Finite and Instantaneous Screw Theory in Robotic Mechanism (Springer Tracts in Mechanical Engineering) 9811519439, 9789811519437

Table of contents :
Preface
Contents
1 Introduction
1.1 Classification of Robotic Mechanism
1.1.1 Open-loop Robotic Mechanism
1.1.2 Closed-loop Robotic Mechanism
1.1.3 Hybrid Robotic Mechanism
1.2 Synthesis, Analysis, Design and Calibration of Robotic Mechanism
1.3 Screw Theory in Robotic Mechanism
1.3.1 Instantaneous Screw
1.3.2 Finite Screw
1.3.3 Relation Between Finite and Instantaneous Screws
1.4 Scope and Organization of This Book
References
2 Finite and Instantaneous Screw Theory
2.1 Introduction
2.2 Finite Screw
2.2.1 Quasi-vector Derived from Dual Quaternion
2.2.2 Screw Triangle Product
2.2.3 Algebraic Structure of Finite Screw
2.3 Instantaneous Screw
2.3.1 Instantaneous Screw in Vector Form
2.3.2 Algebraic Structure of Instantaneous Screw
2.3.3 Twist and Wrench with Reciprocal Product
2.3.4 Classification of Twist Spaces and Wrench Spaces
2.4 Differential Mapping Between the Screws
2.4.1 One-DoF Motion
2.4.2 Multi-DoF Motion
2.5 Discussion on the Algebraic Structures of FIS
2.6 Conclusion
References
3 Topology and Performance Modeling of Robotic Mechanism
3.1 Introduction
3.2 FIS Based Topology and Performance Modeling
3.3 FIS Based Finite and Instantaneous Motion Modeling
3.3.1 The Finite Motion Modeling
3.3.2 The Instantaneous Motion Modeling
3.4 Example
3.4.1 Typical Open-loop Mechanism
3.4.2 Typical Closed-loop Mechanism
3.5 Integrated Framework for Type Synthesis and Performance Analysis
3.6 Conclusion
References
4 Type Synthesis Method and Procedure of Robotic Mechanism
4.1 Introduction
4.2 General Procedure of Finite Screw Based Type Synthesis
4.3 The Commonly Used Motion Pattern
4.3.1 One-DoF Motion Pattern
4.3.2 Multi-DoF Motion Pattern
4.4 Limb Synthesis
4.4.1 Standard Limb Structure
4.4.2 Derivative Limb Structure
4.4.3 Composition Algorithms Among Joint Motions
4.4.4 Equivalent Groups of Joints
4.5 Assembly Condition and Actuation Arrangement
4.5.1 Assembly Condition
4.5.2 Non-redundant Actuation Arrangement
4.5.3 Intersection Algorithms Among Limb Motion
4.6 Type Synthesis of Robotic Mechanism
4.7 Conclusion
References
5 Type Synthesis of Mechanisms with Invariable Rotation Axes
5.1 Introduction
5.2 Mechanism with Invariable Rotation Axes
5.2.1 Mechanism with One Invariable Rotation Axis
5.2.2 Mechanism with Two Invariable Rotation Axes
5.3 Examples with One Invariable Rotation Axis
5.3.1 Open-loop Mechanisms with Schoenfiles Motion
5.3.2 Open-loop Mechanisms with Planar Motion
5.4 Examples with Two Invariable Rotation Axes
5.4.1 Single Closed-loop Mechanisms with Double-Schoenfiles Motion
5.4.2 Closed-loop Mechanisms with Tricept Motion
5.5 Conclusion
References
6 Type Synthesis of Mechanism with Variable Rotation Axes
6.1 Introduction
6.2 Mechanism with Variable Rotation Axes
6.2.1 Mechanism with One Variable Rotation Axis
6.2.2 Mechanism with One Invariable and One Variable Rotation Axes
6.2.3 Mechanism with Two Variable Rotation Axes
6.3 Example with One Variable Rotation Axis
6.3.1 Single Closed-loop Mechanism with 1R1T Motion
6.3.2 Closed-loop Mechanism with 3T1R Motion
6.4 Example with One Invariable and One Variable Rotation Axes
6.4.1 Open-loop Mechanism with 3T2R Motion
6.4.2 Closed-loop Mechanism with Exechon Motion
6.5 Example with Two Variable Rotation Axes
6.5.1 Single Closed-loop Mechanisms with 2R Motion
6.5.2 Closed-loop Mechanisms with Z3 Motion
6.6 Conclusion
References
7 Kinematic Modeling and Analysis of Robotic Mechanism
7.1 Introduction
7.2 Displacement Modeling
7.2.1 Forward Kinematics Modeling
7.2.2 Inverse Kinematics Modeling
7.3 Workspace Analysis
7.3.1 Sub-three-dimensional Orientation Space
7.3.2 Sub-three-dimensional Position Space
7.3.3 Workspace Regardless of Initial Pose
7.4 Velocity Analysis
7.4.1 Jacobian Matrix of Open-loop Mechanism
7.4.2 Jacobian Matrix of Closed-loop Mechanism
7.5 Example
7.5.1 Typical Open-loop Mechanism
7.5.2 Typical Closed-loop Mechanism
7.6 Conclusion
References
8 Static Modeling and Analysis of Robotic Mechanism
8.1 Introduction
8.2 Twist and Wrench Analysis
8.2.1 Open-loop Mechanism
8.2.2 Closed-loop Mechanism
8.3 Stiffness Modeling
8.3.1 m-DoF Virtual Spring
8.3.2 Open-loop Mechanism
8.3.3 Closed-loop Mechanism
8.4 Example
8.4.1 Typical Open-loop Mechanism
8.4.2 Typical Closed-loop Mechanism
8.5 Conclusion
References
9 Dynamic Modeling and Analysis of Robotic Mechanism
9.1 Introduction
9.2 Velocity Modeling
9.2.1 Open-loop Mechanism
9.2.2 Closed-loop Mechanism
9.3 Acceleration Modeling
9.3.1 Open-loop Mechanism
9.3.2 Closed-loop Mechanism
9.4 Dynamic Modeling
9.4.1 Wrench Analysis
9.4.2 Dynamic Modeling
9.5 Example
9.5.1 Typical Open-loop Mechanism
9.5.2 Typical Closed-loop Mechanism
9.6 Conclusion
References
10 Optimal Design of Robotic Mechanism
10.1 Introduction
10.2 Parameter Uncertainty
10.2.1 Statistical Objective
10.2.2 Probabilistic Constraint
10.3 Multi-objective Optimization
10.3.1 Performance Index
10.3.2 Response Surface Method Model
10.3.3 Pareto-based Optimization
10.4 Procedure of Multi-objective Optimization with Parameter Uncertainty
10.5 Example
10.5.1 Typical Open-loop Mechanism
10.5.2 Typical Closed-loop Mechanism
10.6 Conclusion
References
11 Synthesis, Analysis, and Design of Typical Robotic Mechanism
11.1 Introduction
11.2 1T2R Mechanism with Invariable Rotation Axes
11.2.1 Type Synthesis
11.2.2 Performance Modeling
11.2.3 Optimal Design
11.3 2R Mechanism with Variable Rotation Axes
11.3.1 Type Synthesis
11.3.2 Performance Modeling
11.3.3 Optimal Design
11.4 Conclusion
References
12 Kinematic Calibration of Robotic Mechanism
12.1 Introduction
12.2 Error Modeling
12.2.1 Joint
12.2.2 Open-loop Mechanism
12.2.3 Closed-loop Mechanism
12.3 Error Identification
12.3.1 Redundant Error Analysis
12.3.2 Error Identification Algorithm
12.4 Example
12.4.1 Typical Open-loop Mechanism
12.4.2 Typical Closed-loop Mechanism
12.5 Conclusion
References
Appendix A Differentiation of Finite Screw Considering Both Motion Parameters and Screw Axis
References
Appendix B Quasi-differential Function

Citation preview

Springer Tracts in Mechanical Engineering

Tao Sun Shuofei Yang Binbin Lian

Finite and Instantaneous Screw Theory in Robotic Mechanism

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Tao Sun Shuofei Yang Binbin Lian •

•

Finite and Instantaneous Screw Theory in Robotic Mechanism

123

Tao Sun School of Mechanical Engineering Tianjin University Tianjin, China

Shuofei Yang School of Mechanical Engineering Tianjin University Tianjin, China

Binbin Lian School of Mechanical Engineering Tianjin University Tianjin, China

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-15-1943-7 ISBN 978-981-15-1944-4 (eBook) https://doi.org/10.1007/978-981-15-1944-4 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved 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

In recent years, robotic system has attracted intensive attention and is widely applied in many areas, such as aviation, aerospace, transportation, medicine, and health. The successful application of robotic system depends on the robotic mechanism serving as the mechanical execution unit whose type synthesis and performance design are fundamental issues. This book presents a finite and instantaneous screw (FIS) theory and applies on robotic mechanism. The topological model can be analytically expressed and algebraically computed at the finite motion level. In addition, differential mapping between topological and performance models are formulated, with which an integrated analysis and design framework is proposed. This book firstly introduces the FIS theory in Chap. 2 and proposes an integrated design framework in Chap. 3. Finite screw based type synthesis of robotic mechanism is presented in Chap. 4. It can explicitly analyze motion characteristics with analytical expression and algebraic derivation, which is exemplified by the mechanism with invariable and variable rotation axes in Chaps. 5 and 6. Kinematic, stiffness, and dynamic modeling based on FIS theory are discussed in Chaps. 7–9. In Chap. 10, optimal design method is proposed. Chap. 11 illustrates the synthesis, analysis, and design of two typical robotic mechanisms. Based on FIS theory, kinematic calibration is implemented in Chap. 12. The features of this book include: 1. This book presents the FIS theory and proposes an integrated framework including type synthesis, performance modeling, optimal design, and kinematic calibration of robotic mechanism. It solves the long-term challenge on the development of robotic mechanism due to the disconnection between topological and performance models. 2. This book proposes a finite screw based type synthesis method based upon analytical expression and algebraic derivation. The proposed method is a generic one. In particular, motion characteristics of mechanisms with invariable and variable rotation axes can be analyzed, from which the type synthesis of these mechanisms can be performed at the finite motion level.

