Adaptive Hybrid Control of Quadrotor Drones 9811997438, 9789811997433

Table of contents :
Preface
Contents
About the Authors
List of Figures
List of Tables
1 Introduction
1.1 Hybrid Unmanned Aerial Vehicles (UAVs)
1.2 Types of Hybrid UAVs
1.2.1 Tilt-Rotor UAVs
1.2.2 Tilt-Wing UAVs
1.2.3 Rotor-Wing UAVs
1.2.4 Tail-Sitter UAVs
1.3 Propulsion System of Hybrid UAV
1.4 About the Book
References
2 Nonlinear Disturbance Observer-Based Backstepping Control of Tail-Sitter Quadrotors
2.1 Tail-Sitter Quadrotors
2.2 Mathematical Modeling
2.3 Nonlinear Observer Design
2.4 Backstepping Control Design
2.4.1 Quadrotor Mode
2.4.2 Transition Mode
2.4.3 Level-Flight Mode
2.5 Simulation Results
2.5.1 Trajectory Tracking
2.5.2 Quadrotor Mode with External Disturbance
2.6 Conclusions
References
3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor
3.1 Hybrid Controller Design
3.2 Mathematical Model of Biplane Quadrotor
3.3 Hybrid Controller Design
3.3.1 Quadrotor Mode
3.3.2 Transition Mode
3.3.3 Flight Mode
3.4 Results and Discussions
3.5 Conclusions
References
4 Adaptive Controller Design for Biplane Quadrotor
4.1 Biplane Quadrotor: Payload Delivery
4.2 Mathematical Model of Biplane Drone
4.3 Controller Design
4.3.1 Quadrotor Mode
4.3.2 Transition Mode
4.3.3 Level-Flight Mode
4.4 Adaptive Backstepping Controller Design
4.5 Results and Discussions
4.5.1 Autonomous Trajectory Tracking
4.5.2 Packet Delivery Scenario Simulation
4.6 Conclusions
References
5 Multi-observer Based Adaptive Controller for Hybrid UAV
5.1 Nonlinear Observers
5.2 Mathematical Model and Control Architecture
5.3 Observer Design
5.4 Controller Design
5.4.1 Backstepping Controller Design
5.4.2 ITSMC Controller Design
5.5 Adaptive Controller Design
5.5.1 Adaptive Backstepping Controller
5.5.2 Adaptive Hybrid Controller Design
5.6 Stability Analysis
5.7 Results and Discussions
5.8 Conclusions
References
6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load
6.1 Rotor Failure in Biplane Quadrotor
6.2 Dynamical Model of a Biplane with Slung Load
6.2.1 Mathematical Modeling of Slung Load
6.2.2 Dynamics of Slung Load
6.2.3 Mathematical Model of Quadrotor Biplane
6.3 Observer-Based Controller Design
6.4 Results and Discussions
6.4.1 Quadrotor and Transition Modes
6.4.2 Fixed-Wing Mode with Disturbance and Partial Rotor Failure
6.5 Conclusions
References
7 Anti-Swing Control Structure for the Biplane Quadrotor with Slung Load
7.1 Quadrotor with Slung Load
7.2 Biplane Quadrotor with Slung Load Dynamics
7.3 Control Architecture
7.3.1 Anti-Swing Controller (ASC) Design
7.3.2 Trajectory Tracking Controller Design
7.4 Results and Discussions
7.4.1 Case 1: 2.9kg Slung Load
7.4.2 Case 2: 5kg Slung Load
7.5 Conclusions
References
8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor
8.1 Rotor Failures
8.2 Biplane Dynamics and Control Allocation
8.3 Controller Design
8.3.1 Quadrotor Mode
8.3.2 Transition Mode
8.3.3 Fixed Wing Mode
8.4 Results and Discussions
8.5 Conclusions
References
9 Epilogue

Citation preview

Studies in Systems, Decision and Control 461

Nihal Dalwadi Dipankar Deb Stepan Ozana

Adaptive Hybrid Control of Quadrotor Drones

Studies in Systems, Decision and Control Volume 461

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Nihal Dalwadi · Dipankar Deb · Stepan Ozana

Adaptive Hybrid Control of Quadrotor Drones

Nihal Dalwadi Department of Electrical Engineering Institute of Infrastructure, Technology, Research and Management Ahmedabad, Gujarat, India

Dipankar Deb Department of Electrical Engineering Institute of Infrastructure, Technology, Research and Management Ahmedabad, Gujarat, India

Stepan Ozana Department of Cybernetics and Biomedical Engineering Technical University of Ostrava Ostrava-Poruba, Czech Republic

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-19-9743-3 ISBN 978-981-19-9744-0 (eBook) https://doi.org/10.1007/978-981-19-9744-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, 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 publisher, 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 publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains 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

A UAV (Unmanned Aerial Vehicle) is becoming an integral part of human life because of its usage in every sector that directly and indirectly affects our life. For example, UAVs are used in agriculture, traffic monitoring, disaster management, ecommerce, defense, etc. UAVs can be divided into two parts based on their design, (i) rotary-wing and (ii) fixed-wing. Due to the rapid evaluation in electronics technology in the last few decades, numerous types of hybrid UAVs have been developed, which can act as rotary-wing or fixed-wing depending on the mission phase. These hybrid UAVs can take off, hover, and land like rotary-wing UAVs and fly with high velocity like fixed-wing UAVs after performing the transition maneuver. However, hybrid UAVs are highly nonlinear, complex, coupled, and under-actuated systems and are difficult to control. This book focuses on hybrid UAVs called the tail-sitter quadrotor and biplane quadrotor. Tail-sitter quadrotor has only one wing attached to the middle of the quadrotor, while the biplane quadrotor has two wings attached to the quadrotor. Wings generate aerodynamics during the fixed-wing mode. These hybrid UAVs have no deflecting surface, making their construction simple yet effective. However, the speed difference of the actuators performs the transition maneuver, and the whole body rotates, which presents significant challenges in the controller design. A hybrid UAV is a multi-role UAV that can be used for different missions. A controller is designed in such a way that (a) efficiently tracks the autonomous trajectory, (b) handles external disturbances, (c) can adapt to the parameter change during the mission, (d) compensates for a partial or total rotor failure, (e) can track the trajectory when a slung load is attached, and (f) prevents the swinging of the slung load and at the same time it also able to track the desired trajectory tracking. In the initial phase of the book, the classification of hybrid UAVs on their design, advantages and disadvantages, performance evaluation, propulsion system, actuator dynamics, battery model, and electronic speed controller are given briefly. Then the focus is on the different types of nonlinear controller designs of the tail-sitter quadrotor and biplane quadrotor for autonomous trajectory tracking under different conditions like wind gusts. After the nonlinear controller design, this book presents

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different adaptive controller designs with and without estimation algorithms to handle mass change during the payload delivery mission by biplane quadrotor. And the last portion of this book discusses the partial and total rotor failure compensation for the biplane quadrotor during the mission and the design of the anti-swing controller of the biplane quadrotor with slung load system to stabilize the slung load along during autonomous trajectory tracking. Ahmedabad, Gujarat, India Ahmedabad, Gujarat, India Ostrava-Poruba, Czech Republic

Nihal Dalwadi Dipankar Deb Stepan Ozana

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Hybrid Unmanned Aerial Vehicles (UAVs) . . . . . . . . . . . . . . . . . . . . . 1.2 Types of Hybrid UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Tilt-Rotor UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Tilt-Wing UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Rotor-Wing UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Tail-Sitter UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Propulsion System of Hybrid UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 About the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 5 5 6 6 11 14 15

2 Nonlinear Disturbance Observer-Based Backstepping Control of Tail-Sitter Quadrotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Tail-Sitter Quadrotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nonlinear Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Backstepping Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Quadrotor Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Transition Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Level-Flight Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Trajectory Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Quadrotor Mode with External Disturbance . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 23 25 26 29 31 33 33 35 40 41

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor . . . . . . . . 3.1 Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mathematical Model of Biplane Quadrotor . . . . . . . . . . . . . . . . . . . . . 3.3 Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Quadrotor Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Transition Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3.3 Flight Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 58 58

4 Adaptive Controller Design for Biplane Quadrotor . . . . . . . . . . . . . . . . 4.1 Biplane Quadrotor: Payload Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model of Biplane Drone . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Quadrotor Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Transition Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Level-Flight Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Adaptive Backstepping Controller Design . . . . . . . . . . . . . . . . . . . . . . 4.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Autonomous Trajectory Tracking . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Packet Delivery Scenario Simulation . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 64 66 66 68 69 72 76 76 79 84 84

5 Multi-observer Based Adaptive Controller for Hybrid UAV . . . . . . . . 87 5.1 Nonlinear Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Mathematical Model and Control Architecture . . . . . . . . . . . . . . . . . . 89 5.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.1 Backstepping Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 ITSMC Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.5 Adaptive Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.1 Adaptive Backstepping Controller . . . . . . . . . . . . . . . . . . . . . . 95 5.5.2 Adaptive Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . 97 5.6 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Rotor Failure in Biplane Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Dynamical Model of a Biplane with Slung Load . . . . . . . . . . . . . . . . 6.2.1 Mathematical Modeling of Slung Load . . . . . . . . . . . . . . . . . . 6.2.2 Dynamics of Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Mathematical Model of Quadrotor Biplane . . . . . . . . . . . . . . 6.3 Observer-Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Quadrotor and Transition Modes . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Fixed-Wing Mode with Disturbance and Partial Rotor Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 110 110 111 112 113 118 121 125

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6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7 Anti-Swing Control Structure for the Biplane Quadrotor with Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Quadrotor with Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Biplane Quadrotor with Slung Load Dynamics . . . . . . . . . . . . . . . . . . 7.3 Control Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Anti-Swing Controller (ASC) Design . . . . . . . . . . . . . . . . . . . 7.3.2 Trajectory Tracking Controller Design . . . . . . . . . . . . . . . . . . 7.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Case 1: 2.9 kg Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Case 2: 5 kg Slung Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 133 134 135 137 141 141 145 149 149

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Rotor Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Biplane Dynamics and Control Allocation . . . . . . . . . . . . . . . . . . . . . 8.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Quadrotor Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Transition Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Fixed Wing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 155 158 158 160 161 163 170 170

9 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

About the Authors

Nihal Dalwadi was born in Nadiad (Gujarat), in 1992 and is currently pursuing Ph.D. in Department of Electrical Engineering from lnstitute of lnfrastructure Technology Research and Management (ITRAM), Ahmedabad, India. He obtained his Master’s degree in Control and Automation from Institute of Technology, Nirma University, Ahmedabad, India and B. Tech in Instrumentation and control engineering from Sardar Vallabhbhai Patel Institute of Technology (SVIT) Vasad, India. His current research interests include Adaptive control, nonlinear control, stability analysis and UAV Navigation and Control design. Dipankar Deb (M’05–SM’17) is a Senior Member of IEEE and has served a couple of years at IIT Guwahati as an Assistant Professor (AGP 8000) during 2010–2012. He has over 6 years of Industrial experience both in New York (USA) and GE Global Research (Bengaluru) India. From July 2015 to Jan 2019, he has served as an Associate professor, and from Jan 24, 2019, onward he is a Professor in Electrical Engineering at Institute of Infrastructure Technology Research and Management (IITRAM) Ahmedabad. He holds 6 US patents and 1 Indian Patent and has published 44 SCI indexed Journal articles and 40+ International conference papers. He has also authored 11 books with reputed publishers like Springer and Elsevier. He is a Book Series Editor with CRC Press on Control Theory and Applications, and an Associate Editor for IEEE Access. He has also worked extensively in areas such as Adaptive Control, Active flow control, Renewable Energy, Cognitive Robotics, and Machine Learning. He is listed in the top 2% of Researchers worldwide for 2020– 21 as published by Stanford University. Stepan Ozana was born in Bilovec, Czech Republic, on May 16, 1977. He studied Electrical Engineering at VSB-Technical University of Ostrava where he has got M.Sc. degree in Control and Measurement Engineering (2000) and Ph.D. degree in Technical Cybernetics (2004). In 2015, he habilitated in Technical Cybernetics and since he works as an Associate Professor at the Department of Cybernetics and Biomedical Engineering, Faculty of Electrical Engineering and Computer Science

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by VSB-Technical University of Ostrava. At present, he gives lectures on Cybernetics and Control systems. His main area of interest and expertise is modeling and simulation of dynamic systems, control theory, automation, design, implementation, and deployment of control algorithms using softPLC systems.

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 1.13 Fig. 1.14 Fig. 1.15 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11

UAV activities during disasters . . . . . . . . . . . . . . . . . . . . . . . . . . . Hybrid UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of controller design . . . . . . . . . . . . . . . . . . . . . . . . Hybrid UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of tilt-rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of tilt-wing UAV . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of rotor-wing UAV (top view) . . . . . . . . . . . . . Animated picture of MTT UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of CTT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of DDT UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of the dual system . . . . . . . . . . . . . . . . . . . . . . . Classification of hybrid UAV’s propulsion system based on energy types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent circuit of the Li-ion battery . . . . . . . . . . . . . . . . . . . . . Block diagram of electric propulsion system . . . . . . . . . . . . . . . . Schematic diagram of ESC Module . . . . . . . . . . . . . . . . . . . . . . . . Animated picture of quadrotor tail-sitter UAV . . . . . . . . . . . . . . . Animated picture of tail-sitter quadrotor UAV flight regime . . . . Controller block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timeline for trajectory tracking of Tail-sitter quadrotor UAV . . . Altitude and position tracking during quadrotor mode and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking during quadrotor mode and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position tracking by the backstepping controller . . . . . . . . . . . . . Attitude tracking by the backstepping controller . . . . . . . . . . . . . Trajectory tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Altitude and position with disturbance [1 + sin(2t) 1 + sin(2t) 1 + sin(2t)] . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 4 5 6 6 7 8 9 9 11 12 13 13 20 20 21 33 34 34 34 35 35 36 36

xiii

xiv

Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 2.19 Fig. 2.20 Fig. 2.21 Fig. 2.22 Fig. 2.23

Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8

List of Figures

Attitude tracking when disturbance [sin(2t) sin(2t) sin(2t)] applied . . . . . . . . . . . . . . . . . . . . . . . . . . Observer performance for position subsystem with periodic disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observer performance for attitude subsystem with periodic disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position tracking error when periodic disturbance applied . . . . . Attitude tracking error when periodic disturbance applied . . . . . Position tracking with von Karman wind turbulence model . . . . Attitude tracking with von Karman wind turbulence model . . . . Disturbance observer performance for position subsystem with von Karman wind turbulence model . . . . . . . . . . . . . . . . . . . Disturbance observer performance for attitude subsystem with von Karman wind turbulence model . . . . . . . . . . . . . . . . . . . Position tracking error with von Karman wind turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking error with von Karman wind turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear disturbance observer-based backstepping, and nominal backstepping controllers with a periodic disturbance b wind gusts in Quadrotor Mode . . . . . . . . . . . . . . . . Animated picture of biplane quadrotor . . . . . . . . . . . . . . . . . . . . . Block Diagram of controller design . . . . . . . . . . . . . . . . . . . . . . . . Controller design for Fixed-wing mode . . . . . . . . . . . . . . . . . . . . . Timeline for the hybrid control simulation . . . . . . . . . . . . . . . . . . X-axis trajectory tracking during the whole mission . . . . . . . . . . Y-axis trajectory tracking during the whole mission . . . . . . . . . . Z-axis trajectory tracking during the whole mission . . . . . . . . . . Roll angle tracking during the whole mission . . . . . . . . . . . . . . . . Pitch angle tracking during the whole mission . . . . . . . . . . . . . . . Yaw angle tracking during the whole mission . . . . . . . . . . . . . . . . Velocity profile during the whole mission . . . . . . . . . . . . . . . . . . . x−y-position tracking during the mass change . . . . . . . . . . . . . . Altitude tracking during the mass change . . . . . . . . . . . . . . . . . . . Three-dimensional trajectory tracking during the whole mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrotor biplane animated picture . . . . . . . . . . . . . . . . . . . . . . . . Quadrotor biplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cl , Cd , and Cm c versus angle of attack (AoA) . . . . . . . . . . . . . . . Quadrotor mode controller design . . . . . . . . . . . . . . . . . . . . . . . . . Fixed-wing mode controller design . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of adaptive backstepping control . . . . . . . . . . . . . Variable-pitch propulsion system . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation timeline for trajectory tracking with backstepping control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 37 38 38 38 39 39 39 40 40

41 44 47 50 53 54 54 54 55 55 56 56 56 57 57 63 64 66 67 71 73 75 76

List of Figures

xv

Fig. 4.9 Fig. 4.10 Fig. 4.11

77 77

Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. 4.17 Fig. 4.18 Fig. 4.19 Fig. 4.20 Fig. 4.21 Fig. 4.22 Fig. 4.23 Fig. 4.24 Fig. 4.25 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8

Biplane quadrotor’s attitude during trajectory tracking . . . . . . . . Biplane quadrotor’s position during trajectory tracking . . . . . . . . Biplane quadrotor’s angular velocity during trajectory tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AOA, slide slip angle, and vehicle velocity during the trajectory tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust coefficients during trajectory tracking . . . . . . . . . . . . . . . . Control signals during trajectory tracking . . . . . . . . . . . . . . . . . . . Three-dimensional trajectory tracking . . . . . . . . . . . . . . . . . . . . . . Simulation timeline for packet delivery . . . . . . . . . . . . . . . . . . . . . Altitude tracking during packet delivery . . . . . . . . . . . . . . . . . . . . X-axis trajectory tracking during packet delivery . . . . . . . . . . . . . Y-axis trajectory tracking during packet delivery . . . . . . . . . . . . . Roll angle tracking during packet delivery . . . . . . . . . . . . . . . . . . Pitch angle tracking during packet delivery . . . . . . . . . . . . . . . . . Yaw angle tracking during packet delivery . . . . . . . . . . . . . . . . . . Mass tracking by the designed adaptive law . . . . . . . . . . . . . . . . . Thrust change during the mass change . . . . . . . . . . . . . . . . . . . . . Three-dimensional trajectory tracking during payload delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual observer-based control architecture . . . . . . . . . . . . . . . . . . . Block Diagram of an adaptive Backstepping Controller with ESO and DO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block Diagram of an adaptive hybrid controller . . . . . . . . . . . . . . Position subsystem tracking by BSC + ESO with (and without) DO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position tracking by ITSMC + ESO with (and without) DO . . . . Position tracking by different controllers with ESO and DO . . . . Attitude tracking by different Controllers with ESO and DO . . . x − y-position trajectory tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO . . . . . . . . . . . Altitude tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO . . . . . . . . . . . . . . . . . . . . . . . Rotor failure compensation scheme with slung load . . . . . . . . . . Conceptual drawing: A slung load carrying quadrotor biplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Altitude and position tracking with slung load . . . . . . . . . . . . . . . Attitude tracking under slung load . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance generated by slung load in position subsystem . . . . Position and altitude tracking despite wind gusts . . . . . . . . . . . . . Attitude tracking under wind gusts . . . . . . . . . . . . . . . . . . . . . . . . Disturbance due to the slung load under wind gusts in attitude subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 78 78 79 79 80 80 81 81 82 82 83 83 83 84 90 96 98 102 103 103 104 104 105 105 111 112 119 119 120 120 120 121

xvi

Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13 Fig. 6.14 Fig. 6.15 Fig. 6.16 Fig. 6.17 Fig. 6.18 Fig. 6.19 Fig. 6.20 Fig. 6.21 Fig. 6.22 Fig. 6.23 Fig. 6.24 Fig. 6.25 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6

List of Figures

Disturbance due to the slung load under wind gusts in position subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position and altitude tracking under partially failed rotor condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking of biplane with slung load during partial rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance in position subsystem with slung load and partial rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance generated in attitude with slung load and partial rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Altitude and position tracking during quadrotor and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking during quadrotor and transition mode . . . . . . . Disturbance by the wind gust and partial rotor failure in position subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance by wind gusts and partial rotor failure in attitude subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position and altitude tracking in fixed-wing mode with partial rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking in the fixed-wing mode with partial rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance in position, altitude in fixed-wing mode while rotor fails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance about attitude in fixed-wing mode under rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position and altitude tracking under wind gusts and rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking in the presence of wind gust and rotor failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance performance for the position subsystem in fixed-wing mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance performance for the attitude subsystem in fixed-wing mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of anti-swing control scheme . . . . . . . . . . . . . . . . X-axis tracking by different nonlinear controllers with and without anti-swing controller . . . . . . . . . . . . . . . . . . . . . Y-axis tracking by nonlinear with and without anti-swing control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Altitude tracking by nonlinear with and without anti-swing control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roll angle tracking by nonlinear with and without anti-swing control . . . . . . . . . . . . . . . . . . . . . . . Pitch angle tracking by different nonlinear with and without anti-swing controller . . . . . . . . . . . . . . . . . . . . .

121 122 122 122 123 123 124 124 124 125 125 126 126 126 127 127 127 135 142 142 143 143 143

List of Figures

Fig. 7.7 Fig. 7.8 Fig. 7.9 Fig. 7.10 Fig. 7.11 Fig. 7.12 Fig. 7.13 Fig. 7.14 Fig. 7.15 Fig. 7.16 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 8.13 Fig. 8.14 Fig. 8.15 Fig. 8.16 Fig. 8.17 Fig. 8.18

Yaw angle tracking by different nonlinear with and without anti-swing controller . . . . . . . . . . . . . . . . . . . . . X-axis position of slung load during the flight . . . . . . . . . . . . . . . Y-axis position of slung load during the flight . . . . . . . . . . . . . . . Altitude of slung load during the flight . . . . . . . . . . . . . . . . . . . . . X-axis tracking by the nonlinear controller with BSC-based anti-swing controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y-Z axis tracking by the nonlinear controller with BSC-based anti-swing controller . . . . . . . . . . . . . . . . . . . . . . Roll angle tracking by nonlinear controller with BSC based anti-swing controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitch angle tracking by the nonlinear controller with BSC-based anti-swing controller . . . . . . . . . . . . . . . . . . . . . . Yaw angle tracking by the nonlinear controller with BSC-based anti-swing controller . . . . . . . . . . . . . . . . . . . . . . 5 kg slung load position during mission . . . . . . . . . . . . . . . . . . . . Animated picture of biplane quadrotor with deflecting surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor failure in biplane quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . Flow diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control strategy for quadrotor mode and transition mode . . . . . . Control strategy for fixed wing mode . . . . . . . . . . . . . . . . . . . . . . Position and altitude tracking during quadrotor and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking during quadrotor and transition mode . . . . . . . Trust and moments generated during the quadrotor and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motor speeds during the quadrotor and transition modes . . . . . . . Position and altitude tracking during fixed wing and transition modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking during the fixed wing mode and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motor speed during the fixed wing mode and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moments generated during the fixed wing mode and transition mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust, surface angle, and velocity during fixed-wing and transition modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position tracking of the biplane quadrotor during quadrotor mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attitude tracking of biplane quadrotor during the quadrotor mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed of motor during the quadrotor mode . . . . . . . . . . . . . . . . . . Thrust and moments generated during the quadrotor Mode-2 . . .

xvii

144 144 145 145 146 147 147 147 148 148 156 157 157 159 162 164 165 165 165 166 166 167 167 168 168 169 169 169

List of Tables

Table 1.1 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 4.1 Table 6.1 Table 7.1 Table 7.2 Table 7.3 Table 8.1

Performance evaluation of different hybrid UAVs . . . . . . . . . . . . Biplane quadrotor UAV’s specification . . . . . . . . . . . . . . . . . . . . . Hybrid controller gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backstepping controller gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . ITSMC controller Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Time Absolute Error (ITAE) with (i) trajectory tracking and (ii) mass change . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biplane quadrotor parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of the biplane quadrotor . . . . . . . . . . . . . . . . . . . . . . . Specification of biplane quadrotor with slung load . . . . . . . . . . . Integral time absolute error (ITAE) . . . . . . . . . . . . . . . . . . . . . . . . Integral time absolute error (ITAE) for trajectory tracking . . . . . Parameters of biplane quadrotor with deflecting surface . . . . . . .

10 51 52 52 53 57 76 119 141 146 148 163

xix

Chapter 1

Introduction

UAVs are one of the greatest innovations made by human beings in almost all sectors, like e-commerce, defense, agriculture, surveillance, traffic management system, communication in remote areas, human organ transportation, chemical or drug delivery, entertainment sector, wildlife monitoring, etc. In addition, UAVs support disaster management during earthquakes, massive floods, forest fires, and Nuclear/chemical accidents. Figure 1.1 shows different UAV activities for disaster management. UAVs can be used in all three stages of disaster management; for example, in the pre-disaster stage, UAVs are used for surveillance, prevention, and early detection. During a disaster, they might be used for food packet/medicine delivery, monitoring the situation to support critical decision-making and prevent or reduce losses. Finally, after the disaster, UAVs can be used to monitor the situation, assess quickly, and estimate losses. This book’s primary focus is the design of control strategies for hybrid UAVs, like a biplane quadrotor for trajectory tracking, payload delivery/pick up and slung load with partial rotor failure despite wind gusts, and a total rotor failed condition. Nonlinear control strategies are designed to track autonomous trajectories. During the payload delivery/pick up, the system’s overall mass changes, so different adaptive control architectures are developed. In the real world, external disturbances like wind gusts act on the UAV during the mission, requiring a nonlinear disturbance observer (NDO) and a nonlinear controller with complete stability analysis. We also propose a biplane quadrotor UAV with slung load dynamics and an NDO-based control under partial rotor failure and wind gusts effects.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_1

1

2

1 Introduction

Fig. 1.1 UAV activities during disasters

1.1 Hybrid Unmanned Aerial Vehicles (UAVs) A fixed-wing UAVs have a wide range of applications and their usage is mostly in hazardous environments. A fundamental aspect of a controller design is obtaining the dynamics of UAVs that permits us to examine the system performance. The cost of UAVs is high and has a very high risk of damage, and even a small oscillation could lead the UAVs to crash, so to prevent this, the controller needs to be functional experimentally once some simulations have been tested so as to tune the controller parameters. To control the UAVs, there are different control algorithms developed, like LQR (Linear-Quadratic Regulator), PID (Proportional-Integral-Derivative) with gain scheduling control [1], Fuzzy controller [2], BSC (Backstepping Controller), SMC (Sliding Mode Control) [3], ITSMC (Integral Terminal Sliding Mode controller), [4] etc. Based on the dynamics, linear and nonlinear controllers are designed. Kang et al. [5] presented a nonlinear MPC controller while Poksawat et al. [6] established gain scheduled PID controller for fixed-wing UAVs and validated the effectiveness experimentally. A mathematical modeling technique for fixed-wing UAVs based on its robust controller design procedure is proposed in [7]. Espinoza et al. [8] designed five different controllers based on BSC and SMC like BSC, SMC, BSC with SMC, BSC with double SMC, and BSC with higher-order SMC, and performances are evaluated. The rotary-wing quadrotor is widely used in urban areas for traffic monitoring, surveillance, payload delivery, etc. There are many linear and nonlinear controllers developed for the rotary-wing UAVs like PID [9], MPC [10], Self-tuning fuzzy PID controller [11], BSC [12], SMC [13], TSMC [14], H∞ controller [15], INDI (Incremental Nonlinear Dynamic Inversion) [16], and Neural Network-based controller [17].

1.1 Hybrid Unmanned Aerial Vehicles (UAVs)

3

Fig. 1.2 Hybrid UAVs

In the past four decades, different types of UAVs have been developed and are divided based on their structural design into two parts: (i) rotary-wing and (ii) fixedwing UAVs. Both types of UAVs have advantages and disadvantages; rotary-wing UAVs can hover while fixed-wing UAVs do not, but they fly at high velocity and are more energy efficient than rotary-wing UAVs. In addition, rotary-wing UAVs take off and land vertically, while fixed-wing UAVs need a runway. In general, the payload carrying capacity and endurance time of fixed-wing UAVs are more than rotary-wing UAVs. Therefore, many researchers build hybrid UAVs to take off, land, and hover like rotary-wing UAVs and, after the transition, convert them into fixed-wing UAVs to fly at high speed and low power consumption as shown in Fig. 1.2. Rapid growth in electronics, software, and battery technologies leads to the development of different types of hybrid UAVs, mainly tail-sitter, tilt-rotors, tilt-wing, and rotor-wing. UAVs are underactuated, highly nonlinear, and coupled systems, which means less control input and more output. For example, a biplane quadrotor tail-sitter has four control inputs (four actuators) and six output states (altitude, position, and attitude). The role of the control engineer is to design the control law (controller) such that the hardware meets the control requirement. In addition, there are numerous built-in sensors like ultrasonic sensors(distance above the surface), cameras (horizontal motion and speed), pressure sensors (altitude estimation), and IMU (Inertial measurement unit), which contains an accelerometer and three-axis gyroscope to measure acceleration and angular rate. The signals generated by these sensors are further used for the controller design, as shown in Fig. 1.3.

4

1 Introduction

Fig. 1.3 Block diagram of controller design

Onboard sensors measure or estimate the output states to generate tracking errors based on desired signals. The designed controller generates appropriate signals to minimize these errors quickly. These signals are given to the electronic speed controller (ESC) connected to a power source and actuators. ESC gives the signals to the actuators so that required forces and moments are generated.

1.2 Types of Hybrid UAVs Based on the design and operation procedure, hybrid UAVs are divided into several types, as shown in Fig. 1.4. For convertible UAVs, a reference line remains constant (horizontal), which means its body configuration remains constant during the whole mission (take-off, hover, transition, fly, and landing). In contrast, the whole body rotates during the transition, and body configuration is changed in the tail-sitter UAVs. Hybrid UAVs can accomplish complex missions effectively and efficiently, so many researchers have developed dynamics and control architectures for them. Next, we discuss different hybrid UAVs.

Fig. 1.4 Hybrid UAVs

1.2 Types of Hybrid UAVs

5

Fig. 1.5 Animated picture of tilt-rotor

1.2.1 Tilt-Rotor UAVs In tilt-rotor UAVs, numerous rotors are placed on the tilting shaft when the transition from hovering to forward flight is initiated. During the transition, rotors rotate in the front side for forward speed and stop on attaining cruise condition. Tilting machinery is designed in such a way that the longitudinal body axis will not change much, and thrust direction can be switched upward for the take-off, landing, and hover state and forward for fixed-wing mode. Tilt-rotor UAVs have different configurations like tilt two rotors [18], tilt tri-rotor UAV [19], and tilt-four rotor [20]. An animated picture is shown in Fig. 1.5. As discussed earlier, in tilt-rotor UAVs, the rotor tilts to change the flight mode. The detailed dynamics of the bi-rotor-type tilt-rotor are given [21].

1.2.2 Tilt-Wing UAVs In Tilt-wing UAVs [22–24] (with dynamics [25]), both the wing and the rotor rotate in the forward direction during the transition from hover to the fixed-wing phase. Tilt-wing-type UAVs require customized and sophisticated onboard drivers to rotate the wing. An animated picture is shown in Fig. 1.6. A hierarchical adaptive control presented for the quad tilt-wing UAV [26] requires an additional mechanism, just like the tilt-rotors, that increases the complexity and introduces delay. Due to their structure being sensitive to the wind, power consumption is high during the take-off, landing, and hover state, which limits maneuverability. However, overall aerodynamic performance is good and has a simple transition mechanism. This kind of UAV is primarily used in research and industry. It is vulnerable to external disturbances like cross winds, requires powerful actuators, and finds it hard to land in the moving decks.

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1 Introduction

Fig. 1.6 Animated picture of tilt-wing UAV

1.2.3 Rotor-Wing UAVs In vertical flight, the lift force is generated by the rotor wing, while in horizontal flight, it stops, and two sub-rotors provide the required thrust. Figure 1.7 shows the animated picture of rotor-wing UAVs (also called stop-rotor UAVs [27]). Stop rotor UAV is a unique type of UAV that is not much used in industrial applications. This UAV is lightweight and easy to take off and land, but it has a complex transition mechanism and comes with a single rotor for the forward speed.

1.2.4 Tail-Sitter UAVs Tail-sitter UAV’s flight path can be divided into the three-part, (i) quadrotor mode (during take-off, hovering, and landing phase), (ii) transition mode, and (iii) fixedwing mode (fly with higher velocity as conventional fixed-wing UAVs). The rotor or wing rotates the whole airframe during the transition. So these UAVs do not require

Fig. 1.7 Animated picture of rotor-wing UAV (top view)

1.2 Types of Hybrid UAVs

7

different actuators to control the transition and are lightweight and stable. Tail-sitter UAVs can further divide into (a) Mono Thrust Transitioning (MTT), (b) Collective Thrust Transitioning (CTT), and (c) Differential Thrust Transitioning (DTT). Tailsitter UAVs [28–32] land vertically on the tail, achieving cruise flight after tilting the entire body. MPC controller is designed [33] for the hovering flight of cross configuration of quadrotor tail-sitter UAV. A design and controller design of VertiKUL, a tail-sitter quadrotor with no deflecting surface, is presented [34]. Quadrotor tail-sitter with single wing development and PID-based controller is designed in [35]. Six DoF (Degree Of Freedom) dynamics of biplane quadrotor are presented [36], and the inverse controller is designed to track the desired trajectory. No additional mechanism is required, so comparatively, it has low weight and simple construction but is multifaceted to control and sensitive to wind, requiring more power to stabilize in the presence of wind. Numerous advanced algorithms based on disturbance observers (DO) are developed to compensate for the effect of the wind gust-like DO-based H∞ controller [37, 38] and BSC [39] for tail-sitter UAVs. In addition, adaptive algorithms are developed for tail-sitter UAVs to handle the parameter change during the flight. The adaptive control structure for tail-rotor UAV is presented in [40]. While discrete robust adaptive control[41] for tilt-rotor, adaptive BSC [42], and Adaptive H∞ controller [43] for tail-sitter are developed.

1.2.4.1

Mono Thrust Transitioning (MTT)

This type of tail-sitter UAV has one rotor connected to either the rear side or at the nose of the UAV fuselage to generate the required thrust, as shown in Fig. 1.8. The transition from horizontal to vertical and vice versa is achieved using vectored thrust, swashplates, variable-pitch or cycling propeller, or ducted fan vanes. Construction of MTT is simple; no additional actuators are required. However, the mathematical model of the mono thrust transitioning-type tail-sitter UAVs [44] indi-

Fig. 1.8 Animated picture of MTT UAV

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1 Introduction

Fig. 1.9 Animated picture of CTT

cates unstable conditions during vertical flight and at low speeds. Therefore, it has a low payload carrying capacity and less endurance time.

1.2.4.2

Collective Thrust Transitioning (CTT)

In general, it has either one or more than one rotor, and the transition from hover to level flight is done via several control surfaces coupled with thrust. Figure 1.9 shows the animated picture of CTT-type tail-sitter UAV. For the twin-rotor tail-sitter UAV, a real-time hardware-in-loop testing platform is designed and implemented [45, 46] for high-speed and efficient forward flight, commonly used in research and industries because of numerous design options.

1.2.4.3

Differential Thrust Transitioning (DTT)

The design aspect includes rotors connected to both sides of the UAV body. The thrust difference leads to the pitching torque and initiates the transition phase. An animated picture of the DTT-type UAV [39] is shown in Fig. 1.10. Tail-sitter DTT-type UAVs require no extra actuators and have good controllability and stability. However, the design provides reduced efficiency of fixed-wing flight and sensitivity to crosswinds.

1.2.4.4

Dual System

The dual system has dedicated multiple rotors for the take-off, landing, and for cruise flight (called pusher) as shown in Fig. 1.11.

1.2 Types of Hybrid UAVs

9

Fig. 1.10 Animated picture of DDT UAV Fig. 1.11 Animated picture of the dual system

As compared to other configurations of the hybrid UAVs, the design of this system is simple yet effective and no tilting mechanism is required. Dual system-type hybrid UAVs dynamical equations are given as [47]. It has high controllability as well as stability, more option in wing design, and a simple transition mechanism as compared to others. During the curies flight, only the pusher is active and rotors that are used for the vertical force and moments are stopped which leads to more aerodynamic drag and added extra weight that results in more power consumption from the pusher. Overall performance and design-based comparison for the three main types of hybrid UAVs are provided in Table 1.1. The first principles, such as Newton’s laws, help derive the mathematical model of rotary-wing or fixed-wing UAVs. Experimental data of the wind-tunnel test for the low AoA (angle of attack) are considered. Since hybrid UAVs have deflecting surfaces, tilting mechanisms, and additional actuators operating in both low and high AoA during the transition, it only requires additional information to derive the

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1 Introduction

Table 1.1 Performance evaluation of different hybrid UAVs Parameters Tilt-rotor Mechanical structure Parameters Efficiency

Maneuverability Parameters Mechanical structure Parameters Efficiency Maneuverability Parameters Mechanical structure Parameters Efficiency Maneuverability

Complicated tilting mechanism of rotor Additional weight of rotor actuators Aerodynamic effect Actuator delay Lift generation reduced by the download Cruise flight efficiency reduced by the rotor structure Agile maneuvering through thrust vectoring Tilt-wing It has the complex structure to tilt a wing Unnecessary actuator weight is added Sensitive about wind gusts Delay added due to Tilt-actuation More power consumption during the hovering Horizontal flight is efficient Negative impact on agility from a high wing drag Tail-sitter Comparatively less complex design No extra actuator weight Sensitive about wind gusts during hovering High power consumption during hovering Efficient horizontal flight Wing drag reduces an agility reduction

mathematical mode. For example, the interaction between the propeller and wing has to be considered separately from the force and torque generation as calculated based on the momentum theory or lumped vortex model. Lift and drag forces during the entire mission can be calculated using a continuous nonlinear function of force coefficient and AoA. Any CFD (Computational Fluid Dynamics) software, like Ansys fluent, OpenFOAM, or XFOIL, can be used to calculate the lift (cl) and drag (Cd) coefficient for different AoA. Apart from these, the propeller’s rotation generates the gyroscopic effect on the UAVs and should be considered in the mathematical model of the hybrid UAVs. The core structure of the mathematical mode of all hybrid UAVs is the same. Still, force and torque formation, the location of the propeller and its arrangements, whether deflecting surfaces are used or not, the number of actuators and type of actuators used for the tilting mechanism, and the shape and size of the UAVs make a difference.

1.3 Propulsion System of Hybrid UAV

11

There are three types of power sources for small and medium-type hybrid UAVs: (i) battery, (ii) Fuel-based (internal Combustion engine), and (iii) hybrid (battery + fuel).

