯∗2 f¯2

p¯ f¯2 + q¯ˆ pf2

ϑ =1−λ 1 − λw ¯ˆ 1 − λ υ = 2 f¯∗2 n q¯ + 2 f¯∗2 μ( p¯ f¯2 + q) λw λ  2 ¯ ¯   p¯ f + qˆ q¯ + 4 f¯∗2 + 2 f¯∗2 E w0 2 . q pf2

(3.110) (3.111)

(3.112)

From Definition 3.1, it can be concluded that X˜ k is exponentially bounded in terms of the mean-square. 4. Adaptive Improvement of Deterministic Sampling Filtering In practical applications, the statistical parameters of noise are usually time-varying and are difficult to obtain. Inappropriate selection of the parameter would lead to a decrease the filtering performance and even divergence. An efficient method to overcome this issue is to adaptively estimate the statistical parameters online, which yields the adaptive filtering methods [25]. In this section, we introduce a filtering algorithm that can be applied in a general nonlinear system with nonstationary noise. The algorithm is presented in the form of deterministic sampling filtering [26]. The discrete nonlinear system is given as X k+1 = f (X k ) + wk

(3.113)

Z k+1 = h(X k+1 ) + vk+1

(3.114)

where X k ∈ R n is the system state, Zk+1 ∈ R m is the measurement; h(·) is assumed to be continous and differentiable at X k ; wk , vk are respectively the process noise and measurement noise which satisfy Cov(wk , wk ) = Q k , Cov(vk , vk ) = Rk ,  Cov wi , v j = 0.

3.3 Filtering Method for Navigation

103

1) Statistical Parameter Estimation For Stationary Noise The process noise is assumed to be stationary, q k ≡ q, Q k ≡ Q. The estimation equation for linear minimum variance is written as   Xˆ k+1|k = E f (X k )|Z 1:k + q P k+1|k = E

(

)    T f (X k ) − E f (X k ) f (X k ) − E f (X k ) |Z 1:k + Q

(3.115) (3.116)

(1) Statistical Parameter Estimation for Noise Denote the state estimation Xˆ j at j, and introduce the following variable   q j = Xˆ j − E f (X j−1 ) = Xˆ j − ¯f (X j−1 )

(3.117)

Because the noise is stationary and ergodic and N > 0, the noise is approximated by qˆ N =

N 1  q N j=1 j

(3.118)

and the variance is approximated by Cˆ γ =

N N  T  1  1  γ j − γˆ N γ j − γˆ N = γ j γ Tj N − 1 j=1 N j=1

where γ j = Xˆ j − Xˆ j| j−1 , γˆ N =

1 N

N *

(3.119)

γ j.

j=1

Define X˜ j = Xˆ j − X j , and we have "  T # ˜ ˜ E γ j − Xj γ j − Xj (        T ) +Q = E f X j−1 − E f (X j−1 ) f X j−1 − E f (X j−1 )

(3.120)

and " ) ( )  T # T T ˜ ˜ ˜ ˆ ˆ E γ j X j = E X j X j − E X j − X j| j−1 X j − X j| j−1 "  T # ˆ j| j−1 Z j − Zˆ j| j−1 K Tj + E Xj − X (

(3.121)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

According to (3.120) and (3.121), can be rewritten as "  T # ) (   ˜ Tj ˜ jX ˜ ˜ = E γ j γTj − E X E γj − Xj γj − Xj "  T # ˆ ˆ + 2E X j − X j| j−1 X j − X j| j−1 "  T # ˆ ˆ KTj − E X j − X j| j−1 Z j − Z j| j−1 "  T # ˆ ˆ − K j E Z j − Z j| j−1 X j − X j| j−1

(3.122)

According to (3.120), (3.122) is equivalent to " )  T # (   ˆ j| j−1 X j − X ˆ j| j−1 ˜ jX ˜ Tj + 2E X j − X Q = E γ j γTj − E X "  T # ˆ j| j−1 Z j − Z ˆ j| j−1 K Tj − E Xj − X "  T # ˆ j| j−1 − K j E Z j − Zˆ j| j−1 X j − X (        T ) (3.123) − E f X j−1 − f¯ X j−1 f X j−1 − f¯ X j−1 As Q is constant, from (3.119), we have ⎫ ⎧ N   T * ⎪ ⎪ ⎪ ⎪ γ j − γˆ N γ j − γˆ N + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ # " ⎪ ⎪ (   ) ⎡ ⎤ ⎪ ⎪ T T ⎪ ⎪ ⎪ ⎪ ˜ ˆ ˆ ˜ − E X − X − X X X − 2E X ⎪ ⎪ j j| j−1 j j| j−1 j j ⎪ ⎪ ⎢ ⎥ ⎬ ⎨ # 1 ⎢ " ⎥  T QN = ⎢ ⎥ T− ˆ ˆ N K − X − Z Z E X ⎢ ⎥ ⎪ ⎪ N − 1 ⎪ N −1 * j j| j−1 j j| j−1 j ⎢ ⎥⎪ ⎪ ⎪ ⎪ " ⎢ ⎥⎪ ⎪ ⎪  T # N ⎪ ⎥⎪ ⎪ ⎪ j=1 ⎢ ⎪ − ⎢ K j E Z j − Zˆ j| j−1 X j − Xˆ j| j−1 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ( ⎦⎪ ⎪ ⎪ ) ⎪ ⎪         T ⎭ ⎩ f X j−1 − ¯f X j−1 E f X j−1 − ¯f X j−1

(3.124)

(2) Recursive Statistical Parameter Estimation For the mean of noise, it is easy to obtain qˆ k+1 = qˆ k −

 1  qˆ − q k+1 k+1 k

(3.125)

γˆ k+1 = γˆ k −

 1  γˆ k − γ k+1 . k+1

(3.126)

and

3.3 Filtering Method for Navigation

105

When k is too large, (k + 1)(k − 1) ≈ k 2 , thus we have  T k 1  Qk + γ k+1 − γˆ k γ k+1 − γˆ k k+1 k+1 " ⎡ ( )  T # ⎤ T ˜ ˆ ˆ ˜ ⎥ ⎢ −E X k+1 X k+1 + 2E X k+1 − X k+1|k X k+1 − X k+1|k ⎥ ⎢ "  T # ⎥ ⎢ T ⎥ ⎢ ˆ ˆ K k+1 1 ⎢ −E X k+1 − X k+1|k Z k+1 − Z k+1|k ⎥ + ⎥. ⎢ " #  T ⎥ k + 1⎢ ⎥ ⎢ −K k+1 E Z k+1 − Zˆ k+1|k X k+1 − Xˆ k+1|k ⎥ ⎢ ⎦ ⎣ ) (  T −E f (X k ) − ¯f (X k ) f (X k ) − ¯f (X k )

Q k+1 =

(3.127) Then we have Q k+1 = Q k +

)  T 1 ( γ k+1 − γˆ k γ k+1 − γˆ k − K k+1 P Txz,k+1|k . k+1

(3.128)

2) Statistical Parameter Estimation of Non-stationary Noise In practical applications, the noise is usually non-stationary, with the statistical parameters being time-varying. In this section, three types of recursive methods are presented for estimating the time-varying noise. (1) Finite Memory Estimation Algorithm The state variables are estimated using the statistical parameters within a finite time window. The length of time window is set as L, then the recursive formulation for the mean is given as qˆ k+1 =

1 L

k+1 

q j = qˆ k +

j=k+2−L

 1 q − q k−L+1 L k+1

(3.129)

and γˆ k+1 = γˆ k +

 1 γ k+1 − γ k−L+1 L

(3.130)

106

3 Navigation Method and Scheme Design for Rendezvous and Docking

The covariance of the process noise at time k + 1 is given as ˆ k+1 Q

⎡ ⎤  T γ k+1 − γˆ k+1 γ k+1 − γˆ k+1 −  T ⎥  ˆk+ 1 ⎢ = Q ⎣ γ k−L+1 − γˆ k+1 γ k−L+1 − γˆ k+1 − ⎦  T L −1 1 γ k+1 − γ k−L+1 γ k+1 − γ k−L+1 L ) 1( ) 1( ˆ T T ˆ k−L − P x z,k−L+1|k−L K k−L+1 + Q k − P x z,k+1|k K k+1 − Q L L (3.131)

(2) Estimation Algorithm with Fading Factor The fading factor is introduced to adjust the weights of the estimates based on their distances from the current estimate. The fading factor is given as β j = dk+1 b j , j = 0, 1, 2 . . . k

(3.132)

  where dk+1 = (1 − b)/ 1 − bk+1 , 0 < b ≤ 1 is the forgetting rate; then, the mean of the noise can be solved as qˆ k+1 =

k+1 

βk+1− j q j = (1 − dk+1 )qˆ k + dk+1 q k+1

(3.133)

j=1

and γˆ k+1 = (1 − dk+1 )γˆ k + dk+1 γ k+1 .

(3.134)

The covariance matrix is given as

(3.135)

3.3 Filtering Method for Navigation

107

(3) Finite Memory Estimation Algorithm with Fading Factor To utilize the sampling information and track the time-varying parameters, the above two algorithms can be combined. Then, the mean is given as qˆ k+1 =

k+1 

  βk+1− j q j = b qˆ k + d L q k+1 − b L q k+1−L

(3.136)

j=k+2−L

  where d L = (1 − b)/ 1 − b L . Then we have   γˆ k+1 = bγˆ k + d L γ k+1 − b L γ k+1−L ,

(3.137)

with the covariance matrix being    ˆ k+1 = b Q ˆ k − γˆ k+1 − γˆ k γˆ k+1 − γˆ k T Q ⎤ ⎡  T ˆ k− γ k+1 −" γˆ k γ k+1 − γˆ k + Q  T # ⎦  + dL ⎣ K k+1 E Z k+1 − Zˆ k+1|k X k+1 − Xˆ k+1|k ⎤  T ˆ k−L − γ k−L+1 −" γˆ k γ k−L+1 − γˆ k + Q #  T ⎦  − dL b L ⎣ K k−L+1 E Z k−L+1 − Zˆ k−L+1|k−L X k−L+1 − Xˆ k−L+1|k−L ⎡

(3.138) Owing to the subtraction of matrices, the covariance matrix of the process noise may be not semi-definite. This issue can be overcome by taking the eigenvalue ˆ k ≥ 0. ˆ k and setting the negative eigenvalue to be 0, such that Q decomposition of Q 3) Adaptive Deterministic Sampling Filtering The adaptive deterministic sampling filtering algorithm utilizes the estimated mean, variance, and covariance as the approximate to estimate the state variables. The details are stated as follows. (1) Generating Deterministic Sampling Points: The sampling points χ i,k and their weights wi , i = 1 . . . N are generated based on the estimated state and covariance matrix at time k. (2) Time Update: Xˆ k+1|k =

N  i=1

wi f (χ i,k ) + qˆ k

(3.137)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

Pˆ k+1|k =

N 

  T ˆk wi Xˆ k+1|k − f (χ i,k ) Xˆ k+1|k − f (χ i,k ) + Q

(3.140)

i=1

ˆ k are the estimated noise parameters at time k. where qˆ k and Q (3) Re-Generating Deterministic Sampling Points χ i,k+1|k , i = 1 . . . N are re-generated based on Xˆ k+1|k and Pˆ k+1|k in the same manner as in (1). (4) Predicting the Mean Zˆ k+1|k , Variance Pˆ zz,k+1|k and Covariance Pˆ x z,k+1|k of the Measurement. (5) Updating Xˆ k+1 , K k+1 , Pˆ k+1 . (6) Estimating the Noise Mean and Variance using Noise Parameter Estimation Methods: Additional variables are introduced as follows: q k+1 = Xˆ k+1 −

2n 

wi χ i,k+1|k

(3.141)

i=0

γ k+1 = Xˆ k+1 − Xˆ k+1|k .

(3.142)

Then, we have qˆ k+1 = qˆ k +

 1 q k+1 − q k−L+1 L

(3.143)

 1 γ − γ k−L+1 (3.144) L k+1 ⎡ ⎤  T γ k+1 − γˆ k γ k+1 − γˆ k −  T  ⎥ ˆk+ 1 ⎢ = Q ⎣ γ k−L+1 − γˆ k γ k−L+1 − γˆ k − ⎦  T L −1 1 γ k+1 − γ k−L+1 γ k+1 − γ k−L+1 L ) 1( ) 1( ˆ T T ˆ k−L − Pˆ x z,k−L+1|k−L K k−L+1 Q k − Pˆ x z,k+1|k K k+1 Q − + L L (3.145) γˆ k+1 = γˆ k +

ˆ k+1 Q

5. Improving the Robustness of Deterministic Sampling Filtering When a significant modeling error exists, adopting the robust filtering algorithm may be an appropriate solution. Unlike optimal filtering, robust filtering engages to weaken the influence of the modeling error on the estimation accuracy [27]. In this section, a robust UKF is introduced for a class of systems with a linear state equation and nonlinear measurement equation as well as a significant modeling error [28, 29].

3.3 Filtering Method for Navigation

109

(1) Assumption on Uncertain System The discrete stochastic system is given as X k+1 = F k X k + wk

(3.146)

Z k+1 = h(X k+1 ) + Δh + vk+1

(3.147)

where X k ∈ R n is the system state, Zk+1 ∈ R m is the measurement, F k is the state transition matrix, and h(·) is assumed to be continous and differentiable at X k ; noise which satisfy wk and vk , respectively, are the process noise and measurement  Cov(wk , wk ) = Q k , Cov(vk , vk ) = Rk , and Cov wi , v j = 0; Δh is the modeling error. For the system in (3.146) and (3.147), the design of UKF is no longer repeated. The system is assumed to satisfy the following conditions: (1) The modeling error satisfies   0 ≤ E Δh ΔTh ≤ M h

(3.148)

where M h is a semi-definite matrix and known. (2) h(X) is a three-dimensional vector, and the elements in h(X) satisfy the Lipschitz condition, i.e., existing positive constants L h1 , L h2 and L h3 such that          h1 (X k+1 ) − h1 Xˆ k+1|k  ≤ L h1 X k+1 − Xˆ k+1|k 

(3.149)

          h2 (X k+1 ) − h2 Xˆ k+1|k  ≤ L h2  X k+1 − Xˆ k+1|k 

(3.150)

          h3 (X k+1 ) − h3 Xˆ k+1|k  ≤ L h3  X k+1 − Xˆ k+1|k 

(3.151)

A matrix is introduced as ⎞ 0 3L 2h1 0 M l = ⎝ 0 3L 2h2 0 ⎠ 0 0 3L 2h3 ⎛

(3.152)

(3) It is difficult to analyze the sampling points in UKF from a theoretical perspective; we set   h(X k+1 ) = hs X k+1 , Xˆ k+1|k + Δt

(3.153)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

  where hs X k+1 , Xˆ k+1|k is the second order Taylor approximation of h(X k+1 ) at Xˆ k+1|k , and Δt is the higher-order term which is the missing part when using UKF. Then, the measurement estimate is given as  ) (  Zˆ k+1|k = E hs X k+1 , Xˆ k+1|k + vk+1 .

(3.154)

Additionally, only partial second-order information can be obtained in UT, and thus, the time-varying parameter ϑk+1 > 0 should be introduced such that " E

     T # ˆ k+1|k − hs Xk+1 , X ˆ k+1|k − hs Xk+1 , X ˆ k+1|k − vk+1 Z ˆ k+1|k − vk+1 Z

≤ ϑk+1 Pˆ zz,k+1|k

(3.155)

(4) The following equality is assumed to be true: "

Π k+1 Ξ k+1



= E (Δt + Δh ) X k+1 − Xˆ k+1|k

T #

−1 Pˆ k+1|k

(3.156)

where Pˆ k+1|k is the predicted variance of state estimation in the filtering algorithm that satisfies p I ≤ Pˆ k+1|k , Π k+1 is an m × n matrix, and Ξ k+1 is an unknown time-varying n × n matrix satisfying a I ≤ Ξ k+1 Ξ Tk+1 ≤ I with 0 ≤ a < 1. (5) Assuming the densities of prediction and posterior probability are Gaussian, the accuracy of covariance P x z from UT can extend up to the second order. (2) Boundedness Theorem of the State Estimation in UKF Lemma 3.6 Wang and Wu [30]: A is an n-dimensional symmetric matrix, and b is an n-dimensional vector; then, we have A − bbT > 0 ⇔ A > 0, bT A−1 b < 1.

(3.157)

Lemma 3.7 Xie et al. [31]: Given matrices A, H, E, and F, if F F T ≤ I, a −1 I − EU E T > 0 for any symmetric and positive definite matrix U, and any constant a > 0, then we have  −1 T A + a −1 H H T . ( A + H F E)U( A + H F E)T ≤ A U −1 − a E T E

(3.158)

Theorem 3.5 Liu and Xie [28]: For the UKF design of the nonlinear uncertain system in (3.146) and (3.147), (3.164) is utilized to calculate the gain of the filter. If there exists a positive constant γ such that Pˆ 0 ≥ P 0

(3.159)

3.3 Filtering Method for Navigation

111

Pˆ k+1|k = F k Pˆ k F Tk + Q k

(3.160)

) ( −1 T − γ −1 I − Pˆ x z,k+1|k Kk+1 + Kk+1 Pˆ xTz,k+1|k Pˆ k+1 = Pˆ k+1|k " & # $ T T Kk+1 + Kk+1 λk+1 Pˆ zz,k+1|k + ak+1 Ml + 3Mh − 12Rk+1 + γ − pa− k+1 k+1 (

)−1

−

(3.161) when k ≥ 0, there is always a positive solution Pˆ k+1|k > 0 and Pˆ k+1 > 0 satisfying γ I − Pˆ k+1|k > 0.

(3.162)

Then we obtain the following inequality P k+1 = E

"

X k+1 − Xˆ k+1



X k+1 − Xˆ k+1

T #

≤ Pˆ k+1 ,

(3.163)

and the gain is given as  (  )−1 K k+1 = Pˆ x z,k+1|k λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1 + γ − pa Π k+1 Π T k+1

(3.164)

minimizing Pˆ k+1 , where λk+1 = 15ϑk+1

ak+1

 n $% &  12    ˆ k+1|k  = P   n + κ i=1 i

2

  + 6trace P k+1|k .

(3.165)

(3.166)

Proof Step 1: Determining the upper bound of P zz,k+1|k After obtaining Xˆ k+1|k , Pˆ k+1|k , Zˆ k+1|k , and Pˆ zz,k+1|k in UKF, the finding is denoted by Z˜ k+1|k = Z k+1 − Zˆ k+1|k .

(3.167)

From (3.153), we can obtain   Z˜ k+1|k = hs (X k+1 , Xˆ k+1|k ) + vk+1 − Zˆ k+1|k + (Δt + Δh ) = Z˜ s,k+1|k + (Δt + Δh )

(3.168)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

and the covariance matrix of measurement is given as P zz,k+1|k = E

"

Z˜ k+1|k



Z˜ k+1|k

T #

"  T #   T # T ˜ ˜ ˜ ˜ + E Z s,k+1|k Δh + Δh Z s,k+1|k = E Z s,k+1|k Z s,k+1|k "   T #   T ˜ ˜ + E (Δh + Δt )(Δh + Δt )T + E Z s,k+1|k Δt + Δt Z s,k+1|k "

(3.169)

(1) The first term " E

Z˜ s,k+1|k



Z˜ s,k+1|k

T #

≤ ϑk+1 Pˆ zz,k+1|k

(3.170)

(2) The second term ) ( ) ( T T E Z˜ s,k+1|k ΔhT + Δh Z˜ s,k+1|k ≤ E Z˜ s,k+1|k Z˜ s,k+1|k + Δh ΔhT ≤ ϑk+1 Pˆ zz,k+1|k + M h

(3.171)

(3) The third term ) ( ) (   T T E Z˜ s,k+1|k ΔTt + Δt Z˜ s,k+1|k ≤ E Z˜ s,k+1|k Z˜ s,k+1|k + E Δt ΔTt

(3.172)

because "  T #   T ˆ ˆ E Δt Δt ≤ 2E h(X k+1 ) − h( X k+1|k ) h(X k+1 ) − h( X k+1|k ) "  T # ˆ ˆ ˆ ˆ ˆ ˆ + 4 hs ( X k+1|k , X k+1|k ) − Z k+1|k hs ( X k+1|k , X k+1|k ) − Z k+1|k . # "  T + 4E Zˆ k+1|k − hs (X k+1 , Xˆ k+1|k ) Zˆ k+1|k − hs (X k+1 , Xˆ k+1|k ) (3.173) Then, utilizing (3.152) and Lemma 3.6 yields " E

h(X k+1 ) − h( Xˆ k+1|k )



h(X k+1 ) − h( Xˆ k+1|k )

T #

  < M l trace P k+1|k (3.174)

and

3.3 Filtering Method for Navigation

  hs Xˆ k+1|k , Xˆ k+1|k − Zˆ k+1|k =

113

 −1   h(χ i,k+1|k ) − h( Xˆ i,k+1|k ) . 2(n + κ) i=1 2n

(3.175) From (3.149) to (3.151), we have       hs ( Xˆ k+1|k , Xˆ k+1|k ) − Zˆ k+1|k   j  2n      1  ˆ ≤ h(χ i,k+1|k ) − h( X k+1|k )   j 2(n + κ) i=1 & n $%    1  ˆ k+1|k  ≤√ P Lhj   n+κ i i=1

(3.176)

where j = 1, 2, 3. From Lemma 3.6, we have   T ˆ k+1|k , X ˆ k+1|k ) − Zˆ k+1|k hs (X ˆ k+1|k , X ˆ k+1|k ) − Zˆ k+1|k hs (X  n $% & 2   1  ˆ ≤ Pk+1|k    Ml n + κ i=1 i

(3.177)

Substituting (3.155), (3.174), and (3.177) into (3.173) gives ⎡     E t tT ≤ ⎣2trace Pk+1|k +

 n $% &   4   ˆ P k+1|k   n + κ i=1 i 

2

⎤ ⎦Ml

 + 4 ϑk+1 Pˆ zz,k+1|k − Rk+1

(3.178)

and   T # ˜ s,k+1|k tT + t Z˜ s,k+1|k Z ⎧  n $% & ⎨     4  ˆ ˆ ≤ 5ϑk+1 Pzz,k+1|k + 2trace Pk+1|k + Pk+1|k   ⎩ n + κ i=1  i "

E

Ml − 4Rk+1

⎫

2⎬

⎭ (3.179)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

(4) For the last term, from (3.148) and (3.178), we have   E ( h + t )( h + t )T     ≤ 2E h Th + 2E t Tt ⎧  n $% & ⎨     8  ˆ ≤ 2Mh + 4trace Pk+1|k + Pk+1|k   ⎩ n + κ i=1  i   + 8 ϑk+1 Pˆ zz,k+1|k − Rk+1

⎫

2⎬

⎭

Ml (3.180)

Then, substituting the bounds of the above four terms into (3.169), we have Pzz,k+1|k ≤ 15ϑk+1 Pˆ zz,k+1|k +

⎧ ⎨ ⎩

  6trace Pk+1|k +

 n $% &   12  Pˆ k+1|k    n+κ i i=1

+ 3Mh − 12Rk+1

⎫

2⎬

Ml ⎭

(3.181)

For simplicity, the above inequality is rewritten as P zz,k+1|k ≤ λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1

(3.182)

P uzz,k+1|k = λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1

(3.183)

and

Step 2: Determining the upper bound of P k+1 The estimation error is written as   X k+1 − Xˆ k+1 = X k+1 − Xˆ k+1|k − K k+1 Z k+1 − Zˆ k+1|k

(3.184)

Then, we have ⎤ ⎡ " (  T #  T ) T ˆ ˆ − E X k+1 − Xˆ k+1|k Δt + Δh K k+1 ⎥ ⎢ E X k+1 − X k+1|k X k+1 − X k+1|k ⎥ " P k+1 = ⎢ T # ⎦   ⎣ −K k+1 E Δt + Δh X k+1 − Xˆ k+1|k ⎡ "    T # ˆ ˆ ˆ KT k+1 + ⎢ E X k+1 − X k+1|k hs X k+1 , X k+1|k + vk+1 − Z k+1|k . " −⎢     T #/T ⎣ K k+1 E X k+1 − Xˆ k+1|k hs X k+1 , Xˆ k+1|k + vk+1 − Zˆ k+1|k + K k+1 E

"

Z k+1 − Zˆ k+1|k



Z k+1 − Zˆ k+1|k

T #

T K k+1

⎤ ⎥ ⎥ ⎦

(3.185)

3.3 Filtering Method for Navigation

115

(1) For first term if P k ≤ Pˆ k , we have " E

X k+1 − Xˆ k+1|k



X k+1 − Xˆ k+1|k

T #

≤ Pˆ k+1|k

(3.186)

Utilizing (3.156) yields ( E

 )  )T ( T + K k+1 E X k+1 − Xˆ k+1|k (Δt + Δh )T X k+1 − Xˆ k+1|k (Δt + Δh )T K k+1

T = Pˆ k+1|k (Π k+1 Ξ k+1 )T K k+1 + K k+1 Π k+1 Ξ k+1 Pˆ k+1|k

(3.187)

and " (  T #  ) T ˆ ˆ − E X k+1 − Xˆ k+1|k (Δt + Δh )T K k+1 E X k+1 − X k+1|k X k+1 − X k+1|k "  T # ˆ − K k+1 E (Δt + Δh ) X k+1 − X k+1|k    T ≤ I − K k+1 (Π k+1 Ξ k+1 ) Pˆ k+1|k I − K k+1 (Π k+1 Ξ k+1 ) ( ) T − K k+1 (Π k+1 Ξ k+1 ) Pˆ k+1|k (Π k+1 Ξ k+1 )T K k+1

(3.188)

Utilizing (3.162) and Lemma 3.7 gives 

  T I − (K k+1 Π k+1 )Ξ k+1 I Pˆ k+1|k I − (K k+1 Π k+1 )Ξ k+1 I ( −1 )−1 . ≤ Pˆ k+1|k − γ −1 I + γ K k+1 Π k+1 Π Tk+1 K Tk+1

(3.189)

Substituting (3.189) into (3.188), we have (  T #  ) T ˆ ˆ − E X k+1 − Xˆ k+1|k (Δt + Δh )T K k+1 E X k+1 − X k+1|k X k+1 − X k+1|k " T #  ˆ − K k+1 E (Δt + Δh ) X k+1 − X k+1|k "

( −1 )−1 T T ≤ Pˆ k+1|k − γ −1 I + γ K k+1 Π k+1 Π k+1 K k+1 ( ) T T K k+1 − K k+1 paΠ k+1 Π k+1

(3.190)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

(2) From condition (5), we have " E

 T # T X k+1 − Xˆ k+1|k hs (X k+1 , Xˆ k+1|k ) − Zˆ k+1|k K k+1 "  T # + K k+1 E hs (X k+1 , Xˆ k+1|k ) − Zˆ k+1|k X k+1 − Xˆ k+1|k

T T = Pˆ x z,k+1|k K k+1 + K k+1 Pˆ x z,k+1|k

(3.191)

(3) For the third term, " K k+1 E

Z k+1 − Zˆ k+1|k



Z k+1 − Zˆ k+1|k

T #

T T K k+1 ≤ K k+1 P uzz,k+1|k K k+1 .

(3.192) Substituting the above three terms into (3.185) gives P k+1 ≤ Pˆ k+1

(3.193)

where ( −1 )−1 T T Pˆ k+1 = Pˆ k+1|k − γ −1 I − Pˆ x z,k+1|k K k+1 − K k+1 Pˆ x z,k+1|k  (  ) T T + K k+1 λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1 + γ − pa Π k+1 Π k+1 K k+1

(3.194)

Step 3: Determining the gain of the filter Employing the bound to design the filter can ensure uniformity. However, if the bound is overestimated, the accuracy of state estimation will be decreased. Therefore, the gain should be selected to minimize Pˆk+1 .  According to

∂trace Pˆ k+1 ∂ K k+1

= 0 and (3.194), we have

 (  )−1 K k+1 = Pˆ x z,k+1|k λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1 + γ − pa Π k+1 Π T k+1

(3.195)

It is easy to verify that if Xˆ k+1 and Pˆ k+1 are used for a time update at k + 2, then we get

and

Xˆ k+2|k+1 = F k+1 Xˆ k+1

(3.196)

Pˆ k+2|k+1 = F k+1 Pˆ k+1 F k+1 + Q k+1

(3.197)

3.3 Filtering Method for Navigation

P k+2|k+1 = F k+1 E

"

117

X k+1 − Xˆ k+1



X k+1 − Xˆ k+1

T #

T F k+1 + Q k+1

P k+2|k+1 ≤ Pˆ k+2|k+1

(3.198) (3.199)

Based on the inductive method, if the conditions in Theorem 3.5 are satisfied, (3.199) can be ensured, and the gain in (3.198) can result in Pˆ k+1 being minimized. 3) Robust UKF (1) Description of UKF The conclusion in Theorem 3.5 actually presents a design principle for a robust filtering algorithm. In this section, a robust UKF is designed. (1) The state equation is utilized to predict the state estimation and its covariance matrix, i.e., Xˆ k+1|k and Pˆ k+1|k . (2) Sampling points χ i,k+1|k (i = 0 . . . 2n) are selected based on Xˆ k+1|k and Pˆ k+1|k . (3) The mean Zˆ k+1|k , variance Pˆ zz,k+1|k , and covariance Pˆ x z,k+1|k of the measurement are predicted according to the sampling points. (4) Measurement update Xˆ k+1 = Xˆ k+1|k + K k+1 (Z k+1 − Zˆ k+1|k )

(3.200)

( −1 )−1 T T − Pˆ x z,k+1|k K k+1 − K k+1 Pˆ x z,k+1|k Pˆ k+1 = Pˆ k+1|k − γ −1 I  (  ) T T + K k+1 λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1 + γ − pa Π k+1 Π k+1 K k+1

(3.201)

  )−1 T K k+1 = Pˆ x z,k+1|k λk+1 Pˆ zz,k+1|k + ak+1 M l + 3M h − 12 Rk+1 + γ − pa Π k+1 Π k+1

(3.202)

(

where λk+1 = 15ϑk+1  = 6trace Pˆ k+1|k + 

ak+1

 n $% &  12    ˆP k+1|k    n + κ i=1 i

(3.203) 2

(3.204)

(2) The Properties of Robust UKF T (1) In robust UKF, the measurement error is characterized by M h , and Π k+1 Π k+1 describes the effects caused by modeling error. The robust UKF takes the modeling error (both the process modeling error and the measurement modeling error), which is not included in the normal UKF, into full consideration.

118

3 Navigation Method and Scheme Design for Rendezvous and Docking

(2) It is easy to find that if we do not consider the uncertain term of the modeling error in the algorithm and if γ → ∞, the robust UKF would degenerate into a normal UKF. (3) Parameter selection for robust UKF 1) Selection of λ: λ should be chosen such that P uzz,k+1|k is as close to P zz,k+1|k as possible with P uzz,k+1|k ≥ P zz,k+1|k . P zz,k+1|k can be estimated online by the following equation [32]: 0 P zz,k+1|k ≈

δ k+1 δ Tk+1

ρ ∗ P zz,k+1|k +δ k+1 δ Tk+1 1+ρ ∗

k=0 k>0

δ k+1 = Z k+1 − Zˆ k+1|k where 0.95 ≤ ρ ∗ < 1 is the forgetting factor. λ is determined by      λk+1 = max 1, trace P zz,k+1|k − ak+1 M l − 3M h + 12 Rk+1 /trace Pˆ zz,k+1|k

(3.205) If the covariance matrix of measurement is uncertain, Rk+1 can be removed from the above equation. 2) Selection of Π: Assuming Π k+1 Ξ k+1 = M, m ≤ n, M is full column rank. Taking the QR decomposition  of M, and taking  appropriate column transformation such that M = (τ R) τ1 Q with R = I 0 , where τ is a positive constant. Assuming that Π = τ R, Ξ = τ1 Q and selecting appropriate τ can ensure that Ξ Ξ T < I. Thus, we have ΠΠ T = τ 2 I

(3.206)

Generally, the elements in M are small, and 0 ≤ τ  1 can be guaranteed by adjusting τ . If the measurement modeling error is weakly related to the relative state and the high order terms of the measurement model h are sufficiently small, the effect of ΠΠ T will be negligible. 3) Selection of γ : γ should be chosen such that Pˆ k+1 > 0. Clearly, Pˆ k+1 > 0 can be ensured if γ is large, but a larger γ would be too conservative. Therefore, γ is determined online by [33].    γk+1 = ε · max eig Pˆ k+1|k

(3.207)

where ε ≥ 1 is a predesigned constant. 4) a and p can be determined based on a priori, which can be set as a = 0, p = 0 if no a priori is available.

3.3 Filtering Method for Navigation

119

6. Improved Deterministic Sampling Filtering based on Robust Statistics The measurement for space relative motion usually employs optical sensors (such as an infrared sensor, laser radar, and visible light camera). The measurement noise follows a heavy-tailed distribution rather than a Gaussian one. The probability density of heavy-tailed distribution can be modeled as p(x) = (1 − ε) pG (x) + εpt (x), in which 0 ≤ ε ≤ 1, pG (x) is Gaussian and pt (x) can be Laplacian, Cauchy, or Gaussian with large variance. The figure below shows the density function of Gaussian and heavy-tailed distribution. If the heavy-tailed distribution is approximated by the Gaussian function, the outliers will occur frequently in the measurement. Huber created robust statistical methods, such that different loss functions can be used to realize different robust performance. Thereafter, this method has been called the Huber robust statistical method [34]. It can be used to improve the filtering algorithm and thus overcome the issue of outliers [35, 36]. In this section, the Huber statistical method and deterministic sampling filtering will be combined for general nonlinear systems such that the algorithm is robust to the outliers (Fig. 3.7). a) State Estimation Algorithm based on Huber Statistics Denote the predicted state by Xˆ k+1|k and the estimated variance by Pˆ k+1|k ; the state-dependent measurement equation is given as Z k+1 = H k+1 X k+1 + vk+1

(3.208)

where X k ∈ Rn is the system state, Zk+1 ∈ Rm is the measurement, and vk+1 is the measurement noise. If the predicted state is treated as a measurement and combined with the measurement from the sensor, the new measurement equation can be written as Fig. 3.7 The probability density of heavy-tailed distribution and Gaussian one

0.4 Gaussian Heavy-tailed

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −6

−4

−2

0

2

4

6

120

3 Navigation Method and Scheme Design for Rendezvous and Docking

$

Z k+1 Xˆ k+1|k

&

$ =

$ & & vk+1 H k+1 X k+1 + ˜ I X k+1|k

(3.209)

where X˜ k+1|k = Xˆ k+1|k − X k+1 . Let  T k+1 =

−1/2

Rk+1 0

0

−1/2 Pˆ k+1|k

.

(3.210)

Multiplying both sides of (3.209) with T k+1 , the estimated state can be decoupled as $ T k+1

Z k+1 − Zˆ k+1|k + H k+1 Xˆ k+1|k Xˆ k+1|k

&

$ = T k+1

$ & & H k+1 v X k+1 + T k+1 ˜ k+1 . I X k+1|k (3.211)

Equation (3.211) can be rewritten as Y k+1 = G k+1 X k+1 + υ k+1

(3.212)

˜ k+1|k noise follows vk ∼ N (vk |0, Rk ) and X ∼ If the measurement  ˜ k+1|k |0, Pˆ k+1|k , we get υk+1 ∼ N (υk+1 |0, I), i.e., the variables in the decouN X pled state have the same weights. Thus, the least-square method can be utilized to estimate the state. If the measurement noise follows the heavy-tailed distribution and outliers occur frequently, the performance of the least-square method will degrade. The state estimation method based on the Huber robust function utilizes the M estimator Jhuber =

n+m 

λ(ζ i )

(3.213)

i=1

where ζ i = (G k+1 X k+1 − Y k+1 )i   λ ζi =

.1

ζ2 2'i

' γ 'ζ i ' − 21 γ 2

' ' 'ζ i ' 0.

(3.227)

Additionally, the importance density should be selected such that it is easy to sample the points. The Monte Carlo numerical integration with IS is written as  I (ϕ(X 0:k )) =

ϕ(X 0:k )W˜ (X 0:k )q(X 0:k |Z 1:k )d X 0:k  W˜ (X 0:k )q(X 0:k |Z 1:k )d X 0:k

(3.228)

with p(Z 1:k ) p(X 0:k |Z 1:k ) W˜ (X 0:k ) = q(X 0:k |Z 1:k )

(3.229)

being the weight for normalization. Assume that X i0:k ∼q(X 0:k |Z 1:k ), i = 1 . . . N , then (3.228) is approximated by

124

3 Navigation Method and Scheme Design for Rendezvous and Docking

I (ϕ(X 0:k )) ≈

N 

    W X i0:k ϕ X i0:k

(3.230)

i=1

where W



X i0:k



  W˜ X i0:k = N *  j  W˜ X 0:k

(3.231)

j=1

With W being the weight after normalization. The calculation of (3.230) is equivalent to the following discrete approximation of the posterior possibility: p(X 0:k |Z 1:k ) ≈

N 

    W X i0:k δ X 0:k − X i0:k .

(3.232)

i=1

3. Sequential Importance Sampling To overcome the second issue of Monte Carlo numerical integration, the IS method can be improved by recursively calculating the density function, i.e., SIS. To achieve this goal, the importance density function should satisfy qk+1 (X 0:k+1 |Z 1:k+1 ) = qk+1 (X k+1 |X 0:k , Z 1:k+1 )qk (X 0:k |Z 1:k )

(3.233)

  such that X ik+1 can be sampled from qk+1 X k+1 |X i0:k , Z 1:k+1 at time k + 1 without requiring the whole state sequence X 0:k+1 to be sampled. Following the sample, the weights can be obtained by     p(Z k+1 |X i0:k+1 ) p(X ik+1 |X i0:k )   . W˜ k+1 X i0:k+1 = Wk X i0:k q X ik+1 |X i0:k , Z 1:k+1

(3.234)

From the above equation, it can be seen that the computational cost of SIS does not increase with time. The main drawback of SIS lies in that it is only a special case of IS, and therefore, the particle dissipation issue persists. 4. Selection of Importance Density One effective method to overcome the particle dissipation issue in SIS is to select an appropriate importance density function, which should obey the following principles: (1) The support should not be contained in the support of p(X 0:k+1 |Z 1:k+1 ). (2) It should be possible to calculate it recursively. (3) The variance of the weights should be minimized.

3.3 Filtering Method for Navigation

125

' It can be proven that p(X k+1 ' X i0:k , Z k+1 ) can ensure the minimization of the variance of the weights. However, it is generally difficult to obtain the importance density function, and it is an area of focus in particle filtering research. 5. Resampling Another important method to overcome the particle dissipation is to resample the particles. The basic principle is to supress or remove the particles with small weights and copy those with large weights, such that the resource is redistributed according to the particle weights. The resampling methods includes [39] multinomial resampling, residual resampling, minimum variance sampling, and systematic resampling. The last method is most widely used in practical applications. Resampling may decrease the diversity of the particles, which is also an important research direction. 6. Particle Filtering Algorithm Monte Carlo numerical integration and SIS are introduced in Bayesian estimation to establish the particle filtering algorithm. At present, various improved particle filtering algorithms have been proposed, and the main differences lie in [40]: (1) different importance densities, (2) different resampling methods, and (3) the different measures taken to overcome the dissipation issue. Although these algorithms differ in their details, they follow the same principles. A typical particle filtering algorithm is introduced as follows. Assume that the state is a one-order Markov process, the observations are independent of each other when the state is known, and the importance density function satisfies     q X k+1 |X i0:k , Z 1:k+1 = q X k+1 |X ik , Z k+1

(3.235)

A typical step at time k + 1 is given as follows:   (1) Generating particles X ik+1 ∼q X k+1 |X ik , Z k+1 , i = 1 . . . N from the importance density function. (2) The weights can be recursively calculated by ' '  i   i  p(Z k+1 ' X ik+1 ) p(X ik+1 ' X ik ) ˜ ' , i = 1 . . . N. Wk+1 X 0:k+1 = Wk X 0:k q(X ik+1 ' X ik , Z k+1 ) (3.236)   (3) Normalizing the weights according to (3.231) to give Wk+1 X i0:k+1 . (4) The posterior possibility density is approximated by p(X k+1 |Z 1:k+1 ) ≈

N  i=1

    Wk+1 X i0:k+1 δ X k+1 − X ik+1 .

(3.237)

126

3 Navigation Method and Scheme Design for Rendezvous and Docking

3.4 Scheme Design of Navigation 3.4.1 Introduction The task of navigation is to provide the motion parameters estimation for guidance and control. The measured information from the sensors is used to implement the navigation algorithm by considering the motion law of the spacecraft and utilizing the filtering algorithms, such that the motion parameters estimation can be obtained. The design of a navigation scheme should synthetically take the requirements of guidance and control into consideration. The details include the following aspects: (1) Selection of a Reference Frame for Navigation The design of both guidance and control systems requires that the guidance (control) coordinate system is set first, following which the corresponding dynamics/kinematics equations are established for analysis and design. The guidance (control) coordinate system usually determines the form of parameters, provided by the navigation system, that are required by the guidance (control) system. The navigation coordinate system should normally correspond to the guidance (control) system. The choice of the navigation coordinate system should consider the type of target spacecraft, the type of rendezvous orbit, and the characteristics of the guidance law adopted. (2) Selection of State Variables for Navigation The navigation states must generally include the parameters required by the guidance (control) system; they may also include the relevant parameters of the navigation system itself, such as the sensor measurement deviation. (3) Design of the Navigation Algorithm a) State Equation The equation of state describes the motion law of the navigation state variables. To facilitate the filter design, it is necessary to model the navigation state dynamically or kinematically and make necessary approximations and simplifications based on the requirements. b) Measurement Equation The measurement equation establishes the relationship between the states and the measurement information. The establishment of the measurement equation must consider the specific performance of the measurement sensor and the form of measurement and make necessary approximations .

3.4 Scheme Design of Navigation

127

c) Filter Design The navigation algorithm requires the information contained in the equation of state and the measurement quantity to give the statistical estimate value of the state quantity according to some optimal index and certain statistical estimation method. The selection and design of an optimal index and statistical estimation method is the key to filter design.

3.4.2 Orbital Parameters Estimation Lambert guidance and C-W guidance are mainly used in the autonomous control section of the rendezvous approach. Lambert guidance requires the absolute orbit parameters of the two spacecraft, while C-W guidance requires the relative parameters between the two spacecraft. In essence, if the absolute orbit parameters of the two spacecraft are known, it is easy to calculate the relative parameters between them. In the United States, Lambert guidance is usually adopted in the orbit control design of the near-range guidance phase of rendezvous and docking. In general, the absolute orbit parameters of the two spacecraft are directly taken as state variables, and multiple sensors are used to measure the information to design the navigation scheme [41]. 1. Selection of State Variables The state variables are selected as   X = r C vC r T vT bρ bα bβ

(3.238)

where r C is the position vector of the chaser with respect to the inertial frame, vC is the velocity vector of the chaser with respect to the inertial frame, r T is the position vector of the target with respect to the inertial frame, vT is the velocity vector of the target with respect to the inertial frame, bρ is the distance measurement bias of the rendezvous radar, bα is the pitch angle measurement bias of the rendezvous radar, and bβ is the azimuth angle measurement bias of the rendezvous radar. 2. State Equation (1) The orbit dynamics of the chaser are r˙ C = vC v˙ C = g(r C ) + aC +wC

(3.239)

128

3 Navigation Method and Scheme Design for Rendezvous and Docking

where g(r C ) is the gravitational acceleration. At least an eight-order model of the orbit should be taken into consideration for rendezvous and docking around the earth, and at least 16 orders for the moon. (2) The orbit dynamics of the target r˙ T = vT v˙ T = g(r T ) + wT

(3.240)

(3) Measurement modelling error of the sensor The measurement error of the rendezvous radar is given as bρ (k) = e−1/τρ (tk −tk−1 ) bρ (k−1) + wbρ (k − 1) bα (k) = e−1/τα (tk −tk−1 ) bα (k − 1) + wbα (k − 1) bβ (k) = e−1/τβ (tk −tk−1 ) bβ (k − 1) + wbβ (k − 1)

(3.241)

If τ → ∞, then we have bρ (k) = bρ (k − 1) + wbρ (k − 1) bα (k) = bα (k − 1) + wbα (k − 1) bβ (k) = bβ (k − 1) + wbβ (k − 1)

(3.242)

where wC , wT , wbρ , wbα , wbβ are the corresponding process noises. 3. Measurement Equation During rendezvous, the chaser can determine its orbit position and velocity in real time through the satellite navigation equipment. At the same time, relative measurement sensors (such as microwave radar, etc.) can be used to directly measure the relative state between the two spacecraft. The measurement of the satellite navigation equipment and relative measurement sensor are both used in navigation calculation. (1) Measurement of Inertial State The inertial position and velocity of the chaser can be measured using GPS, and the measurement equation is given as " h gps (X) =

rC vC

#

(2) Relative Measurement Employing the rendezvous radar gives the measurement equation

(3.243)

3.4 Scheme Design of Navigation

129

⎤ ⎡  2 2 2+b x$ ⎡ ⎤ ⎡ ⎤ ρ r + yr + z r & vρ ρ ⎥ ⎢ −z r ⎥ ⎢ hr = ⎣ α ⎦ = ⎢ a sin √x 2 +y 2 +z 2 + bα ⎥ + ⎣ vα ⎦ ⎦ ⎣ r r r vβ β a tan xyrr + bβ

(3.244)

⎡

⎤ xr where ⎣ yr ⎦= C r_Cb C Cb_sI (r T −r C )−C r_Cb l r + C r_Cb C Cb_I C TTb_I l Ref . zr C r _Cb is the rotation matrix of the rendezvous radar with respect to the body frame of the chaser, C Cb_I is the measured attitude matrix of the chaser with respect to the inertial frame from the star tracker, l r is the position of the radar with respect to the body frame of the chaser, C Tb_I is the attitude of the body frame of the target with respect to the inertial frame, and l Ref is the position of the cooperative target with respect to the body frame of the target. 4. Design of the Filter Because the orbital dynamics equation and relative measurement equation are nonlinear, and considering the computing power of the onboard computer, the filter can be designed by using EKF or deterministic sampling filter.

3.4.3 Relative Position Parameters Estimation C-W guidance is the most commonly used guidance method for RVD in a nearcircular orbit. The navigation information required by C-W guidance is primarily the relative position and relative velocity between two spacecraft in the guidance coordinate system. The relative position and velocity can be directly taken as states, the C-W equation is adopted as a state equation, and the measurement information of the relative measurement sensors such as the rendezvous radar and satellite navigation equipment can be used to design the relative navigation algorithm. The scheme has been widely used in autonomous and automatic rendezvous and docking missions since the 1990s, including Japan’s EST-VII and HTV, the ESA’s ATV, the USA’s DART and OE, and China’s Shenzhou spacecraft. 1. Selection of the State The relative position and velocity of the chaser with respect to the frame of rendezvous and docking are chosen as the states, i.e.,  T X = x y z x˙ y˙ z˙

(3.245)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

2. State Equation The dynamics of the relative motion is approximated by the C-W equation through linearization. ⎧ ⎨ x¨ + 2ωT o z˙ = u x + wx (3.246) y¨ + ωT2 o y = u y + w y ⎩ 2 z¨ − 2ωT o x˙ − 3ωT o z = u z + wz Considering the relative position and velocity as state variables, the C-W equation is rewritten as ⎛ ⎞ ⎛ ⎞⎛ x ⎞ ⎛ 000⎞ x˙ 0 0 0 1 0 0 ⎜ y ⎟ ⎜ 000⎟⎛ ⎞ ⎜ y˙ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜0 0 0 0 1 0 ⎟ ⎟⎜ ⎟ ux ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜ z˙ ⎟ ⎜ 0 0 000 z 0 0 0 1 ⎟ ⎟⎝ u y ⎠ ⎟+⎜ ⎜ ⎟=⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎜ x¨ ⎟ ⎜ 0 0 0 0 0 −2ωT o ⎟⎜ x˙ ⎟ ⎜ 100⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ uz ⎟ ⎜ ⎜ ⎟ ⎝ 0 −ω2 0 0 0 0 ⎠⎜ ⎝ y˙ ⎠ ⎝ 010⎠ ⎝ y¨ ⎠ To 2 0 0 3ωT o 2ωT o 0 0 001 z˙ z¨ ⎞ ⎛ 000 ⎜ 000⎟⎛ ⎞ ⎟ ⎜ ⎟ wx ⎜ ⎜ 000⎟⎜ ⎟ ⎟⎝ w y ⎠ ⎜ +⎜ (3.247) ⎟ ⎜ 100⎟ ⎟ wz ⎜ ⎝ 010⎠ 001 i.e., X˙ = F X + Bu + Γ w.

(3.248)

The above equation is a linear time-invariant system and can be easily discretized as X(k + 1) = Φ(k + 1, k)X(k) + G(k + 1, k)u(k) + w(k) ⎛

(3.249)

1 0 6(sin φ − φ) (4 sin φ − 3φ)/ωT o 0 −2(1 − cos φ)/ωT o ⎜ cos φ 0 0 sin φ/ωT o 0 ⎜0 ⎜ ⎜0 0 4 − 3 cos φ 2(1 − cos φ)/ωT o 0 sin φ/ωt Φ(k + 1, k) = ⎜ ⎜0 0 −6ωT o (1 − cos φ) 4 cos φ − 3 0 −2 sin φ ⎜ ⎜ ⎝ 0 −ωT o sin φ 0 0 cos φ 0 0 0 3ωt sin φ 2 sin φ 0 cos φ

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.250)

3.4 Scheme Design of Navigation ⎛ ⎜ ⎜ ⎜ ⎜ G(k + 1, k) = ⎜ ⎜ ⎜ ⎜ ⎝

131

0 2 sin φ/ω2T o − 2T/ωT o (−4 cos φ/ωT o + 4/ωT o − 3ωT o T2 /2)/ωt 2 0 −(cos φ − 1)/ωot 0 2T/ωT o − 2 sin φ/ωt2 0 (1 − cos φ)/ω2T o 4 sin φ/ωT o − 3T 0 2(cos φ − 1)/ωT o 0 sin φ/ωT o 0 (2 − 2 cos φ)/ωT o 0 sin φ/ωT o

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.251) where φ = ωT o T , T is the period used for discretization, and the covariance matrix of the process noise is   Q(k) = E w(k)w(k)T ⎡ 2 T3 2 3 σx 3 0 0 σx2 T2 0 2σx2 ωT o T3 2 3 ⎢ σ y2 T3 0 0 σ y2 T2 0 ⎢ 3 3 ⎢ T T 2 2 2 T2 σ −2σ ω 0 σ ⎢ z 3 z To 3 z  2  2 =⎢ 2 2 2 T3 2 ⎢ ωT o T 2 σ T + 4σ ω 0 σ − σ x z x z To 3 ⎢ 2 ⎣ σy T 0 3 2 σz T + 4σx2 ωT2 o T3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ (3.252) ⎥ ⎥ ⎦

with σx , σ y , and σz being the mean-square error along the three axes, determined by the magnititudes of the linearization error, orbital perturbation error, and measurement error of the acceleration. 3. Navigation Scheme Based on GPS For the low-Earth orbit cooperative rendezvous mission, the measurements of the satellite navigation equipment between the two spacecraft are used for differential measurement. The obtained pseudo-range difference or carrier phase difference measurement contains relative position information, which can be used as the measurement for filter. Using satellite navigation equipment for relative navigation allows high navigation accuracy and wide applicability. (1) Measurement Equation using Single Pseudo-Range Assume that the pseudo-range of the chaser and target with respect to the j-th navigation satellite are ρ˜cj (t) = ρcj (t) + Cδtcj (t) + δρion,c (t) + δρtr op,c (t) + vρ,c . j j j ρ˜t (t) = ρt (t) + Cδtt (t) + δρion,c (t) + δρtr op,c (t) + vρ,t

(3.253)

If a single pseudo-range is adopted, then we have   j j j ∇ ρ˜ct (t) = ρcj (t) − ρt (t) + C δtcj (t) − δtt (t) + δρion,c (t) − δρion,c (t) + δρtr op,c (t) − δρtr op,c (t) + vρ,c − vρ,t

(3.254)

132

3 Navigation Method and Scheme Design for Rendezvous and Docking j

j

Thus, δtc (t)−δtt (t) = δtc (t)−δtt (t), i.e., the clock error of the navigation satellite is negated. In addition, the relative positions of the two spacecraft are approximately the same because they are very close to each other. Therefore, the error caused by light propagation in the thermosphere is almost the same, and it can be reduced using differentiation. The position of the target can be determined using the pseudo-range measured from the GPS on the target and transformed to the rendezvous and docking frame, T  which is denoted by xT yT z T . Calculating the orbital position (denoted by T  j j j x g yg z g ) of the j-th satellite with respect to the rendezvous and docking frame based on the GPS navigation ephemeris gives the following measurement equation [42]: % Zj = −

%

j

j

j

(x T + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2 j

j

j

(x T − x g )2 + (yT − yg )2 + (z T − z g )2

+ C(δtc − δtt ) + vρ j = 1 . . . n gps

(3.255)

where vρ = δρion,c (t)−δρion,c (t)+δρtr op,c (t)−δρtr op,c (t) is the measurement noise, and the difference of the clock time C(δtc − δtt ) is the state to be estimated. Assume that C(δtc − δtt ) = d, satisfying .

d˙ = f + wd f˙ = w f

(3.256)

where wd and w f are the Gaussian white noises. (2) Measurement Equation of Single Difference between Carrier Phases The single difference between carrier phase measurements of the chaser and target is given as λ∇ϕ j (t) j

= λϕcj (t) − λϕt (t)   = ρcj (t) + λNcj (t) + Cδtcj + Cδτtr op,c (t) + Cδτion,c (t) + vϕ,c   j j j − ρt (t) + λNt (t) + Cδtt + Cδτtr op,t (t) + Cδτion,t (t) + vϕ,t   j j j = ρcj (t) − ρt (t) + C δtcj − δtt + λ∇ Nct + vϕ . (3.257) Its main distinction from the single-difference pseudo-range lies in the last term j of the above equation λ∇ Nct , i.e., the difference of the ambiguity resolution between the two receivers to the j-th navigation satellite. The ambiguity resolution must be determined by a special algorithm, following which it is defined as a known integer.