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3. This book presents an optimal design method with the consideration of parameter uncertainty. Multiple performances, including kinematics, stiffness and dynamics, are included in the design process. The matching principle among multiple performances are defined, which addresses the difficulty in obtaining optimal result from the multi-objective optimization having objectives with conflict relations. 4. This book proposes a kinematic calibration method based on FIS theory. The complete, continuous, and minimal error model of robotic mechanism is formulated, on which basis the robust parameter identification method is presented. The accuracy of the robotic mechanism can be greatly improved in an efficient manner. The book can be used as graduate textbook in mechanical engineering, or as a research monograph in robotics. It can also be used as a reference by engineers on the robot design and application, researchers on the robotic theory and technology, and students ranging from senior undergraduates to doctoral students. This book gives the readers a deep understanding on the invention, analysis, design, and application of the robotic mechanisms. This book would not have been possible without the help and involvement of many people. In particular, the authors would like to thank Prof. Tian Huang, Prof. J.S. Dai, Prof. Yimin Song from Tianjin University for their useful suggestions, Xinming Huo who worked on the type synthesis, Meng Wang, Dongxing Yang and Ruifeng Guo contributed to the figure drawing and proof reading. The authors are also grateful for the financial support provided by National Key R&D Program of China under the grant 2018YFB1307800, and the National Natural Science Foundation of China (NSFC) under the grant 51875391, 51675366, 51475321, 51205278. Tianjin, China

Tao Sun Shuofei Yang Binbin Lian

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Classification of Robotic Mechanism . . . . . . . . . . . . . . . . . 1.1.1 Open-loop Robotic Mechanism . . . . . . . . . . . . . . . 1.1.2 Closed-loop Robotic Mechanism . . . . . . . . . . . . . . 1.1.3 Hybrid Robotic Mechanism . . . . . . . . . . . . . . . . . 1.2 Synthesis, Analysis, Design and Calibration of Robotic Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Screw Theory in Robotic Mechanism . . . . . . . . . . . . . . . . . 1.3.1 Instantaneous Screw . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Finite Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Relation Between Finite and Instantaneous Screws . 1.4 Scope and Organization of This Book . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Finite and Instantaneous Screw Theory . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Finite Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Quasi-vector Derived from Dual Quaternion . . . . . 2.2.2 Screw Triangle Product . . . . . . . . . . . . . . . . . . . . 2.2.3 Algebraic Structure of Finite Screw . . . . . . . . . . . . 2.3 Instantaneous Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Instantaneous Screw in Vector Form . . . . . . . . . . . 2.3.2 Algebraic Structure of Instantaneous Screw . . . . . . 2.3.3 Twist and Wrench with Reciprocal Product . . . . . . 2.3.4 Classification of Twist Spaces and Wrench Spaces . 2.4 Differential Mapping Between the Screws . . . . . . . . . . . . . 2.4.1 One-DoF Motion . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Multi-DoF Motion . . . . . . . . . . . . . . . . . . . . . . . .

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2.5 Discussion on the Algebraic Structures of FIS . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Topology and Performance Modeling of Robotic Mechanism . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 FIS Based Topology and Performance Modeling . . . . . . . . 3.3 FIS Based Finite and Instantaneous Motion Modeling . . . . . 3.3.1 The Finite Motion Modeling . . . . . . . . . . . . . . . . . 3.3.2 The Instantaneous Motion Modeling . . . . . . . . . . . 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Typical Open-loop Mechanism . . . . . . . . . . . . . . . 3.4.2 Typical Closed-loop Mechanism . . . . . . . . . . . . . . 3.5 Integrated Framework for Type Synthesis and Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Synthesis Method and Procedure of Robotic Mechanism . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Procedure of Finite Screw Based Type Synthesis . . The Commonly Used Motion Pattern . . . . . . . . . . . . . . . . . 4.3.1 One-DoF Motion Pattern . . . . . . . . . . . . . . . . . . . 4.3.2 Multi-DoF Motion Pattern . . . . . . . . . . . . . . . . . . 4.4 Limb Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Standard Limb Structure . . . . . . . . . . . . . . . . . . . . 4.4.2 Derivative Limb Structure . . . . . . . . . . . . . . . . . . . 4.4.3 Composition Algorithms Among Joint Motions . . . 4.4.4 Equivalent Groups of Joints . . . . . . . . . . . . . . . . . 4.5 Assembly Condition and Actuation Arrangement . . . . . . . . 4.5.1 Assembly Condition . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Non-redundant Actuation Arrangement . . . . . . . . . 4.5.3 Intersection Algorithms Among Limb Motion . . . . 4.6 Type Synthesis of Robotic Mechanism . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Type Synthesis of Mechanisms with Invariable Rotation Axes 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mechanism with Invariable Rotation Axes . . . . . . . . . . . . 5.2.1 Mechanism with One Invariable Rotation Axis . . 5.2.2 Mechanism with Two Invariable Rotation Axes . .

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Examples with One Invariable Rotation Axis . . . . . . . . . 5.3.1 Open-loop Mechanisms with Schoenfiles Motion 5.3.2 Open-loop Mechanisms with Planar Motion . . . . 5.4 Examples with Two Invariable Rotation Axes . . . . . . . . 5.4.1 Single Closed-loop Mechanisms with Double-Schoenfiles Motion . . . . . . . . . . . . 5.4.2 Closed-loop Mechanisms with Tricept Motion . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Type Synthesis of Mechanism with Variable Rotation Axes . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Mechanism with Variable Rotation Axes . . . . . . . . . . . . . . 6.2.1 Mechanism with One Variable Rotation Axis . . . . . 6.2.2 Mechanism with One Invariable and One Variable Rotation Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Mechanism with Two Variable Rotation Axes . . . . 6.3 Example with One Variable Rotation Axis . . . . . . . . . . . . . 6.3.1 Single Closed-loop Mechanism with 1R1T Motion . 6.3.2 Closed-loop Mechanism with 3T1R Motion . . . . . . 6.4 Example with One Invariable and One Variable Rotation Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Open-loop Mechanism with 3T2R Motion . . . . . . . 6.4.2 Closed-loop Mechanism with Exechon Motion . . . 6.5 Example with Two Variable Rotation Axes . . . . . . . . . . . . 6.5.1 Single Closed-loop Mechanisms with 2R Motion . . 6.5.2 Closed-loop Mechanisms with Z3 Motion . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Kinematic Modeling and Analysis of Robotic Mechanism . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Displacement Modeling . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Forward Kinematics Modeling . . . . . . . . . . . 7.2.2 Inverse Kinematics Modeling . . . . . . . . . . . . 7.3 Workspace Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Sub-three-dimensional Orientation Space . . . . 7.3.2 Sub-three-dimensional Position Space . . . . . . 7.3.3 Workspace Regardless of Initial Pose . . . . . . 7.4 Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Jacobian Matrix of Open-loop Mechanism . . . 7.4.2 Jacobian Matrix of Closed-loop Mechanism . .

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7.5

Example . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Typical Open-loop Mechanism . 7.5.2 Typical Closed-loop Mechanism 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Static Modeling and Analysis of Robotic Mechanism 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Twist and Wrench Analysis . . . . . . . . . . . . . . . . 8.2.1 Open-loop Mechanism . . . . . . . . . . . . . 8.2.2 Closed-loop Mechanism . . . . . . . . . . . . 8.3 Stiffness Modeling . . . . . . . . . . . . . . . . . . . . . . 8.3.1 m-DoF Virtual Spring . . . . . . . . . . . . . . 8.3.2 Open-loop Mechanism . . . . . . . . . . . . . 8.3.3 Closed-loop Mechanism . . . . . . . . . . . . 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Typical Open-loop Mechanism . . . . . . . 8.4.2 Typical Closed-loop Mechanism . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dynamic Modeling and Analysis of Robotic Mechanism . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Velocity Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Open-loop Mechanism . . . . . . . . . . . . . . . . 9.2.2 Closed-loop Mechanism . . . . . . . . . . . . . . . 9.3 Acceleration Modeling . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Open-loop Mechanism . . . . . . . . . . . . . . . . 9.3.2 Closed-loop Mechanism . . . . . . . . . . . . . . . 9.4 Dynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Wrench Analysis . . . . . . . . . . . . . . . . . . . . 9.4.2 Dynamic Modeling . . . . . . . . . . . . . . . . . . . 9.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Typical Open-loop Mechanism . . . . . . . . . . 9.5.2 Typical Closed-loop Mechanism . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Optimal Design of Robotic Mechanism . 10.1 Introduction . . . . . . . . . . . . . . . . . 10.2 Parameter Uncertainty . . . . . . . . . . 10.2.1 Statistical Objective . . . . . 10.2.2 Probabilistic Constraint . . . 10.3 Multi-objective Optimization . . . . .

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Contents

10.3.1 Performance Index . . . . . . . . . . . 10.3.2 Response Surface Method Model 10.3.3 Pareto-based Optimization . . . . . . 10.4 Procedure of Multi-objective Optimization Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 10.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Typical Open-loop Mechanism . . 10.5.2 Typical Closed-loop Mechanism . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Synthesis, Analysis, and Design of Typical Robotic Mechanism 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 1T2R Mechanism with Invariable Rotation Axes . . . . . . . . 11.2.1 Type Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Performance Modeling . . . . . . . . . . . . . . . . . . . . . 11.2.3 Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 2R Mechanism with Variable Rotation Axes . . . . . . . . . . . 11.3.1 Type Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Performance Modeling . . . . . . . . . . . . . . . . . . . . . 11.3.3 Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Kinematic Calibration of Robotic Mechanism . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 12.2 Error Modeling . . . . . . . . . . . . . . . . . . . . 12.2.1 Joint . . . . . . . . . . . . . . . . . . . . . 12.2.2 Open-loop Mechanism . . . . . . . . 12.2.3 Closed-loop Mechanism . . . . . . . 12.3 Error Identification . . . . . . . . . . . . . . . . . 12.3.1 Redundant Error Analysis . . . . . . 12.3.2 Error Identification Algorithm . . . 12.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Typical Open-loop Mechanism . . 12.4.2 Typical Closed-loop Mechanism . 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Differentiation of Finite Screw Considering Both Motion Parameters and Screw Axis . . . . . . . . . . . . . . . . . . . . . . . . . 397 Appendix B: Quasi-differential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Chapter 1

Introduction

Robots are widely applied in many areas, including but not limited to, automobile engineering, aerospace engineering, port engineering, electronic industry, food industry, surgical operation, housekeeping service. Compared with manual works, robots have advantages in the following perspectives [1, 2]: (1) Robots are more powerful, more efficient, and more accurate in many machining, manufacturing, and assembling tasks; (2) Robots bring no pollution into the process while executing tasks; (3) Robots can perform tasks in extreme conditions, such as in the narrow spaces, and in dangerous environments; (4) Robots are safer in man–robot interaction when curing and serving human because robots will never be affected by exhaustion and emotion; (5) As manual costs increase, applications of robots lead to cost-saving. Attracted by these merits, robot has been through a rapid increment in the last few decades. Robotics has become a key technology in improving the quality of human life. Robot is a complicated system consisting of five subsystems [3], including (1) a mechanical subsystem composed of either rigid bodies, deformable bodies or both, (2) an actuation subsystem acting as the input of the robotic system, (3) a control subsystem achieving the desired outputs by controlling the inputs of the robotic system, (4) a sensing subsystem monitoring the inputs or the outputs or both, (5) an information processing subsystem. This book focuses on the mechanical subsystem that allows a rigid body, namely end-effector serving as the tool directly performing the task, to move with respect to a fixed base. The mechanical subsystem of a robot is called robotic mechanism. It is the execution unit connecting the inputs and outputs of the system. The performances of the robotic mechanism directly determine the behavior of the robot. Development of a robot meeting the application requirements, such as operational accuracy, load capacity, task flexibility, and reliability, relies on the development of a robotic mechanism. © Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_1

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1 Introduction

1.1 Classification of Robotic Mechanism Motion is the property used to identify the robotic mechanism. The independent motion generated by a rigid body is referred to as its Degree of Freedom (DoF). Generally, the fixed base has zero DoF and a free rigid body in space has six DoFs, for instance, three rotations about mutually orthogonal axes and three translations along these axes. The output motion of a robotic mechanism is the result of a set of input motions. The motion transited from the input to the output is realized by connecting rigid bodies with joints. The motion of the joints can be produced from two basic types, namely the rotating pair called revolute (R) joint, and the sliding pair called prismatic (P) joint. From these two basic types of joints, helical (H) joint, cylindrical (C) joint, universal (U) joint, and spherical (S) joint that have practical interest are defined. Among them, R, P, and H joints are one-DoF joints. C and U joints have two DoFs, respectively. S joint has three DoFs. An input motion of the robotic mechanism can be performed by a one-DoF joint while the output motion of the robotic mechanism is produced by different sets of joints. According to the arrangement of joints between the fixed base and the endeffector, the robotic mechanisms are mainly classified into three categories [4–6], i.e., open-loop mechanisms, closed-loop mechanisms, and hybrid mechanisms.