1.3 Propulsion System of Hybrid UAV Figure 1.12 shows the classification of a propulsion system based on energy sources. BLDC (Brushless Direct Current Motor) is widely used for small UAVs due to its high torque/inertia ratio and height efficiency. In the BLDC motor, a hall effectbased sensor helps detect rotor position, and the inverter comprises semiconductor switches in six operating modes. The mathematical model of a BLDC motor is vb (t) = Ri b (t) + K E ω(t), Tf Df d 2 KE i b (t) − − ω (t) − ω(t), ω(t) ˙ = JP JP JP JP

(1.1) (1.2)

where R is the coil resistance, K E is back EMF, angular velocity is ω, J p is the motor inertia, T f and D f are the motor friction and viscous coefficients, and the vb (t) and i b (t) are the voltage and current generated by the ESC module. Around 85% of the battery is expended by the motors. So, we approximate the battery SoC using a mathematical model of the propulsion system and battery dynamics (State of Charge). Many energy sources are available for UAVs, like Batteries, Solar power, Hydro fuel cell, and Combustion engine. Among these, Li-ion batteries play an essential role in life quality as a leading technology for electronic devices. An equivalent circuit of the Li-ion battery is given in Fig. 1.13. The mathematical model for Li-Po and Ni-Mh types of batteries is expressed as

Fig. 1.12 Classification of hybrid UAV’s propulsion system based on energy types

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1 Introduction

Fig. 1.13 Equivalent circuit of the Li-ion battery

Ib (t) , V˙ SoC (t) = CT 1 Ib V˙d (t) = − Vd (t) + , Rd C d Cd VRint (t) = Rint Ib (t), VB (t) = VOC (SoC(t)) − VRint (t) − Vd (t),

(1.3)

where VSoC is the voltage across the capacitance, C T signifies the battery SoC, and Vd sub-model is the reaction of the battery’s voltage, VRint is a voltage drop due to internal resistance Rint of battery, C T is the total capacitance of the battery, and Cd and Rd are parameters related to the battery’s dynamic voltage response. The battery open-circuit voltage signified as VOC (SoC) is a nonlinear function of SoC. The Li-ion battery powers the motor through an electronic speed controller. The speed of BLDCM (Brushless DC Motor) is controlled by the PWM signals given by the ESC which controls the duty cycle as per the input control signal given by the controller. In this study, the dynamics of ESC are given, and the current generated by the BLDCMs and voltage supply by the Li-ion battery is averaged for the duty cycle generated by the control signals. The power consumption of Bi/RW Quadrotor UAV is calculated using the above BLDC dynamic equations and power source dynamics, and the Li-ion battery dynamics are given. The block diagram of the electric propulsion system is shown in Fig. 1.14. The ESC module is connected to the battery, where the battery provides a voltage on demand. This ESC generates PWM signals based on the controller input to drive the BLDC. A propeller is connected to the BLDC motor, generating the thrust and moments. The electric propulsion system is used mainly in small and medium UAVs. Figure 1.15 shows the schematic diagram of the ESC module. A dedicated controller is needed to control these UAVs. Many open-source flight controllers are available for UAVs, like Pixhawk autopilot, Sparky2, and many others. For opensource simulation, (a) gazebo, (b) HackflightSim, and others are available. Hybrid UAVs have certain disadvantages as well. Some of these are as follows: • Hybrid UAVs are expensive because of the complex mechanical/ electrical structure that allows them to have fast, safe, and smooth transitions between level or cruise flight to the hovering state, so the production cost is high and requires regular

1.3 Propulsion System of Hybrid UAV

13

Fig. 1.14 Block diagram of electric propulsion system

Fig. 1.15 Schematic diagram of ESC Module

• •

•

•

maintenance. Depending on size and applications, these structures are fabricated of numerous parts and customizable components. Hybrid UAVs are not as controllable and stable as conventional rotary-wing or fixed-wing UAVs, being a fusion of these two. Hybrid UAVs are less efficient than conventional rotary-wing or fixed-wing configurations because the transition mechanism adds weight and partially reduces the performance. It may force the UAV to consume more energy, increase drag, and underperform in some flight regimes. It is challenging to perfect the transition between level flight to horizontal flight due to significant challenges like payload and external disturbances like wind gusts while transitioning. Any minor mistake in the programming leads to the unavoidable loss of speed and stability. Hybrid UAVs require suitable altitude while initiating the transition between the hovering state to level flight because, during the transition, most of the hybrid UAVs lose a calculated amount of height, and it depends on the weight, speed, and the way it initiates the transition.

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1.4 About the Book Many books are available for UAV control, like Robust discrete-time flight control of UAV with external disturbances [48]. Fault-tolerant control design synopsis extended Kalman filters-based nonlinear fault detection and isolation (FDI) with essential dynamics of small UAVs and nonlinear flight control and guidance system is given [49]. State approximation and control design for a low-cost UAV is discussed [50]. Essential fundamental physics like rigid body dynamics, aerodynamics, stability expansion, and state approximation using available onboard sensors are given [51]. Finally, a diverse flight structure and control system is deliberated for fixed-wing UAVs to rotary-wing UAVs with MATLAB simulation [52]. In the present book, different nonlinear controllers are designed for hybrid drones. Chapter 2 presents autonomous trajectory tracking through a Backstepping controller (BSC) design for tail-sitter quadrotor drones. In the real world, wind gusts act on the drone, so a disturbance observer is also designed and incorporated to handle such external disturbances. Lyapunov-based stability analysis is given for disturbance-based BSC and validated using numerical simulations. Chapter 3 develops many nonlinear control methods and presents several advantages. Sometimes the controlling effect is enhanced when different control methods are combined. A hybrid controller design is presented where the BSC controls a positioning subsystem, and the ITSMC controls the attitude subsystem to track the desired trajectory, which contains all possible maneuver and flight modes of the biplane quadrotor. Chapter 4 presents a biplane quadrotor for payload delivery and pickup. A mathematical model of the biplane quadrotor is discussed, and a BSC design for trajectory tracking is described. Adaptive Backstepping Controller (ABSC) is designed for a variable-pitch quadrotor, and the Lyapunov stability analysis is given to compensate for the effect of the mass change during pickup and drop. Sometimes combining different controllers enhances the performance, as with the observers. For example, Chap. 5 combines two different observers (a) Extended State Observer (ESO) and (b) Nonlinear disturbance Observer (NDO)—ESO for the state estimation and NDO for the estimation of the external disturbances. The main advantages of the dual observer are that it reduces onboard sensors and is easy to implement on the hardware. The payload is connected with the biplane quadrotor via a cable (slung load) swing due to the wind gusts generating disturbance and consequently enhancing the slung load’s swing. A mathematical model of the slung load with a biplane quadrotor is proposed in Chap. 6. There is a chance for failure (partial or total) during the mission. We consider that while the slung load is connected with the biplane quadrotor, a partial rotor failure happens, and at the same time, an external disturbance is also acting. The NDO-based BSC controller is designed for all three modes to compensate for the disturbance effect on the biplane quadrotor. Human organs, sensitive medical drugs, or vaccines in the payload (slung load) can be damaged by the swinging action. So to prevent damage and stabilize the

References

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slung load in finite time, nonlinear anti-swing controllers (BSC and SMC) along with control methods like BSC, ITSMC, and hybrid are designed in Chap. 7, and comparison and effectiveness are validated using numerical study. Drones are expensive; specifically, they become invaluable during critical missions. There is always a chance for failure in any machine. When total rotor failure accrues in the biplane quadrotor, it is difficult to control in the conventional rotor failure method. So in Chap. 8, we propose modifications in the biplane quadrotor’s wing as deflecting surface, which is active only when the total rotor failure occurs. The control methodology is developed in this chapter with stability analysis. However, as expected, there are some limitations of this book, and the following aspects are not covered in it: • Different control algorithms designed for the tail-sitter-type hybrid UAV, called biplane quadrotor, need to be extended to other hybrid UAVs. • Simulation is carried out only using MATLAB, but it can be further enhanced with the use of the combination of ROS, Gazebo, and MATLAB for better understanding as well as the evolution of the designed controller in different circumstances. • Comprehensive mathematical modeling derivation of hybrid UAVs is not given. • The effectiveness of the designed control architectures supported by simulation results needs to be validated by hardware implementation.

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1 Introduction

49. Ducard, G.J.: Fault-tolerant Flight Control and Guidance Systems. Springer London (2009). https://doi.org/10.1007/978-1-84882-561-1 50. Hajiyev, C., Soken, H.E., Vural, S.Y.: State Estimation and Control for Low-cost Unmanned Aerial Vehicles. Springer International Publishing (2015). https://doi.org/10.1007/978-3-31916417-5 51. Beard, R., McLain, T.: Small Unmanned Aircraft: Theory and Practice. Princeton University Press, Princeton (2012). https://books.google.co.in/books?id=YqQtjhPUaNEC 52. Ng, T.S.: Flight Systems and Control. Springer Singapore (2018). https://doi.org/10.1007/978981-10-8721-9

Chapter 2

Nonlinear Disturbance Observer-Based Backstepping Control of Tail-Sitter Quadrotors

A tail-sitter quadrotor is a hybrid vehicle augmenting rotary- and fixed-wing attributes. It is a simple air vehicle capable of operating in helicopter mode and seamlessly transitioning into fixed-wing mode. This multi-mode capability enhances applicability to different missions—package delivery, off-shore inspection, disaster relief, etc. However, due to the aerodynamic parameters diverging in those modes, a robust controller is needed. This chapter develops a nonlinear disturbance observerbased BSC for a tail-sitter quadrotor and the stability analysis for handling parameter variations and wind gusts. Moreover, a BSC provides decent trajectory tracking with minimal altitude shifts.

2.1 Tail-Sitter Quadrotors UAVs are used in all sectors with a high impact on human life to save energy, time, and money for surveillance, exploration, and transportation. Rotary-wing UAVs can hover while fixed-wing UAVs don’t but have higher flight endurance—a hybrid UAV like tail-sitter quadrotor couples both advantages. Hybrid UAVs [1] can accomplish a few missions difficult for rotary-wing or fixed-wing UAVs [2]. The tail-sitter quadrotor is simple as it needs no additional actuators during the transition. Many researchers have developed small-sized tail-sitter UAVs like twin-rotor tail-sitter with elevons [3], twin helicopter rotor tail-sitters [4], quadrotor tail-sitter UAVs [5], and optimal transition strategies [6]. Control actions on tail-sitters include model prediction-based cascade [7] and a hierarchical control for autonomous flight in all modes. Li et al. [8] describe a robust nonlinear controller for the transition mode despite external disturbances and parametric uncertainties. Zhou et al. [9] developed novel algorithms under system positioning constraints. Dynamic modeling, control signals, and hardware design © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_2

19

20

2 Nonlinear Disturbance Observer-Based Backstepping …

are formulated [10]. A low-cost DSP-based flight controller is formulated for hovering mode. Swarnkar et al. presented a 6-DOF flight dynamics model development, describing wing and flight dynamics [12]. Sartori et al. [13] designed a backstepping controller for fixed-wing UAVs on micro-controller and experimental data logging. Dalwadi et al. [14] developed an adaptive backstepping controller for payload delivery using a biplane quadrotor and later a backstepping and integral terminal sliding mode control-based hybrid controller for trajectory tracking [15]. The control architecture handles partial rotor failures despite wind gusts [16].

Fig. 2.1 Animated picture of quadrotor tail-sitter UAV

Fig. 2.2 Animated picture of tail-sitter quadrotor UAV flight regime

2.2 Mathematical Modeling

21

Figure 2.1 demonstrates a quadrotor tail-sitter for ensuring adequate thrust and lift in a horizontal motion. Furthermore, this hybrid UAV switches flight regimes from vertical to level mode (and reverse) by 90◦ pitch angle rotation as shown in Fig. 2.2. This chapter addresses two significant issues for tail-sitter quadrotor UAVs: (i) Trajectory tracking and (ii) Compensating for the effect of external disturbance on a tail-sitter UAV. Nonlinear observer and backstepping control law ensure robustness for all three modes: (i) Quadrotor, (ii) Transition, and (iii) level flight, ensuring a robust approach for flight envelope through the trajectory-tracking simulations.

2.2 Mathematical Modeling Figure 2.3 illustrates a quadrotor tail-sitter with four tilted rotors to give lift force in vertical flight mode and the thrust in level flight. The input signals [U1 U2 U3 U4 ] control the vehicle’s motion. The tail-sitter can switch between hovering and level-flight modes by rotating the aircraft’s pitch angle by about 90◦ degrees. In the transition phase, control over x- and y-positions is

Fig. 2.3 Controller block diagram

22

2 Nonlinear Disturbance Observer-Based Backstepping …

disabled. Assuming that the thrust and the torque produced are proportional to the propeller revolution speed, the inputs are expressed as U1 = k(21 + 22 + 23 + 24 ), U2 = kl(24 − 22 ), U3 = kl(21 − 23 ), U4 = c(−21 + 22 − 23 + 24 )  = (2 + 4 − 1 − 3 ), where i is the propeller revolution speed of ith rotor,  is the overall residual propeller speed, k is thrust constant, c is the torque constant, and l is the arm length. For u, v, and w as the X −, Y -, and Z -directional body-axis velocities, and p, q and r as angular velocities, the flight dynamics are u˙ = r v − qw + g cos θ cos ψ − v˙ = pw − r u + g cos θ sin ψ

L cos α + D sin α m

(2.1) (2.2)

4 L sin α + D sin α  Ti w˙ = qu − pv + g sin θ − − m m i=1

(2.3)

p˙ =

4 qr (I yy − Izz ) Ir r  l(T4 − T2 ) − (−1)i i + Ix x Ix x i=1 Ix x

(2.4)

q˙ =

4 pr (Izz − Ix x ) Ir q  l(T1 − T3 ) + (−1)i i + I yy I yy i=1 I yy

(2.5)

r˙ =

4 pq(Ix x − I yy ) 1  + (−1)i Q i , Izz Izz i=1

(2.6)

where Ix x , I yy , and Izz are the fuselage moment of inertia, Ir is the propeller gyroscopic effect, m is the fuselage mass, g is the gravitational acceleration, (φ, θ, ψ) are the roll, pitch, and yaw angles such that V = L=



 Vx2

+

Vy2

+

Vz2 ,

α = arctan

Vz Vx

 , T = C T ρ2 d 4

1 1 ρV 2 SCl (α), D = ρV 2 SCd (α), Q = C Q ρ2 d 5 , 2 2

where S is the wing area, V is the velocity, ρ is the air density, d is rotor diameter, c is the drag coefficient, k is a constant, α is an Angle of Attack (AoA), (L , D) are the lift and drag forces, T and Q are thrust and torque produced by propellers, and d is the propeller’s diameter. (Cl , Cd , C T , C Q ) are the lift, drag, thrust, and torque coefficients, respectively. C T , C Q , ρ, d are constants for a conventional tail-sitter quadrotor.

2.3 Nonlinear Observer Design

23

For quadrotor mode in the hybrid frame [17], we have

θ¨ ψ¨ x¨ y¨



 I yy − Izz Ir U ˙ + 2 + dφ θ˙ ψ˙ − ψ Ix x Ix x Ix x   Izz − Ix x Ir U3 = φ˙ ψ˙ + θ˙  + + dθ I yy I yy I yy   Ix x − I yy U4 = φ˙ θ˙ + + dψ I yy Izz U1 + dx = (cos φ sin θ cos ψ + sin φ sin ψ) m U1 + dy = (cos φ sin θ sin ψ − sin φ cos ψ) m

φ¨ =

z¨ = −g + cos φ cos θ

U1 + dz . m

(2.7) (2.8) (2.9) (2.10) (2.11)

(2.12)

State vector can be defined for position (2.10)–(2.12) and attitude subsystems ˙ T ∈ R12 , where P = [x y z]T , O = [φ θ ψ]T , and (2.7)–(2.9) as X = [P P˙ O O]   d p = dx d y dz , do = dφ dθ dψ are the external disturbances which affects stability. Next, we develop a nonlinear observer to estimate uncertainties [18, 19].

2.3 Nonlinear Observer Design A limited onboard power source and the high cost of sensors make disturbance measurement complex and unrealistic. So, mathematical algorithms like Nonlinear Disturbance Observer (NDO) are in vogue to estimate the disturbance acting on the system. The NDO can expressively increase disturbance reduction capability and performance. The following assumption holds for the disturbances d. Assumption 1 The disturbance and derivative of disturbance are bounded: ||d˙ p (t)|| ≤ D p ,

||d˙o (t)|| ≤ Do t > 0,

where D p and Do are positive constants. Similarly, a nonlinear disturbance observer proposed by Yang et al. [20] and Viswanath et al. [21] are implemented for the two subsystems: 

 1 ˙ ˙ n˙ p = −L p n p − L p L p P + G + U p , dˆ p = n p + L p P, m  ˙ − Uo , dˆo = n o + L o O, ˙ n˙ o = −L o n o − L o L o O˙ + (O, O)

(2.13) (2.14)

24

2 Nonlinear Disturbance Observer-Based Backstepping …

where U p = R(O)E 3 U1 , Uo = (O)[U2 U3 U4 ]T (Ui , i = 1, . . . , 4) are shown in Fig. 2.3, and dˆ j , j = p, o is the disturbance estimation, n j is the observer state vector, ζ j = L j I3×3 , ζ j > 0 are the tunable gain matrices, G = [0 0 − g]T , m = mass, R(O) = rotation matrix, and E 3 is unit vector basis associated with the earth fixed frame (I ). (O) = [I E M (O)]−1 where E M (O) = Euler matrix. Lemma 2.1 For a smooth system x˙ = f (x), x ∈ R n , with f (0) = 0 and a Lyapunov candidate function V (V (0) = 0), let x(0) ∈ C ⊂ R n . Along any trajectory x : R + → R n , starting in C, the following differential disparity is satisfied with β > 0: d {V (x(t))} < −αV (x(t)) + β, ∀t ≥ 0 with x(0) ∈ C, (2.15) dt where α is a tunable positive parameter [22]. Under Assumption 1, for an adequately large T ∗ , there exist observer gains L j > 0, j = p, o, for prescribed asymptotic estimators (2.13)–(2.14). For every > 0 there exist L ∗j ∀ L j ≥ L ∗j , and the observer errors satisfy ||ed j (t)||2 ≤ , ∀t ≥ T ∗ , j = p, o.

(2.16)

Proof We rewrite (2.7)–(2.12) as Up ˙ + Uo . + d p , O¨ = (O, O) P¨ = G + m

(2.17)

By differentiating (2.13) and using (2.17), we obtain     Up Up d˙ˆ p = n˙ p + L p P¨ = −L p n p − L p L p P˙ + G + + Lp G + + dp m m  ˙ (2.18) = −L p n p + L p P + L p d p = −L p ed p . Similarly, we can show that

d˙ˆo = −L o edo ,

(2.19)

for error terms ed j = dˆ j − d j , j = p, o, and using (2.18) and (2.19), the error derivatives are expressed as (2.20) e˙d j = −d˙ j − L j ed j . For a positive definite function with error term ed j , given by V1 j = edTj ed j ,

(2.21)

and using Assumption 1, (2.20), and −2edTj d˙ j ≤ ||ed j ||2 + ||d˙ j ||2 , the time derivative of V1 j is

2.4 Backstepping Control Design

25

V˙1 j = 2edTj e˙d j = −2edTj L j ed j − 2edTj d˙ j ≤ −2edTj L j ed j + ||ed j ||2 + ||d˙ j ||2 ≤ (−2L j + 1)V1 j + D 2j .

(2.22)

The inequality (2.22) takes the form of (2.15) with α = −2L j + 1 and β = D 2j , i.e., for j = p, o such types of α, β exist. A lower bound on L j given by L ∗j , i.e., L ∗j ≤ L j ensures that (−2L j + 1) ≤ (−2L ∗j + 1), and so V˙1 j (t) ≤ (−2L ∗j + 1)V1 j (t) + D 2j . Therefore, to ensure V1 j (t) = ||ed j ||2 ≤ , ∀t ≥ T ∗ ; j = p, o, we can choose L ∗j such that (−2L ∗j + 1) + D 2j = 0, that is L ∗j

1 = 2



D 2j

 +1 .

(2.23)

Next, through an example, we introduce the backstepping control technique, a recursive method of utilizing states as virtual control signals or intermediate controls for guaranteed global asymptotic stability of strict feedback systems.

2.4 Backstepping Control Design Example: Design a backstepping controller for the second-order strict feedback nonlinear system (electric powered steering system) given as φ˙ p =  p ˙ p = −α1 φ p − α2 φ˙ p + β1 u − β2 Tr + β3 φh 

(2.24)

where u is input (torque) and Tr is the reaction torque and αi , i = 1, 2 and β j , j = 1, 2, 3 are the system parameters. The pinion angle position, as well as velocity, are φ p and  p while φh is the steering wheel angular position. For an error between actual and desired pinion angle position eφ = φ p − φ pd , we choose a Lyapunov positive definite function as V1 =

1 2 e . 2 φ

By taking its time derivative, we get V˙1 = eφ (e˙φ ) = eφ (φ˙ p − φ˙ pd ).

(2.25)

To stabilize (2.25), a virtual control law is selected as  pd = φ pd − k1 e1 , where tunable gain k1 > 0, such that V˙1 = eφ ( p −  pd + k1 eφ ) = eφ e − k1 eφ2 ,

26

2 Nonlinear Disturbance Observer-Based Backstepping …

where e =  p −  pd . Enhancing (2.4), another Lyapunov function is defined as 1 2 V2 = V1 + e 2 ˙ p − ˙ pd ). V˙2 = eφ e − k1 eφ2 + e (

(2.26)

Using (2.24), we get ˙ pd ). V˙2 = eφ e − k1 eφ2 + e (−α1 φ p − α2 φ˙ p + β1 u − β2 Tr + β3 φh − 

(2.27)

The control law for the second-order nonlinear system is chosen as u=

1 ˙ pd − k1 e˙φ ] [−eφ − k2 e +  pd + α1 φ p + α2 φ˙ p + β2 Tr − β3 φh +  β1 (2.28)

2 such that V˙2 = −k1 eφ2 − k2 e , where k2 > 0 is tunable gain. Next, we develop a robust controller through a backstepping formulation [23–25]. The quadrotor tail-sitter is an underactuated formulation where only four control signals are available [6].

2.4.1 Quadrotor Mode Using (2.10)–(2.12), let us consider position subsystem as 1 P˙Q1 = PQ2 , P˙Q2 = −g + U p + d p . m

(2.29)

For position tracking, with error e1 = PQ1d − PQ1 , the time derivative is e˙1 = P˙Q1d − P˙Q1 = P˙Q1d − PQ2 .

(2.30)

Lyapunov candidate function for position subsystem is given as VQ P1 =

1 T e e1 . 2 1

(2.31)

Now, with velocity tracking error e2 = PQ2d − PQ2 , and using (2.30), we get e˙1 = P˙Q1d − PQ2d + e2 ,

(2.32)

2.4 Backstepping Control Design

27

where PQ2d is the virtual input such that PQ2d = P˙Q1d + c1 e1 , P˙Q2d = P¨Q1d + c1 e˙1 , c1 > 0,

(2.33)

and by putting (2.33) in (2.32), we obtain e˙1 = e2 − c1 e1 . For a function candidate VQ P2 =

1 T 1 e1 e1 + e2T e2 , 2 2

(2.34)

(2.35)

the time derivative using (2.33) is V˙ Q P2 = e1T e˙1 + e2T e˙2 = e1T (−c1 e1 + e2 ) + e2T ( P¨Q1d + c1 e˙1 − P˙Q2 ).

(2.36)

Substituting (2.29) into (2.36), we obtain V˙ Q P2 = −e1T c1 e1 + e1T e2 + e2T



  1 ¨ . PQ1d + c1 e˙1 − −g + U p + d p m

(2.37)

Choosing the control signal as

U p = m e1 + c1 e˙1 + g + P¨Q1d + c2 e2 − dˆ p ,

(2.38)

for positive definite matrix c2 , we have (2.12), U P : [U1 , Ux , U y ] given as Ux = (cos φ sin θ cos ψ + sin φ sin ψ)U p , U y = (cos φ sin θ sin ψ − sin φ cos ψ)U p . Up . (2.39) U1 = cos φ cos θ

To compensate for the disturbance d p , we deploy (2.13). Theorem 2.1 For the error subsystem (2.30) with the disturbance observers (2.13) and (2.14), and the control signals (2.38) and (2.39), c1 , c2 > 0 and L p , providing 0 < ||e1 ||2 + ||e2 ||2 ≤ , ∀ t ≥ T ∗

(2.40)

for tracking errors e1 , e2 , and a chosen adequately large . Proof Lyapunov candidate is expressed as VQ P = VQ P1 + VQ P2 .

(2.41)

28

2 Nonlinear Disturbance Observer-Based Backstepping …

Considering (2.36) and (2.38), from (2.41), we obtain V˙ Q P = V˙ Q P1 + V˙ Q P2 = −e1T c1 e1 + e1T e2 + e2T T −edp (L p −



1 P¨Q1d + c1 e˙1 − g + U p + d p m



1 1 I3×3 )edp + D 2p 2 2

T = −e1T c1 e1 − e2T c2 e2 + e2T (dˆ p − d p ) − edp (L p −

1 1 I3×3 )edp + D 2p 2 2

1 1 T ≤ −e1T c1 e1 − e2T c2 e2 − e2T e2 − edp L p edp + D 2p < δ1 VQ P + D 2p , (2.42) 2 2      where δ1 = min 2λmin (c1 ), 2 λmin (c2 ) − 21 , 2 λmin L p − 1 . Reference trajectory for attitude is given as Od = [φd , θd , ψd ]T . The desired angles φd and θd are obtained using (2.39) and (2.4.1) such that Ux Cφd Cθd Cψd = (Cφd Sθd Cψ2 d + Sφd Sψd Cψd )U1 , U y Cφd Cθd Sψd =

(Cφd Sθd Sψ2 d

− Sφd Sψd Cψd )U1 .

(2.43) (2.44)

Adding (2.43) and (2.44) and dividing by Cφd and Cθd , we obtain Ux Cψd + U y Sψd = (tan θd )U1 ,

(2.45)

such that for Ca = cos a and Sa = sin a, φd and θd are obtained from (2.43)–(2.45): θd = tan−1



Ux Cψd + U y Sψd U1



, φd = tan−1



 Cθd (Ux Sψd − U y Cψd ) . (2.46) U1

Similarly, we define the control laws as ˙ + Uo + do . O˙ Q1 = O Q2 , O˙ Q2 = (O, O)

(2.47)

Defining the error in angle as e3 = O Q1d − O Q1 , the time derivative is e˙3 = O˙ Q1d − O Q2 .

(2.48)

Lyapunov candidate function can be V Q O1 =

1 T e e3 . 2 3

(2.49)

For error in angular velocity e4 = O Q2d − O Q2 , the time derivative is e˙4 = O˙ Q2d − O˙ Q2 ,

(2.50)

2.4 Backstepping Control Design

29

and an appropriate Lyapunov function candidate can be 1 VQ O2 = VQ O1 + e4T e4 , 2

(2.51)

and using (2.47) and (2.48), the time derivative of (2.51) is given as   ˙ + Uo + do (.2.52) V˙ Q O2 = −e3T c3 e3 + e3T e4 + e4T O¨ Q1d + c3 e˙3 − (O, O) The control law for the attitude subsystem can be defined as ˙ + O¨ Q1d − dˆo + c4 e4 . Uo = e3 + c3 e˙3 − (O, O)

(2.53)

Next, we prove the attitude counterpart of Theorem 2.1. Theorem 2.2 Consider the attitude error subsystem (2.48) and (2.50) in the closed loop with the disturbance observer designed as in (2.13) and (2.14) and the control law designed according to (2.53). There exist positive definite gain matrices c3 , c4 , and L o , such that the closed-loop attitude error satisfies V˙ Q O = V˙ Q O1 + V˙ Q O2 < δ2 VQ2 + Do2 .

(2.54)

Proof For a Lyapunov function candidate VQ O = VQ O1 + VQ O2 , and considering (2.52) and (2.53), we obtain   ˙ + Uo + do V˙ Q O = V˙ Q O1 + V˙ Q O2 = −e3T c3 e3 + e3T e4 + e4T O¨ Q1d + c3 e˙3 − (O, O) 1 2 T (L − 1 I −edo o 3×3 )edo + Do 2 2 1 T L e + 1 D2 < δ V 2 ≤ −e3T c3 e3 − e4T c4 e4 − e4T e4 − edo o do 2 Q O + Do , 2 2 o

   where δ2 = min 2λmin (c3 ), 2 λmin (c4 ) − 21 , 2 (λmin (L 0 − 1)) .

2.4.2 Transition Mode The Quadrotor mode controller controls the initial phase, and the control law for the pitch angle is changed for direct control of altitude and pitch angle sans x − yposition control. Transfer from quadrotor to level-flight mode happens with αstall , i.e., θ Qsw = αstall - γdes - 90◦ when aerodynamic forces are adequate. The quadrotor mode controller is deployed for pitch angle < λ1 . When the pitch angle exceeds λ2 , the control action is the level-flight controller designed in the following subsection, and the transition controller is for the pitch angle between λ1 and λ2 . For x˙1 = φ˙ = p, ˙ x˙3 = θ˙ = q, x˙4 = θ¨ = q, ˙ x˙5 = ψ˙ = r, x˙6 = ψ¨ = r˙ , x˙7 = z˙ = w, x˙8 = x˙2 = φ¨ = p,

30

2 Nonlinear Disturbance Observer-Based Backstepping …

z¨ = w, ˙ x˙9 = x˙ = u, x˙10 = x¨ = u, ˙ x˙11 = y˙ = v, and x˙12 = y¨ = v, ˙ and a1 = I −I

xx a2 = IIxrx , a3 = IzzI−I , a4 = IIyyr , and a5 = x xIzz yy , b1 = I1x x , b2 = I1yy , b3 =  L cos α+D sin α yy  cos α , and r2 = L sin α+D , using (2.7)–(2.12), we have m m

⎛ ⎞ ⎛ ⎞ φ˙ x2 ⎜ φ¨ ⎟ ⎜ ⎟ x4 x6 a1 − a2 x6  + U2 b1 ⎜ ⎟ ⎜ ⎟ ⎜ θ˙ ⎟ ⎜ ⎟ x 4 ⎜ ⎟ ⎜ ⎟ ⎜ θ¨ ⎟ ⎜ ⎟ x x a + a x  + U b 2 6 3 4 4 3 2 ⎜ ⎟ ⎜ ⎟ ⎜ψ˙ ⎟ ⎜ ⎟ x 6 ⎜ ⎟ ⎜ ⎟ ⎜ψ¨ ⎟ ⎜ ⎟ x x a + U b 2 4 5 4 3 ⎜ ⎟=⎜ ⎟. ⎜ z˙ ⎟ ⎜ ⎟ x 8 ⎜ ⎟ ⎜ ⎟ ⎜ z¨ ⎟ ⎜x4 x10 − x2 x12 + g sin x3 − r1 − U1 /m ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x˙ ⎟ ⎜ ⎟ x10 ⎜ ⎟ ⎜ ⎟ ⎜ x¨ ⎟ ⎜ x6 x12 − x4 x8 − g cos x3 cos x5 − r2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ y˙ ⎠ ⎝ ⎠ x12 x2 x8 − x6 x10 + g cos x3 sin x5 y¨

I yy −Izz , Ix x

1 , r1 Izz

=

(2.55)

Applying backstepping control to the roll angle subsystem, we get x˙1 = x2 , x˙2 = x4 x6 a1 − a2 x6  + U2 b1 . For e1 = x1 – x1d (error between actual and desired roll angle), a positive definite function is defined: VT M1 = 21 e12 , and time derivative is V˙T M1 = e1 e˙1 = e1 (x2 − x˙1d ) ≤ −k1 e12 , k1 > 0. To satisfy this condition, a virtual control x2d = x˙1d − c1 e1 is chosen such that e2 = x2 − x2d = x2 − x˙1d + k1 e1 . Time derivative of e2 is e˙2 = x˙2 − x¨1d + c1 e˙1 . Using these equations, we obtain V˙T M1 = e1 (x2 − x˙1d ) = e1 e2 − k1 e12 .

(2.56)

The next step is to enhance VT M1 with a quadratic term in e2 to obtain a positive definite VT M2 = VT M1 + 21 e22 . With a choice of control law U2 given as U2 =

1 (−x4 x6 a1 + a2 x6  + x¨1 − e1 − k1 e˙1 − e2 k2 ), b1

(2.57)

the time derivative of VT M2 is V˙T M2 ≤ −k1 e12 − k2 e22 which guarantees an asymptotic stable system for appropriately chosen k1 , k2 > 0. Similarly, the control law for altitude, pitch, and roll subsystems is defined as

2.4 Backstepping Control Design

31

U1 = m(−x2 x12 + x4 x10 + g sin x3 − r1 − x¨7d + e7 − k7 e˙7 + k8 e8 ), k7 , k8 > 0, 1 U3 = (−x2 x6 a3 − a4 x4  + x¨3d + k3 e˙3 − e3 − k4 e4 ), k3 , k4 > 0, b2 1 U4 = (−x2 x4 a5 + x¨5d − k6 e6 − e5 + k5 e˙5 ), k5 , k6 > 0. b3 The lift and drag forces are considered in the transition mode. If λ1 > θ , the control laws U1 , . . . , U4 remain same as the quadrotor mode. We have control over x- and y-axis, and so Ux and U y control laws are defined as (2.38). If λ2 < θ , the controller is switched to the level-flight controller, as described next.

2.4.3 Level-Flight Mode Level flight is a fixed-wing mode visualized using the right-hand fixed-wing axis system. Variables are defined in dynamic equations (2.7)–(2.12) of quadrotor tailsitter UAV and can be transformed as shown in Fig. 2.1 and defined as ⎡

⎡ ⎡ ⎤ ⎤ ⎤ ⎡ ⎤ Vx −Vz 0 0 −1 Vx ⎣Vy ⎦ = ⎣ Vy ⎦ = R QW ⎣Vy ⎦ , R QW = ⎣0 1 0 ⎦ . 10 0 Vz W Vx Q Vz Q

(2.58)

To design a controller for level-flight mode, the complete level-flight mathematical model (2.7)–(2.12) is given (2.58). The control law for the yaw subsystem can be defined as shown in transition mode. For the yaw angle subsystem, a positive definite function is defined: VLC1 = 21 e12 and the time derivative of VLC1 is V˙ LC1 = e1 e˙1 = e1 (x2 − x˙1d ) ≤ −s1 e12 , s1 > 0.

(2.59)

A virtual control x2d = x˙1d − c1 e1 is chosen such that e2 = x2 − x˙1d + s1 e1 and time derivative of e2 is e˙2 = x˙2 − x¨1d + c1 e˙1 , and V˙ LC1 = e1 e2 − s1 e12 . Now, VLC1 with a quadratic term in e2 to obtain a positive definite VLC2 as explained in transition mode, and with a choice of control law U4 given as U4 =

1 (x4 x6 a1 − a2 x6  − x¨1d + e1 + s1 e˙1 + e2 s2 ) . b3

(2.60)

Therefore, V˙ LC2 ≤ −s1 e12 − s2 e22 guarantees an asymptotic stable system for an appropriately chosen s1 , s2 > 0. Inputs for controlling the pitch movement U3 , roll movement U2 , and altitude U1 as well as x- and y-positions (Ux and U y ) are given as

32

2 Nonlinear Disturbance Observer-Based Backstepping …

U1 = m (−x6 x12 + x4 x8 + g sin x3 + r2 + x¨7d + s7 e˙7 − e7 − s8 e8 ) 1 U2 = (−x4 x2 a5 + x¨5 + s5 e˙5 − e5 − s6 e6 ) b1 1 U3 = (−x2 x6 a3 − a4 x4  + x¨3d + s3 e˙3 − e3 − e4 s4 ) b2 Ux = (−e9 + x¨9 + s9 e˙9 − s10 e10 − x4 x10 + x2 x12 + r1 ) U y = (−e11 + x¨11 + s11 e˙11 − s12 e12 − x2 x8 + x6 x10 ) . Using these equations, the desired roll φd , pitch θd , and yaw ψd angles are 

Uy φd = arctan Ux





Uy , θd = arctan g cos φd





y , ψd = arctan x

 .

(2.61)

We have shown the development of suitable control signals for all three modes. Theorem 2.3 Let the position error subsystem (2.30) in the closed loop with the disturbance observer (2.13) and (2.14) designed be controlled according to (2.38) and (2.39) and attitude subsystem in the closed loop with the disturbance observer (2.13) and (2.14). Under these conditions, designed according to (2.53), there exists an ensemble of gain matrices c1 , c2 , c3 , c4 , L p , and L o such that the overall closedloop control error vector [e1 e2 e3 e4 ] is bounded as follows: ||e||2 ≤ ∀t ≥ T ∗

(2.62)

with any pre-selected precision > 0 where T ∗ is sufficiently large. Proof Choosing a Lyapunov function candidate V = VQ P + VQ O +

8 

VT Mi +

i=1

8 

VLCi ,

(2.63)

i=1

and differentiating (2.63), and using (2.54) and (2.42), we obtain V˙ = V˙ Q P + V˙ Q O +

8  i=1

V˙T Mi +

8 

V˙ LCi

i=1

1 2 1 D p − δ2 VQ O2 + Do2 − k f e2f − kn en2 − si ei2 − s j e2j 2 2 < −δV + γ − k f e2f − kn en2 − si ei2 − s j e2j , (2.64) ≤ −δ1 VQ P2 +

where δ = min{δ1 , δ1 } and γ = 21 D 2p + 21 Do2 , D P and Do are bounded as per Assumption 1 and k f > 0, f = 1, 3, 5, 7, kn > 0, n = 2, 4, 6, 8, and si > 0, i = 1, 3, 5, 7, 9, 11 and s j > 0, j = 2, 4, 6, 8, 10, 12. Since δ1 and δ2 are both tunable,

2.5 Simulation Results

33

it follows from Lemma 2.1 that for any desired > 0, there exists an ensemble of gain matrices c1 , c2 , c3 , c4 , L p , and L o such that the magnitude of both the position and attitude errors do not exceed on sufficiently long control horizons.