3.4 Scheme Design of Navigation

133

The corresponding measurement equation is written as % Zj =

%

j

j

j

(x T + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2

−

j

j

j

(x T − x g )2 + (yT − yg )2 + (z T − z g )2   j j +C δtcj −δtt + λ∇ Nct +vϕ j = 1 . . . n gps

(3.258)

(3) Measurement Equation of Double-Difference Pseudo-Range The pseudo-range single-difference results of different navigation satellites can be differentiated once more, following which the pseudo-range double difference results can be obtained as

(3.259) In the double-difference result, the clock difference term should not be used for estimation. If the two spacecraft can simultaneously observe n gps navigation satellites, then n gps − 1 double difference of phase measurement can be obtained independently, i.e., % j j j (x T + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2 % j j j − (x T − x g )2 + (yT − yg )2 + (z T − z g )2 & $% % − (x T + x − x gk )2 + (yT + y − ygk )2 + (z T + z − z kg )2 − (x T − x gk )2 + (yT − ygk )2 + (z T − z kg )2

Z jk =

jk

(3.260)

+ vρ j, k = 1 . . . n gps

(4) Measurement Equation of Double-Difference Carrier Phases If the carrier phase difference results of different navigation stars are repeated, the pseudo-range double difference results can be obtained as     j λ∇ϕ j (t) = λϕcj (t) − λϕt (t) − λϕck (t) − λϕtk (t) , with the corresponding measurement equation being Z jk =

% j j j (x T + x − x g )2 + (yT + y − yg )2 + (z T + z − z g )2

(3.261)

134

3 Navigation Method and Scheme Design for Rendezvous and Docking % j j j (x T − x g )2 + (yT − yg )2 + (z T − z g )2 ⎞ ⎛% j 2 j 2 j 2 ⎜ (x T + x − x g ) + (yT + y − yg ) + (z T + z − z g ) ⎟ −⎝ % ⎠ j j j − (x T − x g )2 + (yT − yg )2 + (z T − z g )2 + −

jk

jk

(3.262)

+λ∇ Nct +vϕ,ct

Similar to the pseudo-range double-difference result, the carrier phase doubledifference can eliminate the clock-difference-related items; however, it still has one more ambiguity resolution double-difference than the pseudo-range result. (5) Measurement Equation based on Relative Position and Velocity In the wgs-84 coordinate system, the position measurement information of the two spacecraft relative to the same navigation satellite is used: pseudo-range measurement/carrier phase smoothing pseudo-range and velocity measurement information (pseudo-range rate, single or double difference). The relative position and T  velocity x W G S84 yW G S84 z W G S84 x˙ W G S84 y˙ W G S84 z˙ W G S84 between the chaser and the target can be estimated using the iterative least-square estimation method, which can be transformed to the rendezvous and docking frame and used for observation. ⎤ ⎡ ⎡ ⎤ x W G S84 xˆ T ⎣ yˆ ⎦ = Cr_ To CT o_ I CCT D Cz (θG )⎣ yW G S84 ⎦+Cr_ To CTb_ To lgps,T zˆ

z W G S84 −

T Cr_ To CTo_ I CCb_ I lgps,C

⎡ ⎤ ⎤ ⎡ xˆ˙ x˙ W G S84 − yW G S84 ωe ⎣ yˆ˙ ⎦ = C r_To C T o_I C CT D C z (θG )⎣ y˙ W G S84 + x W G S84 ωe ⎦ z˙ W G S84 zˆ˙

(3.263)

(3.264)

where θG is the Greenwich sidereal hour angle, C C D is the rotation matrix of the J2000 equatorial inertial system with respect to the instantaneous orbital inertial reference frame, C To_I is the rotation matrix of the target’s orbital reference frame with respect to the inertial reference frame, C r_To is the rotation matrix of the target’s orbital reference frame with respect to the rendezvous and docking reference frame, and C z is the rotation matrix around the Z-axis.  T Using Z = xˆ yˆ zˆ xˆ˙ yˆ˙ zˆ˙ as the measurement, the measurement equation is

3.4 Scheme Design of Navigation

135

⎞⎛ ⎞ x ⎜ 1 ⎟⎜ y ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ 1 ⎜ ⎟⎜ z ⎟ Z=⎜ ⎟⎜ ⎟ + vgps ⎜ ⎟⎜ x˙ ⎟ 1 ⎜ ⎟⎜ ⎟ ⎝ 1 ⎠⎝ y˙ ⎠ z˙ 1 ⎛

1

(3.265)

where vgps is the measurement noise. (6) Design of the Filter If the measurement equation based on relative position and velocity is adopted, and it is linear, the classical Kalman filter can be used. In other cases, because the measurement equation is non-linear, a nonlinear filter is required, such as the EKF or deterministic sampling filter methods. 4. Navigation Scheme based on the Laser Radar The laser radar can provide relative distance ρ, elevation α, and azimuth β. The observation equation of the relative navigation filter can be constructed by direct filtering, which considers the measured laser radar data as the measurement. An indirect filtering method, which considers the relative position converted from the measured data as the measurement, can also be used. (1) Direct Measurement Model If the relative distance ρ, elevation α, and azimuth β are taken as the measured information, the measurement equation is obtained as ⎛

%

⎞ xlr2 + ylr2 + zlr2 ⎜ &⎟ $ % ⎜ ⎟ Z = ⎜ a sin −zlr / x 2 + y 2 + z 2 ⎟ + vlr lr lr lr ⎠ ⎝

(3.266)

a tan(ylr /xlr ) where ⎛

⎞ ⎛ ⎡⎛ ⎞ ⎤ ⎞ xlr x ⎝ ylr ⎠ = −⎝ C lr _Cb C Cb_I C rTI ⎣⎝ y ⎠ − C TT b_r l r e f ⎦ + C lr _Cb l lr ⎠ zlr z

(3.267)

C r I is the rotation matrix of the inertial frame to the rendezvous and docking reference frame, C Cb_I is the rotation matrix of the inertial frame with respect to the chaser’s body frame, C lr _Cb is the rotation matrix of the chaser’s body frame with respect to the measurement frame of the laser radar,

136

3 Navigation Method and Scheme Design for Rendezvous and Docking

l r e f is the position of the retro-reflector on the target with respect to the target’s body frame, l lr is the position of the laser radar with respect to the chaser’s body frame, and vlr is the measurement noise of the laser radar, with the covariance matrix being ⎛

⎞ σρ2 0 0 ⎜ ⎟ R = ⎝ 0 σα2 0 ⎠ 0 0 σβ2

(3.268)

where σρ , σα , and σβ are the mean-square variances of the distance and line-of-sight angles. Clearly, the measurement equation is a nonlinear function when the direct measured information is employed. (2) Indirect Measurement Equation By using the measured information from the laser radar, the position of the retroreflector with respect to the laser radar is given as xlr m = ρlr cos αlr cos βlr ylr m = ρlr cos αlr sin βlr zlr m = −ρlr sin αlr

(3.269)

The relative position between the two spacecraft expressed in the rendezvous and docking frame can be calculated according to the alignment of the laser radar, attitude, and orbital data as follows: ⎞ ⎛ ⎛ ⎞ ⎞ xlr m xm T ⎝C lrT _Cb ⎝ ylr m ⎠ + l lr ⎠ + C TT b_r l r e f ⎝ xm ⎠ = −C r I C Cb_I xm zlr m ⎛

(3.270)

where C T b_r is the rotation matrix of the target’s body frame with respect to the rendezvous and docking frame. The indirect measurement equation is given as ⎛ ⎞ x ⎛ ⎞⎜ y⎟ ⎟ 100000 ⎜ ⎜ ⎟ z ⎜ ⎟ Z = ⎝ 0 1 0 0 0 0 ⎠⎜ ⎟ + vlr ⎜ x˙ ⎟ 001000 ⎜ ⎟ ⎝ y˙ ⎠ z˙ T  where Z = xm ym z m , vlr is the measurement noise.

(3.271)

3.4 Scheme Design of Navigation

137

The variance matrix of the noise of indirect observation is written as [43]: R = C r _lr C mραβ Rραβ C Tmραβ C rT_lr

(3.272)

where ⎛

⎞ cαlr · cβlr −ρlr · sαlr · cβlr −ρlr · cαlr · sβlr Cmραβ = ⎝ cαlr · sβlr −ρlr · sαlr · sβlr ρlr · cαlr · cβlr ⎠, −sαlr −ρlr · cαlr 0 ⎞ ⎛ σ2 0 0 ⎟ ⎜ ρ 2 Rραβ (k) = ⎝ 0 σα 0 ⎠ 2 0 0 σβ T C lrT _Cb , cαlr = cos αlr , sαlr = sin αlr , cβlr = cos βlr , sβlr = sin βlr C r _lr = C r I C Cb_I

Using indirect measurement, the measurement equation is linear time-invariant, but the calculation of noise statistics is tedious. (3) Design of Filter The state equation of the system is a linear equation. If the indirect measurement equation is used, the measurement equation is also a linear equation. The Kalman filter can be used for the filter design. If the measurement equation is constructed directly, it is non-linear, and a non-linear filtering algorithm is required to design certain filters, such as EKF and deterministic sampling filters. 5. Navigation Scheme based on Microwave Radar A microwave radar can provide relative distance ρ, elevation α, azimuth β, and the rate of distance ρ. ˙ Similar to the laser radar, the observation equation of the relative navigation filter can be constructed by converting the measured data of a microwave radar into the relative position, which can be used as the indirect filtering method of the observed data. Alternatively, the direct filtering method of taking the measured microwave radar data as the observed data can be adopted. (1) Direct Measurement Equation Micro-wave radar can provide relative distance ρ, elevation α, azimuth β, and the rate of distance ρ. ˙ The measurement equation is given as %

⎛ ⎜ ⎜ ⎜ Z=⎜ ⎜ ⎝

 2 + y2 + z2 xwr wr wr %   2 + y2 + z2 a sin −z wr / xwr wr wr (xwr x˙wr

a tan(ywr /xwr %) 2 + y2 + z2 + ywr y˙wr + z wr z˙ wr )/ xwr wr wr

⎞ ⎟ ⎟ ⎟ ⎟ + vwr ⎟ ⎠

(3.273)

138

3 Navigation Method and Scheme Design for Rendezvous and Docking

where ⎛

⎞ ⎛ ⎡⎛ ⎞ ⎤ ⎞ xwr x T ⎝ ywr ⎠ = −⎝ C wr _Cb C Cb_i C rTI ⎣⎝ y ⎠ − l r ec ⎦ + C Tb_r l wr ⎠ (3.274) z wr z ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ x˙ x x˙wr ⎝ ⎝ y˙wr ⎠ = −Cwr _ Cb CCb_ I CrTI ⎝ y˙ ⎠ − Cwr _ Cb CCb_ I CrTI ω× y⎠ To z˙ wr z˙ z ⎛ ⎞ xwr × × ⎝ l + ω l − ω (3.275) − Cwr _ Cb CCb_ I CrTI ω× ywr ⎠ T o r ec Cb wr Cb z wr where C wr _Cb is the rotation matrix of the chaser’s body frame with respect to the measurement frame of the microwave radar, ωCb is the angular velocity of the chaser with respect to the inertial frame, ω T o is the angular velocity of the rendezvous and docking frame with respect to the inertial frame, l wr is the position of the transmitting antenna with respect to the chaser’s body frame, and l r ec is the position of the answering antenna with respect to the target’s body frame. (2) Indirect Measurement Equation By using the measured information from the microwave radar, the position of the transponder antenna with respect to the measurement frame of the microwave radar can be obtained as xwr m = ρwr cos αwr cos βwr ywr m = ρlr cos αwr sin βwr z wr m = −ρwr sin αwr

(3.276)

Because the rate of the LOS angle cannot be directly obtained by the microwave radar, and the relative velocity cannot be obtained, the rate of relative distance ρ˙ cannot be used accordingly. According to the alignment of the microwave radar and the attitude and orbit data, the relative position between the two spacecraft centroids in the rendezvous and docking frame can be obtained as follows: ⎞ ⎛ ⎛ ⎞ ⎞ xwr m xm T T ⎝C wr ⎝ ywr m ⎠ + l wr ⎠ + C TT b_r l r ec ⎝ xm ⎠ = −C r I C Cb_I _Cb xm z wr m ⎛

(3.277)

3.4 Scheme Design of Navigation

139

Then, the indirect measurement equation is given as ⎛ ⎞ x ⎛ ⎞⎜ y⎟ ⎟ 100000 ⎜ ⎜ ⎟ z ⎜ ⎟ Z = ⎝ 0 1 0 0 0 0 ⎠⎜ ⎟ + vwr ⎜ x˙ ⎟ 001000 ⎜ ⎟ ⎝ y˙ ⎠ z˙

(3.278)

 T where Z = xm ym z m , vwr is the measurement noise. In the same manner, the variance matrix can be calculated in the laser radar. (3) Design of Filter From the above analysis, it can be seen that the state equation of the system is a linear equation. If the indirect measurement equation is adopted, the measurement equation is also a linear equation. The Kalman filter can be used to design the filter. If the measurement equation is constructed directly, the measurement equation is non-linear, and a non-linear filtering algorithm is required to design certain filters, such as EKF and deterministic sampling filters.

3.4.4 Relative Attitude Parameter Estimation 1. Kinematics of Relative Attitude (1) Kinematics of Relative Attitude using Euler Angle The relative angular velocity of the chaser with respect to the target is denoted by ωr = ωC − C r ω T , with ωC and 2ω T being the respective angular velocities of the chaser and the target with respect to the inertial frame. Utilizing the mapping from the rate of the Euler angle to the angular velocity and employing the 3-1-2 transformation sequence gives ⎞ ⎛ ⎞ ⎛ ⎞ θ˙r 0 0 ωr = C 2 (θr )C 1 (ϕr )⎝ 0 ⎠ + C 2 (θr )⎝ ϕ˙r ⎠ + ⎝ 0 ⎠ 0 ψ˙ r 0 ⎛

then we have ⎛ ⎞ ⎛ ⎞ ωr x cos θr cos ϕr + ωr z sin θr cos ϕr ϕ˙r 1 ⎝ ωr x sin θr sin ϕr + ωr y cos ϕr − ωr z cos θr sin ϕr ⎠. ⎝ θ˙r ⎠ = cos ϕr ˙ ψr −ωr x sin θr + ωr z cos θr

(3.279)

(3.280)

140

3 Navigation Method and Scheme Design for Rendezvous and Docking

The target consistently points toward Earth during rendezvous and docking, and the angular velocity of the target’s body frame with respect to the inertial frame is approximated by T  ω T o = 0 −ωT o 0

(3.281)

Thus, we have ⎛

⎞ ⎛ ⎞ ωr x ωC x + cos θr sin ψr ωT o + sin ϕr sin θr cos ψr ωT o ⎝ ωr y ⎠ = ⎝ ⎠ ωC y + cos ϕr cos ψr ωT o ωr z ωC z + sin θr sin ψr ωT o − sin ϕr cos θr cos ψr ωT o

(3.282)

Substituting the above equation in the attitude kinematics shows that it is nonlinear. The higher order terms above the second order are ignored when the relative attitude angles are small, i.e., ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ϕr 0 ωC z ωT o ωcx ϕ˙r ⎝ θ˙r ⎠ = ⎝ −ωC z 0 0 ⎠⎝ θr ⎠ + ⎝ ωcy + ωT o ⎠ ψr ωcz −ωT o −ωC x 0 ψ˙ r ⎛

(3.283)

(2) Relative Attitude Kinematics using Quaternions a) Relative Attitude Described by Quaternions $

& q¯ , with q¯ being the vector part and q4 q4 being the scale part. Denote q C and q T as the respective attitude quaternions of the chaser and target with respect to the inertial frame. The relative attitude quaternion between the chaser and the target is denoted by q r ; then we have The attitude quaternion is denoted by q =

   T   C qr = C qC C q T

(3.284)

b) Equation 1 for the Recursion of Relative Attitude Assuming the error quaternion of the relative attitude   q r,k at time k to time   k + 1 is denoted by δq r,k+1,k which satisfies C q r (k + 1) = C δq r (k + 1, k) C q r (k) ; discretizing the above equation gives q r (k + 1) = M r (k + 1, k) · q r (k).

(3.285)

3.4 Scheme Design of Navigation

141

# −δ q¯ r (k + 1, k)× δ q¯ r (k + 1, k) . where M r (k + 1, k) = δq r,4 (k + 1, k) · I4 + −δ q¯ r (k + 1, k)T 0 When the time step is small, the relation between the angular velocity and Euler angle can be obtained as "

sin θr (k+1,k) θr (k + 1, k) 2 Mr (k + 1, k) = cos · I4 + 2 θr (k + 1, k) " # −θr (k + 1, k)× θr (k + 1, k) −θr (k + 1, k)T 0

(3.286)

where θ r (k + 1, k) = ωr (k + 1, k)t. c) Equation for the Recursion of Relative Attitude From the attitude kinematics, we have & $ 1 q4 I 3 + q¯ × ωr q˙ r = −q¯ T 2

(3.287)

Assuming that & q4 I 3 + q¯ × , Ξ (q) = −q¯ T $ & −ω× ω Ω(ω) = , −ωT 0 & $ q4 I3 − q¯ × , Ψ (q) = −q¯ T $ × & ω ω Γ (ω) = −ωT 0 $

yields 1   Ξ q r ωr 2      1   = Ξ q r ωC − Ξ T q r Ψ q r ω T 2 1 = (Ω(ωC ) − Γ (ω T ))q r 2

q˙ r =

in which the following standard equalities are employed.     Cr = Ξ T qr Ψ qr ,     Ξ q r Ξ T q r = I 4 − q r q rT ,

(3.288)

142

3 Navigation Method and Scheme Design for Rendezvous and Docking

  q rT Ψ q r = 0,   Ψ q r ω T = Γ (ω T )q r The above equation is a time-varying linear system, to design the filter, we need to discretize the equation as follows: & 1 q r (k + 1, k) = exp (Ω(ωC (k)) − Γ (ω T (k)))t q r (k) 2 $ & $ & 1 1 = exp Ω(ωC (k))t exp − Γ (ω T (k))t q r (k) 2 2 $

(3.289)

Then we have [44] $ & 1 ˜ C (k)) = exp (ωC (k))t Ω(ω 2 ⎞ ⎛   sin( 21 ωC (k)t ) × sin( 21 ωC t ) 1 cos 2 ωC t I3 − ωC ωC ωC (k) ω  =⎝  ⎠ (3.290) 1 C sin( 21 ωC t ) T ω t ω cos C C ωC  2 $ & 1 Γ˜ (ωT (k)) = exp − (ωT (k))t 2 ⎞ ⎛   sin( 21 ωT t ) sin( 21 ωT t ) 1 × cos 2 ωT t I3 − ωt (k) − ωT ωT  ω  =⎝  ⎠ 1 T sin( 21 ωT t ) T − ω cos ω t ωT 

T

2

T

(3.291) and ˜ C (k))Γ˜ (ω T (k))q r (k) q r (k + 1) = Ω(ω

(3.292)

d) The Recursion of Error Quaternion The attitude has three degrees of freedom and the quaternion has four parameters which satisfies the constraint of the norm being one, which is not suitable for designing a filter. To overcome this issue, the error quaternion is usually used in practical applications by assuming the scale part is always equal to one, such that the degrees of freedom are reduced to three. The rotation matrix of relative attitude quaternion is given as       C q r = C δq r C qˆ r . The error quaternion is defined as [45].

(3.293)

3.4 Scheme Design of Navigation

143

δq r = q r ⊗ qˆ r−1 .

(3.294)

From the attitude kinematics, we have d qˆ r−1 dq + r ⊗ qˆ r−1 dt dt 1 1 −1 = q r ⊗ qˆ r ⊗ ωˆ r−1 + ωr ⊗ q r ⊗ qˆ r−1 2 2 1 1 −1 = δq r ⊗ ωˆ r + ωr ⊗ δq r 21 2 2 ×    1 − ωˆ r + ωr δ q¯ r + ωr − ωˆ r δq r,4  T = . 2 ωˆ r − ωr δ q¯ r

δ q˙ r = q r ⊗

(3.295)

The relative angular velocity is approximated by ωr ≈ ωˆ r ; then, we have " δ q˙ r =

−ωˆ r× δ q¯ r 0

# (3.296)

Thus, δ q˙ r,4 is approximately constant and does not require updation; only the vector part is updated by the following equation: δ q˙¯ r = −ωˆ r× δ q¯ r .

(3.297)

Once the δ q¯ r is obtained, the relative attitude quaternion is updated as follows: " q r = qˆ r +

# −δ q¯ r× δ q¯ r qˆ r −δ q¯ rT 0

(3.298)

(3) Describing the Relative Attitude using the Modified Rodriguez Parameter The modified Rodriguez parameter (MRP) is derived from the Euler axis/angle, T  which has three variables. If the quaternion is denoted by q r = q¯ rT qr,4 , then the associated MRP is given as [45] q¯ r 1 + qr,4  1 2 pr ' '2 qr = ' '2 ' pr ' ' ' 1 − 1 + pr pr =

(3.299)

(3.300)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

The derivative of MRP is given as  ' '2  1  T  2 ωr pr pr − 2ωr× pr + ωr 1 − ' pr ' 4  ' '2  1 2 pr prT + 2 pr× + 1 − ' pr ' ωr . = 4

p˙ r =

(3.301)

The mapping from the Euler axis/angle to MRP is given by p = tan(ϕ/4)¯e. When ϕ = ±2π, p tends to infinity. For the relative attitude motion between cooperative targets, the relative attitude angle is generally small, and MRP can be directly used to describe the relative attitude motion for filter design in order to estimate the relative attitude. For the case in which the relative attitude varies over a large range, MRP can be used to transform the relative attitude error quaternion, following which the estimated quaternion of the relative attitude can be modified with the estimated MRP parameters. 2. Relative Attitude Determination Scheme using Euler Angle (1) Selection of Estimation State For the cooperative target, because the relative attitude change range between the two spacecraft interface systems is not large, the Euler angle can be conveniently used to describe attitude kinematics, and the camera-type rendezvous and docking sensor can directly solve the relative attitude by using a monocular or binocular vision algorithm. Thus, the relative attitude angles are taken as the estimates as follows. T  X = ϕr θr ψr

(3.302)

(2) Time Update The relative attitude kinematics is a time-varying nonlinear function; the RungeKutta algorithm is utilized to solve the differential equation for the sake of accuracy. In actual application, the equation can also be approximated using the first order term, as the filtering period and relative angular velocity are small. ⎛ ⎞ ⎛ ⎞⎞ ϕ˙r ωr x cos θr cos ϕr + ωr z sin θr cos ϕr 1 ⎝ θ˙r ⎠ = ⎝ ⎝ ωr x sin θr sin ϕr + ωr y cos ϕr − ωr z cos θr sin ϕr ⎠⎠ cos ϕr ˙ ψr k −ωr x sin θr + ωr z cos θr k−1 (3.303) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ϕr ϕ˙r ϕr ⎝ θr ⎠ = ⎝ θr ⎠ + ⎝ θ˙r ⎠ t (3.304) ˙ ψr k ψr k ψr k−1 ⎛

3.4 Scheme Design of Navigation

145

(3) Measurement Equation The relative attitude angle output by the camera can be used as the measurement quantity to design the filter. The following is the measurement equation in this case: ⎡

⎤ ϕ Z = ⎣ 1 ⎦⎣ θ ⎦ + v 1 ψ 1

⎤⎡

(3.305)

(4) Filter Design The nonlinear filtering algorithm should be used because the attitude kinematics is nonlinear. 3. Relative Attitude Determination Scheme using Quaternion (1) State Variable The designed filter directly estimates δ q¯ r to avoid the constraint of the norm being equal to 1. (2) State Equation The relative attitude is updated according to (3.285): qˆ r (k + 1, k) = M r (k + 1, k) · qˆ r (k).

(3.306)

As the state to be estimated, the error quaternion of relative attitude satisfies δ q˙¯ r = −ωˆ r× δ q¯ r .

(3.307)

As the error quaternion is used to correct q r in every cycle, it can be assumed that  T δ q¯ r (k + 1, k) = 0 0 0 . (3) Measurement Equation The direct measurement is the attitude quaternion q catf of the cooperative target with respect to the measurement frame of the camera. As the method adopted in attitude estimation using star sensor [46], considering the attitude alignment of the cooper ative target C t f dt q t f dt and the estimated relative attitude Cˆ dcdt , the measurement frame of the camera can be expressed in the body frame by

146

3 Navigation Method and Scheme Design for Rendezvous and Docking dc tf Zˆ ca = Cˆ dcdt C tTf dt Zˆ ca     = I + 2δ q¯ × C dcdt C tTf dt Z tcaf + Z tcaf     = C dcdt C tTf dt Z tcaf + Z tcaf + 2δ q¯ × C dcdt C tTf dt Z tcaf + Z tcaf

≈ C dcdt C tTf dt Z tcaf + C dcdt C tTf dt Z tcaf + 2δ q¯ × C dcdt C tTf dt Z tcaf

(3.308)

and dc Xˆ ca ≈ C dcdt C tTf dt X tcaf + C dcdt C tTf dt X tcaf + 2δ q¯ × C dcdt C tTf dt X tcaf .

(3.309)

The inner products between the measured axes of the measurement frame of the camera and the actual values are given as T dc  Zˆ ca m 1 = Y dc ca   dc T  C dcdt C tTf dt Z tcaf + C dcdt C tTf dt Z tcaf + 2δ q¯ r× C dcdt C tTf dt Z tcaf = Y ca T T   C dcdt C tTf dt Z tcaf + Y dc 2δ q¯ r× C dcdt C tTf dt Z tcaf = Y dc ca ca   T T = −2 X dc δ q¯ r + Y dc C dcdt C tTf dt Z tcaf (3.310) ca ca T dc  m 2 = X dc Zˆ ca ca T  dc T  = 2 Y ca δ q¯ r + X dc C dcdt C tTf dt Z tcaf ca

(3.311)

T dc  m 3 = Z dc Xˆ ca ca T  dc T  = −2 Y ca δ q¯ r + Z dc C dcdt C tTf dt X tcaf ca

(3.312)

T dc  m 4 = Y dc Xˆ ca ca T  dc T  = 2 Z ca δ q¯ r + Y dc C dcdt C tTf dt X tcaf ca

(3.313)

T  Using Z = m 1 m 2 m 3 m 4 as the measurement gives the following measurement equation: ⎡

 T −2 X dc ca   T ⎢ ⎢ 2 Y dc Z = ⎢  cadc T ⎣ −2 Y ca T  2 Z dc ca

⎡

⎤  dc T 0 Y ca C dcdt C tTf dt  dc T ⎢ ⎥" # ⎥ ⎢ 0 X ca C dcdt C tTf dt ⎥ Z tcaf ⎥ ⎢ ⎥ ⎥δ q¯ r + ⎢  dc T ⎥ X t f T ⎦ 0 ca ⎣  Z ca  C dcdt C t f dt ⎦ T T dc Y ca C dcdt C t f dt 0 (3.314) ⎤

3.4 Scheme Design of Navigation

147

(4) Filter Design The system is a linear time-varying equation; the Kalman filter can be used for the filter design. (5) Update of the Relative Attitude " qˆ r (k + 1) = qˆ r (k + 1, k) +

# −δ q¯ r× δ q¯ r qˆ r (k + 1, k) −δ q¯ rT 0

(3.315)

4. Determination of Relative Attitude using Relative MRP MRP can describe the rotational motion within the angle range of ±2π without the singularity issue, and thus, it can be used directly for the filter design. (1) Selection of State Variable T  The state variables to be estimated are denoted by pr = pr,1 pr,2 pr,3 when using MRP. (2) State Equation p˙ r =

 ' '2  1 2 pr prT + 2 pr× + 1 − ' pr ' ωr 4

(3.316)

(3) Measurement Equation The camera-type rendezvous sensor can provide the relative attitude of the cooperative target with respect to the camera’s measurement frame in the form of the Euler angle, which can be transformed to MRP with the following measurement equation: ⎡

⎤ pr,1 Z = ⎣ 1 ⎦⎣ pr,2 ⎦ + v pr,3 1 1

⎤⎡

(3.317)

(4) Filter Design Because the state equation is a nonlinear equation, a nonlinear filtering method such as EKF or the deterministic sampling filter method should be adopted.

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3 Navigation Method and Scheme Design for Rendezvous and Docking

3.4.5 Relative Motion and Attitude United Estimation This section presents a method to directly take the line of sight measured by the camera-type rendezvous and docking sensor relative to multiple corner-cube reflectors as the measurement, design the filter using the nonlinear filtering method, and estimate the relative state including the relative position and attitude. The direct measurement method requires the design of a filter for a high-dimensional nonlinear system. The filter is relatively complex, but this method avoids the possible numerical pseudo-solution problems in numerical calculation, and the statistical characteristics of the measurement error in the direct measurement information are relatively clear, which is conducive to the design of a high-precision filter [47]. 1. Selection of State Variables The relative position, velocity, and attitude quaternions are chosen as the states to be estimated. T  X = x y z x˙ y˙ z˙ q1 q2 q3 q4 T  = X Tp X qT

(3.318)

2. State Equation The state equation is composed of (3.249) and (3.292) .

X p (k + 1, k) = (k + 1, k)X p (k) + G(k + 1, k)U k + W p (k)     ˜ ωc,k Γ˜ ωt,k q r (k) + W q (k) q r (k + 1, k) = Ω

(3.319)

3. Measurement Equation The principle of the camera-type rendezvous and docking sensor is illustrated in Fig. 3.8, Fig. 3.8 The principle of the camera-type rendezvous and docking sensor

zT

zca

v (vi,ui)

yT

yca

u oca

o

xca

oT

xT Fi(Xi,Yi,Zi)

f

3.4 Scheme Design of Navigation

149

where oca xca yca z ca is the measuring frame of camera, ouv is the image plane, ot xt yt z t is the reference frame fixed on the target, Fi (X i , Yi , Z i ) is the position of the ith (i = 1 . . . n) characteristic point with respect to the body-fixed frame on the target, (vi , u i ) is the image-space position of the corresponding characteristic point, and f is the focal length of the camera. It is easy to obtain the image-space position of the characteristic point BiT = T  − f u i vi with respect to the camera frame oca xca yca z ca . The origin of the camera  T frame with respect to ot xt yt z t is denoted by x y z ; subsequently, we have ⎞ ⎛ ⎞ x − Xi −f ⎝ u i ⎠ = kC ct ⎝ y − Yi ⎠ vi z − Zi ⎛

(3.320)

where k is the proportional factor and C r is the rotation matrix of the chaser with respect to the target. Eliminating k gives c21 (x c11 (x c31 (x vi = − f c11 (x

ui = − f

− X i ) + c22 (y − Yi ) + c23 (z − Z i ) − X i ) + c12 (y − Yi ) + c13 (z − Z i ) − X i ) + c32 (y − Yi ) + c33 (z − Z i ) − X i ) + c12 (y − Yi ) + c13 (z − Z i )

(3.321)

 T Normalizing B iT = − f u i vi yields ⎛

√

⎜ ⎜√ ⎜ ⎝ √

−f f 2 +u i2 +vi2 ui f 2 +u i2 +vi2 vi f 2 +u i2 +vi2

⎞

⎛

√

⎞

x−X i

i ) +(z−Z i ) ⎟ ⎟ ⎜ (x−X i ) +(y−Y y−Yi ⎟ ⎜√ ⎟ ⎟ = C r ⎜ (x−X i )2 +(y−Yi )2 +(z−Z i )2 ⎟ i⎠ ⎠ ⎝ z−Z i √ 2 2 2 2

2

2

(3.322)

(x−X i ) +(y−Yi ) +(z−Z i )

which is denoted by ⎛ ⎜ ⎜ li = ⎜ ⎝

√

−f

f 2 +u i2 +vi2 √ 2 ui 2 2 f +u i +vi √ 2 vi 2 2 f +u i +vi

⎞

⎛

√

x−X i

⎞

⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟, mi = C r ⎜ √(x−X i )2 +(y−Yi )2 +(z−Z i )2 ⎟, (i = 1 . . . n) i ⎠ ⎝ ⎠ z−Z i √ 2 2 2 (x−X i )2 +(y−Yi )2 +(z−Z i )2 y−Yi

(x−X i ) +(y−Yi ) +(z−Z i )

To facilitate the design of the filter, the elevation angle α and azimuth angle β of the normalized line-of-sight can be used as the measured information. Assuming  T that l i = li,x li,y li,z , the line-of-sight angles are defined as   αi = arcsin −li,z /li  (i = 1 . . . n) βi = arctan li,y /li,x

(3.323)

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3 Navigation Method and Scheme Design for Rendezvous and Docking

which gives the following measurement equation:  ⎤ ⎤ ⎡  arcsin −m α1 /m 1,z 1   ⎢ β1 ⎥ ⎢ arctan m 1,y /m 1,x ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ ⎥ .. ⎢ . ⎥=⎢ ⎥+v . ⎢ ⎥ ⎢ ⎥   ⎣ αn ⎦ ⎣ arcsin −m n,z /m n  ⎦   arctan m n,y /m n,x βn ⎡

(3.324)

4. Design of the Filter The state equation of the system is linear, but the measurement equation is non-linear. If EKF is used to design the filter, obtaining the solution of the Jacobian matrix will be a highly complex task. In this section, UKF is used to design the filter. The attitude quaternion must satisfy the module value of one. As UKF uses the weighted sum to estimate the state mean, it is difficult to ensure that the attitude quaternion maintains the unit module value, and thus, the quaternion must be estimated indirectly. The Rodriguez parameter is used to replace the error quaternion. First, the transformation relation between Rodriguez parameter and quaternion is  T given; then, the error quaternion is denoted by δq = δ q¯ T δq4 , following which the Rodriguez parameter can be written as δp =

1 δ q¯ a + δq4

(3.325)

where 0 < a ≤ 1 is a constant and f is the scaling factor satisfying f = 2(a + 1). In this section, we set a = 1 and f = 4. The Rodriguez parameter can be transformed into an error quaternion using 0

δq4 =

−aδ p2 + f

√

f 2 +(1+a 2 )δ p2

f 2 +δ p2

δ q¯ = f −1 (1 + δq4 )δ p

(3.326)

Denoting the Rodriguez parameter and the relative position and velocity by X˜ = T T Xˆ p δ pˆ T , the filtering algorithm for relative state estimation based on UKF is summarized as follows: 

Step 1: Selecting sigma points at time k + 1

T  ˆ˜ , the 19 sigma points X ˆ˜ i = ˆ i T Based on Pˆ X˜ ,k and X X p,k δ pˆ ki T , i = 0 . . . 18 are k k obtained. Denoting qˆ r,k as the estimated relative attitude quaternion at time k, the sigma points for relative attitude quaternion are given as

3.4 Scheme Design of Navigation

151

0 qˆ r,k = qˆ r,k i i qˆ r,k = δq r,k ⊗ qˆ r,k i = 1 . . . 18 i

(3.327)

iT

where δ qˆ r,k is calculated from δ pˆ k using (3.326). Step 2: Prediction and update i i i i into (3.319) gives Xˆ p,k+1 and qˆ r,k+1 , respectively. Substituting Xˆ p,k and qˆ r,k The updated sigma points for the Rodriguez parameter are given as

 −1 i i 0 = qˆ r,k+1 ⊗ qˆ r,k i = 1 . . . 18 δ qˆ r,k

(3.328)

i and δ pˆ ik+1 (i = 1 . . . 18) is calculated from δ qˆ r,k using (3.325), assuming that   T 0 δ pˆ k+1 = 0 0 0 . i ˆ˜ ˆ˜ ˆ ik+1 and Xˆ p,k+1 . X k+1|k and P k+1|k are predicted using δ p

Step 3: Measurement update i ˆ˜ ˆ˜ ˆi ˆ r,k+1 The measurement Zˆ k+1 , K k+1 , X . k+1 and P k+1 is updated using X p,k+1 and q The relative attitude quaternion is updated using 0 qˆ r,k+1 = δ qˆ r,k+1 ⊗ qˆ r,k+1

(3.329)

where δ qˆ r,k+1 is calculated from δ pˆ k+1 according to (3.326).  T Finally, δ pˆ k+1 = 0 0 0 is set for the next calculation cycle. 5. Results from the Numerical Simulation The relative navigation method proposed is applied to the translation consideration section of the lunar orbit rendezvous and docking mission. The corresponding simulation results are presented and analyzed in this section. In the simulation, the initial orbital root number of the target spacecraft is set as: aT = 1938 km, eT = 0.0003, i T = 25◦ , ΩT = 180◦ , wT = 240◦ , f T = 0◦ . The sensor is installed at the center of mass of the chaser, the measurement coordinate system coincides with the chaser, and the measurement noise is zero mean Gaussian white noise with 3σ = 0.01◦ . Four reflectors are installed on the target spacecraft using the four-side cone installation. The respective positions of each reflector in the target body coordinate system are T T  T  , −0.5 0 0 , 0 1.5 cos(30◦ ) −1.5 sin(30◦ ) , T  0 −1.5 cos(30◦ ) −1.5 sin(30◦ ) . 

0 0 1.5

152

3 Navigation Method and Scheme Design for Rendezvous and Docking Estimation error of relative position x/m

1 0 −1

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

y/m

1 0 −1

z/m

1 0 −1

Fig. 3.9 The relative position error

The initial state of the filter is set as  T  T  T Xˆ p,0 = −150 0 0 0 0 0 , qˆ r,0 = 0 0 0 1 , δ pˆ 0 = 0 0 0 ,  ˆ˜ = diag  P 100 100 100 1 1 1 0.001 0.001 0.001 , diag reprents the diagonal 0 matrix. The flight process is as follows: the chaser begins 150 m behind the target in order to approach it in a straight line with a constant speed and approaches 4 m by the end of the simulation (Figs. 3.9, 3.10 and 3.11). The results are presented as follows. It can be seen that the relative position estimation accuracy reaches 0.1 m, the relative speed estimation accuracy reaches 0.01 m/s, and the relative attitude estimation accuracy reaches 0.1°. As the relative distance approaches, the accuracy of relative navigation also improves. Within 5 m of the relative distance, the position error is less than 0.02 m, the velocity error is less than 0.01 m/s, and the attitude error is less than 0.1°.

3.4 Scheme Design of Navigation

153

Estimation error of relative velocity Vx/(m/s)

0.1 0 −0.1

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

Vy/(m/s)

0.1 0 −0.1

Vz/(m/s)

0.1 0 −0.1

Fig. 3.10 The relative velocity error Estimation error of relative attitude roll/(°)

0.5 0 −0.5

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

0

50

100

150 t/s

200

250

300

pitch/(°)

0.5 0 −0.5

yaw/(°)

0.5 0 −0.5

Fig. 3.11 The relative attitude error

154

3 Navigation Method and Scheme Design for Rendezvous and Docking

3.5 Key Points of Navigation System Design The navigation system is a key component of rendezvous and docking system which has significant influence on the performance of the guidance and control system. In the development of the sensor, the trend is to make the sensor smaller and lighter. In addition, because of its redundant measurement information and wide range of action, the laser imaging sensor is suitable for cooperation/non-cooperation and other different types of targets, and relevant studies have been actively carried out in various countries. Owing to the configuration of multiple sensors, the measurement information in the navigation algorithm is abundant and redundant. Using redundant information is expected to improve navigation accuracy as well as the reliability of the navigation system. In order to take full advantage of the redundant information and improve the performance of the navigation system, a redundant information fusion algorithm with better performance should be studied.

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17. Ho, Y.C., Lee, R.C.K.: A Bayesian approach to problems in stochastic estimation and control. IEEE Trans. Autom. Control 9(4), 333–339 (1964) 18. Haug, A.J.: Bayesian Estimation and Tracking: a Practical Guide. Wiley (2012) 19. Lewis, F., Xie, L., Popa, D.: Optimal and Robust Estimation: with an Introduction to Stochastic Control Theory. CRC Press (2017) 20. Julier, S., Uhlmann, J., Durrant-Whyte, H.F.: A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control 45(3), 477–482 (2000) 21. Nørgaard, M., Poulsen, N.K., Ravn, O.: New developments in state estimation for nonlinear systems. Automatica 36(11), 1627–1638 (2000) 22. Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000) 23. Arasaratnam, I., Haykin, S.: Cubature kalman filters. IEEE Trans. Autom. Control 54(6), 1254– 1269 (2009) 24. Liu, T., Xie, Y.: Stability analysis of UKF and its application in relative navigation. J. Astronaut. 31(3), 739–747 (2010) 25. Leathrum, J.: On sequential estimation of state noise variances. IEEE Trans. Autom. Control 26(3), 745–746 (1981) 26. Liu, L., Xie, Y.: Adaptive deterministic sampling filter algorithms. Inf. Control 39(6), 673–680 (2010) 27. Subrahmanya, N., Shin, Y.C.: Adaptive divided difference filtering for simultaneous state and parameter estimation. Automatica 45(7), 1686–1693 (2009) 28. Liu, T., Xie, Y.: Robust unscented kalman filtering for relative navigation of spacecrafts. The academic annual meeting of The Chinese Society of Astronautics, Beijing, p.11 (2012) 29. Liu, T.: New type of filtering and its applications to relative navigation of spacecraft rendezvous and docking. China Academy of Space Technology (2010) 30. Wang, S., Wu, M.: Matrix Inequalities, 2nd edn, p. 167. Beijing: Science Press (2006) 31. Xie, L., Soh, Y.C., De Souza, C.E.: Robust kalman filtering for uncertain discrete-time systems. IEEE Trans. Autom. Control 39(6), 1310–1314 (1994) 32. Zhou, D.: Morden Fault Diagnosis. TsingHua Press, BingJing (2000) 33. Xiong, K.: Research on autonomous navigation based on X-pulsar for sapcecraft. Post doctor thesis, China Academy of Spacecraft Technology (2008) 34. Huber, P.J.: Robust Statistics. Wiley, New York (1981) 35. Karlgaard, C.D., Schaub, H.: Huber-based divided difference filtering. J. Guid. Control Dyn. 30(3), 885–891 (2007) 36. Liu, T., Xie, Y.: A relative navigation algorithm for a chaser tracking a non-cooperative maneuvering target in space. J. Astronaut. 31(5), 1338–1344 (2010) 37. Gordon, N., Salmond, D., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. Proc. Inst. Electr. Eng. 140(2), 107–113 (1993) 38. Chen, Z.: Bayesian filtering: from kalman filters to particle filters, and beyond. Statistics 182(1), 1–69 (2003) 39. Cheng, S.: Review on particle filters. J. Astronaut. 29(4), 1099–1111 (2008) 40. Liu, T., Xie, Y., Hu, H.: Application of particle filtering in relative navigation filter design for spacecraft. Aerosp. Control Appl. 37(6), 19–27 (2011) 41. Hanak, C., Zanetti, R.: Relative navigation for the orion vehicle. AAS 08-306 42. Liu, T., Xie, Y.: Study of relative navigation based on relative differential pseudo range of GPS. 27(1), 1–8 (2007) 43. Yingzi, H., Ying, C., Dong, H.: Rendezvous radar observation-based relative navigation filter. Aerosp. Control 22(6), 17–20 (2004) 44. Son-Goo Kim, J.L., Crassidis, Y.C., Fosbury, A.M.: Kalman filtering for relative spacecraft attitude and position estimation. In: AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California (2005) 45. Shuster, M.D.: A survey of attitude representations. Navigation, 8(9), 439–517 (1993)

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46. Liu, Y., Chen, Y.: Star-sensor measurement model and its application to the spacecraft attitude determination system. J. Astronaut. 24(2), 162–167 (2003) 47. Liu, T., Hu, H., Wang, Y., Wang, D.: A novel relative navigation method using angles observations from vision sensor. Aerosp. Control 30(5), 35–47 (2012)

Chapter 4

Guidance Method and Schematic Design for Rendezvous and Docking

4.1 Introduction In Chap. 1, the process of rendezvous and docking of a flight mission is divided into far range rendezvous phase and near range autonomous control phase based on the distance. The latter can be further divided into homing phase, closing phase, final approach phase, and fly around phase etc. In distinct stages, the requirements for flight time, control accuracy, fuel consumption, and safety are different, and the measurement information provided by the sensor is also different, so the corresponding guidance strategies are not the same. In the far range rendezvous phase, the distance between the two spacecraft is relatively large, and fuel optimization is the main purpose. At the same time, the relative sensors cannot work, so the guidance strategy can be designed using the method of orbit control or the method of relative orbit elements. In the near range autonomous control phase, the importance of control accuracy, flight time, and mission safety increases with approaching distance. Generally, the measurement information provided by relative sensors is used for guidance and control. Apart from position control between two spacecraft, attitude control, i.e., six-DOF control, is also needed in the final approach phase which will be introduced in the following chapters. In addition, in the middle and near range control phases, when the impulsive guidance strategy is adopted, the number of pulses, rendezvous time, fuel consumption, and initial terminal state of rendezvous need to be designed. The theory of optimal impulsive rendezvous mode distribution provides a theoretical basis for these designs. In this chapter, the guidance method of rendezvous and docking and the design of rendezvous scheme in engineering application are introduced. Section 4.2 introduces the rendezvous by orbital maneuver, mainly focusing on the guidance method of single spacecraft. Section 4.3 introduces the method of multiple-impulse optimal rendezvous and gives the distribution of optimal impulsive rendezvous mode. In Sects. 4.4 and 4.5, the common rendezvous and docking guidance strategies, including CW guidance and line of sight guidance, are introduced in combination

© National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_4

157

158

4 Guidance Method and Schematic Design for Rendezvous and Docking

with engineering applications, and the problems in theoretical research and engineering application of these rendezvous strategies are discussed. Sections 4.6, 4.7, and 4.8 discuss the application of the rendezvous strategy in different rendezvous and docking engineering designs according to the stage division of rendezvous and docking in Chap. 1.

4.2 Rendezvous by Orbital Maneuver In the far range rendezvous phase of rendezvous and docking, there is no relative navigation sensor for the two spacecraft being far away. Orbit control strategy is generally adopted to guide the target to a predetermined position. In-plane control and out-of-plane control are independent. In-plane orbit control method adopts Hohmann rendezvous strategy, semi-major axis-eccentricity-perigee angle synergy control, and other methods.

4.2.1 Hohmann Rendezvous Hohmann transfer can be used to realize rendezvous. The characteristics of this method are coplanar circular orbit, double impulse, thrust maneuver in local horizontal direction, and fuel economy. As shown in Fig. 4.1, the orbital altitude of the target is R B and that of the target is R A . The geocentric angle crossed by the target along the Hohmann transfer orbit is 180°, and the time is equal to the half orbital period of the Hohmann orbit, which is  τ A = 0.5TH = π

[0.5(R A + R B )]3 μ

(4.1)

where the parameter μ = 398600.44 km3 /s2 is the gravitational constant of Earth.

Fig. 4.1 Hohmann rendezvous

Target spacecraft

Chaser spacecraft

4.2 Rendezvous by Orbital Maneuver

159

During this period, the geocentric angle crossed by the target is  θt = 0.5ωoT TH = π

RA + RB 2R B

1.5 (4.2)

Therefore, to adopt the Hohmann rendezvous strategy, the initial phase angle difference of the two spacecraft β should satisfy 



β =π 1−

RA + RB 2R B

1.5  (4.3)

Considering R B −R A a a fuel consumption can be induced as

162

4 Guidance Method and Schematic Design for Rendezvous and Docking 



r 1 = −r 2 V1 = ω4o a (e + a ) a ) V2 = ω4o a (−e + a a

(4.8)

Besides, the impulses application time are satisfied: u 1 = a tan 2(e y , ex ) u 2 = a tan 2(−e y , −ex )

(4.9)

The difference between u 1 and u 2 is 180°.