1.1.1 Open-loop Robotic Mechanism An open-loop mechanism consists of several joints connected successively. As twoand three-DoF joints can be regarded as the combination of two and three one-DoF joints, an open-loop mechanism is the connection of several one-DoF joints in serial structure, as shown in Fig. 1.1. The open-loop mechanism can also be denoted by open-loop kinematic chain or open-loop limb. Till now, many open-loop mechanisms have been invented and successfully designed as commercial products by companies like ABB [7], KUKA [8], and FANUC [9]. Among them, the most commonly used ones are those consisting of six R joints and having six DoFs. Sometimes, it is said that they have six controlled Fig. 1.1 The open-loop mechanism

1.1 Classification of Robotic Mechanism

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Fig. 1.2 Six-DoF open-loop mechanism

axes. Designed with different dimensions or actuated in different ways, these sixDoF open-loop mechanisms have been applied in different industry scenarios, from automotive to medical, electronic, and so on. The most typical robots developed upon them are ABB articulated robots [7], KUKA KR QUANTEC robots [8] and FUNUC ARC Mate robots [9], as shown in Fig. 1.2. Another type of open-loop mechanism that is widely used has four DoFs. The four-DoF open-loop mechanisms can realize three-DoF translations and one-DoF rotation. They usually consist of three R joints and one H joint, of which joint axes and directions are parallel. Having less actuated joints than the six-DoF ones, the fourDoF mechanisms have simpler control systems. Thus, they are more cost-effective, more accurate, and faster in most cases. The most typical robots developed upon these mechanisms are ABB SCARA robots [7] and FUNUC SR robots [9], as shown in Fig. 1.3. The motion of these four-DoF open-loop mechanisms is sometimes called SCARA (Schoenfiles) motion. The mechanisms having SCARA motion are named as SCARA mechanisms. Generally, the DoF of a spatial mechanism is no more than six, because the maximum mobility of a mechanism contains at most three translations and three rotations. The mechanisms with less than six DoFs are called lower-mobility mechanisms. It is obvious that the SCARA mechanisms are lower-mobility ones. Some other typical lower-mobility open-loop mechanisms are KUKA KR 40 PA robot [8] and FANUC M-410iC robot [9], which also have four DoFs. As shown in Fig. 1.4, these four-DoF mechanisms consist of two pairs of R joints. The two R joints in each pair have parallel axes at the initial pose. Thus, the mechanisms can realize two-DoF translations and two-DoF rotations. The above six-DoF and lower-mobility robots are mainly used for industrial applications. They don’t have redundant actuation for the purpose of cost-saving. However, for the robots applied in curing and serving human or some other tasks require for flexibility, compliance, and great agility, redundant actuation are assigned to mimic

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1 Introduction

Fig. 1.3 Four-DoF SCARA open-loop mechanism

Fig. 1.4 Other four-DoF open-loop mechanism

human-like movements. For instance, seven R joints are utilized to constitute such a mechanism. The two representatives for this kind of open-loop mechanisms are ABB IRB 14050 single-arm YuMi collaborative robot [4] and KUKA LBR iiwa robot [5], as shown in Fig. 1.5.

1.1 Classification of Robotic Mechanism

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Fig. 1.5 Seven-axis open-loop mechanism

1.1.2 Closed-loop Robotic Mechanism A closed-loop mechanism has at least two kinematic chains forming at least one closed-loop between fixed base and end-effector [6]. The kinematic chain is composed of a set of rigid bodies connected by joints. For distinguishing the output body of open-loop and closed-loop mechanisms, the output motion of the closedloop mechanism is described by the shared body, named as moving platform, of the multiple kinematic chains. The closed-loop mechanism can be further divided into single closed-loop mechanism and multi-closed-loop mechanism according to the number of closed-loops.

1.1.2.1

Single Closed-loop Robotic Mechanism

The single closed-loop mechanism has one closed-loop. As shown in Fig. 1.6, the loop can be separated into two limbs connecting the fixed base to the moving platform. Each limb can be regarded as an open-loop mechanism. The single closed-loop mechanism is usually regarded as the sub-structure of more complicated mechanisms such as multi-closed-loop mechanisms or hybrid mechanisms that will be introduced in the following sections. There are two basic types of single closed-loop mechanisms. The first type is the one-DoF mechanism whose limb is constituted by several R joints with specific geometric relationships among their axes. The second type can be one-DoF to six-DoF mechanisms and each limb consists of several one-DoF joints. The commonly used one-DoF single closed-loop mechanisms are planar 4R mechanism, spherical 4R mechanism, Bennett mechanism, Myard mechanism, Goldberg mechanism, and Bricard mechanism.

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1 Introduction

Fig. 1.6 Single closed-loop mechanism

A planar 4R mechanism consists of four R joints having parallel axes. As shown in Fig. 1.7a, the mechanism generates a one-DoF rotation that has fixed direction and varying position. When the four links of a planar 4R mechanism constitute a parallelogram, it can generate a bifurcated motion which is the union of a one-DoF translation along a circle and a one-DoF rotation that has fixed direction and varying position, as shown in Fig. 1.7b. In a spherical 4R mechanism [10], the four R joints have intersecting axes that share the same rotation center, as shown in Fig. 1.8a. The output motion of the mechanism is a one-DoF rotation whose axis passes through the rotation center and has varying directions. When every two opposite R joints share the same axis, the mechanism generates bifurcated rotation [11] around two axes, as shown in Fig. 1.8b.

Fig. 1.7 Planar 4R mechanism

1.1 Classification of Robotic Mechanism

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Fig. 1.8 Spherical 4R mechanism

Similar to planar and spherical 4R mechanisms, a Bennett mechanism [12] also consists of four R joints. However, the four R joints in the Bennett mechanism are neither parallel nor intersecting to each other. They have skew axes in space as shown in Fig. 1.9. In each pair of opposite links of the mechanism, the two links have the same length. The intersection angle of the two R joints connected to one link is the same as the intersection angle of the two R joints connected to the other link. The ratio between the length and sine of the intersection angle in this pair equals to the ratio in the other pair. The Bennett mechanism generates a one-DoF rotation about the axis with varying direction and position. Both Myard mechanism [13] and Goldberg mechanism [14] are composed of five R joints. They are the combinations of two Bennett mechanisms. As in Fig. 1.10a, the two Bennett mechanisms share two common links and two common R joints, the common links and one common R joint is removed to form the Myard mechanism. As shown in Fig. 1.10b, one common link, and two common R joints are shared Fig. 1.9 Bennett mechanism

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1 Introduction

Fig. 1.10 Myard and Goldberg mechanism

by the two Bennett mechanisms. The Goldberg mechanism is obtained by removing the common link and one common R joint. With the same geometric conditions as the Bennett mechanisms, one-DoF rotation about the axis with varying direction and position is generated by the Myard and Goldberg mechanisms. A threefold-symmetric Bricard mechanism [14] contains six R joints and six links, as shown in Fig. 1.11. All the links have the same length. The intersection angle of a link is defined as the intersection angle of the axes of the two R joints connected to the link. The sum of the intersection angles of any two adjacent links in the mechanism is 2π. The Bricard mechanism generates a one-DoF rotation about the axis with varying direction and position.

Fig. 1.11 Bricard mechanism

1.1 Classification of Robotic Mechanism

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Fig. 1.12 Single closed-loop mechanisms having three-DoF translations

Fig. 1.13 Single closed-loop mechanisms having three-DoF rotations

Besides the above single closed-loop mechanisms with one-DoF and constituted only by R joints, there are many other single closed-loop mechanisms having one to six DoFs, which belong to the second type of single closed-loop mechanism. These mechanisms have fixed translational directions, which are systematically studied by Kong and Gosselin [15]. Here, we list some examples of the second type of single closed mechanisms, as shown in Figs. 1.12, 1.13 and 1.14.

1.1.2.2

Multi-closed-loop Robotic Mechanism

A multi-loop mechanism is composed of more than one closed-loop, as shown in Fig. 1.15. The number of limbs is twice the number of independent closed-loops. The multi-closed-loop mechanism can be divided into two types. As in Fig. 1.15a, the first type has interconnected kinematic chains. For example, Ding et al. [16]

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1 Introduction

Fig. 1.14 Single closed-loop mechanisms having two-DoF translations and one-DoF rotation

Fig. 1.15 Multi-closed-loop mechanism

investigated the kinematic chains consisting of a large number of R joints. All the R joints have parallel axes. Following given adjacency matrix and contracted graphs, a lot of mechanisms with coupled loops were invented. Based on Bennett mechanisms, the multi-loop mobile assemblies were invented by You and Chen [17]. Similarly, more mobile assemblies were presented based on Myard mechanisms [18]. The second type of multi-closed-loop mechanism has no interconnected kinematic chains. It is known as parallel mechanism. There have been many parallel mechanisms designed to be industrial robots and applied in manufacturing. The typical ones are listed in the following. The Gough-Stewart multi-closed-loop mechanism has six DoFs. It consists of six SPS open-loop limbs and can realize any spatial motion. The six limbs are distributed systematically between the fixed base and moving platform. Due to its compact structure, the mechanism does not have a large workspace, but it has high stiffness

1.1 Classification of Robotic Mechanism

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Fig. 1.16 Gough-Stewart multi-closed-loop mechanism

and accuracy. These multi-closed-loop mechanisms have been applied in industry since 1960s for the tasks like astronaut training, radar pointing, and tire testing [19]. A typical commercial robot based upon this mechanism is FANUC F-200iB [9], as shown in Fig. 1.16. The Delta multi-closed-loop mechanism has three DoFs. It consists of three R(SS)2 limbs and can translate along any direction. Herein, (SS)2 is composed of four S joints which form an inner loop in each limb. The R(SS)2 limbs are systematically distributed between fixed base and moving platform. Because of its specific structure, all the actuations can be placed on the fixed base and the moving platform can reach very high speed. The mechanism is mainly applied for fast picking and placing. The typical robots based upon this mechanism are ABB IRB 360 FlexPicker [7] and FANUC M-2iA [9] shown in Fig. 1.17.

Fig. 1.17 Delta multi-closed-loop mechanism

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1 Introduction

Fig. 1.18 Sprint Z3 head multi-closed-loop mechanism

The Sprint Z3 Head multi-closed-loop mechanism [20] has three DoFs, as shown in Fig. 1.18. It is composed of three PRS limbs and described as 3PRS mechanism. It was Hunt [21] who invented the 3PRS and 3RPS mechanisms in 1970s. Both of the 3PRS and 3RPS mechanisms generate the same motions, a translation along the direction perpendicular to the fixed base and two rotations with varying directions and positions. The Sprint Z3 Head has been employed for the machining tasks in the area such as automotive and aviation. There are other two well-known three-DoF multi-closed-loop mechanisms, i.e., 3UPS-UP and 2UPR-SPR mechanisms [22, 23], invented by Neumann [24]. Because the UPS limbs have six DoFs, the motion of the 3UPS-UP mechanism is the same as the UP limb. Thus, it can realize one-DoF translation along with a fixed direction and two rotations about fixed axes. In a 2UPR-SPR mechanism, the axes connecting to the fixed base in the U joint of the UPR limbs are collinear. Hence, the two UPR limbs form a plane. Together with the SPR limb, the mechanism is structured in “T” shape. The 2UPR-SPR mechanism generates one-DOF translation along the fixed direction perpendicular to the fixed base together with two-DoF rotations. Both the axes of the two rotations have fixed positions. However, one of them has fixed direction, while the other has varying direction that is determined by the rotational angle of the first rotation. As shown in Fig. 1.19, these two parallel mechanisms have been developed to be industrial robots by PKM Tricept S. L. [25] and Exechon Enterprises L.L.C. [23], which are called Tricept and Exechon, respectively. They are widely used in machining, welding, and drilling tasks. The Omni-Wrist VI developed by Ross-Hime Designs Inc [26] has four RSR limbs and one SS limb. The SS limb connects to the centers of the fixed base and the moving platform. The four RSR limbs are symmetrically placed with respect to the SS limb as shown in Fig. 1.20. The Omni-Wrist VI has two-DoF rotations about two axes with varying directions and positions. Due to the flexible rotational capability, it has wide applications in target tracking. For instance, it can be applied to track the satellite from the earth or the space station.