2.5 Simulation Results Next, we evaluate the backstepping controller’s performance for trajectory tracking. Simulation parameters are g = 9.8 ms −2 , m = 12 kg, l = 1 m, Ix x = I yy = 7.5 × 10−3 kg · m 2 , Izz = 1.3 × 10−3 kg · m 2 , Ir = 7.5 × 10−5 kg · m 2 , drag coefficient(d) = 7.5 × 10−7 , and lift coefficient(b) = 7.5 × 10−3 . Initial conditions are [x y z φ θ ψ] = [0.1 0.1 0 0 0 0.01]. With the quadrotor tail-sitter in hovering mode, two external disturbances are applied (i) [dx d y dz ] = [1 + sin 2t 1 + sin 2t 1 + sin 2t] and [dφ dθ dψ ] = [sin 2t sin 2t sin 2t] (periodic disturbances) and (ii) von Karman wind gust model. The observer gains are designed as L p = [10 10 10]T and L o = [30 30 30]T .

2.5.1 Trajectory Tracking As shown in Fig. 2.4, simulation for trajectory tracking is carried out for 441 s. Figure 2.5 shows desired position tracking in the quadrotor mode (0–20 s). Transition mode (20–23 s) has no control over the x- and y-position, but altitude is controlled as shown in Fig. 2.6. Yaw angle is measured, and roll and pitch angles are calculated. In quadrotor mode, the desired yaw angle is effectively tracked. The UAV is commanded to rotate 80◦ about pitch angle and transitions sans change in roll and yaw angles. Figures 2.7 and 2.8 show the proposed backstepping controller’s overall position and attitude control. After the transition, the vehicle converts to a conventional fixedwing UAV. For 23 < t < 400 s, the UAV travels at 10 m/s in the level-flight mode without altitude and velocity drop.

Fig. 2.4 Timeline for trajectory tracking of Tail-sitter quadrotor UAV

X Axis (m)

34

2 Nonlinear Disturbance Observer-Based Backstepping … 40

0 0

Y Axis (m)

Transition Mode

Quadrotor Mode

Desired X position Actual X position

20

5

10

15

20

10

15

20

10

15

20

1 Desired Y position Actual Y position

0.5 0

Z Axis (m)

0

5 Desired Z position Actual Z position

40 20 0 0

5

Time (seconds)

Roll (rad)

Fig. 2.5 Altitude and position tracking during quadrotor mode and transition mode 0.1

Transition Mode

Quadrotor Mode

0 Desired Roll angle Actual Roll angle

-0.1 -0.2

Pitch (rad)

0

5

10

15

20

10

15

20

10

15

20

0 Desired Pitch angle Actual Pitch angle

-1 -2

Yaw (rad)

0

5

0.01 Desired Yaw angle Actual Yaw angle 0 0

5

Time (seconds)

X Axis (m)

Fig. 2.6 Attitude tracking during quadrotor mode and transition mode 4000 Desired X position Actual X position

2000 0 0

50

100

150

200

250

300

350

400

Y Axis (m)

30 Desired Y position Actual Y position

20 10 0

Z Axis (m)

0

50

100

150

200

250

300

350

100

400 Desired Z position Actual Z position

50 0 0

50

100

150

200

250

Time (seconds)

Fig. 2.7 Position tracking by the backstepping controller

300

350

400

Roll (rad)

2.5 Simulation Results

35 Desired Roll angle Actual Roll angle

2 0 -2

Pitch (rad)

0

50

100

150

200

250

300

2

350

400

Desired Pitch angle Actual Pitch angle

0 -2

Yaw (rad)

0

50

100

150

200

250

300

350

400

Desired Yaw angle Actual Yaw angle

2 0 -2 0

50

100

150

200

250

300

350

400

Time (seconds)

Fig. 2.8 Attitude tracking by the backstepping controller

Fig. 2.9 Trajectory tracking

During 400 < t < 401s, the UAV rotates about its pitch angle from −80 to 0◦ and transits to quadrotor mode with no position change in x- and y-directions. During 401 < t < 421 s, the UAV holds altitude at 40m and lands during 421 < t < 441 s. Figure 2.9 shows the three-dimensional trajectory tracking, and Fig. 2.10 shows the velocity profile during the same. The proposed controller facilitates effective tracking of the desired trajectory.

2.5.2 Quadrotor Mode with External Disturbance Simulations of the take-off, hovering, and landing phases with external disturbances, during 0 < t < 20 (take-off phase) and 401 < t < 441, are performed. From t = 0−20 s, the UAV is commanded to take off with 2 m/s, t = 20 to t = 40 s in hovering mode and t = 40−60 s for landing. Periodic disturbances [dx d y dz ] = [1 + sin 2t 1 + sin 2t 1 + sin 2t] and [dφ dθ dψ ] = [sin 2t sin 2t sin 2t] are applied.

Z Axis (m/s)

Y Axis (m/s)

X Axis (m/s)

36

2 Nonlinear Disturbance Observer-Based Backstepping … 5 0 -5 0 20

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

10 0 -10 0 4 2 0 -2 -4 0

Time (seconds)

Y Axis (m)

X Axis (m)

Fig. 2.10 Velocity profile 1.5 1 Desired X position Actual X position

0.5 0 0 1.5

20

30

40

50

60

1 Desired Y position Actual Y position

0.5 0 0

Z Axis (m)

10

10

20

30

40

50

60

50 Desired Z position Actual Z position 0 0

10

20

30

40

50

60

Time (seconds)

Fig. 2.11 Altitude and position with disturbance [1 + sin(2t) 1 + sin(2t) 1 + sin(2t)]

Figure 2.11 shows robust position tracking when applying a periodic in-flight disturbance. Figure 2.12 shows attitude tracking with no substantial change in the UAV’s attitude subsystem, validating the proposed backstepping controller performance. Figures 2.13 and 2.14 show the disturbance observer performance. Figure 2.15 shows the trajectory-tracking error with and without the observer. There are minor (0.05 m) steady-state errors in the z-axis during take-off and landing. Tracking error with an observer in the x−y-axis is lesser, thereby validating the observer’s performance under periodic disturbance. Figure 2.16 reveals a minimal tracking error of the attitude subsystem. Next, we apply the von Karman wind gust (turbulence) model. Figure 2.17 shows the position tracking with wind gusts. Similarly, Fig. 2.18 shows small fluctuations in the attitude subsystem. Figures 2.19 and 2.20 show the observer’s performance in disturbance estimation. Figures 2.21 and 2.22 show minor tracking errors in position and attitude subsystems with and without an observer where a small (0.05 m) steady-state error exists in the z-axis with wind gusts acting on the specific subsystems.

Roll (rad)

2 1 0 -1

Pitch (rad)

2.5 Simulation Results

2 1 0 -1

Desired Roll angle Actual Roll angle

0

10

20

30

40

50

60

Desired Pitch angle Actual Pitch angle

0

Yaw (rad)

37

10

20

30

40

50

0.2

60

Desired Yaw angle Actual Yaw angle

0.1 0 0

10

20

30

40

50

60

Time (seconds)

Z Axis (m)

Y Axis (m)

X Axis (m)

Fig. 2.12 Attitude tracking when disturbance [sin(2t) sin(2t) sin(2t)] applied 4 2 0 0

10

20

30

40

50

60

0

10

20

30

40

50

60

10

20

30

40

50

4 2 0

4 2 0 0

60

Time (seconds)

Fig. 2.13 Observer performance for position subsystem with periodic disturbance

Yaw (rad)

Pitch (rad)

Roll (rad)

4 2 0 0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30

40

50

60

4 2 0

4 2 0

Time (seconds)

Fig. 2.14 Observer performance for attitude subsystem with periodic disturbance

38

2 Nonlinear Disturbance Observer-Based Backstepping …

X Axis (m)

1.5 1 0.5 0 -0.5

Tracking error with observer Tracking error without observer

0

10

20

30

40

Y Axis (m)

50

60

Tracking error with observer Tracking error without observer

1 0 -1 0

10

20

30

40

50

60

Z Axis (m)

0.1 Tracking error with observer Tracking error without observer

0 -0.1 0

10

20

30

40

50

60

Time (seconds)

Fig. 2.15 Position tracking error when periodic disturbance applied Tracking error with observer Tracking error without observer

Roll (rad)

2 0 -2 0

10

20

30

40

Pitch (rad)

50

60

Tracking error with observer Tracking error without observer

2 0 -2

Yaw (rad)

0 0.2

10

20

30

40

50

60

Tracking error with observer Tracking error without observer

0.1 0 -0.1 0

10

20

30

40

50

60

Time (seconds)

X Axis (m)

Fig. 2.16 Attitude tracking error when periodic disturbance applied 1 Desired X position Actual X position

0.5 0

Y Axis (m)

0

10

20

30

40

50

60

40

50

60

40

50

60

1 Desired Y position Actual Y position

0.5 0

Z Axis (m)

0

10

20

30

40 Desired Z position Actual Z position

20 0 0

10

20

30

Time (seconds)

Fig. 2.17 Position tracking with von Karman wind turbulence model

2.5 Simulation Results

39

Roll (rad)

4 Desired Roll angle Actual Roll angle

2 0 -2 0

10

20

30

40

50

60

Pitch (rad)

4 Desired Pitch angle Actual Pitch angle

2 0 -2

Yaw (rad)

0

10

20

30

40

50

60

0.1 Desired Yaw angle Actual Yaw angle

0.05 0 0

10

20

30

40

50

60

Time (seconds)

X Axis (m)

2 0 -2 -4

Y Axis (m)

0 6 4 2 0 -2 0

Z Axis (m)

Fig. 2.18 Attitude tracking with von Karman wind turbulence model

2 0 -2 -4 0

10

20

30

40

50

60

10

20

30

40

50

60

10

20

30

40

50

60

Time (seconds)

Yaw (rad)

Fig. 2.19 Disturbance observer performance for position subsystem with von Karman wind turbulence model 0.5 0

Pitch (rad)

-0.5 0

10

20

30

40

50

60

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0.5 0

Yaw (rad)

-0.5 0.5 0 -0.5

Time (seconds)

Fig. 2.20 Disturbance observer performance for attitude subsystem with von Karman wind turbulence model

X Axis (m)

40

2 Nonlinear Disturbance Observer-Based Backstepping … 1 0

Tracking error with observer Tracking error without observer

-1

Y Axis (m)

0

10

20

30

40

0

60

Tracking error with observer Tracking error without observer

-1 0

Z Axis (m)

50

1

10

20

30

40

50

60

0.1 Tracking error with observer Tracking error without observer

0 -0.1 0

10

20

30

40

50

60

Time (seconds)

Roll (rad)

Fig. 2.21 Position tracking error with von Karman wind turbulence model 2 0 Tracking error without observer Tracking error with observer

-2

Yaw (rad)

Pitch (rad)

0

10

20

30

40

50

60

2 0 Tracking error without observer Tracking error with observer

-2 0 0.1

10

20

30

40

50

60

Tracking error without observer Tracking error with observer

0.05 0 -0.05 0

10

20

30

40

50

60

Time (seconds)

Fig. 2.22 Attitude tracking error with von Karman wind turbulence model

Figure 2.23 finds that NDO-BC performs better than BC with periodic disturbance and crosswind while tracking [1 + sin(0.5t) 1 + cos(0.5t) 2t] in quadrotor mode.

2.6 Conclusions A backstepping controller design for quadrotor tail-sitters for autonomous flight in different phases comes with a nonlinear disturbance observer to estimate external disturbances: (i) the von Karman wind gust model and (ii) an appropriate dynamical model considering crosswind. Simulation results reveal a negligible altitude drop during the transition maneuver, and successful tracking of the commanded trajectory happens. (iii) The UAV maintains its position with different external disturbances, demonstrating the proposed controller’s effectiveness.

References

41 Desired path BC NDO-BC

Desired path BC NDO-BC

200

Z Axis (m)

Z Axis (m)

200

100

100

0

0 1

2 1

0 0

Y Axis (m)

-1

(a)

-1

X Axis (m)

1 0

Y Axis (m)

-1

-1

0

1

2

X Axis (m)

(b)

Fig. 2.23 Nonlinear disturbance observer-based backstepping, and nominal backstepping controllers with a periodic disturbance b wind gusts in Quadrotor Mode

References 1. Kita, K., Konno, A., Uchiyama, M.: Transition between level flight and hovering of a tail-sitter vertical takeoff and landing aerial robot. Adv. Robot. 24(5–6), 763–781 (2010) 2. Zhang, F., Lyu, X., Wang, Y., Gu, H., Li, Z.: Modeling and flight control simulation of a quadrotor tailsitter VTOL UAV. In: AIAA Modeling and Simulation Technologies Conference, p. 1561 (2017) 3. Bapst, R., Ritz, R., Meier, L., Pollefeys, M.: Design and implementation of an unmanned tailsitter. In: 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1885–1890 (2015) 4. Forshaw, J.L., Lappas, V.J.: Architecture and systems design of a reusable Martian twin rotor tailsitter. Acta Astronaut. 80, 166–180 (2012) 5. Wang, Y., Lyu, X., Gu, H., Shen, S., Li, Z., Zhang, F.: Design, implementation and verification of a quadrotor tail-sitter VTOL UAV. In: 2017 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 462–471. IEEE (2017) 6. Oosedo, A., Abiko, S., Konno, A., Uchiyama, M.: Optimal transition from hovering to levelflight of a quadrotor tail-sitter UAV. Auton. Robot. 41(5), 1143–1159 (2017) 7. Li, B., Zhou, W., Sun, J., Wen, C., Chen, C.: Model predictive control for path tracking of a VTOL tailsitter UAV in an HIL simulation environment. In: 2018 AIAA Modeling and Simulation Technologies Conference, p. 1919 (2018) 8. Li, Z., Zhang, L., Liu, H., Zuo, Z., Liu, C.: Nonlinear robust control of tail-sitter aircrafts in flight mode transitions. Aerosp. Sci. Technol. 81, 348–361 (2018) 9. Zhou, H., Xiong, H.L., Liu, Y., Tan, N.D., Chen, L.: Trajectory planning algorithm of UAV based on system positioning accuracy constraints. Electronics 9(2), 250 (2020) 10. Flores, G.R., Escareño, J., Lozano, R., Salazar, S.: Quad-tilting rotor convertible MAV: modeling and real-time hover flight control. J. Intell. Robot. Syst. 65(1), 457–471 (2012) 11. Zhou, J., Lyu, X., Li, Z., Shen, S., Zhang, F.: A unified control method for quadrotor tail-sitter UAVs in all flight modes: hover, transition, and level flight. In: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 4835–4841 (2017) 12. Swarnkar, S., Parwana, H., Kothari, M., Abhishek, A.: Biplane-quadrotor tail-sitter UAV: flight dynamics and control. J. Guid. Control Dyn. 41(5), 1049–1067 (2018) 13. Sartori, D., Quagliotti, F., Rutherford, M.J., Valavanis, K.P.: Implementation and testing of a backstepping controller autopilot for fixed-wing UAVs. J. Intell. Robot. Syst. 76(3), 505–525 (2014)

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14. Dalwadi, N., Deb, D., Muyeen, S.M.: Adaptive backstepping controller design of quadrotor biplane for payload delivery. IET Intel. Transp. Syst. (2022). https://doi.org/10.1049/itr2.12171 15. Dalwadi, N., Deb, D., Rath, J.J.: Biplane trajectory tracking using hybrid controller based on backstepping and integral terminal sliding mode control. Drones 6(3), 58 (2022). https://doi. org/10.3390/drones6030058 16. Dalwadi, N., Deb, D., Muyeen, S.: Observer based rotor failure compensation for biplane quadrotor with slung load. Ain Shams Eng. J. 13(6), 101,748 (2022). https://doi.org/10.1016/ j.asej.2022.101748 17. Bouabdallah, S., Siegwart, R.: Backstepping and sliding-mode techniques applied to an indoor micro quadrotor. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pp. 2247–2252. IEEE (2005) 18. Chen, W.H., Ballance, D.J., Gawthrop, P.J., O’Reilly, J.: A nonlinear disturbance observer for robotic manipulators. IEEE Trans. Ind. Electron. 47(4), 932–938 (2000) 19. Liu, C., Chen, W.H., Andrews, J.: Tracking control of small-scale helicopters using explicit nonlinear MPC augmented with disturbance observers. Control Eng. Pract. 20(3), 258–268 (2012) 20. Yang, J., Li, S., Yu, X.: Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans. Ind. Electron. 60(1), 160–169 (2012) 21. Viswanath, D., Deb, D.: Disturbance observer based sliding mode control for proportional navigation guidance. IFAC Proc. Vol. 45(1), 163–168 (2012) 22. Fethalla, N., Saad, M., Michalska, H., Ghommam, J.: Robust observer-based dynamic sliding mode controller for a quadrotor UAV. IEEE Access 6, 45846–45859 (2018) 23. Patel, R., Deb, D., Modi, H., Shah, S.: Adaptive backstepping control scheme with integral action for quanser 2-dof helicopter. In: 2017 International Conference on Advances in Computing, Communications and Informatics (ICACCI), pp. 571–577. IEEE (2017) 24. Patel, R., Deb, D.: Parametrized control-oriented mathematical model and adaptive backstepping control of a single chamber single population microbial fuel cell. J. Power Sources 396, 599–605 (2018) 25. Patel, R., Deb, D.: Adaptive backstepping control of single chamber microbial fuel cell. In: 2017 17th International Conference on Control, Automation and Systems (ICCAS), pp. 574–579. IEEE (2017)

Chapter 3

Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

This chapter is based on the trajectory-tracking controller design for the biplane quadrotor by using three different nonlinear control methods: (i) Backstepping Control (BSC), (ii) Integral Terminal Sliding Mode Control (SMC), and (iii) Hybrid control (ITSMC + BSC) where ITSMC and BSC control the position and attitude subsystems, and the criteria for evaluation is the performance in tracking objectives. Simulation results show that the BSC provides a steady-state error during altitude tracking when in-flight mass change occurs. On the other hand, ITSMC and HC controllers can track the desired altitude without steady-state error. Moreover, the HC controller is faster and generates less ITAE (Integral Time Absolute Error).

3.1 Hybrid Controller Design The deployment of hybrid UAVs for payload, surveillance, defense, and agriculture sectors is effective due to their long flight duration and ability to hover [1, 2]. One of the hybrid UAVs is a Biplane quadrotor with no deflecting surface but the ability to fly as a fixed wing and hover as a rotary wing. Different control strategies based on flight regimes [3, 4], mission planning [5], and trajectory planning [6] are developed. For improved performance under external disturbances like wind gusts, multiple control schemes are proposed [7, 8]. Biplane quadrotor switches from quadrotor to fixedwing mode and vice versa after transitioning, and for the smooth transition, advanced algorithms are developed [6, 8]. The propeller of a biplane quadrotor has either a fixed pitch or a variable-pitch configuration [9]. To control more effectively and efficiently, many researchers came up with different control strategies like PID (Proportional-Integral-Derivative) [10], LQR (LinearQuadratic Regulator) [11], adaptive and nonlinear controllers [12, 13], SMC (Slid© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_3

43

44

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

Fig. 3.1 Animated picture of biplane quadrotor

ing Mode Control) [14, 15], BSC [16], and H∞ control [17]. Trajectory tracking in diverse conditions is witnessed [10, 11] by using basic control schemes. A nonlinear PID [18] and a novel PID for motion control are designed in quadrotor UAV [19]. For airspeed and aerodynamics handling during the transition for the tail-sitter UAVs, a variable and dual fuzzy PID control strategy [20] and a gain scheduling-based PID controller are contributed [21]. For optimal results, many optimization algorithms are used in controller tuning, like intelligent particle swarm optimization [22] for enhanced performance through novel architectures to avoid coordinate switching of tail-sitter UAVs. External disturbances like wind gusts negatively impact the UAV tracking performance, estimated through a nonlinear disturbance observer [16, 17]. Similarly, PD and adaptive IBSC (Integral Backstepping Control) method [23] helps in position control despite wind wake disturbance. For the biplane quadrotor, BSC and ABSC (Adaptive Backstepping Control) are developed for trajectory tracking and payload delivery [24]. A nonlinear disturbance observer-based compensator under a partially failed rotor condition in a biplane quadrotor carrying a slung load is presented [25]. An animated picture of such a vehicle is shown in Fig. 3.1, which takes off, lands, and hovers like a rotary wing but, after transitioning, acts like a fixed-wing UAV. The capability to handle disturbances along with ITSMC’s faster convergence [26] is accomplished through an improved control signal for the quadrotor that is robust to uncertainties [27]. A robust ITSMC controller for attitude control and an ABSC for position control are developed. Integrating SMC techniques with PID under sensor

3.2 Mathematical Model of Biplane Quadrotor

45

faults [28] and augmented design of IBSC, and an ATSMC [29] are proposed. An adaptive backstepping sliding mode control with the swapping gain from a BSC is effective [30] to address disturbance rejection and actuator faults. We design a HC which is more robust, provides faster responses, and has a simple structure to track the desired trajectory with all possible maneuvers. Performance assessment under diverse modes and mass change is undertaken.

3.2 Mathematical Model of Biplane Quadrotor Biplane quadrotor 6-degrees-of-freedom (DOF) dynamics [3] are ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ x¨ −sθ r v − qw Fax 1 ⎣ y¨ ⎦ = ⎣ Fay ⎦ + g ⎣cθ sφ ⎦ + ⎣ pw − r u ⎦ , m −T + F z¨ cθ cφ qu − pv az ⎡ ⎤ ⎡ ⎤ ¨ φ (b1 r + b2 p)q + b3 (L a + L t ) + b4 (Na + Nt ) ⎣ θ¨ ⎦ = ⎣ ⎦, b5 pr − b6 ( p 2 − r 2 ) + b7 (Ma + Mt ) (b8 p − b2 r )q + b4 (L a + L t ) + b9 (Na + Nt ) ψ¨

(3.1)

(3.2)

¨ are linear and angular where c(·) = cos(·), s(·) = sin(·), and [x¨ y¨ z¨ ] and [φ¨ θ¨ ψ] acceleration, and [ p q r ], [u v w] are the velocities, m is the mass, T is the thrust, [L t Mt Nt ] are the moments about roll, pitch, and yaw angles, and [Fax Fay Faz ] and [L a Ma Na ] are the aerodynamic forces and moments, respectively. Inertial terms are defined as constant bi : ⎡ ⎤ b1 ⎢b2 ⎥ ⎢ ⎥ ⎢b3 ⎥ 1 ⎢ ⎥= ⎢b4 ⎥ 2 I I x z − Ix z ⎢ ⎥ ⎣b8 ⎦ b9 For velocity V = ⎡

√

⎡

⎤ (I y − Iz )Iz − Ix2z ⎢ (Ix − I y + Iz )Ix z ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ b5 (Iz − Ix ) ⎢ ⎥ 1 I z ⎢ ⎥ , ⎣b6 ⎦ = ⎣ Ix z ⎦ . ⎢ ⎥ Ix z I y ⎢ ⎥ b7 1 ⎣(Ix − I y )Ix + Ix2z ⎦ Ix

(3.3)

u 2 + v 2 + w2 , the aerodynamic forces and moments are

⎤ ⎡ ⎤⎡ ⎤ Fax sα cα −sα sβ cα −D ⎣ Fay ⎦ = ⎣ sβ cβ 0 ⎦⎣ Y ⎦ −cα cβ cα sβ sα −L Faz

⎡

⎤ ⎡ ⎤ La 0.5ρV 2 SbCl ⎣ Ma ⎦ = ⎣0.5ρ cV ¯ 2 SCm ⎦ , (3.4) Na 0.5ρV 2 SbCn

where Cl , Cm , and Cn are the roll, pitch, and yaw coefficients [3].

46

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

3.3 Hybrid Controller Design Next, a Hybrid controller is designed where the ITSMC controls the position subsystem. Finally, BSC calls the attitude subsystem for the desired trajectory tracking that includes all the modes of interest.

3.3.1 Quadrotor Mode The biplane quadrotor performs take-off, hover, and landing tasks when in the quadrotor mode, and in this period, the angular velocity is minimal. No significant aerodynamic forces and moments act on the vehicle, and the thrust from the four motors balances the weight. The vehicle dynamics in the quadrotor mode, from (3.1) and (3.2), is  T  T  T m x¨ y¨ z¨ = 0 0 mg + RT 0 0 − T ⎡ ⎤ ⎡ ⎤ φ¨ (b1r + b2 p)q + b3 (L a + L t ) + b4 (Na + Nt ) ⎣ θ¨ ⎦ = ⎣ ⎦, b5 pr − b6 ( p 2 − r 2 ) + b7 (Ma + Mt ) (b8 p − b2 r )q + b4 (L a + L t ) + b9 (Na + Nt ) ψ¨

(3.5) (3.6)

where RT is the rotation matrix. The control laws are designed based on the above equations. The block diagram of the controller design is shown in Fig. 3.2. For the BSC-based controller, both controller 1 (position subsystem) and controller 2 (altitude subsystem) blocks are BSC, and same as for the ITSMC controller, both controllers are replaced by the ITSMCs. In the Hybrid case, Controller 1 is substituted by the ITSMC and Controller 2 by the BSC. Additional specifics are discussed later. The position controller determines desired thrust (T ) by using the desired trajectory, while desired roll and pitch angles are determined as per the desired x−yposition, and the desired yaw angle is given directly. The attitude controller ensures attitude tracking and generates roll, pitch, and yaw moments ([L t Mt Nt ]). The preferred signals for the motors [U1 U2 U3 U4 ] are generated based on the thrust and moments. Next, the design of a Hybrid controller with stability analysis is discussed.

3.3.1.1

ITSMC Design

The first step of the ITSMC controller design for the biplane quadrotor is to describe the sliding surface as

S = e˙ +

j

γ e˙ j/i + ζ e 2i−1 ,

(3.7)

3.3 Hybrid Controller Design

47

Fig. 3.2 Block Diagram of controller design

and the reaching law is designated as S˙ = −λS − k sign(S),

(3.8)

where 0 < j/i < 1, γ , ζ, k, λ > 0. Now the error altitude is ezq = z − z d , where z and z d are the actual altitude and desired altitude while suffix q refers to the quadrotor and transition mode. A positive definite function is given as Vzq = 21 Sz2q . By taking the derivative and using (3.5), we get  jz q jzq /i zq 2i z q −1 ˙ + ζz q e Vzq = Szq z¨ − z¨ d + γzq e˙  jz q T jzq /i zq i z q −1 . = Szq g − cφcθ − z¨q d + γzq e˙ + ζz q e m

(3.9)

Using (3.9), the control law is defined as m T = cφ cθ

g − z¨ d +

jz /i z γzq e˙zqq q

jz q 2i z q + jz q

+ ζz q



+ λzq Szq + k zq sign Szq

 

, (3.10)

such that V˙zq = −λzq Sz2q − k zq |Szq | ≤ 0, where λzq , k zq > 0. Control signals for the x−y-position are defined as

48

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor m Ux = T Uy =

m T

jx /i x −x¨d + γxq e˙xqq q

j y /i yq

− y¨d + γ yq e˙ yqq

jx q 2i xq − jxq

+ ζxq

j yq 2i yq − j yq

+ ζ yq



+ λxq Sxq + k xq sign Sxq



 , λxq , k xq > 0,

   + λ yq S yq + k yq sign S yq , λ yq , k yq > 0.(3.11)

Using (3.11), the desired roll and pitch angles are      φd = sin−1 sψd U y + cψd Ux , θd = sin−1 cψd Ux + sψd U y cφd ,

(3.12)

where − π2 < φd , θd < π2 , and −π < ψd < π are realistic bounds ensuring nonsingular situations in (3.10). The position subsystem is controlled by (3.10–3.11), and using these, the preferred roll and pitch angles are given in (3.12).

3.3.1.2

BSC Design

The BSC controls the biplane’s attitude. The roll angle error is expressed as eφq = φ − φd and consequently a positive definite function Vφq = 21 eφ2 q and time derivative V˙φq = eφq e˙φq = eφq e pq − δφq eφ2 q , δφq > 0, where e pq = p − pd (velocity error), are determined. A virtual control pd = φ˙q d − δφq eφq satisfies e pq = p − pd = p − φ˙q d + δφq eφq .

(3.13)

Enhancing Vφq with quadratic term in e pq , to get V pq = 21 e2pq + Vφq , and after taking first time derivative, we get   V˙ pq = e pq (b1r + b2 p) q + b3 L t + b4 Nt − φ¨ d + δφq e˙φq − δφq eφ2 q + eφq e pq . (3.14) Using (3.14), the control law for the roll subsystem is defined as Lt =

 1  −eφq − δ pq e pq + φ¨d − δφq e˙φq − (b1r − b2 p)q − b4 Nt , b3

(3.15)

so that V˙ pq = −δφq eφ2 q − δ pq e2pq , for guaranteed asymptotic stability for appropriate δφq , δ pq > 0. We define control signals for pitch and yaw angles as 1 b7 1 Nt = b9

Mt =

    θ¨d − eθq − δqq eqq − δθq e˙θq + b6 p 2 − r 2 − b5 pr ,

(3.16)

  ψ¨ d − eψq − δrq erq − δψq e˙ψq − (b8 p − b2 r ) q − b4 L t .

(3.17)

3.3 Hybrid Controller Design

49

3.3.2 Transition Mode The transition maneuver helps the biplane quadrotor switch from quadrotor to fixedwing mode and vice versa. The biplane quadrotor rotates gradually ≈ 90 about the pitch axis to initiate this transition and the controller switches when adequate aerodynamic force is available. Only attitude and altitude are controlled during the transition, and there is no control over the x−y-position. The dynamics in the transition mode and the quadrotor mode are the same, and the control signals are T = Lt = Nt = Mt =



jz q   m Faz jzq /i zq 2i z q − jz q − γzq e˙zq − ζz q − λzq Szq − k zq sign Szq , g− cφ cθ m  1  −eφq − δ pq e pq + φ¨ d − δφq e˙φq − (b1r − b2 p)q − b4 (Nt + Na ) − b3 L a , b3  1  ψ¨ d − eψq − δrq erq − δψq e˙ψq − (b8 p − b2 r ) q − b4 (L t + L a ) − b9 Na , b9    1  θ¨d − eθq − δqq eqq − δθq e˙θq + b6 p 2 − r 2 − b5 pr − b7 Ma . (3.18) b7

3.3.3 Flight Mode In the fixed-wing mode, the quadrotor biplane functions as a fixed-wing aircraft and so the dynamics are x˙ = (cθ cψ)u + (sφsθ cψ − cφsψ)v + (cφsθ cψ + sφsψ)w, y˙ = (cθ sψ)u + (sφsθ sψ + cφcψ)v + (cφsθ sψ − sφcψ)w,

(3.19) (3.20)

z˙ = −usθ + vsφcθ + wcφcθ, u˙ = Fax /m − gsθ + pv − qu + T /m,

(3.21) (3.22)

v˙ = Fay /m + gcθ sφ + pw − r u, (3.23) w˙ = Faz /m + gcθ cφ + r v − qw, p˙ = − pq(bw3 + bw9 ) − qr (bw11 − bw12 ) − bw13 (L a + L t ) + bw5 (Na + Nt ), (3.24) q˙ = bw8r 2 + bw9 p 2 + 2 bw10 pr + bw7 (Mt + Ma ), r˙ = pq(bw1 + bw2 ) + qr (bw3 − bw4 ) + bw5 (L t + L a ) − bw6 (Nt + Na ),

(3.25)

where t (·) = tan(·), bw1 = Ix z (I y + Ix z )/B, bw2 = Iz2 /B, bw3 = Ix z (Ix + I y )/B, bw4 = Ix I y /B, bw5 = Ix z /B, bw6 = Iz /B, bw7 = 1/I y , bw8 = Ix /I y , bw9 = Iz /I y , bw10 = Ix z /I y , bw11 = Ix2 /B, bw12 = Ix z (I y − Ix z )/B, bw13 =Ix /B, and B = Ix Iz − Ix2z .

50

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

Fig. 3.3 Controller design for Fixed-wing mode

A controller design applicable in the fixed-wing mode is shown in Fig. 3.3, where the velocity controller generates the required thrust (T ) and the moments [L t Mt Nt ] are generated based on the desired [xd yd z d ] and actual position [x y z]. These generated thrusts and moments are given a variable-pitch propulsion system, and it generated the desired signals [U1 U2 U3 U4 ] to four actuators. The role of Controllers 1 and 2 are the same for the different control methods as explained earlier. For fixed-wing mode, the control law for the required thrust can be derived in the same calculation steps as given in the quadrotor mode and is expressed as

 jz f jz f /i z f Fax 2i z f − jz f T = m g sθ − pv + qu − −ζ − λz f Sz f − k z f sign(Sz f ) . − γz f e˙z f m

(3.26) Similarly, the other control signals during the fixed-wing mode are

3.4 Results and Discussions

51

1

(eφ f + δ p f e p f + δφ f e˙φ f − φ¨ d − pq(bw3 + bw9 ) + qr (bw12 − bw11 ) bw13 +bw5 (Na + Nt ) + L w13 L a ), 1 Mt = (−eθ f − δq f eq f − δθ f e˙θ f + θ¨d − bw8r 2 − bw9 p 2 − 2 bw10 pr − bw7 Ma ), bw7 1 Nt = (eψ f + δr f er f + δψ f e˙ψ f − ψ¨ d + pq(bw1 + bw2 ) + qr (bw3 − bw4 ) bw6 (3.27) +bw5 (L t + L a ) + bw6 Na ). Lt =

Desired roll (φ), pitch (θ ), and yaw (yaw) angle are calculated as

θd = sin

−1





z˙ d − cz (z − z d )

u 2 + (v sφ + w cφ)2  y˙d − k y (yd − y) , ψd = arctan x˙d + k x (xd − x)

+ tan−1



u v sφ + w cφ

, (3.28)

where z d is desired altitude, xd and yd are desired x- and y-positions, and k x , k y , kφ , and k z are gains. Considering roll angle variation to be linear to yaw angle, we have φd = kφ (ψ − ψd ) .

(3.29)

3.4 Results and Discussions A numerical simulation is performed with initial position [x y z] and attitude [φ θ ψ] as [0.5 5 0 0 0 0]. The simulation parameters are given in Table 3.1. The gains of the Hybrid controller, BSC, and ITSMC are shown in Tables 3.2, 3.3, and 3.4, respectively. The simulation timeline is shown in Fig. 3.4, where the total simulation time is 1000 s, and at t = 0−20 s, the biplane takes off with 5 m/s and x−y-position constant. After t = 20 s, the biplane quadrotor is directed to be in to hover state for the next 27 s, and at t = 47 s, transitions till t = 50 s to then remain in fixed-wing mode

Table 3.1 Biplane quadrotor UAV’s specification Parameters Value Parameters g Mass (m) Ix x I yy Izz

9.8 ms−2 12 kg 1.86 kg · m2 2.03 kg · m2 3.617 kg · m2

Wing area (single) Aspect ratio Wing span Gap-to-chord ratio Slung load mass (ml )

Value 0.754 m2 6.9 2.29 m 2.56 2 kg

52

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

Table 3.2 Hybrid controller gains Quadrotor, transition mode Parameter γxq γ yq γz q λxq λ yq λz q pxq p yq pz q

Values 5.70 5.70 8.70 5.3 5.3 5.3 9 9 9

Parameter ζxq ζ yq ζz q k xq k yq kzq qxq q yq qz q

Fixed-wing mode Values 2.317 2.317 3.57 3.66 3.66 2.66 5 5 7

Table 3.3 Backstepping controller gains Quadrotor, transition mode Parameter k xq k yq kzq kφq kθq kψq

Values 01.5 01.5 03 15 15 15

Parameter ku q kvq kwq k pq k qq kr q

Values 2.8 2.8 5 18 18 18

Parameter γx f λx f px f kφ f kθ f kψ f

Values 1.27 2.3 9 113 113 113

Parameter ζx f kx f qx f kpf kq f kr f

Values 3.57 1.66 7 13 13 13

Parameter ku f kpf kq f kr f

Values 5 13 13 13

Fixed-wing mode Parameter kx f kφ f kθ f kψ f

Values 15 113 113 113

for the next 900 s. From t = 50−100 s, it flies at 20 m/s in x-direction and 3 m/s in y-direction, with altitude hold at 100 m. After t = 100 s, for the next 800 s, it flies with curvature trajectory at 10 m/s in the x-axis while maintaining the altitude. Again, it increases velocity to 20 m/s in x- and 3 m/s in y-directions. At t = 950 s, it transitions from fixed-wing to quadrotor mode in just 1 s, and then again, hovers for 29 s before receiving the landing command at t = 980 s. In all the plots, the xaxis shows the simulation time, and states or parameters are given in the y-axis. X-position tracking is shown in Fig. 3.5. The Hybrid controller responds faster to track the desired x-position post-transition. The hybrid controller generates lesser errors and quicker response even when a sudden change in velocity and direction is applied at t = 100 s. Y-axis position tracking during the entire mission is shown in Fig. 3.6. Again, the performance of the hybrid controller is better when a sudden change in direction and velocity is applied at t = 100 s. At t = 950 s, after transitioning to the quadrotor flight, the BSC comes with a higher settling time than the ITSMC, and the ITSMC is marginally slower than the hybrid controller (HC). Figure 3.7 shows the altitude tracking performance. At t = 20 s, the biplane quadrotor holds altitude, and the response of the HC is faster than the other two. At t = 100 s, when a sudden change in direction and velocity is

3.4 Results and Discussions

53

Table 3.4 ITSMC controller Gains Quadrotor, transition mode Parameter γxq γ yq γz q γφq γθq γψq λxq λ yq λz q λφq λθq λψq pxq p yq pz q pφq pθq pψq

Values 2.70 2.70 2.70 1.78 1.78 1.78 3.3 3.3 3.3 3.3 3.3 3.3 9 9 9 7 7 7

Parameter ζxq ζ yq ζz q ζφq ζθq ζψq k xq k yq kzq kφq kθq kψq qxq q yq qz q qφq qθq qψq

Fixed-wing mode Values 0.317 0.317 1.57 1.57 1.57 1.27 2.66 2.66 1.66 5.66 8.66 5.66 5 5 7 5 5 5

Parameter γx f γφ f γθ f γψ f λx f λφ f λθ f λψ f pφ f pθ f pψ f

Values 1.27 2.78 2.78 2.78 2.3 2 2 2 7 7 7

Parameter ζx f ζφ f ζθ f ζψ f kx f kφ f kθ f kψ f qφ f qθ f qψ f

Values 3.57 2.57 2.75 2.57 1.66 2.66 2.66 2.66 5 5 5

Fig. 3.4 Timeline for the hybrid control simulation

applied, it is commanded to hold a constant altitude. The BSC comes with a 6 m error, and the ITSMC has less than a 2 m error in the altitude. Though the HC generates a somewhat larger error than 2 m, the settling time is less than the ITSMC. At t = 980 s, before landing, the HC is slightly faster than the BSC but similar to the ITSMC. Roll angle tracking during the entire flight envelope is shown in Fig. 3.8. During transition (t = 47−50 s), the controllers effectively track the desired roll angle, but the ITSMC response is slower than the other two in the fixed-wing phase. In the second transition at t = 900−901 s, ITSMC’s reaction is again slower. The tracking performance of the pitch angle is shown in Fig. 3.9. All three controllers track pitch angle effectively while decreasing linearly to transition during t = 47−50 s. At t =

54

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor 10000 1200

X Axis (m)

8000 1100

6000

Desired X position BSC ITSMC ITSMC + BSC

1000

4000

100

110

60 40 20 0

120

2000

48

0 0

100

200

300

50

400

500

52

600

700

800

900

1000

900

1000

Time (seconds)

Fig. 3.5 X-axis trajectory tracking during the whole mission

250 Desired Y position BSC ITSMC ITSMC + BSC

Y Axis (m)

200 150

6 160

100 5

140

50 120 100

0 0

100

120

200

300

4 950

140

400

500

600

700

955

800

Time (seconds)

Fig. 3.6 Y-axis trajectory tracking during the whole mission

100 106

Z Axis (m)

80

Desired Altitude BSC ITSMC ITSMC + ABC

104 102

60

100 98

40 101

105

110

95

20 100 0 0

99

100

90 20

100

21

980

981

982

22

200

300

400

500

600

Time (seconds)

Fig. 3.7 Z-axis trajectory tracking during the whole mission

700

800

900

1000

3.4 Results and Discussions

55

Roll angle (deg)

100 Desired Roll angle BSC ITSMC ITSMC + BSC

50

0 20

5

0

-50

0

-20

-100 0

50

100

200

55

300

-5 900 901 902 903

60

400

500

600

700

800

900

1000

Time (seconds)

Fig. 3.8 Roll angle tracking during the whole mission 100 40

Desired Pitch angle BSC ITSMC ITSMC + BSC

Pitch angle (deg)

20

50

0 -20

50

51

52

53

0

100 50

-50 0 951

-100 0

100

200

300

400

500

600

952

700

953

954

800

955

900

1000

Time (seconds)

Fig. 3.9 Pitch angle tracking during the whole mission

951 s, in the post-transition period, the ITSMC response is slower than the other two. Yaw tracking performance is shown in Fig. 3.10. The velocity profile is shown in Fig. 3.11. The ITSMC and HC response is the same because ITSMC is used for the position subsystem in the HC. Position and altitude tracking under mass change is shown in Figs. 3.12 and 3.13. There is no noticeable change in the x−y-position with mass change at t = 50 s. For altitude, BSC produces a 20 cm steady-state error after mass changes to 12 kg. In contrast, ITMSC and HC track altitude, but the HC generates a lesser error and is faster than the ITSMC. The HC outperforms the other methods as validated in Table 3.5 based on ITAE, which integrates the absolute error multiplied by the time over time while weighting the errors appearing after a long time more heavily than those at the start. Consequently, ITAE tuning helps systems settle quicker, with the downside being a sluggish initial response.