, if |V1 | + |V2 | = na , and V1 , V2 have the same (2) When e ≤ a a 2 sign, the solution is optimal. Theoretically, there are infinite groups of optimal solutions, and here are three typical optimal solutions. ➀ Constraint: u 1 is specified. The optimal solution can be obtained as follows: u 1 = a tan 2(e y , ex ), denote V = ωo a, then ⎧ 2 ⎨ ( aa ) −(e2x +e2y ) V1 = V4 a − e cos u 1 +e y sin u 1 ) ( x (4.10) a ⎩ V = V a − V 2

1

2a

cos u 2 = sin u 2 =

V ex 2 V e y 2

−V1 cos u 1 V2 −V1 sin u 1 V2

(4.11)

➁ Constraints: V1 = V2 ; two impulses are applied on both sides of the eccentricity increment. The optimal solution can be obtained as follows: V1 = V2 = ω4o a a a u 1 = u 0 − ar cos ae , u 1 = u 0 + ar cos ae a a u 0 = a tan 2(e y , ex )

(4.12)

➂ Constraints: u 1 is designated at apogee; u 1 = π ; denote u 0 a tan 2(e y , ex ) and V = ωo2a The optimal rendezvous solution can be obtained as follows: V1 =

ωo a ( 4 V 2 4 ωo2 a 2

V2 = V − V1 u2 = u1 +

=

  − e2 )/ ω2o a V − e cos(u 1 − u 0 ) (4.13)

e2 ( ω2o a ) −V12 −V22 cos−1 2V1 V2 2

4.2 Rendezvous by Orbital Maneuver

163 Target spacecraft

O Ω2 Ω1

Δ

u F

Chaser spacecraft

Equatorial plane

X N2

i2

i1 N1

Fig. 4.3 Out-of-plane control diagram

2. Orbit Control for Out-of-Plane The out-of-plane correction of rendezvous and docking mainly refers to make the target’s orbital plane coincide with the target’s orbital plane by using the orbital control on the target out of the plane. The out-of-plane control is relatively independent. The target orbit and target orbit intersect at point F, and the ascending node and the orbital inclination are shown in Fig. 4.3. In the spherical triangle N1 N2 F, there is a spherical cosine theorem. cos  = cos i 1 cos i 2 + sin i 1 sin i 2 cos( 1 − 2 ) The orbital angle  that needs to be adjusted can be obtained from this. According to the spherical triangle sine theorem, there is sin u sin( 1 − 2 ) = sin i 1 sin  The argument of latitude u can be obtained. 1) Modify Only the Orbital Inclination In this situation, Ω1 = Ω2 . So cos Δ = cos i 1 cos i 2 + sin i 1 sin i 2 = cos(i 1 − i 2 ) = cos Δi sin u = 0, that is u = 0 or π , giving the impulse at the ascending node or the descending node. Orbital inclination should be modified at the ascending node or the descending node as i = i 1 − i 2

164

4 Guidance Method and Schematic Design for Rendezvous and Docking

2) Modify Only the longitude of the Ascending Node In this situation, i 1 = i 2 = i, the corresponding formula is  = 1 − 2 where Ω is the difference between the longitude of ascending node of the two orbits at the terminal time of far range rendezvous phase. Because Ω varies under the influence of Earth’s oblateness (the rate of change is related to a, e and i), and it is possible to make in-plane correction after applying Ω correction impulse, the magnitude of Ω must be calculated according to the overall scheme. The law of the longitude of ascending node’s variation is as follows Ω˙ =

 2 Re 3 cos i dΩ = − ωo J2 dt 2 a (1 − e2 )2

The variations of eccentricity and semi-major axis caused by raising the orbit are as follows. 0 −δa Perigee: e1 = aa0 e0 +δa a0 e0 +δa Apogee: e1 = a0 +δa Semi-major axis: a1 = a0 + δa The longitude of ascending node needs to be modified as ˙ T TT − ( 2 − ˙ 0 T0 − ˙ 1 T1 − · · · − ˙ n Tn )  = 1 − where Ω1 and Ω2 are the longitude of ascending node of the current target and target, Ω˙ T is the drift velocity of the target’s longitude of ascending node, TT is the time from current time to the far end, Ω˙ 0 is the current drift rate of the target, T0 is the time from current time to the time before the first in-plane impulse, Ω˙ K is the longitude of ascending node’s drift rate at the kth in-plane impulse (k = 1, . . . , n), and Tk is the time from kth in-plane impulse to the k + 1th one, the time when k = n is the time from n − 1th impulse to the far end. 3) Combination Modify In this situation, we modify both i and Ω, and considering the drift of the longitude of ascending node, Δ, u is solved in turn. In conclusion, the orbital perturbation equation shows that the propellant consumption for adjusting the orbital inclination is less at latitudes 0° and 180°, and the propellant consumption for adjusting the longitude of ascending node at latitudes 90° and 270° is less. For the case in which the orbit inclination deviation and the longitude of ascending node deviation exist at the same time, we can calculate and select the appropriate argument of latitude for an out-of-plane maneuver to correct the above two deviations at the same time.

4.2 Rendezvous by Orbital Maneuver

165

4) Solve for the Velocity Increment Orbital perturbation equations include d di r sin(ω + f ) r cos(ω + f ) an an = = √ √ 2 2 dt dt ωo a 1 − e sin i ωo a 2 1 − e2 According to the above formula, if we only modify the orbital inclination √ iωo a 2 1 − e2 v = r cos(u) If we only modify the longitude of ascending node √  ωo a 2 1 − e2 sin i v = r sin(u) If we modify synthetically, after getting Δ, u, we can obtain the guidance impulse outside the orbit plane as    μ sin2 f + (e + cos f )2  sin . v = 2 a(1 − e2 ) 2

4.3 Multiple-impulse Optimal Rendezvous For a particular optimal rendezvous problem, the optimal rendezvous solution can be obtained by different methods. If the design of rendezvous time and initial and terminal states is unreasonable, even if the optimal rendezvous solution is obtained, more fuel may be consumed. It is necessary to design the rendezvous problem from a global point of view. For example, giving the appropriate initial, terminal states and rendezvous time, and determining how many impulse are needed to solve the rendezvous problem, the distribution of optimal rendezvous modes in multiple impulse optimal guidance can solve the problem [1]. In the 1960s, Prussing’s optimal impulse rendezvous problem of near circular orbit gave the distribution of four-impulse, three-impulse, and two-impulse optimal rendezvous modes in different flight times and initial and terminal states within two circular orbit rendezvous with little difference in the semi-major axis [2]. We developed the theory in the research for Chinese Second Manned Space Project, and the optimal impulse rendezvous distribution of small eccentricity elliptical orbit’s spacecraft to circular orbit’s spacecraft was obtained [3].

166

4 Guidance Method and Schematic Design for Rendezvous and Docking

4.3.1 Solutions of Optimal Impulsive Rendezvous Between Two Near Circular Orbits 1. Problem Description As shown in Fig. 4.4, the target and the target move in a coplanar circular orbit, and the orbital radii of the two spacecraft are not very different. The rendezvous problem in this situation is called the near circular orbit rendezvous problem. In Chap. 2, a dimensionless dynamic Eq. (2.79) based on cylindrical frame is presented. When the spacecraft and the reference frame are in the same orbit plane, i.e., δz = 0, their relative motion is as follows: 

δr¨ = 3δr + 2δ θ˙ + a yr δ θ¨ = −2δr˙ + a yθ

(4.14)

Denote state variable x = [δr δθ δr˙ δ θ˙ ]T ; then, Eq. (4.14) can be written in the form of the state equation: ⎡

⎤ ⎡ δr˙ 0 ⎢ δ θ˙ ⎥ ⎢ 0 ⎢ ⎥=⎢ ⎣ δr¨ ⎦ ⎣ 3 δ θ¨ 0

0 0 0 0

1 0 0 −2

⎤⎡ ⎤ ⎡ 0 δr 0 ⎢ δθ ⎥ ⎢ 0 1⎥ ⎥⎢ ⎥ + ⎢ 2 ⎦⎣ δr˙ ⎦ ⎣ 1 0 δ θ˙ 0

⎤ 0   0⎥ ⎥ a yr 0 ⎦ a yθ 1

(4.15)

⎡

⎤ ⎡ ⎤ 00 1 0 00 ⎢0 0 0 1⎥ ⎢ ⎥ ⎥, B = ⎢ 0 0 ⎥. Equations (4.14) and (4.15) are where we denote A = ⎢ ⎣3 0 0 2⎦ ⎣1 0⎦ 0 0 −2 0 01 used to describe the rendezvous process of near circular orbits. In Fig. 4.4, the orbital radii of the target and the target are R1 and R2 , respectively. The argument of latitude Fig. 4.4 Rendezvous between near circular orbit

Target spacecraft

R2

Chaser spacecraft

β R1

4.3 Multiple-impulse Optimal Rendezvous

167

difference between the target and the target, i.e., phase angle difference is β, and the rendezvous time is τ F . When the intermediate reference orbit is selected to establish the reference frame, the orbit radius of the reference one is Rr = (R1 + R2 )/2, denoted as δ R = (R2 − R1 )/Rr , and the initial origin of the reference frame is on the line between the target and the geocenter. The initial state of the near circular orbit rendezvous problem can be obtained as follows  T x 0 = −0.5δ R 0 0 0.75δ R

(4.16)

Similarly, its terminal state is T  x F = 0.5δ R β − 0.75τ F δ R 0 −0.75δ R

(4.17)

The characteristic phase angle of near circular orbit rendezvous is defined as δθ F , which satisfies the requirement δθ F = β − 0.75τ F δ R

(4.18)

Initial and terminal states determined by kinetic Eqs. (4.14) and (4.16)–(4.18) are used to study the optimal impulse rendezvous problem of near circular orbit. The optimal rendezvous mode distribution determined by rendezvous time, initial and final states is obtained by solving the optimal impulse mode and optimal distribution problem. 2. Multiple impulse optimal rendezvous mode distribution determination By solving two-impulse, two-impulse with drift, three-impulse, three-impulse with drift, four-impulse, and Hohmann rendezvous modes with drift and solving the boundary of optimal rendezvous modes, the distribution of optimal rendezvous



F

, modes can be obtained as shown in Fig. 4.5a. When the ordinate is selected as δθ δR the optimal rendezvous mode distribution as shown in Fig. 4.5b can be obtained by similar analysis and simulation. Figure 4.5a, b show that the distribution of the optimal rendezvous mode is symmetrical.

4.3.2 Solutions of Optimal Impulsive Rendezvous of Ellipse-to-Circle Orbits 1. Problem Description The dimensionless dynamic equation based on cylindrical coordinates in Chap. 2 is used to study the optimal distribution of impulse rendezvous modes from small eccentricity elliptic orbit spacecraft to circular orbit spacecraft. Because the two

168

4 Guidance Method and Schematic Design for Rendezvous and Docking Distribution of optimal rendezvous modes

Distribution of optimal rendezvous modes

40

40

30

10

+2

3

25 Hohmann Coast

0

−10

2+

−20

2

3

20

5

4

Four-Impulse TwoImpulse

ThreeImpulse Three-Impulse with drift

15 10

3+

−30 −40 0

30

+3 |δθF/δR|

δθF/δR

20

35

4

2

Two-Impulse with drift Hohmman rendezvous with drift

0 1.5 2 2.5 0.5 1 Rendezvous time (reference orbital period)

−5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period)

(a)

(b)



δθ F

F Fig. 4.5 Optimal rendezvous mode distribution a = δθ , b =

δR

δR

spacecraft are in the same orbit plane, the reference coordinate can be established by choosing t in the same orbit plane with the two spacecraft as 

δr¨ = 3δr + 2δ θ˙ + a yr δ θ¨ = −2δr˙ + a yθ

(4.19)

Equation (4.19) is written in the form of state space as ⎡

⎤ ⎡ δr˙ 0 ⎢ δ θ˙ ⎥ ⎢ 0 ⎢ ⎥=⎢ ⎣ δr¨ ⎦ ⎣ 3 δ θ¨ 0

0 0 0 0

1 0 0 −2

⎤⎡ ⎤ ⎡ 0 δr 0 ⎢ δθ ⎥ ⎢ 0 1⎥ ⎥⎢ ⎥ + ⎢ 2 ⎦⎣ δr˙ ⎦ ⎣ 1 0 δ θ˙ 0

⎤ 0   0⎥ ⎥ a yr 0 ⎦ a yθ

(4.20)

1

Its state transition matrix is ⎡

4 − 3 cos τ ⎢ −6τ + 6 sin τ Φ(τ ) = ⎢ ⎣ 3 sin τ 6 cos τ − 6

⎤ 0 sin τ 2(1 − cos τ ) 1 −2(1 − cos τ ) 4 sin τ − 3τ ⎥ ⎥ ⎦ 0 cos τ 2 sin τ 0 −2 sin τ −3 + 4 cos τ

(4.21)

In the same orbital plane, the orbital radius of the target is R, the semi-major axis of the target is a, the eccentricity is e, the true proximity angle is f , the argument of latitude difference between the target and the target is β, and the rendezvous time is τ F . The reference frame is established at the center of mass of the target. And the difference of the two spacecraft’s semi-major axis is denoted as δa = (R − a)/R. According to Eqs. (2.91) and (2.92), the initial and terminal states of the optimal rendezvous are obtained as follows

4.3 Multiple-impulse Optimal Rendezvous

x 0 = [−δa − e cos f x F = [0 0 0 0]T

169

− β e sin f 1.5δa + 2e cos f ]

T

(4.22)

In this chapter, the optimal impulse rendezvous problem of elliptic orbit spacecraft will be researched using the dynamic equation described by Eqs. (4.19) and (4.22). The necessary conditions for optimal rendezvous between elliptic orbit spacecraft and circular orbit spacecraft can be obtained as follows: (1) the primer vector p must be continuous and the first derivative continuous; (2) during the whole rendezvous process, | p| ≤ 1 and | p| = 1 at the time of applying the impulse; (3) the direction of applying the impulse u corresponds to the direction of the primer vector p; and (4) except for the initial and terminal impulse, | p| ˙ = 0 in other impulse moments. Because the linear Eq. (4.19) is used to inspect the rendezvous problem, the following two conclusions are established in Ref. [4]: (1) Linear Eq. (4.19) is used to describe the optimal impulsive rendezvous process. The impulsive solution satisfying the necessary conditions of optimal rendezvous is the optimal rendezvous solution of the problem; therefore, the necessary conditions of optimal rendezvous are also the sufficient conditions of optimal rendezvous. (2) In the optimal impulse rendezvous described by linear Eq. (4.19), because there are only four state variables, the maximum number of impulses in the optimal rendezvous mode is 4. From the above two conclusions, we can see that there are two impulses, two impulse with drifts, three impulses, three impulse with drifts, and four impulse modes in the optimal impulse rendezvous described by Eqs. (4.19) and (4.22), and their distribution is independent. The characteristic phase angle related to rendezvous time is defined as follows: δθ F = β − 0.75τ F δa − e sin(τ F + f ) + e sin f

(4.23)

The boundary value problem of optimal impulse rendezvous satisfies: x F = Φ(τ F )x 0 +



Φ(τ Fi )Bui Vi

(4.24)

i

where τ Fi = τ F −τi0 , τi0 = τi −τ0 is the ith impulse’s time, τ0 is the initial time, Vi is the ith impulse’s amplitude, δx F = x F − Φ(τ F )x 0 , V = [V1 V2 V3 V4 ]T , and W = [Φ(τ F1 )Bu 1 Φ(τ F2 )Bu 2 Φ(τ F3 )Bu 3 Φ(τ F4 )Bu 4 ]. 2. Multiple Impulse Optimal Rendezvous Mode Computation The process of solving the optimal rendezvous mode is divided into two-impulse, two-impulse with drift, three-impulse, three-impulse with drift, and four-impulse rendezvous mode. Because of the symmetry of the four-impulse optimal rendezvous, the primer vector can be obtained independently. First, the phase angle corresponding

170

4 Guidance Method and Schematic Design for Rendezvous and Docking

to the impulse time and primer vector information can be obtained. Then, the boundary value problem can be constructed using the information of the intermediate time to obtain the amplitude of the impulse. The solution of three-impulse optimal rendezvous is obtained by solving a complex non-linear system of equations that integrates the primer vector information and boundary value problem. Because there are only four information of two-impulse optimal rendezvous, it can be obtained directly by boundary value problem. Then, the primer vector information can be obtained, and the necessary conditions of optimal rendezvous can be used to verify whether the optimal solution of two-impulse is the optimal rendezvous solution. The solution of the rendezvous mode with drift can be obtained on the basis of the corresponding solution of the mode without drift. 1) Four-Impulse Optimal Rendezvous Mode Because of the symmetry of the four-impulse optimal rendezvous mode, when the rendezvous time τ F is known, the time τ1 , τ2 , τ3 , τ4 and the corresponding impulse direction u i can be obtained using the primer vector equation. The boundary value problem of the four-impulse optimal rendezvous mode can be solved using the information of the intermediate time as δx H = HΔV

(4.25)

δx H = Φ(τ H − τ F )x F − Φ(τ H − τ F )Φ(τ F )x 0

(4.26)

H = Φ(τ H − τ F )W

(4.27)

where

τ H = 0.5τ F . From Eqs. (4.21)–(4.23) and (4.25), we can obtain that ⎤ δa + e cos(τ H + f ) ⎢ β − 1.5τ H δa − 2e sin(τ H + f ) + 2e sin f ⎥ ⎥ =⎢ ⎦ ⎣ −e sin(τ H + f ) −1.5δa − 2e cos(τ H + f ) ⎡

δx H

(4.28)

Denote l = h 11 h 41 − h 21 h 41 , h = h 22 h 31 − h 21 h 32 , q j = 2h 4 j + 3h 1 j ( j = 1, 2), and δθβ as δθβ = β − 1.5τ H δa − 2e sin(τ H + f ) + 2e sin f From Eqs. (4.25) and (4.27), we can get

(4.29)

4.3 Multiple-impulse Optimal Rendezvous ⎤ −2 hh32 δθβ + ql2 δa + 2hl42 e cos(τ H + f ) + 4hl12 e cos(τ H + f ) − 2hh22 e sin(τ H + f ) ⎢ q h 2h 4h 2h 1 ⎢ 2 31 δθ − l1 δa − l41 e cos(τ H + f ) − l11 e cos(τ H + f ) + h21 e sin(τ H + f ) ⎥ ⎥ V = ⎢ hh31 β ⎥ 4 ⎣ 2 h δθβ + ql1 δa + 2hl41 e cos(τ H + f ) + 4hl11 e cos(τ H + f ) + 2hh21 e sin(τ H + f ) ⎦ −2 hh32 δθβ − ql2 δa − 2hl42 e cos(τ H + f ) − 4hl12 e cos(τ H + f ) − 2hh22 e sin(τ H + f )

171

⎡

(4.30)

The four-impulse optimal rendezvous solution exists when Vi ≥ 0. 2) Three-Impulse Optimal Rendezvous Mode When the rendezvous time is τ F , the three-impulse optimal rendezvous without drift satisfies the following equations: δx F = W ΔV

(4.31)

 

δxF w1 w2 w3 = 0

(4.32)



pi = 1 i = 1, 2, 3

(4.33)



p˙ 2 = 0

(4.34)

τ3 − τ1 = τ F

(4.35)

The above five equations (groups) contain 3, 1, 3, 1, and 1 equations, respectively. The Eqs. (4.31) and (4.32) are derived from the boundary value problems (4.24) and (4.25), and w1 , w2 and w3 are the three column vectors of W in the threeimpulse optimal rendezvous mode. The Eqs. (4.33) and (4.34) are determined by the necessary conditions. Equations (4.35) is determined by the rendezvous time τ F . Three undetermined coefficients A, B and C, time τi , and impulse amplitude V corresponding to the three impulse can be obtained from the complex non-linear equations composed of the above nine equations. 3) Two-Impulse Optimal Rendezvous Mode Denote Sτ = sin τ F , Cτ = cos τ F . From Eq. (4.24) to (4.25), the boundary value problem of two-impulse optimal rendezvous satisfies    V 1  δx F = B 21 B 22 V 2

(4.36)

      0 2(1 − Cτ ) N21 Sτ where B22 = , N21 = , T21 = , B21 = T21 −2(1 − Cτ ) 4Sτ − 3τ F I   2Sτ Cτ . −2Sτ −3 + 4Cτ

172

4 Guidance Method and Schematic Design for Rendezvous and Docking

When N21 is nonsingular, the four components of two sets of impulse can be obtained as follows 

V 1 V 2



⎡ 1 ⎢ ⎢ = ⎢ |N 21 | ⎣

4Sτ − 3τ F 2(Cτ − 1) 0 0 2(1 − Cτ ) Sτ 0 0 3τ F Cτ − 4Sτ 2(Cτ − 1) 8(1 − Cτ ) − 3τ F Cτ 0 14(1 − Cτ ) − 6τ F Cτ Sτ 0 8(1 − Cτ ) − 3τ F Cτ

⎤ ⎥ ⎥ ⎥δx F ⎦

(4.37) The primer vectors are obtained by two groups of increments of velocity determined by Eq. (4.37) λv1i =

Vir Viθ , λv2i = , i = 1, 2 Vi Vi

(4.38)

The primer vector is solved in the following form: 

λv1i = A# cos τ + B # sin τ + 2C # λv2i = 2B # cos τ − 2 A# sin τ − 3C # τ + D #

(4.39)

From Eqs. (4.37)–(4.38), we get ⎡ ⎤⎡ ⎤ ⎤ λv11 4(1 − Cτ ) − 3τ F Cτ A# 2Sτ −4(1 − Cτ ) −2Sτ ⎢ ⎢ ⎥ ⎢ #⎥ ⎥ 1 ⎢ 3τ F Cτ − 4Sτ 2(1 − Cτ ) 4Sτ − 3τ F 2(Cτ − 1) ⎥⎢ λv21 ⎥ ⎢B ⎥ ⎢ ⎥⎢ ⎢ #⎥= ⎥ ⎦⎣ λv12 ⎦ ⎣ C ⎦ |N 21 | ⎣ 2(1 − Cτ ) −Sτ 2(1 − Cτ ) Sτ D# 8Sτ − 6τ F Cτ 4(1 − Cτ ) − 3τ F Cτ 6τ F − 8τ F Sτ 4(1 − Cτ ) λv22 ⎡

(4.40)

After calculating the coefficients of the primer vector solution, we can verify whether the primer vector composed of these coefficients satisfies the necessary conditions for optimal rendezvous. That is to judge whether the maximum value of the primer vector modulus is less than 1 in the (0, τ F ) interval. If all the primer vector amplitude is less than 1, the solution is the optimal two-impulse rendezvous solution; otherwise, it is not the optimal solution. 4) Optimal Rendezvous Mode with Drift The solution of the optimal two-impulse rendezvous mode with drift is based on the optimal two-impulse rendezvous mode. If the primer vector amplitudes at the time τ− before the first impulse time is equal to 1, and all the primer vector modulus between (τ− , τ1 ) is less than 1, there exists an optimal rendezvous mode with initial drift of the two-impulse mode. Accordingly, when the primer vector amplitudes at the time τ+ after the second impulse time are equal to 1, and the value of the primer vector modulus between (τ F , τ+ ) is less than 1; then, there is an optimal rendezvous mode with the end drift of the two impulses. If both of these situations are satisfied, the optimal two-impulse rendezvous mode with drift at both the initial and the terminal is obtained.

4.3 Multiple-impulse Optimal Rendezvous

173

The solution of three-impulse optimal rendezvous mode with drift is based on the four-impulse optimal rendezvous mode with zero amplitude of the first impulse or the fourth impulse. When the amplitude of the first impulse is zero and the rendezvous time is between τ F4 + τ1 − τ2 and τ F4 (where τ F4 is the rendezvous time in the fourimpulse optimal rendezvous mode with the first impulse amplitude being zero, τ1 , τ2 are the rendezvous time corresponding to the first two impulse moments), there exists a three-impulse optimal rendezvous mode with initial drift. Correspondingly, when the fourth impulse amplitude is zero and the rendezvous time is between τ F4 +τ3 −τ4 and τ F4 (where τ F4 is the rendezvous time in the four-impulse optimal rendezvous mode with the fourth impulse amplitude being zero, τ3 , τ4 are the rendezvous time corresponding to the last two impulse moments), there exists a three-impulse optimal rendezvous mode with terminal drift. 3. Multiple-Impulse Optimal Rendezvous Mode Distribution Determination In the optimal rendezvous of near circular orbits, the distribution of optimal rendezvous modes is discussed with the rendezvous time τ F as abscissa and δθ F /δθ R = β/δ R−0.75τ F as the ordinate, and the distribution of optimal rendezvous modes is only related to τ F and β/δ R. In the rendezvous between elliptical orbit spacecraft and circular orbit spacecraft, we choose the rendezvous τ F as abscissa and δθ F /δa as ordinate, and δθ F is expressed by Eq. (4.23). If E_δa = e/δa, the distribution of optimal rendezvous modes in the rendezvous of elliptical orbit spacecraft to circular orbit spacecraft is related not only to τ F and β/δ R, but also to f and E_δa. The simulation results show that when E_δa is small, the distribution of the optimal rendezvous mode is shown in Fig. 4.6. From Fig. 4.6, it can be seen that the distribution of the optimal rendezvous mode is similar to that of the optimal rendezvous mode of the near circular orbit shown in Fig. 4.5. Because the process of solving the optimal rendezvous mode distribution in the case of δa > 0 is similar to that in the case of δa < 0, the determination of the boundary of the optimal rendezvous mode distribution area in Fig. 4.6 is analyzed by taking δa > 0 as an example. Fig. 4.6 Optimal rendezvous mode distribution

Distribution of optimal rendezvous modes 40 30

δθF/δR

20

2

3

10 +2 0

+2+

−10 2+ −20

4 +3

2

3

3+ 4

−30 −40 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period)

174

4 Guidance Method and Schematic Design for Rendezvous and Docking

1) Four-Impulse Optimal Rendezvous Mode Denote E f _bu = − sin(τ F + f ) + 2 sin(τ H + f ) − sin f , the relation between δθ F and δθβ can be obtained by Eqs. (4.23) and (4.29) δθ F = δθβ + e ∗ E f _bu

(4.41)

From Vi ≥ 0, we can get ⎧ ⎪ −2 h 32 δθβ + ql2 δa + 2hl42 e cos(τ H + f ) + 4hl12 e cos(τ H + f ) − 2hh22 e sin(τ H + f ) ≥ 0 ⎪ ⎪ ⎨ h 31 h 2 h δθβ − ql1 δa − 2hl41 e cos(τ H + f ) − 4hl11 e cos(τ H + f ) + 2hh21 e sin(τ H + f ) ≥ 0 q1 h 31 2h 41 4h 11 2h 21 ⎪ ⎪ 2 h δθβ + l δa + l e cos(τ H + f ) + l e cos(τ H + f ) + h e sin(τ H + f ) ≥ 0 ⎪ ⎩ q2 h 32 2h 42 4h 12 −2 h δθβ − l δa − l e cos(τ H + f ) − l e cos(τ H + f ) − 2hh22 e sin(τ H + f ) ≥ 0

⎧ 2h 42 4h 12 2h 22 ⎪ ⎪ L L1 = l 2h cos(τ H + f ) + l 4h cos(τ H + f ) − h2h sin(τ H + f ) ⎨ L L2 = − l 41 cos(τ H + f ) − l 11 cos(τ H + f ) + h21 sin(τ H + f ) , Denote 2h 41 4h 11 2h 21 ⎪ ⎪ L L3 = l cos(τ H + f ) + l cos(τ H + f ) + h sin(τ H + f ) ⎩ L L4 = − 2hl 42 cos(τ H + f ) − 4hl12 cos(τ H + f ) − 2hh22 sin(τ H + f ) then ⎧ −2 h 32 [δθ F − e ∗ E f _bu] + ql2 δa + e ∗ L L1 ≥ 0 ⎪ ⎪ ⎨ h 31 h 2 h [δθ F − e ∗ E f _bu] − ql1 δa + e ∗ L L2 ≥ 0 ⎪ 2 h 31 [δθ − e ∗ E f _bu] + ql1 δa + e ∗ L L3 ≥ 0 ⎪ ⎩ h h 32 F −2 h [δθ F − e ∗ E f _bu] − ql2 δa + e ∗ L L4 ≥ 0

⎧ −2 h 32 δθ F ≥ − q2 − 2 h 32 E f _bu ∗ E_δa − E_δa ∗ L L1 ⎪ ⎪ ⎨ h 31 hδθ Fδa q1 l h 31 h 2 h δa ≥ l + 2 h E f _bu ∗ E_δa − E_δa ∗ L L2 h 31 δθ F ⎪ ≥ − ql1 + 2 hh31 E f _bu ∗ E_δa − E_δa ∗ L L3 2 ⎪ ⎩ h h 32δaδθ F −2 h δa ≥ ql2 − 2 hh32 E f _bu ∗ E_δa − E_δa ∗ L L4

(4.42)

It can be verified that the primer vector of four-impulse optimized rendezvous mode satisfies h 31 h 32 0, then,

h 32 h

< 0, from (4.42)

⎧ δθ E_δa F ≥ 2hq232h l + E f _bu ∗ E_δa + h2h ∗ L L1 ⎪ δa ⎪ 32 ⎪ ⎨ δθ F ≥ q1 h + E f _bu ∗ E_δa − h E_δa ∗ L L2 δa 2h 31 l 2h 31 q1 h δθ F h E_δa ⎪ ≥ − 2h 31 l + E f _bu ∗ E_δa − 2h 31 ∗ L L3 ⎪ δa ⎪ ⎩ δθ F E_δa ≥ − 2hq232h l + E f _bu ∗ E_δa + h2h ∗ L L4 δa 32 Similarly, when

h 31 h

(4.43)

< 0, then,

h 32 h

> 0, from (4.42)

(4.44)

4.3 Multiple-impulse Optimal Rendezvous

⎧ δθ E_δa F ≤ 2hq232h l + E f _bu ∗ E_δa + h2h ∗ L L1 ⎪ ⎪ 32 ⎪ δθδaF q1 h ⎨ h E_δa ≤ 2h 31 l + E f _bu ∗ E_δa − 2h 31 ∗ L L2 δa q1 h δθ F h E_δa ⎪ ⎪ δa ≤ − 2h 31 l + E f _bu ∗ E_δa − 2h 31 ∗ L L3 ⎪ ⎩ δθ F q2 h E_δa ≤ − 2h 32 l + E f _bu ∗ E_δa + h2h ∗ L L4 δa 32

175

(4.45)

The distribution region satisfying the optimal four-impulse rendezvous mode can be determined by Eqs. (4.44) and (4.45). When satisfying Eq. (4.44), the boundary of the optimal rendezvous mode is determined by the maximum value of the four subformulas on the right side of Eq. (4.44). When satisfying Eq. (4.45), the boundary of the optimal rendezvous mode is determined by the minimum value of the four sub-formulas on the right side of Eq. (4.45). 2) Three-Impulse Optimal Rendezvous Mode with Drift For the optimal four-impulse rendezvous mode with first impulse amplitude being zero, the characteristic phase angle is recorded as δθ F42 , the rendezvous time corresponding to the first two impulses are τ1 and τ2 , and the rendezvous time is τ F4 . When the rendezvous time is τ F3 , denoting τ43 = τ F4 −τ F3 . The three-impulse optimal rendezvous mode with initial drift exists when τ F3 is between τ F4 + τ1 − τ2 and τ F4 , and the characteristic phase angle satisfies δθ F = δθ F42 − 0.75τ43 δa − e sin(τ43 + f ) + e sin f . The left boundary of three impulse optimal rendezvous mode with initial drift is τ F4 + τ1 − τ2 in abscissa, and the ordinate satisfies δθ F42 δθ F = − 0.75(τ2 − τ1 ) − E_δa sin(τ2 − τ1 + f ) + E_δa sin f δa δa

(4.46)

Its right boundary is the left boundary of the four-impulse optimal rendezvous mode. The lower boundary of the three-impulse optimal rendezvous mode with initial drift is a curve starting from the lowest point (the characteristic phase angle is δθ F4+ ) of the four-impulse optimal rendezvous mode above the abscissa axis as δθ F δθ F4+ = − 0.75(τ F4 − τ F3+ ) − E_δa sin(τ F4 − τ F3+ + f ) + E_δa sin f δa δa (4.47) where τ F3+ lies between τ F4 + τ1 − τ2 and τ F4 . Similarly, for the three-impulse optimal rendezvous mode with terminal drift, the characteristic phase angle of the corresponding four-impulse optimal rendezvous mode with the fourth impulse amplitude being zero is δθ F43 . For the third and fourth impulses in this rendezvous mode, the corresponding rendezvous time are τ3 and τ4 . Then, the right boundary of the distribution is the four-impulse optimal rendezvous mode, and the left boundary whose abscissa of τ F4 +τ3 −τ4 . And its ordinate satisfies δθ F δθ F43 = + 0.75(τ4 − τ3 ) + E_δa sin(τ4 − τ3 + f ) − E_δa sin f δa δa

(4.48)

176

4 Guidance Method and Schematic Design for Rendezvous and Docking

The upper boundary is a curve starting from the highest point in the four-impulse optimal rendezvous mode below the abscissa axis, and it satisfies δθ F δθ F4− = + 0.75(τ F4 − τ F3− ) + E_δa sin(τ F4 − τ F3− + f ) − E_δa sin f δa δa (4.49) where δθ F4− is the characteristic phase angle corresponding to the highest point of the four-impulse optimal rendezvous mode under the abscissa axis, and τ F3− lies between τ F4 + τ3 − τ4 and τ F4 . 3) Three-Impulse Optimal Rendezvous Mode The calculation of three-impulse optimal rendezvous mode is complex, and its distribution area can be determined by other optimal rendezvous modes. The left boundary of the three-impulse optimal rendezvous mode is the right boundary of the twoimpulse optimal rendezvous mode. The right boundary is the left boundary of the three-impulse optimal rendezvous mode with drift. The lower boundary above the abscissa axis is the two-impulse optimal rendezvous mode with initial drift, and the upper boundary below the abscissa axis is the two-impulse optimal rendezvous mode with terminal drift. 4) Two-Impulse Optimal Rendezvous Mode with and without Drift From the simulation, we can see that the two-impulse optimal rendezvous mode only holds for a short rendezvous time, and for given δθδaF , there exists a maximum rendezvous time satisfying the two-impulse optimal rendezvous mode. By searching the maximum rendezvous time, the distribution area of the two-impulse optimal rendezvous can be determined. The solution of optimal two-impulse rendezvous mode distribution with drift is based on the solution of optimal two-impulse rendezvous mode. For the optimal twoimpulse rendezvous mode with initial drift, it is necessary to add the judgment of the primer vector at the first impulse time on the basis of the judgment of the optimal two-pulse rendezvous mode, that is, to judge whether the derivative of the primer vector modulus at that time is zero. If the derivative is zero, there exists a two-pulse optimal rendezvous mode with initial drift. By solving the maximum rendezvous time which satisfies | p| = 1 and is less than τ1 , the corresponding rendezvous time τ− , at the beginning of initial drift, can be obtained. Then, the region where the optimal rendezvous mode with initial drift is established satisfies the curve beam: δθ F δθ F2 = + 0.75(τ1 − τ ) + E_δa sin(τ1 − τ + f ) − E_δa sin f δa δa

(4.50)

where δθ F2 is the characteristic phase angle in the corresponding two-impulse optimal rendezvous mode τ ∈ (τ− , τ1 ). For the optimal two-impulse rendezvous mode with terminal drift, it is necessary to add the judgment of the primer vector of the second impulse time on the basis of

4.3 Multiple-impulse Optimal Rendezvous

177

the judgment of the optimal two-impulse rendezvous mode, that is, to judge whether the derivative of the primer vector modulus of the second impulse time is zero. If the derivative is zero, there exists a two-impulse optimal rendezvous mode with terminal drift. By finding the minimum rendezvous time which satisfies | p| = 1 and is larger than τ2 , the corresponding rendezvous time is τ+ at the end of terminal drift can be obtained. Then, the region where the two-impulse optimal rendezvous mode with terminal drift is established satisfies the curve beam: δθ F2 δθ F = − 0.75(τ − τ2 ) − E_δa sin(τ − τ2 + f ) + E_δa sin f δa δa

(4.51)

where δθ F2 is the characteristic phase angle in the corresponding two-impulse optimal rendezvous mode τ ∈ (τ2 , τ+ ). If drift exists both at the beginning and the end, it satisfies optimal rendezvous mode with drift at both beginning and ending. The two-impulse optimal rendezvous mode with drift at both beginning and ending is between the two-impulse optimal rendezvous modes with initial drift and terminal drift. 5) Optimal Rendezvous Mode Distribution The distribution of optimal rendezvous modes in the rendezvous of an elliptical orbit spacecraft to a circular orbit spacecraft is simulated and analyzed, which consists of two parts: ➀ Comparing the distribution of optimal rendezvous modes between an elliptical orbit spacecraft and an near circular orbit spacecraft. ➁ Comparing the similarities and differences of optimal rendezvous modes’ distribution at different true anomalies under the same E_δa. (1) Comparing with the Mode Distribution in Near Circular Orbits Optimal Rendezvous The distribution of optimal rendezvous mode between elliptical orbit spacecraft and circular orbit spacecraft is compared with the mode distribution in near circular orbit optimal rendezvous shown in Fig. 4.5. Figure 4.7a, b give the optimal rendezvous mode distribution of elliptical orbit spacecraft to circular orbit spacecraft rendezvous when f = 0◦ , E_δa = 0.1 and mode distribution in near circular orbits optimal rendezvous. The abscissa of the two distributions are the rendezvous time, and the unit is the period of the reference orbit. β − 0.75τ F − E_δa sin(τ F + f ) + E_δa sin f , The ordinate of Fig. 4.7a is δθδaF = δa δθ F and the ordinate of Fig. 4.7b is δ R = δβR − 0.75τ F . Figure 4.7a, b show the optimal rendezvous modes of two, three, and four impulse for the number 2, 3, and 4, respectively. The + sign before digital indicates the optimal rendezvous mode with initial drift, and the one after indicates the optimal rendezvous mode with terminal drift. From the optimal rendezvous mode distributions of Fig. 4.7a, b, it can be seen that ➀ the overall distribution structure of two optimal rendezvous modes is roughly the same, and there exist four-impulse, threeimpulse, two-impulse and two-impulse optimal rendezvous modes, whose distributions are also roughly the same; ➁ by enlarging Fig. 4.7a, we can see that the

178

4 Guidance Method and Schematic Design for Rendezvous and Docking Distribution of optimal rendezvous modes

Distribution of optimal rendezvous modes 40

40

30

30 2

4

3

+3

δθF/δR

10 +2

10

0

+2+

−10 2+ −20 2 −30

20 δθF/δR

20

3

3+ 4

4

2 +2

3

+3 Hohmman rendezvous

0

−10

2+

−20

2

3

3+

4

−30

−40 −40 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period) (a)

0.5 1 1.5 2 2.5 Rendezvous time (reference orbital period) (b)

Fig. 4.7 Rendezvous mode distribution of near circular orbits, a f = 0◦ , E_δa = 0.1, rendezvous mode distribution, b Rendezvous mode distribution of near circular orbit

distribution of optimal rendezvous modes between elliptical orbit spacecraft and circular orbit spacecraft is not symmetric as the near circular orbit rendezvous; ➂ in the region between the optimal rendezvous mode of two-impulse with initial drift and the optimal rendezvous mode of two-impulse with terminal drift, it is Hohmann type optimal rendezvous mode in near circular orbit rendezvous, while in the rendezvous of elliptical orbit spacecraft to circular orbit spacecraft, it is the optimal rendezvous mode of two-impulse with both initial and terminal drift. The consuming fuel is not all 0.5δa, the time interval between the two impulses is not all half of the orbital period, and there is no symmetry about the abscissa axis; ➃ in terms of free-flying of trajectory, the curve of variation of ordinate with time in the rendezvous of near circular orbits is one of the two lines determined by Eqs. (4.52) and (4.53): δθ F N δθ F = − 0.75τ F N δR δR

(4.52)

δθ F N δθ F = + 0.75τ F N δR δR

(4.53)

where δθ F N is the characteristic phase angle before drift in the rendezvous of close circular orbits, and τ F N is the drift time. In the rendezvous of an elliptical orbit spacecraft to a circular orbit spacecraft, the drift curve is one of the curves determined by Eqs. (4.54) and (4.55): δθ F e e δθ F N = − 0.75(τ F − τ N ) − sin(τ F − τ N + f ) + sin f δa δa δa δa

(4.54)

δθ F N e e δθ F = + 0.75(τ F − τ N ) + sin(τ F − τ N + f ) − sin f δa δa δa δa

(4.55)

4.3 Multiple-impulse Optimal Rendezvous

179

where δθ F N is the characteristic phase angle before drift in the rendezvous of elliptical orbit spacecraft to circular orbit spacecraft, and τ N is the drift time. ➀In the two types of rendezvous, the boundary of each optimal rendezvous mode is also different. In the rendezvous of elliptical orbit spacecraft to circular orbit spacecraft, the distribution boundary is related to the true anomaly of the elliptical orbit spacecraft. (2) Comparison between Elliptical Orbit Spacecraft’s Optimal Rendezvous Mode Distribution with Different True Anomaly Figure 4.8 gives the optimal rendezvous mode distributions of elliptical orbit spacecraft to circular orbit spacecraft when E_δa = 0.1 and the true anomalies f of elliptical orbit spacecraft are f = 30◦ , f = 90◦ , f = 180◦ and f = 330◦ ,. In these figures, the abscissa is the rendezvous time, whose unit is the period of the reference orbit, and the ordinate is δθ F β = − 0.75τ F − E_δa sin(τ F + f ) + E_δa sin f δa δa 40

Distribution of optimal rendezvous modes

30

2

δθF/δR

20 B

10 0 −10 −20 −30

+2

3

+3

4

F D E +2+ G 3+

4

−40 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period) (a) Distribution of optimal rendezvous modes 50 40 2 10 +2

δθF/δR

3

+3

4

10

2 +2

3

4

+3

+2+

0 −10 2+ −20 2 −30

3

3+

4

−40 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period) (b) Distribution of optimal rendezvous modes 40

3

3+

2 10 +2 20

3

+3

−10 4

−40 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period) (c)

Fig. 4.8 Distribution of optimal rendezvous mode

−20

4 +2+

0

+2+

0 −10 2+ −20 2 −30

20

30

30 20

Distribution of optimal rendezvous modes

30 3

A C 2+ 2

40

2+ 2

3

3+ 4

−30 −40 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Rendezvous time (reference orbital period) (d)

180

4 Guidance Method and Schematic Design for Rendezvous and Docking

Figure 4.8a–d show the optimal rendezvous modes of two, three, and four impulse with numbers 2, 3, and 4, respectively. The + sign before digital indicates the optimal rendezvous mode with initial drift, and the one after indicates the optimal rendezvous mode with terminal drift. From Figs. 4.6 and 4.8, it can be seen that when E_δa = 0.1, the distribution of optimal rendezvous modes of spacecraft in the same elliptical orbit is roughly the same at different true anomalies. However, comparing the specific boundaries of different optimal rendezvous modes, we can see that the boundaries are different for the rendezvous problems at different true anomalies. As shown in Fig. 4.8a, points A, B, and C are three special points on the boundary of two-impulse optimal rendezvous mode with and without drift. Points D and E are the intersections of three optimal rendezvous modes: three-impulse with drift, two-impulse with drift and three-impulse. Points F and G are the two vertices of four-impulse optimal rendezvous mode. Table 4.1 gives the coordinate of the above special points in the optimal rendezvous mode distribution, when E_δa = 0.1 and f , respectively, satisfy f = 30◦ , f = 90◦ , f = 180◦ , f = 330◦ . From Table 4.1, it can be seen that under the same E_δa = 0.1, the distribution of the optimal rendezvous modes of elliptical orbit spacecraft to circular orbit spacecraft is very similar when the true anomalies f are different. However, the boundaries of each mode are different, for example, for different f , the symbols of B’s ordinates on the distribution may change. The distribution of optimal rendezvous modes in the rendezvous of elliptical orbit spacecraft to circular orbit spacecraft is related not only to τ F and β/δa but also to f and E_δa as show in Figs. 4.6 and 4.8 Similar optimal rendezvous modes can be obtained when E_δa < 0.1. When E_δa > 0.1, the distribution may be different from the above-mentioned distribution modes. It is necessary to further study the distribution of optimal rendezvous mode in elliptic orbit rendezvous with large eccentricity.

4.3.3 Guidance Law from Multiple-impulse Optimal Rendezvous According to the previous analysis, Hohmann rendezvous is a fuel-saving rendezvous mode when the initial, terminal states and rendezvous time can be freely designed. In the following, based on the distribution of optimal impulse rendezvous modes, a design method of Hohmann rendezvous guidance law with drift is presented. 1. Analysis of Fuel Consumed by Hohmann Rendezvous If the orbits before and after the orbit transfer are circular, whose radius ratio is not more than 11.4, Hohmann orbit transfer is the most fuel-saving.

A

(0.42, 0.43)

(0.43, 0.46)

(0.44, 0.74)

(0.43, 1)

(0.41, 0.5)

f \points

f = 0°

f = 30°

f = 90°

f = 180°

f = 330°

(0.48, −0.19)

(0.5, 0.2)

(0.53, 0.06)

(0.52, −0.2)

(0.49, −0.27)

B

(0.41, −0.67)

(0.44, −0.62)

(0.44, −0.96)

(0.42, −0.94)

(0.41, −0.81)

C

(1.16, 3.27)

(1.15, 3.6)

(1.16, 3.42)

(1.17, 3.2)

(1.16, 3.27)

D

Table 4.1 Coordinate of special points in distribution with different f as E_δa = 0.1 E

(1.15, −3.73)

(1.16, −3)

(1.17, −3.1)

(1.16, −3.5)

(1.15, −3.72)

F

(1.66, 5.3)

(1.61, 5.8)

(1.66, 5.9)

(1.68, 5.7)

(1.67, 5.5)

G

(1.61, −5.82)

(1.64, −5.4)

(1.68, −5.6)

(1.65, −5.95)

(1.63, −5.92)

4.3 Multiple-impulse Optimal Rendezvous 181

182

4 Guidance Method and Schematic Design for Rendezvous and Docking

2. Guidance Law of Hohmann Rendezvous with Drift Among the optimal rendezvous mode distribution, there is a class of optimal rendezvous modes related to Hohmann orbit transfer. When the rendezvous time is longer than half of the orbital period, i.e. τ F > π , drift is necessary to make the phase angle difference between the two spacecraft satisfy the requirement β = 0.75π δ R. When β = 0.75π δ R and the rendezvous time τ F > π , the Hohmann rendezvous mode with terminal drift is satisfied. The two impulse times are 0 and π , and the extra time τ F − π is the terminal drift time. There is a linear relationship between the characteristic phase angle δθ F and the rendezvous time τ F : δθ F = −0.75δ R(τ F − π )

(4.56)

When β = 1.5τ F δ R − 0.75π δ R and τ F > π , after target drift τ F − π , the phase angle difference between the target and the target becomes β = 0.75π δ R. This case satisfies the Hohmann rendezvous mode with initial drift, and the two impulse times are τ1 = τ F − π, τ2 = τ F . There is a linear relationship between the characteristic phase angle δθ F and the rendezvous time τ F : δθ F = 0.75δ R(τ F − π )

(4.57)

When the rendezvous time τ F > π and the initial phase angle difference satisfies 0.75π δ R < β < 1.5τ F δ R −0.75π δ R, the Hohmann optimal rendezvous mode with both initial and terminal drift is the optimal one. As shown in Fig. 4.9, the mode lies between the lines represented by Eqs. (4.56) and (4.57) (Shadowed part). In these rendezvous modes, the amplitude of two impulses are V1 = V2 = 0.25δ R, whose time interval is π , and the phase angle difference between the two spacecraft is 0.75π δ R when the first impulse is applied. The Hohmann rendezvous is performed by means of adding initial drift, terminal drift, and both initial and terminal drift to realize the rendezvous task. Hohmann optimal rendezvous mode has four kinds of rendezvous modes, in which the amplitude of two impulse is V1 = V2 = 0.25δ R, whose direction is tangential. Fig. 4.9 Hohmann rendezvous mode

δθF δR

with initial drift Hohmann rendezvous π τF with terminal drift

4.3 Multiple-impulse Optimal Rendezvous

183

(1) Hohmann rendezvous mode: β = 0.75π δ R, τ F = π , the times of two impulse are 0 and π . (2) Hohmann rendezvous mode with terminal drift: β = 0.75π δ R, τ F > π , the times of two impulse are 0 and π , and the extra time τ F − π is the terminal drift time. There is a linear relationship between the characteristic phase angle δθ F and the rendezvous time τ F as showed in Eq. (4.56). (3) Hohmann rendezvous mode with initial drift: β = 1.5τ F δ R − 0.75π δ R, τ F > π , after target’s drift τ F − π , the phase angle difference between the target and the target becomes β = 0.75π δ R. The times of two impulses are τ F −π and τ F . There is a linear relationship between the characteristic phase angle δθ F and the rendezvous time τ F as showed in Eq. (4.57). (4) Hohmann rendezvous mode with both initial and terminal drift:τ F > π , 0.75π δ R < β < 1.5τ F δ R − 0.75π δ R. The time corresponding to the intersection of the line determined by Eq. (4.56) and the line determined by Eq. (4.57) is denoted as τ F0 ; then, it is known that τ F0 − π is the initial drift time in the rendezvous mode from Fig. 4.9. When τ F > τ F0 , τ F − τ F0 is the terminal drift time, τ F0 and times of two impulse τ1 , τ2 can be obtained: τ F0 =

β + 0.75π δ R , τ1 = τ F0 − π, τ2 = τ F0 1.5δ R

(4.58)

The optimal Hohmann rendezvous with drift is to obtain the phase angle difference and rendezvous time that Hohmann rendezvous should satisfy through free-flying. The time interval between two impulses is half the period of reference orbit. When the first impulse is applied, the phase angle difference between two spacecraft is fixed, and satisfies β H = 0.75π δ R, and the fuel consumption is 0.75δ R.

4.4 CW Guidance 4.4.1 Two-Impulse CW Guidance Law CW guidance is based on the CW equation, and usually adopts the two-impulse CW guidance strategy. In near range, the CW equation is used to describe the relative motion with high accuracy in short period. CW equation is linear, and its in-plane motion and out-of-plane motion are decoupled. Thus, simple and effective guidance law suitable for in-orbit operation can be designed, which has been widely used in engineering. While analyzing, the design of in-plane guidance law and out-of-plane guidance law are generally separated. 1. In-Plane Guidance Law When the initial conditions are known, the analytic solution of motion in the plane of the orbit can be obtained using two-impulse control.

184

4 Guidance Method and Schematic Design for Rendezvous and Docking



   x(t) ∂ 2 Ω x(t) ˙ Denote ρ(t) = ˙ = 2 , ρ(t) z(t) ∂u z˙ (t) Solution to CW equation: 

        A(t) B(t) ρ(t0 ) B(t) 0 ρ(t f ) = + v(t0 ) + ˙ f) ˙ 0) ρ(t D(t) C(t) D(t) ρ(t v(t f )

(4.59)

where 

 1 6 sin(ωoT t) − 6ωoT t 0 4 − 3 cos(ωoT t)   4 sin(ωoT t)/ωoT − 3t −2/ωoT + 2 cos(ωoT t)/ωoT B(t) = 2/ωoT − 2 cos(ωoT t)/ωoT sin(ωoT t)/ωoT   0 −6ωoT + 6ωoT cos(ωoT t) C(t) = 0 3ωoT sin(ωoT t)   4 cos(ωoT t) − 3 −2 sin(ωoT t) D(t) = cos(ωoT t) 2 sin(ωoT t) A(t) =

˙ 0 )], the control impulses With the known initial position and velocity [ρ(t0 ), ρ(t are sought to be imposed on the initial and termination time, respectively, to make ˙ 0 )] within a given time t = t f − t0 . the relative position and velocity reach [ρ(t0 ), ρ(t When 8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t) = 0,B(t) is inversible, the solution of two-impulse control is ˙ 0) v(t0 ) = B −1 (t)[ρ(t f ) − A(t)ρ(t0 )] − ρ(t

(4.60)

˙ f ) - C(t)ρ(t0 ) - D(t)ρ(t ˙ 0 ) - D(t)v(t0 ) v(t f ) - ρ(t

(4.61)

where 1 B−1 (t) = 8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t)   −2ωoT (cos(ωoT t) − 1) ωoT sin(ωoT t) 2ωoT (cos(ωoT t) − 1) ωoT (4 sin(ωoT t) − 3ωoT t) It can be further inferred that  v(t0 ) =

Vx Vz 

ωoT +  v(t f ) =

x˙ f z˙ f



 =−

x˙0 z˙ 0



(x f − x0 ) sin(ωoT t) + z 0 (14 cos(ωoT t) + 6ωoT t sin(ωoT t) − 14) + 2z f (1 − cos(ωoT t)) 2(x f − x0 )(cos(ωoT t) − 1) + z 0 (3ωoT t cos(ωoT t) − 4 sin(ωoT t)) + z f (4 sin(ωoT t) − 3ωoT t)



8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t)



4.4 CW Guidance  ωoT −

185

(x f − x0 ) sin(ωoT t) + z f (14 cos(ωoT t) + 6ωoT t sin(ωoT t) − 14) + 2z 0 (1 − cos(ωoT t)) 2(x f − x0 )(1 − cos(ωoT t)) + z f (4 sin(ωoT t) − 3ωoT t cos(ωoT t)) + z 0 (3ωoT t − 4 sin(ωoT t)) 8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t)

When 8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t) 0, 2π, 2.814π, 4π, 4.891π, 6π, 6.923π . . ..