1.1 Classification of Robotic Mechanism

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Fig. 1.19 Tricept and Exechon multi-closed-loop mechanisms

Fig. 1.20 Omni-wrist VI multi-closed-loop mechanism

Besides the above well-known ones, many multi-closed-loop mechanisms, that generate three-translational and two-rotational (3T2R) five-DoF motions, 3T1R fourDoF motions, 2T1R three-DoF motions, etc., have been synthesized and analyzed in academia and have great potentials to be applied as commercial robots.

1.1.3 Hybrid Robotic Mechanism A hybrid mechanism consists of several parts in serial structure. Each part is either an open-loop mechanism or a multi-closed-loop mechanism, as shown in Fig. 1.21. In most cases, the multi-closed-loop parts are parallel mechanisms. The conventional hybrid mechanisms are commonly constituted by two parts, i.e., a parallel part and an open-loop part (serial part). The parallel part is connected to

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1 Introduction

Fig. 1.21 Hybrid mechanism

the fixed base while the serial part contains the end-effector. Usually, two- and threeDoF open-loop mechanisms are connected to the moving platforms of the three-DoF multi-closed-loop mechanisms, resulting in five- and six-DoF hybrid mechanisms. For example, the five-axis Exechon machine tool [23] is the combination of an Exechon multi-closed-loop mechanism and a two-DoF RR open-loop mechanism. The six-DoF FANUC M-3iA [9] is the combination of a Delta multi-closed-loop mechanism and a three-DoF RRR open-loop mechanism, as shown in Fig. 1.22. These hybrid mechanisms have the advantages of both the open-loop and closedloop parts in terms of large workspace, flexible orientations, high stiffness, and high accuracy. Recently, some general hybrid mechanisms were invented by Lu and Hu [27, 28]. These mechanisms are the combinations of several closed-loop mechanisms, as

Fig. 1.22 Typical hybrid mechanism

1.1 Classification of Robotic Mechanism

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Fig. 1.23 The hybrid mechanisms consisting of several closed-loop mechanisms

shown in Fig. 1.23. Similarly, more hybrid mechanisms can be synthesized as the combinations of several open-loop and closed-loop mechanisms in serial structure following specific sequences.

1.2 Synthesis, Analysis, Design and Calibration of Robotic Mechanism The development of a robotic mechanism undergoes type synthesis [15], performance analysis [4], optimal design [29], and kinematic calibration [30]. (1) Type synthesis: Type synthesis focuses on obtaining all possible robotic mechanisms that can realize the expected motion characteristics. The motion characteristics, including the number and types of motions, are given by the task, from which all the kinematic architectures of the robotic mechanism are generated. (2) Performance analysis: Performance analysis involves kinematic, stiffness, and dynamic modeling of the obtained robotic mechanism from the previous study. The relations of inputs and outputs, motions and forces are thoroughly investigated. Performance analysis and modeling lead to a comprehensive understanding of the robotic mechanism. (3) Optimal design: Optimal design determines the structural parameters, including dimensional and sectional parameters of the robotic mechanism, which lead to the desired performances. The expected performances are given by the application. Based on the performance models in the previous study, the optimal parameters can be obtained by the optimal design. (4) Kinematic calibration: Kinematic calibration compensates the errors of the robotic mechanism to achieve high precision in task operation. The generation

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1 Introduction

of kinematic errors is inevitable during the construction of the physical prototype guided by the optimal design results. Kinematic calibration, including error modeling, measurement, identification, and compensation, can improve the accuracy of the robotic mechanism economically. Above mentioned processes are necessary steps to the development of a robotic mechanism for specific application requirements. The key issue of carrying out the synthesis, analysis, design, and calibration of the robotic mechanism is the description and computation of its topological, motion, and performance characteristics and properties. Hence, mathematical modeling of the topology and performance models are the prerequisites for the development of robotic mechanism. It is the foundation of this book to select a mathematical tool that is rigorous and suitable for topology and performance modeling. Compared with matrix Lie group and Lie algebra, dual quaternions, finite and instantaneous screws have simpler and non-redundant formats. The composition algorithms of screws are easier to be implemented. In addition, the interrelations of the topology and performances of the robotic mechanism can be revealed and calculated. Therefore, finite and instantaneous screws are selected as the mathematical tool that is consistently employed throughout this book.

1.3 Screw Theory in Robotic Mechanism The literature review on the screw theory applied in robotic mechanisms is given in this section. Based upon the authors’ and other researchers’ works, the sound and thorough finite and instantaneous screw theory will be proposed in this book (Chap. 2), which is named as FIS theory.

1.3.1 Instantaneous Screw The most well-known and commonly used screw is an instantaneous screw. It originates from Mozzi’s theorem that describes an angular velocity of a rigid body about a line followed by a linear velocity along that line. This line is referred to as instantaneous screw axis. The amplitude of instantaneous screw is defined as the value of angular velocity, while the pitch is defined as the ratio between values of linear and angular velocities. Hence, instantaneous motion can be expressed by an instantaneous screw in a six-dimensional vector form. Similarly, a force exerted on a rigid body can also be described by an instantaneous screw defining its intensity as the value of force, and its pitch as the ratio between values of moment and force, respectively. It is based upon Poinsot’s theorem in which a force can be equivalent to a force along and a moment about an axis. In order to distinguish the instantaneous screws describing instantaneous motion and force, the former is called twist, and the latter is named as wrench.

1.3 Screw Theory in Robotic Mechanism

17

The systematic theory on screw has been proposed by Ball [31] in his treatise more than one hundred years ago. However, this theory had not been paid much attention for a quite long time. Its first application in robotic mechanisms can be traced back to 1978 by Hunt [32]. He put forward a new and more effective method to build the velocity Jacobian matrix of the robotic mechanism through writing the velocity of moving platform as the linear combination of the twists generated by all actuation joints. In this way, the Jacobian mapping between joint parameters and velocity of the mechanism is formulated clearly. The Jacobian mapping can be written into matrix form easily, which was followed by Angeles and Tsai [33, 34], resulting in force Jacobian and overall Jacobian, respectively. Hunt’s outstanding work sparked subsequent thorough and systematical researches on instantaneous screws. Hunt, Duffy, and Angeles [32, 35, 36] spent much efforts on the definitions of twists generated by and wrenches exerted on robotic mechanisms. Based on Ball’s description, they gave clearly physical meanings of the concepts of instantaneous screws, screw systems, and reciprocal products. Their work made screws to be visual and concrete physical tools instead of purely abstract mathematical ones. The instantaneous screw can be directly used to describe velocities, forces, powers, and analyze performances of various robotic mechanisms including open-loop and closed-loop mechanisms as well as hybrid ones. Taking the orders of mechanisms with different DoFs into account, the classification of screw systems which are vector spaces spanned by no more than six screws was carried out by Gibson and Hunt, Martínez and Duffy [37], Dai and Jones [38]. They presented a comprehensive enumeration of possible linear combinations of given instantaneous screws that are central to the analysis of multi-DoF mechanisms and established normal form for each screw system in terms of base screws. Based upon these work, Huang [39], Dai [40], and their colleagues implemented mobility analysis of numerous mechanisms through determining the orders and characteristics of mechanisms’ instantaneous screw systems and corresponding reciprocal systems. By employing instantaneous screw systems and their reciprocal products to describe instantaneous motions of a multi-closed-loop mechanism, its limbs, joints and their relationships, type synthesis of multi-closed-loop mechanisms was carried out in an instantaneous motion level by Huang and Li [41, 42], Fang and Tsai [43, 44], Kong and Gosselin [45, 46], as well as the authors of this book [47–53]. This is actually the reverse process of mobility analysis. As for any given robotic mechanism, the velocity, force, stiffness, accuracy, acceleration, and dynamic modeling and performance evaluation can be done using the instantaneous screw based Jacobian matrices or Hessian matrices. In this way, performance analysis, optimal design, and kinematic calibration of different categories of mechanisms can be carried out.

18

1 Introduction

1.3.2 Finite Screw As the counterpart of instantaneous motion, finite motion can be defined by Chasles’ theorem that any rigid body finite motion can be produced by a translation along a line followed by a rotation about that line. This line is called finite screw axis in the whole book. It has long been desired to describe and calculate finite motions by employing finite screw with non-redundant and the simplest form, just as instantaneous motions. This work can be dated back to 1965. It is Dimentberg and Dai [54, 55] who proposed a concept of finite displacement screw matrix to describe finite motion. Parkin [56] defined its pitch in 1992 as the ratio between half the translational distance and tangent of half the rotational angle, which is called quasi-pitch by Hunt and Parkin [57]. Two years later, Huang defined the amplitude as twice the tangent of half the angle. So far, the finite motion can be described by a six-dimensional quasi-vector. This quasi-vector is referred to as finite screw. Recently, the authors of this book strictly proved that the finite screw is derived from dual quaternion and gave a more general quasi-vector form by considering the two situations that the rotational angle equals zero and does not equal zero. Utilizing finite screw, the finite motions generated by one-DoF joints could be described. Huang and Chen [58] rewrote the composition of two finite screws expressing finite motions of one-DoF joints as the screw triangle product that equals to linear sum of five terms. This leads to the easy acquisition of the finite motion expression of simple open-loop mechanisms, RR, PR, RP for instance. Huang’s work was extended by the authors of this book. They proposed general procedures to do finite motion analysis and kinematics of robotic mechanisms in an algebraic manner. Based on the finite motion described and calculated by finite screw, the authors proposed an analytical approach of type synthesis for parallel mechanisms by systematically investigating properties of screw triangle products [59–62]. Compared with the approach based on instantaneous screw, the finite screw-based approach can synthesize parallel mechanisms algebraically in finite motion level, resulting in that the full cycle DoFs of the obtained mechanisms do not need to be verified.

1.3.3 Relation Between Finite and Instantaneous Screws Finite and instantaneous screws are proved to be powerful mathematical tools respectively in describing and calculating finite and instantaneous motions. The finite motion described by finite screw enables the type synthesis of robotic mechanism with analytical expression and algebraic derivation. The instantaneous motion denoted by the instantaneous screw is able to model the motions and forces that correspond to the kinematic, stiffness and dynamic of the robotic mechanism. In order to enable type synthesis and performance analysis of robotic mechanisms to be

1.3 Screw Theory in Robotic Mechanism

19

integrated into a unified framework, it has long been a desire to build the connection between these two kinds of screw. To handle this problem, Dai [63, 64] demonstrated the relationship between finite displacement screw operation and the matrix representations of finite motions by solving the eigenscrew and derivative of finite displacement screw matrix, leading to the eigen and differential mappings between finite and instantaneous screws. Huang and Roth [65] discussed the correspondence between vector subspaces and screw systems to distinguish finite screw systems from instantaneous ones. Through investigating associativity and derivative properties of the screw triangle product, the differential mapping between finite and instantaneous screws was built by the authors of this book [59], allowing the algebraic structures of finite screws in quasi-vector form and instantaneous screws in vector form to be revealed, which correspond to the finite motion-based type synthesis and instantaneous motion-based performance analysis. Hence, type synthesis and performance analysis of robotic mechanism can be unified by the screw theory. In addition, the differential relationship of the finite and instantaneous screws are further investigated considering kinematic errors of the robotic mechanism. The generation of the kinematic errors are defined and their accumulations are analytically revealed, which leads to the kinematic calibration of robotic mechanism with clear physical interpretation and high accuracy.