56

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

Yaw angle (deg)

100 Desired Yaw angle BSC ITSMC ITSMC + BSC

50

0

-50

30

0

20

-5

10

-10

0

-15 50

-100 0

100

200

52

54

300

56

58

400

900 902 904 906 908

500

600

700

800

900

1000

900

1000

Time (seconds)

Velocity (m/s)

Fig. 3.10 Yaw angle tracking during the whole mission

BSC ITSMC ITSMC + BSC

20

10

0 0

100

200

300

400

500

600

700

800

Time (seconds)

Fig. 3.11 Velocity profile during the whole mission

X Axis (m)

1.5 Desired X Axis

BSC

ITSMC

ITSMC + BSC

1

Y Axis (m)

0.5 0 0 8

10

20

30

40

50

60

Desired Y Axis

70 BSC

80 ITSMC

90

100

ITSMC + BSC

6 4 0

10

20

30

40

50

60

Time (seconds)

Fig. 3.12 x−y-position tracking during the mass change

70

80

90

100

3.4 Results and Discussions

57

Fig. 3.13 Altitude tracking during the mass change Table 3.5 Integral Time Absolute Error (ITAE) with (i) trajectory tracking and (ii) mass change Sr. No Time (S) BSC ITSMC ITSMC + BSC (i) (ii)

0–50 50–950 950–1000 0–100

390 1.57 × 104 132 1305

395 1.56 × 104 128 230

382 1.52 × 104 122 133

Fig. 3.14 Three-dimensional trajectory tracking during the whole mission

Figure 3.14 shows three-dimensional trajectory tracking by all three controllers. In the hybrid controller, if the position subsystem is controlled by the BSC and the attitude subsystem by the ITSMC, then after mass change, a steady-state error is generated in altitude, and BSC is less robust compared to the ITSMC.

58

3 Nonlinear Controllers for Hybrid UAV: Biplane Quadrotor

3.5 Conclusions A hybrid control design based on the ITSMC and BSC results provides some interesting insights. The ITSMC takes longer to track along the y-axis, and the BSC results in a maximum error among the three. The hybrid controller efficiently achieves altitude tracking. At t = 20 s, when directed to hover after take-off at 5 m/s, the ITSMC gives a sluggish response, taking an excess of 3 s to hold the desired altitude. The Hybrid controller takes only 2 s for the intended altitude tracking. In attitude tracking, ITSMC is the most sluggish among the three. The performance of these controllers is also assessed for sudden mass change scenarios wherein the BSC generates a 20 cm steady-state error, and the ITSMC has a slow response. The BSC and ITSMC ensure reference trajectory tracking, but a hybrid controller is an understandable choice for more accurate and fast control.

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12. Zhong, J., Song, B., Li, Y., Xuan, J.: L 1 adaptive control of a dual-rotor tail-sitter unmanned aerial vehicle with input constraints during hover flight. IEEE Access 7, 51312–51328 (2019). https://doi.org/10.1109/ACCESS.2019.2911897 13. Liu, H., Peng, F., Lewis, F.L., Wan, Y.: Robust tracking control for tail-sitters in flight mode transitions. IEEE Trans. Aerosp. Electron. Syst. 55(4), 2023–2035 (2019). https://doi.org/10. 1109/TAES.2018.2880888 14. Xi, L., Zhu, Q., Zhang, D.: Sliding mode control design based on fuzzy reaching law for yaw angle of a tail-sitter UAV, pp. 238–243 (2016). https://doi.org/10.1109/IConAC.2016.7604925 15. Gambhire, S.J., Kishore, D.R., Londhe, P.S., Pawar, S.N.: Review of sliding mode based control techniques for control system applications. Int. J. Dyn. Control 9(1), 363–378 (2020). https:// doi.org/10.1007/s40435-020-00638-7 16. Dalwadi, N., Deb, D., Kothari, M., Ozana, S.: Disturbance observer-based backstepping control of tail-sitter UAVs. Actuators 10(6), 119 (2021). https://doi.org/10.3390/act10060119 17. Lyu, X., Zhou, J., Gu, H., Li, Z., Shen, S., Zhang, F.: Disturbance observer based hovering control of quadrotor tail-sitter VTOL UAVs using H∞ synthesis. IEEE Robot. Autom. Lett. 3(4), 2910–2917 (2018). https://doi.org/10.1109/LRA.2018.2847405 18. Abdul Salam, A., Ibraheem, I.: Nonlinear PID controller design for a 6-DOF UAV quadrotor system. Eng. Sci. Technol., Int. J. 22 (2019). https://doi.org/10.1016/j.jestch.2019.02.005 19. Moreno-Valenzuela, J., Pérez-Alcocer, R., Guerrero-Medina, M., Dzul, A.: Nonlinear PID-type controller for quadrotor trajectory tracking. IEEE/ASME Trans. Mechatron. 23(5), 2436–2447 (2018). https://doi.org/10.1109/TMECH.2018.2855161 20. Zhang, D., Chen, Z., Xi, L.: Adaptive dual fuzzy PID control method for longitudinal attitude control of tail-sitter UAV. In: 2016 22nd International Conference on Automation and Computing (ICAC), pp. 378–382 (2016). https://doi.org/10.1109/IConAC.2016.7604949 21. Jung, Y., Shim, D.H.: Development and application of controller for transition flight of tailsitter UAV. J. Intell. Robot. Syst. 65(1–4), 137–152 (2011). https://doi.org/10.1007/s10846011-9585-1 22. Khodja, M.A., Tadjine, M., Boucherit, M.S., Benzaoui, M.: Tuning PID attitude stabilization of a quadrotor using particle swarm optimization (experimental). Int. J. Simul. Multidiscip. Des. Optim. 8, A8 (2017). https://doi.org/10.1051/smdo/2017001 23. Raza, S.A., Etele, J., Fusina, G.: Hybrid controller for improved position control of quadrotors in urban wind conditions. J. Aircr. 55(3), 1014–1023 (2018). https://doi.org/10.2514/1.c034573 24. Dalwadi, N., Deb, D., Muyeen, S.M.: Adaptive backstepping controller design of quadrotor biplane for payload delivery. IET Intel. Transp. Syst. (2022). https://doi.org/10.1049/itr2.12171 25. Dalwadi, N., Deb, D., Muyeen, S.: Observer based rotor failure compensation for biplane quadrotor with slung load. Ain Shams Eng. J. 13(6), 101,748 (2022). https://doi.org/10.1016/ j.asej.2022.101748 26. Su, Y., Zheng, C.: A new nonsingular integral terminal sliding mode control for robot manipulators. Int. J. Syst. Sci. 51(8), 1418–1428 (2020). https://doi.org/10.1080/00207721.2020. 1764658 27. Ullah, S., Khan, Q., Mehmood, A., Kirmani, S.A.M., Mechali, O.: Neuro-adaptive fast integral terminal sliding mode control design with variable gain robust exact differentiator for underactuated quadcopter UAV. ISA Trans. (2021). https://doi.org/10.1016/j.isatra.2021.02. 045 28. Mofid, O., Mobayen, S., Wong, W.K.: Adaptive terminal sliding mode control for attitude and position tracking control of quadrotor UAVs in the existence of external disturbance. IEEE Access 9, 3428–3440 (2021). https://doi.org/10.1109/ACCESS.2020.3047659 29. Modirrousta, A., Khodabandeh, M.: A novel nonlinear hybrid controller design for an uncertain quadrotor with disturbances. Aerosp. Sci. Technol. 45, 294–308 (2015). https://doi.org/10. 1016/j.ast.2015.05.022 30. Huang, S., Huang, J., Cai, Z., Cui, H.: Adaptive backstepping sliding mode control for quadrotor UAV. Sci. Program. 2021, 1–13 (2021). https://doi.org/10.1155/2021/3997648

Chapter 4

Adaptive Controller Design for Biplane Quadrotor

If augmented with fixed-wing design, air vehicles with the capability to land and take off vertically, such as quadcopters, become promising candidates for missions needing longer hover endurance, faster travel times, and higher payload capacities. Therefore, a backstepping controller (BSC) design of a biplane quadrotor during payload delivery, like vaccines in a remote area, and an adaptive backstepping controller (ABC) to handle mass changes despite wind gusts during the mission. Furthermore, we compare BSC, Integral Terminal Sliding Mode (ITSMC), and ABC for payload delivery to demonstrate effective tracking of the desired trajectory through ABC. The ITSMC also tracks the desired trajectory with mass change but provides a sluggish response, and BSC produces a steady-state altitude error whenever a significant mass change happens.

4.1 Biplane Quadrotor: Payload Delivery Hybrid development of drones like the biplane quadrotor facilitates improved efficiency and impact through innovative control configurations. A step-by-step process of development of a biplane quadrotor is offered [1]. While Sridharan et al. [2] developed a range and load approximation technique on a biplane tail-sitter quadrotor. Dawkins et al. developed a mathematical model and PID control of a micro quadrotor [3]. Task scheduling and a path planning problem are addressed by Mathew et al. for cooperating vehicles enabling autonomous shipment in metropolitan landmarks [4], while Liu et al. presented formation control of a swarm of tail-sitters [5]. A robust nonlinear control action is developed wherein the coordinate system and the controller structures don’t need to switch [6]. Wagter et al. developed a Linear-Quadratic Regulator (LQR) controller [7]. Mofid et al. addressed a sensor failure scenario where © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_4

61

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4 Adaptive Controller Design for Biplane Quadrotor

a PID-SMC controller is used for a known upper bounded disturbance, and an adaptive PID-SMC for an unknown disturbance [8]. Finally, Muthusamy et al. proposed a new fuzzy brain emotional learning control for a quadrotor UAV trajectory tracking for real-time payload uncertainties [9]. An observer assesses the unmeasured states or when sensors are not readily available. First, Chen et al. formulate a quadrotor UAV model which uses an SMC and BSC for desired trajectory-tracking control and adaptive observer-based failure estimator [10]. Next, Luo et al. analyze control challenges for UAVs with payload slung by a wire [11]. Finally, Dalwadi et al. [12] propose a nonlinear observer-based backstepping controller for tail-sitters under crosswind. A novel control strategy [13] associates backstepping and the dynamic inversion control methods to control fixed wings. A nonlinear observer-based backstepping controller is designed for trajectory tracking despite wind gusts [14]. Mofid et al. [15] proposed an AITSMC (Adaptive Integral Terminal Sliding Mode) scheme for finite-time position and attitude tracking of a quadrotor UAV in the presence of the model uncertainties and exterior turbulences and also an adaptive backstepping global sliding mode control method for position and attitude despite input saturation, model uncertainties, and wind perturbations [16]. An adaptive super-twisting terminal sliding mode control is developed for quadrotor UAVs with the upper bound of model uncertainty unknown, and wind disturbances present [17]. Based on the recursive least square algorithm, an accurate real-time approximation of the quadrotor’s varying mass and inertia tensor element with payload is developed [18]. Navabi et al. [19] propose an optimum ASMC (Adaptive Sliding Mode Controller) enhanced by particle swarm optimization (PSO) algorithm for trajectory tracking. Liu et al. proposed [20] an adaptive control approach for tail-sitter UAVs without having to change the coordinate system or the controller arrangement. Oshman et al. [21] proposed a strategy of using the outputs of gyro rates and available supplementary data of onboard sensors for independent attitude approximation of a UAV with a payload. A path-following controller with uncertainty and disturbance estimator is developed [22] for a quadrotor with a cable-suspended payload. In contrast, Yang et al. [23] design a nonlinear controller to track a quadrotor’s preferred suspended payload position. Lee et al. [24] propose a new trajectory generation strategy for cable-suspended quadrotor UAV to minimize the swing motion and an effective tracking based on the Linear-quadratic control technique. A novel online anti-swing trajectory generation and planning for a quadrotor with slung load system contain mainly two parts: position tracking and anti-swing [25]. Outeiro et al. presented a scheme for altitude and yaw angle control with an unknown load [26]. Due to the wings, the biplane quadrotor has better stability in long flight that guarantees safer transport of critical items. The biplane quadrotor is powered by Li-ion batteries or IC (Internal combustion) engines. Phillips et al. [27] present a design of a quadrotor biplane and test it in hover flight mode for package delivery capability. Chipade et al. [28] present a hypothetical design and proof-of-concept flight demonstration of a variable-pitch quadrotor biplane for payload delivery. A

4.1 Biplane Quadrotor: Payload Delivery

63

Fig. 4.1 Quadrotor biplane animated picture

hybrid controller based on the ITSMC and backstepping control is developed [29] with an observer-based partial rotor failure control scheme [30]. In this chapter, we focus on designing a BSC (Backstepping Controller) for autonomous trajectory tracking and an ABSC (Adaptive Backstepping Controller) for payload delivery. During the payload delivery mission, 6 kg is dropped at location A, and 3 kg is picked from location B and location c each while picking the payload, the altitude of the biplane quadrotor is reduced by 2 m and maintained at 20 m for trajectory tracking. Furthermore, we design an ABSC that helps the biplane quadrotor sustain its altitude during changes in mass. For instance, a biplane quadrotor is used to deliver and pick up vaccines in a remote area. The animated picture of the biplane quadrotor is shown in Fig. 4.1 for better visualization of the three modes and transition between modes. The enclosed box holds a Li-ion battery, embedded system, and payload. In the beginning, the overall weight of the biplane quadrotor with a 6 kg payload is 18 kg. Figure 4.2 shows that holding a x−y-position gains desired altitude (20 m) during the take-off. Then after performing the transition maneuver, it switches to level-flight mode, flies with 15 m/s velocity, and travels to a height of 1022 m (delivery point A). While hovering at 20 m altitude, it is commanded to drop a 6 kg weight payload, and the biplane performs a transition maneuver to switch to the level-flight mode and flies at 15 m/s like a fixed-wing UAV. On reaching point B, it again reduces the altitude from 20 to 2 m and picks up a 3 kg payload, and the same repeats at location C.

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4 Adaptive Controller Design for Biplane Quadrotor

Fig. 4.2 Quadrotor biplane

4.2 Mathematical Model of Biplane Drone A Li-ion battery [27] or small IC engine [28] can be a power plant for the biplane quadrotor. The dynamics of the biplane quadrotor are given as ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ x¨ −sθ r v − qw Fax ⎣ y¨ ⎦ = 1 ⎣ Fay ⎦ + g ⎣cθ sφ ⎦ + ⎣ pw − r u ⎦ , m −T + F z¨ cθ cφ qu − pv az ⎡ ⎤ ⎡ ⎤ φ¨ (b1 r + b2 p)q + b3 (L a + L t ) + b4 (Na + Nt ) ⎣ θ¨ ⎦ = ⎣ ⎦, b5 pr − b6 ( p 2 − r 2 ) + b7 (Ma + Mt ) ¨ p − b r )q + b (L + L ) + b (N + N ) (b ψ 8 2 4 a t 9 a t

(4.1)

(4.2)

where the linear velocity is [u v w], [ p q r ] is angular velocity, [L t Mt Nt ] are moments, and [L a Ma Na ] are aerodynamics moments about roll, pitch, and yaw angles applied on the biplane quadrotor such that the aerodynamics force is

4.2 Mathematical Model of Biplane Drone

65

⎡

⎤ ⎡ ⎤⎡ ⎤ Fax sα cα −sα sβ cα −D ⎣ Fay ⎦ = ⎣ sβ cβ 0 ⎦⎣ Y ⎦, −cα cβ cα sβ sα −L Faz

(4.3)

where angle of attack is α, slide slip angle is β, and [D Y L] are the aerodynamics forces acting about a respective axis. The inertial terms are defined as constants bi : ⎡ ⎤ b1 ⎢b2 ⎥ ⎢ ⎥ ⎢b3 ⎥ 1 ⎢ ⎥= ⎢b4 ⎥ Ix Iz − Ix2z ⎢ ⎥ ⎣b8 ⎦ b9

⎡

⎤ (I y − Iz )Iz − Ix2z ⎢ (Ix − I y + Iz )Ix z ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ b5 (Iz − Ix ) ⎢ ⎥ 1 I z ⎢ ⎥ , ⎣b6 ⎦ = ⎣ Ix z ⎦ . ⎢ ⎥ Ix z I y ⎢ ⎥ b7 1 ⎣(Ix − I y )Ix + Ix2z ⎦ Ix

The biplane quadrotor airspeed (V) and the pressure (Q ∞ ) are given as V =

 u 2 + v2 + w2 ,

Q∞ =

1 ρV 2 . 2

(4.4)

The moments due to aerodynamic forces are defined as ⎡

⎤ ⎡ ⎤ La 0.5ρV 2 SbClc ⎣ Ma ⎦ = ⎣0.5ρ cV ¯ 2 SCm c ⎦ , Na 0.5ρV 2 SbCn c

(4.5)

where Clc , Cm c , and Cn c are the roll, pitch, and yaw moments coefficients, S is wing area (m 2 ), b is wing span (m), c is the chord of the rotor blade, c¯ is the mean aerodynamics chord (m), and Pw b rw b + C lr 2V 2V pw b rw b + C nr , = Cnβ β + Cn p 2V 2V

Clc = Clβ β + Cl p

(4.6)

Cnc

(4.7)

such that Clβ , Cl p , Clr , Cn β , Cn p , and Cnr values are obtained after studying the physical representation of the system. Subscript w is for wing-body axis. The XFOIL code helps predict the aerodynamic response of NACA 0012 aerofoil at Reynolds number 0.25 × 106 . XFOIL is simple and easy to use, while other CFD analysis software like ANSYS Fluent is more complex; XFOIL is also faster and more accurate [31]. After receiving the lift, drag, and pitch moment coefficients from XFOIL, using MATLAB, a polynomial function is generated for the aerodynamics coefficients lift (Cl ), drag (Cd ), and pitch moment (Cm c ). Figure 4.3 shows the graph of these coefficients versus AoA (α).

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4 Adaptive Controller Design for Biplane Quadrotor

Fig. 4.3 Cl , Cd , and Cm c versus angle of attack (AoA)

4.3 Controller Design Based on the mathematical model of the biplane quadrotor next, we discuss BSC design for autonomous trajectory tracking.

4.3.1 Quadrotor Mode In the quadrotor mode, due to almost zero angular velocity, no substantial aerodynamic forces [Fax Fay Faz ] and moments [ L a Ma Na ] are acting on it. So its dynamics can be considered the conventional rotary-wing UAV. The dynamical equation of the biplane quadrotor in the quadrotor mode (Fig. 4.4) is given as

T

T

T m x¨ y¨ z¨ = 0 0 mg + R 0 0 −T , ⎡ ⎤ cφcψ sφsθ cψ − cφsψ cφsθ cψ + sφsψ R = ⎣ cθ sψ sφsθ sψ + cφcψ cφsθ sψ − sφcψ ⎦ . −sθ sφcθ cφcθ

(4.8) (4.9)

The desired three-dimensional trajectory [xd yd z d ] and desired yaw angle are given. The position controller generates the required thrust T as well as the desired roll and pitch angles based on the x−y-trajectory tracking. These desired angles are used by the attitude controller to generate the required moments [L t Mt Nt ] about the respected axis. First, we design a BSC for the attitude subsystem, and based on the same calculation procedure, the controller is designed for the position subsystem. Using (4.2) without aerodynamic moments, the error between the roll angle is eφ = φ − φd .

(4.10)

4.3 Controller Design

67

Fig. 4.4 Quadrotor mode controller design

A positive definite function VQ Mφ = 21 eφ2 and time derivative V˙ Q Mφ = eφ e˙φ = eφ e p − kφ eφ2 , kφ > 0, where e p = p − pd , help satisfy this condition with a virtual control pd = φ˙ d − kφ eφ chosen such that e p = p − pd = p − φ˙ d + kφ eφ .

(4.11)

The next step is to enhance VQ Mφ with quadratic term in e p , to get positive definite function as VQ M p = 21 e2p + VQ Mφ , and time derivative is  V˙ Q M p = e p (b1r + b2 p) q + b3 L t + b4 Nt − φ¨ d + kφ e˙φ − kφ eφ2 + eφ e p . Therefore, the control law for a roll subsystem can be defined as Lt =

1  −eφ − k p e p + φ¨ d − kφ e˙φ − (b1 r − b2 p)q − b4 Nt b3

(4.12)

so that V˙ Q M p = −kφ eφ2 − k p e2p which guarantees asymptotic stable system for appropriately chosen kφ , k p > 0. Similarly, the control laws for pitch, yaw, and position subsystems are

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4 Adaptive Controller Design for Biplane Quadrotor

Mt = Nt = T = ux = uy =

 1  θ¨d − eθ − kq eq − kθ e˙θ + b6 p 2 − r 2 − b5 pr , b7 1  ψ¨ d − eψ − kr er − kψ e˙ψ − (b8 p − b2 r ) q − b4 L t , b9 m (ez + ew kw − z¨ d + k z e˙z + g) , cφ cθ m (ex + ku eu − x¨d + k x e˙x ) , T m e y + kv ev − y¨d + k y e˙ y . T

(4.13) (4.14) (4.15) (4.16) (4.17)

By using (4.12) and (4.14), the control law for the roll subsystem is   b9 −eφ − k p e p − kφ e˙φ + φ¨ d − b1rq − b2 pq b3 b9 − b4 b4  −eψ − kr er − kψ e˙ψ + ψ¨ d − b8 pq + b2 qr . − b9

Lt =

(4.18)

Using (4.16)–(4.17), the desired roll and pitch angle are calculated as

ux − u y φd = arcsin sψ + cψ



 u x − u y − sφ(sψ − cψ) , θd = arcsin . 2cφcψ

4.3.2 Transition Mode The biplane quadrotor is commanded to gradually rotate about the pitch axis up to ≈90◦ . During the transition maneuver, while decreasing the pitch angle, the vertical component of thrust generated by the rotors starts falling and the two wings mainly produce no considerable lift force. When the lift force is enough to balance the weight, the biplane quadrotor switches from quadrotor to fixed-wing mode. The transition maneuver has two phases: (i) primary transition and (ii) final transition phase; in the primary phase, the biplane quadrotor is directed to reduce its pitch angle linearly from θ Q h (pitch angle at hovering) to an interior pitch angle, θ Qsw that is governed by the AoA and required flight path angle (γ ) [32]. The duration of the transition from quadrotor mode to fixed-wing mode is about 3 s and fixed-wing mode to quadrotor mode is 1 s. Here also note that during the transition maneuver, the altitude and attitude of the biplane quadrotor are controlled. There is no control over the x−y-position. Control laws for the transition mode are the same as the quadrotor mode but in this, we considered aerodynamics forces and moments.

4.3 Controller Design

69

Using (4.2), the roll angle error is defined as eφ = φ − φd . A positive definite function is defined based on the error VT Mφ = 21 eφ2 and V˙T Mφ = eφ e p − δφ eφ2 , δφ > 0, where e p = p − pd . A virtual control pd = φ˙ d − δφ eφ is chosen such that e p = p − pd = p − φ˙ d + kφ eφ .

(4.19)

Now, to enhance VT Mφ with quadratic term in e p , to get positive function as VT M p = 1 2 e + VT Mφ and time derivative is 2 p V˙T M p = e p ((b1r + b2 p)q + b3 (L t + L a ) + b4 (Nt + Na ) − φ¨d + kφ e˙φ ) − δφ eφ2 + eφ e p .

(4.20)

By using (4.20), the control signal is expressed as  b9 (−eφ − k p e p − kφ e˙φ + φ¨ d − b1rq − b2 pq b3 b9 − b4 b4  −eψ − kr er − kψ e˙ψ + ψ¨ d − b8 pq + b2 qr ) − L a , − b9

Lt =

(4.21)

so that VT M p = −k12 eφ2 − k22 e2p ≤ 0, and similarly, the other control signals are  1  b5 θ¨d − eθ − kq eq − kθ e˙θ + b6 p 2 − r 2 − Ma − pr, b7 b7 1 Nt = (−eψ − kr er + ψ¨ d − kψ e˙ψ − (b8 p − b2 r )q − b4 (L t + L a ) − b9 Na ), b9

 m Faz T = ez + ew kw − z¨ d + k z e˙z + g − . (4.22) cφ cθ m

Mt =

4.3.3 Level-Flight Mode In the fixed-wing mode, the dynamics are the same as the conventional fixed-wing UAVs [33, 34]. The biplane quadrotor dynamics (4.1) and (4.2) are defined based on the body frame to transform with respect to fixed-wing axis and it can be done by ⎡ ⎤ ⎡ ⎤⎡ ⎤ vx 0 0 −1 vx ⎣v y ⎦ = ⎣0 1 0 ⎦ ⎣v y ⎦ . 10 0 vz W vz Q

(4.23)

Based on the above equation, the biplane quadrotor dynamics for the fixed-wing mode are given as

70

4 Adaptive Controller Design for Biplane Quadrotor

x˙ = cθ cψu + (sφsθ cψ − cφsψ)v + (cφsθ cψ + sφ sψ)w,

(4.24)

y˙ = cθ sψu + (sφsθ sψ + cφcψ) v + (cφsθ sψ − sφcψ) w, (4.25) z˙ = −u sθ + v sφcθ + w cφcθ, (4.26) Fax T u˙ = − g sθ + pv − qu + , (4.27) m m Fay + g cθ sφ + pw − r u, (4.28) v˙ = m Faz w˙ = + g cθ cφ + r v − qw, (4.29) m p˙ = − pq(bw3 + bw9 ) − qr (bw11 − bw12 ) − bw13 (L a + L t ) + bw5 (Na + Nt ), (4.30) q˙ = bw8 r 2 + bw9 p 2 + 2 bw10 pr + bw7 (Mt + Ma ),

(4.31)

r˙ = pq(bw1 + bw2 ) + qr (bw3 − bw4 ) + bw5 (L t + L a ) − bw6 (Nt + Na ), (4.32) φ˙ = p + q sφ tθ + r cφ tθ, (4.33) θ˙ = q cφ − r sφ, (4.34) sφ cφ ψ˙ = q +r , (4.35) cθ cθ where bw1 = Ix z (I y + Ix z )/B, bw2 = Iz2 /B, bw3 = Ix z (Ix + I y )/B, bw4 = Ix I y /B, bw5 = Ix z /B, bw6 = Iz /B, bw7 = 1/I y , bw8 = Ix /I y , bw9 = Iz /I y , bw10 = Ix z /I y , bw11 = Ix2 /B, bw12 = Ix z (I y − Ix z )/B, bw13 = Ix /B and B = Ix Iz − Ix2z . For the fixed-wing mode, the controller design is shown in Fig. 4.5. The velocity controller generates the enquired thrust T based on the desired trajectory [xd yd z d ]. The desired signal generator block generates the required angles [ψd θd ψd ] by using the desired input and position of the biplane quadrotor. The attitude controller uses these signals to generate the required moments [L t Mt Nt ] which along with thrust when given to the variable-pitch propulsion system help generate the signals for four actuators. Using (4.24), the error in the x-axis is given as ex = x − x d .

(4.36)

A positive definite function is defined as VF Mx = 21 ex2 by tracking first time derivative, and using Eq. (4.24), we get V˙ F Mx = ex (cθ cψ u d + (sφsθ cψ − cφsψ)v + (cφsθ cψ + sφsψ) w − x˙d ) , so for desired u d velocity, control law can be designed as ud =

−1 (k x ex + (sφsθ cψ + cφsψ)v + (cφsθ cψ + sφsψ)w + x˙d ) , cθ cψ

4.3 Controller Design

71

Fig. 4.5 Fixed-wing mode controller design

so that V˙ F M = −k x ex2 guarantees asymptotic stability for k x > 0. Using (4.3.3), we calculate the desired thrust. The error in x-direction velocity is defined as eu = u − u d .

(4.37)

For a positive definite function VF Mu = 21 eu2 , using (4.27), the time derivative is V˙ F Mu = eu

 Fax T − g sθ + pv − qu + − u˙ d . m m

The control law for the desired thrust can be defined as 

Fax + g sin θ − pv + qu + u˙ d , T = m ku eu − m

(4.38)

(4.39)

so that V˙ F Mu = −ku eu2 , which guarantees an asymptotic stable system for appropriately chosen ku > 0. Desired pitch and yaw angle can be calculated [35] as

72

4 Adaptive Controller Design for Biplane Quadrotor

 θd = sin

−1



z˙ d − cz (z − z d )

u 2 + (v sφ + w cφ)2

 y˙d − k y (yd − y) −1 , ψd = tan x˙d + k x (xd − x)

 + tan−1

u v sφ + w cφ

 , (4.40)

where z d is desired altitude, xd and yd are desired x- and y-positions, and k x , k y , kφ and k z are tunable gains. Considering roll angle as a linear function of yaw angle, we get (4.41) φd = kφ (ψ − ψd ) . Using the same procedure, we define control laws for the attitude subsystem as Lt =

1  k p e p − pq(bw3 + bw9 ) + qr (bw12 − bw11 )

bw13 + bw5 (Na + Nt ) − p˙ d ) , (4.42) −1  Mt = kq eq + bw8 r 2 + bw9 p 2 + 2 pr bw10 +bw7 Ma − q˙d ) , (4.43) bw7 1 Nt = (kr er + pq(bw1 + bw2 ) + qr (bw3 − bw4 ) + bw5 (L t + L a ) bw6 − bw6 Na − r˙d . (4.44) Using (4.44), we rewrite (4.42) as   bw6 − pq(bw3 + bw4 ) − qr (bw11 − bw12 ) + k p e p − p˙ d Lt = 2 bw6 bw13 − bw5  

bw5 + ( pq (bw1 + bw2 ) + qr (bw3 + bw4 ) + kr er − r˙d ) − L a . (4.45) bw6

4.4 Adaptive Backstepping Controller Design The biplane quadrotor is only able to drop or pick up the payload while in the hovering state, so when it is directed to pick up or drop off the payload, it has to perform the transition to switch to the quadrotor mode. Now to handle the mass change, an ABSC control architecture is shown in Fig. 4.6. When mass changes occur, the adaptive laws update the controller to maintain the desired altitude and attitude. In the initial stage, the weight of the biplane quadrotor with a payload is 18 kg, it drops 6 kg weight at the height of 20 m and picked a 3 kg payload twice at a 2 m attitude from a different location. The mass of the biplane quadrotor is considered an uncertain parameter and the adaptive control law is designed in such a way that it adapts to the mass changes in

4.4 Adaptive Backstepping Controller Design

73

Fig. 4.6 Block diagram of adaptive backstepping control

finite time. To design the adaptive law, we take an attitude subsystem of the quadrotor mode. Now we can rewrite Eq. (4.8) z¨ = g − T λ∗ cθ cφ,

(4.46)

where λ∗ = m1∗ . During the payload drop or pick, the biplane should hold the desired altitude. The error in altitude is given as ez = z − z d .

(4.47)

A positive definite function is defined as VQ Mz = 21 ez2 and time derivative is V˙ Q Mz = ez e˙z = ez ew − k z ez2 , k z > 0 with ew = w − wd . To satisfy this condition, a virtual control wd = z˙ d − k z ez is chosen such that ew = w − z˙ d + k z ez . The next step is to enhance VQ Mz with quadratic term in ew and to have a positive definite VQ Mw = 1 2 e + VQ Mz + 2γ1 λ˜ 2 , where λ˜ = λ∗ − λˆ for an estimate λˆ and an adaptation gain 2 w γ > 0. By differentiating, we get     1 V˙ Q Mw = −k z ez2 + ew −T λ˜ + λˆ cφ cθ + g − z¨ d + k z e˙z + ez ew − λ˜ λ˙ˆ . γ (4.48) Observing (4.48), a choice of λ˙ˆ = −γ ew T cφ cθ leads to cancelation of terms. However, for robustness, considering the drone’s physical lower bound of λ = m1 , we design the adaptive update law as ⎧ ⎪ ⎨−γ ew T cφ cθ, if |λˆ (t)| > λ ˙λˆ = −γ ew T cφ cθ, if |λˆ (t)| = λ& ew T cφcθ sgn(λˆ (t)) ≥ 0, ⎪ ⎩ 0 otherwise.

(4.49)

The control law for the required thrust (T ) is chosen as T =

1 λˆ cφ cθ

(ez + ew kw + g − z¨ d + k z e˙z ) ,

(4.50)

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4 Adaptive Controller Design for Biplane Quadrotor

so that V˙ Q Mw = −k z ez2 − kw ew2 which guarantees asymptotic stable system for appropriately chosen k z , kw > 0. By using the adaptive law (4.49), the control laws for roll, pitch, yaw, u x , and u y defined by using the same procedure are

Lt =

b9 b3 b9 − b4



(−eφ − k p e p − kφ e˙φ + φ¨ d − b1rq − b2 pq

b4 (−eψ − kr er − kψ e˙ψ + ψ¨ d − b8 pq + b2 qr )) − L a , b9 1 = (θ¨d − eθ − kq eq − kθ e˙θ + b6 ( p 2 − r 2 ) − b5 pr ), b7 1 = (ψ¨ d − eψ − kr er − kψ e˙ψ − (b8 p − b2 r )q − b4 L t ), b9 1 = (ex + ku eu − x¨d + k x e˙x ), T λˆ 1 = (e y + kv ev − y¨d + k y e˙ y ). T λˆ

− Mt Nt ux uy

(4.51) (4.52) (4.53) (4.54) (4.55)

A propulsion system of the UAVs comes with Electronics Speed Controller (ESC), BLDC motor, and propellers for maximum efficiency in different speed ranges [36]. The blade pitch angle varies for variable-pitch propulsion to get the desired thrust. The advantages of this mechanism are (i) thrust in both upward and downward directions employing positive and negative pitch, (ii) compact power depletion by optimizing pitch and rotational speed, and (iii) improved rotational speed for maximal thrust. The block diagram is shown in Fig. 4.7. The pitch of the rotor adjusts to the servo mechanism connected to ball-bearing joints. The mathematical expression of the blade pitch angle for rotor i is 6C Ti 3 θ0i = + σ Clα 2



C Ti , 2

where Ti is the trust produced by the i th rotor, C Ti =

(4.56) Ti ρπ R 2 Vti2p

the thrust coefficient,

σ = is solidity ratio, Clα is the lift curve’s slope, air density ρ, blade tip’s angular velocity Vti p = ω R, rotor radius R, Nb is number of blades at each rotor, chord length c, and blade’s AoA is αb . To control the position and altitude of the biplane quadrotor for all three modes, the control laws are defined in (4.12)–(4.15) and in fixed-wing mode (4.39), (4.45), (4.43), and (4.44). The total thrust and moments for variable-pitch propeller system [37] are given as cNb πR

4.4 Adaptive Backstepping Controller Design

75

Fig. 4.7 Variable-pitch propulsion system

⎡

⎤ ⎡ ⎤⎡ ⎤ T C T1 k k k k ⎢ L t ⎥ ⎢k · d −k · d −k · d k · d ⎥ ⎢C T2 ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ Mt ⎦ ⎣k · d k · d −k · d −k · d ⎦ ⎣C T3 ⎦ , Nt C T4 C1 −C2 C3 −C4

(4.57)

where k = ρπ R 2 Vti p , Vti p =  · R,  = 3200 for hovering mode, and  = 2000 √ (|C Ti |1/2 , i = 1, . . . , 4. In the first iteration, we for fixed-wing mode [28], Ci = k·R 2 calculate the thrust coefficient by taking the inverse of (4.57), given as ⎤ ⎡ ⎤−1 ⎡ ⎤ ⎡ k k k k T C T1 ⎢C T2 ⎥ ⎢k · d −k · d −k · d k · d ⎥ ⎢ L t ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣C T3 ⎦ = ⎣k · d k · d −k · d −k · d ⎦ ⎣ Mt ⎦ . C T4 Nt C1 −C2 C3 −C4

(4.58)

Using these thrust coefficients, the inputs are intended for successive iterations to find U1 = T , U2 = L t , U3 = Mt , and U4 = Nt as given in (4.57).