(4.62) =

0,

ωoT t

=

2. Out-of-Plane Guidance Law Out-of-plane equations are ( y˙ +V )

y f = 0 ωoT y0 sin(ωoT t) + y0 cos(ωoT t) y˙ f = ( y˙0 + Vy0 ) cos(ωoT t) − y0 ωoT sin(ωoT t) + Vy f When sin(ωoT t) = 0, we can get ω (y −y cos(ω t))

Vy0 = oT fsin(ω0 oT t) oT − y˙0 ω (y −y cos(ω t)) cos(ωoT t) Vy f = y˙ f − oT f 0 sin(ωoToTt) + y0 ωoT sin(ωoT t) When, sin(ωoT t) = 0, it satisfies ωoT t = π, 2π, 3π, 4π . . ..

4.4.2 Rendezvous Time and Fuel Consumption Section 4.4.1 shows that the guidance impulse of CW two-impulse guidance is related not only to the initial relative state and the terminal relative state but also to the rendezvous time. When the rendezvous time is close enough to satisfy 8 − 3ωoT t sin(ωoT t) − 8 cos(ωoT t) = 0 or sin(ωoT t) = 0, the guidance impulse is very large, and the closer to zero, the faster the amplification is. Outside the orbit plane, when the rendezvous time satisfies sin(ωoT t) = 0, the guidance impulse is very large with the denominator being zero. 1. Analysis of Fuel Consumption of the In-Plane Guidance ˙ 0 )] = [3600 13500 − 23.23 0]T , When the initial and terminal states are [ρ(t0 ), ρ(t T ˙ f )] = [5000 0 0 0] , respectively, the relation between the fuel[ρ(t f ), ρ(t consumption (expressed as increment of velocity) and rendezvous time is shown in Fig. 4.10. = When the rendezvous time satisfies ωoT t 0, 2π, 2.814π, 4π, 4.891π, 6π, 6.923π or is near these points, the velocity increment increases sharply. When the value of the Z axis is zero in the initial and terminal state, the velocity increment is nonsingular when ωoT t = 2π, 4π, 6π . . .. For example, the relationship between fuel consumption (expressed as increment of velocity) and rendezvous time

186

4 Guidance Method and Schematic Design for Rendezvous and Docking

Velocity increment (m/s)

250

200

150

100

50

2

4

6

8 10 12 Rendezvous time (rad)

14

16

18

Fig. 4.10 Relation between velocity increment and time within CW two-impulse guidance

is shown in Fig. 4.11, with the initial and terminal state being [ρ(t0 ), ρ(t ˙ 0 )] = ˙ f )] = [400 0 0 0]T . When ωoT t = 0, 2.814π, 4.891π [5000 0 0 0]T and [ρ(t f ), ρ(t or is close (Here ωoT t = 2π ), the increment of velocity increases sharply.

50

Velocity increment (m/s)

45 40 35 30 25 20 15 10 5 0 2

4

6

8 10 12 14 Rendezvous time (rad)

16

18

Fig. 4.11 Relation between velocity increment and time within CW two-impulse guidance

4.4 CW Guidance

187

45

Velocity increment (m/s)

40 35 30 25 20 15 10 5 0 2

4

6

14 12 10 8 Rendezvous time (rad)

16

18

Fig. 4.12 Relation between velocity increment and time of CW two-impulse guidance

For the same example, when the rendezvous time is selected as an integer orbital period, it is easy to make the computed impulse very large when there are errors in ˙ 0 )] = [5000 10 0 0]T navigation. As shown in Fig. 4.12, with the initial [ρ(t0 ), ρ(t T and terminal state [ρ(t f ), ρ(t ˙ f )] = [400 0 0 0] , as long as the Z-axis direction of the initial or terminal state has position deviation, when the rendezvous time is ωoT t = 2π, 4π, 6π . . ., the increment of velocity will increase sharply. From Eq. (4.62), we can see that when ωoT t = 2π, 4π, 6π . . ., the denominator of the velocity increment is zero, and the molecule is also zero when the values of Z-axis at the initial and terminal state are zero. As long as the values of Z-axis at the initial or the terminal state are not zero, the molecule must be non-zero and the velocity increment will increase sharply. 2. Analysis of fueL Consumption of the Out-of-Plane Guidance When the initial and terminal states are [ y0 y˙0 ] = [ 1000 0.2 ]T and [ y f y˙ f ] = [ 0 0 ]T , respectively, the relation between the fuel consumption (expresses as the increment of velocity) and rendezvous is shown in Fig. 4.13. The velocity increment increases sharply when ωoT t = π, 2π, 3π, 4π . . .

4.4.3 Selection of Rendezvous Time From the analysis of the previous two sections, it can be seen that the CW twoimpulse guidance strategy is simple and feasible. For near-range rendezvous, the accuracy of the description of CW equation is relatively high, and the computational

188

4 Guidance Method and Schematic Design for Rendezvous and Docking 20 18 Velocity increment (m/s)

16 14 12 10 8 6 4 2 0 2

4

6

8 10 12 14 Rendezvous time (rad)

16

18

Fig. 4.13 Relation between velocity increment and time of CW two-impulse guidance (out-of-plane guidance)

complexity of using the CW two-impulse guidance strategy is small, which is very suitable for autonomous computation on satellite. However, matrix inversion or division by the sinusoidal function are involved in the process of CW two-impulse guidance, which makes it singular when some parameters are selected as the rendezvous time. In the near-range rendezvous phase, the rendezvous time is generally required to not be long, for example, it should be no more than one orbital period or two orbital periods. To improve the accuracy of CW two-impulse guidance, the correction impulse is usually added. To reduce the singularity probability of solving the correction impulse, the rendezvous time is usually chosen not to exceed one orbit period if there only exists in-plane control at that time. If the out-of-plane control is included, the rendezvous time is chosen not more than half of the orbit period. From Figs. 4.10, 4.12 to 4.13, we can see that, when CW two-impulse guidance is adopted, the increment of velocity will increase sharply with rendezvous time being close to zero.

4.5 Light of Slight Guidance

189

4.5 Light of Slight Guidance 4.5.1 Guidance Law Light of slight guidance ensures that the target is always facing the target, which is suitable for autonomous rendezvous and docking in close range. Two purposes of light of sight guidance are (1) to make the rotational velocity of relative distance zero in space, and (2) to make the velocity of relative distance negative in light of sight direction. The control of the angular velocity along the light of sight rotation is called lateral control, and the control along the light of sight is called longitudinal control. Line of sight guidance designs control law for dynamic equation based on line-of sight frame. The dynamic equation can be seen in Eqs. (2.76) and (4.63). 1. Lateral Control Lateral control uses switch control, which aims at eliminating the angular velocity of light of sight rotation. In Fig. 4.14, ωon and ωo f f are thresholds of switch, which are determined by measurement accuracy, switching-on times, propellant consumption, and minimum working time. 1. Longitudinal Control Based on the lateral control, the purpose of longitudinal control is to make the two spacecraft approach each other step by step according to the given trajectory or corridor. After eliminating the line of sight rotation, the first formula of Eq. (4.63) can be approximately obtained as follows: ρ¨ = aξ , which is a cascade of second-order integral. Because the actuator is the switching mode, the sliding mode is designed with variable structure control strategy as ˙ + k1 ρ + k2 ρ˙ Sρ = ρ˙ 2 sgn(ρ)

(4.63)

Fig. 4.14 Dead zone control of rotating angular velocity F ωoff

ωon ω

−ωon −ωoff −F

190

4 Guidance Method and Schematic Design for Rendezvous and Docking

aξ = −Aξ sgn(Sρ )

(4.64)

4.5.2 Stability Analysis 1. Sliding Mode’s Existing Region of Lateral Control and Stability Analysis 1) Sliding Mode’s Existing Region of Lateral Control In the lateral control system, the sliding mode is Sβ = kρ β˙ + β, k > 0, and the control law is aη = −Aη sgn(Sβ ). Based on the accessibility condition Sβ S˙β < 0, conditions of the sliding mode existence can be found. The derivative of Sβ is ˙ β˙ + ωoT )) S˙β = k ρ˙ β˙ + kρ β¨ + β˙ = (k ρ˙ + 1)β˙ + k(aη − 2ρ(

(4.65)

(1) When Sβ > 0, S˙β < 0, aη = −Aη , there is ˙ β˙ + ωoT )) < 0 S˙β = (k ρ˙ + 1)β˙ + k(−Aη − 2ρ( Since k > 0, there is Aη >

k ρ˙ + 1 β˙ − 2ρ( ˙ β˙ + ωoT ) k

(4.66)

(2) When Sβ < 0, S˙β > 0, aη = Aη , there is ˙ β˙ + ωoT )) > 0 S˙β = (k ρ˙ + 1)β˙ + k(Aη − 2ρ( Then, Aη > −

k ρ˙ + 1 β˙ + 2ρ( ˙ β˙ + ωoT ) k

(4.67)

From Eq. (4.67) to (4.68), when



k ρ˙ + 1



Aη > − β˙ + 2ρ( ˙ β˙ + ωoT )

k

(4.68)

is satisfied, the sliding mode of lateral control system exists. 2) Stability Analysis of Lateral Control (1) The stability of the closed-loop system outside sliding mode is analyzed at first. Let the Lyapunov function be

4.5 Light of Slight Guidance

191

V = 1 + Sβ2

(4.69)

V˙ = 2Sβ S˙β

(4.70)

Then V > 0, and

Because when Sβ < 0, S˙β > 0, and when Sβ > 0, S˙β < 0, therefore V˙ = 2Sβ S˙β < 0 always holds. Based on the Lyapunov theorem, the lateral control system outside the sliding mode is asymptotically stable. (2) On the sliding mode, Sβ = 0, S˙β = 0, the equivalent control can be acquired, as follows u eg = aη = −

k ρ˙ + 1 β˙ + 2ρ( ˙ β˙ + ωoT ) k

(4.71)

and the equivalent system is ˙ =0 ρ β¨ + (k ρ˙ + 1)β/k

(4.72)

If Lyapunov function is selected as 1 ˙ 2 Vβ = 1 + k(ρ β) 2

(4.73)

Then Vβ > 0, and we can further obtain ˙ β¨ + ρ˙ β) ˙ = ρ β(− ˙ β) ˙ = −ρ(β) ˙ 2 V˙β = kρ β(ρ

(4.74)

Because ρ ≥ 0, V˙β ≤ 0, the equal situation holds only when ρ = 0 or β˙ = 0. Thus, the equivalent closed-loop system is asymptotically stable. 2. Sliding Mode’s Existing Region of Longitudinal Control and Stability Analysis 1) Sliding Mode’s Existing Region of Longitudinal Control ˙ + k1 ρ + k2 ρ, ˙ The sliding mode of the longitudinal control system is Sρ = ρ˙ 2 sgn(ρ) k1 > 0, k2 > 0, and the control law is aξ = −Aξ sgn(Sρ ). Based on the accessibility conditions Sρ S˙ρ < 0, conditions of the sliding mode existence can be found. The derivative of Sβ is ˙ + k2 )(ρωζ2 + aξ ) + k1 ρ˙ S˙ρ = (2ρ˙ sgn(ρ)

(4.75)

192

4 Guidance Method and Schematic Design for Rendezvous and Docking

(1) When Sρ > 0, S˙ρ < 0, aξ = −Aξ , there is ˙ + k2 )(ρωζ2 + aξ ) + k1 ρ˙ < 0 S˙ρ = (2ρ˙ sgn(ρ) Because 2ρ˙ sgn(ρ) ˙ + k2 > 0, we get aξ < −

k1 ρ˙ − ρως2 (2ρ˙ sgn(ρ) ˙ + k2 )

(4.76)

Then, Aξ >

k1 ρ˙ + ρως2 (2ρ˙ sgn(ρ) ˙ + k2 )

(2) When Sρ < 0, S˙ρ > 0, aξ = Aξ , there is ˙ + k2 )(ρωζ2 + aξ ) + k1 ρ˙ > 0 S˙ρ = (2ρ˙ sgn(ρ) Because 2ρ˙ sgn(ρ) ˙ + k2 > 0, there is Aξ > −

k1 ρ˙ − ρως2 (2ρ˙ sgn(ρ) ˙ + k2 )

(4.77)

From Eqs. (4.76) to (4.77), when





k1 ρ˙ 2

+ ρως

Aξ >

(2ρ˙ sgn(ρ) ˙ + k2 )

(4.78)

is satisfied, the sliding mode of longitudinal control system exists. 2) Stability Analysis of Longitudinal Control (1) The stability of the closed-loop system outside sliding mode is analyzed at first. Let the Lyapunov function be V = 1 + Sρ2

(4.79)

V˙ = 2Sρ S˙ρ

(4.80)

Then V > 0, and

Because when Sρ < 0, S˙ρ > 0, and when Sρ > 0, Sρ < 0, therefore V˙ = 2Sρ S˙ρ < 0 always holds. Based on Lyapunov theorem, the longitudinal control system outside the sliding mode is asymptotically stable.

4.5 Light of Slight Guidance

193

(2) In the sliding mode, Sρ = 0, S˙ρ = 0, the equivalent control can be acquired u eg = aς = −

k1 ρ˙ − ρως2 (2ρ˙ sgn(ρ) ˙ + k2 )

(4.81)

k1 ρ˙ (2ρ˙ sgn(ρ) ˙ + k2 )

(4.82)

and the equivalent system is ρ¨ = −

If Lyapunov function is selected as 1 Vρ = 1 + ρ˙ 2 > 0 2

(4.83)

Then, V˙ρ = ρ˙ ρ¨ = −

k1 ρ˙ 2 (2ρ˙ sgn(ρ) ˙ + k2 )

(4.84)

Because k1 > 0, 2ρ˙ sgn(ρ) ˙ + k2 > 0, V˙ρ ≤ 0 and the equal situation holds only when ρ˙ = 0. Thus, the equivalent closed-loop system is asymptotically stable. Considering the above two points, the longitudinal control closed-loop system is asymptotically stable.

4.6 Scheme Design for Far Range Rendezvous Phase In the far range rendezvous phase, the trajectory control method is usually used to guide the target to a predetermined position near the target. The main tasks of the far range rendezvous phase are to improve the orbital height of the target, to adjust the phase angles difference between the two spacecraft, to shorten the relative distance, and to eliminate the deviation of the orbital plane of the two spacecraft. From the development of rendezvous and docking, the development of far range rendezvous strategy has gone through three stages: small phase angle docking guaranteed by rocket launching in the early stage, 2–3 days far range rendezvous strategy in mature stage, and short rendezvous guidance in quick rendezvous and docking in these years. A 2–3 day guidance scheme and a short rendezvous and docking guidance scheme are given.

194

4 Guidance Method and Schematic Design for Rendezvous and Docking

4.6.1 2–3 Day Guidance Scheme 1. Guidance Scheme Five orbital transfer impulses are adopted to conduct far range rendezvous, which guide the target from the initial orbit to the orbit required by the starting point of autonomous control phase, and they are described in Table 4.2. The above maneuver can also be divided into in-plane and out-of-plane one. 1) In-Plane Maneuver The orbit adjustment in the orbital plane (Fig. 4.15) includes raising the orbital height and adjusting the phase angle difference between the two spacecraft. The target transfers through one or more intermediate transitional orbits to improve the orbital height and adjusts the phase angle by adjusting the flight time in the different Table 4.2 Two-to three-day guidance scheme Number

Orbit transfer point

Maneuvering direction

Goal

1

Apogee

Tangential

Ascend perigee

2

The argument of latitude not fixed

Normal

Orbital plane correction

3

Perigee

Tangential

Ascend apogee

4

Apogee

Tangential

Orbit roundness

5

The argument of latitude not fixed

Tangential

Combination correction

Third impulse

Orbit-insertion point

Earth

First impulse Fourth impulse

Fig. 4.15 In-plane maneuver

4.6 Scheme Design for Far Range Rendezvous Phase Fig. 4.16 Orbital plane correction

195 N Real orbit plane

Desired orbit plane δ Node 2

O

u

Node 1 i0

i1

Equatorial plane

intermediate transitional orbits. At the time all the requirements are satisfied, the target reaches the predetermined orbital position by rounding the orbit. Twice perigee elevation and once apogee elevation maneuvers are involved as follows: 2) Orbital Plane Correction The purpose of orbital plane correction is to correct the initial orbit deviation of the target and target and to meet the accuracy requirement of the coplanar alignment between the two orbits at the starting point of the near range autonomous control phase. The main corrections include the launching aiming deviation of orbital plane and the initial orbital plane deviation of target when finished launching by the rocket. The deviation of orbital plane includes the deviation of orbital inclination and the deviation of the longitude of ascending node. To save propellant, an integrated orbit maneuver is used to modify the two deviation elements. The maneuvering execution point is the intersection of the target orbit and the target orbit, as shown in Fig. 4.16. 3) Integrated Correction Considering the deviation of each orbit determination and the deviation of engine execution, after in-plane maneuver and orbital plane correction being respectively carried out, an integrated correction maneuver is designed to eliminate the orbital maneuver deviation, improve the control accuracy, and meet the requirements of far range phase terminal conditions. The theoretical maneuvering point of the integrated correction is the same as that of the orbital plane control and the control velocity is include the in orbit plane correction and out-of plane correction. The orbit determination time between current maneuver and the last maneuver must be designed longer enough for higher orbit determination accuracy for integrated correction maneuvering., and. If the actual orbit control meets the far range phase terminal control requirements, this maneuver will not be implemented.

196

4 Guidance Method and Schematic Design for Rendezvous and Docking

2. Simulation Analysis During the far range rendezvous phase, the target flies 65 h 14 min. The original orbital information of the target and the target are listed in Table 4.3. Orbital control strategy of the far range rendezvous phase is listed in Table 4.4. At the end of far range rendezvous phase, orbital elements of two spacecraft are listed in Table 4.5. At the end of far range rendezvous phase, the relative position and velocity of two spacecraft (RVD frame) are listed in Table 4.6. The trend of important variables’ variation in the far range rendezvous process is shown in Figs. 4.17, 4.18, 4.19, and 4.20. Table 4.3 Original orbit of two spacecraft Spacecraft

Semi-major axis a (km)

Eccentricity e

Orbit inclination i (°)

Longitude ascending node (°)

Argument of True perigee ω (°) anomaly f (°)

Target

6770.48

0.000942

42.299

98.031

171.873

45.0

Target

6646.02

0.011285

42.35

98.3

0.1

0.1

Table 4.4 Orbital control strategy of the far range rendezvous

Circle Ascended orbit height (km) Velocity increment (m/s) 5

54.9871

32.005

30

21.174

11.953

33

35.359

19.882

12

19.048

−3.881

38

Table 4.5 Orbital elements of spacecraft after the far range rendezvous phase Spacecraft

Semi-major axis (km)

Eccentricity e

Orbit inclination i (°)

Longitude ascending node (°)

Argument of True perigee ω (°) anomaly f (°)

Target

6769.883

0.000814

42.295

82.286

181.980

38.842

Target

6751.739

0.000579

42.295

82.289

245.360

334.890

Table 4.6 Relative position and velocity of spacecraft after the far range rendezvous phase Relative position (km)

Relative velocity (m/s)

x

y

z

dx

dy

dz

67.068

0.1616

17.721

−29.229

−0.204

1.5822

4.6 Scheme Design for Far Range Rendezvous Phase

6.76

197

Semimajor axis

×106

6.74

a (m)

6.72 6.70 6.68 6.66 6.64 6.62

0

0.5

1.5 1 Rendezvous time (s)

2.5 ×105

2

Fig. 4.17 Trend of semi-major axis during far range rendezvous phase

Eccentricity 0.012

0.010

e

0.008

0.006

0.004

0.002

0

0

0.5

1 1.5 Rendezvous time (s)

Fig. 4.18 Trend of eccentricity during far range rendezvous phase

2

2.5 ×105

198

4 Guidance Method and Schematic Design for Rendezvous and Docking Inclination 42.35 42.34 42.33

i (deg)

42.32 42.31 42.30 42.29 42.28 42.27

0

0.5

2

1.5 1 Rendezvous time (s)

2.5 ×105

Fig. 4.19 Trend of orbital inclination during far range rendezvous phase

Longitude ascending node 100 98 96

Ω (deg)

94 92 90 88 86 84 82

0

0.5

1 1.5 Rendezvous time (s)

2

2.5 ×105

Fig. 4.20 Trend of the longitude ascending node during far range rendezvous phase

From the simulation, the key parameters such as orbital inclination and the longitude ascending node of the target are close to the target through five impulses transfers. The semi-major axis and eccentricity are within the predetermined design range, and the relative position and velocity meet the design requirements.

4.6 Scheme Design for Far Range Rendezvous Phase

199

4.6.2 Short Rendezvous Guidance Scheme In August 2012, rendezvous and docking task was finished by the Russian Progress M-16M to the ISS in less than the 6 h. Its fast rendezvous and docking scheme was based on the normal 2–3 day scheme, but with short time. In this section, a short rendezvous and docking scheme with 1.8 circles (about 10088 s) during far range is introduced. The flight scheme includes in-plane orbit modulation strategy and out-of-plane correction strategy. The orbit modulation strategy in orbit plane adopts four-impulse orbital transfer strategy, and out-of-plane orbital transfer needs one impulse and adopts the guidance strategy in Sect. 4.2.3. There are three apsis, which are apogee, perigee, and apogee in far range phase. The four-impulse strategy (first, third, fourth, and fifth pulses) is adopted for in-plane phase modulation. The main phase modulation impulse is the first three impulses, and the fifth impulse is near the fourth impulse to correct the argument of perigee, as shown in Table 4.7. 1. In-Plane Orbit Modulation 1) The four-Impulse Strategy (first, third, fourth and fifth pulses) is Adopted for In-Plane Orbit Modulation. The first impulse is imposed on the apogee to elevate the height of perigee. It has the following steps: (1) Estimate the mean orbital elements of two spacecraft at apogee and at perigee; (2) Calculate the phase angle difference θ of two spacecraft. (3) Calculate the phase angle that the target can track after apogee maneuver, including ➀ The phase angle θ1 that can be tracked after the first impulse maneuver and before the second impulse maneuver is Table 4.7 Short rendezvous and docking scheme for far range rendezvous Number

Orbit transfer point

Maneuvering direction

Goal

1st impulse

Apogee at 1st circle

Tangential

Ascend perigee

2nd impulse

1st orbit period after 1500 s

Out of orbit plane

Correct deviation out of orbit plane

3rd Impulse

Perigee at 2nd circle

Tangential

Ascend apogee

4th Impulse

Near apogee at 2nd circle

Tangential

Orbit roundness

5th Impulse

Around 4th impulse

Tangential

Correct argument of perigee

200

4 Guidance Method and Schematic Design for Rendezvous and Docking





θ1 = 1 −

(a2 + x)3 a13

 × 180

➁ The phase angle θ2 that can be tracked after the third impulse maneuver and before the fourth impulse maneuver is δa2 =(a1 (1+ e1 ) − a2 (1  + e2 ) − h 0 )/2 3 2) × 180 θ2 = 1 − (a2 +x+δa a3 1

➂ The phase angle θ3 that can be tracked after the fourth impulse maneuver and until the end of free-flying far range rendezvous phase is      (a1 − h 0 )3 x0 × 1000 180 θ3 = u 1_aim + 360 − u 2 × 1 − × × 180 + a1 π a13 ➃ Semi-major axis x needed to be raised by the first impulse can be obtained by iterative optimization by minimizing (θ − θ1 − θ2 − θ3 ). Where, h 0 is the radial height that target under the target at the terminal time of far range rendezvous phase, x0 is the track position at the terminal time of far range rendezvous phase. 2) The Third Impulse The third impulse is imposed on the perigee to elevate the height of apogee, which is set to h 0 under the target. It has the following steps: (1) Estimate the mean orbital elements of two spacecraft at apogee and at perigee. (2) Calculate the phase angle difference θ of the two spacecraft. (3) Calculate the phase angle in which the target can track after apogee maneuver, including ➀ The phase angle θ2 that can be tracked after the third impulse maneuver and before the fourth impulse maneuver is  θ2 = 1 −



(a2 + x)3 a13

 × 180

➁ The phase angle θ3 that can be tracked after the fourth impulse maneuver and until the end of free-flying (about half of period) far range rendezvous phase is      (a2 (1 + e2 ) + 2x)3 θ3 = u 1_aim + 180 − w2 × 1 − a13

4.6 Scheme Design for Far Range Rendezvous Phase

× 180 +

201

x0 × 1000 180 × a1 π

Here, +2x denotes the circular orbit elevated at current apogee position + after the fourth impulse. (4) Semi-major axis x needed to be raised by the third impulse can be obtained by iterative optimization by minimizing (θ − θ2 − θ3 ). 3) The Fourth Impulse The fourth impulse is imposed on the apogee to elevate the height of perigee of circle orbit, which is set to h 0 under the target. It has the following steps: (1) Estimate the mean orbital elements of the two spacecraft at apogee and at perigee. (2) Calculate the phase angle difference θ of the two spacecraft. (3) Calculate the phase angle that the target can track after apogee maneuver, including: The phase angle θ3 that can be tracked after the fourth impulse maneuver and until the end of free-flying (about half of period) far range rendezvous phase is      (a2 + x)3 x0 × 1000 180 θ3 = u 1_aim + 360 − u 2 × 1 − × × 180 + 3 a1 π a1 (4) Semi-major axis x needed to be raised by the fourth impulse can be obtained by iterative optimization by minimizing (θ − θ3 ). 4) The Fifth Impulse The fifth impulse is solved by the combined control method of semi-major axis, eccentricity, and perigee in Sect. 4.2.3. 2. Out-of-Plane Correction The second impulse is for out-of-plane orbital transfer, and adopts the guidance strategy in Sect. 4.2.3. 3. Simulation Analysis The flight time of the far range rendezvous phase is 10088.96 s, and the initial phase angle difference between the two spacecraft is 10.9892°. The initial orbits of the two spacecraft are listed in Table 4.8. Orbit control strategy in far range rendezvous phase is listed in Table 4.9.

202

4 Guidance Method and Schematic Design for Rendezvous and Docking

Table 4.8 Initial orbits of the two spacecraft Spacecraft

Semi-major axis (km)

Eccentricity e

Orbit inclination i (°)

Longitude of ascending node (°)

Argument of True perigee ω (°) Anomaly f (°)

Target

6761.29

0.000599

42.749

9.601

202.94

9.999

Target

6655.80

0.000119

42.758

9.6118

201.94

0.00979

Table 4.9 Orbit control strategy in far range rendezvous Number

Impulse time (s)

Velocity increment (m/s)

1st impulse

2583.52

16.398

2nd impulse

3714.3999

1.4368

3rd impulse

5363.8398

6.4765

4th impulse

7895.2002

35.5876

5th impulse

8600.9600

−1.5136

Table 4.10 Orbital elements of spacecraft after the far range rendezvous phase Spacecraft

Semi-major axis (km)

Eccentricity e

Orbit inclination i (°)

Longitude of ascending node (°)

Argument of True perigee ω (°) anomaly f (°)

Target

6761.810

0.00104

42.751

8.8493

127.78

22.981

Target

6754.829

0.00181

42.751

8.8408

133.21

17.153

Table 4.11 Relative position and velocity of the spacecraft after the far range rendezvous phase Relative position (km)

Relative velocity (m/s)

x

y

z

dx

dy

dz

48.018

−0.565

12.372

−23.772

−0.425

−1.150

At the end of far range rendezvous phase, orbital elements of the two spacecraft are listed in Table 4.10. At the end of far range rendezvous phase, the relative position and velocities of the two spacecraft (RVD frame) are listed in Table 4.11. The trend of important variables’ variation in the far range rendezvous process is shown in Figs. 4.21, 4.22, 4.23 and 4.24. From the simulation, the key parameters such as orbital inclination and the longitude of ascending node of the target are basically close to the target through five pulse transfers. The semi-major axis and eccentricity are within the predetermined design range, and the relative position and velocity meet the design requirements.

4.6 Scheme Design for Far Range Rendezvous Phase Semimajor axis

×106

6.78

203

6.76

6.74

a (m)

Target spacecraft Chaser spacecraft 6.72

6.70

6.68

6.66

6.64

0

2000

4000

8000 6000 Rendezvous time (s)

10000

12000

Fig. 4.21 Trend of semi-major axis’s variation during far range rendezvous phase

Eccentricity 0.012 Target spacecraft Chaser spacecraft

0.010

e

0.008

0.006

0.004

0.002

0

2000

4000

6000 8000 Rendezvous time (s)

10000

Fig. 4.22 Trend of eccentricity’s variation during far range rendezvous phase

12000

204

4 Guidance Method and Schematic Design for Rendezvous and Docking Inclination 42.765 Target spacecraft 42.760

Chaser spacecraft

42.755 42.750 i (deg)

42.745 42.740 42.735 42.730 42.725 42.720 42.715

0

2000

4000

6000 8000 Rendezvous time (s)

10000

12000

Fig. 4.23 Trend of orbital inclination’s variation during far range rendezvous phase

4.7 Scheme Design for Homing Phase The homing phase is the transition from the far range rendezvous phase to the near range approach phase. Relative measurement information is used to eliminate the errors caused by the orbit control in the far range rendezvous phase. The characteristics of orbit control and relative control should be considered in the design of flight trajectory and guidance scheme in the homing phase. The former needs to optimize fuel consumption and select better initial and terminal states and guidance strategy to ensure that fuel consumption is as low as possible. The latter needs to consider the accuracy at the end of the control. Generally, a simple guidance model is used to conduct the multi-impulse control. In this section, the initial and final states of the homing phase are designed by using Hohmann rendezvous mode with drift in Sect. 4.2.3. Then, the multi-impulse method based on CW guidance is given, and the accuracy of the two methods is analyzed. Finally, the flight trajectory of the homing phase is simulated and analyzed.

4.7 Scheme Design for Homing Phase

205

Longitude ascending node 9.8 Target spacecraft

9.7

Chaser spacecraft

9.6

Ω (deg)

9.5 9.4 9.3 9.2 9.1 9.0 8.9 8.8

0

2000

4000

8000 6000 Rendezvous time (s)

10000

12000

Fig. 4.24 Trend of the longitude of ascending node’s variation during far range rendezvous phase

4.7.1 Determination of Initial and Final States, and Rendezvous Time The application of Hohmann rendezvous with drift is discussed in this section for the design of initial and final states of the homing phase. If the homing mission is completed within one orbit period, fome Sect. 4.2.3, denoting βMin = 0.75π δ R, βMax = 2.25π δ R, and then, the initial phase difference β of Hohmann optimal rendezvous satisfies βMin ≤ β ≤ βMax

(4.85)

The relationship between the orbital altitude difference and the relative distance Re_Dis of the two spacecraft is analyzed below. As shown in Fig. 4.25, point A and point C are the location of the target and target, respectively, and the phase angle difference between them is β. When β is small, triangle ABC can be approximated as a equicrural triangle, while β can be approximated as AB/Rc ; then  β=

Re_Dis R

2 −1

206

4 Guidance Method and Schematic Design for Rendezvous and Docking

Fig. 4.25 Analysis of relative motion

B

A

C O

It can be further obtained that  R = Re_Dis/

β δR

2 +1

(4.86)

From Eq. (4.86), the following linear relationship between maximum height difference, minimum height difference, and relative distance can be obtained: RMax Re_Dis RMin Re_Dis



2 = 1/ βδMin +1 R   = 1/

βMax δR

2

(4.87)

+1

Equation (4.87) can be used to obtain the maximum and minimum change of orbital altitude that satisfies the Hohmann transfer with drift at a given relative distance, and the maximum and minimum orbital altitude difference is proportional to the relative distance.

4.7.2 Multiple-impulse Optimal Guidance Law Due to the errors of measurement, navigation, guidance and control, the terminal of the far range rendezvous phase may not be exactly the nominal position of the homing phase, and the target and the target are not in circular orbits. In the homing phase, there are generally relative measurement sensors, which provide more accurately navigation results of the relative position and relative velocity than that of the orbital elements. Therefore, although the initial and terminal states and the flight time of the homing phase are determined using the Hohmann orbit transfer with drift strategy, the algorithm for guidance impulse calculation is not Hohmann guidance strategy. A multi-impulse guidance strategy combining Hohmann orbit transfer with drift and CW guidance is adopted. First, the initial and terminal states and flight time of the homing phase are obtained using the Hohmann orbit transfer strategy with drift; then, the first maneuver time th0 and the last maneuver time th f are obtained by using the Hohmann orbit transfer strategy with drift; finally, the CW multi-impulse

4.7 Scheme Design for Homing Phase

207

guidance strategy is adopted, and the flight time of the CW guidance is th f − th0 . Two-impulse CW guidance strategy is adopted, and in the case of high precision requirement, 1–3 correction impulse can be additionally added.

4.7.3 Analysis of Guidance Accuracy Differences and the accuracy of orbit transfer of CW guidance and Hohmann orbit transfer in Hohmann orbit condition are analyzed. 1. Hohmann Orbit Transfer Two spacecraft satisfying Hohmann orbital condition are moving in circular orbit, in which the orbital radius of the target is r B , and the orbital radius of the target is r A . The phase angle difference β between the target and the target satisfies Eq. (4.3), and the rendezvous time t H satisfies Eq. (4.1). Two impulses of Hohmann orbit transfer can be obtained as Eq. (4.4). 2. Solution of Relative Motion in Orbital Plane The orbital radii of the target and the target are r A and r B , respectively, and the phase angle difference between them is β. The relative position and velocity in the orbit plane can be obtained as follows:  

x = r A sin β z = r B − r A cos β

(4.88)

˙ A cos β = (ω B − ω A )r A cos β x˙ = βr ˙ z˙ = βr A sin β = (ω B − ω A )r A sin β

(4.89)

3. Approximate Representation of Two Impulse of CW Guidance Whether the two impulse of CW guidance is horizontal under the condition of Hohmann orbit transfer is analyzed. First, the relative state before the first impulse of Hohmann orbit transfer is regarded as the initial relative state of CW guidance. Then, two impulses are obtained using CW two-impulse guidance method, and the properties of the impulse are determined. The two impulse of CW guidance can be calculated approximately by the following two equations: 

   −28ω(r B − r A cos β) (ω B − ω A )r A cos β − V1 ≈ 4ωr A sin β − 3π ω(r B − r A cos β) (ω B − ω A )r A sin β       70 12ω(r B − r A cos β) 7(ω B − ω A )r A cos β + + V2 ≈ V1 0 (ω B − ω A )r A sin β 01 1 16

208

4 Guidance Method and Schematic Design for Rendezvous and Docking

For Hohmann rendezvous, it approximately holds that cos β = 1, then 

   −28ω(r B − r A ) (ω B − ω A )r A − V1 ≈ (ω B − ω A )r A sin β 4ωr A sin β − 3π ω(r B − r A )       70 12ω(r B − r A ) 7(ω B − ω A )r A + + V2 ≈ V1 (ω B − ω A )r A sin β 0 01 1 16

Further, 

− 41 ωh 2 15 π ω rhB 16

V 1 ≈  V 2 ≈



3 ωh 2 2 9 − 8 π ω rhB



 +

− 47 ωh 2 15 π ω rhB 16



 =

− 14 ωh 2 3 − 16 π ω rhB



Equation (4.4) can be simplified as 

 2R B μ h 1 R A ≈ ωh −1 ≈ v1 = 3 RA + RB 4 R A 2(R A + R B )    μ 2R A μ h 1 R B ≈ ωh v2 = 1− ≈ RB RA + RB 4 R 3B 2(R A + R B )

μ RA

The velocity of X-axis direction obtained by CW guidance is consistent with the impulse expression in Hohmann optimal impulse rendezvous. The order of magnitude ratio of X-direction and Z-direction components of two CW guidance impulse is ωh/(π ω

h2 ) = r B /(π h). rB

The result is about 100, and the corresponding attitude angle is about 0.57°. 4. Calculation Analysis The orbital radii of the target and the target are r A = 6689 km and r B = 6710 km, respectively, and the initial phase angle difference between the target and the target is β = 0.007371 rad. This condition satisfies the Hohmann rendezvous, and the rendezvous time is t H = 2728.6 s. Then, the relative states are x = 49.3054 km/s, z = 21.18172 km/s, x˙ = −36.2096 m/s z˙ = −0.26691 m/s. Using CW two-impulse guidance, two impulse can be obtained as Vx1 = −6.368 m/s,Vz1 = 0.082 m/s Vx2 = −6.0828 m/s,Vz2 = −0.095 m/s. The angles between the two impulse’ horizontal components and the velocity direction are 0.73779o and 0.89699o , respectively, and

4.7 Scheme Design for Homing Phase

209

two impulses are basically horizontal, which do not need to be adjusted the attitude when executing the impulse. This is equivalent to the result of Hohmann orbit transfer.

4.7.4 Simulations The orbits of the two spacecraft at the initial time of the homing phase are listed in Table 4.12. The relative position and velocity of the two spacecraft at the initial time (in RVD frame) are listed in Table 4.13. The homing guidance strategy is listed in Table 4.14. The relative position and velocity of the two spacecraft at the end of homing phase (in RVD frame) are listed in Table 4.15. The typical simulation curves of the homing phase are shown in Figs. 4.26, 4.27 and 4.28. Table 4.12 Orbits of the two spacecraft at the initial time of the homing phase Spacecraft

Semi-major axis (km)

Eccentricity e

Target

6716.482

0.002

Target

6702.551

0.00180

Orbit Inclination i (°)

Longitude of ascending node (°)

Argument of True perigee ω (°) anomaly f (°)

42.7378

359.4392

329.899

90.00

42.7444

359.343

318.028

101.382

Table 4.13 Relative position and velocity of the two spacecraft at the initial time (in RVD frame) X (km)

58.913

Y (km)

0.900

Z (km)

1.18

dX (m/s)

−18.373

dY (m/s)

0

dZ (m/s)

1.6010

Range (km)

58.9248

Table 4.14 Homing guidance strategy Characteristic points

Relative position (km)

1st impulse

39.980

Velocity increment (m/s) −3.8206

4.2382

0.0071

2nd impulse

8.5882

−1.0814

−0.3294

Correctness impulse

2.5998

0.5441

0.2209

0.654

3rd impulse

4.7490

4.4306

0.9576

−1.2112

0.1638

210

4 Guidance Method and Schematic Design for Rendezvous and Docking

Table 4.15 Relative position and velocity of the two spacecraft at the end of the homing phase (in RVD frame)

−2000

0

0.5

X (m)

5118.79872

Y (m)

53.300

Z (m)

43.544

dX (m/s)

−0.4660389

dY (m/s)

−0.020901

dZ (m/s)

0.86397

1

1.5

2.5

2

3

3.5

4

4.5

5

0 2000

Z/m

4000 6000 8000 10000 12000 14000

X/m

×104

Fig. 4.26 Transfer curve of x–z plane in the homing phase RVD frame −500

0

0.5

1

1.5

2.5

2

3

3.5

4

4.5

5

Y/m

0

500

1000

×104 X/m

Fig. 4.27 Transfer curve of x–y plane in the homing phase RVD frame

4.7 Scheme Design for Homing Phase

211

dx/(m/s)

20 0 −20 −40

0

500

1000

1500

2000

2500 t/s (a)

3000

3500

4000

4500

5000

0

500

1000

1500

2000

2500 t/s (b)

3000

3500

4000

4500

5000

0

500

1000

1500

2000

2500 t/s (c)

3000

3500

4000

4500

5000

dy/(m/s)

1 0 −1 −2

dz/(m/s)

5 0 −5 −10

Fig. 4.28 Relative velocity change curve of homing phase

From the result of simulation, under the designed initial conditions of the homing phase, the four-impulse CW guidance can guide the target to a predetermined position accurately.

4.8 Scheme Design for Closing Phase In the closing phase, the target is gradually approaching the target with relative navigation to capture the docking corridor and providing an appropriate entry for the final approach phase. Because the two spacecraft are close, the CW equation can describe the relative motion between the two spacecraft accurately, so the CW guidance strategy is generally used in the closing phase. In the process of rendezvous and docking, especially in the backward rendezvous and docking process, CW-guided trajectories tend to be backward, which easily makes the trajectories enter the keep-out-zone. In addition, the final impulse of CW guidance may be relatively large, which will bring some potential safety hazards. Line of sight guidance approaches the target step by step according to the planned trajectory, and the closer it approaches the target, the smaller

212

4 Guidance Method and Schematic Design for Rendezvous and Docking

the correction impulse is, which can better overcome the problems caused by CW guidance. In the project application of closing phase, CW guidance and line-of-sight guidance are combined.

4.8.1 CW Guidance Law The rendezvous time of the CW two-impulse guidance strategy, such as 5 km–400 m approach process, is slightly less than half of the orbit period. At 400 km orbital altitude, the orbital period is about 5553.6 s, and the rendezvous time is 2600 s slightly less than that half of the orbital period. The first and last impulse in the orbital plane are [−0.0297 1.3692] and [0.0297 1.3692] m/s, respectively. The relative position change in the orbital plane is shown in Fig. 4.29. The terminal impulse of CW guidance reaches 1.3692 m/s with the acceleration of 0.01 m/s2 . It takes about 137 s to execute this impulse, and the braking time is relatively long.

0

500

1000

1500

2000

X/m 2500

3000

200

Z/m

400

600

800

1000

1200

Fig. 4.29 Curve of relative position change in the orbital plane

3500

4000

4500

5000

4.8 Scheme Design for Closing Phase

213

4.8.2 Light of Sight Guidance Law The t guidance law can be seen in Sects. 4.5.1 and 4.5.2.

4.8.3 United Guidance LAW CW guidance has high precision, but it may be unsafe when approaching the target, and the braking time is longer when the second impulse is applied. Line of sight guidance consumes more fuel. Therefore, combing line of sight guidance and CW guidance can solve the above two problems. While using the combining control method, the CW guidance is adopted at first and then switched to line of sight guidance when the distance between the two spacecraft is less than a certain value or the rendezvous time reaches a certain value. Two marking points ⊗ in Figs. 4.30 and 4.31 are marked at 2000 s and at a position less than 1300 m. The trajectory is produced by CW guidance and line of sight guidance separated by the first ⊗. It can be seen that the CW guidance and line of sight guidance are well connected, and the flight trajectory is smooth and there is no sudden change, when the control parameters are well designed.

−200

0

1000

2000

0 200

Z/m

400 600 800 1000 1200

Fig. 4.30 Relative position in orbital plane

X/m 3000

4000

5000

6000

214

4 Guidance Method and Schematic Design for Rendezvous and Docking

−60

0

X/m 3000

2000

1000

5000

4000

6000

−50

Y/m

−40 −30 −20 −10 0 10

Fig. 4.31 Relative position out of orbital plane

4.8.4 Simulations The flight trajectory of closing phase starts from the 5 km hold point to the 400 m hold point, and then, to the 140 m hold point. The initial orbits of the two spacecraft at the initial time of the homing phase are listed in Table 4.16. The relative position and velocity of the two spacecraft at the initial time (in RVD frame) are listed in Table 4.17. Table 4.16 Initial orbits of two spacecraft Spacecraft

Semi-major axis (km)

Eccentricity e

Orbit inclinationi (°)

Longitude of ascending node (°)

Argument of perigee ω (°)

True anomaly f (°)

Target

6718.787

0.00101

42.79

359.065

133.072

18.000

Target

6716.858

0.00079

42.793

359.068

131.677

19.350

Table 4.17 Relative position and velocity of the two spacecraft at the initial time

X (m)

5118.79872

Y (m)

53.300

Z (m)

43.544

dX (m/s)

−0.4660389

dY (m/s)

−0.020901

dZ (m/s)

0.86397

4.8 Scheme Design for Closing Phase

215

The typical simulation curves of the closing phase are shown in Figs. 4.32, 4.33, 4.34 and 4.35. The relative states at the end of each stage (in RVD frame) are listed in Table 4.18. From the simulation, 400 m and 140 m approach are adopted in the closing phase, which makes the target approach the target step by step. The flight trajectory is smooth and controllable, and the position accuracy at the end of the flight is high.

−200

0

X/m 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0 200

Z/m

400 600 800 1000 1200

Fig. 4.32 Flight trajectory of the x–z plane in closing phase

−600

0

X/m 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

−400

Y/m

−200 0 200 400 600

Fig. 4.33 Flight trajectory of the x–y plane in closing phase

216

4 Guidance Method and Schematic Design for Rendezvous and Docking 6000 x/m

4000 2000 0

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

500 y/m

0 −500 −1000 2000

z/m

1000 0 −1000

Fig. 4.34 Relative position change curve of single simulations in closing phase

dx/(m/s)

2 0 −2 −4

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

dy/(m/s)

2 1 0 −1

dz/(m/s)

2 0 −2

Fig. 4.35 Relative velocity change curve of single simulations in closing phase

4.9 Key Points of Guidance System Design

217

Table 4.18 Relative states at the end of each flight phase in the approaching process Phase

Relative position (m)

Relative velocity (m/s)

x

y

z

dx

400 m approaching

421.161

−30.394

3.306

400 m remain

378.260

18.556

−21.527

140 m approaching

151.300

−3.996

2.118

dy

dz

−0.3418

0.418

−0.083

−0.00175

−0.0418

−0.0203

−0.2820

0.0382

−0.049

4.9 Key Points of Guidance System Design The choice of rendezvous and docking guidance strategy mainly depends on the flight phase determined by the distance between the two spacecraft, the relative measurement, and the guidance purpose to be achieved in the flight phase. In the far range rendezvous phase, the two spacecraft are relatively far away, and the relative sensors cannot provide relative navigation information. Orbital control rendezvous guidance strategy is generally adopted to guide the target to the vicinity of the target. In the far range rendezvous phase, fuel consumption and control accuracy are the main assessment indicators. In the middle and near range, the relative navigation sensors can provide more accurate measurement information than that of orbital elements. Generally, the relative states are regarded as control parameters, and the target is gradually and accurately guided to the target at a closer distance until docking. The design of guidance scheme includes the design of initial and terminal states of different flight stages based on fuel consumption, stability, and robustness. Fuel consumption, control accuracy and initial condition adaptability are the main assessment indicators in the homing phase, and the appropriate initial and terminal states need to be selected. In the closing phase, safety, robustness, guidance accuracy, and working conditions for relative measurement sensors are the main evaluation indexes. The design of guidance scheme ensures that the target approaches the target gradually along the nominal trajectory, and the guidance impulse during flight should become smaller and smaller as the two spacecraft approach gradually.

References 1. Changqing, C.: Trajectory Safety and Optimization of Rendezvous and Docking. Beijing Institute of Control Engineering (2008) (In Chinese) 2. Prussing, J E.: Optimal Multiple - Impulse Orbital Rendezvous. Sc. D. thesis, Department of Aeronautics and Astronautics Massachusetts Institute of Technology (1967) 3. Changqing, C, Yongchun X.: Optimal Impulsive Ellipse-to-Circle Coplanar Rendezvous[J]. Science in China (Series E), 52(5), 1135–1143 (2009) 4. Prussing, J E.: Optimal Impulsive Linear Systems: Sufficient Conditions and Maximum Number of Impulse[J]. The Journal of the Astronautical Science, 43(2), 195–206 (1995)

Chapter 5

Automatic Control Method and Scheme Design for Rendezvous and Docking

5.1 Introduction For a chaser, the difficulties in design of rendezvous and docking controller are mainly reflected by three aspects—stable attitude control of the chaser in the process of orbit control, control of the six degrees of freedom of high-precision relative position and relative attitude in the final approach phase, and the long-term relative position maintenance control at the hold points. Stable attitude control of the chaser during orbit control is mainly to solve the attitude stability problem under conditions of large disturbance moment during the start-up of the orbit control jets. The control of 6° of freedom of the relative position and relative attitude is mainly used in the final approach phase to achieve the main objective of high-precision control. Because the hold time is usually relatively long, relative position control of the hold points is mainly associated with how to control the chaser to minimize propellant consumption on the premise of meeting the requirements of relative position maintenance accuracy. Many methods for spacecraft attitude and position control exist, among which PID control and phase plane control are widely used in engineering. In addition,  there are also application examples of H∞ H2 robust control, LQR/LQG control, and adaptive control [1, 2]. The intelligent adaptive control method based on characteristic model [3] is developed from the all-coefficient adaptive control method and has proven to be a practical adaptive control method. The method is to establish a characteristic model for the plant to be controlled, based on the physical mechanism, the dynamic characteristics, and environmental characteristics of spacecraft or industrial plant, combined with the control objectives and performance requirements. Design of the controller is based on this model, combined with intelligent control and adaptive control theory, according to the complexity of the plant and performance requirements. Theoretical research results have proved that under certain conditions, the closed-loop system composed of the controller designed according to this method and the original plant to be controlled is stable [4]. © National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_5

219

220

5 Automatic Control Method and Scheme Design …

The main feature of the intelligent adaptive control method based on characteristic model is that a characteristic model with the sum of the all coefficients equal to 1 is established to determine the parameter range, and the adaptive controller is designed from the engineering application. The controller mainly includes golden section adaptive control law, logical differential control law, and logic integral control law. The method has a simple design and is easy to use, and normally debugging and testing are not required. The practical application results show that this method is not only effective for linear time-invariant systems with unknown parameters and orders but is also applicable to some nonlinear systems and slow time-varying systems. The characteristic model provides a theoretical basis for controller design of high order plants with unknown parameters and makes the practical application of engineering significantly convenient [5–11]. This chapter mainly introduces the characteristic modeling of the plant to be controlled of rendezvous and docking, and the application of intelligent adaptive control method based on characteristic model in rendezvous and docking tasks. By combining golden section control and logical differential control with phase plane control, the problems of attitude stability control during chaser orbit control, and six degree-of-freedom control of high-precision relative position and relative attitude control in the final approach phase are solved. The problem of long-term position maintenance in the rendezvous and docking process is mainly related to the relative orbital motion characteristics. Based on the research results of the relative orbital motion characteristics, this chapter provides a strategy of relative position control at the hold points for a long time, which has the advantages of less propellant consumption.

5.2 Characteristic Modeling for the Plant to be Controlled 5.2.1 Important Lemma First, we introduce the following important lemma. Lemma 1 Wang [8, 12]. Consider the following two systems x˙ = f (t, x) + ϕ(t, x) + g(t, x)u

(5.1)

f (t, x), g(t, x) are smooth functions. ϕ(t, x) satisfies condition ϕ(t, x) ≤ where t −λ(t−s) x(s)ds, x ∈ Rn 1 , u ∈ Rn 2 are any control inputs, m > 0 and λ > 0. 0 me v˙ = f (t, v) + ψ(t, v) + g(t, v)u

(5.2)

where v ∈ Rn 1 , u ∈ Rn 2 . Function ψ(t, v) satisfies the condition ψ(t, v) ≤ √ m√ c˜2 v, c˜2 ≥ c˜1 > 0. λ c˜ 1

5.2 Characteristic Modeling for the Plant to be Controlled

221

Set the compact set X 1 ⊂ Rn 1 as invariant. If there is a controller u to make the system (5.2) exponentially stable in the compact set X 1 , then the system (5.1) is also exponentially stable in the compact set X 1 .