1.4 Scope and Organization of This Book The type synthesis, performance analysis, optimal design and kinematic calibration of robotic mechanism, including open-loop mechanism, closed-loop mechanism and hybrid mechanism, have attracted much attention and the researches on one or some of the topics have been growing. Although a number of new robotic mechanisms were proposed, none seems to be obtained by the finite motion-based type synthesis process that allows analytical expression and algebraic derivation. Although various performance modeling methods were presented, the study for the unified framework that can reflect the connection from the topological structures to the performances is still a challenge. Although the promising architecture can be obtained by the optimization methods, none seems to take the effects of the prototype construction into account. The aim of this book is to present a systematic framework for synthesis, analysis, design, and calibration of robotic mechanisms using FIS theory as the mathematical tool. To reach this goal, the scope and organization of this book are as follows: In Chap. 2, Finite and Instantaneous Screw Theory, presents fundamental concepts, expressions, operations and properties of the FIS theory. The finite screw in six-dimensional quasi-vector form and its composition operation is derived from dual quaternion. The instantaneous screw in six-dimensional vector form is introduced and the classification of instantaneous screw space is discussed. The differential mapping between finite and instantaneous screws is rigorously proved, leading to

20

1 Introduction

a direct connection between the modeling and analysis of finite motion and that of instantaneous motion. At the end of this study, the algebraic structures of FIS, matrix Lie group and Lie algebra, and dual quaternions are compared. It is shown that FIS is the simplest and most concise mathematical tool for finite and instantaneous motion description and composition. In Chap. 3, Topology and Performance Modeling of Robotic Mechanism, proposes an integrated framework that can unify the finite motion based topology model and instantaneous motion based performance model using the FIS theory presented in the previous study. The finite motions of robotic mechanisms are expressed by finite screws, leading to the topology models that can also be applied as the displacement model. The instantaneous motion models are directly obtained by the differential mapping of finite and instantaneous screws, corresponding to the velocity and acceleration models closely linked to performances of robotic mechanism. Based on the integrated framework proposed in Chap. 3, the methodologies for type synthesis and performance analysis of robotic mechanisms are illustrated from Chaps. 4–9, where type synthesis is shown from Chaps. 4–6 and the performance analysis is from Chaps. 7–9. In Chap. 4, Type Synthesis Method and Procedures of Robotic Mechanism, summarize the general steps for the topological invention of robotic mechanisms, including motion pattern description, limb synthesis, mechanism assembly, and actuation arrangement. The objective of type synthesis is to search for the mechanism having desired number and types of DoF. It can be interpreted as the continuous set of poses of the end-effector or moving platform, which is called motion pattern. The fundamental issue using analytical expressions, i.e., finite screw, to define, distinguish and express motion patters are investigated. Based on the algebraic composition and intersection algorithms of finite screws, the procedure of obtaining the whole set of feasible mechanical structures having the expected motion pattern is discussed. FIS theory is capable of describing the expected DoFs of the robotic mechanism in type synthesis. What’s more, motion characteristics of the expected DoFs can be further analyzed due to the analytical expression and algebraic derivation of finite screws, which leads to a comprehensive understanding of the motion capability of the robotic mechanism thus is beneficial for the performance analysis. To illustrate the detailed motion features, robotic mechanisms are divided into mechanisms with invariable rotation axes and variable rotation axes, and the type synthesis of them is carried out in Chaps. 5 and 6, respectively. In Chap. 5, Type synthesis of Mechanisms with Invariable Rotation Axes, gives the synthesis examples of mechanisms with one invariable rotation axis and two invariable rotation axes. Invariable rotation axes indicate that the mechanism has fixed rotation directions, which means that the motion parameters in the motion expression, i.e., the translational distances and rotational angles, have no influence on the direction vector. Open-loop mechanism with SCARA (Schoenfiles) motion, closed-loop mechanism with planar motion, single closed-loop mechanism with Double-Schoenfiles motion, and multi-closed-loop mechanism with Tricept motion are synthesized. In Chap. 6, Type Synthesis of Mechanisms with Variable Rotation Axes, discusses the motion characteristics of mechanism with variable rotation axes and implements

1.4 Scope and Organization of This Book

21

their type synthesis. The phase “variable rotation axes” only denotes that the mechanisms have unfixed rotation directions. Unlike the mechanism with invariable ration axes whose position can only be changed by the translational parameters, the mechanism with variable rotation axes have more complex expressions and mechanical structures. Synthesis examples on the mechanism with one variable rotation axis, one invariable and one variable rotation axes, and two variable rotation axes are given. Especially, the mechanisms with Exechon motion and Z3 motion are synthesized. In Chap. 7, Kinematic Modeling and Analysis of Robotic Mechanism, carry out the displacement and velocity modeling. The forward kinematic and inverse kinematic problems are solved by the finite screw. On this basis, the workspaces, including position and orientation workspaces, are analyzed. By differentiating the displacement model formulated by finite screw, the velocity model described by instantaneous screw is obtained, in which the Jacobian matrix reflecting the mapping between input and output of twist and wrench is analyzed. In Chap. 8, Static Modeling and Analysis of Robotic Mechanism, addresses the static stiffness modeling method. The elastic deformations of the robotic mechanism, defined as deformation twist, under the static external wrenches are considered. The twist and wrench mapping models considering the deformation twist are reformulated. Taking the effects of the passive joints into account, an m-DoF virtual spring is applied to denote the complete and accurate elastic feature of the component. The stiffness modeling is implemented by adding the deformation twist of m-DoF virtual springs within the limb and applying virtual work at the mechanism level. In Chap. 9, Dynamic Modeling and Analysis of Robotic Mechanism, discusses the mapping between force, velocity and acceleration of robotic mechanisms. By directly differentiating the topology model, the Jacobian matrix for the velocity model and the Hessian matrix for the acceleration model are obtained. On this basis, the velocities, accelerations and forces of the joints, components and end-effector (or moving platform) are easily obtained, with which dynamic model of robotic mechanism at joint space and operated space can be formulated. The performance models from Chaps. 7–9 can be applied to analyze the behavior of a robotic mechanism with known topology and structural parameters. The parametric models of the performances can also be adopted to the optimal design determining the most promising architecture for specific applications. In Chap. 10, Optimal Design of Robotic Mechanism, presents the optimization procedure considering parameter uncertainty and multiple performances. Parameter uncertainty refers to the parametric deviations when applying the optimized parameters to build the physical prototype of the robotic mechanism. They are regarded as random variables centered on the deterministic parameter values visited by the optimizer. Statistical objectives and probabilistic constraints are formulated to consider the parameter uncertainty. Further concerning the multiple performance requirements, a Pareto-based multi-objective optimization method is applied to the optimal design, resulting in a set of non-dominant solutions. A cooperative equilibrium searching method is proposed to search for the best compromise among multiple performances.

22

1 Introduction

In Chap. 11, Synthesis, Analysis and Design of typical robotic mechanism, give two examples on the development of typical robotic mechanisms. One is the robotic mechanism with invariable rotational axes having the Tricept motion, and the other one is the robotic mechanism with variable rotational axes having the same motion pattern as the Omni-wrist. The mobility and performance requirements are given by the application scenarios, from which the type synthesis, performance modeling and optimal design are implemented following the methods proposed in the previous chapters. In Chap. 12, Kinematic Calibration of Robotic Mechanism, focuses on the kinematic accuracy improvement. The calibration process, including error modeling, measurement, identification and compensation, is implemented by the FIS theory. Especially, error model meeting the criteria in terms of completeness, continuity, and minimality is investigated, and the efficient, robust, and accurate error identification is proposed. Calibration examples on the typical open-loop mechanisms and closedloop mechanisms show the FIS based calibration method can effectively improve kinematic accuracy of the robotic mechanism.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Mordechai BA, Francesco M (2018) Element of robotics. Springer, Switzerland Siciliano B, Khatib O (2008) Element of robotics. Springer, Berlin Angeles J (2007) Fundamentals of robotic mechanical systems. Springer, New York Liu XJ, Wang JS (2014) Parallel mechanism: type, kinematics, and optimal design. Springer, Berlin Li QC, Hervé JM, Ye W (2020) Geometric method for type synthesis of parallel manipulators. Springer, Singapore Merlet JP (2006) Parallel robots. Springer, Netherlands Asea Brown Boveri Ltd. https://new.abb.com. Accessed 22 Aug 2019 KUKA ROBOT. https://www.kuka.com. Accessed 22 Aug 2019 FANUC. https://www.fanuc.com. Accessed 22 Aug 2019 Farhang K, Zargar YS (1999) Design of spherical 4R mechanisms: function generation for the entire motion cycle. J Mech Des 121(4):521–528 Wang B, Fang YF (2018) Structural constraint and motion mode analysis on parallel mechanism with bifurcated motion. J Xi’an Jiaotong Univ 52(6):62–68 Bennett Geoffrey T (1903) A new mechanism. Engineering 76(1903):777 Liu SY, Chen Y (2009) Myard linkage and its mobile assemblies. Mech Mach Theory 44(10):1950–1963 Chen Y, You Z (2007) Spatial 6R linkages based on the combination of two Goldberg 5R linkages. Mech Mach Theory 42(11):1484–1498 Kong X, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Heidelberg Ding WH, Deng H, Li QM et al (2014) Control-orientated dynamic modeling of forging manipulators with multi-closed kinematic chains. Robot Comput Integr Manuf 30(5):421–431 You Z, Chen Y (2012) Motion structures: deployable structural assemblies of mechanisms. Spon, London Chen Y, You Z (2008) An extended Myard linkage and its derived 6R linkage. J Mech Des 130(5):052301 (8 pages) Gough VE, Whitehall SG (1962) Universal type testing machine. In: Proceedings of the 9th international automobile technical congress, vol 1962. London, pp 117–137