76

4 Adaptive Controller Design for Biplane Quadrotor

4.5 Results and Discussions We design BSC for autonomous trajectory tracking and ABSC to handle the mass changes during the payload mission. Simulation is carried out using MATLAB Simulink by using the parameter given in Table 4.1.

4.5.1 Autonomous Trajectory Tracking The simulation timeline for the autonomous trajectory tracking is shown in Fig. 4.8. For the first 40 s, the biplane quadrotor is commanded to take off with 5 m/s velocity while holding the x−y-position constant, and then after, it will be in the hover state till t = 60 s and that time altitude would be 200 m. Then the transition phase was initiated for the next 3 s and smoothly switched to the fixed-wing mode. During the fixed-wing mode, the overall velocity is around 15 m/s during the t = 63–400 s. After 400 s, the biplane quadrotor is directed to again change its fixed-wing mode to quadrotor after performing the transition maneuver that took only 1 s time. For the next 20 s, it will be in hover state, and then finally at t = 421 s, it starts landing with 5 m/s velocity.

Table 4.1 Biplane quadrotor parameters Parameters g Mass (m) Ix x I yy Izz Wing area (single) Aspect ratio Wing Span Gap-to-chord ratio

Value 9.8 ms−2 12 kg 1.86 kg · m2 2.03 kg · m2 3.617 kg · m2 0.754 m2 6.9 2.29 m 2.56

Fig. 4.8 Simulation timeline for trajectory tracking with backstepping control

Yaw (deg) Pitch (deg) Roll (deg)

4.5 Results and Discussions

77

50 0

-50 0

Desired Roll angle

50

50 0 -50 0

150

200

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50

20 0 -20 0

100

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100

150

Desired Yaw angle

50

100

150

250

300

350

400

450

300

350

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450

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350

400

450

350

400

450

350

400

450

350

400

450

Actual Pitch angle

200

250

Actual Yaw angle

200

250

Time (seconds)

Z Axis (m)Y Axis (m) X Axis (m)

Fig. 4.9 Biplane quadrotor’s attitude during trajectory tracking 6000 4000 2000 0 0 100 50 0 0 200 100 0 0

Desired X position

Actual X position

50

100

150

50

100

150

200

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200

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50

100

150

200

250

300

Actual Y position

250

300

Actual Altitude

250

300

Time (seconds) Fig. 4.10 Biplane quadrotor’s position during trajectory tracking

Attitude tracking during the trajectory tracking with BSC is shown in Fig. 4.9. The BSC controller effectively tracks all the desired states with all three modes. However, it generates fluctuation within a short period while switching the modes. Altitude and x−y-position tracking during the mission is shown in Fig. 4.10. The BSC has a good tracking performance, generating a 2 m error after the transition maneuver in altitude and 6 m error in the x-axis, and a 0.5 m error in the y-axis during the transition. Figure 4.11 demonstrates the angular velocity tracking with BSC. The biplane quadrotor velocity profile is provided in Fig. 4.12 that flies at 15 m/s in the fixed-wing mode, the α is around 3◦ , and β is 0◦ and takes off and lands at 5 m/s. Figure 4.13 demonstrates the thrust coefficients during the flight. Thrust and moments generated by the actuators are provided in Fig. 4.14. The thrust generated during the fixed-wing mode is lower than that needed in the quadrotor mode. The three-dimensional trajectory tracking is shown in Fig. 4.15, and it reveals

4 Adaptive Controller Design for Biplane Quadrotor

q (deg/s) p (deg/s)

78 100

pd

p

0

-100 0 100

50

100

150

200

250

300

0

r (deg/s)

-100 0 100

50

100

150

200

250

100

150

200

250

450

400

450

400

450

350 rd

50

400

q

300

0 -100 0

350 qd

300

r

350

Time (seconds) Fig. 4.11 Biplane quadrotor’s angular velocity during trajectory tracking

, (deg)

100

0

V (m/s)

-100

0

50

100

150

50

100

150

200

250

300

350

400

450

200

250

300

350

400

450

20 10 0 0

Time (seconds)

Fig. 4.12 AOA, slide slip angle, and vehicle velocity during the trajectory tracking

0.1

Thrust Coefficients

CT

1

CT

2

CT

3

CT

4

0.05

0

-0.05

0.1

0.1

0

0

-0.1 63

-0.1 0

50

100

63.05

150

63.1

200

-0.1 401

250

Time (seconds)

Fig. 4.13 Thrust coefficients during trajectory tracking

401.05

300

401.1

350

400

450

4.5 Results and Discussions

79

Thrust (N)

200 U1

100

Moments (N-m)

0 0 100

50

100

150 U2

50

U3

200 U4

250 20 0 -20

0 -50 0

50

100

150

200

250

300

405

300

350

400

450

400

450

410

350

Time (seconds) Fig. 4.14 Control signals during trajectory tracking

Fig. 4.15 Three-dimensional trajectory tracking

that the designed BSC is able to efficiently track the autonomous trajectory and there is a minor fluctuation during the transition mode.

4.5.2 Packet Delivery Scenario Simulation Atmospheric turbulence is critical in finding aircraft performance and management qualities [38]. A real-time scenario for payload delivery is shown in Fig. 4.16. The biplane quadrotor of 18 kg payload is commanded to take off with 1 m/s at the position [0.5 5 0] for 20 s. It is then directed to transition up to t = 23 s and switch to the fixed-wing mode for t = 100 s when it transitions in 1 s and changes to quadrotor mode, and at t = 105 s payload is dropped. It again performs the transition to fixedwing mode at t = 193 s and t = 283s. It picks up a 3 kg payload at an altitude of

80

4 Adaptive Controller Design for Biplane Quadrotor

Fig. 4.16 Simulation timeline for packet delivery 25

Z Axis (m)

20 Desired Z Position Backstepping Intergral Termial Sliding Mode Adaptive Backstepping

15 20.1

10

20

5

19.9 104

106

108

21

110

20

0

284 286 288 290 292

-5

0

50

100

150

200

250

300

350

400

450

500

Time (seconds) Fig. 4.17 Altitude tracking during packet delivery

2m. For a more realistic scenario, wind gusts are applied at t = 299–t = 537 s. At t = 516 s, it performs a transition maneuver, and finally, after 20 s it lands. The response of the BSC, ITSMC, and ABSC for altitude tracking during the payload delivery mission is shown in Fig. 4.17. BSC has generated an 8 cm steadystate error in altitude while ITSMC took 4 s to track again the desired altitude. ABSC effectively and quickly adapts to the mass change and tracks desired altitude. It can be also observed at t = 284−292 s that the ABSC controller took less time than ITSMC to reach the required altitude after the transition maneuver. There is no noticeable change in the altitude while wind gusts act on the biplane quadrotor. The performance of the controllers for x-axis trajectory tracking is shown in Fig. 4.18. All three controllers: BSC, ITSMC, and ABSC track the x trajectory effectively and there is no significant impact of the wind gusts on the performance of the x-axis trajectory tracking but while transitioning, a small error ensues. The velocity component of the x-axis is 15 m/s.

4.5 Results and Discussions

81

3500 3000

X Axis (m)

2500 2000 1500

Desired X Position Backstepping Integral Terminal Sliding Mode Adaptive Backstepping

1000 500 0 0

50

100

150

200

250

300

350

400

450

500

Time (seconds) Fig. 4.18 X-axis trajectory tracking during packet delivery 100

Desired Y Position Backstepping Integral Terminal Sliding mode Adaptive Backstepping

Y Axis (m)

80 60 40 20 0 0

50

100

150

200

250

300

350

400

450

500

Time (seconds) Fig. 4.19 Y-axis trajectory tracking during packet delivery

Figure 4.19 shows the y-axis trajectory-tracking response by these three controllers. Despite significant fluctuations due to wind gusts in the y-axis trajectory, no significant change occurs due to mass change. The roll angle tracking during the mission is shown in Fig. 4.20. BSC, ITSMC, and ABSC controllers track the desired roll angle but the response of the ITSMC is slower than BSC and ABSC. Figure 4.21 demonstrates the pitch angle tracking performance of BSC, ITSMC, and ABSC. In the quadrotor mode, the pitch angle is about 0◦ , but on transition, gradually decreases to ≈−90◦ and on switching to fixed-wing mode it increases ≈+90◦ while returning to the quadrotor mode. The ABSC performs a little better than BSC and ITSMC during the mission. Wind gusts have no significant impact on the pitch angle during the landing phase. Yaw angle tracking by BSC, ITSMC, and ABSC is shown in Fig. 4.22. The ITSMC encounters a short interval spike in the yaw angle throughout state change, while

82

4 Adaptive Controller Design for Biplane Quadrotor

100 50 0

Roll (deg)

50

-50 275

0

285

290

Desired Roll Angle Backstepping Integral Terminal Sliding Mode Adaptive Backstepping

-50

-100 0

280

50

100

150

200

250

300

350

400

450

500

Time (seconds) Fig. 4.20 Roll angle tracking during packet delivery 100 Desired Pitch angle Backstepping Integral Terminal Sliding Mode Adaptive Backstepping

Pitch (deg)

50

0

10 0

-50 -10

-100 0

289

50

100

150

200

250

300

350

290

291

400

292

450

293

500

Time (seconds) Fig. 4.21 Pitch angle tracking during packet delivery

BSC and ABSC see comparatively small spikes. All three controllers are able to track the yaw angle effectively. The performance of the adaptive law is shown in Fig. 4.23, which indicates effective tracking of the mass changes within 1 s during payload drop and pick up. The thrust profile during the payload delivery mission is shown in Fig. 4.24. At t = 105 s, produced thrust is 177 N, and after dropping off a 6 kg payload, the thrust is 117 N. Wind gusts have a noticeable impact on thrust generation in both fixedwing and quadrotor modes. Figure 4.25 shows three-dimensional trajectory tracking by the BSC, ITSMC, and ABSC with mass changes and wind gusts.

4.5 Results and Discussions

83

Yaw (deg)

50

0 Desired Yaw angle Backstepping Integral Terminal Sliding Mode Adaptive Backstepping

-50

0

50

100

150

200

250

300

350

400

450

500

Time (seconds) Fig. 4.22 Yaw angle tracking during packet delivery 20 18

Mass (kg)

20

16 14 18 16 12 14 12

15 280.5 281 281.5

15

193

Desired Mass Adaptive Backstepping

10

193.5

10

105 105.5 106

0

50

100

150

200

250

300

Time (seconds) Fig. 4.23 Mass tracking by the designed adaptive law

Fig. 4.24 Thrust change during the mass change

350

400

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4 Adaptive Controller Design for Biplane Quadrotor

Fig. 4.25 Three-dimensional trajectory tracking during payload delivery

4.6 Conclusions The simulation is carried out for a biplane quadrotor’s autonomous trajectory tracking and payload delivery mission. This mission includes all three modes with take-off, landing, and hovering phases. For trajectory tracking, the response of the BSC controller while BSC, ITSMC, and ABSC response for the payload delivery mission is evaluated. We also applied fixed-wing and quadrotor modes during the payload delivery to make the simulation more realistic. The BSC tracks the autonomous trajectory containing quadrotor, transition, and fixed-wing modes with take-off, landing, and hovering phases. The yaw angle fluctuates in both fixed-wing and quadrotor modes, while the biplane quadrotor faces crosswinds. During mass changes, the ABSC takes around 1 s to compensate for the sudden mass change and gives a faster response than ITSMC with zero steady-state error.

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Chapter 5

Multi-observer Based Adaptive Controller for Hybrid UAV

A hybrid UAV, like a biplane quadrotor, has many applications in agriculture, disaster management, and relief operation. In this chapter, we designed a dual observer (i) an Extended state observer (ESO) for the state approximation, and (ii) a Nonlinear Disturbance Observer (DO) for the exterior disturbance estimation. There are three different nonlinear controllers; (i) Backstepping Controller (BSC), (ii) Integral Terminal Sliding Mode Controller (ITSMC), and Hybrid Controller (ITSMC + BSC) are designed, and ESO (with and without DO) are applied for the trajectory tracking to evaluate the results. Mass change during the flight despite wind gusts is also handled using the Adaptive Backstepping controller (ABSC) and Adaptive hybrid controller with ESO and DO. The Simulation results show that the hybrid controller with ESO and DO is most efficient compared to BSC and ITSMC and can eliminate external disturbances. Furthermore, an adaptive hybrid controller with ESO and DO is more efficient than the adaptive BSC with ESO and DO.

5.1 Nonlinear Observers Controller design for drones or Unmanned Ariel Vehicles (UAVs) gets significant attention because of a wide range of applications in civil, agriculture, military, surveillance, and e-commerce sectors. There are mainly two types of drones: (i) Rotary-wing UAVs and (ii) Fixed-wing UAVs. A biplane quadrotor is a hybrid type of UAV with the advantages of rotary-wing and fixed-wing UAVs. It has two wings connected with the quadrotor, providing the lift force for flying at high velocity like fixedwing UAVs. Oosedo et al. [1] introduced a quadrotor tail-sitter with strategies for optimal transition [2]. Swarnkar et al. [3] present a comprehensive six degrees of freedom mathematical modeling of the biplane quadrotor and a nonlinear dynamic inverse controller. Phillips et al. [4] presented the design and development of the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_5

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biplane quadrotor and successfully tested it in hovering mode for packet delivery. Yeo et al. [5] showed initial results of onboard flow measurement to expand the longitudinal steadiness of a biplane quadrotor under perpendicular gusts. Diverse modeling approaches are deliberated [6] with blade element theory (BET) and dynamic inflow to originate the reduced ordered model that is used for trajectory planning, and Optimization-based trajectory planner for the autonomous transition of the biplane quadrotor is introduced [7]. A real-time onboard algorithm for the biplane quadrotor is introduced to iteratively learn a forward conversion maneuver via periodic flight trials to demonstrate the robustness and rapid convergence [8]. Finally, Dalwadi et al. [9] presented BSC with DO for the trajectory tracking of a tail-sitter, and then BSC for trajectory tracking and ABSC (Adaptive Backstepping controller) for the payload delivery are designed for the biplane quadrotor [10]. An NDO-based nonlinear controller handles partial rotor failure despite wind gusts and a slung load [11]. Accurate mathematical modeling for an underactuated system like a biplane quadrotor is challenging, and many assumptions are considered while modeling. In the real-time scenario, there are unmolded parameters, actuator saturation, and external disturbance acting on the UAVs, and many algorithms have been developed to handle it. Observers estimate the unmeasurable states of the system by using the known control input and measurable output to improve closed-loop stability. An extended DO-centered sliding mode controller [12] is proposed for the underactuated system, while a higher-order DO for robotic systems with mismatched uncertainties [13] is proposed. Rojsiraphisal et al. [14] developed a DO-based FTSMC (Fast Terminal Sliding Mode Control) method for steadying underactuated robotic systems. Castillo et al. [15] developed a DO-based attitude controller for quadrotor UAVs and an observer technique-based super-twisting sliding mode controller [16]. Based on the DO, standoff tracking guidance for the multiple small fixed-wing UAVs is presented [17], where the Lyapunov guidance vector field strategy is used to balance the wind effect and track the ground target. Dhaybi et al. [18] offered a precise instantaneous approximation of mass and inertia tensors. Boss et al. [19] proposed a robust feedback controller for trajectory tracking with a high gain observer (EHGO) assessment framework to approximate the unmeasured state of multirotor UAV, modeling error, and external disturbances. For rotary-wing UAV trajectory tracking with unmeasurable states, external disturbance, and parameterized uncertainties, many researchers have developed different controllers with the combination of the different types of observers. For example, Shao et al. [20] designed robust BSC with ESO, robust BSC with ESO [21], backstepping sliding mode control with ESO [22], Nonlinear mapping (NM) with ESO [23], and a novel ADRC requiring only output state information [24]. Guo et al. [25] presented MOBADC (Multiple Observer-Based Anti Disturbance Control) algorithms that contain DO-based controllers with ESO for the multiple disturbances acting on the quadrotor UAVs. Lyu et al. [26] developed a DO-based H∞ synthesis technique to enhance the hovering accuracy of tail-sitter UAVs despite crosswind. Liu et al. [27] developed a robust nonlinear control method to achieve the desired trajectory without switching the coordinate. A Model Predictive controller (MPC) is proposed for position control of the tail-sitters [28].

5.2 Mathematical Model and Control Architecture

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Researchers have also combined two control methods with observers to achieve more accurate results for the trajectory-tracking control problem. Two closed-loop control framework is proposed by Yang et al. [29] with ADRC (Active Disturbance Rejection Control) for the inner loop and PD (Proportional-Derivative) controller for the outer loop. Based on a combination of backstepping and dynamic inversion control, a novel control strategy with disturbance observer is developed [30], while Zhou et al. [31] proposed a hybrid adaptive controller that contains mass observer and robust controller for the quadrotor UAV. In addition, adaptive control strategies with observers are developed to adapt to parameter changes in underactuated systems. ABC with ESO is presented [32]. All such observers are based on adaptive controllers for either rotary or fixedwing UAVs. We developed a dual observer-based controller for a hybrid UAV-biplane quadrotor. All the states of the biplane quadrotor are not measurable due to the limitations of onboard sensors. Information about the signals is required for controller design. The presence of error, noise, or external disturbance weakens system performance and even averts safe and trustworthy operation. So to address this issue, in this chapter, we consider only the quadrotor mode of the biplane quadrotor. Based on it, we designed ESO (Extended State Observer) to estimate the position, attitude, and linear and angular velocity. The main advantage of the ESO is the ease of implementation on actual hardware and less information required of the system model. Based on these estimated parameters, DO is designed to estimate external disturbances like wind gusts for improved closed-loop stability and overall system robustness. Finally, based on ESO and DO, Nonlinear and adaptive nonlinear controllers are designed to handle mass change during trajectory tracking.

5.2 Mathematical Model and Control Architecture For the quadrotor mode, the state space representation of a biplane quadrotor [3] is ⎡ ⎤ ⎤ x2 x˙1 ⎢ ⎥ ⎢ x˙2 ⎥ ⎢ (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙3 ⎥ ⎢ x4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ ⎥ ⎢ ⎢ 4 ⎥ ⎢ b5 x2 x6 − b6 (x22 − x62 ) + b7 Mt + dθ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙5 ⎥ ⎢ x6 ⎥ ⎢ ⎥ ⎢ ⎢ x˙6 ⎥ ⎢(b x − b x )x + b L + b N + d ⎥ ⎢ ⎥ ⎢ 8 2 2 6 4 4 t 9 t ψ⎥ ⎥, ⎢ ⎥=⎢ ⎥ ⎢ x˙7 ⎥ ⎢ x8 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙8 ⎥ ⎢ T g − m cx1 cx3 + dz ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙9 ⎥ ⎢ x 10 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢x˙10 ⎥ ⎢ T − m Ux + dx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢x˙ ⎥ ⎢ ⎥ ⎣ 11 ⎦ ⎢ x 12 ⎣ ⎦ T x˙12 − m Uy + dy ⎡

(5.1)

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Fig. 5.1 Dual observer-based control architecture

where c(·) = cos(·) and s(·) = sin(·). [L t Mt Nt ] are the moments and T is thrust. [dφ dθ dψ ] and [dx d y dz ] are the external disturbance acting on the biplane quadrotor and g is gravitation force. Ux = sx1 sx5 + cx1 sx3 cx5 , U y = −sx1 cx5 + cx1 sx3 cx5 . with inertial constants defined as bi : ⎡ ⎤ ⎡ ⎤ b1 (I y − Iz )Iz − Ix2z ⎢b2 ⎥ ⎢ (Ix − I y + Iz )Ix z ⎥ ⎢ ⎥ ⎢ ⎥ ⎢b3 ⎥ ⎢ ⎥ 1 Iz ⎢ ⎥= ⎢ ⎥, ⎢b4 ⎥ ⎥ 2 ⎢ I I I − I xz x z ⎢ ⎥ ⎥ xz ⎢ ⎣b8 ⎦ ⎣(Ix − I y )Ix + Ix2z ⎦ b9 Ix ⎡ ⎤ ⎡ ⎤ b5 (Iz − Ix ) ⎣b6 ⎦ = 1 ⎣ Ix z ⎦ . Iy b7 1 Based on these biplane quadrotor dynamics, we design a multiple observer-based controller as shown in Fig. 5.1. In this control architecture, ESO (Extended State Observer) is used to estimate position, attitude, and linear and angular velocities. The nonlinear disturbance observer (DO) estimates the external disturbances by using the estimated state by ESO and known control inputs (L). Controller gives command (L) to the variablepitch propulsion system, and based on it, signal U is generated for the four motors. The main advantage of this control scheme is it requires less knowledge of the system and can estimate both internal and external disturbances.

5.3 Observer Design In this section, first, we will design Extended State Observer [9] intended to approximate and observe the disturbances and plant uncertainties. In this simulation study, we introduce an ESO to estimate the linear and angular position and velocity of

5.3 Observer Design

91

the biplane quadrotor. The core advantage of the ESO is it does not require much information about the plant and is still able to estimate immeasurable variables and to implement it in the actual hardware is comparatively easy so that required less onboard sensors and ultimately energy saved. The ESO for the second-order system can be designed as l˙1 = l2 l˙2 = f (xn ) + d + b0 Us , where f (xn ) is the nonlinear function, n = 1 . . . 12, and d is the external disturbance acting on the attitude and position subsystem of the biplane quadrotor. b0 is the controlling factor. s is [Mt Nt Nt T ]. For the above second-order system, ESO is designed as ⎧ em ⎪ ⎪ ⎪ ⎨ζ˙ 1ι Ek = ˙ ⎪ ζ2ι ⎪ ⎪ ⎩˙ ζ3ι

= ζ1ι − xm = ζ2ι − η1ι em = ζ3ι − η2ι · l f (em , χ1 , μ) + b0 Us = −η3ι · l f (em , χ2 , μ)

(5.2)

where η1 , η2 , η3 , χ1 , χ2 , and μ are the set parameter, ι = [φ θ ψ z x y], and m = [x1 x3 x5 x7 x9 x11 ]. l f (exm , χ , μ) is the saturation function that regulates signal chattering and this function is given by l f (em , χ , μ) =

, |em | ≤ μ |em |χ · sign(em ), |em | > μ. em μ1−χ

Using the above equation, we design ESO for the roll subsystem as x˙1 = x2 x˙2 = (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ . ESO is designed for the roll subsystem as ⎧ ex1 ⎪ ⎪ ⎪ ⎨ζ˙ 1φ Eφ = ˙ ⎪ ζ 2φ ⎪ ⎪ ⎩˙ ζ3φ e l f (ex1 , χ , μ) =

= ζ1φ − x1 = ζ2φ − η1φ ex1 = ζ3φ − η2φ · l f (ex1 , χ1 , μ) + b0 U2 = −η3φ · l f (ex1 , χ2 , μ),

, |ex1 | ≤ μ |ex1 |χ · sign(ex1 ), |ex1 | > μ. x1

μ1−χ

(5.3)

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

Similarly, we can design the ESO for the pitch, yaw, altitude, and x − y-position subsystems. Next, we design a Nonlinear disturbance observer (NDO) to estimate disturbances with the derivative also slow and bounded [9] such that ||d˙ p (t)|| ≤ D p , ||d˙o (t)|| ≤ Do t > 0:

 1 Up , n˙ p = −L p n p − L p L p P˙ + G + ma ˙ dˆ p = n p + L p P,

(5.4)

where U p = R(O)E 3 U1 , dˆ p is the disturbance estimation, n p is the observer state vector, L p > 0 are the tunable gain matrices, and G = [0 0 − g]T .

5.4 Controller Design In this section, we design BSC and ITSMC controller based on the block diagram shown in Fig. 5.1 for the biplane quadrotor.

5.4.1 Backstepping Controller Design Biplane quadrotor dynamics can be divided into six subsystems. Let’s consider the dynamics of the roll subsystem ζ˙1φ = ζ2φ ζ˙2φ = (b1 ζ2ψ + b2 ζ2φ )ζ2θ + b3 L t + b4 Nt + dφ .

(5.5)

As in (5.4), a nonlinear observer for the roll subsystem can be designed as n˙ φ = −L φ (n φ + L φ ζ˙1φ + (b1 ζ2ψ + b2 ζ2φ )ζ2θ + b3 L t +b4 Nt ), ˆ dφ = n φ + L φ ζ2φ .

(5.6)

Differentiating dˆφ , we get d˙ˆφ = n˙ φ + L φ x˙2 = −L φ n φ − L φ (L φ ζ˙1φ + (b1 ζ2ψ + b2 ζ2φ )x4 + b3 L t + b4 Nt ) +L φ ((b1 ζ2ψ + b2 ζ2φ )ζ2θ + b3 L t + b4 Nt + dφ ), = −L φ (n φ + L φ ζ2φ ) + L φ dφ = −L φ d˜φ , (5.7)

5.4 Controller Design

93

where d˜φ = dφ − dˆφ is the error, and dˆφ is the estimated disturbance in roll subsystem and L φ > 0 is a tune-able observer gain. For roll subsystem, error in roll angle is defined as e1 = ζ1φ − x1d and for a positive definite function V1 = 21 e12 , we get V˙1 = e1 e˙1 = e1 (ζ˙1φ − x˙1d ) = e1 (ζ2φ − x˙1d ), with k1 > 0. To satisfy this condition, a virtual control x2d = x˙1d − k1 e1 is designed such that e2 = ζ2φ − x2d = ζ2φ − x˙1d + k1 e1 , and so we get V˙1 = e1 ((e2 + x˙1d − k1 e1 ) − x˙1d ) = e1 e2 − k1 e12 .

(5.8)

The next step is to enhance the function V1 with e2 = x2 − x2d and the error dynamics e˙2 = x˙2 − x¨1d + k1 e˙1 . Lyapunov positive definite function is given as 1 V2 = V1 + e22 . 2 Taking the first derivative, we get V˙2 = e1 e2 − k1 e12 + e2 ((b1 ζ2ψ + b2 ζ2φ )ζ2θ + b3 L t + b4 Nt + dφ ).

(5.9)

By using (5.7) and (5.9), the control law is defined as Lt =

  1 (−e1 − e2 k2 + x¨2d − k1 e˙1 − b1 ζ2ψ − b2 ζ2φ ζ2θ − b4 Nt − dˆφ ), b3

(5.10)

so that V˙2 = −k1 e12 − k2 e22 − d˜φ2 ≤ 0, k1 , k2 > 0. Using a similar procedure, for ki > 0, i = 3, . . . , 12 tunable gains, the control laws for the pitch and yaw subsystems are Mt = Nt = T = Ux = Uy =

  1 (−e3 − k4 e4 + x¨3d − e˙3 k3 − b5 ζ2φ ζ2ψ + b6 ζ22φ − ζ22ψ − dˆθ ), b7 1 (−e5 − k6 e6 − e˙5 k5 − (b8 ζ2φ − b2 ζ2ψ )ζ2θ − b4 L t − dˆψ + x¨5d ), b9 m (e7 + k8 e8 − x¨7d + e˙7 k7 + g + dˆx − k7 e˙7 ), cζ1φ cζ1θ  m e9 + k10 e10 − x¨9d + dˆx + e˙9 k9 , T  m e11 + k12 e12 − x¨11d + dˆy + e˙11 k11 . T

(5.11) (5.12) (5.13) (5.14) (5.15)

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5.4.2 ITSMC Controller Design In this section, we design the Integral Terminal Sliding Mode controller for the biplane quadrotor. Let’s define the sliding surface as [33],  S = e˙I +

b

ωe˙b/a + τ e 2a−1 ,

(5.16)

and the reaching law is selected as S˙ = −∇ S − ε sign(S),

(5.17)

where 0 < b/a < 1, ω, τ, ε, ∇ > 0, and e I is a state error term. Now let us consider the altitude subsystem: ζ˙1z = ζ2z , ζ˙2z = g −

T cζ1 cζ1 + dz . m φ θ

(5.18)

Based on the block diagram given in Fig. 5.1, the error between the actual altitude and desired altitude ez = ζ1z − z d with ζ1z as the ESO estimated altitude, a positive definite function Vz = 21 Sz2 when differentiated provides

 bz V˙z = Sz ζ¨1z − z¨ d + ωz e˙zbz /az + τz ez2az −1

 bz T = Sz g − cζ1φ cζ1θ + dˆz − z¨ d + ωz e˙zbz /az +τz ezaz −1 . m

(5.19)

Using (5.19), the control law is defined as m T = ζ1φ cζ1θ

 bz 2az +bz bz /az ˆ g + dz − z¨ d + ωz e˙z + τz ez + ∇z Sz + εz sign (Sz ) , (5.20)

such that V˙z = −∇z Sz2 − εz |Sz | ≤ 0, where ∇z , εz > 0. By using the same calculation procedure, we can define Lt =

1 b3

− (b1 ζ2ψ + b2 ζ2φ )ζ2θ − b4 Nt − dˆφ + φ¨d

b /aφ

− ωφ e˙φφ

bφ 2a +bφ

− τφ eφ φ

 , ∇φ , εφ > 0,

(5.21)

5.5 Adaptive Controller Design

1 Mt = b7 − Nt =

  − b5 ζ2φ ζ2ψ + b6 ζ22φ − ζ22ψ − dˆθ + θ¨d

b /a ωθ e˙θ θ θ

1 b9

− τθ eθ

b /aψ

T Uy = m

bθ 2aθ +bθ

bψ 2a +bψ

− τψ eψ ψ

− x¨d +

 , ∇θ , εθ > 0,

(5.22)

− (b8 ζ2φ − b2 ζ2ψ )ζ2θ − b4 L t + (x¨5d − dˆψ )

− ωψ e˙ψψ T Ux = m

95

ωx e˙xbx /ax

b /a y

− y¨d + ω y e˙ y y

 , ∇ψ , εψ > 0, bx 2a x −bx

+ τx ex

by 2a y −b y

+ τy ey

(5.23)

 ˆ + ∇x Sx dx + εx sign (Sx ) , ∇x , εx > 0, (5.24)    + dˆy + ∇ y S y + ε y sign S y , ∇ y , ε y > 0. (5.25)

5.5 Adaptive Controller Design Next, we derive the adaptive backstepping and adaptive hybrid controllers. For the design of adaptive law to handle mass change during the flight. As per the dynamics of the biplane quadrotor, the position subsystem is directly affected during the mass change so we consider the position subsystem to derive the adaptive law for the BSC and hybrid controller with ESO and DO. Control laws for the attitude subsystem remain the same as derived earlier.

5.5.1 Adaptive Backstepping Controller The adaptive backstepping controller given in Fig. 5.2 handles mass change during the flight and despite wind gusts. Let us consider the position subsystem given as ⎤ ⎤ ⎡ x8 x˙7 ⎥ ⎢ x˙8 ⎥ ⎢ g − T λ∗ cζ1φ cζ1θ + dz ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x10 ⎢ x˙9 ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥, ∗ ⎢x˙10 ⎥ ⎢ −T λ U + d ⎥ x x ⎢ ⎥ ⎢ ⎥ ⎢x˙ ⎥ ⎢ ⎥ x12 ⎣ 11 ⎦ ⎣ ⎦ x˙12 −T λ∗ U y + d y ⎡

(5.26)

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

Fig. 5.2 Block Diagram of an adaptive Backstepping Controller with ESO and DO

ˆ First, we define error in where λ∗ = 1/m ∗ , m ∗ uncertain mass, and λ˜ = λ∗ − λ. altitude, x-, and y-position as e7 = ζ1z − x7d , e9 = ζ1x − x9d , e11 = ζ1 y − x11d , and the error in velocity as e8 = ζ2z − x8d , e10 = ζ2x − x10d , e12 = ζ2 y − x12d . Lyapunov positive definite function for the position subsystem is defined as Vp =

 1 ei2 , 2 i=7,9,11

(5.27)

and the time derivative is        ei e˙i = e7 ζ˙1z − x˙7d + e9 ζ˙1x − x˙9d + e11 ζ˙1 y − x11d ˙ . V˙ p = i=7,9,11

For stabilization, virtual control laws are defined as x8d = x˙7d − k7 e7 , x10d = x˙9d − k9 e9 , x12d = x˙11d − k11 e11 , so that 2 . V˙ p = e7 e8 − k7 e72 + e9 e10 − k9 e92 + e11 e12 − k11 e11

(5.28)

The next step is to enhance the V p with velocity error dynamics with error in mass and disturbances given as Vv =



1 2 1 2 1 ˜2 λ˜ + ei + dj . 2 2δ 2L b j i=8,10,12

(5.29)

5.5 Adaptive Controller Design

97

The time derivative of the above function is 1 ˜ ˙ˆ 1 djdj V˙v = e8 (ζ˙2z − x˙8d ) + e10 (ζ˙2x − x˙10d ) + e12 (ζ˙2 y − x˙12d ) − λ˜ λˆ − δb Lj = e8 (g − T λ∗ cζ1φ cζ1θ + dz − x¨7d + k7 e˙7 ) + e10 (−T λ∗ Ux + dx − x¨9d + k9 e˙9 ) 1 ˜ ˙ˆ 1 + e12 (−T λ∗ U y + d y − x¨11d + k11 e˙11 ) − λ˜ λ˙ˆ − (5.30) djdj. δb Lj The adaptive law is defined as λ˙ˆ = {−e8 (T cζ1φ cζ1θ ) − e10 (T Ux ) − e12 (T U y )}δb ,

(5.31)

and the control laws for the position subsystem are T =

1

  e7 + k8 e8 − x¨7d + e˙7 k7 + g + dˆz − k7 e˙7 ,

λˆ cζ1φ cζ1θ  1  Ux = e9 + k10 e10 − x¨9d + dˆx + e˙9 k9 , ˆ λT  1  Uy = e11 + k12 e12 − x¨11d + dˆy + e˙11 k11 λˆ T

(5.32) (5.33) (5.34)

so that for j = x, y, z and ki > 0, we have V˙v = −

12 

ki ei2 − d˜ 2j ≤ 0.

i=7

5.5.2 Adaptive Hybrid Controller Design Next, an adaptive hybrid controller handles the mass change and in-flight wind gust disturbance. Let’s consider the Eq. (5.26). The adaptive hybrid controller is shown in Fig. 5.3, where desired signals [xd yd z d ] and [ψd ] are given to the ITSMC and BSC Controller, respectively. ESO estimates [ζo ] for the attitude subsystem and [ζ p ] for the position subsystem controlled by the Adaptive ITSMC. Nonlinear disturbance observers estimate the external disturbances acting on the quadrotor biplane. BSC controller generates desired roll, pitch, and yaw moments [L T Mt Nt ], and ITSMC generates the desired thrust T for the variable-pitch propulsion system. Accordingly, the signal [U ] is given to the respective actuators. Adaptive law is designed according to the sliding surface, the roll and pitch angle’s estimated value, and the ITSMC’s output. To design an adaptive hybrid controller, the Sliding surface and reaching law are defined in Eq. (5.16) and we added suffix h for the hybrid controller:

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

Fig. 5.3 Block Diagram of an adaptive hybrid controller

 d/c

Sh = e˙h +

d

ωh e˙h + τh eh2c−1 ,

(5.35)

and reaching law is selected as we get S˙h = −∇h Sh − εh sign(Sh ),

(5.36)

where 0 < d/c < 1, ωh , τh , εh , ∇h > 0. We define error in altitude, x-, and yposition: eh 7 = ζ1z − x7d , eh 9 = ζ1x − x9d , eh 11 = ζ1 y − x11d

(5.37)

Based on the Eqs. (5.35) and (5.36), Lyapunov positive function for the position subsystem is defined as Vp =

 i=7,9,11

Si +

1 2 λ˜ . 2δh

5.6 Stability Analysis

99

By taking the time derivative of the above equation, we get λ˜ λ˙ˆ V˙ p = S7 S˙7 + S9 S˙9 + S11 S˙11 − . δh

(5.38)

By using the Eqs. (5.26), (5.35), and (5.36), the adaptive law can be designed as λ˙ˆ = {−S7 (T cζ1φ cζ1θ ) − S9 (T Ux ) − S11 (T U y )}δh

(5.39)

and control law for the position subsystem is T =

1 λˆ cζ1φ cζ1θ

dh

7

2ch +dh dh /ch g + dˆz − x¨7d + ωh 7 e˙h 77 7 + τh 7 eh 7 7 7



+ ∇h 7 Sh 7 + εh 7 sign(Sh 7 )

∇h 7 , εh 7 > 0,

(5.40)

dh

9 2ch −dh dh /ch Ux = T λˆ − x¨9d + dˆx + ωh 9 e˙h 99 9 + τh 9 eh 9 9 7

 + ∇h 9 Sh 9 + εh 9 sign(Sh 9 )

∇h 9 , εh 9 > 0,

(5.41)

dh

11 2ch −dh dh 11 /ch 11 11 11 ˆ ˆ U y = T λ − x¨11d + d y + ωh 11 e˙h 11 + τh 11 eh 11

 + ∇h 11 Sh 11 + εh 11 sign(Sh 11 )

∇h 11 .εh 11 > 0.

(5.42)

As explained earlier, with the adaptive hybrid controller, the attitude subsystem is controlled by the BSC, and so control laws will remain the same as derived earlier.

5.6 Stability Analysis In this section, we discuss stability analysis, and for that, the Lyapunov positive definite function Ve of the Extended State Observer (ESO) is deliberated and the error dynamics are defined as ⎧ es ⎪ ⎪ ⎪ 1 ⎨ e˙s1 ⎪ e˙s2 ⎪ ⎪ ⎩ e˙s3

=ζ −x = es2 − η1 es1 = es3 − η2 l f (es1 ) = −η2 l f (es1 ).