5.2.2 Characteristic Model The so-called characteristic modeling refers to modeling based on the dynamic characteristics and control performance requirements of a plant to be controlled, rather than on the accurate dynamic analysis of the plant. The modeling is performed for plants with high orders and unknown parameters, which is of great significance for the modeling of a flexible body. The features of the characteristic model are as follows [4]. (1) Under the same input control, the outputs of the characteristic model and the actual plant are equivalent (that is, it can be kept within the allowable output error in the dynamic process), and the output is equal in the stable condition. (2) The form and order of the characteristic model mainly depend on the control performance requirements except for the characteristics of the plant. (3) The form of the characteristic model should be simpler than the original plant dynamics equation, and the engineering implementation should be easy and convenient. (4) The characteristic model is different from the reduced-order model of the higher-order system. In the characteristic model, the relevant information of the higher-order model is compressed into several feature parameters without loss of information. In general, the characteristic model is described by the slow time-varying difference equation. The chaser usually has two flexible solar panels, which can be described by the flexible dynamic model of complex spacecraft with a rigid center body and flexible accessories, as shown in (2.35)–(2.40). With a chaser such as the controlled plant, the establishment process of its characteristic model is divided into three steps as follows. 1. Turn it into the Interconnection Form of the Center Body and Flexible Solar Panels We take the three attitude angles of the chaser center body as the output and the control torque as the main input and consider the effect of external forces on the chaser center body and external torque acting on the solar panels. The original dynamic model (2.35)–(2.40) can be transformed into a interconnection model of input/output center body model and flexible panels. The specific process as described below. Take statement (2.35), (2.37), and (2.38) into (2.39) and (2.40), we obtain

222

5 Automatic Control Method and Scheme Design …

A11 η¨ ls + A12 η˙ ls + A13 ηls + A14 η¨ r s + A15 ω˙ s + T 1 = 0

(5.3)

A21 η¨ r s + A22 η˙ r s + A23 ηr s + A24 η¨ ls + A25 ω˙ s + T 2 = 0

(5.4)

−1 2 T T where A11 = I m×m − F tls M −1 F tls − F als I als F als , A12 = 2ξ ls Ω als , A13 = Ω als , −1 T −1 T −1 T T T A14 = −F tls M F tr s , A15 = F sls − F als I als Rasls , T 1 = F als I als T als + T T −1 M −1 P s , A21 = I m×m − F trT s M −1 F tr s − F ar F tls s I ar s F ar s , A22 = 2ξ r s Ω ar s , 2 T −1 T T −1 T A23 = Ω ar s , A24 = −F tr s M F tls , A25 = F sr s − F ar s I ar s R asr s , T 2 = T −1 T −1 F ar s I ar s T ar s + F tr s M P s . Take (2.37) and (2.38) into (2.36), we get

A31 ω˙ s + A32 ωs + A33 η¨ ls + A34 η¨ r s + T 3 − T s = 0

(5.5)

−1 T −1 T ˜ s I s , A33 = F sls − where A31 = I s − Rasls I als Rasls − Rasr s I ar s R asr s , A32 = ω −1 −1 −1 −1 T als + Rasr s I ar Rasls I als F als , A34 = F sr s − Rasr s I ar s F ar s , T 3 = Rasls I als s T ar s . Formulas (5.3) and (5.4) could be further expressed as

 ¯1 A

       T1 η¨ ls η˙ ls ηls ¯2 ¯3 ¯ 4 ω˙ s + +A +A +A =0 η¨ r s η˙ r s ηr s T2

where    A11 A14 ¯ 2 = A12 0m×m , A A24 A21 0m×m A22     A15 0 A 13 m×m ¯4 = ¯3 = , A A 0m×m A23 A25 ¯1 = A



i.e. 

η¨ ls η¨ r s



       −1 η˙ ls ηls T1 ¯ ¯ ¯ ¯ = − A1 A2 + A3 + A4 ω˙ s + η˙ r s ηr s T2

Putting (5.6) into (5.5), ¯ ω˙ + A32 ωs ¯ −1 ( A31 − [ A33 A34 ] A 1 A4 ) s     η˙ ls ηls −1 −1 ¯ 2) ¯ 3) ¯1 A ¯1 A + (−[ A33 A34 ] A + (−[ A33 A34 ] A η˙ r s ηr s   T1 −1 ¯1 ) + (−[ A33 A34 ] A + T3 − Ts = 0 T2 which can be simply written as

(5.6)

5.2 Characteristic Modeling for the Plant to be Controlled

 P 1 ω˙ s + P 2 ωs + P 4

223

   η˙ ls ηls + P3 + T4 − Ts = 0 η˙ r s ηr s

(5.7)

where, ¯ , ¯ −1 P 1 = A31 − [ A33 A34 ] A 1 A4 P 2 = A32 , ¯ , ¯ −1 P 3 = −[ A33 A34 ] A 1 A3 ¯ , ¯ −1 P 4 = −[ A33 A34 ] A 1 A2  T 4 = (−[

¯ −1 A33 A34 ] A 1 )

 T1 + T3 T2

From Eq. (5.7), it can be seen that the original dynamics Eq. (5.4) is transformed into an input-output model with ωs as the output and T s (attitude control torque) and T 4 (other minor control forces and torques) as the input, which includes the disturbance of flexible solar panels. The dynamics of the flexible solar panels is further considered follows.    as  T

ηls η˙ ls Define X 1 = , X2 = , X = X 1T X 2T , then, (5.6) could be written η˙ r s ηr s as ⎧ ˙ X = X2 ⎪ ⎨ 1



⎪ ¯ ¯ −1 ¯ −1 ¯ ] ⎩ X˙ 2 = [− A 1 A3 − A1 A2

X1 X2



 ¯ ˙s − A ¯ −1 ¯ −1 −A 1 A4 ω 1

T1



T2

Which can be further simplified as  X˙ = P 5 X + B 1 ω˙ s + B 2

T1



T2

where 

 02m×2m I 2m×2m P5 = , ¯ ¯ −1 ¯ −1 ¯ −A 1 A3 − A1 A2   02m×3 , B1 = ¯ ¯ −1 −A 1 A4   02m×3 B2 = ¯ −1 −A 1

(5.8)

224

5 Automatic Control Method and Scheme Design …

Then, (5.8) is the state equation of the flexible solar panels. Owing to θ˙ = T (θ )ωs where

T θ= ϕθ ψ , ⎡

⎤ sin θ 0 cos ψ ⎥ ⎢ T (θ) = ⎣ tan ψ cos θ 1 tan ψ sin θ ⎦, − sin θ 0 cos θ T

ω s = ω x ω y ωz cos θ cos ψ

Then, the dynamic model of the chaser center body can be written as ⎧ ⎪ ⎨ P 1 ω˙ s + P 2 ωs + P 3 X 1 + P 4 X 2 + T 4 − T s = 0 θ˙ = T (θ)ωs ⎪ ⎩ y=θ

(5.9)

From the above equation, it can be seen that (5.8) is the state equation of flexible solar panels and (5.9) is the input-output equation of the chaser center body. The two systems are coupled with each other. (5.8) is affected by ω˙ s , and (5.9) is affected by X 1 , X 2 . Equations (5.8) and (5.9) constitute an interconnected structure. 2. Transfer Higher Order System to Lower Order System Because the system is essentially a high-order system, and the characteristic model is a low-order system, it must be transformed. We solve (5.8) and get 

  t    t X 1 (t) X T1 10 + e P 5 (t−τ ) B 1 ω˙ s (τ )dτ + e P 5 (t−τ ) B 2 dτ = e P5t X 2 (t) X 20 T2 0

0

By means of fractional integration 

   X X 1 (t) 10 + B 1 ωs (t) − e P 5 t B 1 ωs (0) = e P5t X 2 (t) X 20   .  t  t T1 P 5 (t−τ ) P 5 (t−τ ) + P5 e B 1 ωs (τ )dτ + e B2 dτ T2 0 0

Therefore, we put them into the first equation of (5.9) and get

5.2 Characteristic Modeling for the Plant to be Controlled

225

P 1 ω˙ s + P 2 ωs + P 3 X 1 + P 4 X 2 + T 4 − T s t = P 1 ω˙ s + ( P 2 + [ P 3 P 4 ] B 1 )ωs + [ P 3 P 4 ] P 5 

t

+ [ P 3 P 4]

 e P 5 (t−τ ) B 2

0

e P 5 (t−τ ) B 1 ωs (τ )dτ

0   . T1 X 10 dτ + T 4 − T s + [ P 3 P 4 ] e P 5 t T2 X 20

− [ P 3 P 4 ] e P 5 t B 1 ωs (0) = 0 Because T als and T ar s are generally constant values and in general have little influence on the attitudes, it can be assumed that T 1 and T 2 are constant,  so    t P (t−τ )  t P (t−τ ) T1 T1 5 5 B2 B 2 dτ dτ can be reduced to a simpler form 0 e . 0 e T2 T2 The above equation can be abbreviated as P¯ 1 ω˙ s + P¯ 2 ωs + P¯ 3





t

e

P 5 (t−τ )

B 1 ωs (τ )dτ + P¯ 4

0

T1 T2



+ P¯ 5 X 0 + P¯ 6 ωs (0) + T 4 − T s = 0

(5.10)

where P¯ 1 = P 1 , P¯ 2 = P 2 + [ P 3 P 4 ] B 1 , P¯ 3 = [ P 3 P 4 ] P 5 , P¯ 4 = t [ P 3 P 4 ] 0 e P 5 (t−τ ) B 2 dτ , P¯ 5 = [ P 3 P 4 ] e P 5 t , and P¯ 6 = −[ P 3 P 4 ] e P 5 t B 1 . Considering that the system of flexible solar panels itself is a stable system, there are diagonal matrices Λ1 = −λ1 I, Λ2 = −λ2 I, where λ1 > λ2 > 0, to make Λ1 ≤ P 5 ≤ Λ2 , thus, eΛ1 (t−τ ) ≤ e P 5 (t−τ ) ≤ eΛ2 (t−τ ) . T

3 3 3 Set P¯ 3 B 1 ωs (τ ) = , so i=1 b1i ωs,i (τ ) i=1 b2i ωs,i (τ ) i=1 b3i ωs,i (τ ) ⎡

⎤ −λ1 (t−τ ) e b ω (τ )dτ 1i s,i ⎢ i=1 0 ⎥ ⎢ 3 ⎥ t ⎢   t −λ (t−τ ) ⎥ ⎢ ⎥ 1 b2i ωs,i (τ )dτ ⎥ ≤ P¯ 3 e P 5 (t−τ ) B 1 ωs (τ )dτ 0 e ⎢ ⎢ i=1 ⎥ 0 ⎣ ⎦ 3  t −λ1 (t−τ ) b3i ωs,i (τ )dτ 0 e 3   t

i=1

⎡

⎤ −λ2 (t−τ ) e b ω (τ )dτ 1i s,i ⎢ i=1 0 ⎥ ⎢ 3 ⎥ ⎢   t −λ (t−τ ) ⎥ ⎥ 2 ≤⎢ e b ω (τ )dτ 2i s,i 0 ⎢ ⎥ ⎢ i=1 ⎥ ⎣ ⎦ 3  t −λ2 (t−τ ) b3i ωs,i (τ )dτ 0 e 3   t

i=1

Because of where |b| =

   t    ¯ P 5 (t−τ ) B 1 ωs (τ )dτ ≤ 3|b| t e−λ2 (t−τ ) 3 ωs,i (τ )dτ , i=1 0 P 3e 0 max{|b1i |, |b2i |, |b3i |}, ∀i = 1, 2, 3. According to Lemma

226

5 Automatic Control Method and Scheme Design …

 3  ωs,i (t) of 5.1, we can construct the upper bounds of equivalence 3 |b| i=1 λ 2  t   P¯ 3 e P 5 (t−τ ) B 1 ωs (τ )dτ . According to the processing method in reference [12], 0     3   3 |b| ωs,i (t), where k(υ) = i=1 ωs,i (t) belongs to a closed convex set k λ2 |b| |b| −3 λ2 + 6(1 − υ) λ2 , υ ∈ [0, 1]. Therefore, (5.10) can be further written as  P¯ 1 ω˙ s + ( P¯ 2 + K (υ))ωs + P¯ 4

T1 T2

 + P¯ 5 X 0 + P¯ 6 ωs (0) + T 4 − T s = 0

where K (υ) = (k(υ))3×3 Because θ˙ = T (θ )ωs , therefore we have 2 −1 ω˙ s = T −1 (θ )θ¨ + T˙ (θ)θ˙

It can be seen from (5.10) that e P 5 t → 0, ∀t → ∞; therefore, the final convergence of P¯ 5 , P¯ 6 is 0, and the initial value term P¯ 5 X 0 , P¯ 6 ωs (0) is ignored  P¯ 1 ω˙ s + ( P¯ 2 + K (υ))ωs + P¯ 4

 T1 + T4 − Ts = 0 T2

which can be further written as  P¯ 1 T −1 (θ)θ¨ + (( P¯ 2 + K (υ))T −1 (θ ) + P¯ 1 T˙

−1

˙ θ˙ + P¯ 4 (θ )θ)

 T1 + T4 − Ts = 0 T2

This can be simply written as  −1 ˙ ˙ ¯ θ¨ + F −1 1 (θ)F 2 (θ , θ , υ)θ + F 1 (θ) P 4

 T1 + F −1 1 (T 4 − T s ) = 0 T2

(5.11)

−1 ˙ where F 1 (θ ) = P¯ 1 T −1 (θ ), F 2 (θ , θ˙ , υ) = ( P¯ 2 + K (υ))T −1 (θ ) + P¯ 1 T˙ (θ )θ. Based on the above transformation, the original dynamics model (5.8) and (5.9) is further converted to the equivalent low-order input-output model (5.11), which includes the influence of the high-order system (solar panel flexibility) on the chaser center body. In terms of control input, T s represents the main control quantity, and T 4 integrates the other secondary control quantities.

3. Precise Discretization The state equation of the system (5.11) can be written as

5.2 Characteristic Modeling for the Plant to be Controlled



z˙ = A(θ , θ˙ , υ)z + B 1 (θ )u1 + B 2 (θ , θ˙ , υ)u2 y = z1

227

(5.12)

  T 0 I T ˙ , a2 (θ, θ˙ , υ) = , A(θ , υ) = θ θ 0 a2 (θ, θ˙ , υ) T

˙ ˙ F −1 0 b1T (θ ) , b1 (θ) = −F −1 1 (θ )F 2 (θ , θ , υ), B 1 (θ) = 1 (θ), B 2 (θ , θ , υ) =  



T −1 ˙ , u1 = T s , and u2 = 0 bT (θ, θ˙ , υ) , b2 (θ, θ˙ , υ) = F −1 1 (θ)F 2 (θ , θ , υ) F 1

T 2 T T T T1 T2 T4 If it is assumed that the state of the system (5.11) changes within a closed set, i.e. θ , θ˙ ∈ D ⊂ R3 , υ ∈ [0, 1], then the above coefficient matrices all have definite intervals. Therefore, according to Refs. [13, 17], its vertex system can be uniformly written as  z˙ = Ai z + B 1,i u1 + B 2,i u2 (5.13) y = z1 where z =



z 1T z 2T

T

=



T

where Ai , B 1,i and B 2,i , i = 1, . . . , 8, are respectively the matrices corresponding to a2 (θ , θ˙ , υ), b1 (θ), and b2 (θ , θ˙ , υ) taking the minimum and maximum values. Now, to easily build up the characteristic model, we write Ai , B 1,i , and B 2,i in terms of A, B 1 , and B 2 and a2 (θ, θ˙ , υ), b1 (θ), and b2 (θ, θ˙ , υ) in terms of a2 , b1 and b2 . Firstly, discretize (5.13) and then construct the characteristic model. The exact solution of (5.13) can be written as z k+1 = e

Ah

tk+1 zk + e A(tk+1 −s) ds(B 1 u 1,k + B 2 u 2,k ) tk

For

 τk+1 τk

e A(τk+1 −s) ds, we decompose and integrate e A(τk+1 −s) and get τk+1 1 1 e A(τk+1 −s) ds = I h + Ah 2 + A2 h 3 + · · · 2 3!

τk

Besides, the series expansion e Ah k can be substituted into (5.14) 1 2 2 h A + · · · )z k 2! 1 1 + h(I + Ah + A2 h 2 + · · · )(B 1 u 1,k + B 2 u 2,k ) 2 3!

z k+1 = (I + h A +

Further arrangement gives us

(5.14)

228

5 Automatic Control Method and Scheme Design …

      0 0 z 1,k I Ih z k+1 = + u1,k + u2,k 0 I + ha2 hb2 hb1 z 2,k     z 1,k 2 + h Uk + Sk B 1 u1,k + Sk B 2 u2,k z 2,k 

where 1 1 U k = A2 + h A3 + · · · = 2! 3!



   1 1 2 2 U 1k U 2k S1k S2k , Sk = Ah + A h + · · · = , 2 3! U 3k U 4k S3k S4k

e Ah k is constant and computable; therefore, U k , Sk is constant too and is infinitesimal in the same order with h. Further, we write the above relation as  z 1,k+1 = (I + h 2 U 1k )z 1,k + (I h + h 2 U 2k )z 2,k + h 2 S2k b1 u1,k + h 2 S2k b2 u2,k z 2,k+1 = h 2 U 3k z 1,k + (I + ha2 + h 2 U 4k )z 2,k + (I h + h 2 S4k )(b1 u1,k + b2 u2,k ) Therefore, if we shift z 1,k+1 and put z 1,k+1 , z 2,k+1 , we get z 1,k+2 = (2I + ha2 + O(h 2 ))z 1,k+1 + (−I − ha2 + O(h 2 ))z 1,k + (I h 2 + O(h 3 ))(b1 u1,k + b2 u2,k ) + h 2 S2,k+1 (b1 u1,k+1 + b2 u2,k+1 ) Considering y = z 1 , characteristic variables z 1 , u1 , u2 were taken to construct the characteristic model yk+2 = F 1 yk+1 + F 2 yk + G 1 u1,k+1 + G 2 u1,k + G 3 u2,k+1 + G 4 u2,k

(5.15)

where F 1 = 2I + ha2 + O(h 2 ), F 2 = −I − ha2 + O(h 2 ), G 1 = h 2 b1 S2,k+1 , G 2 = I b1 h 2 + O(h 3 ), G 3 = h 2 b2 S2,k+1 , and G 4 = I b2 h 2 + O(h 3 ). It is considered that the upper and lower bounds of a2 , b1 , and b2 can construct a closed convex set Ds independent of z, that is, when h  1, there are continuous functions εi , which is infinitesimal in the same order with h. |F 1 − 2I| < ε1 (h) |F 2 + I| < ε2 (h) Iε3 (h 2 ) < G 1 < Iε4 (h 2 ) Iε5 (h 2 ) < G 2 < Iε6 (h 2 ) Iε7 (h 2 ) < G 3 < Iε8 (h 2 ) Iε9 (h 2 ) < G 4 < Iε10 (h 2 ) The above characteristic model includes the influence of solar panel dynamics on the input-output equations of the spacecraft system. u1 represents the control

5.2 Characteristic Modeling for the Plant to be Controlled

229

moment that plays a primary role in the attitude control of the spacecraft, and u2 is the synthesis of other control forces and moments that play a secondary role.

5.2.3 Simulation Verification This section takes the Shenzhou-manned spacecraft as an example to verify the variation of characteristic parameters in the characteristic model (5.15) through mathematical simulation. According to the expression of the characteristic parameters F 1 , F 2 , G 1 in type (5.15), which is the rendezvous and docking process from the 5.km hold point to the 140-m hold point, parameters F 1 , F 2 , G 1 , change with the simulation time, as shown in Figs. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8 and 5.9. Figure 5.10 shows the entire process of attitude angle and angular velocity curve of the chaser, where the sampling period, h, is 0.16 s. It can be seen from the mathematical simulation results that the values of the characteristic parameters F 1 , F 2 , and G 1 are slowly time-varying, and the main diagonal element values of F 1 and F 2 fluctuate between 2 and −1. The non-diagonal element values are small relative to the main diagonal element values, which can be approximated to zero in engineering. The original high-order flexible dynamics

F1[1,1]

1.988 1.987 1.986

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

F1[1,2]

−0.005 −0.01 −0.015 −0.02

F1[1,3]

−0.005 −0.01 −0.015

Fig. 5.1 The first row of the parameter matrix F1

230

5 Automatic Control Method and Scheme Design …

F1[2,1]

−1

×10−3

−2 −3

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

F1[2,2]

1.9985 1.998 1.9975

F1[2,3]

0

×10−3

−2 −4

0

Fig. 5.2 The second row of the parameter matrix F1

F1[3,1]

−1

×10−3

−2 −3 0

F1[3,2]

0

1000

1500

2000 t/s (a)

2500

3000

3500

4000

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

×10−3

−2 −4 0 2

F1[3,3]

500

×10−3

1.998 1.996 0

Fig. 5.3 The third row of the parameter matrix F1

5.2 Characteristic Modeling for the Plant to be Controlled

231

F2[1,1]

−0.986 −0.9865 −0.987 −0.9875

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

F2[1,2]

0.02 0.015 0.01 0.005

F2[1,3]

0.015 0.01

0.005

Fig. 5.4 The first row of the parameter matrix F2

model can be represented by characteristic model in type (5.15). Various secondary information is compressed into the characteristic parameters.

5.3 Intelligent Adaptive Control Method Based on Characteristic Model According to the analysis in Sect. 5.2.3, for each channel of roll, pitch, and yaw, the characteristic model shown in Eq. (5.15) can be described by a second-order slowly time-varying difference equation y(k + 1) = f 1 (k)y(k) + f 2 (k)y(k − 1) + g0 (k)u(k) + g1 (k)u(k − 1)

(5.16)

When the plant is a minimum phase system or some weak non-minimum phase system, to simplify and facilitate engineering design, the control input of its characteristic model is generally only one term u(k), namely y(k + 1) = f 1 (k)y(k) + f 2 (k)y(k − 1) + g0 (k)u(k)

(5.17)

232

5 Automatic Control Method and Scheme Design …

F2[2,1]

3

×10−3

2 1

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

F2[2,2]

−0.997 −0.998 −0.999

0 −3

F2[2,3]

4

×10

2 0

0

Fig. 5.5 The second row of the parameter matrix F2

F2[3,1]

3

×10−3

2 1

0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

0

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

−3

F2[3,2]

4

×10

2 0

F2[3,3]

−0.996 −0.998 −1

Fig. 5.6 The third row of the parameter matrix F2

5.3 Intelligent Adaptive Control Method Based on Characteristic …

G0[1,1]

−4.0

233

×10−6

−4.1 −4.2 0

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

−8

G0[1,2]

1.95

×10

1.9 1.85

G0[1,3]

−4

0 ×10−8

−6 −8

0

Fig. 5.7 The first row of the parameter matrix G1 ×10−7

G0[2,1]

1 0 −1

0

G0[2,2]

−5.8

500

1000

1500

2000 t/s (a)

2500

3000

3500

4000

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

×10−7

−6 −6.2 0 −9

G0[2,3]

2

×10

0 −2 0

Fig. 5.8 The second row of the parameter matrix G1

234

5 Automatic Control Method and Scheme Design …

G0[3,1]

0 −1 −2 1 G0[3,2]

×10−7

0

1000

1500

2000 t/s (a)

2500

3000

3500

4000

500

1000

1500

2000 t/s (b)

2500

3000

3500

4000

500

1000

1500

2000 t/s (c)

2500

3000

3500

4000

×10−9

0.5 00

−5.5 G0[3,3]

500

×10−7

−6

0

Fig. 5.9 The third row of the parameter matrix G1 0.5 droll ((°)/s)

roll (°)

2 1 0 −1

0

1000

2000 t/s

3000

0 −0.5

4000

0 −1 −2

0

1000

2000 t/s

3000

4000

3000

4000

−0.5

0

1000

2000 t/s

3000

4000

0

1000

2000 t/s

3000

4000

0.5 dyaw ((°)/s)

yaw (°)

2000 t/s

0

2 1 0 −1

1000

0.5 dpitch ((°)/s)

pitch (°)

1

0

0

1000

2000 t/s

3000

4000

0 −0.5

Fig. 5.10 Attitude angle and angular velocity curves

5.3 Intelligent Adaptive Control Method Based on Characteristic …

235

This section uses the least square method to estimate the parameters f 1 , f 2 , and g0 . Define the observation vector Φ ls (k) = [y(k), y(k − 1), u(k)]T

(5.18)

The parameters to be estimated are θ ls (k) = [ f 1 (k), f 2 (k), g0 (k)]T

(5.19)

Recursive least square method is used to estimate unknown parameters θ ls . θˆ ls (k) = θˆ ls (k − 1) + K ls (k)[y(k) − Φ T (k)θˆ ls (k − 1)]

(5.20)

K ls (k) = P ls (k − 1)Φ ls (k)[λ + Φ T (k) P ls (k − 1)Φ ls (k)]−1

(5.21)

where

P ls (k) =

1 [I − K ls (k)Φ T (k)] P ls (k − 1) λ

(5.22)

λ is the forgetting factor, 0.95 ≤ λ ≤ 0.99, and K ls (k) is the gain matrix of parameter estimation. According to the all-coefficient adaptive control theory [3], the parameter g0 of the nominal plant characteristic model can be set in the range g0 ∈[0.003, 0.3] in engineering. It has been proved that when the ratio of the sampling period  t to the minimum equivalent time constant T satisfies the condition

t Tmin ∈ min

   1 10, 1 3 , there is a definite range for stable plant parameters f 1 (k) and f 2 (k), as per reference [5]. f 1 (k) ∈ [1.4431, 1.9974] f 2 (k) ∈ [−0.9999, − 0.5134] f 1 (k) + f 2 (k) ∈ [0.9196, 0.9999]. Based on this, it is reasonable to limit the range of f 1 , f 2 , and g0 correspondingly when estimating the parameters f 1 , f 2 , and g0 . The intelligent adaptive controller based on characteristic model is composed of a golden section adaptive control law, logical differential control law, and logic integral control law.

236

5 Automatic Control Method and Scheme Design …

5.3.1 Golden Section Adaptive Control We set the discretized difference equation of a linear time-invariant second-order plant with unknown parameters as follows y(k) = α1 y(k − 1) + α2 y(k − 2) + β0 u(k − 1) + e(k)

(5.23)

where e(k) is the zero average white noise. The design control law is given as follows. u(k) = −[L 1 αˆ 1 y(k) + L 2 αˆ 2 y(k − 1)]/βˆ0

(5.24)

where 0 < L 1 ≤ 1, 0 < L 2 ≤ 1. When L 1 = 1 and L 2 = 1, Eq. (5.24) is the minimum variance self-tuning control law. When L 1 = 0.382 and L 2 = 0.618, which are the golden section coefficients, Eq. (5.24) is the golden section adaptive control law. Although the minimum variance self-tuning control law is adopted as the characteristic root of the closed-loop system z 1,2 = 0 when the parameters estimated are equal to the true values, the closed-loop system may be unstable when the parameters estimated are not equal to the true values, so it is difficult to apply in the starting process of the adaptive control system. By adopting the golden section adaptive control law, although the variance of the steady-state output error of the closed-loop system cannot be made minimum when the estimated values of the parameters are equal to the truth values, it can be guaranteed that the closed-loop system has a good dynamic quality, and when the estimated values of the parameters are not equal to the truth values, the closed-loop system is stable. Therefore, as a suboptimal controller, the golden section adaptive controller has important practical value. To prove the robustness of the golden section adaptive controller, two lemmas are given below. Lemma 5.2 Let the discretized difference equation of linear time-invariant secondorder plants with unknown parameters be y(k) = α1 y(k − 1) + α2 y(k − 2) + β0 u(k − 1) + e(k)

(5.25)

where α1 ∈ [1.4, 2], α2 ∈ [−1, −0.5], and α1 and α2 satisfy the condition 0.9 ≤ α1 + α2 < 1, 0.5β0 ≤ βˆ0 < ∞. The controller is designed as follows. u(k) = −[L 1 αˆ 1 y(k) + L 2 αˆ 2 y(k − 1)]/βˆ0

(5.26)

When values of L 1 and L 2 are in the triangle ABC shown in Fig. 5.11, the closed-loop system is stable. Proof The characteristic equation of the closed-loop system is given by the problem.  z + 2

 β0 β0 L 1 αˆ 1 − α1 z + L 2 αˆ 2 − α2 = 0 βˆ0 βˆ0

(5.27)

5.3 Intelligent Adaptive Control Method Based on Characteristic … Fig. 5.11 The value range of controller parameters L 1 , L 2

237

L2 L2=1.9196L1 1

B

N G

0.5

M

C

A 0

0.5

1

L1

L2=1.45−2L1

According to the Jury stability criterion [14], the closed loop system is stable if and only if the following inequalities are true.     β0  L 2 αˆ 2 − α2  < 1  ˆ β0   β0 β0 1+ L 1 αˆ 1 − α1 + L 2 αˆ 2 − α2 > 0 βˆ0 βˆ0   β0 β0 1− L 1 αˆ 1 − α1 + L 2 αˆ 2 − α2 > 0 βˆ0 βˆ0

(5.28) (5.29) (5.30)

Presuming 0.5β0 ≤ βˆ0 < ∞ and 0 < ββˆ0 ≤ 2 is true. 0 According to the value range of each quantity, by solving the compatible solution of linear inequality (5.28)–(5.30), it can be deduced that, the stability of the closedloop system is guaranteed. L 1 and L 2 should satisfy L 2 < 1.9196L 1

(5.31)

L 2 < 1.45 − 2L 1

(5.32)

L2 > 0

(5.33)

On the plane L 1 − L 2 , the values of L 1 and L 2 that meet the requirements are located in the triangle ABC, as shown in Fig. 5.11.

238

5 Automatic Control Method and Scheme Design …

Lemma 5.3 Let the difference equation of discretized linear time-invariant secondorder plants with unknown parameters be Eq. (5.23), and the controller be Eq. (5.24). When L 1 ∈ (0, 1), L 2 = 1 − (1 − L 1 )2 , then the estimated parameters converge to the truth values. We get he following. (1) in the plane Z , the closed-loop poles z i and the open-loop poles z oi are satisfied z i = (1 − L 1 )z oi (i = 1, 2) The closed-loop poles are all in a circle of radius 1 − L 1 . (2) In the plane S, the open-loop poles of the system are closed-loop poles when 1 |ln(1 − L 1 )| to the left. If the open loop system is an underdamped shifted t T system, write g = min ∈ [8, 15], and then

t

ts ≤

tso 1 − 2g ln(1 − L 1 )

where M po is the maximum overshoot of the open-loop system, and tso is the adjustment time of the open-loop system; M p is the maximum overshoot of the closed-loop system, and ts is the closed-loop adjustment time. Proof we know from the problem that when the estimated parameters converge to the true values, the characteristic equation of the closed-loop system is z 2 + (L 1 − 1)α1 z + (L 2 − 1)α2 = 0

(5.34)

If the two poles of the second-order continuous stable plant are p1 and p2 , then the coefficients of the discretized difference Eq. (5.23) can be obtained as α1 = e p1 t + e p2 t , α2 = −e( p1 + p2 ) t . A corresponding open-loop pole as z oi = e p1 t , i = 1, 2. If L 1 , L 2 satisfy the condition L 2 = 1 − (1 − L 1 )2 , where L 1 ∈ (0, 1), then the root of the closed-loop system characteristic Eq. (5.34) is z i = (1 − L 1 )z oi , (i = 1, 2)

(5.35)

For stable plants, |z oi | < 1(i = 1, 2), so |z i | < 1 − L 1 , (i = 1, 2)

(5.36)

1 By transformation equation s = t ln z, the closed loop poles z 1 and z 2 in the corresponding domain S can be calculated as

si = pi +

1 ln(1 − L 1 ), (i = 1, 2)

t

(5.37)

5.3 Intelligent Adaptive Control Method Based on Characteristic …

239

Because 0 < 1 − L 1 < 1, ln(1 − L 1 ) < 0. Therefore, in the plane S, the system’s 1 |ln(1 − L 1 )| to the left and become closed-loop poles. open-loop poles shift t If the open-loop system is underdamped, according to the definition of the maximum overshoot and system adjustment time, it can be deduced that M p ≤ M po e2gπ ln(1−L 1 )

(5.38)

tso 1 − 2g ln(1 − L 1 )

(5.39)

ts ≤

According to Lemmas 5.2 and 5.3, it can be proved that the golden section adaptive control law has the following characteristics [5]. Theorem 5.1 Let the discretized difference equation of the stable linear timeinvariant second-order plants with unknown parameters be (5.23), where, α1 ∈ [1.4331, 1.9975), α2 ∈ (−1, −0.5134], and α1 , α2 satisfy the condition 0.9 ≤ α1 + α2 < 1, 0.5β0 ≤ βˆ0 < ∞, and the controller is shown as Eq. (5.24). When L 1 = 0.382 and L 2 = 0.618, then (1) the closed-loop system is stable. (2) after the estimated parameters converge to the true values, in the plane Z , the closed-loop poles z i and open-loop poles z oi satisfy z i = 0.618z oi (i = 1, 2), and the closed-loop poles are all within the circle with radius of 0.618. On the field S, the open-loop poles of the system are closed-loop poles when shifted   ln(0.618)   t  to the left. In particular, when the open-loop system is an underT damped system, define g = min ∈ [3, 10], then the maximum overshoot of the

t closed-loop is M p ≤ e2gπ ln(0.618) M po , and the closed-loop adjustment time is

ts ≤

tso 1 − 2g ln(0.618)

Proof Lemmas 5.2 and 5.3 show that for the second order plant given as Theorem 5.1, the control law as (5.24), when, L 1 and L 2 take parabolic L 2 = 1 − (1 − L 1 )2 within the triangle ABC part 0.080 < L 2 < 0.403 in Fig. 5.11, i.e., take the points on the curve of arc MN shown in Fig. 5.11, the closed-loop system is stable. After the parameters converge to the true values, for all the stable second-order systems with unknown parameters, the performance of the closed-loop system gets the same degree of improvement, and as L 1 becomes bigger, the control law (5.24) and minimum variance control law become closer, and the performance of the system becomes better. This means that the closed loop poles as close to the origin, the closed-loop maximum overshoot is smaller, and the closed-loop adjustment time is shorter. When L 1 → 0.403, L 1 + L 2 → 1.047. Considering the stability margin for parameter estimation in the worst case, take L 1 + L 2 = 1. Combining L 1 + L 2 = 1 and L 2 = 1 − (1 − L 1 )2 , we get the following solution.

240

5 Automatic Control Method and Scheme Design …

√ 3− 5 L1 = ≈ 0.382 √ 2 5−1 ≈ 0.618 L2 = 2 The point G in Fig. 5.11 is the golden section point. When L 1 = 0.382 and L 2 = 0.618, the conditions satisfy Lemmas 5.2 and 5.3. According to Lemmas 5.2 and 5.3, the conclusion of Theorem 5.1 can be drawn. The design method and robustness conclusion of the golden section adaptive tracking controller for stable plants are presented. Design a tracking controller for the stable plant in Theorem 5.1 u 0 (k) = [TM (k + 1) − αˆ 1 TM (k) − αˆ 2 TM (k − 1)]/βˆ0

(5.40)

u L (k) = −[L 1 αˆ 1 y(k) + L 2 αˆ 2 y(k − 1)]/βˆ0

(5.41)

u(k) = u 0 (k) + u L (k)

(5.42)

where T (k) is the actual output, TM (k) is the output of the tracking model, y(k) = T (k) − TM (k) is the output error, u(k) is the total control quantity, u 0 (k) is the maintenance control quantity, and u L (k) is the feedback control quantity. At this time, the characteristic equation of the closed-loop system is still Eq. (5.25). According to Lemmas 5.2 and 5.3, and Theorem 5.1, when L 1 = 0.382 and L 2 = 0.618, the closed-loop system is stable, and after the estimated parameters converge to the true value, the closed-loop poles z i and open-loop poles z oi satisfy z i = 0.618z oi (i = 1, 2). When reaching a steady state, T (k + 1) = TM (k + 1) + y(k), . 1 y(k) = e(k) 1 − (1 − L 1 )α1 − (1 − L 2 )α2 This means T (k) can track TM (k), and the variance of the output error is not the minimum variance because of 0 < 1 − (1 − L 1 )α1 − (1 − L 2 )α2 < 1. At this time, L 1 and L 2 can smooth switch along L 2 = 1 − (1 − L 1 )2 (0 < L 1 ≤ 1) to L 1 = 1, L 2 = 1, which is the minimum variance control. However, there is no switching problem for parameter slow time varying systems.

5.3 Intelligent Adaptive Control Method Based on Characteristic …

241

5.3.2 Logical Differential Control The function of the logical differential control law is mainly to provide negative velocity feedback in the system transient process and to change the differential strategy according to the system working state and control requirements, so as to increase the system damping, suppress overshoot, and suppress the steady state micro-oscillation. 1. Expression Form of Control Law The logical differential control law has various forms and can be designed according to the needs of the control tasks in different occasions. Reference [18] introduces several commonly used logical differential forms. Here, the following logical differential control laws I and II are introduced. (1) Logical Differential Control I u d1 (k) = −kd1 | y˙ (k)|ed1

(5.43)

where  kd1 =

∀τ ∈ [t − Ts , t], |y(τ )| ≤ r c1 sgn( y˙ ) c1 (0.5|y| + cd1 )−1/2 sgn( y˙ ) ∃τ ∈ [t − Ts , t], |y(τ )| > r  ed1 = ρ max [ y˙ (τ )] t−T ≤τ mT , in which m is a natural number, T is the time interval between two zero crossings of the system output, c1 is a normal number, cd1 is a small normal number added to avoid singularity when |y| = 0, and ρ(x) indicates the limit of x. (2) Logical Differential Control II u d2 (k) = −kd2 | y˙ (k)|ed2

(5.44)

√ where kd2 = c2 |y y˙ |sgn( y˙ ), and c2 is a normal number; 2. Analysis of Noise Suppression Ability To facilitate analysis, the expression of the logical differential control law is written as a continuous function of time t; then, the input signal of the logical differential control law is y(t), its derivative is y˙ (t), and the output is the differential control quantity ud (t). In general, the input signal is obtained through the measurement sensor. When the input signal contains several interferences and measurement noises, the control

242

5 Automatic Control Method and Scheme Design …

quantity may change. To study the robustness of the logical differential control law against interference and measurement noise, this chapter regards these interference and noise as the uncertainty of system input, as follows. y = y0 + δ y , y˙ = y˙0 + δ y˙

(5.45)

where δ y and δ y˙ denote the uncertain quantity. We expect the output ud (t) to be unaffected or less affected by the uncertainty of the input, that is, the logical differential control law to be robust. For the uncertainty of input y(t) and y˙ (t), the sensitivity function is defined as follows. ⎧ ∂u d

u d ⎪ ⎪ ⎨ S y = y = ∂ y (5.46) ∂u d

u d ⎪ ⎪ ⎩ S y˙ = =

y˙ ∂ y˙ The sensitivity function reflects the degree to which the output ud is affected by the input uncertainty. The smaller S y and S y˙ are, the smaller is the output affected by the input uncertainty. Next, we analyze the suppression effect on noise by the sensitivity function to the input uncertainty of the logical differential control law. (1) Logical Differential Control I We discuss the Logical Differential Control I with the following forms, and write it as the following continuous function, u d1 = −kd1 | y˙ (t)|ed1 sgn( y˙ )

(5.47)



c ∀τ ∈ [t − Ts , t], |y(τ )| ≤ r c(0.5|y| + c1 )−1/2 ∃τ ∈ [t − Ts , t], |y(τ )| > r To facilitate the calculation of partial derivative function, Eq. (5.47) can be written

where kd1 = as

u d1 = −kd1 | y˙ (t)|ed1 −1 y˙

(5.48)

In the following analysis, to obtain the influence of different gain coefficients on the sensitivity function, the temporal correlation of y was not considered. Thus,  kd1 =

c |y| ≤ r c(0.5|y| + c1 )−1/2 |y| > r

(5.49)

When |y| ≤ r , if there is uncertainty for y and y˙ , the sensitivity functions of the output ud1 corresponding to the input y and y˙ are

5.3 Intelligent Adaptive Control Method Based on Characteristic …

243

⎧ ∂u ⎪ ⎪ ⎨ Sy = ∂ y = 0 ∂u ⎪ ⎪ ⎩ S y˙ = = −c| y˙ |ed −1 − c(ed − 1) y˙ | y˙ |ed −2 sgn( y˙ ) ∂ y˙

(5.50)

When |y| > r , if there is uncertainty for y and y˙ , the sensitivity functions of the output ud1 corresponding to the input y and y˙ are ⎧ 1 ∂u −1.5 ed −1 ⎪ ⎪ y˙ sgn(y) ⎨ S y = ∂ y = 4 c(0.5|y| + c1 ) | y˙ |

 ∂u ⎪ ⎪ ⎩ S y˙ = = −c(0.5|y| + c1 )−0.5 | y˙ |ed −1 + (ed − 1) y˙ | y˙ |ed −2 sgn( y˙ ) ∂ y˙

(5.51)

(1) When ed1 < 1, we get images of ud1 , S y and S y˙ as shown in the figures below. (We take ed1 = 0.85 and c = 1 here. Because of the limitation of drawing tools, the coordinate dy represents y˙ ). In the following, continuous discussion is given. Because the expressions of S y and S y˙ contain the terms of | y˙ |ed1 −1 and | y˙ |ed1 −2 , the domain of S y˙ is y˙ = 0; the continuity of S y˙ at y˙ = 0 is discussed below. When |y| ≤ r , we get lim S y = 0

y˙ →0

lim S y˙ = −c lim | y˙ |ed −1 − c(ed − 1) lim y˙ | y˙ |ed −2 sgn( y˙ )

y˙ →0

y˙ →0

y˙ →0

= −c lim | y˙ |

ed −1

= −c lim | y˙ |

ed −1

y˙ →0

− c(ed − 1) lim y˙ 2 | y˙ |ed −3 y˙ →0

y˙ →0

− c(ed − 1) lim | y˙ |ed −1

(5.52)

y˙ →0

Since ed1 < 1, ed1 − 1 < 0, lim S y˙ = −∞

(5.53)

y˙ →0

When |y| > r 1 c(0.5|y| + c1 )−1.5 sgn(y) lim | y˙ |ed1 −1 y˙ y˙ →0 4 1 = c(0.5|y| + c1 )−1.5 sgn(y) sgn( y˙ ) lim | y˙ |ed1 y˙ →0 4 =0

lim S y =

y˙ →0

lim S y˙ = −c(0.5|y| + c1 )

y˙ →0

−0.5

 lim | y˙ |

y˙ →0

ed −1

ed −2

+ (ed − 1) lim y˙ | y˙ | y˙ →0

(5.54)  sgn( y˙ )

244

5 Automatic Control Method and Scheme Design …

  = −c(0.5|y| + c1 )−0.5 lim | y˙ |ed −1 + (ed − 1) lim y˙ 2 | y˙ |ed −3 y˙ →0 y˙ →0   = −c(0.5|y| + c1 )−0.5 lim | y˙ |ed −1 + (ed − 1) lim | y˙ |ed −1 y˙ →0

y˙ →0

= −∞

(5.55)

In conclusion, when ed1 < 1, S y , which is defined on R, is continuos, and the domain of S y˙ is y˙ = 0. When y˙ = 0, S y˙ is infinite, which has no continuity. (2) When ed1 > 1, we get images of ud1 , S y and S y˙ , as shown in the figures below. (we take ed1 = 1.85). In the following, continuous discussion is given. Because ed1 > 1, ed1–1 > 0, then When |y| ≤ r lim S y = 0

y˙ →0

lim S y˙ = −c lim | y˙ |ed −1 − c(ed − 1) lim y˙ | y˙ |ed −2 sgn( y˙ )

y˙ →0

y˙ →0

= −c lim | y˙ | y˙ →0

y˙ →0

ed −1

− c(ed − 1) lim y˙ 2 | y˙ |ed −3 y˙ →0

= −c lim | y˙ |ed −1 − c(ed − 1) lim | y˙ |ed −1 y˙ →0

y˙ →0

=0

(5.56)

When |y| > r 1 c(0.5|y| + c1 )−1.5 sgn(y) lim | y˙ |ed1 −1 y˙ y˙ →0 4 1 = c(0.5|y| + c1 )−1.5 sgn(y)sgn( y˙ ) lim | y˙ |ed1 y˙ →0 4 =0

lim S y =

y˙ →0

(5.57)

  lim S y˙ = −c(0.5|y| + c1 )−0.5 lim | y˙ |ed −1 + (ed − 1) lim y˙ | y˙ |ed −2 sgn( y˙ ) y˙ →0 y˙ →0 y˙ →0   = −c(0.5|y| + c1 )−0.5 lim | y˙ |ed −1 + (ed − 1) lim y˙ 2 | y˙ |ed −3 y˙ →0 y˙ →0   = −c(0.5|y| + c1 )−0.5 lim | y˙ |ed −1 + (ed − 1) lim | y˙ |ed −1 y˙ →0

=0

y˙ →0

(5.58)

In conclusion, when ed1 > 1, S y is continuous in ∀y, y˙ ∈ R. When y˙ = 0, S y˙ makes lim S y˙ = 0. y˙ →0

5.3 Intelligent Adaptive Control Method Based on Characteristic …

245

(2) Logical Differential Control II We discuss the Logical Differential Control II with the following forms, and write the continuous function as follows. u d2 (t) = −kd2 | y˙ (t)|ed2 sgn( y˙ )

(5.59)

√ where kd2 = c |y y˙ | = c|y|a | y˙ |d , a = 0.5, and d = 0.5. Transform the form of Logical Differential Control II into [6] u d2 = −c|y|a | y˙ |b y˙

(5.60)

where b = ed2 + d − 1. When there is uncertainty with y (y = 0), the sensitivity function of output ud2 to input y is as follows. Sy =

∂u d2 = −ac|y|a−1 | y˙ |b y˙ sgn(y) ∂y

(5.61)

When there is uncertainty with y˙ ( y˙ = 0), the sensitivity function of the output ud2 to the input y˙ is as follows. S y˙ =



∂u d2 = −c|y|a | y˙ |b + y˙ | y˙ |b−1 sgn( y˙ ) ∂ y˙

(5.62)

(1) When ed2 < 1, we get images of ud2 , S y , and S y˙ , as shown in the figure below (we take ed2 = 0.85, c = 1, here). (2) When ed2 > 1, we also get images of ud2 , S y , and S y˙ , as shown in the figure below (we take ed2 = 1.85). (3) Continuous Discussion Irrespective of ed2 < 1 or ed2 > 1, because the domain of S y is y = 0, there is no continuity, and the following only exists, lim S y = −ac lim |y|a−1 | y˙ |b y˙ sgn(y)

y→0 y˙ →0

y→0 y˙ →0

=0

(5.63)

For S y˙ , the domain is y˙ = 0, so there is no continuity, and only the following exists,

 lim S y˙ = −c lim |y|a | y˙ |b + y˙ | y˙ |b−1 sgn( y˙ )

y→0 y˙ →0

y→0 y˙ →0

=0

(5.64)

246

5 Automatic Control Method and Scheme Design …

(4) Comprehensive Analysis Whether we use the logical differential control I or II in Figs. 5.12 and 5.15 (logical differential control I) and Figs. 5.18 and 5.21 (logical differential control II), a difference can be seen in that the curved surface of the control law is related with the index value of ed , that is, when the index ed < 1, the surface becomes relatively flat, and the logical differential control law changes slightly, which is good to improve the accuracy via slow adjustment and is suitable for the small near balance adjustmen. When exponential value ed > 1, the control law changes fast, so it can provide a negative rate feedback quickly, which is conducive to increasing the damping of the system and returning to the equilibrium position quickly. Similarly, regardless of the form of logical differential control law, the sensitivity functions S y and S y˙ also have a similar trend with the control law surface. Therefore, it can be seen that exponential ed in the logical differential control law has an important influence on the change speed and the noise suppression ability. Sensitivity function Sy . When ed ≤ 1, it can be seen from the comparison of Figs. 5.13 and 5.19 that S y , the sensitivity function of two kinds of logical differential control laws corresponding to y, has different trends. In Fig. 5.13, the S y surface of the logical differential control I appears as a patchwork of several surfaces, but has continuity, and its value within the neighborhood of the origin of coordinates is zero. But in Figs. 5.5, 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16, 5.17, 5.18 and 5.19, the S y surface of logic differential control II does not have continuity, and its value changes quickly with y˙ . When ed > 1, the same conclusion can be drawn by comparing Figs. 5.16 and 5.22. Sensitivity function S y˙ . When ed ≤ 1, it can be seen from the comparison of Figs. 5.14 and 5.20 that the sensitivity functions of the two kinds of logical differential control laws corresponding to y˙ have different shapes. In Fig. 5.14, the value of S y˙ of logic differential control I tends to infinite within the neighborhood y˙ = 0. In Fig. 5.20, the surface of S y˙ for logical differential control II is continuous, and its Fig. 5.12 ud1 of the logical differential control I(ed1 < 1)

8 6 4

ud1

2 0 −2 −4 −6 −8 2 0 y

−2 2

1

0 y·

−1

−2

5.3 Intelligent Adaptive Control Method Based on Characteristic …

247

10 10

8

5

4

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

6 2 Sy

Sy

0

0 −2

−5

−4 −6

−10 −2 0 2 −2

y

−1

1

0

−8

2

−10 −2 −1.5 −1 −0.5

y·

(a)

0 0.5 y· (b)

1

1.5

2

1

1.5

2

Fig. 5.13 S y of the logical differential control I (ed1 < 1)

0 −1

0

−2

−2

−3 −4 Sy·

Sy·

−4 −6

−6 −7

−8 −10 2

−5

−8 2 1

0 y·

−1

0 −2 −2

y

(a)

−9

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

−10 −2 −1.5 −1 −0.5 0 0.5 y· (b)

Fig. 5.14 S y˙ of the logical differential I (ed1 < 1)

value is small. When ed > 1, for the logical differential control I, as shown in Fig. 5.17, the value of S y˙ dramatically changes with y˙ and changes relatively stably with y. For the logical differential control II, as shown in Fig. 5.23, the value of S y˙ changes quickly with y and y˙ , which means, they are sensitive to the uncertainty of y and y˙ (Figs. 5.21 and 5.22). In the discussion, we found that in some cases, the sensitivity function value tends to be infinite, which shows that when y˙ tends to zero ( y˙ is small), the signal-to-noise ratio is small, and the uncertainty caused by noise is not negligible. This kind of situation is unfavorable for use (Figs. 5.24 and 5.25).

248

5 Automatic Control Method and Scheme Design …

Fig. 5.15 ud1 of the logical differential control I (ed1 > 1)

8 6 4 2 ud1

0 −2 −4 −6 −8 2 0 −2 y

y·

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

6 4 2

5 Sy

0

−2

−1

0

1

10 8

10

Sy

2

0 −2 −4

−5 −10 2 0 y

−2 2 (a)

1

−1

0 y·

−2

−6 −8 −10 −2 −1.5 −1 −0.5 0 0.5 y· (b)

1

1.5

2

Fig. 5.16 S y of the logical differential control I (ed1 > 1)

According to the definition of sensitivity function, if the logical differential control law is expected to be robust to uncertainty, it is better when the values of S y and S y˙ are relatively small in the certain range of y and y˙ . Let (y, y˙ ) ∈ [−2, 2], the maximum sensitivity function values of the logical differential control I and II are compared in Table 5.1. The following conclusions can be drawn from the Table 5.1. ➀ Except for the situation of ed < 1, the expression of logical differential control I contains the items | y˙ |ed −1 and | y˙ |ed −2 , which tends to infinity when y˙ = 0. In other cases, the sensitivity function values of the logic differential II are 1.7–3.0 times bigger than that of the logic differential I logic. From this perspective,

5.3 Intelligent Adaptive Control Method Based on Characteristic …

249

0 0

−1 −2

−2

−3 −4 Sy·

Sy·

−4 −6

−6 −7

−8 −10 2

−5

−8 0 y·

−2 −2

0

−1

1

−9

2

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

−10 −2 −1.5 −1 −0.5 0 0.5 y· (b)

y

(a)

1

1.5

2

Fig. 5.17 S y˙ of the logical differential control I (ed1 > 1)

8 6 4

ud2

2 0 −2 −4 −6 −8 2

1

0

−1

−2

2

0

1

y

−1

−2

y·

Fig. 5.18 ud1 of the logical differential control II (ed2 < 1

logical differential control I is superior to logical differential II in the aspect of noise suppression effect. ➁ For the two forms of logical differential control law, the values of sensitivity functions S y and S y˙ are smaller when ed < 1 than that when ed > 1. Therefore, from the perspective of noise suppression, to obtain a better noise suppression effect when using logical differentiation, the parameter ed < 1 can be selected.