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20. Sprint Z3 Head. https://www.ctemag.com. Accessed 22 Aug 2019 21. Hunt KH (1983) Structural kinematics of in-parallel-actuated robot-arms. J Mech Trans Autom Des 105(4):705–712 22. CHNROBOT. http://www.chnrobot.com. Accessed 22 Aug 2019 23. EXECHON. http://www.exechon.com. Accessed 22 Aug 2019 24. Neumann KE (2002) Tricept application. In: 3rd chemnitz parallel kinematics seminar, vol 2002. Zwickau, pp 547–551 25. Tricept. http://www.pkmtricept.com. Accessed 22 Aug 2019 26. Omni-Wrist VI. http://www.anthrobot.com. Accessed 22 Aug 2019 27. Lu Y, Dai ZH, Wang P (2018) Full forward kinematics of redundant kinematic hybrid. Appl Math Model 62:134–144 28. Hu B (2014) Complete kinematics of a serial-parallel manipulator formed by two Tricept parallel manipulators connected in serials. Nonlinear Dyn 78:2685–2698 29. McCarthy JM, Gim SS (2011) Geometric design of linkages. Springer, New York 30. Zhang D (2010) Parallel robotic machine tools. Springer, New York 31. Ball RS (1875) The theory of screws: a geometrical study of kinematics, equilibrium and small oscillations of a rigid body. Trans R Irish Acad 25:157–218 32. Hunt KH (1978) Kinematic geometry of mechanisms. Oxford University Press, USA 33. Angeles J (2012) Spatial kinematic chains: analysis-synthesis-optimization. Springer Science & Business Media, Heidelberg 34. Tsai LW, Roth B (1972) Design of dyads with helical, cylindrical, spherical, revolute and prismatic joints. Mech Mach Theory 7(1):85–102 35. Crane C, Rico J, Duffy J (2009) Screw theory and its application to spatial robot manipulators. Center for Intelligent Machines and Robotics, University of Florida, Gainesville, FL, Technical Report 36. Angeles J (2007) Fundamentals of robotic mechanical systems: theory, methods and algorithms, 3rd edn. Springer, New York 37. Martínez JMR, Duffy J (1993) The principle of transference: history, statement and proof. Mech Mach Theory 28:165–177 38. Dai JS, Jones JR (2001) Interrelationship between screw systems and corresponding reciprocal systems and applications. Mech Mach Theory 36(5):633–651 39. Lu WJ, Zhang LJ, Xie P et al (2017) Review on the mobility development history with the understanding of overconstraints. J Mech Eng 53(15):81–92 40. Leal ER, Dai JS, Pennock G (2013) Screw-system-based mobility analysis of a family of fully translational parallel manipulators. Math Probl Eng 3:1–9 41. Huang Z, Li QC (2002) General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. Int J Robot Res 21(2):131–145 42. Huang Z, Li QC (2003) Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method. Int J Robot Res 22:59–79 43. Fang YF, Tsai LW (2002) Structure synthesis of a class of 4-DoF and 5-DoF parallel manipulators with identical limb structures. Int J Robot Res 21(9):799–810 44. Fang YF, Tsai LW (2004) Structure synthesis of a class of 3-DoF rotational parallel manipulators. IEEE Trans Robot Autom 20(1):117–121 45. Kong XW, Gosselin CM (2004) Type synthesis of 3-DoF translational parallel manipulators based on screw theory. ASME J Mech Des 126(1):83–92 46. Kong XW, Gosselin CM (2006) Type synthesis of 4-DoF SP-equivalent parallel manipulators: a virtual chain approach. Mech Mach Theory 41(11):1306–1319 47. Song YM, Gao H, Sun T et al (2014) Kinematic analysis and optimal design of a novel 1T3R parallel manipulator with an articulated travelling plate. Robot Comput Integr Manuf 30(5):508–551 48. Song YM, Lian BB, Sun T et al (2014) A novel five-degree-of-freedom parallel manipulator and its kinematic optimization. ASME Trans J Mech Robot 6(4):410081–410089 49. Lian BB, Sun T, Song YM (2017) Stiffness modeling, analysis and evaluation of a 5 degree of freedom hybrid manipulator for friction stir welding. Proc Inst Mech Eng Part C J Mech Eng Sci 231(23):4441–4456

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50. Lian BB, Sun T, Song YM (2016) Stiffness analysis of a 2-DoF over-constrained RPM with an articulated traveling platform. Mech Mach Theory 96:165–178 51. Lian BB, Sun T, Song YM (2015) Stiffness analysis and experiment of a novel 5-DoF parallel kinematic machine considering gravitational effects. Int J Mach Tools Manuf 95:82–96 52. Sun T (2012) Performance evaluation index framework of lower mobility parallel manipulators. Dissertation, Tianjin University 53. Lian BB (2017) Methodology of multi-objective optimization for a five degree-of-freedom parallel manipulator. Dissertation, Tianjin University 54. Dimentberg FM (1965) The screw calculus and its applications in mechanics. Izdat, Mauda Moscow 55. Dai JS (2015) Historical relation between mechanisms and screw theory and the development of finite displacement screws. J Mech Eng 51(13):13–26 56. Parkin IA (1992) A third conformation with the screw systems: finite twist displacements of a directed line and point. Mech Mach Theory 27(2):177–188 57. Hunt KH, Parkin IA (1995) Finite displacement of points, planes and lines via screw theory. Mech Mach Theory 30(2):177–192 58. Huang CT, Chen CM (1995) The linear representation of the screw triangle-a unification of finite and infinitesimal kinematics. ASME J Mech Des 117(4):554–560 59. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 60. Yang SF, Sun T, Huang T et al (2016) A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech Mach Theory 104:405–419 61. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 62. Yang SF (2017) Type synthesis of parallel mechanisms based upon finite screw theory. Dissertation, Tianjin University 63. Dai JS (2006) An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist. Mech Mach Theory 41(1):41–52 64. Dai JS, Holland N, Kerr DR (1995) Finite twist mapping and its application to planar serial manipulators with revolute joints. Proc Inst Mech Eng Part C J Mech Eng Sci 209(4):263–271 65. Huang CT, Roth B (1994) Analytic expressions for the finite screw systems. Mech Mach Theory 29(2):207–222

Chapter 2

Finite and Instantaneous Screw Theory

2.1 Introduction In general, according to motion status and properties, rigid body motions can be divided into two categories, finite motion and instantaneous motion, which can be expressed by finite and instantaneous screw theory (FIS theory) [1]. This chapter presents fundamental concepts, expressions, operations, and properties of FIS theory. Firstly, the finite screw in quasi-vector form and its composition operation, i.e., the screw triangle product, is derived from dual quaternion for the first time. The entire set of finite screws is strictly proved to be a Lie group under the screw triangle product. Secondly, the algebraic structure of instantaneous screws in vector form is presented, leading to that the entire set of the instantaneous screws is proved to form a Lie algebra under the screw product. The reciprocal product between twist and wrench as well as the classification of instantaneous screw spaces are discussed. Then, the differential mapping between finite and instantaneous screws is revealed for the first time in the situations of both one-DoF and multi-DoF rigid body motions. This mapping leads to a direct connection between the modeling and analysis of finite motion and that of instantaneous motion [2]. Finally, further issues on the algebraic structures of FIS in comparison with matrix and dual quaternion representations of Special Euclidean group SE(3) and its Lie algebra se(3) are investigated. In this way, a unified framework of FIS theory for motion description, modeling, and analysis is proposed and established [3]. This framework also provides an effective and concise mathematical tool to solve the long-term problems about type synthesis [4–6], performance analysis and design [7–10] of robotic mechanisms in a general and integrated way.

© Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_2

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26

2 Finite and Instantaneous Screw Theory

2.2 Finite Screw When a rigid body moves along a continuous path, two categories of motions are involved at each pose of the body during its movement. The first category of motions is the total movement of the body with respect to its initial pose, which is described by the motion transformation between its current pose and initial pose. This category of motions is called finite motion. The second one does not concern about the body’s initial pose. It only takes the velocity (and acceleration, jerk, etc.) of the body at the current pose into account. Thus, this category of motions is called instantaneous motion. It has long-term been desired that finite motion and instantaneous motion could be expressed by a unified framework that can provide a simple mathematical tool in a straight forward manner. After many years of research and comparison on the methods of describing rigid body motions by many scholars around the world, we find that FIS theory containing finite screw and instantaneous screw meets the requirement of a simple and unified framework. Under this framework, finite motion and instantaneous motion can be, respectively, expressed by finite screw in quasi-vector form and instantaneous screw in vector form. In this way, all rigid body motions are described algebraically in the most concise forms. As an important part of FIS theory, finite screw for finite motion description will be introduced firstly. The expression, operations, and properties of finite screw will also be presented in this section.

2.2.1 Quasi-vector Derived from Dual Quaternion Finite motion description originates from the Chasles’ theorem, that any finite motion of a rigid body can be equivalently regarded as a rotation about an axis followed by a translation along the same axis [11]. As shown in Fig. 2.1, the finite motion should be represented by three basic eleFig. 2.1 Basic elements of finite motion

2.2 Finite Screw

27

ments. They are the Chasles’ axis which will be called finite motion axis hereinafter, the rotational angle about that axis, and the translational distance along the same axis. Thus, these elements should be analytically expressed and completely included in any description method of finite motion. The traditional methods of describing finite motion are mainly based upon matrix Lie group and dual quaternions. An element of the matrix Lie group for finite motion description has at least twelve items. A dual quaternion has eight items. As the most well-known mathematical tool for finite motion description, dual quaternion provides an analytical expression of the three basic elements of finite motion. Hereinafter, we use L f , Lf =



 T T , sTf r f × s f

(2.1)

to denote the Plücker coordinates of the finite motion axis. t and θ are used to denote the  translational   distance and rotational angle along and about that axis. In Eq. (2.1), s f  s f  = 1 is the unit direction vector of the axis, and r f is the axis’ position vector pointing from the original point O of the fixed reference frame O−xyz to an arbitrary point on the axis. Through using the Plücker coordinates of the finite motion axis, the dual quaternion [12] can be expressed as 



θ θ D = cos + sin S, 2 2

(2.2)

where a dual unit ε with the property ε2 = 0 is introduced for putting rotation and 

translation into one formula to simplify expressions and operations. Hence, S =    s f + ε r f × s f denotes the dual vector form of finite motion axis, and θ = θ + εt indicates a dual angle. In this way, a dual quaternion is represented as D = cos

  θ + εt  θ + εt + sin sf +ε rf × sf . 2 2

(2.3)

Utilizing Taylor expansions of sine and cosine functions leads to cos

θ t θ θ + εt = cos − ε sin , 2 2 2 2

(2.4)

sin

θ + εt θ t θ = sin + ε cos . 2 2 2 2

(2.5)

and

28

2 Finite and Instantaneous Screw Theory

Substituting Eqs. (2.4)–(2.5) into Eq. (2.3) results in the expanded format of the dual quaternion, which is constructed by the dual scalar part and dual vector part. The former part contains two terms, and the latter contains six terms, as       t t θ θ θ θ θ + sin s f + ε sin r f × s f + cos s f D = cos − ε sin 2 2 2 2 2 2 2 = Ds + D v , (2.6) where Ds and Dv denote the dual scalar and dual vector parts of the dual quaternion, respectively, as Ds = cos

t θ θ − ε sin , 2 2 2

(2.7)

and    t θ θ θ Dv = sin s f + ε sin r f × s f + cos s f . 2 2 2 2

(2.8)

It can be clearly seen that the six-dimensional Dv contains all the three basic elements of finite motion. Without losing the elements, it can be rewritten into a θ six-dimensional quasi-vector form through being divided by half of cos , as 2     θ 0 sf S f = 2 tan +t , (2.9) sf 2 rf × sf which, named as finite screw [1], has a similar format with instantaneous screw. It has a general format as Eq. (2.9) in which the parts before and after “+” are, respectively, called its rotational and translational parts. The finite screw also has the following two formats depending on the rotational angle. When θ = 0, Eq. (2.9) can be rewritten into ⎛

sf

θ⎜ S f = 2 tan ⎝ r f × s f + 2

⎞ t θ 2 tan 2

which denotes a finite screw with amplitude 2 tan

⎟ s f ⎠, θ = 0,

θ and pitch 2

(2.10)

t

. In order to θ 2 tan 2 distinguish this pitch with that of the instantaneous screw, it is called quasi-pitch [1, 13]. Equation (2.10) expresses a pure rotation under the condition of t = 0, otherwise a helical motion.