(5.43)

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

For es = [es1 , es2 , es3 ]T , we can rewrite above equation in the matrix format as ⎡

⎤ η1 −1 0 e˙s = −(es )es ,  = ⎣η2 ρ 0 −1⎦ η3 ρ 0 0,

(5.44)

where ρ = l f (es1 )/e1 > 0 and it is bounded. Then, an adequate constraint for the stability of the ESO in Eq. (5.43) is detailed by the theorem given below. Theorem 5.1 The third-order dynamics of the ESO in the given Eq. (5.43) with adopting the observer gains which satisfy ηi > 0(i = 1 . . . 3), and η1 , η2 > η3 . There exists a matrix  in which all main diagonal elements are positive and  ·  is a symmetric non-negative definite matrix such that the zero equilibrium point of the ESO is asymptotic stability [34]. Proof: Matrix  is chosen as follows: ⎡

⎤ γ11 γ12 γ13  = ⎣−γ12 γ22 γ23 ⎦ . −γ13 −γ23 γ33

(5.45)

For the analysis simplification, the elements value of the main diagonal is allocated as γ11 = 1, γ22 = γ33 = , Lyapunov function for the Eq. (5.44) is defined as Ve  Ve =

t

( · (es )es .e˙s )dT.

(5.46)

0

If the matrix  ·  is positive definite, Ve also becomes positive definite. Using (5.44) and (5.45), matrix H is translated to ⎤ h 11 −1 −γ12 H = ⎣h 21 γ12 − ⎦ , h 31 γ13 γ23 ⎡

(5.47)

h 11 = η1 + γ12 η2 ρ + γ13 η3 ρ,

(5.48)

h 21 = −γ12 η1 + η2 ρ + γ23 η3 ρ, h 31 = −γ13 η1 − γ23 η2 ρ + η3 ρ.

(5.49)

The elements of matrix H are defined as h 21 = −1, h 31 = −γ12 , h 13 = −.

(5.50)

By combining (5.48) and (5.50), we calculate the two elements of matrix  as

5.6 Stability Analysis

η2  η12 + η2 ρ + η1 η3 ρ + − (η1 + η3 ρ), η1 η2 − η3 ρ η1 η2 − η3 1  η12 + η2 ρ + η1 η3 ρ 1 = · − , ρ η1 η2 − η3 ρ η1 η2 − η3

101

γ12 =

(5.51)

γ23

(5.52)

since ηi > 0, i = 1, 2, 3 and η1 η2 > η3 , if  → 0+ so the principal minor determinants of matrix H are calculated as 

1 + η12 + η2 ρ + η1 η2 ρ − η3 ρ h 11 = η1 + η2 ρ − η1 + η2 η1 η2 − η3

η22 ρ = η1 + −  ρ(η1 η2 + η3 ) + ρ 2 η2 η3 η1 η2 − η3  2 η22 ρ 2 η1 + η2 ρ + η1 η3 ρ ≈ η1 + − η2 ρ > 0. (5.53) η1 η2 − η3 η1 η2 − η3   h 11 −1   h 21 γ12  = h 11 · γ21 − 1  

1 + η12 + η2 ρ + η1 η2 ρ η2 ρ − η1 + η2 − 1, = η1 + − σ1 η1 η3 − η3 η1 η2 − η3

 η2 η22 ρ η3 + η23 ρ ≈ η1 + −1= > 0. (5.54) η1 η2 − η3 η1 η2 − η3 η1 η2 − η3   h 11 −1 −γ12    h 12 γ12 −  = h 11 (γ12 γ23 − 2 ) − 2γ12 − γ23 − γ 3 , 12   h 31 γ13 γ23  2 η2 ρ − γ23 − (γ23 η2 ρ)3 , ≈ h 11 γ23 3

η2 1 η2 η3 − = h 11 − = > 0. (η1 η2 − η3 )2 ρ η1 η2 − η3 (η1 η2 − η3 )ρ (η1 η2 − η3 )2 ρ

Using (5.53) and (5.54), we observe that the principal minor determinants of matrix H remain positive and are symmetric positive definite, and so ∃ a  fulfilling Theorem 5.1. Replacing matrix  into (5.46), we get  Ve =

t

−((es )es )T (es )es dT 0  t  2 2     2  η1 es1 − es2 +  η2 l f es1 − es3 +  η3l f (es1 ) dT. =− 0

Based on the above equations, the time derivative is

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

2  V˙e = −(η1 es1 − es2 )2 −  η2 l f (es1 ) − es3 − (η3l f (es1 ))2 ≤ 0.

(5.55)

The above analysis depends on the central diagonal component of , and V˙e is negative semi-definite. So if Ve (es1 , es2 , es3 ) is bounded, es1 , es2 , and es3 are bounded and V˙e is also bounded. Next, we show the results of the different controllers plus ESO and with and without DO for trajectory tracking despite wind gusts.

5.7 Results and Discussions Next, we show the results of the different controllers’ pulse ESO with and without DO for trajectory tracking despite wind gusts. Figure 5.4 shows the x-axis position tracking by the BSC with ESO and BSC with ESO and DO. The response of the BSC with ESO and DO is faster with less overshoot, indicating that the disturbance observer is efficient. Figure 5.4 also shows that in y-axis tracking, the BSC with ESO has more fluctuations and higher overshoot than BSC with ESO and DO. The effect of disturbance is significant in the y-axis. Altitude tracking reveals that BSC with ESO generates a significant error than the BSC with ESO and DO, ensuring an efficient nonlinear disturbance observer. Position subsystem tracking by the ITSMC controller + ESO with and without DO is shown in Fig. 5.5. ITSMC + ESO with DO can handle wind gusts more efficiently. x-axis tracking by ITSMC with ESO and DO generates less overshoot with less settling time and the same type of response as y- and z-axis tracking. The biplane is commanded to track an autonomous trajectory with aggressive maneuver, and the responses of the BSC, ITSMC, and the Hybrid controller + ESO + DO are compared, and results are shown in Fig. 5.6. HC with ESO and DO has less overshoot than the BSC with ESO and DO and has a faster response than the ITSMC with ESO and DO while tracking

Fig. 5.4 Position subsystem tracking by BSC + ESO with (and without) DO

5.7 Results and Discussions

103

Fig. 5.5 Position tracking by ITSMC + ESO with (and without) DO

Fig. 5.6 Position tracking by different controllers with ESO and DO

the x- and y-axes. The hybrid controller with ESO and DO generates less overshoot in altitude tracking than the BSC with ESO and DO. Figure 5.7 shows a comparison between tracking the attitude of the biplane quadrotor by the Hybrid controller, BSC and ITSMC with ESO and DO. The desired roll and pitch angle are determined using Ux , U y and desired ψ angle. Roll, pitch, and yaw angle tracking by the hybrid controller with ESO and DO is better than the BSC with ESO + DO and ITSMC with ESO + DO. Simulation results reveal that the hybrid controller has the advantage of both BSC and ITSMC. Next, we compare the response of the adaptive BSC with ESO + DO and the adaptive hybrid controller with ESO and DO during in-flight mass change despite wind gusts. Initially, the net mass is 18 kg. At t = 50 s, 6 kg is dropped, and again the exact weight is dropped at t = 80 s. The response of the adaptive BSC and adaptive hybrid controller with ESO and DO for x − y-axis trajectory tracking is shown in Fig. 5.8.

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X Axis (m)

Fig. 5.7 Attitude tracking by different Controllers with ESO and DO 0.8

Adaptive HC + ESO + DO

0.6 0.4 0 6

Y Axis (m)

Desired X position Adaptive BSC + ESO + DO

20

40

60

80

100

120

100

120

5 Desired Y position Adaptive BSC + ESO + DO

4 3 0

20

40

Adaptive HC + ESO + DO

60

80

Time (seconds) Fig. 5.8 x − y-position trajectory tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO

The adaptive hybrid controller with ESO and DO produces a minor fluctuation, but a significant overshoot is generated by the adaptive BSC + ESO + DO controller in x- and y-axes tracking. There is a 0.44 m steady-state error generated by the BSC + ESO + DO. In contrast, the hybrid controller with ESO and DO more effectively tracks the y-axis trajectory despite wind gusts and the mass change with no steadystate error. Altitude tracking by the adaptive BSC and adaptive hybrid controller with ESO and DO is shown in Fig. 5.9. With BSC + ESO + DO, a change in the mass induces an error of 0.1 m. The hybrid controller tracks the desired altitude after mass changes, showing the effectiveness of the adaptive hybrid controller with ESO + DO. Attitude tracking by the adaptive controller with ESO and DO is shown in Fig. 5.10. The response of the adaptive hybrid controller with ESO and DO is better than the adaptive BSC with ESO and DO for roll and pitch angle tracking and lesser error for Yaw angle tracking.

5.8 Conclusions

105

Fig. 5.9 Altitude tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO

Fig. 5.10 Attitude tracking by the Adaptive BSC and Adaptive Hybrid controller with ESO and DO

5.8 Conclusions We design ESO to estimate the state and a Nonlinear disturbance observer (DO) to approximate wind gusts (as represented by the von Karman model). The control actions with ESO (and with and without DO) reveal that the BSC, ITSMC, and hybrid controllers with ESO and DO are more efficient in handling wind gusts faster and generate reduced tracking errors. Next, we compare the BSC, ITSMC, and Hybrid controller + ESO with and without DO. For the same trajectory, the results of these controllers on evaluation show that x-axis trajectory tracking by the BSC + ESO with DO has a faster response but a significant overshoot. In comparison, ITSMC + ESO with DO has a sluggish response, but Hybrid + ESO + DO has a faster

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5 Multi-observer Based Adaptive Controller for Hybrid UAV

response than the ITSMC + ESO + DO and less overshoot than the BSC + ESO + DO. Therefore, the hybrid + ESO + DO is the faster and most effective controller. Furthermore, ITSMC + ESO + DO has less overshoot than the other two in altitude tracking. However, attitude tracking by the hybrid controller with ESO + DO is better than BSC and ITSMC with ESO and DO. Based on the evaluation of different controllers in trajectory tracking, we infer that the ESO estimates the position, altitude, and velocity using only position and attitude signals. DO estimates the disturbance signal applied on a biplane quadrotor. A dual observer-based adaptive hybrid controller tracks the desired altitude trajectory despite crosswind and mass change. When mass changes, a small error is generated in the attitude, and a slight fluctuation happens due to external disturbances. Therefore, the proposed control architecture is adequate.

References 1. Oosedo, A., Abiko, S., Konno, A., Koizumi, T., Furui, T., Uchiyama, M.: Development of a quad rotor tail-sitter VTOL UAV without control surfaces and experimental verification. In: 2013 IEEE International Conference on Robotics and Automation, pp. 317–322 (2013) 2. Oosedo, A., Abiko, S., Konno, A., Uchiyama, M.: Optimal transition from hovering to level flight of a quadrotor tail-sitter UAV. Auton. Robot. 41(5), 1143–1159 (2016). https://doi.org/ 10.1007/s10514-016-9599-4 3. Swarnkar, S., Parwana, H., Kothari, M., Abhishek, A.: Biplane-quadrotor tail-sitter UAV: flight dynamics and control. J. Guid. Control Dyn. 41(5), 1049–1067 (2018). https://doi.org/10.2514/ 1.g003201 4. Phillips, P., Hrishikeshavan, V., Rand, O., Chopra, I.: Design and development of a scaled quadrotor biplane with variable pitch proprotors for rapid payload delivery. In: Proceedings of the American Helicopter Society 72nd Annual Forum, West Palm Beach, FL, USA, pp. 17–19 (2016) 5. Yeo, D., Hrishikeshavan, V., Chopra, I.: Gust detection and mitigation on a quad rotor biplane. In: AIAA Atmospheric Flight Mechanics Conference, p. 1531 (2016) 6. Reddinger, J.P., McIntosh, K., Zhao, D., Mishra, S.: Modeling and trajectory control of a transitioning quadrotor biplane tailsitter. In: Vertical Flight Society 75th Annual Forum, vol. 93 (2019) 7. McIntosh, K., Mishra, S.: Optimal trajectory generation for a quadrotor biplane tailsitter (2020) 8. Raj, N., Simha, A., Kothari, M., Banavar, R.N., et al.: Iterative learning based feedforward control for transition of a biplane-quadrotor tailsitter UAS. In: 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 321–327. IEEE (2020) 9. Dalwadi, N., Deb, D., Kothari, M., Ozana, S.: Disturbance observer-based backstepping control of tail-sitter UAVs. Actuators 10(6), 119 (2021). https://doi.org/10.3390/act10060119 10. Dalwadi, N., Deb, D., Muyeen, S.M.: Adaptive backstepping controller design of quadrotor biplane for payload delivery. IET Intell. Transp. Syst. (2022). https://doi.org/10.1049/itr2. 12171 11. Dalwadi, N., Deb, D., Muyeen, S.: Observer based rotor failure compensation for biplane quadrotor with slung load. Ain Shams Eng. J. 13(6), 101,748 (2022). https://doi.org/10.1016/ j.asej.2022.101748 12. Ding, F., Huang, J., Wang, Y., Zhang, J., He, S.: Sliding mode control with an extended disturbance observer for a class of underactuated system in cascaded form. Nonlinear Dyn. 90(4), 2571–2582 (2017). https://doi.org/10.1007/s11071-017-3824-3

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32. Liu, J., Gai, W., Zhang, J., Li, Y.: Nonlinear adaptive backstepping with ESO for the quadrotor trajectory tracking control in the multiple disturbances. Int. J. Control Autom. Syst. 17(11), 2754–2768 (2019). https://doi.org/10.1007/s12555-018-0909-9 33. Dalwadi, N., Deb, D., Rath, J.J.: Biplane trajectory tracking using hybrid controller based on backstepping and integral terminal sliding mode control. Drones 6(3), 58 (2022). https://doi. org/10.3390/drones6030058 34. Dou, J., Kong, X., Wen, B.: Altitude and attitude active disturbance rejection controller design of a quadrotor unmanned aerial vehicle. Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 231(9), 1732–1745 (2017)

Chapter 6

Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

The most common application of the biplane quadrotor is delivering a package connected with a cable. However, it is evident that the performance is degraded when there is a swinging slung load, and it can be considered an external disturbance. A backstepping controller with a nonlinear disturbance observer is intended to nullify its effect. The proposed nonlinear disturbance-based controller tackles the slung load’s swinging and handles the disturbance caused by the partial rotor failure and the wind gusts. Simulation is carried out using MATLAB Simulink, considering slung load swinging, partial rotor failure, and crosswind to validate the proposed control scheme. Simulation results reveal that the proposed control arrangement effectively handles the disturbances involved.

6.1 Rotor Failure in Biplane Quadrotor To control biplane quadrotors, many researchers have developed different control architectures. Swarnkar et al. [1] developed the dynamics of the biplane quadrotor, and a nonlinear controller was designed, while Hrishikeshavan et al. [2] designed the quadrotor biplane. Mission planning methods are assessed for QBiT vehicles [3] with a folding winglet system. McIntosh et al. [4] developed and simulated an optimization-based trajectory planner for an autonomous transition between hover and level flight. The biplane quadrotor is energy efficient, takes less time to accomplish a mission, and also can fly with high velocity than the conventional rotary wing quadrotor, so it will be the better choice for payload delivery. A slung load can be connected with the biplane quadrotor with cables while it operates in the quadrotor mode and after delivery switch to the fixed-wing mode to get to the origin at high speed while consuming less energy. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_6

109

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6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

Many researchers developed the aircraft dynamics with the slung. A small helicopter with slung load dynamics [5] and another one containing multi-rotors [6] are discussed. For the sensitive task deliveries with slung loads, a distributed controller [7] and a backstepping controller [8] are offered. The performance of the nonlinear and highly coupled system has been impacted negatively in the presence of external disturbances, and uncertainties [9]. Enns and Si introduced an actuator geometry to facilitate reconfiguration in helicopter flight control for the sake of accommodation of failures in main rotors [10]. SMC-based sliding mode disturbance observer is designed [11] to handle the external disturbance and model uncertainties, while a novel nonlinear disturbance-based BSC controller is presented [12]. DO-based H∞ [13] and NDO-based H∞ [14] help enhance accurate hovering despite crosswind. Recently, Dalwadi et al. [15] offered an NDO-based BSC for tail-sitter quadrotors to assure tracking despite wind gusts and a terminal sliding mode controller-based failure compensator despite failed rotors, and crosswind [16]. A DO, and ISM (Integral sliding mode) method based on FTC subject to actuator failure and external disturbances are also explored [17]. Lien et al. [18] applied a flight control technique relying upon SMC for quadrotors with a rotor failure event and included a dual observer strategy. An emergency PID-based rotor failure management strategy for quadrotors using control reallocation on the active ones is developed [19]. Rabasa et al. [20] presented an adaptive control-based failure diagnostic strategy for quadrotors under partial actuator faults [21]. Finally, Lippiello et al. proposed a BSC method [22] and PID scheme [23] to enable safe landing under propeller failures by converting to a bi-rotor configuration. Next, we present the dynamics of the biplane quadrotor (only quadrotor mode) using an NDO to approximate the disturbances caused by a slung load.

6.2 Dynamical Model of a Biplane with Slung Load Figure 6.1 shows the vehicle (with slung load) control architecture. The reference trajectory is given to a mode selection section and an appropriate signal to the BSC. We considered partial rotor failure measured using reference and actual speeds, wind gusts, and slung load swinging as the external disturbance. NDO uses the known ˆ This estimated value is given control output and states to estimate disturbance (d). to the BSC controller to generate signal U , and based on this signal, the propulsion system generates [U1 U2 U3 U4 ] for the biplane quadrotor actuators. For this study, we considered slung load a point mass [24].

6.2.1 Mathematical Modeling of Slung Load To make this model simple yet realistic, some assumptions like vehicle symmetry and rigidity, and negligible cable mass but the non-negligible force of the cable are

6.2 Dynamical Model of a Biplane with Slung Load

111

Fig. 6.1 Rotor failure compensation scheme with slung load

made. As presented in Fig. 6.2, the slung load remains below the vehicle’s lowest point √ despite swinging. The load location about z-axis is given as ζ = L 2 − r 2 − s 2 > 0, where l is length of the cable. The slung load position about x and y axes of the inertial frame are r and s respectively.

6.2.2 Dynamics of Slung Load Using Newton’s second law, the slung load dynamics are given as ⎛ ⎞ ⎛ ⎞ x¨ + r¨ dx ⎠, y¨ + s¨ d = ⎝d y ⎠ = −m l ⎝ ¨ dz z¨ + ζ − g(ζ /L)

(6.1)

where slung load mass is m l , the vehicle’s acceleration components are [x¨ y¨ z¨ ], and cable force components are [dx d y dz ]. Acceleration due to gravity g, slung load position and accelerations are [r s ζ ] and [¨r s¨ ζ¨ ]. The slung load swing hinders the translational movement but leaves the rotational movement unaffected.

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6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

Fig. 6.2 Conceptual drawing: A slung load carrying quadrotor biplane

6.2.3 Mathematical Model of Quadrotor Biplane Biplane dynamics [1] are ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ x¨ −sθ r v − qw Fax 1 ⎣ y¨ ⎦ = ⎣ Fay ⎦ + g ⎣cθ sφ ⎦ + ⎣ pw − r u ⎦ , m −T + F z¨ cθ cφ qu − pv az ⎡ ⎤ ⎡ ⎤ ¨ φ (b1 r + b2 p)q + b3 (L a + L t ) + b4 (Na + Nt ) ⎣ θ¨ ⎦ = ⎣ ⎦, b5 pr − b6 ( p 2 − r 2 ) + b7 (Ma + Mt ) ¨ (b8 p − b2 r )q + b4 (L a + L t ) + b9 (Na + Nt ) ψ

(6.2)

(6.3)

where linear as well as rotational acceleration and velocity of the biplane quadrotor ¨ [u v w] and [ p q r ], and the inertial terms are defined as are [x¨ y¨ z¨ ], [φ¨ θ¨ ψ], ⎡ ⎤ b1 ⎢b2 ⎥ ⎢ ⎥ ⎢b3 ⎥ 1 ⎢ ⎥= ⎢b4 ⎥ Ix Iz − Ix2z ⎢ ⎥ ⎣b8 ⎦ b9

⎡

⎤ (I y − Iz )Iz − Ix2z ⎢ (Ix − I y + Iz )Ix z ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ b5 (Iz − Ix ) ⎢ ⎥ 1 I z ⎢ ⎥ , ⎣b6 ⎦ = ⎣ Ix z ⎦ , ⎢ ⎥ Ix z I y ⎢ ⎥ b7 1 ⎣(Ix − I y )Ix + Ix2z ⎦ Ix

6.3 Observer-Based Controller Design

113

Aerodynamic forces and moments are [Fax Fay Faz ], [L a Ma Na ], generated thrust T and moments are [L t Mt Nt ]. The mass of the biplane quadrotor is m and gravitational force g. Based on Newton’s second law, the rotational matrix based on Euler (ψ, θ, φ) for the biplane quadrotor is given as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ x¨ 0 cφcψ sφsθ cψ − cφsψ cφsθ cψ + sφsψ 0 m ⎣ y¨ ⎦ = ⎣ 0 ⎦ + ⎣ cθ sψ sφsθ sψ + cφcψ cφsθ sψ − sφcψ ⎦ ⎣ 0 ⎦ (.6.4) z¨ mg −sθ sφcθ cφcθ −T Expanding (6.1), (6.3) and (6.4), we get T + dx ma T + dy y¨ = −U y ma gm T + + dz z¨ = −cφcθ ma ma φ¨ = (b1r + b2 p)q + b3 L t + b4 Nt θ¨ = b5 pr − b6 ( p 2 − r 2 ) + b7 Mt x¨ = −Ux

ψ¨ = (b8 p − b2 r )q + b4 L t + b9 Nt ,

(6.5) (6.6) (6.7) (6.8) (6.9) (6.10)

where Ux = cφsθ cψ + sφsψ and U y = cφsθ sψ + sφcψ, dx = −¨r mmal , d y =

/L) s s˙ )2 ¨ +˙r 2 +¨s s+˙s 2 , the combined mass with s¨ mmal and dz = g m l (ζ + mma rr + (¨r r˙ +¨ ma ζ ζ3 slung load is m + m l = m a , Slung load velocity is [˙s r˙ ζ˙ ]. To design the NDO, some assumptions like the boundedness of the disturbance and the first derivative of disturbance are bounded and slow: ||d˙ p (t)|| ≤ D p ,

||d˙o (t)|| ≤ Do t > 0.

6.3 Observer-Based Controller Design The design of the BSC based on NDO for trajectory tracking is discussed next. Quadrotor Mode No considerable aerodynamic forces and moments act during the quadrotor mode. Using (6.5)–(6.10), we have

114

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

⎤ ⎡ ⎤ x2 x˙1 ⎢ x˙2 ⎥ ⎢ (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙3 ⎥ ⎢ ⎥ x4 ⎢ ⎥ ⎢ ⎥ ⎢ x˙4 ⎥ ⎢ b5 x2 x6 − b6 (x22 − x62 ) + b7 Mt + dθ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙5 ⎥ ⎢ ⎥ x6 ⎢ ⎥ ⎢ ⎥ ⎢ x˙6 ⎥ ⎢(b8 x2 − b2 x6 )x4 + b4 L t + b9 Nt + dψ ⎥ ⎢ ⎥=⎢ ⎥. ⎢ x˙7 ⎥ ⎢ ⎥ x8 ⎢ ⎥ ⎢ ⎥ ⎢ x˙8 ⎥ ⎢ ⎥ g mma − mTa cx1 cx3 + dz ⎢ ⎥ ⎢ ⎥ ⎢ x˙9 ⎥ ⎢ ⎥ x 10 ⎢ ⎥ ⎢ ⎥ T ⎢x˙10 ⎥ ⎢ ⎥ − U + d x x ma ⎢ ⎥ ⎢ ⎥ ⎣x˙11 ⎦ ⎣ ⎦ x12 x˙12 − mTa U y + d y ⎡

(6.11)

There are three external disturbances as shown in Fig. 6.1: (i) wind gust disturbance, (ii) due to swinging load, and (iii) failed (or reduced in effectiveness) rotor condition. Among these three, disturbance due to slung load swing is active in the quadrotor mode and the other two act in all three modes. To design the BSC, we divide the above dynamics into six subsystems. The roll subsystem is expressed as x˙1 = x2 x˙2 = (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ .

(6.12)

Formulating an NDO [15] for the roll subsystem, we have estimates given as   n˙ φ = −L φ n φ + L φ x˙1 + (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt , dˆφ = n φ + L φ x2 .

(6.13)

Differentiating the estimated roll disturbance, dˆφ , we have d˙ˆφ = n˙ φ + L φ x˙2 = −L φ n φ − L φ (L φ x˙1 + (b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt ) +L φ ((b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ ) = −L φ (n φ + L φ x2 ) + L φ dφ = −L φ dˆφ + L φ dφ = −L φ d˜φ ,

(6.14)

for error d˜φ = dφ − dˆφ , and a tuneable observer gain L φ > 0, and for error in roll angle e1 = x1 − x1d and positive definite V1 = 21 e12 , we get V˙1 = e1 e˙1 = e1 (x˙1 − x˙1d ) = e1 (x2 − x˙1d ), A virtual control x2d = x˙1d − k1 e1 (for k1 > 0) is designed such that e2 = x2 − x2d = x2 − x˙1d + c1 e1 , and so we get V˙1 = e1 ((e2 + x˙1d − k1 e1 ) − x˙1d ) = e1 e2 − k1 e12 .

(6.15)

6.3 Observer-Based Controller Design

115

For e˙2 = x˙2 − x¨1d + k1 e˙1 , the Lyapunov positive definite function is given as V2 = V1 + 21 e22 . Taking the time derivative, and using (6.15), we get V˙2 = e1 e2 − k1 e12 + e2 ((b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt + dφ ).

(6.16)

Using (6.14) and (6.16), the control signal is Lt =

1 −e1 − e2 k2 + x¨2d − k1 e˙1 − (b1 x6 − b2 x2 ) x4 − b4 Nt − dˆφ , b3

(6.17)

so that V˙2 = −k1 e12 − k2 e22 ≤ 0, k1 , k2 > 0. Using the same steps, control laws for the other subsystems are Mt = Nt = T = Ux = Uy =

1 −e3 − k4 e4 + x¨3d − e˙3 k3 − b5 x2 x6 + b6 (x22 − x62 ) − dˆθ , b7

1 −e5 − k6 e6 + x¨5d − e˙5 k5 − (b8 x2 − b2 x6 )x4 − b4 L t − dˆψ , b9   ma m ˆ e7 + k8 e8 − x¨7d + e˙7 k7 + g − dz , cx1 cx3 ma

ma e9 + k10 e10 − x¨9d + e˙9 k9 − dˆx , T

ma e11 + k12 e12 − x¨11d + e˙11 k11 − dˆy , T

(6.18) (6.19) (6.20) (6.21) (6.22)

where ki > 0, i = 1, . . . , 12. The slung load is only connected in quadrotor mode. After delivery, biplane quadrotor performs transition maneuver to switch into fixedwing UAVs and fly at high velocity, consuming lesser energy and time. Transition Mode The transition maneuver permits change of mode from quadrotor to fixed-wing and vice versa. With a command to rotate about the pitch axis at ≈90◦ and sufficient aerodynamic forces, the controller switches from quadrotor to fixed-wing. With the same calculation process, and for ki > 0, i = 1, . . . , 12, the control laws for the transition mode with aerodynamic forces [Fax Fay Faz ]T and moments [L a Ma Na ]T are defined as 1 (−e1 − e2 k2 + x¨2d − k1 e˙1 − f 1 + b3 L a − b4 (Na + Nt ) − dˆφ ), b3

1 x¨3d − e3 − k4 e4 − e˙3 k3 − b5 x2 x6 + f 2 − dˆθ − Ma , Mt = b7

1 −e5 − k6 e6 + x¨5d − e˙5 k5 − f 3 − b4 (L a + L t ) − b9 Na − dˆψ , Nt = b9   m Faz T = e7 + k8 e8 − x¨7d + e˙7 k7 + g − − dˆz , cx1 cx3 m Lt =

116

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

  m Fax e9 + k10 e10 − x¨9d + e˙9 k9 − − dˆx , T m   Fay m e11 + k12 e12 − x¨11d + e˙11 k11 − − dˆy . Uy = T m Ux =

Fixed-Wing Mode During this mode, the biplane quadrotor dynamics replicates the conventional fixedwings. We need to transform the variables, and express as ⎡

⎡ ⎡ ⎤ ⎤ ⎤ ⎡ ⎤ Vx −Vz 0 0 −1 Vx ⎣Vy ⎦ = ⎣ Vy ⎦ = R QW ⎣Vy ⎦ , R QW = ⎣0 1 0 ⎦ . 10 0 Vz W Vx Q Vz Q Based on the above equations, the dynamics [25] are x˙ = cθ cψu + (sφsθ cψ − cφsψ)v + (cφsθ cψ + sφ sψ)w, y˙ = cθ sψu + (sφsθ sψ + cφcψ) v + (cφsθ sψ − sφcψ) w, z˙ = −u sθ + v sφcθ + w cφcθ, Fax T u˙ = − g sθ + pv − qu + + dx , ma ma Fay + g cθ sφ + pw − r u + d y , v˙ = m Faz + g cθ cφ + r v − qw + dz , w˙ = m p˙ = − pq(bw3 + bw9 ) − qr (bw11 − bw12 ) − bw13 (L a + L t ) + bw5 (Na + Nt ) + dφ , q˙ = bw8r 2 + bw9 p 2 + 2 bw10 pr + bw7 (Mt + Ma ) + dθ , r˙ = pq(bw1 + bw2 ) + qr (bw3 − bw4 ) + bw5 (L t + L a ) − bw6 (Nt + Na ) + dψ , φ˙ = p + q sφ tθ + r cφ tθ, θ˙ = q cφ − r sφ, sφ cφ ψ˙ = q +r , cθ cθ

(6.23)

where bw1 = Ix z (I y + Ix z )/B, bw2 = Iz2 /B, bw3 = Ix z (Ix + I y )/B, bw4 = Ix I y /B, bw5 = Ix z /B, bw6 = Iz /B, bw7 = 1/I y , bw8 = Ix /I y , bw9 = Iz /I y , bw10 = Ix z /I y , bw11 = Ix2 /B, bw12 = Ix z (I y − Ix z )/B, bw13 = Ix /B and B = Ix Iz − Ix2z . The error between actual and desired roll angle is eφ = φ − φd .

(6.24)

6.3 Observer-Based Controller Design

117

Based on the error, a Lyapunov function is defined as VF Mφ = 21 eφ2 and from the time derivative using (6.23), we get   V˙ F Mφ = eφ p + q sφtθ + r cφ tθ − φ˙ d ,

(6.25)

and the control of the roll angle rated pd is defined as pd = −kφ eφ − q sφ tθ − r cφ tθ + φ˙ d .

(6.26)

For e p = p − pd , and positive definite function defined as VF M p = VF Mφ + 21 e2p + 1 ˜2 d , taking time derivative, we get 2L φ φ V˙ F M p = e p (−qr (bw11 − bw12 ) − bw13 (L a + L t ) + bw5 (Na + Nt ))   1 ˜ ˙ˆ dφ dφ . +e p − pq (bw3 + bw9 ) + dφ − p˙ d − kφ eφ2 + eφ e p − Lφ

(6.27)

Similar to (6.14), (6.17), a new control signal is 1  k p e p − pq (bw3 + bw9 ) + qr (bw12 − bw11 ) + bw5 (Na + Nt ) bw13

(6.28) + dˆφ + eφ − p˙ d ,

Lt =

such that V˙ F M p = −kφ2 eφ2 − k p e2p − d˜φ2 which guarantees asymptotic stability for appropriate k p > 0. The remaining control laws are

−1 kq eq + bw8r 2 + bw9 p 2 + 2 pr bw10 + bw7 Ma − q˙d − eθ + dˆθ , bw7 (6.29) 1 Nt = (kr er + pq (bw1 + bw2 ) + qr (bw3 − bw4 ) + bw5 (L t + L a ) bw6 (6.30) − bw6 Na + eψ − r˙d + dˆψ ),   Fax (6.31) + g sθ − pv + qu − dˆz − eu + u˙ d . T = m a ku eu − ma

Mt =

Using (6.30) for tunable gains kq , kr and ku > 0, we rewrite (6.28) as Lt =



bw6 k p e p − pq (bw3 + bw4 ) − qr (bw11 − bw12 ) − p˙ d + eφ dˆφ 2 bw6 bw13 − bw5

bw5 + pq (bw1 + bw2 ) + qr (bw3 + bw4 ) + kr er − r˙d + eψ dˆψ 2 bw6 bw13 − bw5 −L a . (6.32)

118

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

For desired altitude z d , and desired x, y-positions xd and yd , and tunable gains k x , k y , kφ and k z , the desired pitch and yaw angles [25] are  θd = sin

−1





z˙ d − k z (z − z d )

+ tan

u 2 + (vsφ + wcφ)2   y˙d − k y (yd − y) , ψd = tan−1 x˙d + k x (xd − x)

−1



u v sφ + w cφ

 , (6.33)

Considering roll angle linearly varying with yaw angle, we have φd = kφ (ψ − ψd ) .

(6.34)

For a fixed pitch propulsion system with d, k, and l as the drag and lift coefficients, and arm length, we have ⎤ ⎡ k T ⎢ L t ⎥ ⎢−kl ⎢ ⎥=⎢ ⎣ Mt ⎦ ⎣−kl Nt −d  ⎡

k k kl kl −kl kl d −d  δ

⎤ ⎡2 ⎤ 1 k ⎢2 ⎥ −kl ⎥ 2⎥ ⎥⎢ ⎢ ⎥. −kl ⎦ ⎣23 ⎦ d 2  4

(6.35)

By taking the inverse of the δ matrix, we calculate the motor speed as ⎡

⎤ ⎡ 21 k ⎢2 ⎥ ⎢−kl ⎢ 22 ⎥ = ⎢ ⎣3 ⎦ ⎣−kl 24 −d

k kl −kl d

k kl kl −d

⎤−1 ⎡ ⎤ k T ⎢ Lt ⎥ −kl ⎥ ⎥ ⎢ ⎥. −kl ⎦ ⎣ Mt ⎦ Nt d

(6.36)

The difference between the speed of the nominal and reference model (6.36) is called partial rotor failure, which the NDO and BSC controller estimate by adjusting thrust and moments using (6.35).

6.4 Results and Discussions The vehicle with slung load under crosswind and partial failed rotor condition is simulated using MATLAB Simulink and parameters used are given in Table 6.1. For an initial position [x y z] and attitude [0.5 5 0.1 0 0 0], and take-off time 20 s, the vehicle is commanded to fly at 4 m/s up to t = 294 s, and then directed to land within t = 294 − 314 s. The position and altitude tracking during the mission is shown in Fig. 6.3. A small oscillation is generated due to the slung load swing in the y − z axis. The designed NDO-based BSC tracks the desired path effectively. Attitude tracking is provided in Fig. 6.4.

6.4 Results and Discussions

119

Table 6.1 Parameters of the biplane quadrotor Parameters Value ms−2

Z Axis (m) Y axis (m) X Axis (m)

g Mass (m) Ix x I yy Izz

9.8 12 kg 1.86 kg · m2 2.03 kg · m2 3.617 kg · m2

1000 500 0

Desired X Axis 50

15 10

Parameters

Value

Wing area (single) Aspect ratio Wing span Gap-to-chord ratio Slung load mass (m l )

0.754 m2 6.9 2.29 m 2.56 2 kg

Actual X Axis 100

150

Desired Y Axis

200

250

300

250

300

250

300

250

300

250

300

250

300

Actual Y Axis

5 60 40 20 0

0

50

100

150

200

Desired altitude 0

50

100

Actual Altitude

150

200

Time (seconds)

Yaw (rad) Pitch (rad) Roll (rad)

Fig. 6.3 Altitude and position tracking with slung load 2

Desired Roll angle

Actual Roll angle

0 -2

0

50

2 0 -2

100

150

Desired Pitch angle 0

50

0.4 0.2 0 -0.2

100

50

100

Actual Pitch angle

150

Desired Yaw angle 0

200

150

200

Actual Yaw angle

Time (seconds)

200

Fig. 6.4 Attitude tracking under slung load

Desired roll and pitch angles calculated as per x − y position have slight fluctuation, but the roll and pitch angles fluctuate more. BSC autonomously controls the yaw angle. The NDO performance is shown in Fig. 6.5 which reveals efficient tracking of the position disturbance despite swinging in the slung. Figure 6.6 shows the x − y position and altitude tracking despite wind gusts. Figure 6.7 shows the attitude tracking during the mission despite wind gusts. Due to oscillation in the position tracking, a fluctuation is generated in the roll and pitch angles, but over attitude is tracked effectively by the proposed control scheme.

120

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load X Axis

1.5 1 0.5 0 -0.5 50

100

150

200

250

300

50

100

150

200

250

300

50

100

150

200

250

300

250

300

250

300

250

300

250

300

250

300

250

300

Z Axis

Y Axis

0 1.5 1 0.5 0 -0.5 0 2 1 0 0

Time (seconds)

Z Axis (m) Y Axis (m) X Axis (m)

Fig. 6.5 Disturbance generated by slung load in position subsystem 1000

Desired X position

Actual X position

500 0 25 20 15 10 5

0

50

100

150

Desired Y position 0

50

100

200

Actual Y position

150

200

50

Desired Altitude 0

0

50

100

Actual Altitude

150

Time (seconds)

200

Yaw (rad) Pitch (rad) Roll (rad)

Fig. 6.6 Position and altitude tracking despite wind gusts 2

Desired Roll angle

Actual Roll angle

0 -2

0 2

50

100

150

Desired Pitch angle

200

Actual Pitch angle

0 -2

0 0.1 0 -0.1 -0.2 0

50

100

150

Desired Yaw angle 50

100

Actual Yaw angle 150

Time (seconds)

Fig. 6.7 Attitude tracking under wind gusts

200

200

6.4 Results and Discussions Yaw (rad) Pitch (rad) Roll (rad)

1 0.5 0 -0.5

0 1 0.5 0 -0.5 0 1 0.5 0 -0.5 0

121

50

100

150

200

250

300

50

100

150

200

250

300

50

100

150

200

250

300

Time (seconds)

Fig. 6.8 Disturbance due to the slung load under wind gusts in attitude subsystem

Z Axis

Y Axis

X Axis

10 5 0 -5 0 10 5 0 -5 0 10 5 0

50

100

150

200

250

300

50

100

150

200

250

300

0

50

100

150

200

250

300

Time (seconds)

Fig. 6.9 Disturbance due to the slung load under wind gusts in position subsystem

The performance of NDO for disturbance estimation that is generated by the wind gust is shown in Figs. 6.8 and 6.9. Figure 6.10 shows a slung load carrying biplane quadrotor’s position and altitude tracking in partial rotor failure condition. Due to partial rotor failure, a noticeable tracking error is found in the y − z axis. During landing, the altitude error is quite large, and attitude tracking is shown in Fig. 6.11. However, a small error is generated in roll and pitch angle tracking while yaw angle is effectively tracked, describing an effective NOD-based BSC. NOD’s output for altitude, position and altitude are shown in Figs. 6.12 and 6.13. Next, the simulation is performed for quadrotor and transition modes under crosswind and partially failed rotor conditions, assuming the payload is delivered.