250

5 Automatic Control Method and Scheme Design … 10 10

8

5

4 2

0

Sy

Sy

6

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

0 −2

−5

−4 −6

−10 2 0 −2 −2

y·

(a)

0

−1

1

−8

2

−10 −2 −1.5 −1 −0.5 0 (b)

y

0.5 y·

1

1.5

2

0.5

1

1.5

2

Fig. 5.19 S y of the logical differential control II (ed2 < 1

0 −1

0

−2

−2

−3 −4 Sy·

Sy·

−4 −6

−5 −6

−8

−7

−10 2 0 y·

−2 −2

0

−1

1

y

(a)

2

−8 −9

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

−10 −2 −1.5 −1 −0.5 0 (b) y·

Fig. 5.20 S y˙ of the logical differential control II (ed2 < 1)

3. Improvement of the Control Law This section will attempt to improve the parameter selecting method for the logical differential control law II properly to enhance its ability to suppress noise and improve the dynamic performance. The logical differential II control law is given as u d2 = −c|y|a · | y˙ |ed +d−1 y˙

(5.65)

Based on the conclusion obtained from the analysis of logical differential index parameter ed in Sect. 5.3.2, in the transition process, we hope a larger index ed can provide a larger negative feedback for logical differential control law, which is conducive to the dynamic process to enter the steady state quickly and the vibration

5.3 Intelligent Adaptive Control Method Based on Characteristic …

251

8 6 4

ud2

2 0 −2 −4 −6 −8 2 0 y

−2

0

1

2

−1

−2

y·

Fig. 5.21 ud1 of the logical differential control II (ed2 > 1)

10

0

Sy

Sy

5

−5 −10 2 0 y·

−2 (a)

−1 y 0

1

2

10 8 6 4 2 0

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

−2 −4 −6 −8 −10 −2 −1.5 −1 −0.5 0 0.5 y· (b)

1

1.5

2

Fig. 5.22 S y of the logical differential control II (ed2 > 1)

of flexible body can be suppressed quickly. After entering the steady-state process, to suppress the interference of uncertain factors such as noise on the attitude stability, the value of ed is expected to be smaller, to get a smaller value for the sensitivity function of the logical differential control law. Therefore, in this section, ed is designed to be a variable parameter with different values in different response stages to improve the performance of the whole system. According to these principles, this section attempts to make the following improvements in the selection of logical differential II index parameter ed . Take a larger value for ed during the transition process, and after entering the steady state,

252

5 Automatic Control Method and Scheme Design … 0 −1

0

−2

−2

−3 −4

−6

Sy·

Sy·

−4

−5 −6

−8

y= −2 y= −0.7 y= 0 y= 0.7 y= 2

−7

−10 2

−8 0 y·

−1

−2

0

1

−9

2

−10 −2 −1.5 −1 −0.5 0· 0.5 y (b)

y

(a)

1

1.5

maximum values of sensitivity function

Fig. 5.23 S y˙ of the logical differential control II (ed2 > 1) 12 10

max(Sy) max(Sdy)

8 6 4 2 0 0

0.2

0.4

0.6

0.8

1 d

1.2

1.4

1.6

1.8

2

maximum values of sensitivity function

Fig. 5.24 Curve of relationship between the maximum sensitivity function and parameter D 12 11 10 9 8 7 6 5 4 3 2 0

max(Sy) max(Sdy)

0.2

0.4

0.6

0.8

1 a

1.2

1.4

1.6

1.8

2

Fig. 5.25 Curve of relationship between the maximum sensitivity function and parameter a

2

5.3 Intelligent Adaptive Control Method Based on Characteristic … Table 5.1 Comparision of the maximum sensitivity function values

Sensitivity function   max( S y )

  max( S y˙ )

253

Control forms

ed < 1

ed > 1

Logical differential control I

1.4444

2.8887

Logical differential control II

4.5063

9.0125

Logical differential control I

∞

4.9166

Logical differential control II

2.4334

8.4718

take a smaller ed . However, the sudden change of parameters will cause drastic changes in the control law, so a softening tracking curve is adopted to make the change stable. ed (t) = ed∞ − (ed∞ − ed )e−σ t

(5.66)

where ed0 and ed∞ are the values of the index in the transition process and the steady state process, respectively. Parameter σ can control the speed of change of the index. Next, we discuss the influence of different a and d values on the sensitivity functions S y and S y , where a and d are variables, We set the changing range of input (y, y˙ ) ∈ [−2, 2]. When a and d change within the range [0.1, 2], the relationship between the maximum values max(S y ) and max(S y ) of the sensitivity functions in this region and values of a and d are shown by the curve below. According to the principle of taking the value of sensitivity functions as small as optimal, as can be seen from the figures, when the values of a and d are small, the maximum values of the sensitivity functions in the determined region are also small. So we take a = 0.1 and d = 0.1 here. After improvement, the Logical Differential Control II can be written as u d2 = −c|y|0.1 | y˙ |ed (t)+0.1−1 y˙

(5.67)

4. Simulation and Comparison To verify the conclusion of the above analysis on the noise suppression ability of logical differential control law, this section takes the flexible spacecraft as the control plant for mathematical simulation. The control goal is to achieve stable attitude in the condition of large measurement noise, which requires fast control process and high steady-state accuracy. It can suppress the vibration of flexible body and avoid the excitation of the flexible modes. Take the logical differential control law I or the logical differential control II separately, and compose a single loop controller together with the golden section adaptive control law and logic integral control law (hereinafter referred to as the

254

5 Automatic Control Method and Scheme Design …

logical differential I controller and the logical differential II controller, respectively) in order to compare the performance of both controllers. In addition, the double-loop controller is used for comparison. Simulation conditions and requirements are listed in the follows. (1) The initial attitude angles are all zero degree, achieving an attitude maneuver of 2°. (2) Simulation time t is 200 s. (3) To highlight the influence of measurement noise on the control system, the interference moment is not considered. Separately, with three conditions of small noise, middle noise, and large noise, the noise suppression capabilities of the three types of controllers—the logical differential I controller, the logical differential II controller, and the double-loop controller— are discussed through the simulation and compared according to the performance of attitude control and the flexible vibration suppression. (1) Condition of Small Noise Set the mean variance of measurement noise by attitude sensor as 0.02°, and that by gyro as 0.00067°/s. The simulation results of the three controllers under the same conditions are shown in Figs. 5.26, 5.27, and 5.28. The steady-state time and steady-state error values of the control process for each axis are shown in Table 5.2. Figures 5.27, 5.28, and 5.29 show the control effect of the three controllers on a flexible satellite. As can be seen from the figures, the control effect of the three controllers under the condition of small noise is satisfactory. From the data

0

50

100 t/s

150

200

0

50

100 t/s

150

4 2 0 −2 −4

200

4 2 0 −2 −4

×10−3

0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s

150

200

×10−3

0 ×10−3

qz1

ψ/(°)

3 2 1 0 −1

4 2 0 −2 −4

qy1

θ/(°)

3 2 1 0 −1

qx1

φ /(°)

3 2 1 0 −1

0

50

100 t/s (a)

150

200

0

(b)

Fig. 5.26 Noise suppression capability of the logical differential I controller

3 2 1 0 −1

0

50

100 t/s

150

4 2 0 -2 −4

0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

×10−3

qy1 0

50

100 t/s

150

200

0 −3

4 2 0 −2 −4

×10

qz1

ψ/(°)

3 2 1 0 −1

×10−3

4 2 0 −2 −4

200

θ/(°)

3 2 1 0 −1

255

qx1

φ /(°)

5.3 Intelligent Adaptive Control Method Based on Characteristic …

0

50

100 t/s (a)

150

200

0

3 2 1 0 −1 0

50

100 t/s

150

100 t/s

150

200

50

200

qz1

ψ/(°)

3 2 1 0 −1 0

qy1

θ/(°)

3 2 1 0 −1 0

qx1

φ /(°)

Fig. 5.27 Noise suppression capability of the logical differential II controller

50

100 t/s (a)

150

200

4 2 0 −2 −4 4 2 0 −2 −4 4 2 0 −2 −4

×10−3

0 ×10−3

50

100 t/s

150

200

0 ×10−3

50

100 t/s

150

200

0

50

100 t/s (b)

150

200

Fig. 5.28 Noise suppression capability of the double-loop controller

in Table 5.2, the double-loop controller and the logical differential I controller are superior to the logical differential II controller in terms of attitude control time and the flexible vibration suppression.

256

5 Automatic Control Method and Scheme Design …

Table 5.2 Performance comparison when mean variance is 0.02° Accuracy requirement: 0.05°

Logical differential I controller

Logical differential II Double-loop controller controller

Roll

Steady state time (/s)

23.1250

42.0240

18.6430

Steady-state error (/°)

0.0043

0.0058

0.0032

Steady state time (/s)

21.2580

40.5120

19.2150

Steady-state error (/°)

0.0045

0.0048

0.0040

Steady state time (/s)

23.2860

40.1790

20.6520

Steady-state error (/°)

0.0047

0.0073

0.0046

Pitch

3 2 1 0 −1

0

50

100 t/s

150

200

4 2 0 −2 −4

3 2 1 0 −1

×10−3

50

100 t/s

150

200

0

50

100 t/s

150

200

50

100 t/s (b)

150

200

qy1

0 ×10−3

0

50

100 t/s

150

×10−3

4 2 0 −2 −4 0

qz1

ψ/(°)

θ /(°)

3 2 1 0 −1

200

4 2 0 −2 −4

qx1

φ /(°)

Yaw

0

50

100 t/s (a)

150

200

Fig. 5.29 Noise suppression capability of the logical differential I controller

(2) Condition of Middle Noise Set the mean variance of measurement noise by attitude sensor as 0.50°, and that by gyro as 0.01675°/s. The simulation results of the differential I logic controller and the double-loop controller under the same conditions are as follows. Figures 5.30 and 5.31 show the posture control effect of the logic differential I controller and the double-loop controller when the sensor measurement noise variance is 0.50° and the gyro measurement noise variance is 0.01675°/s, respectively. The figures show that the two controllers have better effects on attitude control and

100 t/s

150

50

100 t/s

150

0

50

100 t/s (a)

150

4 2 0 −2 −4 4 2 0 −2 −4

200

200

ψ/(°)

3 2 1 0 −1

0

qx1 50

θ/(°)

3 2 1 0 −1

0

4 2 0 −2 −4

qy1

φ /(°)

3 2 1 0 −1

qz1

5.3 Intelligent Adaptive Control Method Based on Characteristic …

200

257

×10−3

0 ×10−3

50

100 t/s

150

200

0 ×10−3

50

100 t/s

150

200

0

50

100 t/s (b)

150

200

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

3 2 1 0 −1 3 2 1 0 −1

×10−3

qx1

4 2 0 −2 −4

0

50

100 t/s

150

200

4 2 0 −2 −4

0 ×10−3

qy1

3 2 1 0 −1

0

50

100 t/s

150

200

4 2 0 −2 −4

0 ×10−3

qz1

ψ/(°)

θ/(°)

φ /(°)

Fig. 5.30 Noise suppression capability of the double-loop controller

0

50

100 t/s (a)

150

200

0

Fig. 5.31 Noise suppression capability of the logical differential I controller

flexible vibration suppression, and the control process of the double-loop controller enters the steady state a little faster, with a shorter steady state time and higher accuracy. Steady-state time and steady-state error values are shown in Table 5.3. In this condition, the logical differential II controller is invalid, and the simulation result is out of control. Through the simulation test, the tolerance level of the logic

258

5 Automatic Control Method and Scheme Design …

Table 5.3 Performance comparison when mean variance is 0.50° Accuracy requirement: 0.10°

Logical differential I controller

Double-loop controller

Roll

Steady state time (/s)

25.8450

22.1320

Steady-state error (/°)

0.0179

0.0121

Pitch

Steady state time (/s)

24.8950

20.6380

Steady-state error (/°)

0.0412

0.0211

Steady state time (/s)

31.1720

24.6540

Steady-state error (/°)

0.0102

0.0092

Yaw

differential II controller to measurement noise is as follows. The attitude measurement noise variance should be below 0.2° and the gyro measurement noise variance should be under 0.013°/s. (3) Condition of Large Noise

3 2 1 0 −1 3 2 1 0 −1

qx1 0

50

100 t/s

150

200

qy1

3 2 1 0 −1

0

50

100 t/s

150

200

qz1

ψ/(°)

θ/(°)

φ /(°)

Set the mean variance of measurement noise by attitude sensor as 2.00° and that by gyro as 0.067°/s. The simulation results of the differential I logic controller and the double-loop controller under the same conditions are as follows. Figures 5.32 and 5.33 respectively show posture control effect on the flexible satellite of the logic differential I controller and the double-loop controller when the sensor measurement noise variance is 2.00° and the gyro measurement noise variance is 0.0670°/s. Simulation results show that in terms of attitude control rapidity, two

0

50

100 t/s (a)

150

200

4 2 0 −2 −4 4 2 0 −2 −4 4 2 0 −2 −4

×10−3

0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

×10−3

0 ×10−3

0

Fig. 5.32 Noise suppression capability of the double-loop controller

5.3 Intelligent Adaptive Control Method Based on Characteristic …

50

100 t/s

150

200

qy1 100 t/s

150

200

qz1

ψ/(°)

3 2 1 0 −1 0

50

50

100 t/s (a)

4 2 0 −2 −4

×10−3

0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

−3

θ/(°)

3 2 1 0 −1 0

qx1

φ /(°)

3 2 1 0 −1 0

150

200

4 2 0 −2 −4 4 2 0 −2 −4

259

×10

0 ×10−3

0

Fig. 5.33 Noise suppression capability of the double-loop controller

kinds of control strategy perform well, but in terms of the flexible vibration suppression, the double-loop controller is superior to the logical differential I controller. Steady-state time and steady-state error values are shown in Table 5.4. Through the simulation test, the tolerance level of the logic differential I controller to the measurement noise is as follows. The attitude measurement noise variance should be below 2.00° and the gyro measurement noise variance should be under 0.17°/s. When the measurement noise continues to increase, the double-loop controller is superior and can still achieve better control effect. When the measurement noise by attitude sensor is set to be with a mean variance of 3.00° and that by gyro is 0.100°/s, the simulation results of the control with the double-loop controller are shown in the following figure. The last part of 50 s of the simulation process is taken for local amplification, as shown in Fig. 5.34. Table 5.4 Performance comparison when mean variance is 2.00° Accuracy requirement: 0.20°

Logical differential I controller

Double-loop controller

Steady state time (/s)

92.0280

90.6400

Steady-state error (/°)

0.0421

0.0106

Pitch

Steady-state time (/s)

97.3320

88.1490

Steady-state error (/°)

0.0308

0.0336

Yaw

Steady-state time (/s)

85.6320

78.2370

Steady-state error (/°)

0.0662

0.0521

Roll

260

5 Automatic Control Method and Scheme Design …

Table 5.5 The maximum mean variance of measurement noise that the three controllers can bear

Logical differential II controller

Logical differential I controller

Double-loop controller

Attitude measurement noise (°)

0.2

2.0

5.1

Gyro measurement noise (°/S)

0.013

0.17

0.31

It can be seen that in the last 50 s, the attitude angles changed within the expected value range of ±0.1◦ , and the double-loop controller could basically meet the requirements of control accuracy in the environment of large noise. Through a lot of simulation, it is proved that the double-loop controller has a good effect on big noise suppression. When the mean variance of the measurement noise of the attitude sensor is over 3.00°, it can guarantee good attitude stability and accuracy. The maximum mean variance of measurement noise that the three controllers can bear is shown in Table 5.5. With respect to the above typical control effect in measurement noise case, the conclusion can be draw that the various controllers have a certain ability of noise suppression, and the double-loop controller is obviously better than the logical differential I controller and the logic differential II controller with respect to the mean variance of noise restrained. The steady-state time and steady-state accuracy of the double-loop controller are better than those of the other two controllers under the same condition of the noise mean variance. The logic differential I controller is relatively optimal than the logic differential II controller. It can be seen that different forms of logical differential control law have important effects on the control process rapidity, steady-state accuracy, and measurement noise suppression. (4) Noise Suppression Effect Of Improved Logical Differential Control Law For Eq. (5.66), we take the initial index ed0 = 1.28, the final value ed∞ = 0.85, and σ = 0.005 for the logical differential II. According to the Eq. (5.67), the parameter selection improvement is designed for the logic differential II controller. The simulation results are as follows. (1) In middle noise cases, the control effect of the improved logic differential II controller is given below. When there is a large mean variance noise (posture the sensor measurement noise variance is 0.50°, and the gyro measurement noise variance is 0.01675°/s), the original logic differential II controller is invalid, and the improved logic differential II controller can suppress the noise and achieve the better satisfactory control effect. The simulation results are as follows.

5.3 Intelligent Adaptive Control Method Based on Characteristic …

261

The last part of 50 s of the simulation process is taken for local amplification, as shown in the Figs. 5.34, 5.35, 5.36 and 5.37. From the simulation results, the improved logic differential II controller can achieve high control precision and good control effect in the larger measurement noise environment. (2) On the same noise level, the effect comparison of the improved and original logic differential II controller Set the mean variance of measurement noise by attitude sensor as 0.15°, and that by gyro as 0.0067°/s. Under the same measurement noise environment, the control effects of the original and the improved logic differential II were compared. Figures 5.38 and 5.39 respectively show the control effect of the improved logic differential II controller and the original logic differential II controller. Figures 5.40 and 5.41 are the local amplification images of attitude angle curve and flexible modal coordinate curve in the last 50 s, respectively. It can be seen from the results that the improved logic differential II controller is superior to the original logic differential II controller in the control precision and flexible vibration suppression. Steady-state time and steady-state error values are shown in Table 5.6. Based on the discussion above, the conclusion can be made that through the simulation, when the attitude sensor measurement noise variance is 0.9° and the gyro measurement noise variance is 0.028°/s, the logical differential II controller can achieve good control effect. From the simulation results, the improved differential II controller has a certain ability of noise suppression, and under the same noise environment, the improved logic differential II control law, compared with the original logic differential II control law, can improve the system performance, reduce the time

φ /(°)

2.1 2 1.9 150

155

160

165

170

175 t/s

180

185

190

195

200

155

160

165

170

175 t/s

180

185

190

195

200

155

160

165

170

175 t/s

180

185

190

195

200

θ/(°)

2.1 2 1.9 150 ψ/(°)

2.1 2 1.9 150

Fig. 5.34 Local amplification of attitude angles

3 2 1 0 −1 0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (a)

150

200

−3 4 ×10 2 0 −2 −4 0

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

−3

qy

3 2 1 0 −1 0 3 2 1 0 −1 0

qx

5 Automatic Control Method and Scheme Design …

qz

ψ/(°)

θ/(°)

φ /(°)

262

4 ×10 2 0 −2 −4 0 ×10−3 4 2 0 −2 −4 0

Fig. 5.35 Noise suppression capability of the improved logic differential II controller

φ /(°)

2.1 2 1.9 150

155

160

165

170

175 t/s

180

185

190

195

200

155

160

165

170

175 t/s

180

185

190

195

200

155

160

165

170

175 t/s

180

185

190

195

200

θ/(°)

2.1 2 1.9 150

ψ/(°)

2.1 2 1.9 150

Fig. 5.36 Local amplification of attitude angles

of the steady state, and improve the precision to some extent, while suppressing the effect of flexible vibration which is obviously improved.

3 2 1 0 −1

0

50

100 t/s

150

qy1 50

100 t/s

150

100 t/s (a)

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

50

100 t/s

150

200

50

100 t/s

150

200

50

100 t/s (b)

150

200

×10

−5 0

200

−3

qz1 50

100 t/s

0

5

0

50 −3

ψ/(°)

3 2 1 0 −1

0

5

0

150

×10

0 −5 0

200

263

×10−3

−5 0

200

θ/(°)

3 2 1 0 −1

5 qx1

φ /(°)

5.3 Intelligent Adaptive Control Method Based on Characteristic …

3 2 1 0 −1 3 2 1 0 −1

qx1

5

0

50

100 t/s

150

50

100 t/s

150

−5

200

50

100 t/s (a)

×10

0 0 −3

5

0

0 −3

5

0

×10−3

0 −5

200

qy1

3 2 1 0 −1

qz1

ψ/(°)

θ/(°)

φ /(°)

Fig. 5.37 Control effect of the improved logic differential II controller

150

200

×10

0 −5

0

Fig. 5.38 Control effect of the original logic differential II controller

264

5 Automatic Control Method and Scheme Design … 2.1 φ /(°)

φ /(°)

2.1 2 1.9 150

160

170

t/s

180

190

1.9 150

200

θ/(°)

θ/(°)

2 160

170

190

170

160

170

160

170

t/s

180

190

200

180

190

200

170

t/s (a)

t/s

180

190

200

2.1

2 160

2 1.9 150

200 ψ/(°)

ψ/(°)

180 t/s

2.1

1.9 150

160

2.1

2.1

1.9 150

2

180

190

2 1.9 150

200

t/s (b)

Fig. 5.39 Local amplification of attitude angles

×10−3

5 qx1

qx1

5 0

−5 150

160

170

t/s

180

190

200

qy1 160

170

180

190

×10−3

5

0

−5 150

160

170

180

190

200

170

160

170

160

170

t/s

180

190

200

180

190

200

180

190

200

−3

−5 150

200

160

0

t/s qz1

qy1 qz1

5 ×10

0

5

0 −5 150

−3 5 ×10

−5 150

×10−3

t/s

×10−3

0 −5 150

t/s (a)

t/s (b)

Fig. 5.40 Local amplification of flexible modal coordinates

5.3.3 Golden Section Phase Plane Control

1. Phase Plane Control The phase plane method is a graphical method for solving first-order and second-order ordinary differential equations and was first proposed by Poincare. H in 1885. The phase plane control method involves dividing the phase plane composed of output deviation and its change rate of the plant to be controlled into different regions

5.3 Intelligent Adaptive Control Method Based on Characteristic … θ

ˆ

G

R1

R7 F'

265

E' R'6

θS

H'

θV

R'5

B' R'4 I'

R'3

θL

A

C1 C2

O

D'

−θL

C2'

R'2

C1'

A'

−θV −θS

I R3

θD

D

K

R2 θB

R5

H

B E

R4 θˆ

R6 F

R7 R1 G'

Fig. 5.41 Phase plane control partition diagram

Table 5.6 Control effect comparison of the improved and the original logic differential II controller

Original control Improved control law law Roll

Steady state time (/s)

23.6280

18.9560

Steady-state error (/°)

0.0066

0.0056

Pitch Steady state time (/s)

24.3870

19.8070

Steady-state error (/°)

0.0060

0.0019

Steady state time (/s)

25.9050

22.8860

Steady-state error (/°)

0.0223

0.0105

Yaw

and design different control laws according to the motion characteristics of the phase trajectory in different regions to achieve control. With a different partition, the different control laws corresponding to the phase plane control method are also different. Because the phase plane control is not strict to the plant to be controlled, its robustness is good, and the anti-interference ability is strong. Therefore, this model has a very widespread application in industrial control and the spacecraft attitude control.

266

5 Automatic Control Method and Scheme Design …

Figure 5.41 shows a control partition [14] of a kind of attitude phase plane control. The x-axis is the attitude angle estimation, and the y-axis is the attitude angular velocity estimation. The phase plane in the figure is divided into 14 control regions by several switch lines. Because the switch lines are symmetric about the center of the origin O, the right half plane is taken as an example to illustrate the phase plane control parameters and control strategy. Generally, it is assumed that there are attitude control engines with large or small thrust. We define TN as the thruster injection length within a control period, Tmin is the minimum jet injection length, Tmax is the maximum jet injection length. Phase plane control parameters include dead zone threshold θ D , large thrust zone threshold θB , large thrust control parameter ac1 , small thrust control parameter ac2 , maximum angular velocity in step zone θ˙V , maximum angular velocity in small thrust zone θ˙S , speed limit value θ˙L , step zone parameter k j , and step zone threshold θv . The switch θˆ˙ 2

line GC1 C2 D is also designed to satisfy the equation θˆ + 2ac1 k X = θ D , in which k X is also a control parameter. The corresponding jet injection control strategy of Fig. 5.41 can be designed as follows: in region R1 (High Thrust Full Open Zone), a high thrust is adopted, TN = Tmax . In R2 (Low Thrust Full Open Zone), a low thrust engine is operated TN = Tmax . In R3 (Small Thrust   Stepping Zone),   small thrust engines are run, TN = TN 1 + TN 2 , where, TN 1 =

 ˆ˙  θ 

ac2

and TN 2 = K j

 ˆ θ −θe ac2

. In region R4, (High Thrust    ˆ˙  ˙ θ +θ L

Speed Limit Zone), high thrust engines are operated, TN = a1 + Tmin . In region R6 (High R5 (Anti-Slip Zone), low thrust engines are operated, TN = Tmin . In region     θ˙L −θˆ˙ 

Thrust Speed Limit Zone), high thrust engines are operated, TN = a1 + Tmin . All engines are shutdown in region R7, TN = 0. The stability and performance of phase plane control depend on the parameters of phase plane control. Usually, the above control parameters can be determined by trial and experience. With the guidance of golden section adaptive control theory based on characteristic model control, a design method of phase plane control parameters is given. 2. Golden Section Phase Plane Control According to the characteristic model shown in Eq. (5.25), the control law can be designed as u li (k) = −[L 1 αˆ i1 yi (k) + L 2 αˆ i2 yi (k − 1)]/βˆi0 



(5.68)

where α i j (k) and β i0 (k) (i = 1, 2, 3, j = 1, 2) are the estimated characteristic parameters. When L 1 = 1, L 2 = 1, the control law shown in Eq. (5.68) is the minimum variance adaptive control law. When the parameter estimates are equal to the truth values, the minimum variance adaptive control law can guarantee the closed-loop

5.3 Intelligent Adaptive Control Method Based on Characteristic …

267

system has the best control performance. Otherwise, the minimum variance adaptive control law can easily cause system divergence. Therefore, this kind of control law is difficult to be applied in practical engineering. When L 1 = 0.382, L 2 = 0.618, the control law shown in Eq. (5.68) is the golden section phase plane adaptive control law. It can be proved that under certain conditions, the golden section adaptive control law can guarantee stability of the closed-loop system and the robust stability of the unmodeled error [2, 16] when the parameter estimates do not converge to the ‘truth value’, and the closed-loop system has sub-optimal performance when the parameter estimate converges to the ‘truth value’. Inspired by this result, we apply the golden section ratio to the calculation of phase plane control parameters and propose the golden section phase plane adaptive control method. (1) Design the speed limit value θ˙L in the jet injection control law according to the requirements of system delay Tdelay and dynamic performance. If the system delay is Tdelay , a large thrust angular acceleration is a J L , and the range

of angular  velocity allowed by the control performance requirements is θ˙min , θ˙max , then the speed limit value θ˙L in the jet injection control law

 can be designed to meet the condition θ˙L > Tdelay a J L and θ˙L ∈ θ˙min , θ˙max . (2) According to the dynamic performance requirements of the system and considering the speed limit value θ˙L designed in step (1), the maximum angular velocity θ˙V of the stepping zone and the maximum angular velocity θ˙S of the small thrust zone in the jet injection control law are designed. The dead zone threshold θ D and step threshold θv in the jet injection control law are designed according to measurement error θerror and system delay Tdelay . The maximum angular velocity θ˙V in the stepping zone and the maximum angular velocity θ˙S in the small thrust zone determine the heights of regions R2 and R3. The design principle is to raise the heights of regions R2 and R3 as high as possible to avoid triggering flexible vibration on the premise of satisfying the requirements of system response speed. On this basis, it is possible to design θ˙V > θ˙L or θ˙S > θ˙L to make the phase track enter region R3 directly from region R7 after entering the left half plane into the right half plane, or enter region R3 after entering region R2 from region R7. If the measurement error is θerror , the dead zone threshold θ D in the jet injection control law can be designed to meet the condition θ D > θerror + Tdelay θ˙V , θv > θerror . (3) Design the large thrust zone threshold θB in the jet injection control law according to the control accuracy requirements and the dead zone threshold θ D determined in step (2). If the control accuracy requirement is set as θdesired , the large thrust zone threshold θB in the jet injection control law can be designed to make it as large as possible under the premise of meeting the requirements θ D < θB < θdesired .

268

5 Automatic Control Method and Scheme Design …

(4) According to the golden section coefficient, we calculate the small thrust angular acceleration ac2 and large thrust angular acceleration ac1 in the jet injection control law.  The small thrust angular acceleration parameter ac2 = θ˙V (k2 T ), and the  large thrust angular acceleration parameter ac1 = θ˙L (k1 T ), where k2 ∈ 0.8], k1 ∈ [0.2, 0.5], T√is the sampling control period, the optimal k2 is [0.5, √ 5−1 , and the optimal k1 is 3−2 5 . 2 (5) We calculate the step zone parameter k j in the jet injection control law according to the small thrust angular acceleration parameter ac2 . The parabolic coefficient K X in the jet control law is calculated according to the large thrust other phase plane parameters. k j = anglar acceleration  parameter ac1 and  (1 − k2 )ac2 T (θ B − θe ), K X = γ a J L ac1 . γ takes the value range of [1, 4], and θe is a constant slightly less than θ D . If the horizontal and vertical axes of the phase plane are replaced by relative position and relative velocity, the phase plane control method can also be applied to the position translation control. In the rendezvous and docking final approach phase, the chaser approaches the target along the docking corridor. With the decrease of longitudinal relative distance, the control precision of transverse position is required to improve gradually. In this case, the longitudinal relative distance can be selected as the characteristic parameter, and the phase plane parameters that affect the control accuracy can be adjusted adaptively according to the characteristic parameter, so as to achieve the control goal.

5.3.4 Logical Differential Phase Plane Control This section presents a control law design method combining logical differential and phase plane. Figure 5.42 shows a simplified phase plane diagram of an axis (relative position or relative attitude). As shown in Fig. 5.42, the phase plane is divided into nine regions: R0, R11, R12, R13, R14, R21, R22, R23, and R24. The switching lines of these nine zones are symmetric with respect to the center of the origin. Table 5.7 shows the phase plane switching lines equation of the right half plane. In Table 5.7, a J is the thrust acceleration of the engine or angular acceleration, θc is the navigation output of the relative position or attitude angle of the docking frame of the target, and θ˙c is the navigation output of the relative speed or attitude angular velocity. The switch line parameters of the phase planes θ D , θ B , θ˙ s , and θ˙ L are designed according to the control requirements of initial docking conditions. The control law corresponding to Fig. 5.42 is as follows. Region R11: u θ Region R12: u θ Region R13: u θ Region R14: u θ

= −u max = u d1 − k p1 θc = u d2 − k p2 θc = −k p3 θc

5.3 Intelligent Adaptive Control Method Based on Characteristic …

269

 θc G

R0

H′ R24

E′

F′ −θB B′

R11

R23 R22

A′

 θs

−θD D′ C′

C

A R12 θc

 −θL

D θD R13 E

B θB

R14 H′

F

R0

R21

G′

Fig. 5.42 Phase plant

Table 5.7 Switching line equations

Right half plane Line

Equation

GCD

θc +

θ˙c2 2a J

= θD

BF

θc −

θ˙c2 2a J

= θB

AB

θc = θ B

DE

θc = θ D θ˙c = θ˙ s θ˙c = − θ˙ L

CA EFH

Region R0: u θ = 0 Region R21: u θ = u max Region R22: u θ = u d1 − k p1 θc Region R23: u θ = u d2 − k p2 θc Region R24: u θ = −k p3 θc Limit the control quantity. If u θ > u max , u θ = u max . If u θ < −u max , u θ = −u max . k p1 , k p2 , and k p3 are the proportional control coefficient, usually, 0 < k p3 < k p1 , 0 < k p3 < k p2 . u d1 and u d2 are the logical differential terms. The general expression ˙ of logical differential control law is u d = −kd (θc , θ˙c ) f (θ˙c )ed (θc ,θc ) , where f (θ˙c ) is a function of θ˙c , whose function is to normalize θ˙c . kd (θc , θ˙c ) and ed (θc , θ˙c ) are

270

5 Automatic Control Method and Scheme Design …

functions of θc and θ˙c , kd (θc , θ˙c ) representing the equivalent gain, respectively, and ed (θc , θ˙c ) determines the waveform of u d . These parameters are designed to ensure stability of the system. The features of this control law design are that the system overshoot is small and the transition time is short.

5.4 Control Scheme Design for Rendezvous and Docking As described in the introduction, this section specifically introduces the design of control scheme for rendezvous and docking, including the attitude stability control of the chaser in the process of orbit control, the six degrees of freedom control of high-precision relative position, attitude control in the final approach phase, and the long-term relative position maintenance at the hold point.

5.4.1 Attitude Stable Control in Orbit Maneuver The function block diagram of rendezvous and docking closed-loop control system of chaser using jet control is shown in Fig. 5.43.

position controller

guidance control law

target attitude +

control error

−

measurement and navigation

position and attitude estimator

measurement sensors

Force instructions

desired velocity increment

attitude controller

Controller

thruster instruction assignment algorithm

torque instructions

desired force and torque

target actual force and torque

orbit and attitude dynamics

thruster dynamic interference

Fig. 5.43 Function block diagram of rendezvous and docking closed-loop control system

5.4 Control Scheme Design for Rendezvous and Docking

271

In the figure, orbit and attitude dynamics include not only the absolute orbit and attitude dynamics of the tracker but also the relative position and attitude dynamics relative to the target, which are controlled plants in the control system. The dynamic disturbance includes not only the environmental disturbance forces and torques such as orbit perturbation and atmospheric disturbance but also the plume disturbance forces and torques generated when the engine is switched on. Due to the uncertainty of engine switch-up and solar panel rotation, plume interference force and torque are also uncertain. In actual control systems, because of the time required for sensor response, signal transmission, data processing, guidance, navigation and control law calculation, the time delay from the sensor measurement output to the thrusters generating control force and torque is generally large under the current technical conditions. In the process of orbit control, the adaptive control method combining golden section control and phase plane control introduced in Sect. 5.3.3 can ensure the stability of the control system under the condition of a large delay time and large interference. The curves of attitude angles and angular velocity of the chaser with time in the process of orbit control are given in Figs. 5.44 and 5.45. In addition, the curves of multiple orbit control in the rendezvous and docking approaching process are shown in Figs. 5.46 and 5.47. Results show that the control scheme is effective.

roll (°)

5 0 −5 2.24

2.2405

2.241

2.2415

2.242

time/s

2.2425 ×105

pitch (°)

5 0 −5 2.24

2.2405

2.241

2.2415

2.242

time/s

2.2425 ×105

yaw (°)

5 0 −5 2.24

2.2405

2.241

2.2415 time/s

Fig. 5.44 Curves of attitude angles (Orbit control process)

2.242

2.2425 ×105

5 Automatic Control Method and Scheme Design … droll ((°)/s)

272 0.5 0 −0.5 2.24

2.2405

2.241

2.2415

dpitch ((°)/s)

×105

0.5 0 −0.5 2.24

2.2405

2.241

2.2415

2.242

time/s dyaw ((°)/s)

2.2425

2.242

time/s

2.2425 ×105

0.2 0 −0.2 2.24

2.2405

2.241

2.2415

2.242

2.2425 ×105

time/s

Fig. 5.45 Curves of angular velocity (Orbit control process)

roll (°)

1 0

pitch (°)

−1

0

100

200

300

400 500 time/s

600

700

800

900

0

100

200

300

400 500 time/s

600

700

800

900

0

100

200

300

400 500 time/s

600

700

800

900

5 0 −5

yaw (°)

5 0 −5

Fig. 5.46 Curves of attitude angles (Approaching process)

5.4 Control Scheme Design for Rendezvous and Docking

273

droll ((°)/s)

0.05 0 −0.05

0

100

200

300

400

500

600

700

800

900

dpitch ((°)/s)

time/s 0.2 0 −0.2 0

100

200

300

400 500 time/s

600

700

800

900

0

100

200

300

400 500 time/s

600

700

800

900

dyaw ((°)/s)

0.2 0 −0.2

Fig. 5.47 Curves of angular velocity (Approaching process)

5.4.2 Accurate Six-Degree-of-Freedom Relative Position and Attitude Control The control of the translational approach process is relatively complex. It is necessary to consider the control of the relative position and relative attitude at the same time and overcome the coupling between attitude and orbit control. The goal of control is to meet the initial docking conditions. The initial docking conditions are defined including the relative positions, relative velocity, relative attitudes, and angular velocity when the first mechanical contact between the docking mechanisms of two spacecraft, which are defined in the docking frame of the chaser respect to that of the target. Before the docking mechanism contact, the controller should ensure that the chaser meets the initial docking conditions shown in Table 5.8. Based on the navigation outputs relative to the target docking frameframe, namely the estimate of the relative position and relative attitude of the chaser docking frame relative to that of the target, the relative position and relative attitude are controlled. Six degree-of-freedom control is adopted, that is, the relative position control of the three axes and relative attitude control of three axes. The six controllers can adopt the same control law and different control parameters. The control method combining logical differential and phase plane is adopted, which can ensure a higher control accuracy when the control system is stable.

274

5 Automatic Control Method and Scheme Design …

Table 5.8 Initial docking conditions

Project

Requirements √

Y 2 + Z 2 < 0.15 m

Lateral displacement Lateral velocity

Vy2 + Vz2 ≤ 0.1 m/s

Axial velocity

0.05 m/s ≤ V x ≤ 0.3 m/s

Pitch and yaw angle

| θ|, | ψ| ≤ 4°

Pitch and yaw angular velocity

|ωy| , |ωz | ≤ 0.8°/s

Roll angle

| ϕ| ≤ 4°

Roll angular velocity

|ωx | ≤ 0.8°/s

X/m −2

0

50

100

150

200

250

0 2

Z/m

4 6 8 10 12 14 16

Fig. 5.48 Relative position in the X-Z plane

The curves of relative positions of in and outside the orbital plane with longitudinal relative distance in the translational approach process are shown in Figs. 5.48 and 5.49. Figures 5.50 and 5.51 show the curves of the attitude angles and angular velocity with time. The target distribution diagram of the transverse relative position at the contact time is given in Fig. 5.52. Results show the control scheme is effective (Figs. 5.53 and 5.54).

5.4.3 Relative Position Control at the Hold Points for Long Period The balance between the precision of position control and the consumption of propellant should be considered when designing control parameters of the relative position of the hold point.

5.4 Control Scheme Design for Rendezvous and Docking

275

X/m −4

0

50

100

150

200

250

−2 0

Y/m

2 4 6 8 10 12

Fig. 5.49 Relative position outside the X-Y plane

roll (°)

5 0 −5

0

100

200

300

400 time/s

500

600

700

800

0

100

200

300

400 time/s

500

600

700

800

0

100

200

300

400 time/s

500

600

700

800

pitch (°)

2 0 −2

yaw (°)

5 0 −5

Fig. 5.50 Attitude angles of the chaser

dyaw ((°)/s)

dpitch ((°)/s)

droll ((°)/s)

276

5 Automatic Control Method and Scheme Design … 0.2 0 −0.2 100

200

300

400 time/s

500

100

200

300

400 time/s

500

100

200

600

700

0.1 0 −0.1 600

700

0.1 0 −0.1 300

400 time/s

500

600

700

Fig. 5.51 Attitude angular velocity of the chaser −0.2

−0.1

Y/m 0

0.1

0.2

−0.2

Z/m

−0.1

0

0.1

0.2

0.3

Fig. 5.52 the target distribution diagram of relative position Y-Z at the contact time

0.3

5.4 Control Scheme Design for Rendezvous and Docking

277

1.5 1

y·

0.5 0

−0.5 −1 −1.5 −1000

−500

0 y

500

1000

Fig. 5.53 Phase plane trajectory outside the plane

0.8 0.6 0.4

z·

0.2 0

−0.2 −0.4 −0.6 −0.8 −50

0

50

100

150

200

250

300

350

400

z

Fig. 5.54 Phase plane trajectory of Z-direction control

(1) Outside the Orbital Plane The relative motion dynamics equation outside the orbital plane is as follows: y¨ + ω02 y = a y Its trajectory change trend is

278

5 Automatic Control Method and Scheme Design …

⎧ ⎨ y(t) = y˙0 sin(ω t) + y cos(ω t) 0 0 0 ω0 ⎩ y˙ (t) = y˙0 cos(ω0 t) − y0 ω0 sin(ω0 t) Thus, the change trend of the curve outside the orbital plane is a sinusoidal curve, which is not coupled with the curve inside the orbital plane, and the maximum deviation outside the plane can be calculated as !

 y02

+

y˙0 ω

2

For example, if the position error (3σ ) is 1 km and the speed error (3σ ) is 1 m/s, the maximum deviation is about 450 m. For the hold point with a relatively distant position, such as 5 km, if no control is applied outside the orbital plane during an orbital period, it will not cause unsafe problems, and it is also close to the control requirements of the holding position, which can reduce propellant consumption. The uncontrolled phase plane trajectory outside the plane is shown in Fig. 5.48. If longer hold time or higher accuracy are expected, the control outside the orbital plane is better applied near the zero position, which can reduce propellant consumption. (2) In the Orbital Plane The relative motion dynamics equation in the orbital plane is 

x¨ + 2ω0 z˙ = ax z¨ − 2ω0 x˙ − 3ω02 z = az

It can be seen that a constant interference term 3ω02 z in Z-direction will not be neglected after a certain period of time. This means that in the design of control parameters, if you want to save propellant by widening the position and maintaining error, you may have a negative effect. On the contrary, under the premise of not causing frequent jet injection, proper improvement of control accuracy in the Zdirection can reduce propellant consumption. The phase plant method can be adopted in the Z relative position control. The phase plane trajectory of the Z-direction control in the orbital plane is shown in Fig. 5.49.

5.5 Key Points of Control System Design The control system is the core part of the rendezvous and docking system, whose performance and stability directly determine the success or failure of the rendezvous and docking missions. The design of rendezvous and docking control system should

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focus on three issues. First, the initial docking conditions are the most important performance requirements of the rendezvous and docking control system, which mainly depend on the geometric size and performance of the docking mechanism, mass and inertia characteristics of the two spacecraft to be docked, and the impact resistance of important components such as solar panels. The realization of the docking initial conditions is related not only to the force of the tracker engines, mass, inertia, flexibility of the solar panels, liquid sloshing, and other characteristics but also to the geometric dimension of the two spacecraft to be docked, the time delay of the control system, plume interference characteristics, navigation accuracy, and control law design. Therefore, in the design of rendezvous and docking control system, it is required to take all aspects into consideration to reach the highest control precision. Second, if the tracker is with flexible solar panels or serious liquid sloshing, or the disturbance force or the disturbance moment of the control system is large and the time delay of the control system is large, special focus is needed to ensure closed-loop system stability by the design of the control law. Finally, design of rendezvous and docking control system needs to consider the reduction of the propellant consumption as much as possible on the premise of satisfying the performance and stability requirements.

References 1. Charbonnel, C.H.: Controller design and M-analysis: powerful tools for flexible satellite attitude control. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, Ontatio, Canada (2010) 2. Grocott, S., How, J., Miller, D., et al.: Robust Control design and implementation on the Middeck active control experiment. J. Guid. Contr. Dyn. 17(6), 1163–1170 (1994) 3. Wu, H.: Theory and Application of the All-Coefficient Adaptive Control Method. National Defense Industry Press, Beijing (1990). (In Chinese) 4. Wu, H., Hu, J., Xie, Y.C.: Intelligent Adaptive Control Method Based on Characteristic model. Chinese Science And Technology Press, Beijing (2001). (In Chinese) 5. Xie, Y., Wu, H.: The application of the golden section in adaptive robust controller design. Chin. J. Autom. 4(3), 197–205 (1992) 6. Xie, Y.: Spacecraft robust adaptive control method and its application. Ph.D. thesis, Beijing Institute of Control Engineering (1994) (In Chinese) 7. Xie, Y., Wu, H.: Robustness of the all-coefficient adaptive control method. J. Autom. 23(2), 151–159 (1997). (In Chinese) 8. Wang, Y.: Stability analysis of characteristic model based adaptive control method for a kind of minimum phase nonlinear system. Control Theory Appl. 29(9), 1097–1107 (2012). (In Chinese) 9. Huang, J., Kang, Y., Meng, B., Zhao, Y., Ji, H.: Characteristic model based adaptive controller design and analysis for a class of SISO systems. Sci. China Inf. Sci. 59(5), 1–15 (2016) 10. Jiang, T., Wu, H.: Sampled-data feedback and stability for a class of uncertain nonlinear systems based on characteristic modeling method. Sci. China Inf. Sci. 59(092205), 1–15 (2016) 11. Jiang, T., Wu, H.: A framework for stability analysis of high-order nonlinear systems based on CMAC method. Sci. China Inf. Sci. 59(112201), 1–20 (2016) 12. Wang, Y.: Research on attitude control of hypersonic vehicle based on characteristic model. Ph.D. thesis. Beijing Institute Of Control Engineering (2012) (In Chinese)

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13. Mikael, J.: Piecewise Linear Control System. Springer (2003) 14. Jury, E.I.: On the roots of a real polynomial inside the unit circle and a stability criterion for linear discrete system. 2th IFAC, paper 413 (1963) 15. Bukley, A.P.: Hubble space telescope pointing control system design improvement study results. Guid. Contr. Dyn. 18(2), 194–199 (1995) 16. How, J., Glaese, R., Grocott, S., et al.: Finite element model-based robust controllers for the Middeck active control experiment (MACE). IEEE Trans. Contr. Syst. Technol. 5(1), 110–118 (1996) 17. Kolonic, F., Poljugan, A., Petrovic, I.: Tensor product model transformation-based controller design for gantry crane control system-an application approach. Acta Polytech. Hung. 3(4), 95–112 (2006) 18. Yu, X., Xie, Y.: Analysis and design of logical differential control law. In: Guadalajara: 67th International Astronautical Congress (2016)

Chapter 6

Manual Control Method and Scheme Design for Rendezvous and Docking

6.1 Introduction The manual control system is a system for astronauts to complete the control tasks of attitude control, rendezvous and docking control, and return and reentry control of the chaser, and monitor the automatic control process with the cooperation of the Instrument and Lighting Subsystem. The basic parts of a manual control system include manual controller, display instruments, control panels, observation windows, TV camera, attitude control handle, position control handle, inertial measurement unit for manual control, control circuit, inertial measurement unit, optical attitude sensors (sun sensors, infrared Earth sensor, etc.), and executive components (propulsion system), which are common in automatic control systems. In addition, manual control systems also include the measurement information of laser lidar to assist the astronaut in judgment. The composition of the manual control system is shown in Fig. 6.1. In general, the astronaut and the equipments mentioned above are located in the chaser, and the signs for the television cameras are installed in the target. The astronauts by observing the images of the target and/or the signs displayed on the instruments, estimate the relative motion state of the chaser to the target, and operate the handles to complete the manual rendezvous and docking mission. In another case, astronauts, control handles, display instruments, and signs are on the target, and other components are on the chaser. In this case, the camera image data and the controller display data are compressed and transferred to the coding and decoding unit in the target via a wireless link channel. The extracted display data are directly sent to the display instrument, and the compress image data are extracted and transformed to the display instrument. The astronaut determines the attitude and position of the chaser according to the images and data, controls the handles and panel instruments, and sends the control instructions. The control instructions are encoded by the communication machine again and transferred via a wireless link channel to the track spacecraft. The controller on the chaser receives the control instructions and samples the sensor measurement data for orbit and attitude calculation to generate © National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_6

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Analogy sun sensors

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Control panel Attitude control handle

Infrared earth sensor

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Fig. 6.1 The composition of the manual control system

the control commands, which control the attitude and position engines, to finish the manual rendezvous and docking task. This situation is also known as remote operation rendezvous and docking. The principle of remote operation rendezvous and docking is shown in Fig. 6.2. In the rendezvous and docking process, the manual control system is an auxiliary control and backup means for the final translational and closing section. There are generally several situations when the automatic control system is transferred to the manual control system: ➀ both the automatic and manual control system system are normal, and rendezvous and docking experiment with the manual control system needs to be carried out; ➁ the automatic control system has failed (such as there is an issue with the automatic measurement sensors), and the manual control system is needed by ground orders. ➂ the astronauts discover that the automatic control system is working abnormally through visual observation and switch to the manual control system by ground orders. Considering the above task requirements, the design principles of the manual control system are as follows: (1) the manual rendezvous and docking control system should be designed as a relatively independent control system as far as possible. When the components of the automatic control system fail, it can act as a backup system for the automatic control system. The switching between each other should be reliable and the faults can be isolated. (2) the configuration of the manual rendezvous and docking control system should be simple and reliable to achieve the goal of completing the rendezvous and

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Engines

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Signal compression and decompression devices

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Fig. 6.2 Schematic diagram of the manual remote operation

docking task and ensuring safety with the least measurement sensors and execution components; (3) the special conditions of human factors should be taken into account when designing the manual rendezvous and docking control system, such as mental state, training fatigue, the influence of space factors, and the delay of human reaction, so astronauts should operate the system as little as possible. (4) the manual rendezvous and docking control system should be able to provide complete and reliable display information for astronauts. Based on the basic composition of the manual control system, this chapter mainly introduces the navigation and control principles, design methods and schemes of the manual rendezvous and docking control system, and gives the corresponding man-controlled rendezvous and docking operation method.

6.2 Measurement Principle and Scheme Design 6.2.1 Sensors for Manual Rendezvous and Docking

1. Camera and Sign TV cameras and signs are the main optical measurement equipments for chaser docking with the target. A TV camera is installed on the chaser, and the matching sign is installed on the target.

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The TV camera provides the image of observing the target and signs from the chaser and transmits it to the monitor on the chaser, as shown in Fig. 6.3. The sign is generally composed of a chassis and a cross mark with a certain distance from the horizontal plane of the chassis. According to the relative position of the image, as well as the size and position of the cross mark from the chassis on the screen, astronauts can estimate the relative state of the longitudinal distance, transverse position, and relative attitude. The sign of the international space station is installed in the center of the docking port [1]. The radius of the circle is 76 mm. The cross mark is a little bigger than the circle, with 305 mm from the chassis plane. The sign of the Russian manual control system is installed near the docking port. The chassis is a diamond, which was changed to a circle [2]. To cooperate with the manual rendezvous and docking task of the ShenZhou chaser, the sign shown in Fig. 6.4 is installed on the Chinese Space Laboratory. The background is painted black with white lines, including long cross lines and short angle lines. The cross extends the background plane to a certain height, and the cross is painted white. For convenience of later descriptions, the two frames are given as follows. The sign frame ot xt yt z t is denoted as {ot }, whose origin ot is the center of the sign’s background plane. The axis ot xt is perpendicular to the surface of the background plane and points to the opposite direction of the cross. The axis ot yt is parallel to the middle line of the cross on the surface of the background plane and points to the

Fig. 6.3 TV image display screen

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D

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L

Xt

yt

H

Zt (a)

Zt (b)

Fig. 6.4 An example of a sign

right when viewed along with the direction of the axis ot xt . The axis ot z t forms the right hand system together with the axes ot xt and ot yt . The TV frame oT V x T V yT V z T V is denoted as {oT V }, whose origin oT V is the optical center of the TV camera. The axis oT V x T V is the main optical axis of the TV camera, pointing outwards. The axis oT V yT V is perpendicular to the axis oT V x T V and points to the right when viewed along the direction of the axis oT V x T V . The axis oT V z T V forms the right hand system together with the axis oT V x T V and oT V yT V . To enable astronauts to use a TV camera and sign for manual rendezvous and docking measurement and to meet the requirements of high precision measurement under normal conditions and safety judgment under abnormal control conditions, it is necessary to design the geometric size and scale of the sign, as well as the camera’s view field and lines. Specific methods are as follows. (1) Design of the sign geometry The background plane diameter D of the sign is designed to be between 1/8 and 1/12 of the maximum diameter of the outer contour of the target. The longitudinal distance H between the cross and the background plane is about 1.5 times the diameter D. The length L of the cross is 0.45 D, and the width is designed to be 0.05 D. (2) Design of the TV camera view field The TV camera view field is designed according to the requirements of docking corridor, initial docking conditions, target geometry, installation conditions, and measurement accuracy. ➀ Determine the design conditions The docking corridor is a cone with the docking axis as the symmetry axis and the angle γ AC as the half angle. The vertex of the cone is determined by the transverse position deviation in the initial docking condition.