2.2 Finite Screw

29

When θ = 0, Eq. (2.9) turns to be 

0 Sf = t sf

 , θ = 0,

(2.11)

which denotes a finite screw with amplitude t and infinite pitch. Different from Eq. (2.10), the Eq. (2.11) expresses a pure translation. It is noted that the quasi-vector form of finite screw together with the definition of its amplitude and quasi-pitch has been put forward by Hunt, Parkin, and Huang [13, 14]. However, the relationship between it and dual quaternion is discovered by the authors of this book for the first time. The rigorous derivations to obtain a finite screw from dual quaternion clearly prove the correctness of the quasi-vector form. Compared with other mathematical tools used to describe finite motion including dual quaternion, finite screw has a simpler and more concise format. In addition, the finite screw is not a true vector although it looks like one because it has nonlinear characteristics and its corresponding sets cannot form linear vector spaces. The algebraic structure of finite screws will be further discussed in the following sections of this chapter.

2.2.2 Screw Triangle Product When we describe finite motions with finite screws, the main problem encountered will be how to composite finite screws that express successive finite motions generated by the same rigid body. Since the finite screw has been derived from dual quaternion, we will derive the composition algorithm of finite screws from the dual quaternion multiplication. Suppose that the following two dual quaternions in the form of Eq. (2.6) are used to describe two successive finite motions,   ta θa θa − ε sin Da = cos 2 2 2     ta θa θa θa  r f,a × s f,a + cos s f,a + sin s f,a + ε sin 2 2 2 2 = Ds,a + Dv,a ,

(2.12)

  tb θb θb Db = cos − ε sin 2 2 2     tb θb θb  θb r f,b × s f,b + cos s f,b + sin s f,b + ε sin 2 2 2 2 = Ds,b + Dv,b .

(2.13)

and

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2 Finite and Instantaneous Screw Theory

In the meantime, their compositional motion is described as Dab

  tab θab θab −ε sin = cos 2 2 2     tab θab θab  θab r f,ab × s f,ab + + sin s f,ab + ε sin cos s f,ab 2 2 2 2 = Ds,ab + Dv,ab . (2.14)

The multiplication of the two dual quaternions for finite motion composition is implemented in the following way, Dab = Db Da ⇒ Ds,ab + Dv,ab = Ds,a Ds,b − Dv,a · Dv,b + Ds,b Dv,a + Ds,a Dv,b + Dv,b × Dv,a . (2.15) Substitute Eqs. (2.12)–(2.14) into Eq. (2.15), and compare the origin and dual parts of the dual scalar and dual vectors on both sides of Eq. (2.15). Equations (2.16)–(2.18) are obtained, cos sin

θa θb θa θb θab = cos cos − sin sin sTf,a s f,b , 2 2 2 2 2

(2.16)

 θab θa θa θa θb θb θb  s f,b × s f,a , s f,ab = sin cos s f,a + cos sin s f,b + sin sin 2 2 2 2 2 2 2 (2.17)  tab θab θab  r f,ab × s f,ab + cos s f,ab 2 2 2   θa θb tb θa θb θb  ta θa cos cos − sin sin s f,a + sin cos r f,a × s f,a = 2 2 2 2 2 2 2 2    θa θb ta θa θb θb  θa tb cos cos − sin sin s f,b + cos sin r f,b × s f,b + 2 2 2 2 2 2 2 2     θa θb ta θa θb tb sin cos + cos sin s f,b × s f,a + 2 2 2 2 2 2      θb  θa s f,b × r f,a × s f,a + r f,b × s f,b × s f,a . + sin sin (2.18) 2 2

sin

Dividing Eqs. (2.17)–(2.18) by half of cos

θab in Eq. (2.16) results in 2

 θb  θa θb θa s f,b × s f,a 2 tan s f,a + 2 tan s f,b + 2 tan tan θab 2 2 2 2 2 tan , s f,ab = θa θb T 2 1 − tan tan s f,a s f,b 2 2 (2.19)

2.2 Finite Screw

31

 θab  r f,ab × s f,ab + tab s f,ab 2    ⎞ ⎛ θa θa θb θb ta − tb tan tan s f,a + tb − ta tan tan s f,b ⎜ ⎟ 2 2 2 2 ⎜ ⎟ ⎜ ⎟     θ θ b ⎜ + 2 tan a r × s ⎟ r f,b × s f,b f,a f,a + 2 tan ⎜ ⎟ 2 2 ⎜  ⎟  ⎜ ⎟   θa θb ⎜ ⎟ + ta tan s f,b × s f,a ⎜ + tb tan ⎟ ⎜ ⎟ 2 2 ⎜ ⎟ ⎜ ⎟ θ θ ⎜ + 2 tan a tan b ⎟ ⎝ ⎠ 2     2  s f,b × r f,a × s f,a + r f,b × s f,b × s f,a = . θb θa 1 − tan tan sTf,a s f,b 2 2

2 tan

(2.20)

Based upon the relationship between dual quaternion and finite screw, the corresponding finite screws of Da , Db and Dab are S f,a = 2 tan

θa 2



s f,a r f,a × s f,a



 + ta

0 s f,a

 ,

    θb s f,b 0 S f,b = 2 tan + tb , s f,b 2 r f,b × s f,b     θab 0 s f,ab S f,ab = 2 tan + tab . s f,ab 2 r f,ab × s f,ab

(2.21) (2.22) (2.23)

In this way, rewriting Eqs. (2.19)–(2.20) respectively, as the first three terms and last three terms of a quasi-vector leads to 1 θb θa 1 − tan tan sTf,a s f,b 2 2    S f,b × S f,a θa θb 0 , − tan tan × S f,a + S f,b + 2 2 2 tb s f,a + ta s f,b (2.24)

S f,ab =

where the cross product “×” is known as the screw product [15], which computes out the screw along the common perpendicular of two screws, ⎛ S f,b × S f,a

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

⎞ θb θa s f,b × 2 tan s f,a 2 ⎟ 2 ⎟  θb θa  r f,a × s f,a + ta s f,a ⎟ 2 tan s f,b × 2 tan ⎟. 2 2 ⎟   ⎠   θa θb r f,b × s f,b + tb s f,b × 2 tan s f,a + 2 tan 2 2 2 tan

(2.25)

32

2 Finite and Instantaneous Screw Theory

It should be pointed out that the screw product is also applicable to instantaneous screws, which will be further discussed in the following sections. Equation (2.24) shows the composition algorithm of two finite screws, S f,a and S f,b . S f,ab is the composition screw. This algorithm is known as the screw triangle product [1], which is denoted by the operation symbol “”, as S f,ab = S f,a S f,b .

(2.26)

The screw triangle product is a binary operation. It is raised from the dual quaternion multiplication or Euler-Rodrigue’s formula with dual angles for dual quaternion composition. As shown in Eq. (2.24), the product is written as a semi-linear formula. The composition screw of two finite screws is the linear combination of five terms, i.e., two original screws, the screw along their common perpendicular and their translational parts. Based upon Eq. (2.24), the finite motion axis, the rotational angle and translational distance of the composition finite screw in Eq. (2.23) can be computed as θb θa θb θa s f,a + tan s f,b + tan tan s f,b × s f,a 2 2 2 2 , (2.27) s f,ab =    θ θ θ θ a b a tan s f,a + tan s f,b + tan tan b s f,b × s f,a    2 2 2 2     A s f,ab × s f,b + B s f,a × s f,ab   r f,ab = (2.28) ± ks f,ab , k ∈ R, sTf,a s f,ab × s f,b 2      T s f,a × s f,ab  T A = r f,a s f,ab × s f,a × s f,ab +  2 s f,b × s f,a × s f,b s f,a × s f,b   ta  2 s f,a × s f,ab , r f,b − r f,a − 2 2    T   s f,b × s f,ab  T B = r f,b s f,ab × s f,b × s f,ab +  2 s f,a × s f,b × s f,a s f,a × s f,b   tb  2 s f,b × s f,ab , r f,a + r f,b + 2 ⎞ ⎛   tan θa s f,a + tan θb s f,b + tan θa tan θb s f,b × s f,a  ⎟  ⎜ 2 2 2 2 ⎟, (2.29) θab = 2 arctan⎜ ⎠ ⎝ θb T θa 1 − tan tan s f,a s f,b 2 2 ⎞ ⎛ T   θb  θa tan r s − r × s tan f,a f,b f,a f,b ⎟ ⎜ 2 2   ⎟ 2⎜ ⎝ θa tb θb θb θa ⎠ ta tan + sTf,a s f,b tan + tan + sTf,a s f,b tan + 2 2 2 2 2 2   . tab =   tan θa s f,a + tan θb s f,b + tan θa tan θb s f,b × s f,a    2 2 2 2 (2.30) tan

2.2 Finite Screw

33

Fig. 2.2 Screw triangle product of finite screws

It can be summarized from Eq. (2.26) that the screw triangle product is a closure operation. The geometrical meaning of this product is illustrated in Fig. 2.2, in p p p which S f,a , S f,b , and S f,ab denote the translational parts of S f,a , S f,b , and S f,ab , respectively. Two finite screws and their composition are drawn as three directed lines with dual angles. These three lines constitute a screw triangle. Like a triangle constructed by three vectors, any two finite screws in the triangle can be composited into the third one [14]. The relationships among the three screws in a screw triangle can be expressed by Eq. (2.26) together with Eqs. (2.31)–(2.32),   S f,ab  −S f,b = S f,a ,

(2.31)

−S f,a S f,ab = S f,b .

(2.32)

and

In Eq. (2.26), we use the symbol “” to denote the screw triangle product with S f,a on the left of the operation symbol “” and S f,b on the right of “”. In other words, Eq. (2.26) is used to denote the algorithm in Eq. (2.24) for compositing any two finite screws, S f,a and S f,b , whose composition result is S f,ab . Since the screw triangle product is a binary operation, we can also use anther symbol “∇” to denote the composition product between finite screws, leading to another denotation of Eq. (2.24) with an inverse computation order between S f,a and S f,b , as S f,ab = S f,b ∇ S f,a .

(2.33)

Equations (2.26) and (2.33) are equivalent to each other, because they denote the same algorithm of the screw triangle product shown in Eq. (2.24) and Fig. 2.2. This equivalence is expressed as S f,a S f,b = S f,b ∇ S f,a .

(2.34)

34

2 Finite and Instantaneous Screw Theory

Similarly, the equations that correspond to the relationships shown in Eqs. (2.31)– (2.32) are −S f,b ∇ S f,ab = S f,a ,

(2.35)

  S f,ab ∇ −S f,a = S f,b ,

(2.36)

and

For any two successive finite motions generated by the same rigid body, it should be noted that the motion expressed by S f,a happens before the motion expressed by S f,b . This means that, for several motions from the one generated earliest to the one generated latest, the composition order led by “” is from the left to the right, while the composition order led by “∇” is from the right to the left. Following this manner, we name “” and “∇” as “the right screw triangle operation” and “the left screw triangle operation”, respectively. Suppose that the finite motions of a rigid body are generated by n motion generators. Hence, its resultant motion is the composition of the n motions generated by these motion generators. The generators are connected in serial structure. Commonly, they are numbered from the one fixed to the ground to the one connected to the rigid body, as the 1st, the 2nd, …, the nth joints. The n motions generated with their directions and/or axes at the initial poses are expressed by S f,1 , S f,2 , . . . , S f,n , respectively. It is easy to see that the movement of a joint has influence on the directions and/or axes of the joints with bigger numbers. In other words, the ith joint will change the directions and/or axes of the (i + 1)th, …, nth joints from their initial poses. Therefore, the motion generators should move from the nth one to the 1st one in order to generate the motions S f,n , . . . , S f,1 , which means that these motions are generated from S f,n to S f,1 . We use S f,n···21 to denote the resultant motion of the rigid body. It is the composition of S f,1 , S f,2 , . . . , S f,n . It can be computed in any of the following two ways using “the right screw triangle operation” and “the left screw triangle operation”, as S f,n···21 = S f,n  · · · S f,2 S f,1 ,

(2.37)

S f,n···21 = S f,1 ∇ S f,2 ∇ · · · ∇ S f,n .