6.4.1 Quadrotor and Transition Modes For this simulation, the biplane quadrotor is commanded to take off at t = 0 − 30 s and then hover from 30 to 50 s before transitioning for the next 3 s. Crosswind is

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load Z Axis (m) Y Axis (m) X Axis (m)

122 1000 500 0

Desired X position

0

50

0

50

Actual X position

100

150

200

250

300

250

300

250

300

15 10

Desired Y position

5 40 20 0

100

Desired Altitude 0

50

100

Actual Y position

150

200 Actual Altitude

150

Time (seconds)

200

Yaw (rad) Pitch (rad) Roll (rad)

Fig. 6.10 Position and altitude tracking under partially failed rotor condition 2

Desired Roll angle

Actual Roll angle

0 -2

0 2

50

0 -2

0

0.2 0 -0.2 -0.4

100

150

Desired Pitch angle

50

100

150

Desired Yaw angle 0

50

100

200

250

300

Actual Pitch angle

200

250

300

250

300

Actual Yaw angle

150

Time (seconds)

200

Fig. 6.11 Attitude tracking of biplane with slung load during partial rotor failure X Axis

1 0.5 0 -0.5

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

Z Axis

Y Axis

1 0.5 0 -0.5 10 5 0

Time (seconds)

Fig. 6.12 Disturbance in position subsystem with slung load and partial rotor failure

Roll

6.4 Results and Discussions 0 -5 -10

Pitch

0 10

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

5 0 0.5

Yaw

123

0

-0.5

Time (seconds)

Z Axis (m) Y Axis (m) X Axis (m)

Fig. 6.13 Disturbance generated in attitude with slung load and partial rotor failure 20 Desired X position

10

Actual X position

0 0 6 4 2 0 0

5

10

15

Desired Y position

5

10

20

25

Transition Mode Quadrotor Mode

30

35

40

45

50

30

35

40

45

50

Actual Y position

15

20

25

50 Desired Z position

0

0

5

10

15

20

25

30

Time (seconds)

35

Actual Z position

40

45

50

Fig. 6.14 Altitude and position tracking during quadrotor and transition mode

applied at t = 0 with partially failed rotor condition while transitioning, and the position tracking is shown in Fig. 6.14. While altitude is effectively tracked, the x − y axis has minor crosswind-induced errors. As explained, no control action is available in the x − y axis during the transition. Therefore, the desired NDO and BSC are effective in this system. Figure 6.15 shows the attitude tracking under the same conditions. Minor oscillations are observed in all three angles despite crosswind. At t = 50 s, the biplane is commanded to decrease pitch angle gradually. The NDObased controller tracks pitch angle but slightly fluctuates in the other two angles. Disturbance estimation by the NDO are shown in Figs. 6.16 and 6.17. The NDO effectively tracks applied disturbances.

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load Yaw (rad) Pitch (rad) Roll (rad)

124 6 4 2 0 -2

Desired Roll angle

Actual Roll angle

Transition Mode Quadrotor Mode

0 2

5

10

15

20

25

30

35

Desired Pitch angle

40

45

50

Actual Pitch angle

0 -2

0 1

5

10

15

20

25

5

10

15

20

25

30

35

40

45

50

0 Desired Yaw angle

-1

0

30

Time (seconds)

35

Actual Yaw angle

40

45

50

Fig. 6.15 Attitude tracking during quadrotor and transition mode

Transition Mode Quadrotor Mode

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

Z Axis

Y Axis

X Axis

15 10 5 0 -5 0 4 2 0 -2 0 10 5 0 0

Time (seconds)

Fig. 6.16 Disturbance by the wind gust and partial rotor failure in position subsystem 20 10 0 -10

Roll

Transition Mode Quadrotor mode

5

10

15

20

25

30

35

40

45

50

0 2 1 0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

Yaw

Pitch

0 40 20 0

Time (seconds)

Fig. 6.17 Disturbance by wind gusts and partial rotor failure in attitude subsystem

6.4 Results and Discussions

125

6.4.2 Fixed-Wing Mode with Disturbance and Partial Rotor Failure

Z Axis (m) Y Axis (m) X Axis (m)

We validate the designed NDO-based BSC through a simulation of 100 s with wind gusts throughout, but partially failed rotor condition only in the last 50 s. The biplane quadrotor is directed to fly at 15 m/s with a 30 m altitude while holding the y-axis constant. Figures 6.18 and 6.19 show the altitude, position, and attitude tracking. There is a minor variation in the y-axis while no significant change in the x − z axis, while a small error is generated in the yaw and roll angle. NDO performance for the position as well as altitude subsystem are shown in Figs. 6.20 and 6.21. The dexterity and efficiency of NDO are decent. The responses of position and attitude subsystem tracking by the NDO-based BSC are shown in Figs. 6.22 and 6.23. As explained earlier, wind gusts are applied at the beginning, while after 50 s, a partially failed rotor condition ensues. The response of position and attitude subsystem tracking by the NDO-based BSC is shown in Figs. 6.22 and 6.23. The NDO outcome in fixed-wing mode is shown in Figs. 6.24 and 6.25. Partial rotor failure results in a minor spike in elevation at t = 50 s, but tracking is attained.

1500 1000 500 0

Desired X position

0 5.5

35 30 25 20

20

30

Desired Y position

5 4.5

10

Actual X position

0

10

20

30

Desired Z position 0

10

20

30

40

50

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

Actual Y position

40

50

Actual Z position 40

50

Time (seconds)

Yaw (rad) Pitch (rad) Roll (rad)

Fig. 6.18 Position and altitude tracking in fixed-wing mode with partial rotor failure 1 0 -1

Desired Roll angle 20

Actual Roll angle

0 1 0 -1

10

30

0 1 0 -1 0

10

20

30

10

20

30

40

50

60

70

80

90

100

Desired Pitch angle

Actual Pitch angle

40

50 60 70 Desired Yaw angle

80 90 100 Actual Yaw angle

40

50

60

Time(seconds)

70

80

Fig. 6.19 Attitude tracking in the fixed-wing mode with partial rotor failure

90

100

6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load 1 0 -1

0 1

10

20

30

40

50

60

70

80

90

100

0 10 5 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

Z Axis

Y Axis

X Axis

126

0 -1

Time (seconds)

Fig. 6.20 Performance in position, altitude in fixed-wing mode while rotor fails

Roll

10 0

-10

10

20

30

40

50

60

70

80

90

100

10

20

30

40

50

60

70

80

90

100

10

20

30

40

50

60

70

80

90

100

Yaw

Pitch

0 10 5 0 -5 0 0.5 0

-0.5

0

Time (seconds)

Z Axis (m) Y Axis (m) X Axis (m)

Fig. 6.21 Performance about attitude in fixed-wing mode under rotor failure 1500 1000 500 0

Desired X position

0 15 10 5 0 40 35 30 25

10

20

Actual X position

30

Desired Y position 0

10

20

30

Desired Z position 0

10

20

30

40

50

60

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6.4 Results and Discussions 1 0 -1

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6 Partial Rotor Failure Compensation for Biplane Quadrotor with Slung Load

6.5 Conclusions The simulations are carried out under von Karman wind gust disturbance and failure in a rotor condition using NDO-based BSC in three parts: (a) vehicle with slung load, (b) Quadrotor and transition mode, and (c) fixed-wing mode. The proposed control scheme handles the disturbance applied on the vehicle with slung load system and is effective even in the transition phase. In the fixed-wing mode, minor errors in trajectory tracking are observed.

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16. Hou, Z., Lu, P., Tu, Z.: Nonsingular terminal sliding mode control for a quadrotor UAV with a total rotor failure 98, 105,716 (2020). https://doi.org/10.1016/j.ast.2020.105716 17. Li, B., Hu, Q., Yang, Y., Postolache, O.A.: Finite-time disturbance observer based integral sliding mode control for attitude stabilisation under actuator failure 13(1), 50–58 (2019). https:// doi.org/10.1049/iet-cta.2018.5477 18. Lien, Y.H., Peng, C.C., Chen, Y.H.: Adaptive observer-based fault detection and fault-tolerant control of quadrotors under rotor failure conditions 10(10), 3503 (2020). https://doi.org/10. 3390/app10103503 19. Merheb, A.R., Noura, H., Bateman, F.: Emergency control of AR drone quadrotor UAV suffering a total loss of one rotor. IEEE/ASME Trans. Mechatron. 22(2), 961–971 (2017). https:// doi.org/10.1109/TMECH.2017.2652399 20. Guzmán-Rabasa, J.A., López-Estrada, F.R., González-Contreras, B.M., Valencia-Palomo, G., Chadli, M., Pérez-Patricio, M.: Actuator fault detection and isolation on a quadrotor unmanned aerial vehicle modeled as a linear parameter-varying system 52(9–10), 1228–1239 (2019). https://doi.org/10.1177/0020294018824764 21. Sadeghzadeh, I., Mehta, A., Zhang, Y.: Fault/damage tolerant control of a quadrotor helicopter UAV using model reference adaptive control and gain-scheduled PID. American Institute of Aeronautics and Astronautics (2011). https://doi.org/10.2514/6.2011-6716 22. Lippiello, V., Ruggiero, F., Serra, D.: Emergency landing for a quadrotor in case of a propeller failure: a backstepping approach. In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4782–4788 (2014). https://doi.org/10.1109/IROS.2014.6943242 23. Lippiello, V., Ruggiero, F., Serra, D.: Emergency landing for a quadrotor in case of a propeller failure: a PID based approach. In: 2014 IEEE International Symposium on Safety, Security, and Rescue Robotics (2014), pp. 1–7 (2014). https://doi.org/10.1109/SSRR.2014.7017647 24. Zhou, X., Liu, R., Zhang, J., Zhang, X.: Stabilization of a quadrotor with uncertain suspended load using sliding mode control. American Society of Mechanical Engineers (2016). https:// doi.org/10.1115/detc2016-60060 25. Ambati, P.R., Padhi, R.: A neuro-adaptive augmented dynamic inversion design for robust auto-landing. IFAC Proc. Vol. 47(3), 12202–12207 (2014)

Chapter 7

Anti-Swing Control Structure for the Biplane Quadrotor with Slung Load

One of the core ideas behind the development of UAVs is payload delivery in remote areas. For such applications, a biplane quadrotor is a better choice than the rotarywing UAVs because after the payload delivery biplane quadrotor can fly like a fixedwing UAV and get back to the origin, which saves energy and time. This chapter proposes a novel control structure to stabilize a slung load attached to the vehicle. This control structure has two controllers, (i) the Main controller (for trajectory tracking) and (ii) the Anti-swing Controller (ASC), to stabilize the slung load swinging. There are multiple nonlinear control methods like Integral Terminal Sliding Mode Control (ITSMC), Backstepping control (BSC), and a Hybrid control (ITSMC+BSC) method-based central controller designed and the same with ASC. It is designed based on BSC and Sliding Mode Control (SMC). MATLAB Simulink used for the simulation show that offered control structure can stabilize the slung load swinging while trajectory tracking produces considerably fewer oscillations. Moreover, a Hybrid controller with BSC-based ASC has the lowest IATE (Integral Absolute Time Error). It is also observed that the slung load-carrying capacity can be increased from 2.9 to 5 kg with ASC.

7.1 Quadrotor with Slung Load UAVs have a role in surveillance and defense, agriculture, surveying, photography, e-commerce, and transport of medical items. The selection for a particular application mainly depends on endurance time, speed, altitude, and payload. For example, Fixed-wing UAVs are used in surveillance, while rotary-wing UAVs help monitor the wildlife in remote areas. A hybrid UAV-like biplane quadrotor with a long flight duration with vertical take-off, landing, and hovering capabilities, has the upper hand over the rotary-wing quadrotors for payload delivery. Biplanes used to deliver/pick © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_7

131

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7 Anti-Swing Control Structure for the Biplane …

up a payload can fly like a fixed-wing UAV after a drop or before pick up, saving time, energy, and money. Many researchers have formulated the biplane dynamics and associated efficient control methodology. Hrishikeshavan et al. [1] introduced the quadrotor biplane. Six degrees of freedom dynamics of the biplane quadrotor are presented [2], and proof-of-concept of the variable pitch biplane quadrotor is demonstrated for payload delivery [3]. Biplane quadrotor modeling approaches employ blade element theory to approve a reduced-order model of biplane quadrotor in terms of trajectory planning and tracking [4]. Consequently, strategies for mission planning are evaluated for QBiT vehicles [5] while an optimized trajectory planner is developed and simulated [6]. UAVs with slung load systems can be assumed to be the pendulum connected with a moving body, and control is quite an exciting control challenge. Many researchers have developed different algorithms for a system like cranes and UAVs. As verified by experiments, Sun et al. [7] developed an energy optimization solution for the double pendulum dynamics with a transportation crane. Slung load is connected with the biplane quadrotor via cable while in the quadrotor mode, and with swinging, the quadrotor performance gets degraded. Mathematical modeling of quadrotor UAV with slung load and control strategies to handle the UAV slung load system is developed [8, 9]. Modeling and control approaches are examined for a helicopter with slung load [10] for a small-scale helicopter [11]. El-Ferik et al. [12] developed a backstepping-based trajectory tracking control for a helicopter slung load system, and a trajectory tracking method based on the energy function is proposed [13]. L 2 –L ∞ controller is developed for the helicopter slung load system with unknown disturbances [14]. A nonlinear deterministic control relying on a cascade approach for a hovering helicopter with a slung load is proposed [15]. Researchers have developed controllers for a quadrotor with slung load system despite uncertainties. A static and dynamic mathematical model of a pendulum is placed at the top of the quadrotor UAV [16], while a 3D inverted pendulum is put on the quadrotor, and optimal stabilization law is designed for a bilinear system [17]. Backstepping controller design for a spherical pendulum mounted on a moving quadrotor [18] and for the quadrotor with slung load [19] are presented. Mathematical modeling and control of quadrotor UAV with a double pendulum is developed [20, 21], and related work is furthered by Qi et al. [22] for explored precise trajectory planning. Finally, a hybrid control approach based on the MPC and PD control for swing-free transportation of a quadrotor with a double pendulum is discussed [23]. Qian et al. [24] proposed a dynamic model based on a nonlinear path-following flight controller for multiple quadrotors carrying a slung load via cables. An RMPC method is developed for the trajectory tracking of a swarm of autonomous multicopters connected with suspended payload via wires [25]. Arab et al. [26] developed a new control and planning procedure for supportive non-uniform slung load transportation using multiple quadrotors. A hierarchical control algorithm is implemented for a pair of quadrotors carrying a slung load [27]. Further, Li et al. [28] developed a trajectory planning method based on a value function estimation algorithm for a cable

7.2 Biplane Quadrotor with Slung Load Dynamics

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suspended load transportation with three quadrotors. Different control approaches are developed to restrict the slung load’s swing. Finally, Omar [29] presented a dual controller, (i) trajectory tracking and (ii) the anti-wing (FZC), for a helicopter with a slung system despite external disturbances while a recursive equation-based filtering method is applied to approximate the swing angle of the slung load for a multi-rotor system [30]. An online anti-swinging method is proposed to eliminate the swinging of the payloads connected to quadrotors [31, 32]. In previous works, we propose trajectory control of a hybrid tail-sitter through a nonlinear observer-based backstepping controller to handle external disturbances like wind gusts [33]. For payload deliveries at multiple points using a biplane quadrotor, an adaptive backstepping controller is deployed [34], and a BSC and ITSMC-based hybrid control architecture is proposed for the designed trajectory tracking [35], and for handling a partial rotor failure in the vehicle with slung load [36]. Till now, antiwing control algorithms are available only for quadrotors. While delivering a human organ or sensitive chemicals or drugs through a biplane quadrotor, the chance of damage is thwarted by a control architecture for the vehicle with slung load stabilization. There are two control sections: (i) the primary controller for trajectory tracking and (ii) slung load stabilization with trajectory tracking. Two different control schemes (BSC & ITSMC) and the hybrid (ITSMC + BSC) are used to design the central controller, while BSC and SMC help stabilize the slung load.

7.2 Biplane Quadrotor with Slung Load Dynamics Next, we present the relevant portions of the slung dynamics with the biplane quadrotor which is presented in detail in our previous work [36]: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x¨ + r¨ Uxs dx ⎠ + ⎝U ys ⎠ , y¨ + s¨ d = ⎝d y ⎠ = −m l ⎝ dz Ts z¨ + ζ¨ − g(ζ /L)

(7.1)

where m l is slung load mass, acceleration components of biplane quadrotor are [x, ¨ y¨ z¨ ], and dx , d y , and dz are the cable force components in x, y, and z axes respectively, g is the gravitational constant, [r s ζ ] and [¨r s¨ ζ¨ ] are the position and acceleration of slung load about [x y z] axis. As per the Pythagoras theorem, √ ζ = L 2 − r 2 − s 2 is always non-negative value, and Uxs , U ys and T s are the control signals generated by the anti-swing controller to stabilize slung load. BSC and SMC control-based anti-swing controllers are designed. The slung load only affects the translational motion. Overall biplane quadrotor dynamics with slung load [36] are given as

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⎡ ⎤ ⎡ ⎤ x2 x˙1 ⎢x˙2 ⎥ ⎢(b1 x6 + b2 x2 )x4 + b3 L t + b4 Nt ⎥ ⎢ ⎥ ⎢ ⎥ ⎢x˙3 ⎥ ⎢ ⎥ x4 ⎢ ⎥=⎢ ⎥ ⎢x˙4 ⎥ ⎢ b5 x2 x6 − b6 (x 2 − x 2 ) + b7 Mt ⎥ , 2 6 ⎢ ⎥ ⎢ ⎥ ⎣x˙5 ⎦ ⎣ ⎦ x6 x˙6 (b8 x2 − b2 x6 )x4 + b4 L t + b9 Nt

(7.2)

⎤ ⎡ ⎤ x8 x˙7 mb Tm ⎢ x˙8 ⎥ ⎢g m − m cx1 cx3 + dz ⎥ t ⎢ ⎥ ⎢ t ⎥ ⎢ x˙9 ⎥ ⎢ ⎥ x10 ⎢ ⎥=⎢ ⎥, T ⎢x˙10 ⎥ ⎢ ⎥ − U + d x m t xm ⎢ ⎥ ⎢ ⎥ ⎣x˙11 ⎦ ⎣ ⎦ x12 T x˙12 − m U ym + d y

(7.3)

⎡

t

where c(·) = cos(·) and s(·) = sin(·), [L t Mt Nt ] are the moments and Tm , Uxm , U ym are total thrust and forces generated by the controller to track the altitude and position, [dx d y dz ] are the external disturbance due to the slung load acting on the vehicle, m b + m l = m t is the biplane quadrotor’s collective mass, g is the acceleration due to gravity, Ux1 = sx1 sx5 + cx1 sx3 cx5 , U y1 = −sx1 cx5 + cx1 sx3 cx5 , and inertial constants are defined as bi : ⎡ ⎤ b1 ⎢b2 ⎥ ⎢ ⎥ ⎢b3 ⎥ 1 ⎢ ⎥= ⎢b4 ⎥ Ix Iz − Ix2z ⎢ ⎥ ⎣b8 ⎦ b9

⎡

⎤ (I y − Iz )Iz − Ix2z ⎢ (Ix − I y + Iz )Ix z ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ b5 (Iz − Ix ) ⎢ ⎥ 1 Iz ⎢ ⎥ , ⎣b6 ⎦ = ⎣ Ix z ⎦ . ⎢ ⎥ Ix z Iy ⎢ ⎥ b 1 7 ⎣(Ix − I y )Ix + Ix2z ⎦ Ix

7.3 Control Architecture Next, we present a control architecture to stabilize the slung load with biplane quadrotor system for trajectory tracking as described in Fig. 7.1. The mode selector gives the desired signal to controllers based on the mode for the desired trajectory. The slung load is connected with the biplane quadrotor while it operates in the quadrotor mode, so the anti-swing controller came into action. The desired position of the slung load [rd sd ζd ] is set as [0 0 0]. Slung load controller generates the control signals [Ts Uxs U ys ] which are added along with the control signals generated by the position controller. Controller 1 (position) and Controller 2 (attitude) are either based on the backstepping controller or the ITSMC. But for the hybrid controller, Controller 1 is based on the ITSMC, and Controller 2 is based on the BSC. Controller 1 generates the signals [Tm Uxm U ym ]. Using these signals and slung load controller, [φd θd ] are generated by the desired angle generator, and based

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Fig. 7.1 Block diagram of anti-swing control scheme

on that, Controller 2 generates the roll, pitch, and yaw moments [L t Mt Nt ]. Finally, based on these signals, the biplane quadrotor propulsion system generates the desired signals for the actuators. Next, the design of slung load controller, Controller 1, and Controller 2 are discussed. First, we look at the slung load controller design.

7.3.1 Anti-Swing Controller (ASC) Design Anti-swing controller is designed based on the BSC and SMC and is augmented by the central controller for performance evaluation of the overall control architecture.

7.3.1.1

BSC-Based ASC Design

There is a chance of damage to the contents of the vehicle payload with a swinging slung load. Next, we design an ASC to stabilize the slung load using the backstepping control method. Slung load dynamics (7.1) are

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dx Uxs − x¨ + , ml ml U ys dy − y¨ + , s¨ = − ml ml dz Ts ζ ζ¨ = − − z¨ + g + . ml L ml r¨ = −

(7.4) (7.5) (7.6)

The state space representation of (7.4)–(7.6) is ⎤ ⎡ ⎤ ⎡ r2 r˙1 − mdxl − x¨ + Umxsl ⎥ ⎢r˙2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢s˙1 ⎥ ⎢ s2 ⎢ ⎥ ⎢ ⎥=⎢ , ⎢s˙2 ⎥ ⎢ − d y − y¨ + U ys ⎥ ⎥ ⎢ ⎥ ⎢ ml ml ⎥ ⎣ζ˙1 ⎦ ⎣ ⎦ ζ1 dz gζ Ts ˙ζ2 − m l − z¨ + L + m l

(7.7)

where the error between the desired slung load position and the actual position is defined as er1 = (r1 − r1d ), es1 = (s1 − s1d ), eζ1 = (ζ1 − ζ1d ). Based on these errors, and a Lyapunov function defined as VSL 1 = 21 er21 + 21 es21 + 1 2 e , the first derivative is expressed as 2 ζ1 V˙ SL 1 = er1 (˙r1 − r˙1d ) + es1 (˙s1 − s˙1d ) + eζ1 (ζ˙1 − ζ˙1d ). With virtual control laws selected as r2d = r˙1d − α1 er1 , s2d = s˙1d − α2 es1 and ζ2d = ζ˙1d − α3 eζ1 for αi > 0, i = 1, 2, 3, we get V˙ SL 1 = er1 er2 − α1 er21 + es1 es2 − α1 es21 + eζ1 eζ2 − α3 eζ21 .

(7.8)

The next step is to improve V˙ SL 1 with errors er2 = r2 − r2d , es2 = s2 − s2d and eζ2 = ζ2 − ζ2d and using the virtual control laws, determine the time-derivative of another positive definite function VSL 2 = VSL 1 + 21 er22 + 21 es22 + 21 eζ22 given as V˙ SL 2 = V˙ SL 1 + er2 e˙r2 + es2 e˙s2 + eζ2 e˙ζ2

dx Uxs = er1 er2 − α1 er21 + er2 − − x¨ + − r¨2d + α1 e˙r1 ml ml

dy U ys +es1 es2 − α1 es21 + es2 − − y¨ + − s¨2d + α2 e˙s1 ml ml

Ts dz gζ + +eζ1 eζ2 − α3 eζ21 + eζ2 − − z¨ + − ζ¨2d + α3 e˙ζ1 . (7.9) ml L ml For increased slung load-carrying capacity of the biplane quadrotor, choosing the control laws as

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dx + x¨ + r¨2d − α1 e˙r1 ), ml dy + y¨ + s¨2d − α2 e˙s1 ), U ys = m l (−es1 − α5 es2 + ml dz gζ + z¨ + ζ¨2d − α3 e˙ζ1 ), − Ts = m l (−eζ1 − α6 eζ2 + ml L

Uxs = m l (−er1 − α4 er2 +

(7.10) (7.11) (7.12)

and αi > 0, i = 1, . . . 6, (7.9) reduces to V˙ SL 2 = −α1 er21 − α4 er22 − α2 es21 − α5 es22 − α3 eζ21 − α6 eζ22 ≤ 0. 7.3.1.2

(7.13)

SMC-Based ASC Design

To design an SMC based ASC controller, first, let us define the error of the slung load position as esr = r − rd , ess = s − sd , esζ = ζ − ζd , (7.14) and the sliding surface is defined as S f = χ esi + e˙si ,

(7.15)

where i = s, r, ζ . We choose a Lyapunov stability function given as VS M = 21 S 2f . Using (7.7), the errors defined in (7.14) and the sliding surface (7.15), the control laws for the slung load are given as

  dz gζ ¨ − χζ e˙sζ − μζ sign S fζ , − Ts = m l ζd + z¨ + ml L

  dx − χr e˙sr − μr sign S fr , Uxs = m l r¨d + x¨ + ml

  dy − χs e˙ss − μs sign S fs . U ys = m l s¨d + s¨ + ml

(7.16) (7.17) (7.18)

where χi , μi > 0. i = r, s, ζ.

7.3.2 Trajectory Tracking Controller Design This section designs Controller 1 and Controller 2 based on BSC, ITSMC, and a hybrid control scheme.

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7 Anti-Swing Control Structure for the Biplane …

BSC Controller Design

Controller 1 and Controller 2 are designed based on the BSC. Using (7.3), the error in the position subsystem is defined as e7 = x7 − x7d for altitude, e9 = x9 − x9d and e11 = x11 − x11d for x−y position, and based on these errors by defining a positive 2 , and by taking time-derivative, we get definite function V P = 21 e72 + 21 e92 + 21 e11 V˙ P = e7 (x8 − x˙7d ) + e9 (x10 − x˙9d ) + e11 (x12 − x˙11d ) . To satisfy this condition, virtual controls for the altitude and x−y position are designed as x8d = x˙7d − k7 e7 , x10d = x˙9d − k9 e9 , x12d = x˙11d − k11 e11 , so that 2 . V˙ P = e7 e8 − k7 e72 + e9 e10 − k9 e92 + e11 e12 − k11 e11

(7.19)

For e8 = x8 − x8d , e10 = x10 − x10d and e12 = x12 − x12d , Lyapunov positive definite function V P is defined as 1 1 2 1 2 Va = V P + e82 + e10 + e12 , 2 2 2 by taking the first time derivative, and then using (7.3) and (7.19), we get

mb Tm V˙a = e7 e8 − k7 e72 + e8 −x¨7d + k7 e˙7 + g − cx1 cx3 + dz mt mt

T +e9 e10 − k9 e92 + e10 −x¨9d + k9 e˙9 − Uxm + dx mt

T 2 +e11 e12 − k11 e11 + e12 −x¨11d + k11 e˙11 − U ym + d y . mt

(7.20)

Using (7.20), the control laws for the altitude subsystem are defined as

mt mb e7 + k8 e8 − x¨7d + e˙7 k7 + g Tm = + dz , cx1 cx3 mt mt Uxm = (e9 + k10 e10 − x¨9d + e˙9 k9 + dx ) , T  mt  e11 + k12 e12 − x¨11d + e˙11 k11 + d y , U ym = T

(7.21) (7.22) (7.23)

such that for ki > 0, i = 7 . . . , 12, we get 2 2 − k12 e12 ≤ 0. V˙a = −k7 e72 − k8 e82 − k9 e92 − k11 e11

(7.24)

Let us consider the attitude subsystem (7.2), the error in roll, pitch, and yaw angle given as e1 = x1 − x1d , e3 = x3 − x3d and e5 = x5 − x5d , and the positive definite function Vq = 21 e12 + e32 + e52 , and time-derivative given as

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V˙q = e1 e2 + e3 e4 + e5 e6 = e1 (x2 − x˙2d ) + e3 (x4 − x˙4d ) + e5 (x6 − x˙6d ) . (7.25) By designing virtual control laws designed as x2d = x˙1d − k1 e1 , x4d = x˙3d − k3 e3 and x6d = x˙5d − k5 e5 for k1 , k2 , k3 > 0, we get V˙q = e1 e2 − k1 e12 + e3 e4 − k3 e32 + e5 e6 − k5 e52 .

(7.26)

Enhancing Vq with e2 = x2 − x2d , e4 = x4 − x4d and e6 = x6 − x6d , and a Positive definite function Vr = Vq + 21 e22 + e42 + e62 , we get the time-derivative V˙r given as V˙r = e1 e2 − k1 e12 + e2 (−x¨2d + k1 e˙1 + ((b1 x6 + b2 x2 ) x4 + b3 L t + b4 Nt ))     + e3 e4 − k3 e32 + e4 −x¨4d + k3 e˙3 + b5 x2 x6 − b6 x22 − x62 + b7 Mt + e5 e6 − k5 e52 + e6 (−x¨6d + k5 e˙5 + (b8 x2 − b2 x6 )x4 + b4 L t + b9 Nt ) , and with control laws for the roll, pitch, and yaw angle designed as 1 (−e1 − e2 k2 + x¨2d − k1 e˙1 − (b1 x6 − b2 x2 )x4 − b4 Nt ), b3 1 Mt = (−e3 − k4 e4 + x¨3d − e˙3 k3 − b5 x2 x6 + b6 (x22 − x62 )), b7 1 Nt = (−e5 − k6 e6 + x¨5d − e˙5 k5 − (b8 x2 − b2 x6 )x4 − b4 L t ), b9 Lt =

(7.27) (7.28) (7.29)

for ki > 0, i = 1 . . . , 6, we get V˙r = −

6 

ki ei2 ≤ 0.

(7.30)

i=1

Now, using (7.13), (7.24), and (7.30) to ascertain the overall closed-loop asymptotic stability, we have V˙ = V˙ SL 1 + V˙ SL 2 + V˙ p + V˙a + V˙q + V˙r ≤ 0.

7.3.2.2

(7.31)

ITSMC Controller

To design the ITSMC controller, a sliding surface is designed as  S = e˙ +

j

γ e˙ j/i + ηe 2i−1 ,

(7.32)

140

7 Anti-Swing Control Structure for the Biplane …

where 0 < j/i < 1, 0 < γ , η, and the reaching law is selected as S˙ = −λS − τ sign(S), τ, λ > 0,

(7.33)

such that the control laws for the position and attitude subsystem are

jz gm b mt j /i − z¨ d + γz e˙z z z + ηz e 2i z + jz + λz Sz + τz sign (Sz ) , (7.34) cφ cθ mt   jx mt j /i 2i − j Uxm = (7.35) −x¨d + γx e˙xx x + ηx x x + λx Sx + τxq sign (Sx ) , T   jy   mt j y /i y 2i y − j y U ym = + ηy + λ y S y + τ y sign S y , (7.36) − y¨d + γ y e˙ y T jφ 1 j /i − (b1 r + b2 p)q − b4 (Na + Nt ) − b3 L a + φ¨d − γφ e˙φφ φ − ηφ e 2iφ + jφ , Lt = b3

  − λφ Sφ − τφ sign Sφ , (7.37) Tm =

1 b7



jθ

j /i − b5 pr + b6 ( p 2 − r 2 ) − b7 Ma + θ¨d − γθ e˙θθ θ − ηθ e 2iθ + jθ

− λθ Sθ − τθ sign (Sθ ) ,

Mt =



(7.38) jψ

j /i − (b8 p − b2 r )q − b4 (L a + L t ) − b9 Na + ψ¨ d − γψ e˙ψψ ψ − ηψ e 2iψ + jψ

  − λψ Sψ − τψ sign Sψ . (7.39)

1 Nt = b9

where τiq , λ jq > 0, i, j = x, y, z, φ, θ, ψ.

7.3.2.3

Hybrid Controller

Following similar steps, the control laws for the hybrid controller are developed as

jz gm b mt − z¨ d + γz e˙zjz /i z + ηz e 2iz + jz + λz Sz + τz sign (Sz ) , cφ cθ mt

jx mt Uxm = −x¨d + γx e˙xjx /i x + ηx2i x − jx + λx Sx + τxq sign (Sx ) , T

jy   mt j y /i y 2i y − j y U ym = − y¨d + γ y e˙ y + ηy + λ y S y + τ y sign S y , T 1 L t = (−e1 − e2 k2 + x¨2d − k1 e˙1 − (b1 x6 − b2 x2 )x4 − b4 Nt ), b3 Tm =

(7.40) (7.41) (7.42) (7.43)

7.4 Results and Discussions

141

1 (−e3 − k4 e4 + x¨3d − e˙3 k3 − b5 x2 x6 + b6 (x22 − x62 )), b7 1 Nt = (−e5 − k6 e6 + x¨5d − e˙5 k5 − (b8 x2 − b2 x6 )x4 − b4 L t ). b9

Mt =

(7.44) (7.45)

where kiq , λ jq , k0 0. i, j = x, y, z, o = 1, . . . , 6.

7.4 Results and Discussions MATLAB Simulink-based simulation for the vehicle with slung load system is performed for two trajectory tracking cases: (i) with a slung load weight of 2.9 kg, and (ii) with a slung load weight of 5 kg. Table 7.1 provides the requisite parameters.

7.4.1 Case 1: 2.9 kg Slung Load Initially, the biplane quadrotor position [x y z] is [0.1 0.1 20] m and attitude [φ θ ψ] is [0 0 0], and is commanded to travel at 2 m/s velocity in x direction while holding y−z position at [0.1 20] and 0◦ yaw angle, for t = 0−100 s. Then, it hovers for the next 100 s, and at t = 200 s again, it is commanded to fly at 3 m/s velocity for the next 100 s. We consider nine possible combinations with ASC and without ASC (WASC), BSC, ITSMC, and Hybrid controller. Figure 7.2 shows the X-axis trajectory tracking of a biplane quadrotor with a slung load. A hybrid controller with BSC-based ASC takes less settling time and less overshoot, while BSC + AWSC has an ample settling time with high overshoot. Y-axis trajectory tracking of biplane quadrotor with slung load is shown in Fig. 7.3. BSC + WASC generates significant errors while BSC with SMC gives a second-worst performance. Hybrid + BSC generates a minimum error in Y-axis trajectory tracking. Similar outcomes are observed for altitude tracking as shown in Fig. 7.4. ITSMC + AWSC generates more significant errors compared to BSC + AWSC.

Table 7.1 Specification of biplane quadrotor with slung load Parameters Value Parameters g Mass (m) Ix x I yy Izz

s−2

9.8 m 12 kg 1.86 kg · m2 2.03 kg · m2 3.617 kg · m2

Wing area (single) Aspect ratio Wing Span Gap-to-chord ratio

Value 0.754 m2 6.9 2.29 m 2.56

142

7 Anti-Swing Control Structure for the Biplane …

Fig. 7.2 X-axis tracking by different nonlinear controllers with and without anti-swing controller Desired Y position BSC + WASC BSC + SMC BSC + BSC ITSMC + WASC

1.5

Y Axis (m)

1

ITSMC + SMC ITSMC + BSC Hybrid + WASC Hybrid + SMC hybrid + BSC

0.5 0

-0.5 -1

0

50

100

150

200

250

300

Time (seconds)

Fig. 7.3 Y-axis tracking by nonlinear with and without anti-swing control

When we need to track the altitude for this system precisely, a Hybrid with BSC would be the best choice among all these combinations. Roll angle tracking during the flight of the biplane quadrotor is shown in Fig. 7.5. BSC + SMC generates a constant but slight fluctuation of ±12◦ while BSC + WASC generates a much larger error in roll angle tracking. The biplane quadrotor’s desired roll and pitch angle is based on the x−y position tracking and desired yaw angle. The constant oscillations are because one encounters minimal periodic fluctuations during x−y position tracking. There are also constant oscillations in pitch angle tracking, shown in Fig. 7.6. The tracking performance of the different controllers is the same. Hybrid + BSC has the best, while BSC + WASC has the worst performance. All these controllers can track the desired yaw angle directly from the desired signal as shown in Fig. 7.7, with a minimal error in the yaw angle when the state changes.