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 Initial docking conditions are listed here, transverse position y 2 + z 2 ≤ Rdock , relative roll angle |ϕ| ≤ ϕ, relative pitch angle |θ | ≤ θ , and relative yaw angle |ψ| ≤ ψ. Set the three-axis installation deviation of the sign relative to the target’s docking mechanism as ϕT A , θT A , and ψT A , and the three-axis installation deviation of the TV camera relative to the tracker’s docking mechanism as ϕC A , θC A , and ψC A . ⎡ ⎤ xTV Set the installation position of the TV camera in the tracking frame as ⎣ yTV ⎦ , z TV {d } 2 and the installation matrix is the unit matrix. The installation position of the sign in ⎡ ⎤ xtarget the target frame is ⎣ ytarget ⎦ , and the installation matrix is the unit matrix. z target {d1 } Then, we require the TV camera to be mounted directly with the sign. Therefore, yTV = ytarget and z TV = z target . ➁ Calculate the relative attitude According to the requirements of initial docking conditions, C{d2 }{d1 } C y (θ )C x (ϕ)C z (ψ)

=

C{Ot }{OT V } = C{Ot }{d1 } C{d1 }{d2 } C{d2 }{OT V } ⎡ ⎤⎡ ⎤T ⎤T ⎡ 1 ψT A −θT A 1 ψC A −θC A 1 ψ −θ ≈ ⎣ −ψT A 1 ϕT A ⎦⎣ −ψ 1 ϕ ⎦ · ⎣ −ψC A 1 ϕC A ⎦ θT A −ϕT A 1 θC A −ϕC A 1 θ −ϕ 1 ⎡ ⎤ 1 ψT A − ψC A − ψ −(θT A − θC A − θ ) = ⎣ −(ψT A − ψC A − ψ) 1 ϕT A − ϕC A − ϕ ⎦ −(ϕT A − ϕC A − ϕ) 1 θT A − θC A − θ ➂ Calculate the relative position When the relative longitudinal distance between the tracker and the target is R and the transverse distance is on the edge of the docking corridor (ϕ = 0◦ − 360◦ ), the position coordinate of any point on the sign relative to the TV camera is

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➃ Determine the size of the view field The boundary point on the sign and longitudinal distance R are traversed within the working range of the sensor, and the maximum angle of view field is deter⎡ ⎤ x mined according to the angle between ⎣ y ⎦ and the main optical z Body{O }/{O } t TV ⎡ ⎤ 1 axis ⎣ 0 ⎦ . 0 {O } TV

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(3) Design of background plane lines The carved lines on the sign background plane are designed according to the measurement accuracy and key points interpretation requirements, as shown in Fig. 6.4. The color of the background plane should be in strong contrast with the color of and the lines and the cross for on-orbit identification. For example, the background plane in Fig. 6.4 is black and the lines are white. The design of the lines ensures that the width of the cross is the same as the width of the lines on the background plane when seeing from the TV camera at the docking time, and the cross mark and cross line on the chassis are overlapping during accurate docking. (4) Design of lines superposition on TV camera images Depending on the size of the camera’s view field and taking into account the sharpness of the image, it is possible to design the superimposed lines on the TV camera image shown in Fig. 6.3. The design of calibration lines is considered from the intuitive point of view, and it should be ensured that the outer contour of the target or the background plane in the image is exactly on the calibration line at the key moments (such as parking and docking points) to ensure that astronauts can quickly judge whether the control of the tracker is normal at those key moments. 2. Lidar The lidar is installed on the chaser, and the measurement schematic diagram is shown in Fig. 6.5. It is assumed that the lidar measurement frame coincides with the chaser system, the lidar is installed at the center of mass of the chaser, and the transponder (angular reflector) is installed at the center of mass of the target. The measurement ρr

αr

Orad

Xrad βr

yrad Zrad

Fig. 6.5 Schematic diagram of Lidar measurement

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output of the lidar is the sight distance ρr of the target relative to the chaser, the elevation angle αr of the target relative to the chaser, and the azimuth angle βr of the target relative to the chaser. The relative position of the center of mass of the target relative to the center of mass of the chaser can be calculated using the measurement information of the lidar and the attitude information of the chaser.  ⎧ ⎪ ⎪ ρr = xr2 + yr2 + zr2 + vρr ⎨ αr = sin−1 (−zr /ρr ) + vαr ⎪ ⎪ ⎩ βr = tan−1 (yr /xr ) + vβr Based on the above measurement information, the relevant frame transformation and relative navigation calculation can be carried out to calculate the relative position and relative velocity between the docking interfaces of the two spacecraft, which can be displayed on the instrument interface for astronauts to observe, mainly for controlling the relative velocity.

6.2.2 Relative Motion State Determination The determination of the relative motion state of the manual rendezvous and docking includes attitude determination, relative position estimation, and relative velocity determination. Attitude determination is sorted into several methods, determining the orientation attitude of the chaser relative to the target, determining the attitude to the ground, and judging the relative attitude according to the images observed by TV camera. Relative position estimation is based on the image observed by the TV camera. The determination of relative speed is mainly based on the measurement information of the distance measuring equipments or the estimation of the size of the target. The following is a detailed introduction of the methods for determining the motion status for astronauts by using various information. During the manual rendezvous and docking process, the measurement information of the lidar, the image of the target, sign observed by the TV camera, and other available information can be used to judge and determine the relative motion state between the two spacecraft. When the relative distance is large enough, all images of the target are displayed on the screen, and the astronauts estimate the relative position and attitude through the complete image of the target. As the distance decreases, a portion of the image of the target is displayed on the screen, at which time, only the most effective features can be used for navigation. In the final translational and closing section, just a small part near the docking port can be observed, and at this time, the special sign installed on the target can mainly be used. At the same time, it should be ensured that in a certain relative distance and attitude range, the target of the image will not exceed the scope of the screen, which means the target is in the TV camera’s field of view for docking.

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1. Definition of Relative Motion Parameters The relative motion parameters of the manual rendezvous and docking include relative distance, rendezvous speed, angle of the target relative to the sight line (pitch angle and yaw angle), angle of the TV camera relative to the sight line (pitch angle and yaw angle), and roll angle of the target relative to TV and sight angular velocity, which are defined as follows: (1) Relative distance refers to the distance between the origin of the docking frame of the target and that of chaser. Because the TV camera is installed fixed on the chaser with certain focal length and the sign is installed fixed on the target, the image size of the docking mechanism and the sign on the TV camera shows a determined relationship with the relative distance. (2) Rendezvous speed is the rate of change of relative distance with time. (3) Sight line begins at the origin of TV frame and ends at the observation point. In the manual control rendezvous and docking process, the observation target is constantly adjusted with the distance. For example, when the distance is relatively large, the target is almost a bright spot, and the observation target is the whole target. As the distance reduces and the docking mechanism becomes clear, the center of the docking mechanism becomes the observation target. When the sign is clear, the center of the sign background plane becomes the target. (4) Angle of the target relative to the sight line (pitch angle and yaw angle) is defined as the angular deviation of the target body coordinate axis x relative to the sight line. At a large distance, it is the angular deviation of the target body coordinate axis x relative to the sight line. It is the angle of the sign frame axis x relative to the sight line when it is near, that is the angle between the line from the center of the cross mark to the origin of the sign frame and the sight line. It is defined as the yaw angle along the projection of the sign frame axis z, and as the pitch angle along the projection of the sign frame axis y. The angle of the target relative to the sight line is mainly determined by the lateral position deviation of the chaser relative to the target and the attitude of the target itself. When the attitude of the target is oriented to the ground, the angle of the target relative to the sight line is mainly caused by the lateral position deviation of the tracking vehicle relative to the target. This angle causes the image of the sign cross mark and the cross line on the background plane to not coincide in the image seen by the TV camera. (5) Angle of the TV camera relative to the sight line (pitch angle and yaw angle) is defined as the angle of the TV frame axis x relative to the sight line. It is defined as the yaw angle along the projection of sign frame axis z and as the pitch angle along the projection of the sign frame axis y. The angle of the TV camera relative to the sight line is mainly determined by the attitude of the chaser itself. When the chaser is moving along the line of sight orientation, even when the transverse position deviates from the chaser to the target, the sight line of the

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spacecraft is zero. In this case, the target image will always be in the middle of the display image. (6) Roll of the target relative to TV refers to the angle of target frame relative to TV frame along the axis x. 2. Estimation of Relative Motion Parameters The method to estimate relative motion parameters by using manual control measurement equipment is introduced as follows. (1) Relative distance. Taking the cylindrical segment, docking mechanism, and sign of the target as the reference size, the relative distance parameters are determined by their imaging size on the TV camera. The measurement output of lidar ρ reflects the relative distance information, which can also provide reference for astronauts. (2) Rendezvous speed, which is estimated according to the change of distance in a certain time. The measurement output of lidar ρ˙ reflects the relative velocity information, which can provide reference for astronauts. (3) The angular deviation of the target relative to the sight line, which is determined by the relative position of the cross mark on the imaging surface relative to the cross line of the target background plane. The pitch angular deviation causes the cross mark to move up and down relative to the cross line. When the target is tilted upward, the cross mark is tilted relative to the line, and vice versa. The yaw angular deviation causes the cross mark to move left and right relative to the cross line. When the target deviates to the left, the cross mark also deviates to the left relative to the background cross line, and vice versa. (4) The angular deviation of the TV relative to the sight line, which is determined according to the moving distance of the target or sign on the imaging surface of the TV camera relative to the cross line of the display. Pitch angular deviation causes the up and down motion of the target or sign image on the LCD. When the chaser moves up, the image moves down, and vice versa. Yaw angle causes deviation left-right movement of target or sign image. When the tChaser moves to the left, the image moves to the right, and vice versa. The measurement output of lidar α reflects the deviation information of pitch angle, and the measurement output of lidar β reflects the deviation information of yaw angle. (5) Relative roll angle, which is determined by the angle of the horizontal line of the target, solar panels, and sign bracket on the imaging plane relative to the reference horizontal line. For the chaser, if the image is rotated clockwise relative to the display cross, the target is rotated clockwise relative to the chaser, and the chaser is rotated counterclockwise relative to the target, and vice versa. (6) Angular velocity of sight line: estimated by the variation of the sight line angular deviation of the target in a certain period of time.

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6.2.3 Display of Image and Data Instrument lighting and display devices include integrated display screens, TV camera display screens, various indicators, and lights for chaser manual motion control and manned rendezvous and docking tasks. The parameters of the rendezvous and docking control system can be displayed on the integrated display screens in the form of attitude graph, trajectory diagram, relative attitude position diagram, number, and text. The image of the TV camera is displayed on a special display screen, during which the distance and the velocity measured by the lidar can be superimposed on the transmitted TV image as required. Human-computer interaction design of the Apollo manned lunar landing project of the United States is an early and relatively successful example of human-computer interaction and image-data display [3]. Special devices for manual control include the display and keyboard components, attitude control handle, position control handle, landing point indicator, and attitude indicator. The Apollo lunar modules mainly rely on human eyes to observe the terrain of the outer landing zone through a porthole with a coordinate frame as a sighting device. The display and keyboard components of the Apollo are used to send instructions to the control computer and monitor the calculation data of the landing process for the astronauts. The attitude indicator can provide display information, such as the attitude angle, attitude deviation, and attitude angular velocity of the roll, pitch, and yaw axes of the lander. The moon project of the United States at the beginning of the century used the Altair for landing on the moon, which adopted a more abundant and complicated display page, mainly including the flight home page, the landing point resetting page, and the landing trajectory display page [4]. ➀ The flight home page. The background of this page is the real-time lunar surface topography image generated by data processing and rendering. The flight phase and time are displayed on the top of the page, the scale and the current height instructions are shown on the left side, while the propellant, attitude, and radar state information are displayed below. ➁ The landing point resetting page. Information of three recommended landing points is superimposed on the landing area topographic map, and other data are the same as the flight home page. ➂ The landing trajectory display page. The predicted landing track and the upper and lower boundary lines of the landing corridor are displayed in the center of the page. The distance from the target point and altitude data are superimposed. The scale on the left shows the current speed and acceleration. The rest is the same as the flight main page. Orion is a new generation of the U.S. manned spacecraft that will rendezvous and dock with the Space Station [1]. In addition to the operating handles, the display interface consists of three color flat panel monitors. The screen in the middle displays the camera image, and the auxiliary driving tools are superimposed on the image including the dead zone indicator, flight path markers, translational control indicator, and feedforward controller to help astronauts decouple the position error caused by the attitude, and forecast the control input for translational control and the docking precision based on the current deviation. The screen on the right displays attitude

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information and range, and the left screen displays data related to the rendezvous and docking process.

6.3 Control Method and Scheme Design 6.3.1 Control Handles 1. The Attitude Control Handle The attitude control handle is the main tool for astronauts to directly control and track the attitude of the spacecraft. It has an independent output of three degrees of freedom. Under the action of a certain external force, it can rotate independently or simultaneously around two or three orthogonal axes and output the voltage corresponding to the rotation angle of each axis. The triaxial output voltage is used to control the pitch, roll, and yaw attitude of the tracking vehicle. The attitude control handle is mounted on the armrest bracket of the astronaut seat. Considering the ergonomic requirements, it is mounted on the astronaut’s right hand position. 2. Position Control Handle The position control handle is the main tool for astronauts to directly control the position of the chaser, with three independent degrees of freedom of the output. It can independently or simultaneously control the vertical and horizontal movement, while the front and rear direction movement have a separate control. The triaxial output is used to control the forward, vertical, and lateral motion of the mass center of the chaser. The position control handle is also mounted on the armrest bracket of the astronaut seat. Considering the ergonomic requirements, it is mounted on the astronaut’s left side. The direction of each axis is consistent with the chaser body frame.

6.3.2 Control Method of Relative Motion The manual control rendezvous and docking task is finished by astronauts by observing the TV camera images of the target and the sign and manipulating the attitude and position control handles to track the movement of the chaser to make the chaser docking frame relative to the target docking frame meet with the initial docking conditions, which is expressed by the relative position and velocity, relative attitude angle, and the relative attitude angular velocity.

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The manual control rendezvous and docking task is realized through the attitude and position control handles. The control rules from the outputs of the two handles to the engine torque and thrust are given here. 1. The Attitude Control law Attitude control requires the measurement information of attitude angular velocity by the gyros. The attitude control law is as follows: (1) When the attitude control handle has an output signal, open the actuator to make the channel reach the desired angular velocity if the absolute value of the difference between the handle output and the measured angular velocity value is greater than the angular velocity threshold δ; otherwise, close the actuator. (2) When the attitude control handle stops signal output, the angle ϕ is calculated through angular velocity integral. According to ϕ and the measured angular velocity ωi by gyro, the phase plane control law is used to make the angle and angular velocity meet the control accuracy requirements. The above control law assumes that the target is of the inertial orientation; if it is of the ground orientation, we need to consider the orbital angular velocity. 2. The Translation Control Law For the near-earth  round orbit with an orbital height of 400 km, the orbital angular μ = 0.0011 rad/s, where the radius of Earth is Re = 6378 km velocity ωo = (Re +H )3 and the gravitational constant μ = G M = 3.986005 × 105 km3 /s2 . In the relative motion Eq. (2.47) of the chaser relative to the target, when the final translational and closing section, y, z, x, ˙ z˙ is small, that is, when the distance is close and the relative speed is slow, the coupling term caused by coriolis force and gravitational difference can be ignored, and we obtain ⎧ ⎪ x¨ = ax = ⎪ ⎪ ⎪ ⎪ ⎨ y¨ = a y = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z¨ = a = z

Fx m Fy m Fz m

From the above equation, it can be seen that the relative motion of the chaser and the target can be approximately described by Newton’s second law. Under the condition that the target does not change orbit, the relative position change of the chaser with respect to the target directly depends on the position change caused by the thrust received by the chaser on the three axes. Astronauts as controller are included in the whole closed-loop control circuit. The control goal is to make the chaser docking along the axis at a certain speed

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according to the required docking velocity of the docking mechanism, and in the process of closing, to gradually eliminate the horizontal position deviation, and when the position deviation is eliminated, as far as possible to maintain transverse velocity near zero. After estimating the motion parameters from the images observed by the astronaut from the television cameras, the relative relationship between the chaser and the target is judged. Then, astronauts control the handles’ voltage output to eliminate the deviation. The output can be classified according to the large or small value of the voltage. The engine in the corresponding direction can be directly assigned the corresponding startup time. 3. The Remote Operation Control Under Large Time Delay As mentioned in Sect. 6.1, compared with the conventional manned rendezvous and docking tasks, the remote operation adds the process of data compression, packaging, transmission, decompression, and display. Therefore, the focus of the remote operation rendezvous and docking control scheme is to determine the time delay of the whole loop, as well as the adaptability and optimization of the control scheme under large time delay. The time delay of the closed-loop control of remote operation rendezvous and docking includes TV imaging, image compression, image and measurement data transmission, image decompression, astronaut judgment and operation, handle voltage acquisition, control command transmission, backup controller calculation, and propulsion execution delay. According to the information transmission direction, it is divided into forward transmission link, backward transmission link, and execution link. (1) the time delay of the forward transmission link mainly includes image shooting by camera, image data compression and packaging, transmission by communication device, image decompression, and image display. (2) the time-delay of the backward transmission link mainly includes handle action, handle signal packaging, communication device transmission, signal decompression, and instruction output. (3) the time delay of the execution link includes propulsion control driver delay and thrust delay. Large time delay can easily cause the control to be out of time or excessive. Therefore, the design of control parameters should fully consider the amount of time delay and enhance the robustness of the closed-loop system against time delay. For details, please refer to Chap. 5. In addition, some prediction results can be superimposed on the display page to play the role of feedforward and assist astronauts in judgment, so as to avoid out of time control or excessive control. For example, the flight path indicator designed by the American Orion manned spacecraft [1] is used to indicate the possible position of the interface center in the final contact time at the current speed. The translation indicator is used to indicate the number of control operations in each direction required to eliminate deviations. All these can provide

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6 Manual Control Method and Scheme Design …

convenience for high precision manual control operation in the case of large time delay.

6.4 Operation Method 6.4.1 Approaching Velocity Control Astronauts use the image information from the TV camera and the measurement information from the lidar to judge the relative position and speed. Automatic measurement of distance and speed is more accurate than visual measurement. When the measurement data of the laser radar are available, the astronaut can control the rendezvous speed according to the distance and speed information provided by the lidar, by turning the actuator on and off through the position control handle. In addition, when controlling the X-channel rendezvous speed, the spacecraft can also visually estimate the distance and rendezvous speed according to the reference image size and its changes.

6.4.2 Lateral Position and Attitude Control When the target is far, astronauts can estimate the angular deviation of the tracking and target along the sight line through the observation of target images. Then, the attitude and lateral deviation control are conducted together, and the control strategy is to make the images of the target at the center of the field through attitude control, so as to make the angular deviation of the chaser relative to the sight line as small as possible. By translational control, the angular deviation of the target relative to the sight line is reduced. In the process, the angular deviation of the chaser relative to the sight line is likely to increase. Therefore, it is necessary to keep the chaser angular deviation as small as possible through fine attitude control. When the distance is relatively close, the angular deviation of the target along the sight line reflects the transverse position deviation. The astronauts estimate the angular deviation of the chaser and the target along the sight line by observing the image of the docking sign and then control it. The angular deviation of the target along the sight line is determined by the deviation between the cross mark on the sign and the lines on the sign background plane. To correct the angular deviation, the astronauts should move the mass center of the chaser laterally until the cross mark is aligned with the cross line. For different angular deviation of target along the sight line, the images of the docking sign and the shifting direction of the control handle to eliminate the angular deviation are shown in Fig. 6.6. As the chaser continues to approach the target, we compared the geometric center of the chaser, the geometric center of the docking mechanism, and then, place the

6.4 Operation Method

(a)

297

(b)

(e)

(c)

(d)

Fig. 6.6 Images of the docking sign and the shifting direction of the control handle when angular deviation exists. a Handle to left; b handle to right; c handle upward; d handle downward; e no angular deviation exists

cross mark center to the position of the cross line on the television camera screen to determine the angular deviation of the chaser along the sight line. Mutual scrolling angle is determined by the asymmetry between the symmetry axis of the docking sign and the cross lines on the TV camera screen. To correct the angular deviation, the astronauts should rotate the chaser around the center of mass. When the approach is completed at a distance of about 10 m, the angular deviation of the chaser and the target along the sight line can be precisely eliminated by hovering. According to the completion of the correction, the astronauts should select the necessary docking speed, which should meet the requirements of initial docking conditions at the contact time. In the process of waiting for contact, if there is angular deviation, astronauts need to continue to adjust angular velocity of the sight line.

6.4.3 Simulations Using the simulation controllers, the electronic handle devices and the visual simulation rendezvous and docking system, various cases and boundary conditions are set up to carry out simulation tests to verify the manual rendezvous and docking control scheme and strategy. Simulation results of a group of typical manual docking process are shown in figures bellow (Fig. 6.7, 6.8, 6.9, 6.10, 6.11 and 6.12). The initial attitudes of the chaser and the initial relative positions of the two spacecraft are shown in Tables 6.1 and 6.2.

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6 Manual Control Method and Scheme Design …

x/m

200 100 0 1.477

1.4775

1.478

1.4785

1.479

1.4795

1.48 5

time/s

×10

y/m

2 0 −2 1.477

1.4775

1.478

1.4785

1.479

1.4795

1.48 ×105

time/s

z/m

0.5 0

−0.5 1.477

1.4775

1.478

1.4785

1.479

1.4795

1.48 ×105

time/s

dx/(m/s)

Fig. 6.7 Relative positions of the three axes 0 −0.5 −1 1.477

1.4775

1.478

1.4785 time/s

1.479

1.4795

1.48 ×105

dy/(m/s)

0.05 0

−0.05 1.477

1.4775

1.478

1.4785 time/s

1.479

1.4795

1.48 ×105

1.4775

1.478

1.4785 time/s

1.479

1.4795

1.48 ×105

dz/(m/s)

0.05 0

−0.05 1.477

Fig. 6.8 Relative velocity of the three axes

6.4 Operation Method

299

roll (°)

5 0 −5 1.477

1.4775

1.478

1.4785 time/s

1.479

1.4785

1.479

1.4795

1.48 ×10

5

pitch (°)

5 0 −5 1.477

1.4775

1.478

1.4795

time/s

1.48 5

×10

yaw (°)

5 0 −5 1.477

1.4775

1.478

1.4785 time/s

1.479

1.4795

1.48 ×105

Fig. 6.9 The attitudes of the three axes of the chaser

dpitch ((°)/s)

droll ((°)/s)

0.5 0 −0.5 1.477

1.4775

1.478

1.4785 time/s

1.479

1.4785

1.479

1.4795

×105

0.5 0 −0.5 1.477

1.4775

1.478

1.4795

time/s dyaw ((°)/s)

1.48

1.48 ×105

1 0 −1 1.477

1.4775

1.478

1.4785 time/s

Fig. 6.10 Angular velocity of the three axes of the chaser

1.479

1.4795

1.48 ×105

300

6 Manual Control Method and Scheme Design … x y z

5 4 3

FxFyFz (V)

2 1 0 −1 −2 −3 −4 −5

0

50

100

150 time/s

200

250

300

Fig. 6.11 Voltage output of the position control handle 5

φ θ ψ

MomentxMomentyMomentz (V)

4 3 2 1 0 −1 −2 −3 −4 −5

0

50

100

150 time/s

Fig. 6.12 Voltage output of the attitude control handle

200

250

300

6.4 Operation Method Table 6.1 The initial attitudes of the chaser

301 Scroll (°)

2.00

Pitch (°)

−4.00

Pitch (°)

−3.00

Roll angular velocity (°/s)

Table 6.2 Initial relative positions of two spacecraft

0.2

Pitch angular velocity (°/s)

−0.1

Yaw angular velocity (°/s)

0.6

x(m)

140.0

y(m)

−0.880

z(m)

−0.120

dx(m/s)

−0.00

dy(m/s)

0.022

dz(m/s)

−0.014

References 1. Karl, D.B., Eric, R.M., Chad, R.F.: Handling qualities evaluation of piloting tools for spacecraft docking in earth orbit. J. Space Craft Rockets 48(5), 846–855 (2011) 2. Jin, Y., Guohua, J., Jiangang, C.: Manual RVD methods for astronauts based on the docking sign images. J. Aerosp. 31(5), 1398–1404 (2010) (In Chinese) 3. John, L.F.: Apollo 13 guidance, navigation, and control challenges. In: AIAA Space 2009 Conference & Exposition, Pasadena, California 4. Robert, L.H., Zarrin, K.C.: Developing a prototype ALHAT human system interface for landing. In: Aerospace, IEEE Conference, pp. 1–14 (2011)

Chapter 7

Theory and Design of Thruster Configuration and Control Allocation

7.1 Introduction We introduced navigation, guidance, and control methods of RVD in Chaps. 3, 4, and 5, respectively. However, actuators play an important role in implementing RVD control. In complex space missions that involve RVD [1–3], the momentum exchange control system and reaction control system (RCS) are often utilized because of the various needs of control tasks. RCS is a critical subsystem in a spacecraft control system because of its adaptability to various space missions. Besides, the completeness and versatility of the mission rely much on the performance of RCS [4]. RCS uses reaction thrusters as actuators. The deliverable force/torque produced by a single thruster, which is mechanically fixed to the vehicle reference frame, is definite. Thruster configuration deals with problems such as how to determine the number of required thrusters, thruster specifications like rated thrust and minimum impulse bit (MIB), and installation designs such as position and inclination of each thruster. With a definite thruster configuration, the force/torque commands derived from the controller of a spacecraft are implemented by control allocation, which calculates the suitable firing duration for each thruster. Within each control cycle, the momentum or angular momentum increments are calculated by the control law, which are then transformed into the firing duration of each thruster through a control allocation algorithm. Thereafter, the thrust or moment required in a certain direction can be accurately implemented. A schematic diagram of the closed-loop control allocation is shown in Fig. 7.1. For spacecraft such as earth observation satellites, whose main task is three-axis attitude control, the thruster configuration and control allocation are straightforward: they can be carried out independently according to the direction of rolling, pitching, and yawing. For RVD missions, because of the need for simultaneous high-precision position and attitude control, the number of thrusters is generally large and the geometry is complex, thereby risking giving lower control accuracy and higher propellant consumption if control allocation is not aptly designed. © National Defense Industry Press 2021 Y. Xie et al., Guidance, Navigation, and Control for Spacecraft Rendezvous and Docking: Theory and Methods, https://doi.org/10.1007/978-981-15-6990-6_7

303

304

7 Theory and Design of Thruster Configuration … Controller function

Guidance

+

Controller

−

Thruster control allocation

State estimation Measurement and navigation

Sensor

Desired force/torque

Force/torque

Spacecraft dynamics

Thruster on/off command

Force/torque requirement

Thrusters Disturbance

Fig. 7.1 A closed-loop view of control allocation

In this chapter, we study the thruster configuration for attitude and orbit control, and the thruster control allocation for complex spaceflight missions. We begin by reviewing some typical engineering solutions and formulate the mathematical description of thruster configuration and control allocation. Considering that the performance of thruster configuration is ultimately embodied by the control allocation algorithm, several common control allocation methods in engineering and theoretical research are introduced. Their performances are analyzed and compared. A comprehensive design method available in engineering is provided. Then, according to the performance requirements of RVD tasks, the performance of thruster configuration is analyzed based on control allocation strategies. Finally, the design method of thruster configuration based on performance requirements is given.

7.1.1 Thruster Configuration for Typical RVD Missions Next, we examine the thruster configuration of the chaser in some typical RVD missions. Soyuz The Soyuz spacecraft is the most mature manned spacecraft in the former Soviet Union. It mainly carries out the mission of transporting astronauts to the Mir Space Station and the International Space Station (ISS). The Soyuz spacecraft has 27 thrusters [5]: 1 orbit control thruster with a thrust of approximately 2940 N and a rocking half-cone angle of 5°, which is used for the implementation of orbit

7.1 Introduction

305

maneuver in the far range rendezvous phase; 26 attitude-control thrusters in total, with 12 thrusters around the tail of Soyuz and 14 thrusters around the center of mass, which are divided into two asymmetric groups: main group and backups. There are 14 large attitude control thrusters with 130 N thrust and 6 small attitude control thrusters with 26 N thrust in the main group. Furthermore, the other 6 small attitude control thrusters act as backups. The large attitude control thruster is mainly used for forward, backward, and translation control in RVD, whereas the small attitude control thrusters are mainly used for three-axis attitude control in roll, pitch, and yaw. Apollo The Apollo spacecraft is divided into three parts: service module, command module, and lunar module. The lunar module, as an active spacecraft, can perform a RVD mission with the service-command module on the lunar orbit. The reaction control system of the lunar module consists of 4 pods comprises 4 thrusters. [6, 7]; there are two thrusters along each unit parallel to the direction of the X-axis, one positive and one reverse; the other two thrusters are perpendicular to the X-axis, thrusting in the Y and Z directions. Each of the 16 thrusters offers 445 N pushing force. The pipeline is divided into two ways, each with eight thrusters. Two of each of the quaternion unit attributes to one way, and the other two attributes to another way. Space Shuttle US space shuttles have repeatedly docked with ISS, transporting astronauts and cargo like life supply, supplementary components, and science instruments. Reaction control system of the space shuttle consists of two orbital maneuver thrusters, 38 large thrusters, and 6 small thrusters [8–10]. The large maneuver thruster is mounted at the rear with a thrust of 26.7 kN and with a half cone swing angle of 15. 9°/6.5° along the pitching/yaw direction. There are 38 large thrusters with 3.87 kN thrust for three-axis position control and three-axis attitude control: 14 units are installed at the front end, providing forces along the ±Y, ±Z, and −X directions; 12 units are on the left and right sides of the rear end, providing forces along the ±Y, ±Z, and + X directions. Six small thrusters, whose thrusts are 111 N, are installed at the front end, left rear end, and right rear end. If one thruster fails, the system can complete the task. In the case of two failures, security can still be ensured. Automated Transfer Vehicle The Automated Transfer Vehicle (ATV) cargo ship developed by ESA is used to refuel the ISS, transport supplies, and remove waste. The ATV’s main propulsion system consists of four 490 N thrusters and twenty-eight 220 N attitude-controlled thrusters [11]. Four 490 N thrusters are used for their own orbit and ISS orbital lifting, which is installed at the bottom of the propulsion module. Among the twenty-eight 220 N attitude-controlled thrusters, 20 are installed at the rear end of the propulsion module for ATV attitude control, rendezvous maneuvers, pre-docking braking, split maneuvers, and collision avoidance maneuvers. The 20 thrusters, divided into 4 pods comprises 5 thrusters, are symmetrically installed on the outer wall of the propulsion

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7 Theory and Design of Thruster Configuration …

module. Two thrusters on each unit are also used for braking. The other eight attitude control thrusters are divided into four pods comprises 2 thrusters, symmetrically mounted on the front cone section of the load module. Each pods contains a main thruster and a backup thruster, which are used to enhance the translation control of the spacecraft. The entire propulsion system can be divided into two parts: the main part and the backup. At the beginning of the task, the main part is selected and once it fails, the backup comes to work. H-II Transfer Vehicle Japan’s H-II Transfer Vehicle (HTV) cargo ship is also an unmanned service and transportation vehicle that transports goods to the ISS (but does not carry propellant). When it is off-track, it loads waste from the ISS and then destroys it into the atmosphere [12]. HTV’s main propulsion system consists of four 490 N main thrusters [13] and twently-eight 110 N attitude control thrusters [14–17]. The main thrusters are located at the rear of the propulsion module for orbital maneuvers: off-track and anti-collision maneuvers. Attitude control thrusters are used for attitude control like rolling, pitching, and yawing and translation control in rendezvous, braking before docking, separating maneuvers, and collision avoidance maneuvers. There are 16 sets installed in the rear of the propulsion cabin, divided into 2 pods comprises 5 thrusters and 2 pods comprises 3 thrusters [17]. There are 12 thrusters installed on the outer wall of the front pressurized cargo, divided into 4 pods comprises 2 thrusters and 4 single thruster pods [17]. Shenzhou The Shenzhou series manned spacecraft’s (Fig. 7.2 [18]) propulsion system is mainly used to complete the orbit and attitude control of the whole flight and the braking and attitude control before returning. Twenty-eight thrusters are equipped in total. Four 2500 N thrusters are used for orbit maneuver, braking before return, and braking or acceleration during emergency rescue control outside the atmosphere; Eight 150 N large attitude control thrusters and sixteen 25 N small attitude control thrusters, through different combinations, generate different levels of control torques for attitude control. Twently-eight thrusters are divided into completely symmetrical main and backup groups. The types of thrusters and the number of main and backup thrusters are identical. The switching between the main and backup can be carried out through the self-locking valve.

16 25 N small attitude control thrusters

4 2500 N orbit maneuver thrusters

Fig. 7.2 Thruster configuration of shenzhou spacecraft

8 150 N large attitude control thrusters

7.1 Introduction

307

The Shenzhou 8th, 9th, and 10th spacecraft that perform RVD missions are based on the thruster configuration of the Shenzhou spacecraft propulsion module. Eight 120 N translation thrusters and four 150 N reverse thrust thrusters are added near the center of mass, completing translation control, separation maneuver, and collision avoidance of RVD. Dragon Dragon spacecraft is a reusable cargo spacecraft for delivering cargo to ISS, which successfully docked with the ISS in 2012. The propulsion system of the Dragon spacecraft has 18 thrusters [19], which are used for attitude and translation control, orbital maneuver, and derailment of RVD. The 18 thrusters are divided into 2 pods comprises 4 thrusters and 2 pods comprises 5 thrusters, which are cross-mounted in the service module [20]. Its thruster configuration is similar to Apollo’s design. The thruster configuration of typical RVD missions discussed are summarized in Table 7.1 as follows. Table 7.1 shows that the number of thrusters for the RVD task is more than that of general satellites [21]. To realize the same RVD function, the thruster configuration can be designed flexibly because of the different mission planning and dynamic characteristics, such as mass and inertia of spacecraft. With an appropriate thruster control allocation algorithm, the total number of thrusters shared by attitude control and translation control can be reduced, improving thruster efficiency. Therefore, for Table 7.1 Comparison of thruster configuration of typical RVD missions Soyuz

Space shuttle

Apollo-Lunar Module

ATV

HTV

Shenzhou

Dragon

Orbital control

1× 2940 N

2 × 26.7 kN

1 × 46.7 kN 1 × 15.5 kN

4× 490 N

4× 490 N

4× 2500 N

18 shared

Pitch

4 × 26 N

5 × 3.87 16 × 445 N kN shared 2 × 111 N

16 × 220 N shared

20 × 110 N shared

4 × 25 N

Yaw

4 × 26 N

4 × 3.87 kN 2 × 111 N

4 × 25 N

Roll

4 × 26 N

2 × 111 N

8 × 25 N

Translation

8 × 130 N (rolling included)

22 × 3.87 kN (rolling included)

8 × 117 N

Forward thrust

4 × 130 N

4 × 3.87 kN

4× 220 N

4× 110 N

8 × 150 N

Reverse thrust

2 × 130 N

3 × 3.87 kN

8× 220 N

4× 110 N

4 × 150 N

32

32

40

Total pieces 27

46

18

18

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7 Theory and Design of Thruster Configuration …

the RVD task, it is valuable to carry out systematic theoretical research on the thruster configuration and control allocation algorithm.

7.1.2 Control Allocation The control allocation algorithm for multi-actuator high-redundancy systems has long been studied in the fields like aircraft, ships, and submarines. However, the actuators considered in these fields are mostly two-way actuators, that is, the positive and negative control can be generated. The particularity of control allocation for the spacecraft thruster lies in the unidirectionality of the thruster as the actuator: a fixed mounted thruster can only produce control in one direction. Currently, the control allocation methods for spacecraft thrusters can be summarized into the following categories. (1) Lookup table method It is a traditional control allocation algorithm that has been commonly used in early and current engineering [22]. This method requires designing a thruster configuration so that the thrust/torque along each axis can be generated by a special thruster (group). It has a lookup table to find which thruster (group) to be used and then divides the direction control force/torque by the total thrust or total torque generated by the corresponding thruster combination, finally yielding the firing duration of the thruster (group) and also the firing duration of each thruster according to inner firing duration ratio [19, 23, 24]. This method is called “decoupled look-up table method” in this book. This method gives a faster online calculation speed and puts a smaller load on the CPU. However, for thruster efficiency and accuracy, these lookup tables provide only suboptimal solutions. An improved look-up table method is proposed in [23]. By combining thrusters with similar characteristics, each group provides greater thrust or torque in as many directions as possible than a single thruster, thereby increasing the control command generation capability of the look-up table method. However, essentially as decoupling and reducing the original 6-dimensional problem, the method obtains a suboptimal solution and can only be applied to the two-dimensional problem. In recent years, ESA has proposed a new look-up table method in developing the ATV [11, 25–32]. This method first divides each of the six thrusters into one group by offline optimization and calculates the configuration inverse matrix corresponding to each combination, which is pre-stored in the onboard computer. After the online command is assigned to select a group of thrusters during the on-orbit state, the corresponding configuration inverse matrix can be found in the inverse matrix table; then, the operating time of each thruster can be calculated according to the product of the inverse matrix and the control command vector. The goal of this method is to obtain an optimal command allocation solution with a shorter online calculation time. The key is the design of the thruster combination table. Such a look-up table

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method has been applied on ATV, but the grouping method of its thruster is not specified in the literature. (2) Pseudo inverse method The pseudo-inverse method is often mentioned in the literature for studying the control allocation of actuators, especially in aircrafts. It solves the control allocation by introducing the Moore inverse of the thruster configuration matrix. However, this simple matrix multiplication does not guarantee the non-negativity of each thruster’s firing duration. Ref. [33] solved this problem by a symmetrical thruster configuration, that is, when the corresponding firing duration of a thruster is less than zero, the corresponding firing duration of the thruster in the opposite direction will be opened. Ref. [34] guaranteed the non-negative nature of the boot time by introducing a kernel space vector as the offset. Since the vector is fixed, it is difficult to satisfy the constraints of each thruster. In Ref. [35], the dynamic bias method was obtained by multiplying the offset by a varying coefficient, whereas Ref. [36] used iteration to limit the generation of negative solutions and make the output closer to the required amount of control commands. Ref. [37] proposed a least squares allocation method, which divided the control commands into positive and negative parts, and found the inverse matrix. This method can guarantee that the solution is always non-negative, but it is only suitable for the case where the control force and torque command change is not very large. The advantages of the pseudo-inverse method are the simplicity of the algorithm, multiplication of matrix and vector only, no loop iteration, and ease of theoretical analysis of the performance of the thruster configuration. Ref. [34] provided some analytical expressions regarding the range of thruster configuration control capabilities based on the commandal assignment results of the pseudo-inverse method. However, this algorithm always limits the solution to the allocation problem to a small extent in the solution set and cannot obtain the optimal solution. (3) Linear/nonlinear programming method In the reaction control system of a complex spacecraft, the number of thrusters is typically much larger than the dimension of the task, so there are infinite solutions to the problem. If we want to solve the thruster firing duration to meet a certain optimization goal, we can find an optimal solution in its solution set. With the optimization goal as the objective function, the optimal control allocation of the thruster can be described as a linear/nonlinear programming problem [34], which can be solved by classical linear/nonlinear programming algorithm. In 1969, Crowford [38] solved the control allocation problem with classical Dantzig’s simplex algorithm and showed its geometric meaning based on the simplified linear programming description. In contrast, Ref. [39] explained the linear programming solution for control allocation problems. In Ref. [40], the solution of the pseudo-inverse method is used as the initial value of the linear programming method to speed up the iteration. Ref. [41] proposed a mixed integer linear programming description of the control allocation problem and used it to solve the

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control allocation problem of reentry vehicle. Ref. [42] introduced multi-objective programming to establish a new linear programming model for the control allocation problem, which can compromise between the realization of control quality and the consumption of propellant, and obtained an optimal solution if the two goals were assigned with different weights. Refs. [43, 44] further proposed a nonlinear programming algorithm for control allocation. The advantage of the programming method is that it can obtain the optimal solution of the command allocation problem, and the algorithm is more versatile and robust. Fehse [22] predicted that with the increasing processing power of on-board computers, the inherent flexibility of the algorithm will make it commonly applicable to thruster control of diverse spacecraft. However, as far as the current situation is concerned, the programming method will still impose a large burden on the CPU, and its performance on real-time and on-orbit computing still needs to be improved. In Ref. [45], the linear programming method was combined with the lookup table method, and an optimal look-up table method was proposed. According to the linear programming algorithm, the optimal thruster combination table is obtained offline, and the input is calculated according to the control command online. The optimal thruster combination and thruster firing duration are obtained, thus resolving the contradiction between optimality and real-time. It will be described in more detail in the following chapters. (4) Other methods Another idea of suboptimal control allocation to calculate the dot product of each control vector was proposed in Refs. [39, 46]. These resulting values of dot products characterize the proximity of each thrust vector to the control command within the thruster “thrust vector space”. Thrusters with the highest value or higher than some threshold value are set to start working. For this method, more thrusters yield improved accuracy. However, the method also needs to traverse when selecting the thruster, and only the approximate solution is obtained. In particular, when the thrusters are not enough, the accuracy and the optimality of the solution are not guaranteed. Ref. [47] combined the pseudo-inverse method with linear programming. They first solved the pseudo-inverse problem. If the solution propellant consumption exceeds some threshold value, it is solved online by linear programming. When the value is lower than the threshold value, the iteration is stopped, thereby avoiding slow linear programming during online computing.

7.1.3 Thruster Configuration Design The design of the thruster configuration is generally carried out for specific space missions [34, 36, 48–51], and most designs are configured along the main axis in the orthogonal, symmetrical, and centroid mounting configurations. The disadvantage of this configuration is that the efficiency of the thruster is relatively low, and to ensure

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311

system redundancy, the number of thrusters becomes very large. Non-orthogonal symmetrical thruster configuration design is a complex problem, especially considering some practical engineering constraints such as the orthogonality of control torque and control force [50, 52, 53], the limited amplitude of force and its arm, and protection from plume pollution. At present, there is a lack of theoretical research in this area. The relationship between the geometry of the thruster configuration and system redundancy is studied in Ref. [38], where the minimum number of thrusters required to complete the m-dimensional control task was found to be m + 1, which provides a reference to determine the number of thrusters. Some of the factors that should be considered in the thruster configuration design of the space station are given in Ref. [46], which states that the translation control capability is the highest requirement for thruster configuration. The minimum thruster configuration unit with only four thrusters is also given for attitude control of the space station. Ref. [54] proposed a uniform thruster configuration for a general one-way actuator system to ensure that each actuator unit has the same load during operation and focused on the minimum uniform thruster configuration with the few actuator units. The structure, also based on a co-directional thruster configuration, gives a thruster configuration design method for completing a 6-dimensional mission with seven thrusters in a special thruster configuration matrix. Ref. [52] proposed a feasibility analysis method for analysis of complex thruster configuration control capability, which can be used to guide thruster configuration design. For the optimal design of the thruster configuration, Ref. [38] proposed the concept of the minimum redundant structure, that is, the minimum number of thrusters required for tasks with different dimensions at a certain degree of redundancy. Ref. [46] provided an eight-thruster configuration with the highest robustness, considering the centroid variation, which is used to complete the 3D attitude control problem. In Refs. [36, 55], the optimization target is to realize largest safety margin with minimum propellant consumption. The installation angle of the thruster is optimized by linear programming provided the thruster installation positions are given. Ref. [56] also addressed a simpler thruster configuration and discussed how to optimize the thruster installation pitch to achieve the most provincial propellant control. For the case where the control commands are evenly distributed in the command space, Ref. [54] provided the design method of the minimum uniform thruster configuration matrix through cyclic matrix [57]. Ref. [58] designed a minimum uniform thruster configuration matrix to ensure that propellant consumption is statistically minimal. However, this thruster configuration can only be applied in the case of a threedimensional four-thruster configuration but not to a six-dimensional task. Based on the analysis of the control ability of the literature [52], Ref. [59] proposed the performance index to measure the control ability of complex thruster configuration and optimized the thruster configuration accordingly.

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7 Theory and Design of Thruster Configuration …

7.2 Problem Description 7.2.1 Thruster Configuration Matrix The thrusters of a spacecraft are generally fixed at a certain distance from the spacecraft’s center of mass. They generate three-dimensional thrust vector F by blowing out the working medium. Suppose the position vector relative to the spacecraft’s centroid is l; then, the control torque generated by the thruster can be expressed as T = l × F, which is shown in Fig. 7.3. It can be seen that the magnitude of each component of F relates to the thrust of the thruster and the installation inclination, and the magnitude of each component of T is related to the magnitude of the thrust of the thruster and the position and inclination of installation. Assume that there are n thrusters on a spacecraft; F i and T i (i = 1, 2 . . . n) respectively represent the control force vector and control torque vector generated by the i-th thruster; and l i denotes the position vector of the i-th thruster. With the control force and torque generated by the i-th thruster defined as the i-th column, the overall control generated by these n thrusters can be formulated as matrix A, as follows:     F2 · · · Fn F1 F1 F2 · · · Fn = A= T1 T2 · · · Tn l 1 × F1 l 2 × F2 · · · l n × Fn The matrix A is referred to as the configuration matrix of the spacecraft’s thrusters. Obviously, A is determined once the thruster configuration is fixed. The number of columns in the matrix A reflects the number of thrusters in the control system. The modulus value of F i reflects the thrust of each thruster. The vector direction of F i reflects the installation inclination. F i together with T i reflects the installation position of the thruster. Fig. 7.3 Thruster configuration and force/torque vectors T l

O F

7.2 Problem Description

313

7.2.2 Mathematical Model of Thruster Control Allocation Given the configuration matrix of thrusters A, the total control u composed of the combined momentum increment and the combined angular momentum increment vector, that act on the spacecraft body, can be expressed as ⎤ t1 ⎢ ⎥  F 1 F 2 · · · F n ⎢ t2 ⎥ u= ⎢ . ⎥ = At T 1 T 2 · · · T n ⎣ .. ⎦ ⎡

(7.1)

tn

T where t = t1 t2 · · · tn and ti ≥ 0 denotes the firing duration of the i-th thruster. T

∈ R m is the vector of input control command derived from the u = u1 · · · um controller, and m is the dimension of vector u, which is the dimension of the control task. Then, A can be expressed as an m × n matrix, each column of which represents the deliverable force and torque produced by each thruster. It can be seen from Eq. (7.1) that the solution of the control allocation problem can be obtained by solving linear equations, that is, the firing duration t of each thruster can be solved under the condition that u and A are known. According to the knowledge of solving linear equations, we know • If m > n, the equations have no solution, that is, when the task dimension is greater than the number of thrusters, the control commands in some directions cannot be realized irrespective of the type of thruster combination selected; • If m = n, A−1 is non-singular; then, the equations have a unique solution, but a non-negative value of the solution can not guaranteed, that is, given that control commands of a combination of thrusters can be found, but it may be the situation where negative firing duration is obtained and makes no sense; • If m < n, the equations have a large number of feasible solutions. For any control command, various thruster combination schemes are available for selection, and a physically meaningful thruster firing duration is obtained. Thereafter, one of the most suitable thruster combinations can be selected according to the task requirements. Therefore, spacecraft can generally be equipped with multiple (m » n) thrusters to ensure proper execution of control commands. However, how to determine the most suitable one from diverse feasible combinations is the goal of the thruster control allocation algorithm.

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7 Theory and Design of Thruster Configuration …

7.3 Theory and Design of Thruster Control Allocation When the thruster is mounted symmetrically along the main axis of the spacecraft, its command distribution algorithm is relatively simple because the thrust/torque vector generated by the thruster is decoupled along the direction of each body axis. One example can be shown in Fig. 7.4a. However, because of the various factors such as the installation envelope and plume pollution, the real installation of the thruster typically exhibits a complex configuration, as shown in Fig. 7.4b. The higher the task dimension, the more the number of thrusters. Furthermore, the more irregular the layout, the stronger the coupling between the components of the control amount generated by the thruster. For example, the space shuttle depicted in Sect. 7.1.1 and the ATV cargo spacecraft of ESA installed more than 30 thrusters of various thrust levels, and there are many cases of diagonal installation. As a result, for the complex space missions such as RVD, which require extremely high control precision, designing a thruster control allocation algorithm with high performance is notably important [60, 61].

7.3.1 Thruster Control Allocation Method This chapter mainly introduces the decoupled lookup table method commonly used in engineering, the planning optimization algorithm commonly used in theoretical research, and the optimal lookup table method. Other control allocation algorithms such as pseudo-inverse can be found in the references given in Sect. 7.1.2. 1. Decoupled Lookup Table Method 1) Theory Each thruster installed on the spacecraft may generate force and torque in multiple directions simultaneously according to the position and inclination of its installation. The basic principle of the decoupled lookup table method is to divide all thrusters into several groups according to the specific thruster configuration. The thrusters in the group work according to a certain time ratio, so that each group is only responsible Fig. 7.4 2D view of an example thruster configuration

(a)

(b)

7.3 Theory and Design of Thruster Control Allocation

315

for generating force/torque in one direction. The force/torque values in the other directions cancel each other. 2) Design of the Thruster Combination Table In typical engineering design, thrusters are installed as orthogonally and symmetrically as possible in the direction of multiple control command. In this way, thrusters in each of the control channel are naturally divided into a group, and the unnecessary force/torque is offset by the symmetric installation so that the decoupled lookup table method can be conveniently used for control allocation. Indeed, because of other engineering constraints such as installation errors, the final installation will still cause some disturbances in other directions. However, their magnitudes are relatively small and have little effect on closed-loop control. 3) Online Algorithm The online control allocation is as follows: divide the directional control input u = [u 1 u 2 . . . u m ]T , whose elements are obtained from the control law, by the joint force/torque value of the corresponding thruster combination in the direction, thereby obtaining the combined firing duration. Then, the firing duration of each thruster is obtained according to the firing duration ratio of each thruster in the group. In addition, if the thruster firing durations obtained according to the control frequency (for example, the output of the phase plane control law) are directly set as control inputs, the combination table can be directly applied to the corresponding thruster. 2. Programming Method (1) Linear programming 1) Theory The mathematical model of linear programming contains three components: decision variables, objective functions, and constraints [62]. The correspondence between them and the mathematical description of the control allocation problem is as follows: ➀ The decision variables determine the results of the programming problem, corresponding to the variables to be solved, that is, firing duration of each thruster ti (i = 1, . . . n). Because the command is allocated within one control period, the range of ti (i = 1, . . . N ) values is D1 = {x|0 ≤ x ≤ Tmax }, where the upper bound Tmax is generally the control period, and the non-negative constraint reflects the unidirectionality of the thruster. ➁ The objective function is the optimization criterion of the programming problem. It must be a function of the decision variable. The goal of optimization is to minimize/maximize the objective function. If the control allocation requirements

316

7 Theory and Design of Thruster Configuration …

minimize propellant consumption, and the propellant consumption is a function of firing durations ti (i = 1, . . . n), it can be set as an objective function, given as Min goal =

ti ∈D1

N

ci ti

i=1

where ci is the propellant consumed by the i-th thruster per unit firing duration. ➂ Constraints specify the range of variation of decision variables. With respect to the control allocation problem, the most import constraint is that firing durations ti (i = 1, . . . N ) of thrusters finally generate control outputs required by the control law. The relative position and attitude control of RVD need control outputs in 6 dimensions, expressed as momentum increment and angular momentum increment along the X, Y and Z axes in the chaser’s body coordinate system, i.e. u j ( j = 1, . . . 6). Then, Eq. (7.1) becomes an equality constraint for the linear programming model to control allocation. In summary, the thruster’s control allocation problem can be transformed into the following linear programming problem (s.t. stands for subject to): min

n

ci ti

(7.2)

i=1

s.t.

At=u Tmax ≥ t ≥ 0

That is, in the case where all the constraints are satisfied, look for ti (i = 1, . . . n)

N ci ti . This problem can be solved directly that minimizes the objective function i=1 by various classical algorithms for solving linear programming problems, thereby obtaining the firing duration of each thruster in the control cycle. 2) Online Algorithm A lookup table is not required. Instead, by knowing the thruster configuration matrix A and the weight coefficient c, we can calculate the firing duration of each thruster according to the control law output u within each control cycle. Simplex method, the interior point method and the ellipsoid algorithm [62–66] can be used to solve linear programming. (2) Multi-Objective Programming 1) Theory The linear programming (LP) method only optimizes the propellant consumption and incorporates control into constraints. The treatment makes it difficult to compromise between these two design goals. In addition, it does not consider the limitations of the thruster’s minimum firing duration that results in control error. The

7.3 Theory and Design of Thruster Control Allocation

317

multi-objective programming approach solves the above problems by compromising between multiple design goals. The mathematical model is p

− − Min goal = P j+ d + j + Pj d j − d+ ,d ≥0 j=1 j j ⎧ − + ⎨ f j (x) + d j − d j = fˆj ( j = 1, · · · p) s.t. Ax ≤ b ⎩ x ≥0

(7.3)

where fˆj is the target value, f j (x) is the actual value, and P j is the weight coefficient of the j-th target deviation. Different from LP method, Eq. (7.3) introduces deviation + variables d − j and d j . We have + ˆ d− j − d j = f j − f j (x)

➀ If f j (x) ≥ fˆj , then + d− j − dj ≤ 0 + + − − d− j = 0 minimizes P j d j + P j d j , and

ˆ d+ j = f j (x) − f j

(7.4)

+ ˆ that is, d + j represents how much f j (x) exceeds f j . We therefore call d j a positive deviation variable. ➁ Similarly, if f j (x) ≤ fˆj , then + d− j − dj ≥ 0

and + ˆ d− j = f j − f j (x) d j = 0 − ˆ That is, d − j represents how much f j (x) is lower than f j . We therefore call d j a negative deviation variable. Overall, it always holds that

    − ˆ f + d = (x) − f d+  j j j j

(7.5)

+ Or d − j plus d j stands for the absolute value of deviation between the actual and target values.

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7 Theory and Design of Thruster Configuration …

For the convenience of problem solving, deviation variables can be used in the following ways:     − ➀ To minimize  f j (x) − fˆj , adding P j (d + j + d j ) to the objective function − − ➁ To minimize d − j , adding P j d j to the objective function + ➂ To minimize d j , adding P j+ d + j to the objective function.

Now, we apply the idea of multi-objective programming to construct a mathematical model for control allocation. Our two design goals are ➀ To achieve the required control, the target value should be equal to control command u j (1 ≤ j ≤ 6): fˆj = u j ( j = 1, . . . 6) The actual value is f j (x) = A j,1 t1 + A j,2 t2 + · · · A j,n tn ( j = 1, . . . 6) + ˆ Then, the constraint of Eq. (7.3), f j (x) + d − j − d j = f j , becomes + A j,1 t1 + A j,2 t2 + · · · A j,n tn + d − j − d j = u j ( j = 1, . . . 6)

(7.6)

    − Because we want to minimize  f j (x) − fˆj , P j (d + j +d j ) is added to the objective function. ➁ To minimize propellant consumption, the target value should be zero, i.e., all thrusters should have zero firing durations: fˆ6+i = 0 (i = 1, . . . n) The actual firing durations are f 6+i (x) = ti (i = 1, . . . n) Then, the constraint becomes − + − di+6 = 0 (i = 1, . . . n) ti + di+6

(7.7)

+ Because we want f 6+i (x) to not exceed fˆ6+i , n+6 m=7 Pm dm is added to the objective function. Based on the above analysis, the multi-objective programming model for the control allocation problem is reformulated as

7.3 Theory and Design of Thruster Control Allocation

319

6 n+6

+ − Min goal = P (d + d ) + Pm dm+ m m m dm+ ,dm− ≥0 m=1 m=7  + A j,1 t1 + A j,2 t2 + · · · A j,n tn + d − j − d j = u j ( j = 1, . . . 6) s.t. − + ti + di+6 − di+6 = 0 (i = 1, . . . n)

(7.8)

+ As can be seen from the constraint (7.6), once the deviation variables d − j and d j are added, the momentum increment and angular momentum increment generated by the thrusters are required to strictly be equal to the required controls. The deviation is reflected in the deviation variable, so it can be optimized along with the propellant consumption. For the constraint (7.7), because ti ≥ 0, from Eq. (7.4) we know that after opti− + = 0. We move di+6 to the right-hand side of the equation, mization is completed, di+6 and then, + ti = di+6

(7.9)

That is, the deviation variable equals to the firing duration. Then, Eq. (7.8) can be simplified to Min

− d+ j ,d j ,ti ≥0

goal =

6

j=1

− P j (d + j + dj ) +

s.t. A j,1 t1 + A j,2 t2 + · · · A j,N t N +

d− j

n

ci ti

i=1

−

d+ j

(7.10)

= u j ( j = 1, . . . 6)

Considering the limitations of the thruster’s shortest firing time, we assume that the minimum firing time of all thrusters is Tmin . We define D3 = {x|Tmin ≤ x ≤ Tmax } and D4 = {x|x = 0}. Then, the range of ti is D2 = D3 ∪D4 . Then, Eq. (7.10) becomes Min

− d+ j ,d j ≥0

ti ∈D2

z=

6

j=1

+ P j (d − j + dj ) +

N

ci ti (7.11)

i=1

s.t. A j,1 t1 + A j,2 t2 + · · · A j,n tn +

d− j

−

d+ j

= u j ( j = 1, . . . 6)

By adjusting the weights P j and Ci , it is possible to achieve a good compro+ mise between control realization and propellant consumption.ti , d − j , and d j (i = 1, . . . n; j = 1, . . . 6) together form the new decision variables. 2) Online Algorithm Equation (7.11) contains a non-continuous range of values: ti ∈ D2 (i = 1, . . . n) and D2 = D3 ∪ D4 , where D3 = {x|Tmin ≤ x ≤ Tmax }, D4 = {x|x = 0}. Hence, we cannot directly use the simplex method. This range can be divided into two continuous parts D3 and D4 , which are very similar to the case of integer programming, in which a decision variable is assigned with two discrete integer values. The branch and bound method can be used to solve problem (7.11). For more details, the reader can refer to ref. [62, 63].

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7 Theory and Design of Thruster Configuration …

3. Optimal Thruster Combinations Lookup Table Several basic concepts (such as relaxation variable, artificial variable, basic solution, basic feasible solution, optimal solution, basic variable, nonbasic variable, base, entering variable, leaving variable, base matrix, simplex table, right-hand-side term, and pivot column), steps, and mathematical concepts concerning the simplex method and linear programming that will be used in the optimal lookup table can be found in Refs. [62–64]. (1) Theory When using linear programming to solve control allocation, the same combination is called to fire, although the firing duration is different in some cases. This gives us the possibility to find a fixed optimal thruster combination determined by the optimization objective function for a given thruster configuration. By finding this new optimal thruster combination, a new thruster combination table can be created to quickly find the optimal solution to the control allocation problem in the online look-up. Section 7.3.1 provides a general linear programming description of the control allocation problem. According to classical simplex method theory, in the process of solving Eq. (7.2), each basic feasible solution t contains m basic variables whose values are solved by m constraint equations. Furthermore, the remaining n-m nonbasic variables’ values are zero. The constraint in the model can be written as follows:  ˆ   ˆ 2 t1 = u ˆA1 A ˆt 2 m where ˆt 1 ∈ R+ is the basic vector comprising m basic variables and ˆt 2 = 0n−m is a ˆ 1 ∈ Rm×m zero vector with n-m zero-valued nonbasic variables. The base matrix A is a subsection of thruster configuration matrix A, and its columns correspond to m basic variables. Then, a basic feasible solution can be obtained:

ˆ −1 ˆt 1 = A 1 u

(7.12)

The key steps for solving linear programming by the simplex method include: selecting the entering and leaving variables in each iteration, using Gaussian elimination to solve linear equations, and moving from the current to a better-neighboring basic feasible solution until optimum is reached. The m basic variables in the optimal solution are called optimal bases, and the remaining n-m nonbasic variables’ values are zero. For a given control command u, there is an optimal base corresponding to it. If we know the optimal base in advance, the optimization iterations can be skipped and the optimal solution of Eq. (7.2) can be directly obtained using Eq. (7.12). In the offline case, according to all possible values of u, all the optimal bases of the thruster

7.3 Theory and Design of Thruster Control Allocation

321

configuration matrix A are predetermined, such that the optimal thruster combination table is obtained. With respect to the online case, according to the designed optimal table, the real-time control command u given by the controller is realized, reaching optimality with a high computing efficiency. In the following section, according to the model of Eq. (7.2) and combined with the linear programming simplex method, we will specify the optimal table lookup method, including the design of the offline optimal thruster combination table and the online control allocation algorithm. (2) Design of the optimal thruster combinations table There are two differences between the standard LP problem modeled as Eq. (7.2) and the problem we need to solve: (1) the uncertainty of the control command u; (2) the solutions we seek for are not the values of t (the firing durations), but the optimal basis, i.e. thrusters with nonzero values in an optimal solution. According to the simplex method, the uncertainty of u will affect the optimal value of t. Because we only required the basic variables in the optimal solution instead of their values, the uncertainty of u at this point may not be considered. Besides, the uncertainty of u also affects the determination of leaving variables. Because the nonnegative ratios of the right-hand side of the equations to the corresponding constraint coefficients under the entering variable in a simplex tableau cannot be computed without a definite value of u, there may exist more than one possible leaving variable. However, it is this uncertainty that makes it possible to obtain all the optimal bases of matrix A. The algebraic algorithm for generating the optimal thruster combinations is summarized as follows: Step 1 Initialization 

Add artificial variables r = r1 . . . rm to the equations in problem (7.2) and penalize them in the objective function with M (M = constant, and M must be sufficiently large relative to the original objective coefficients). The resulting new model is given as max − Z goal s.t.

− Z goal + cexp t exp = 0 Aexp t exp = u t exp ≥ 0

T



  where t exp = t r , Aexp = A I m×m , cexp =

m c0 01×m , and c0 is the coefficient matrix of nonbasic variables. c0 j = c j − M i=1 ai, j is an element of c0 where ai, j is the element of row i and column j of matrix A (i = 1,…, m; j = 1,…, n). Select the artificial variables as the initial basic variables (t B = r T ) and the original

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7 Theory and Design of Thruster Configuration …

decision variable t as the initial nonbasic variable. The basic matrix is initialized as ˆ 1 = B= I m×m . A Step 2 Iteration • Determine an entering variable: Select a nonbasic variable that corresponds to the highest negative coefficient in cexp ; • Determine leaving variables: we call a column of Aexp a pivot column if it is associated with the basic variable. Select all the basic variables associated with the strictly positive constraint coefficients pivot column of leaving variables. • Swap the entering variable with every possible leaving variable respectively by the Gauss-Jordan row operations to produce a set of new basic feasible solutions t Bk and the corresponding basic matrix B k (k = 1, . . . , w, w stands for number of leaving variables). According to the matrix theory of simplex algorithm [62], the new objective coefficients and constraint coefficients associated with every new BFS can be obtained by 

−1 c ex p = c B B −1 k A + c0 c B B k 

−1 A ex p = B −1 k A Bk where c B is the objective coefficients vector associated with the basic variables in the new basic feasible solution (BFS). Step 3 Check the Optimality Condition If the coefficients of the nonbasic variables c ex p are nonnegative, stop iteration of the current branch, and the corresponding t Bk is one of the optimal bases. Thus, the thrusters associated with this t Bk constitute an optimal thruster combination. Otherwise, go to step 2, cex p ← c ex p and start a new iteration. It must be noted that the right-hand side of the constraint equations is required to be nonnegative in a standard LP model. If the control command vector u has negative elements, the constraint coefficients (elements of matrix A) will be standardized by multiplying by −1 on both sides of the constraint equations. As a result, the objective coefficients vector cex p will be changed. That means all the standardized forms of matrix A should be found before we start the algorithm proposed above. Different standardized forms of A with respect to vector u in different quadrants of the command space ensure that all the optimal thruster combinations could be obtained. Given task dimension m, there are 2m different standardized forms of A. Because m is generally no more than 6 (considering translation and rotation) and the optimal thruster combinations are computed offline, the algorithm proposed above is feasible. The above process is technically an appropriate modification of the simplex method, outputting all optimal thruster combinations provided the control command

7.3 Theory and Design of Thruster Control Allocation Fig. 7.5 Thrust vectors in 2-D command space

323 y

J4 J2

x

o

u

J1

J3

u is uncertain in an offline situation. All we need to know is the configuration matrix A. A 2-dimension example presented below will give some geometrical interpretation of the optimal thruster combinations, which can bring us a deeper understanding of the algebraic method proposed above. T

Example 7.1 Suppose that u2d = u x u y , and the configuration matrix A2d with 4 thrusters is given as follows: 

A2d

200 −200 60 −60 = −100 100 −150 150



Thrust vectors of the 4 thrusters J 1 –J 4 in the 2-D command space are illustrated in Fig. 7.5. Suppose that the four

thrusters  have the same efficiency, which means the cost efficiency vector c = 1 1 1 1 . The process of seeking all the optimal thruster combinations according to the algorithm described previously can be summarized to a branching tree displayed in Fig. 7.6. The basic feasible solution in every iteration step is represented by a bracket with thruster number 1–4, and the initial basic variables are two artificial variables with number 5 and 6, which are not real thrusters. According to different control command u2d , the algorithm branches to different nodes (thruster combination) as a result of selecting different entering and leaving variables at each stage based on the minimum nonnegative ratio test. At the termination of each branch, an optimal thruster combination is obtained. All the optimal thruster combinations in Fig. 7.6 can be summarized into four combinations below: (2 3) (1 3) (1 4) (2 4) Observing Figs. 7.5 and 7.6, we find that every two adjacent thrust vectors form an optimal thruster combination, and these four combinations divide the whole command space into four independent areas. By searching for the optimal combinations along the branch in Fig. 7.6 according to the conditions above, we can obtain the optimal thruster combination with respect to vector u2d , which is (1 3). We generalize the method to any m dimensions: each optimal thrust combination consists of

324

7 Theory and Design of Thruster Configuration … ux uy ux uy




1,3

e.v.4 (5,1)

4,1

l.v.5

1,4

l.v.6 e.v.3 l.v.6 e.v.2 l.v.6

2,3

(5,2)

uy≥0 ux 200 < uy 100

60 150

(1,6)

uy

2,4

e.v. stands for entering variable and l.v. for leaving variable”

Fig. 7.6 Branch figure for solving the 2-D problem. Here, e.v. stands for entering variable and l.v. for leaving variable

m thrusters that divide the m-dimensional control command space into several subregions, the division method being determined by the objective function in Eq. (7.2). Each combination is responsible for generating the m-dimensional control command within the scope of the region. When the command u lies in a certain area, its optimal base is the several thrusters that form the area. As such, a relationship between the optimal thruster combination and u is generated. It can be interpreted as a geometrical reflection of the algebraic algorithm for generating the optimal thruster combinations by precomputing all the optimal bases. It also indicates the geometrical nature of the simplex method. (3) Online Algorithm First, it is necessary to pre-store the optimal thruster combination, which is calculated offline according to the previous algorithm, onto the onboard computer. It should include two main parts: (1) the optimal thruster combination table: all the optimal thruster combinations calculated offline with each containing m thrusters; (2) optimal thruster combination inverse matrix table: a table comprising the inverse of every m × m-dimensional thruster sub-configuration matrix related to the corresponding optimal thruster combination.

7.3 Theory and Design of Thruster Control Allocation

325

The specific steps of the on-board real-time control allocation algorithm based on the stored overall optimal thruster combination table are listed as follows: (1) Calculate the firing duration of thrusters using Eq. (7.12) according to the inverse matrix table and control commands derived from the controller. (2) Select the thruster combination whose firing duration is nonnegative as the optimal thruster combination. (3) Calculate the turn on/off time of each thruster in the selected combination according to the calculated firing duration from step (1). Other thrusters are not turned on. Because this real time control allocation algorithm only involves the multiplication between matrices and vectors, its computation speed is much faster than that of the simplex iterative algorithm. Furthermore, it proves to be compatible with today’s processing power of an on-board computer. One property of the algorithm is simultaneous feasibility and optimality: through group-by-group calculations, once a feasible solution of Eq. (7.12) is found, it is also an optimal solution for Eq. (7.2) [67]. Shared solution is another property of the online algorithm: for the control allocation problem (7.2), the linear programming method and the optimal lookup table method always give the same solution. If the former gives multiple feasible solutions, the latter also gives multiple feasible solutions. In this case, we can take any feasible solution therein. 4. Deviation Optimization Method The controls generated by the above various control allocation methods will have a range. When the amplitude of control exceeds this range, some firing duration ti does not satisfy the range requirement D2 = D3 ∪ D4 , where D3 = {x|tmin ≤ x ≤ Tmax }, D4 = {x|x = 0}. Several solutions to this problem are given in this section. (1) Scale reduction method After t is obtained, we first calculate ¯t = t × Tmax /max{Tmax , ti (1 ≤ i ≤ n)} Its effect is to reduce all ti (1 ≤ i ≤ n) by the same scale when the maximum value of ti (1 ≤ i ≤ n) is greater than Tmax . As a result, its maximum value is exactly equal to Tmax . t Secondly, examine t¯i (1 ≤ i ≤ n), if min ≤ t¯i < tmin , let t¯i = tmin ; if 2 t min 0 < t¯i < 2 , let t¯i =0. The above method is simple, and it does not use prior thruster configuration information. Thus, in most cases, the deviation of the control caused by the working time limit cannot be optimized sufficiently.

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7 Theory and Design of Thruster Configuration …

(2) Two-dimensional control deviation optimization method Based on the idea of multi-objective programming, we introduce the deviation variable d = [d1 d2 ]T , which represents control deviation. Let b∗a be the two-dimensional control required by subproblems. Let A∗a ∈ R2×2 be the force matrix determined by the optimal lookup table and its elements are along the above two control channels. We define t ∗a ∈ R2×1 as the firing durations of these two thrusters. Then, the target controls can be expressed as A∗a t ∗a + d = b∗a We move d to the right-hand side of the equation and then multiply A∗−1 on both a sides: ∗ ∗−1 t ∗a = A∗−1 a ba − Aa d

(7.13)

The first term on the right-hand side of (7.13) can be regarded as the result of the optimal look-up table, whereas the second term can be viewed as a deviation variable exerted on the thruster’s firing duration. Equation (7.13) can also be written as ∗ t ∗a = A∗−1 a (ba − d)

It shows that introducing deviation variables leads to an adjustment of the control requirement in the algorithm. As a result, the control can be implemented within the required working time. Because we want to minimize control deviation, we define the optimal deviation variables as follows: |d1 | + |d2 | is minimized if and only if the minimum deviation variable is reached. The goal of deviation optimization is to solve d = [d1 d2 ]T according to (7.13), such that of t ∗a satisfy D2 and meanwhile |d1 | + |d2 | is minimized.  elements  tx1 ∗ ∗ Let = A∗−1 a ba be the result of optimal look-up table. Define Aa = tx2       a11 a12 a¯ 11 a¯ 12 t∗ ∗−1 ∗ , Aa = , and t a = a1 ∗ . Then, Eq. (7.13) becomes a21 a22 a¯ 21 a¯ 22 ta2 

∗ ta1



∗ ta2

 Define

tc1 tc2



 =

∗ ta1 ∗ ta2



 =

 −

tc1 and tc2 are adjustments.

tx1 tx2

tx1 tx2





 −

a¯ 11 a¯ 12 a¯ 21 a¯ 22



d1 d2



7.3 Theory and Design of Thruster Control Allocation

327

7.3.2 Performance Analysis 1. Performance Index The thruster’s control allocation algorithm is diverse, and the specific design depends on control capability, thruster configuration characteristics, propellant consumption, and time and space complexity for online computing. We should synthetically consider the existing engineering conditions and task requirements. The control capability generally has two implications: First, the completeness in the direction, which means the control command in all control channels can be synthesized; And second, the amplitude of control command that can be generated in a required control channel. An appropriate control allocation algorithm can use the thrust generated by thrusters as much as possible in the required control channel. This allows us to achieve a larger amplitude in this direction. Propellant consumption reflects the efficiency of the thruster. A good control allocation gives higher efficiency, that is, thrusters are used to generate the required control forces/torques. On the contrary, if part of the thrusters generates unwanted interference force/torque, and other thrusters are needed to offset the interference, the propellant consumption will be relatively large and the thruster efficiency will be low. The algorithm complexity refers to the online computing time required by the control allocation algorithm to solve the firing duration of each thruster. Considering the actual computing power of the current onboard computer, the algorithm that should not be iterative or complex in case that the computing time exceeds the control period. In contrast, the complexity of software coding logic caused by complex algorithms also needs to be considered. The storage space mainly involves pre-loading parameters and tables needed by the control allocation algorithm, such as thruster combination table, thruster configuration matrix, inverse matrix, and weight coefficient matrix. Typically, the storage space is not too large. However, if the optimal lookup table method is used or the task dimension is large, resulting in numerous thruster combinations, it may take more storage space. 2. Simulation Study The simulation of the control allocation algorithm is carried out for the 6-DOF control task of the final approach phase of RVD, using the decoupled lookup table (DLT) method, linear programming (LP) method, and 6-D optimal lookup table (OLT) method, and the performance of various algorithms is compared. (1) Simulation Initialization In the relative coordinate system to the target spacecraft, we initialize the simulation by

328

7 Theory and Design of Thruster Configuration …

Table 7.2 Thrusts and torques produced by the 22 thrusters in the body reference frame No Fx

1

2

3

150

150

Fy

0

Fz

50

Tx

0

Ty Tz

4

5

6

9

10

0

7 0

0

−150

−150

0

0

120

−120

−100

100

0

−120

120

0

0

0

0

−30

0

0

0

0

0

0

0

0

−100

100

0

0

0

0

−100

−350

0

0

−100

100

−150

150

0

16

17

18

19

20

21

22

0

8

11

150

150

0

−50

0

50

0

−50

0

0

0

0

350

0

−350

0

350

0

No

12

13

14

15

Fx

0

0

0

0

0

0

0

0

0

0

0

Fy

0

30

30

0

0

−30

−30

0

0

0

0

Fz

−30

0

0

30

30

0

0

−60

60

60

−60

Tx

0

0

0

0

0

0

0

60

60

−60

−60

Ty

−100

0

0

100

100

0

0

−60

60

60

−60

Tz

0

−100

−100

0

0

100

100

0

0

0

0

• • • •

Target spacecraft orbit: a circular orbit with a height of 400 km; Initial relative position of chaser spacecraft: [50 0 0] m; Initial relative speed between the chaser and the target: [−0.45 −0.13 −0.18] m/s; Initial chaser attitude angle and angular velocity: 0.5◦ , 0.5◦ /s.

The control target is set to [0 0 0] m. The relative position and relative attitude are controlled by the phase plane control law. The dimension of the control command m is 6, considering translation and rotation in the body reference frame. The control cycle is limited to 1 s. The propulsion system includes 22 thrusters numbered 1–22, which generate thrusts Fx , Fy , and Fz under the body coordinate system, as well as rolling, pitching, and yawing torques Tx , Ty , and Tz , respectively. The details are provided in Table 7.2, which forms the 6 × 22 configuration matrix A. All thrusters are supposed to have the same cost efficiency. (2) Thruster Combination Table Design ➀ Grouping thrusters for DLT method The DLT method divides the 22 thrusters into 12 groups according to the six control channels, as shown in Table 7.3. We can figure out that when the desired torques are generated in the direction of Ty and Tz . The disturbance forces are involved in the direction of Fz and Fy . ➁ The LP method does not need grouping thrusters. It requires the initialization of the thruster configuration matrix. ➂ Grouping thrusters for the OLT method Because thrusters 12, 14, 16, and 18 are backups for thrusters 11, 13, 15, and 17, respectively, we consider each of these pairs as one thruster. Then, a configuration

7.3 Theory and Design of Thruster Control Allocation

329

Table 7.3 Thruster combinations for decoupled lookup table method Force/torque direction

Force/torque amplitude

Thruster combination

Fx

+

300 N

1 + 3 or 2 + 4

−

300 N

9 + 10

+

90 N

7 + 17

−

90 N

8 + 13

+

90 N

6 + 11

−

90 N

5 + 15

+

120 Nm

19 + 20

−

120 Nm

21 + 22

+

100 Nm

16

−

100 Nm

12

+

100 Nm

18

−

100 Nm

14

Fy Fz Tx Ty Tz

matrix A6d with 18 thrusters is obtained. Based on the optimal thruster grouping method, set c j = 1( j = 1, . . . , 18) and follow the algebra described in Sect. 7.3.1. Thus, we can derive 248 combinations. A shortcut view of these combinations is shown in Table 7.4. According to the six thrusters in each combination in the table, a sub-matrix of the six thrusters can be gained. All the inverse matrices corresponding to the combination are obtained, which can be used for the online real-time control allocation algorithm. We summarize that: using the DLT method, thrusters are grouped according to the direction of control, and the interference is cancelled pairwise so that force/torque by each combination generate decoupled control in one direction. In the case of using the OLT method, the thrusters are grouped according to its performance, and each combination can produce a certain range of control in the six directions in the 6-DOF command space. The LP method is more flexible and does not require pre-grouping.

Table 7.4 Six-dimensional optimal thruster combination table Combination 1 Thruster no.

1

2

3

4

5

6

7

… … 242 243 244 245 246 247 248

1

1

1

1

1

18

18

11

5

8

8

10 10

5

5

7

4 12

12

12

12

4

4

18

18

9 14 14 10

7 10

7

3

3

7

7

11

3

14

1 … … 18

15 15 15 15 15 15 15

5

7

5

18

18

5

11

5

8

9

5

5

9

11

11

11

12

12

14

14

2 14

2

8

2

2

5

10

10

7

3

11

2

2

5

330

7 Theory and Design of Thruster Configuration …

(3) Performance Comparison According to the above conditions, the DLT method, LP method, and OLT method are used in simulation. The results are listed in Table 7.5 (Pentium4 CPU 3. 40 GHz, 1 GB memory). ➀ Comparison between OLT and DLT It can be seen from Table 7.5 that the control error of OLT is significantly less than that for DLT. For each control cycle of the two algorithms, the control errors in the six control channels are shown in Fig. 7.7. The main factor causing the control error is the difference in the control capability of different algorithms. In addition, the DLT method not only produces the required control torque in the T y and T z directions but also involves the interference force in the F z and F y directions, which causes a large control error. Table 7.5 Performance comparison of control allocation algorithms Control error (%) Momentum increment error

Angular momentum increment error

Total firing duration (s)

Computing time (s)

26.29

110.56

0.0123

0.97

82.09

0.0479

LP

0.00

0.00

82.01

4.0589

My/(N·m)

20 0 −20 0

Mz/(N·m)

20 0 −20 0

20

20

20

20

40

40

40

40

40

60 80 Time /s

100

60 80 Time /s

100

60 80 Time /s

100

60 80 Time /s

100

60 80 Time /s

60 80 Time /s (a)

100

100

Fig. 7.7 Control error comparison

120

20

40

60 80 Time /s

100

120

140

140

10 0 −10 0

120

20

40

60 80 Time /s

100

120

140

140

10 0 −10 0

120

20

40

60 80 Time /s

100

120

140

120

0 −5 0

20

40

60 80 Time /s

100

120

140

10 0 −10 0

120

20

40

60 80 Time /s

100

120

140

10 0 −10 0

120

20

40

60 80 Time /s (b)

100

120

140

140

140

140

Mx/(N·m)

Mx/(N·m)

10 0 −10 0

20

40

My/(N·m)

Fz/N

50 0 −50 0

20

Mz/(N·m)

Fy/N

50 0 −50 0

Fx/N 140

0.05 0 −0.05 0

Fx/N

0.1 0 −0.1 0

Fy/N

9.74 0.23

Fz/N

DLT OLT-6

−16 5 ×10

7.3 Theory and Design of Thruster Control Allocation

331

5 DLT 6-DOF OLT

Thruster total firing duration/s

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

20

40

60

80

100

120

140

Time/s

Fig. 7.8 Firing durations of the DLT and OLT-6 methods

Table 7.5 shows that propellant consumption and firing duration of the OLT method are much less than those of the DLT method. Figure 7.8 compares the total thruster operating time of the two algorithms. Most of the time the curve of OLT lies under the curve of DLT. This is mainly because the thruster combination of the DLT method generates force/torque in a single direction. The forces/torques in other directions are cancelled. In comparison, the OLT method divides the thrusters with similar performances into one group to minimize cancellation, and the control in each direction is utilized sufficiently, thereby saving propellant consumption. The online computing speeds of the two algorithms are approximate, and the OLT method is slightly faster. ➁ Comparison between OLT-6 and LP The control error of the OLT method based on the optimal basis is slightly larger than that of the LP method. Figure 7.9 shows the control errors in the six control channels changing with control cycle. This is because the OLT opens six thrusters per control cycle, and some control command with large amplitude makes the thruster working time exceed the maximum value in the control period, resulting in control error. In comparison, the LP method can find more than six thrusters in this situation, so that the thruster working time is not exceeded, and the control error is smaller. To verify that without the constraints of thruster operating time, the OLT and LP methods have the same solution in each control cycle. We synchronously solve LP in the simulation of OLT to ensure they have the same control command input in every control cycle. Figure 7.10 draws out the difference between the total firing durations

332

20

40

60 80 Time /s

100

120

140

Fy /N 140

10 0 −10 0

20

40

60 80 Time /s

100

120

140

140

10 0 −10 0

20

40

60 80 Time /s

100

120

140

−16 5 ×10 0 −5 0 20

40

60 80 Time /s

100

120

140

40

100

120

Fy /N

60 80 Time /s

40

60 80 Time /s

100

120

Fz /N

−12 5 ×10 0 −5 0 20

40

My /(N·m)

Mx /(N·m)

−14 2 ×10 0 −2 0 20

Mz /(N·m)

2 0 −2 5 0 −5

40

×10−11

0

20

40

×10−11

0

20

40

60 80 Time /s

100

60 80 Time /s

100

60 80 Time /s

60 80 Time /s (a)

100

100

120

120

120

120

140

140

140

My /(N·m)

20

−11 5 ×10 0 −5 0 20

10 0 −10 0

20

40

60 80 Time /s

100

120

140

Mz /(N·m)

0

Mx /(N·m)

Fx /N 140

0.05 0 −0.05 0

Fz /N

×10−11

Fx /N

5 0 −5

7 Theory and Design of Thruster Configuration …

10 0 −10 0

20

40

60 80 Time /s (b)

100

120

140

Fig. 7.9 Control error comparison

7

×10−9

Thruster firing duration difference/s

6 5 4 3 2 1 0

20

40

60

80

100

120

140

Time /s

Fig. 7.10 The relative error of working time between the LP and OLT-6 methods

7.3 Theory and Design of Thruster Control Allocation

333

4 LP 6-DOF OLT

3.5

Thruster total firing duration /s

3 2.5 2 1.5 1 0.5 0

0

20

40

60

80

100

120

140

Time /s

Fig. 7.11 Firing durations of the LP and OLT-6 method

obtained by these two algorithms, whose magnitude is about 10−9 s. This verifies the homogeneity of the two algorithms’ solution and the optimality of the OLT method. Table 7.5 also shows that the thrusters’ total firing durations calculated by the OLT and LP methods are approximate. Their firing durations w.r.t. control cycles are shown in Fig. 7.11. Compared with Fig. 7.8, it can be clearly seen that for the LP and OLT-6 method, the overall trends of propellant consumption are relatively close. In addition, from Table 7.5 we can see that the OLT-6 method helps improve the online computing speed and therefore has higher engineering realizability. 3. Summary From the previous analysis, we can see (1) Decoupled lookup table Pros: the algorithm is simple and less computation is required. Cons: (i) The requirements for thruster configuration are very high and cannot be decoupled in some cases; (ii) because of the fixed function and working time ratio for each thruster, it is hard to flexibly apply certain amount of control to each thruster; thus, the thruster configuration cannot be efficiently utilized and cannot reduce propellant consumption. (2) Linear programming Pros: strong versatility, flexible use of each thruster, high thruster efficiency, and strong control capability;

334

7 Theory and Design of Thruster Configuration …

Table 7.6 Pros and cons of control allocation algorithms

DLT

LP

OLT

Universality Optimality Control capability Complexity Storage space

Cons: iterative optimization costs lots of computation resources, thus overburdening the on-board CPU given too many thrusters. (3) Optimal lookup table Pros: it is similar to LP and less computation is required. Cons: the six-dimensional optimal combination table is huge; the dimension can be reduced appropriately at the expense of optimality. The following table sums up the above information (Table 7.6). The essence of the DLT method is to reduce the original six-dimensional control allocation problem into six one-dimensional sub-problems, whereas the LP and OLT methods can solve either the original problem or sub-problems. There is a trade-off between dimension reduction: the more close to 6-DOF the dimension of the subproblems is, the higher degree of optimality the solution will reach, but the solution process will be more complicated. The more close to 1-DOF the dimension of subproblems is, the lower degree of optimality the solution will reach, and the solution process will be simpler.

7.3.3 Synthetical Design of Thruster Control Allocation Algorithm 1. Design Flow In the specific engineering design, it is necessary to consider the different characteristics of different thruster configurations combined with the computing power and storage space. Therefore, a universally applicable design method cannot be obtained. However, we can summarize a general design idea as follows: (1) Analyze the thruster configuration matrix to find out the control channel with strong coupling interference; (2) Combine these control channels to formulate a sub-problem so as to reduce the dimensions of the original six-dimensional problem;

7.3 Theory and Design of Thruster Control Allocation

335

(3) For each reduced sub-problem, ignore the control force/torque of the thruster in the other directions and list the thruster configuration matrix for each subproblem. (4) Design a corresponding control allocation algorithm according to each subquestion. For example, for one-DOF sub-problem, we can use the DLT method; for the two-DOF sub-problem, the OLT method can be used. (5) According to the control allocation algorithm of each sub-question, we design the corresponding thruster combination table and an online control allocation algorithm and provide a data table to be pre-loaded. A synthetic control allocation algorithm based on the above idea has been applied to China’s manned spacecraft RVD design and has achieved good results. The following is a simulation example for a further comprehensive illustration on the control allocation method, its design process, and specific applications. 2. Simulation Study Using the initial conditions for the simulation in Sect. 7.3.2, we might as well remove thrusters 11, 13, 15, and 17 from the original configuration. Considering the numerous thrusters and limited storage space, we do not use the six-DOF OLT method. Besides, the thruster grouping policy of the DLT method cannot be designed as shown in Table 7.3 because thrusters 12, 14, 16, and 18 may substitute thrusters 11, 13, 15, and 17 to generate F y and F z . As a result, to prevent weakening of the control capability for translation, torques T y and T z need to be generated by other thrusters. (1) Analysis on Dimension Reduction for Thruster Configuration Matrix Observe the thruster configuration table (Table 7.2), we know that thrusters 1, 2, 3, and 4 can not only provide F x control but also give a large component (350 N m) in both the positive and negative directions of T y and T z , satisfying the control requirement, whereas the force produced by them in the F y and F z directions is relatively small. Therefore, we consider combining the three directions of F x , T y, and T z to establish a three-DOF control allocation model. The OLT method is used to solve this sub-problem. The F y , F z , and T x directions are set as three one-DOF sub-problems and solved by the DLT method. In summation, the control allocation after the dimension reduction is listed in Table 7.7. (2) Modeling 3-DOF Control Allocation From Table 7.7, we know that thruster 1–4, 9, and 10 generate controls in the F x , T y , and T z directions. Let the control force and torque generated by these thrusters in the F x , T y , and T z directions be column vectors. We can list a thruster configuration matrix corresponding to the three-DOF F x + T y + T z problem as follows: ⎡

A f xmymz

⎤ 150 150 150 150 −300 = ⎣ 350 0 −350 0 0 ⎦ 0 350 0 −350 0

336

7 Theory and Design of Thruster Configuration …

Table 7.7 Control allocation with reduced dimensions

Subproblem with reduced dimension

Thrusters used

3-dimension

Fx + T y + T z

1 2 3 4 9 10

1-dimension

Fy

+

7 + 18

−

8 + 14

1-dimension

Fz

+

6 + 12

−

5 + 16

1-dimension

Tx

+

19 + 20

−

21 + 22

Then, the model for the 3-DOF control allocation is A f xmymz t f xmymz = u3d T T



5 where t f xmymz = t1 t2 t3 t4 t9+10 ∈ R+ and u3d = u F x u M y u M z . Herein, thruster 9 and 10 are combined to output one thrust vector. Otherwise, if these two thrusters are used separately, it will cause a large disturbance to the F y and F z directions, increasing the coupled effect. (3) Optimal Lookup Table Design for the 3-D of Sub-Problem The thrust vector distribution of thrusters 1–4, 9, and 10 in the three dimensions (i.e., column vectors in A f xmymz ) is shown in Fig. 7.12. According to the general grouping

400 J2

200 Tz /(N·m)

J1 0

J3

−200

J9+J10 J4

−400 400 200 0 Ty /(N·m)

−200 −400 −300

−200

−100 Fx /N

Fig. 7.12 Thrust vectors viewed in the 3-dimensional command space

0

100

200

7.3 Theory and Design of Thruster Control Allocation Table 7.8 The optimal thruster combination table for the 3-DOF subproblem F x + Ty + Tz

337 Thruster no.

Combination 1

(1 2 3)

Combination 2

(1 2 9 + 10)

Combination 3

(1 2 4)

Combination 4

(1 4 9 + 10)

Combination 5

(2 3 9 + 10)

Combination 6

(3 4 9 + 10)

algorithm described in Sect. 7.3, the six thruster combinations can be obtained, as shown in Table 7.8. We notice that the resulting 6 thruster combinations completely contain the entire 3D command space, in which the combinations 1, 2, and 5 constitute the upper half space, and the combinations 3, 4, and 6 constitute the lower half space. Therefore, there is a non-negative feasible solution for any control command in the threeDOF control command space; the regions governed by each combination have no intersections, so there is only one feasible solution for each control command. (4) Performance Comparison on Control Allocation Algorithms According to the above dimension reduction design, we the denote solution to the reduced problem similar to the 3-DOF OLT method. At the same time, the six-DOF OLT method is designed to solve the same problem, and the comparison results of the performance indexes of the two algorithms are shown in Table 7.9. The 6-DOF OLT method outperforms the 3-DOF OLT method in the overall control error, which is mainly because the 6-DOF OLT method can flexibly use the force/torque generated by each thruster in all six directions, achieving larger control capability so that the case of working time exceeding the limit is less. The control capabilities in the F x -T y -T z directions for two algorithms are shown in Fig. 7.13. The 6-DOF OLT method gives less propellant consumption. The total firing durations with respect to the control cycle of the 3-DOF OLT method and the 6-DOF OLT method are shown in Fig. 7.14. It indicates that for the 6-DOF orbit and attitude control, the global optimal solution can be reached only if we consider the problem within the 6-DOF space. Table 7.9 Performance comparison of control allocation algorithms Control error (%)

Total firing duration (s)

Computing time (s)

13.83

114.09

0.0254

890

1.75

83.13

0.0481

17712

Momentum increment error

Angular momentum increment error

3-DOF OLT

6.75

6-DOF OLT

0.36

Storage space (B)

338

7 Theory and Design of Thruster Configuration … 600

600

400 Tz/(N·m)

Tz /(N·m)

400 200 0

0

−200

−200

−400

−400 −600 500

200

−600 500 0 Ty /(N·m)

0 Fx /N

−500

500

0 Ty /(N·m)

500 0 Fx /N

−500

Fig. 7.13 Control capability along F x -T y -T z direction: comparison between OLT-3 and OLT-6

6 3-DOF OLT 6-DOF OLT Thruster total firing duration/s

5

4

3

2

1

0

0

20

40

60

80

100

120

140

Time/s

Fig. 7.14 Firing durations of 3-DOF OLT and 6-DOF OLT methods

According to the storage space required for pre-loaded data, the 3-DOF OLT method is better than the 6-DOF OLT method, dropping from 17 kB to less than 1 kB. This is because the inverse table of the 6-DOF OLT method contains 248 6 × 6 square matrices, while the 3-DOF OLT method only needs to store 6 3 × 3 square matrices. In summary, the results are consistent with the analysis in Sect. 7.3.2. For a specific task, the control allocation algorithm suitable for the thruster configuration characteristics can be designed according to the design flow provided in Sect. 7.3.3.

7.4 Theory and Design of Thruster Configuration

339

7.4 Theory and Design of Thruster Configuration The design of the thruster configuration is a process that requires repeated iterations: propose a preliminary feasible thruster configuration, perform some simulations for specific space missions, analyze the performance of the thruster configuration, find problems, and optimize the parameter settings of thruster configuration until the solution that matches the task best is finally determined. In this section, we first discuss the general task requirements for thruster configuration, based on which we introduce analysis method for thruster configuration performance. Then, we formulate the mathematical description for thruster configuration design problems and provide a preliminary discussion of theoretical methods for thruster configuration parameter design and optimization.

7.4.1 Performance Requirements The purpose of configuring the thruster is to output the control force/torque that meets the needs of the space mission under normal or even faulty conditions. Therefore, the performance requirements of thruster configuration mainly include control capability, control accuracy, efficiency, redundancy, and plume impact. 1. Control Capability The control capability requirements mainly refer to two aspects. First, the completeness that the required control commands can be generated in all directions. Second, the amplitude of the control command in each direction should be sufficient to meet the task’s needed to overcome the interference and to achieve the required amount of control. For RVD tasks, the high requirements for control capability are mainly reflected in the six-DOF control during the closing phase, in which the relative position and relative attitude are simultaneously controlled. The thruster configuration determines the amplitude and direction of the force/torque generated by each thruster in the spacecraft’s body coordinate system, that is, a vector in the command space. To achieve control capability requirements, it is necessary to analyze the interference in the process of the mission, select the appropriate size of the thruster, and design a reasonable layout. Figure 7.15 shows the thruster configuration layout using the simplest twodimensional thruster configuration as an example, obtaining the relationship between control capability and thruster layouts. The two thrusters shown in Fig. 7.15a, J 1 and J 2 , can only output the control along the X-axis. We add one thruster J 1 in the y-axial direction, as shown in Fig. 7.15b; then, the thrusters can produce a force within a certain range on the upper half-plane. By changing vector direction of J 1 and J 2 , as shown in Fig. 7.15c, the thruster configuration now can produce forces in all directions in the entire x–y plane. When the amplitude of the thrust changes, the scope of control capability will change correspondingly, as shown in Fig. 7.15d.

340

7 Theory and Design of Thruster Configuration … y

y

J3

y

J3

y

J3 J2

J1 o

x

J2

o

o J2

(a)

(b)

o

x

J1 x J1

x

J2

J1

(c)

(d)

Fig. 7.15 Control capability for diverse thruster configurations

In contrast, the control allocation algorithm will also impact control capability, which has been discussed in detail in Subsection 2 of Sect. 7.3.3. 2. Control Accuracy The control accuracy requirement of the thruster configuration is reflected in the minimum velocity changes in each control channel. For the six-DOF control of the relative position and relative attitude of RVD, the velocity changes are defined as the minimum translation velocity increments and angular velocity increments. The control accuracy requirements are caused by the position and attitude control need of tasks, such as docking conditions for RVD tasks. With respect to the thruster, the realization of control accuracy requirements is mainly determined by the thruster’s thrust and minimum firing duration. The layout and control allocation method of the thruster configuration will affect the nominal control force/torque in the corresponding control channel. 3. Efficiency The efficiency of the thruster configuration is reflected in how low the propellant cost can be offset by the thruster configuration, achieving the required control commands. The efficiency of the thruster configuration is closely related to the configuration of the thruster. As shown in Fig. 7.16, if the control law gives a set of control commands distributed in the shadow area along the x-axis, installing thruster J along the x-axis will result in less propellant consumption compared to other installation directions such as in the J direction because most propellants of J are used to achieve u, Fig. 7.16 Control distribution affects propellant consumption

y u J

J′

x

7.4 Theory and Design of Thruster Configuration uy

uy

ux

(a)

341 uy

ux

(b)

ux

(c)

Fig. 7.17 Thruster configuration affects redundancy

whereas some propellants of J are wasted for generating unnecessary control in the y-direction. Therefore, we can induct two main factors affecting the consumption of propellant: (i) distribution of control command provided by control law; and (ii) distribution of thrust vector given by each thruster. The former mainly depends on the task and control algorithm, whereas the latter is closely related to the thruster configuration. The more similar the two distributions are, the higher efficiency will be reached. 4. Redundancy The redundancy of the thruster configuration was first proposed by Crawford in 1969 [38], who defined it as the maximum number of thruster failures allowed to complete certain control tasks. The redundancy of the thruster configuration can be divided into multiple levels according to specific control tasks, for example, redundancy requirements for attitude control, redundancy requirements for six-DOF RVD control, and redundancy requirements to ensure safety. The redundancy of the thruster configuration is directly related to the configuration layout. For example, for 2D thrusters shown in Fig. 7.17, there are three thrusters in Fig. 7.17a, and the redundancy is 0. There are six thrusters in Fig. 7.17b, and the redundancy is 1. Because of the different installation, the redundancy in Fig. 7.17c is zero, despite it having the same thruster number as the configuration in Fig. 7.17b. More information regarding the relationship between the number, distribution, and redundancy of the thruster can be found in Ref. [38]. 5. Plume Impact Generally, the spacecraft is equipped with solar arrays, which are very sensitive to heat, pressure, and pollution. If the plume produced by the thruster hits the solar arrays, it will bring the risk of heat, pressure, and pollution and will also produce disturbance forces and torques on the spacecraft itself. Therefore, to minimize plume impact, certain constraints will be put forward for the configuration and installation of the thruster according to the actual installation of the solar arrays in engineering.

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7.4.2 Performance Analysis Because the control accuracy requirements are easier to judge, we focus on control capability, efficiency, and redundancy requirements. The above factors are closely related to thruster configuration and the control allocation algorithm. When the number of thrusters is large and the configuration is complicated, it is more difficult to conduct analysis, which will be shown in more detail as follows. 1. Control Capability Analysis Control capability of a thruster configuration should be analyzed in the context of specific task requirement. We define a configuration as feasible when the control capability of a thruster configuration can accomplish the requested task. We call this configuration a feasible one w.r.t. this task. In the following section, we first provide some related definitions and then provide theorems to judge whether a thruster configuration is feasible. 1) Definitions on feasibility Definition 7.1 (Control set) The control set Z c w.r.t. the thruster configuration matrix A ∈ Rm×n is defined as follows:   Z c = uc |uc = At, t ∈ Rn+ , t ∞ ≤ tm where tm is a positive constant and represents the upper bound of the thruster firing duration. Physical interpretation: the control set is a collection of deliverable controls produced by the thruster configuration in the whole control space Rm , such as the area encircled by the dashed lines in Fig. 7.15b, c. Obviously, different configurations correspond to different control sets. Furthermore, the constraint t ∞ ≤ tmax is a limitation on the magnitude of thrust vectors. Definition 7.2 (Command set) The command set Z u w.r.t. the given task is defined as follows:   Z u = x u | x u ≤ v max , x u ∈ Rm where v max corresponds to the control command vectors produced by the control law with the largest magnitude. Physical interpretation: the command set is a collection of m-DOF control commands derived from the control law while performing the m-DOF task. Definition 7.3 (Feasible under constraints (FC)) Given an m-dimensional task equipped with n thrusters, and its configuration matrix A ∈ Rm×n , if its command set and control set have a relation Zu ⊆ Zc ,

7.4 Theory and Design of Thruster Configuration

343

then, we say this configuration matrix A is feasible under constraints. Physical interpretation: Matrix A is feasible under constraints means that this thruster configuration is capable of generating any control command requested by the control law. For example, if the command set Z u and control set Z c are depicted as Fig. 7.18a, then any control command in Z u could be realized by the linear combination of the three thrust vectors. Definition 7.4 (Feasible under non-constraints (FNC)) Given the configuration matrix A ∈ Rm×n , we define its control set without any constraints compared with Definition 7.1 as follows:   n Z cn = ucn |ucn = At, t ∈ R+ If the command set of the task satisfies Z u ⊆ Z cn , then, we say this configuration matrix A is feasible under non-constraints. Physical interpretation: Feasible under non-constraints is defined on the assumption that there is no limit to the magnitude of the thrusts. A in FNC only guarantees that any direction of the required control command in the command set could be realized by this thruster configuration. For example, if some control commands in the command set Z u have exceeded the boundary of the control set Z c , such as v max in Fig. 7.18b, then, the thruster configuration can only try its best to generate a command umax using J 1 and J 3 that preserves the direction of the desired command v max and attains the maximum feasible modulus along it. However, the magnitude of v max will not be reached. This situation often happens in a space mission. If the magnitude of the control command given by the control law is too large, the corresponding thrusters may not be able to completely realize this command even if they keep on firing during the whole control cycle. Definition 7.5 (Infeasible (IF)) If a configuration matrix A ∈ Rm×n is not feasible with non-constraints, i.e. ∃u ∈ Z c , such that there is no solution t to the problem u = At, t ∈ Rn+ , then we say A is infeasible. y Zc

J3

y J3

vmax Zc

Zu J2

J3

x

y

vmax J3

umax

Zc

Zu

J2

J1

x

Zu

J2 vmax

(a)

(b)

Fig. 7.18 Feasibility of thruster configuration: a FC; b FNC; and c IF

(c)

J1

x

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7 Theory and Design of Thruster Configuration …

Physical interpretation: A is infeasible means that directions of some control commands derived from the control law could not be realized by the given thruster configuration. Taking Fig. 7.18c as an example, we see that the control commands in the negative y-axis half-plane within the command set (such as v max ) could not be realized by the combination of the three thrusters depicted in this figure. In this case, the configuration matrix is called infeasible. 2) Criterions on FNC FNC essentially judges the completeness of the thruster configuration control capability according to all directions. The judgements already have some useful criterions for judging whether a configuration matrix A is FNC [38, 54], which are summarized as follows: Theorem 7.1 The configuration matrix A is FNC if and only if A has full rank, and for every m − 1 dimensional hyperplane which divides the command space in half, there must be at least one thrust vector in each half, not counting vectors in the hyperplane. ˜ = P A is FNC, where P is an m × m matrix with Theorem 7.2 If A is FNC, then A full rank. Definition 7.6 Zhang [54] For the configuration matrix A ∈ Rm× j ( j ≥ 1), if the inequality equations j

Ax < 0, x ∈ R+ ¯ is minus, where ‘