7.4 Results and Discussions

143

Z Axis (m)

20.2

20 20

Desired Altitude BSC + WASC BSC + SMC BSC + BSC ITSMC + WASC

19.9

19.8

19.8 19.7

19.6

19.6 0

0

ITSMC + SMC ITSMC + BSC Hybrid + WASC Hybrid + SMC Hybrid + BSC

50

2

4

100

6

150

200

250

300

Time (seconds)

Fig. 7.4 Altitude tracking by nonlinear with and without anti-swing control 40

80

BSC + WASC BSC + SMC BSC + BSC ITSMC + WASC ITSMC + SMC

Roll angle (deg)

60 40

ITSMC + BSC Hybrid + WASC Hybrid + SMC Hybrid + BSC

20 0 -20 -40 200

205

210

215

220

20 0 -20 -40 0

50

100

150

200

250

300

Time (seconds)

Fig. 7.5 Roll angle tracking by nonlinear with and without anti-swing control 120 BSC + WASC BSC + SMC BSC + BSC ITSMC + AWSC ITSMC + SMC

100

Pitch angle (deg)

80 60

20

ITSMC + BSC Hybrid + WASC Hybrid + SMC Hybrid + BSC

0 -20 -40

40

200

205

210

215

220

20 0 -20 -40 -60 0

50

100

150

200

250

Time (seconds)

Fig. 7.6 Pitch angle tracking by different nonlinear with and without anti-swing controller

300

144

7 Anti-Swing Control Structure for the Biplane …

Yaw angle (deg)

0.5 BSC + WASC BSC + SMC BSC + BSC ITSMC + WASC ITSMC + SMC

ITSMC + BSC Hybrid + WASC Hybrid + SMC Hybrid + BSC

0

-0.5

0

50

100

150

200

250

300

Time (seconds)

Fig. 7.7 Yaw angle tracking by different nonlinear with and without anti-swing controller 1.5 0.2 0 -0.2 -0.4

0.2

1

0 -0.2 -0.4

0.5

r (m)

0

20

200

40

220

240

0 BSC + SMC BSC + BSC ITSMC + SMC

-0.5

-1

0

50

100

ITSMC + BSC Hybrid + SMC Hybrid + BSC

150

200

250

300

Time (seconds)

Fig. 7.8 X-axis position of slung load during the flight

We discuss slung load position during trajectory tracking. The desired slung load position is zero. Figures 7.8, 7.9, and 7.10 show the slung load position about x−y and z axis. Simulation results of the main controller with a combination of two different control methods (BSC & SMC) based ASC is shown. ITSMC + BSC generates a significant overshoot during the x-position tracking of the slung load. At the same time, the BSC + SMC generates small but constant oscillations during the slung load y-axis position tracking, and ITSMC + BSC also generates a large undershoot while tracking the z-axis position of slung load. As a result, the hybrid + BSC controller takes significantly less settling time than the other controllers while tracking the slung load position. Table 7.2 provides the ITAE values of the whole system with trajectory tracking and slung load position. These analytical data show that to stabilize the slung load, the x−y position has to adjust such that the slung load moments are zero. The ASC generates the command for stabilizing the slung load, and based on these signals,

7.4 Results and Discussions

145

0.3

s (m)

0.2

BSC + SMC BSC + BSC ITSMC + SMC

0.2

0

0.1

-0.2 100

110

120

ITSMC + BSC Hybrid + SMC Hybrid + BSC

130

0

-0.1

-0.2

0

50

100

150

200

250

300

Time (seconds)

Fig. 7.9 Y-axis position of slung load during the flight 2.1 2

2

1.98

2.05

1.98

1.96

(m)

1.94

0

5

10

15

20

1.96

25

100

105

110

115

120

2

1.95

0

BSC + SMC BSC + BSC ITSMC + SMC

ITSMC + BSC Hybrid + SMC hybrid + BSC

50

100

150

200

250

300

Time (seconds)

Fig. 7.10 Altitude of slung load during the flight

trajectory tracking is achieved. A hybrid controller with BSC-based ASC efficiently stabilizes the slung load position with the best trajectory tracking performance.

7.4.2 Case 2: 5 kg Slung Load The performance of BSC-based ASC is better than the SMC-based ASC. Next, simulation is carried out for trajectory tracking with a slung load of 5 kg using two control methods (BSC and ITSMC) and one hybrid control scheme. Again the total time for the simulation is 300 s, in which t = 0–100 s biplane is commanded to fly at 4 m/s velocity while holding y−z axis position [0.1 20] m and after that to hover for next 100 s and from t = 200 to t = 300 s, it will be commanded to fly at 3 m/s velocity in the x direction. During the flight, the desired yaw angle is 0◦ , the cable

146

7 Anti-Swing Control Structure for the Biplane …

Table 7.2 Integral time absolute error (ITAE) Controller Slung load position BSC + BSC BSC + SMC BSC + WASC ITSMC + BSC ITSMC + SMC ITSMC + WASC Hybrid + BSC Hybrid + SMC Hybrid + WASC

Desired X position BSC ITSMC Hybrid Controller

150 100

500

X Axis (m)

1.20 × 104 1.45 × 104 5347 4695 1.8 × 104 5001 1.6 × 104 1.76 × 104 1760

810 1708 3.40 × 104 1261 600 5.40 × 104 313 523 4.40 × 104

700 600

Trajectory tracking

50

400 0 0

300

10

20

30

410

460 440

200 400

420

100

400 390 100

0 0

50

100

120

200

140

150

200

210

220

250

300

Time (seconds)

Fig. 7.11 X-axis tracking by the nonlinear controller with BSC-based anti-swing controller

length is 1.5 m, and the slung load weight is 5 kg. The response of the BSC, ITSCM, and hybrid controllers with ASC are compared for trajectory tracking. Figure 7.11 shows the x-axis position tracking. The response of the hybrid controller with ASC is observed to outperform BSC and ITSMC with ASC. Figure 7.12 shows the y−z position tracking. BSC with ASC generated a large overshoot and took a long time the settling, while ITSMC and the hybrid controllers have a lesser overshoot. Figures 7.13, 7.14 and 7.15 show attitude tracking by these controllers. Roll and pitch angles are efficiently tracked by the hybrid controller, while the yaw angle is tracked correctly by all controllers. Slung load position during the trajectory tracking is shown in Fig. 7.16. The slung load stabilizes faster than ITSMC and BSC with the hybrid controller, showing higher effectiveness. ITAE of the entire system for the trajectory tracking is given in Table 7.3. This simulation-based study was carried out for (i) 2.9 kg and (ii) 5 kg slung loads. Simulation results and analytical data suggest that a hybrid controller with BSC-based ASC is more efficient for Cases (i) and (ii). A slung-load-carrying quadrotor biplane

7.4 Results and Discussions

147

Y Axis (m)

1.5 Desired Y position

1

BSC

ITSMC

Hybrid Controller

0.5 0 -0.5

Z Axis (m)

0 20.1

50

100

150

200

250

300

250

300

20 19.9

Desired Altitude

19.8 0

50

BSC

100

ITSMC

Hybrid Controller

150

200

Time (seconds)

Fig. 7.12 Y-Z axis tracking by the nonlinear controller with BSC-based anti-swing controller 15 BSC ITSMC Hybrid

10

10

0 -10 100

Roll (deg)

5

120

140

0 5

10

-5

0 0

-5

-10 -10

-15

200 0

20

0

220

240

40

50

100

150

200

250

300

Time (seconds)

Fig. 7.13 Roll angle tracking by nonlinear controller with BSC based anti-swing controller

Pitch angle (deg)

100

20

20

10

20

0

0

50 -10 0

10

20

0

-20 100

30

110

120

-20 200

130

210

220

230

0 BSC ITSMC Hybrid Controller

-50 0

50

100

150

200

250

300

Time (seconds)

Fig. 7.14 Pitch angle tracking by the nonlinear controller with BSC-based anti-swing controller

148

7 Anti-Swing Control Structure for the Biplane … 6 BSC ITSMC Hybrid Controller

Yaw angle (deg)

4 2 0 -2 -4 -6

0

50

100

150

200

250

300

Time (seconds)

r (m)

0.6 0.4 0.2 0 -0.2

s (m)

Fig. 7.15 Yaw angle tracking by the nonlinear controller with BSC-based anti-swing controller

0.06 0.04 0.02 0 -0.02 0

50

100

50

BSC

ITSMC

BSC

ITMSC

150

100

Hybrid Controller

200

150

200

250

300

Hybrid controller

250

300

1.5 1.45

BSC

ITSMC

Hybrid Controller

1.4 0

50

100

150

200

250

300

Time (seconds)

Fig. 7.16 5 kg slung load position during mission Table 7.3 Integral time absolute error (ITAE) for trajectory tracking Controller ITAE Controller ITSMC + AWSC ITSMC + BSC BSC + WASC

5014 4695 6874

BSC + BSC Hybrid + WASC Hybrid + BSC

ITAE 5386 4685 3353

increases power consumption for two main reasons: the added weight requires the rotors to provide additional thrust to ensure a specific altitude, causing reduced flight time. In addition, the position controller’s demand for a quick adjustment from the attitude subsystem caused increased thrust requirement from the rotors. Therefore, a slung load swing-free controller aims to increase flight time by damping the loadintroduced oscillations.

References

149

7.5 Conclusions In this simulation study, control architecture is designed to stabilize a biplane quadrotor’s slung load for trajectory tracking. BSC, ITSMC, and hybrid control schemes augmented with SMC and BSC-based anti-swing controllers are evaluated. Simulations are carried out for two trajectory tracking cases with two slung load weights (in Kg): (i) 2.9 and (ii) 5. A Hybrid controller with BSC-based ASC is the most efficient among all contenders. Without ASC, the biplane quadrotor handles swinging of 2.9 kg slung load with 2 m cable length, but with ASC, this capacity is enhanced up to 5 kg weight and 1.5 m cable length. In addition, the position of the slung load is effectively stabilized by the ASC. Also, the hybrid controller tracks the trajectory most efficiently. Both anti-swing and central controllers are synchronized so that the Hybrid + BSC control architecture follows the desired trajectory while effectively stabilizing the slung load.

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Chapter 8

Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

It is challenging to control a UAV during the actuator failure because it is a highly nonlinear, coupled, underactuated system. Partial UAV control is achievable by reallocating the control signals, but control over one state has to be compromised, which would be the yaw angle for the quadrotor UAVs. In a biplane quadrotor yaw and roll, angles are coupled, so in this chapter, we propose a redundancy in the form of a supplementary deflecting surface connected to both wings. After successfully detecting the failure in the biplane quadrotor, it switches to fixed-wing mode. Using the three rotors and deflecting surface, it will navigate to the same area and switch again into quadrotor mode before landing.

8.1 Rotor Failures Nowadays, Unmanned Aerial vehicles are used in all sectors that directly impact human life, like surveillance and monitoring, defense and rescue mission, emergency medical things delivery, payload delivery in flood areas, etc. While in operation, it may be possible that there is a total rotor failure that occurs in the biplane quadrotor. In this situation, the first aim should be to protect the biplane from damage, navigate it into a safe area, and then land. There are mainly four categories of UAVs: (i) Fixed-wing UAVs, (ii) Rotary-wing UAVs, (iii) Flapper wing UAVs, and (iv) Hybrid UAVs. A biplane quadrotor (Hybrid UAV) can land, take off, and hover similar to a rotary-wing but also flies like a conventional fixed-wing, thereby getting significant attention from researchers [1]. Biplane quadrotor [2] with 6-DoF (degree of freedom) dynamical model is introduced. Mission scheduling methods are assessed for QBiT vehicles [3]. Dalwadi et al. [4] designed a backstepping controller with a nonlinear disturbance observer for a tail sitter quadrotor’s trajectory tracking. An adaptive controller for payload delivery is introduced [5] based on © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_8

153

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8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

backstepping control and ITSMC (Integral Terminal Sliding Mode Control), and a hybrid controller is developed for a biplane quadrotor’s trajectory tracking. Numerous control strategies have been developed to handle rotor failure in rotary wing UAVs. Zhiwei et al. [6] proposed a Nonsingular Terminal Sliding Mode Control (NTSMC) for a quadrotor UAV with total rotor failure where the flight controller reconfigures when a fault occurs and also incorporates an approximation method for external disturbance and model nonlinearities, while introducing an emergency controller [7] for quadrotors facing loss of one actuator. Fault-tolerant method for a coaxial octocopter regarding rotor failure based on error detection, isolation, and system recovery is introduced [8]. However, motor speed and current measurements based on active fault-tolerant control for an octorotor are presented [9], and the nonlinear model predictive control method with an active fault-tolerant controller [10] designed for a Y6 coaxial tricopter with actuator constraints and one rotor failure [11]. Robust feedback linearization control [12] for the rotor failure is done by sacrificing yaw state controllability. For the actuator redundancy management, active fault tolerance (Re-configurable control) and passive fault tolerance control (Robust control) methods are proposed for UAV actuator failure in [13]. Guo et al. [14] introduced operational control of UAV where the rotor is forced down and resumed at a specific location based on the balance and control necessities. Partial or total rotor actuator failure for the quadrotor UAV based on the robust linear parameter varying observer and analyzing the displacements of the roll and pitch angles [15], while Michieletto et al. [16] provided a set of new universal algebraic situations to assurance static hover for any multi-rotor with any number of commonly oriented rotors and a Fault-tolerant control created on the gain scheduling control technique in the framework of the structure of H∞ . Dalwadi et al. developed an observercentric partial rotor failure compensator using the nonlinear disturbance observebased backstepping controller [17]. Many control methods handle actuator failure in fixed-wing UAVs. Yun et al. [18] proposed a novel control distribution strategy to actuate the flaps at the same speed as other control surfaces while providing a supplementary way of varying lift directly. Nonlinear aircraft dynamics with the presence of wind is proposed [19] and based on this sliding mode controller is developed. Venkataraman et al. [20] developed a faulttolerant controller for UAVs armed with dual aerodynamic control surfaces plus one tractor-type propeller. Jana et al. [21] offered a novel technique for the fixed-wing biplane micro-air vehicle that includes the load carrying capacity and steadiness, as well as onboard processing obligatory for the vision-supported independent navigation and a finite time fault-tolerant control (FTC) based on the frictional order backstepping method for a fixed-wing UAV with actuator faults and input saturation. Event-triggered based FTC is designed [22] for multiple fixed-wing UAVs with six DOF in the presence of the actuator fault. Till now, significant development has not happened for handling rotor failure in hybrid UAVs. This chapter proposes a redundancy-based novel concept to handle total rotor failure. When failure is detected, the control allocation system is updated to maintain altitude and controllable pitch angle. One deflecting surface in each wing is activated only when a failure occurs to help navigate the vehicle to a safe destination for

8.2 Biplane Dynamics and Control Allocation

155

landing. Such deflecting surfaces allow pitch control such that the biplane quadrotor is controllable even if a rotor stops working. A hybrid UAV (e.g., biplane quadrotor) is an energy-efficient multi-role UAV that can replace conventional quadrotors and fixed-wing UAVs in many applications. The backstepping approach is used for the controller design, and signals generated by the controller are given to the control allocation block—a mathematical structure based on which we calculate the rotor speed (rotation per min) using basic arithmetic operations.

8.2 Biplane Dynamics and Control Allocation This section discusses the biplane quadrotor’s dynamics and control allocation with and without rotor failure. Biplane quadrotor dynamics are ⎤ ⎡ ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ r v − qw −sθ Fax x¨ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ y¨ ⎦ = ⎣ Fay ⎦ + g ⎣cθ sφ ⎦ + ⎣ pw − r u ⎦ , m qu − pv cθ cφ z¨ −T + Faz ⎡ ¨⎤ ⎡ ⎤ φ (b1 r + b2 p)q + b3 (L a + L t ) + b4 (Na + Nt ) ⎢ ¨⎥ ⎢ ⎥ b5 pr − b6 ( p 2 − r 2 ) + b7 (Ma + Mt ) ⎣θ ⎦ = ⎣ ⎦, (b8 p − b2 r )q + b4 (L a + L t ) + b9 (Na + Nt ) ψ¨

(8.1)

(8.2)

where s(·) = sin(·) and c(·) = cos(·), [x y z], [u, v, w] and [x¨ y¨ z¨ ] are the position, ¨ and angle, angular linear velocity and acceleration and [φ θ ψ], [ p q r ] and [φ¨ θ¨ ψ] velocity and acceleration, m is the mass of the biplane, T is the thrust generated by four motors, the moments and forces are √ [L t Mt Nt ], [L a Ma Na ] are [Fax Fay Faz ] respectively, velocity is given as V = u 2 + v2 + w2 and bi is defined as ⎡ ⎤ b1 ⎢ ⎥ ⎢b2 ⎥ ⎢ ⎥ ⎢b3 ⎥ 1 ⎢ ⎥ ⎢ ⎥= ⎢b4 ⎥ Ix Iz − Ix2z ⎢ ⎥ ⎢ ⎥ ⎣b8 ⎦ b9

⎡

(I y − Iz )Iz − Ix2z

⎤

⎢ ⎥ ⎢ (Ix − I y + Iz )Ix z ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ b5 (Iz − Ix ) ⎢ ⎥ Iz 1 ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ , ⎣b6 ⎦ = ⎣ Ix z ⎦ . ⎢ ⎥ Iy I x z ⎢ ⎥ b7 1 ⎢ ⎥ ⎣(Ix − I y )Ix + Ix2z ⎦

(8.3)

Ix

Control allocation sans rotor failure, considering kb , db as the motor parameters, lb as the motor distance from center of mass, and i as ith motor speed, is given as

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8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

Fig. 8.1 Animated picture of biplane quadrotor with deflecting surface

⎡

T

⎡

⎤

kb

kb

kb

⎤⎡

21

kb

⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ L t ⎥ ⎢ kb lb −kb lb −kb lb kb lb ⎥ ⎢22 ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ M ⎥ ⎢ k l k l −k l −k l ⎥ ⎢2 ⎥ . b b b b⎦ ⎣ 3⎦ ⎣ t⎦ ⎣ b b b b Nt −db db −db db 24

(8.4)

Suppose Motor-1 stops working during the flight with the biplane quadrotor in the quadrotor mode, then (8.4) is no longer feasible, and so one has to compromise and let go one of the four signals, thereby leading to ⎡

T

⎤

⎡

kb

kb

kb

⎤⎡

22

⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ L t ⎦ = ⎣kb lb −kb lb kb lb ⎦ ⎣23 ⎦ . Nt db −db db 24

(8.5)

Such a configuration is valid only for one motor failure. If more than one rotor fails at a time, the biplane quadrotor will become uncontrollable. Here we choose pitch angle since (8.2) indicates that roll and yaw angles are coupled. During rotor failure in a biplane quadrotor, the priority is to ensure that there is no damage to the UAV. The cost of the biplane quadrotor type hybrid UAVs is relatively high, so it is beneficial to have built-in redundancy. We suggest a deflecting surface [23–25] attached to a biplane quadrotor’s wings as shown in Fig. 8.1. The main objective of this additional structure in the wings is to provide good pitching moments under rotor failure to prevent damage to the UAV and enable a safe landing. The rotating angle of the deflecting surface lies within [−20◦ , 20◦ ]. Figure 8.2 shows the rotor failure in the biplane quadrotor. An operational flow diagram is shown in Fig. 8.3. Without rotor failure, the biplane quadrotor takes off in quadrotor mode and switches to fixed-wing mode after the transition maneuver. Before reaching the destination, it again performs the changeover maneuver and converts to the quadrotor mode to land. For a quadrotor mode failure, after detecting the rotor failure in the

8.2 Biplane Dynamics and Control Allocation

Fig. 8.2 Rotor failure in biplane quadrotor

Fig. 8.3 Flow diagram

157

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8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

first instance, the vehicle is commanded to reallocate control signals while leaving the yaw angle free. Then it executes the transition maneuver and changes to the fixed wing mode where with control allocation, the remaining three motors control the yaw and roll angle. In contrast, the deflecting surface controls the pitch angle. Under rotor failure, the priority is to prevent damage to the UAV and ensure a safe landing. With this additional structure, the biplane can fly in fixed-wing mode and navigate to a safe area. While reaching the safe area, the biplane is commanded to decrease its altitude gradually. With an altitude of around 4 m, it switches to quadrotor mode from fixed-wing mode. Note that biplane velocity is around 5 m/s in the fixed-wing mode. When in the quadrotor mode, the control signal reallocates so that the pitch angle is free before landing. This minor modification allows a journey to the safe zone during the rotor failure and land without damage. Force (F) and moments (M) generated by the deflecting surface are F=

AρV 2 sin δ , 2

M = 2 F L,

(8.6)

where L is the distance from the center of gravity, A is an area of the wing, ρ is the air density, δ is the angle of the deflecting surface, and the V is the velocity. For instance, if the vehicle speed is 20 m/s and the moment generated sans actuator failure is −7.56 N.m, and L = 0.11 m, the angle of the deflecting surface is 10◦ . Using this force and moments, we can control the biplane and navigate it to a safe place before transitioning to the quadrotor mode.

8.3 Controller Design Next, the controller is designed by using the backstepping control method for all the possible modes of the biplane quadrotor.

8.3.1 Quadrotor Mode In the quadrotor mode of the biplane quadrotor, there are no considerable aerodynamic forces and moments produced by the wings so its dynamics can be considered as the convectional quadrotor [4], and that is given as  T  T  T m x¨ y¨ z¨ = 0 0 mg + R 0 0 −T .

(8.7)

where R is the rotational matrix [4]. Block diagram of the backstepping-based controller design for the quadrotor mode is shown in Fig. 8.4, where desired position and yaw angle are given to the position and attitude controller. Desired thrust T and moments [L t Mt Nt ] are generated by the position and attitude controller. Desired

8.3 Controller Design

159

Fig. 8.4 Control strategy for quadrotor mode and transition mode

pitch (θd ) and roll (φd ) angle is generate based on the x−y position control signals. The control allocation block generates the signals [U1 U2 U3 U4 ] according to the thrust and moments generated by the backstepping controller. Using (8.2), the pitch subsystem is given as θ˙ = q



  q˙ = b5 pr − b6 p 2 − r 2 + b7 Mt .

(8.8)

There are no aerodynamic moments generated during the quadrotor mode. Let’s define the pitch angle error as eθ = θ − θd , a Lyapunov function and derivative as 1 2 e , 2 θ V˙θ = eθ e˙θ = eθ ( p − θ˙d ) = eθ e p − K θ eθ2 .

Vθ =

(8.9)

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8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

A virtual control input is designed as qd = θ˙d − kθ eθ . Now to improve the Vθ with quadratic term in eq , for that positive definite function is design Vq = 21 eq2 + Vθ and it’s time derivative and using (8.9) and virtual control laws is V˙q = eq e˙q + eθ eq − kθ eθ2 , = eq (b5 pr − b6 ( p 2 − r 2 ) + b7 Mt ) − θ¨d + kθ e˙θ ) + eθ eq − kθ eθ2 . (8.10) Based on (8.10), the control law for the pitch subsystem is designed as Mt =

1 (−eθ − kq eq − kθ e˙θ + θ¨d − b5 pr + b6 ( p 2 − r 2 )), b7

(8.11)

so that V˙q ≤ 0 where kθ , kq > 0. Similarly, other control laws are Lt = Nt = T = ux = uy =

 1

−eφ − k p e p + φ¨ d − kφ e˙φ − b1rq − b2 pq − b4 Nt , b3  1

ψ¨ d − eψ − kr er − kψ e˙ψ − (b8 p − b2 r ) q − b4 L t , b9 m (ez + ew kw − z¨ d + k z e˙z + g) , cφ cθ m (ex + ku eu − x¨d + k x e˙x ) , T  m

e y + kv ev − y¨d + k y e˙ y . T

(8.12) (8.13) (8.14) (8.15) (8.16)

where ki > 0, i = φ, ψ, p, r, u, v, z, p, r . Now using (8.15)–(8.16), the desired roll and pitch angle calculate as φd = ar csin

ux − u y sψ + cψ

, θd = ar csin

u x − u y − sφ(sψ − cψ) . (8.17) 2cφcψ

8.3.2 Transition Mode In the transition mode, the vehicle is ordered to rotate about pitch angle ≈90◦ gradually while roll and yaw angles are set to 0◦ and holding a constant altitude. It can be observed from Eq. (8.17) that desired roll angle is calculated based on the position control laws so in the transition mode there is no control over the x−y position. Aerodynamic forces and moments are considered to design the control laws as

8.3 Controller Design

161

Lt =

1 (−eφ − k p e p + φ¨d − kφ e˙φ − b1 rq − b2 pq − b3 L a − b4 (Nt + Na )), b3

Mt =

 1

b5 θ¨d − eθ − kq eq − kθ e˙θ + b6 p 2 − r 2 − Ma − pr, b7 b7

1 (−eψ − kr er + ψ¨ d − kψ e˙ψ − (b8 p − b2 r )q − b4 (L t + L a ) − b9 Na ), b9

m Faz T = ez + ew kw − z¨ d + k z e˙z + g − . (8.18) cφ cθ m

Nt =

8.3.3 Fixed Wing Mode During fixed-wing mode, a biplane quadrotor is considered as conventional fixedwing UAV. Axis rotation between quadrotor and fixed-wing modes [4] is ⎡ ⎤ ⎡ ⎤⎡ ⎤ x 0 0 −1 x ⎣ y ⎦ = ⎣0 1 0 ⎦ ⎣ y ⎦ . z W 10 0 z Q

(8.19)

Using (8.19), the inertial matrix (8.3) also changes, and so the fixed-wing mode dynamics for bwi , i = 1, . . . , 12, are given as T + Fax , (8.20) m Fay y¨ = pw − r u + g cθ sφ + , m Faz z¨ = r v − qw + g cθ cφ + , m φ¨ = bw5 (Na + Nt ) − bw13 (L a + L t ) − qr (bw11 − bw12 ) − pq(bw3 + bw9 ), x¨ = pv − qu − g sθ +

θ¨ = bw7 (Mt + Ma ) + bw8 r 2 + bw9 p 2 + 2 bw10 pr, ψ¨ = bw5 (L t + L a ) − bw6 (Nt + Na ) + qr (bw3 − bw4 ) + pq(bw1 + bw2 ). The controller for fixed-wing mode is described in Fig. 8.5 where the desired angle and velocity are given to the velocity controller and desired signal generator block. The signal generator block generates desired roll, pitch, and yaw angle and the attitude controller generated the required moments [L t Mt Nt ] based on it while the velocity controller generates the required thrust T . Control allocation block generates the signals [U1 U2 U3 U4 ] based on the thrust and moments and gives it to the biplane quadrotor actuators.

162

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

Fig. 8.5 Control strategy for fixed wing mode

Now again using same calculation process, the pitch angle error is e f θ = θ − θd and Lyapunov function and derivative are 1 2 e , 2 θ = eθ e˙θ = eθ ( p − θ˙d ) = eθ e p − k f θ eθ2 .

Vfθ = V˙ f θ

(8.21)

A virtual control input is chosen as qd = θ˙d − k f θ e f θ . Now to improve the V f θ with quadratic term in e f q , for that positive definite function is design V f q = 21 e2f q + V f θ and time derivative using (8.21) and virtual control laws are V˙ f q = e f q e˙ f q + e f θ e f q − k f θ e2f θ ,

 V˙q = eq bw8r 2 + bw9 p 2 + 2 bw10 pr + bw7 (Mt + Ma ) + e f θ e f q − k f θ e2f θ . (8.22) Based on the above equation, control is chosen as Mt =

1 (θ¨d − e f θ − k f q e f q − k f θ e˙ f θ − bw8r 2 − bw9 p 2 − 2 pr bw10 − bw7 Ma ), bw7 (8.23)

8.4 Results and Discussions

163

such that V f q ≤ 0 where k f θ , k f q > 0. Now based on the same calculation process, the Control signals are T = m(−e f x − k f u e f u − k f x e˙ f x + x¨d −

Fax + g sin θ − pv + qu), m

(8.24)

1 (e f φ + k f p e f p + k f φ e˙ f φ − φ¨d + bw5 (Na + Nt ) bw13 + qr (bw12 − bw11 ) − pq(bw3 + bw9 )), 1 Nt = (−e f ψ + k f r e f r + k f ψ e˙ f ψ − ψ¨ d + bw5 (L t + L a ) − bw6 Na bw6 + qr (bw3 − bw4 ) + pq(bw1 + bw2 )). Lt =

(8.25)

(8.26)

required yaw and pitch angle is calculated as equation gives in [4] as  θd = ar csin



z˙ d − cz (z − z d )



+ ar ctan

u v sφ + w cφ

u 2 + (v sφ + w cφ)2

y˙d − c y (yd − y) , φd = cφ (ψ − ψd ) . ψd = ar ctan x˙d + k x (xd − x)

, (8.27)

where z d , xd and yd are the desired positions, and x and y are the quadrotor positions, and cz , c y and cφ are tunable gains. Control allocation with and without rotor failure are performed.

8.4 Results and Discussions Simulation is carried out with simulation parameters of biplane quadrotor with deflecting surface given in Table 8.1. The biplane quadrotor is commanded to take off with 5 m/s velocity during t = 0−20 s (an altitude of 100 m) before being in hover state. Total rotor failure

Table 8.1 Parameters of biplane quadrotor with deflecting surface Parameters Value Parameters g Mass (m) Ix x I yy Izz

ms−2

9.8 12 kg 1.86 kg · m2 2.03 kg · m2 3.617 kg · m2

Wing area (single) Aspect ratio Wing span Gap-to-chord ratio slung load mass (ml )

Value 0.754m2 6.9 2.29 m 2.56 2 kg

X Axis (m)

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor 20

Y Axis (m)

164

0

10

Desired x position

0

Z Axis (m)

Actual x position

0 5

10

15

-2

20

25

Desired y position

30 Actual y position

-4 0 100

5

10

15

20

50 0

25

Desired altitude

0

5

10

15

20

25

30

Actual altitude

30

Time (seconds)

Fig. 8.6 Position and altitude tracking during quadrotor and transition mode

is introduced into the system at t = 30 s, and the after biplane control allocation is reconfigured to handle the failure. It will rotate about the pitch axis with constant angular velocity for the next 3 s, and the controller will switch to the fixed-wing controller. Figure 8.6 shows the position and altitude tracking of the biplane quadrotor. When total rotor failure occurs due to the new control allocation scheme, only a 1 m error is generated in the altitude. In contrast, performing the transition maneuver, as per the new control allocation, only altitude, roll, and pitch angle are controlled and yaw angle is compromised due to rotor failure, resulting in a 16 m error in the x-direction and 3 m error in the y-direction. Attitude tracking is shown in Fig. 8.7. After the total rotor failure, the pitch angle is controlled effectively. Still, an error is generated in the roll angle due to the new control allocation scheme where only pitch and roll angle are controlled. The roll and yaw angles of the biplane quadrotor are coupled, resulting in a 4◦ error generated in the roll angle. Figure 8.8 shows the generated thrust and moments, and Fig. 8.9 depicts the speed of the BLDC motor. Under total rotor failure, the speed of Motor-1 momentarily becomes zero, and due to the reallocation of the control signal, the speed of the remaining three motors is in their physical bounds while performing the transition maneuver. As a result, the speed of Motor-3 becomes zero. After the transition maneuver, the biplane quadrotor switches to the fixed wing mode. One deflecting surface is added to handle the rotor failure by providing redundancy against fault/failure. The main objective is to navigate to a safe place that may be far from its current position using the three working rotors and one surface. Position tracking with one rotor failure and deflecting surface is shown in Fig. 8.10. The biplane quadrotor slowly descends at 2 m/s about the z-axis and 7 m/s about the x-axis, and almost zero velocity about the y-axis. When the altitude is 6 m, then for the next 2 s, the biplane is commanded to reduce 1 m/s about the z-axis and 5 m/s about the x-axis and transition to quadrotor mode 4 m away from the landing surface.

Roll (deg)

8.4 Results and Discussions 0 Desired roll angle Actual roll angle

-2 -4 0

5

10

15

20

25

30

0 Desired pitch angle Actual pitch angle

-50 -100

Yaw (deg)

Pitch (deg)

165

0

5

10

15

20

25

30

0 Desired yaw angle Actual yaw angle

-1 -2 0

5

10

15

20

25

30

Time (seconds)

Fig. 8.7 Attitude tracking during quadrotor and transition mode

T (N)

400 200

Lt (N-m)

0

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

20 0

Mt (N-m)

-20 20 0 -20 -40 -60

Time (seconds)

Fig. 8.8 Trust and moments generated during the quadrotor and transition mode 6000

6000

5000

4000 2000

rmp

4000

0

3000

30

30.5

31

31.5

32

32.5

33

2000 1000 M1

M2

M3

M4

0 0

5

10

15

20

Time (seconds)

Fig. 8.9 Motor speeds during the quadrotor and transition modes

25

30

166

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

X Axis (m)

400 200 Desired x position

Actual x position

0

Y Axis (m)

0

5

10

15

20

25

Z Axis (m)

0 -1 -2 -3

30

35

Desired y position

0 100

5

10

15

20

25

30

40

Actual y position

35

40

Desired alitutde

50

45

45

Actual altitude

0 0

5

10

15

20

25

30

35

40

45

Time (seconds)

Pitch (deg)

Roll (deg)

Fig. 8.10 Position and altitude tracking during fixed wing and transition modes 20 Desired roll angle

10 -10 0 100

5

10

15

20

25

Desired pitch angle

30

35

40

45

40

45

40

45

Actual pitch angle

0

-100

Yaw (deg)

Actual roll angle

0

0 20 0 -20 -40

5

0

5

10

15

20

25

Desired yaw angle

10

15

20

30

35

Actual yaw angle

25

30

35

Time (seconds)

Fig. 8.11 Attitude tracking during the fixed wing mode and transition mode

The quadrotor mode controller handles the error generated in the last two seconds. Figure 8.11 shows attitude tracking during the fixed-wing mode. Initially, an error is seen in all three angles because of x−y position error during the transition. However, when the biplane reaches a safe place, the command to rotate about the pitch axis (≈ 90◦ ) is executed using the deflecting surface and the three rotors. The simulation results show that the deflecting surface helps in this maneuver. Figure 8.12 shows the speed of three motors. At the initial time, all three motors’ speeds are around 5500 because of an error generated in the biplane’s position during the transition. However, when the biplane again performs the transition maneuver, the motor speed increases, and between these two transitions, the speed of Motor-2 the one beside the failed motor is low. Figure 8.13 shows the Moments which indicate the effectiveness of the deflecting surface.

8.4 Results and Discussions

167

6000 M2

M3

M4

5000

rpm

4000 3000 2000 1000 0

0

5

10

15

20

25

30

35

40

45

50

Time (seconds)

Lt (N-m)

Fig. 8.12 Motor speed during the fixed wing mode and transition mode 5 0 -5

Nt (N-m)

Mt (N-m)

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

2 0 -2 5 0 -5

Time (seconds)

Fig. 8.13 Moments generated during the fixed wing mode and transition mode

Thrust, angle of the surface, and vehicle velocity during the fixed wing mode are shown in Fig. 8.14. Due to the deflecting surface, the motor is reacquired to generate low thrust around 11 N. and with 7 m/s velocity and with −15◦ deflecting surface biplane is able to travel safe zone and altitude that it can again switch into quadrotor mode and then finally land without damage. Figure 8.15 shows the position tracking and Fig. 8.16 shows the attitude tracking during the quadrotor mode of the biplane where it is going to land. While switching to quadrotor mode again, it generated a 1.5 m error in the altitude and it is taken care of by the quadrotor mode controller. Here note that roll, pitch, and yaw angle remain constant during the landing.

168

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

T (N)

400 200 0 0

5

10

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Fig. 8.14 Thrust, surface angle, and velocity during fixed-wing and transition modes

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Fig. 8.15 Position tracking of the biplane quadrotor during quadrotor mode

While landing, only two motors operate, and the other two on the opposite side remain zero such that attitude is controlled effectively, as shown in Fig. 8.17. Figure 8.18 describes the moments and thrust obtained from quadrotor Mode-2. Again, no moments are required because the attitude remains constant during landing.

8.4 Results and Discussions Roll (deg)

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Fig. 8.16 Attitude tracking of biplane quadrotor during the quadrotor mode 5000 4000

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Fig. 8.18 Thrust and moments generated during the quadrotor Mode-2

170

8 Deflecting Surface-Based Total Rotor Failure Compensation for Biplane Quadrotor

8.5 Conclusions This chapter proposes minor structural modifications in the biplane quadrotor wings to sustain total rotor failure. The control scheme handles the situation of a single rotor failure in the hovering state with the help of the deflecting surface. The biplane quadrotor cannot survive sudden total rotor failure during the hover state and may become unstable and crash. However, if a deflecting surface is attached, using the proposed control scheme, the biplane can land sans damage in a nearby safe location. When rotor failure occurs, the biplane switches to fixed-wing mode and transitions in about three seconds, after which it requires less thrust and navigates (7 m/s) using three rotors and deflecting surface, while the passive deflecting surface angle is about 15◦ . The motor speed is found to be within allowable bounds during the flight. After performing the transition maneuver, the vehicle lands safely without damage, showing the suitability of the deployed control action.

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Chapter 9

Epilogue

Drones can deliver the online packages in just minutes. Whether it’s Amazon, Google, or local startups, drone delivery is revolutionizing e-commerce. It is an ongoing pursuit of many researchers to find ways and means to ensure safer, resource (including energy) efficient, and reliable delivery. In many places, wild terrain, vast mountains, deep jungles, hinder speedy delivery of medical supplies to remote hospitals that need them—so drone delivery companies seek to revolutionize those requests with a fleet of drones that can revolutionize medical requests. In the near future drones could be regularly delivering packages to our doorstep. In this book, we presented control challenges encountered in the process of dronebased deliveries like wind gusts, rotor failures, and issues caused due to change in mass and drop/pickup of the slung load. It has been demonstrated through extensive simulation study that the quadrotor biplane is energy efficient over the rotary-wing quadrotor, especially for long haul flights. Quadrotor biplanes are super easy to fly and exhibit smooth transition from hover to forward flight and back, while providing fully autonomous flight. The key advantages of a quadrotor biplane are required to be reiterated again: (1) The biplane configuration provides the necessary lift in a small footprint. (2) Since the prop-rotors provide all the thrust in hover as well as cruise, there are no separate lift or cruise propellers. This feature enables shedding of ‘dead-weight’ in exchange for a higher useful load fraction. (3) No aerodynamic control surfaces imply lesser susceptibility to gusts as well as smooth transition from hover to forward flight and vice-versa by utilizing the superior pitching moment offered by quad-rotor RPM control. Additionally, part of the wing surface immersed in the propeller wake affords superior (delayed) stall characteristics during transition. In the relentless pursuit of delivering payloads, researchers are working on control algorithms to allow a group of small drones to together lift and deliver heavier payloads. Although not covered in this book, such objectives could be the natural © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 N. Dalwadi et al., Adaptive Hybrid Control of Quadrotor Drones, Studies in Systems, Decision and Control 461, https://doi.org/10.1007/978-981-19-9744-0_9

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progression of the work presented in this book. In fact, such deliveries under wind gusts, rotor failures and demands for energy efficiency would add additional challenge. The measurement of thrust and monitoring altitude would be used to estimate the weight of the package in such cases. These collaborative drones could serve other functions apart from delivering and picking up heavy packages. Such a system might be able to resupply small groups of soldiers in the field. Add to these the demands to pick up or drop heavy items collaboratively but travel long distances, the need for collaborative quadrotor biplanes could be envisaged in the near future. Although new cargo drones (and by extension quadrotor biplanes specifically) are a fast and eco-friendly alternative to traditional package delivery that could save one time and money. However, before this trend can take off, customers have to be comfortable with giant and swarm of drones flying over their neighborhoods. Machine learning will allow swarms of drones learn new environments on the fly, navigating cluttered spaces without bumping into each other or anything else. However, it is not at all clear yet what would be the comfort level for large-scale deliveries in the near future.