(2.38)

and

The above two equations are equivalent to each other. From the comparisons between their physical meanings, it can be concluded that (1) The computation order of “” in Eq. (2.37) from left to right obeys the movement order in which the motions are generated. (2) The computation order of “∇” in Eq. (2.38) from left to right obeys the connection order in which the motion generators are numbered.

2.2 Finite Screw

35

Due to the equivalence between Eqs. (2.37)–(2.38), and without loss of generality, “” (“the right screw triangle operation”) is consistently used to denote the screw triangle product hereinafter for readers’ convenience.

2.2.3 Algebraic Structure of Finite Screw After the format of finite screw is proved to be closed under the screw triangle product, the entire set of finite screws is supposed to be a specific algebraic structure. When the finite motion axis, the translational distance, and the rotational angle are arbitrarily selected, the entire set of finite screws is defined as θ S f = 2 tan 2



sf rf × sf





0 +t sf

 , s f , r f ∈ R3 , θ ∈ [0, 2π], t ∈ R.

(2.39)

As a mathematical tool to describe finite motions, the finite screw set in Eq. (2.39) should form a Lie group which is homomorphism with the Special Euclidean group SE(3). In order to verify this conjecture, it will be proved whether this finite screw set satisfies the properties of Lie group under the screw triangle product, i.e., the closure, the associativity, the existence of an identity element, and the existence of an inverse of any element. (a) The closure As proved in Sect. 2.2.2, the composition of any two finite screws can be rewritten into a finite screw in the same format as these two screws. The detailed proof is shown in Eqs. (2.27)–(2.30). Hence, it can be concluded that the finite screw set is closed under the screw triangle product. (b) The associativity Suppose that S f,i (i = a, b, c) are three arbitrarily selected elements in the finite screw set, each is in the form of Eq. (2.9) as S f,i = 2 tan

θi 2



s f,i r f,i × s f,i



 + ti

0 s f,i

 , i = a, b, c.

(2.40)

The composition result of these three elements can be computed in the following two different sequences,   S f,(ab)c = S f,a S f,b S f,c ,

(2.41)

  S f,a(bc) = S f,a  S f,b S f,c .

(2.42)

and

In the former computation sequence in Eq. (2.41), the composition is computed as

36

2 Finite and Instantaneous Screw Theory

S f,(ab)c = S f,ab S f,c ⎞ ⎛ 1 1 1 S f,a + S f,b + S f,c + S f,b × S f,a + S f,c × S f,a + S f,c × S f,b ⎟ ⎜ 2 2 2 ⎟   1⎜ 1 ⎟ ⎜ = ⎜ + S f,c × S f,b × S f,a − (1 − D)S f,c ⎟, ⎟ C⎜ 4 ⎠ ⎝ 1 − S f,c × S f p,ab − S f p,ab − S f p,(ab)c 2 (2.43) where θa θb θc θc θa θb tan sTf,a s f,b − tan tan sTf,a s f,c − tan tan sTf,b s f,c 2 2 2 2 2 2  θb θc T  θa − tan tan tan s f,c s f,b × s f,a , 2 2 2

C = 1 − tan

D = 1 − tan ⎛ S f p,ab = ⎝

0

tan

⎞

⎠, θa θb θb θa tan tb s f,a + tan tan ta s f,b 2 2 2 2

⎞ 0  ⎟ ⎜ ⎟ ⎜ tan θa tan θc tc s f,a + tan θb tan θc tc s f,b ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜  θb θc  θa ⎟ ⎜ + tan tan tan tc s f,b × s f,a ⎟ ⎜ 2 2 2 ⎟ ⎜ T   θa θb θc  =⎜ ⎟. + 2 tan tan tan r f,a − r f,b s f,a × s f,b s f,c ⎟ ⎜ ⎟ ⎜ 2 2 2   ⎟ ⎜ θc θb θc T θa ⎟ ⎜ + tan tan + tan tan s f,a s f,b ta s f,c ⎟ ⎜ 2 2 2 2 ⎟ ⎜    ⎠ ⎝ θb θc θa θc T + tan tan + tan tan s f,a s f,b tb s f,c 2 2 2 2 ⎛

S f p,(ab)c

θb θa tan sTf,a s f,b , 2 2

In the other computation sequence in Eq. (2.42), the composition is computed similarly in the following manner, S f,a(bc) = S f,a S f,bc ⎞ ⎛ 1 1 1 S f,a + S f,b + S f,c + S f,b × S f,a + S f,c × S f,a + S f,c × S f,b ⎟ ⎜ 2 2 2 ⎟  1⎜ 1 ⎟ ⎜ = ⎜ + S f,c × S f,b × S f,a − (1 − E)S f,a ⎟, ⎟ C⎜ 4 ⎠ ⎝ 1 − S f p,bc × S f,a − S f p,bc − S f p,a(bc) 2 (2.44) where

2.2 Finite Screw

37

E = 1 − tan ⎛ S f p,bc = ⎝

0

tan

⎛

S f p,a(bc)

θc θb tan sTf,b s f,c , 2 2

⎞

⎠, θb θc θc θb tan tc s f,b + tan tan tb s f,c 2 2 2 2

⎞ 0  ⎜ ⎟ ⎜ tan θa tan θb ta s f,b + tan θa tan θc ta s f,c ⎟ ⎜ ⎟ 2 2 2 2 ⎜ ⎟ ⎜ ⎟   θa θb θc ⎜ ⎟ s + tan × s tan tan t a f,c f,b ⎜ ⎟ 2 2 2 ⎜ ⎟ T   θa θb θc  =⎜ ⎟. + 2 tan tan tan r f,b − r f,c s f,b × s f,c s f,a ⎟ ⎜ ⎜ ⎟ 2 2 2   ⎜ ⎟ θb θa θc T θa ⎜ ⎟ + tan tan + tan tan s f,b s f,c tb s f,a ⎜ ⎟ 2 2 2 2 ⎜ ⎟    ⎝ ⎠ θc θa θb T θa + tan tan + tan tan s f,b s f,c tc s f,a 2 2 2 2

Through analytical derivations, it is easy to verify that Eq. (2.45) holds,   1 1 S f,c × S f,b × S f,a − (1 − D)S f,c − S f,c × S f p,ab − S f p,ab − S f p,(ab)c 4 2  1 1 = S f,c × S f,b × S f,a − (1 − E)S f,a − S f p,bc × S f,a − S f p,bc − S f p,a(bc) . 4 2 (2.45) Substituting Eq. (2.45) into Eqs. (2.43)–(2.44) results in     S f,(ab)c = S f,a(bc) ⇒ S f,a S f,b S f,c = S f,a  S f,b S f,c .

(2.46)

The above equation indicates that the computation sequence has no influence on the composition result of several finite screws. In other words, the screw triangle product satisfies the associative law with regards to any three or more elements in the finite screw set. (c) The identity element The identity element in a finite motion transformation group denotes the null transformation. In the finite screw set, the null transformation is defined as a screw with both the rotational angle and the translational distance set to be zero. Thus, the identity element in this set is     0 0 sf +0 , (2.47) S f 0 = 2 tan sf 2 rf × sf which can be simplified as

38

2 Finite and Instantaneous Screw Theory

Sf0

  0 = . 0

(2.48)

The screw triangle product of the identity element and an arbitrary screw in the finite screw set satisfies the communicative law, as S f,a S f 0 = S f,a ,

(2.49)

S f 0 S f,a = S f,a .

(2.50)

(d) The inverse of an element According to the definition of the finite screw set in Eq. (2.39), if S f,a is an arbitrarily selected screw in the set, −S f,a must be in the set. −S f,a is defined to be the inverse of S f,a , as − S f,a

   0 s f,a − ta r f,a × s f,a s f,a     −θa 0 s f,a + (−ta ) = 2 tan r f,a × s f,a s f,a 2     θa 0 −sf,a  , + ta = 2 tan −s f,a 2 r f,a × −s f,a

θa = −2 tan 2

⇒ −S f,a ⇒ −S f,a



(2.51)

which has the same finite motion axis with S f,a , but has the opposite rotational angle and translational distance. −S f,a can also be regarded as a screw having the reverse axis but the same rotational angle and translational distance with S f,a . A finite screw and its inverse are communicative in the screw triangle product, and they satisfy the annihilation relationship. Their composition is the identity element,   S f,a  −S f,a = S f 0 ,

(2.52)

−S f,a S f,a = S f 0 .

(2.53)

Through verifying the closure, the associativity, the existence of an identity element, and the existence of an inverse of any element, we find that the algebraic structure of the finite screw set in Eq. (2.39) is strictly proved to be a group. Since the expression of the set is continuous and smooth, the finite screw set is also a differentiable manifold. Thus, it is a true Lie group named as the finite screw Lie group. The differential properties of this Lie group will be further discussed in Sect. 2.4.

2.3 Instantaneous Screw

39

2.3 Instantaneous Screw As the counterpart of finite screw, the instantaneous screw describes instantaneous motion of a rigid body in vector form. It means that any velocity or infinitesimal displacement of a rigid body obtained at a given pose can be described by an instantaneous screw. Here, velocity includes both linear velocity and angular velocity. Similar to those of finite motion, the three basic elements of an instantaneous motion are concluded based upon Mozzi’s theorem [16]. The three basic elements are: (a) the Mozzi’s axis which will be called instantaneous motion axis here-in-after, (b) the magnitude of the angular velocity about that axis, and (c) the magnitude of the linear velocity along the same axis, as shown in Fig. 2.3. Thus, these elements should be analytically expressed and completely included in any description method of instantaneous motion. Because velocity can be amplified by multiplying scalar and several velocities can be linearly added together, the described methods of instantaneous motion are all in the form of vectors in nature [17, 18]. In this way, the sets of instantaneous motions are described by vector spaces.

2.3.1 Instantaneous Screw in Vector Form Among various vector-based mathematical tools that are used to describe instantaneous motion, instantaneous screw has the most concise format. Six-dimensional Plücker ray coordinates are employed to express the instantaneous motion axis, which is also called the unit instantaneous screw along the axis as shown in Eq. (2.54), T  L t = sTt (r t × st )T ,

(2.54)

where L t denotes the Plücker coordinates of the instantaneous motion axis, st (|st | = 1) is the unit direction vector of the axis, and r t is the axis’ position vector Fig. 2.3 Basic elements of instantaneous motion

40

2 Finite and Instantaneous Screw Theory

pointing from the original point O of the fixed reference frame to an arbitrary point on the axis. The instantaneous screw can be regarded as the combination of a unit instantaneous motion axis, the amplitude of the motion and a pitch. The pitch is defined as the ratio between the magnitude of the linear velocity and that of the angular velocity. It is expressed as pt =

v , ω

(2.55)

where pt denotes the pitch of instantaneous motion, v and ω are the magnitude of the linear velocity along the instantaneous motion axis and the angular velocity about that axis, respectively. The amplitude of instantaneous screw is defined correspondingly as  at =

ω ω = 0 , v ω=0

(2.56)

where at is used to denote the amplitude of instantaneous screw, which is given in two situations including velocities of pure rotation and helical motion (ω = 0) as well as pure translation (ω = 0). Based upon the pitch in Eq. (2.55) and the amplitude in Eq. (2.56), the standard format of the instantaneous screw is given as  ⎧  st ⎪ ⎪ ⎪ω v ω = 0 ⎨ r t × st + st , St = ω   ⎪ ⎪ 0 ⎪ ⎩ ω=0 v st

(2.57)

where St denotes instantaneous screw. This format can be further rewritten into a general format. The general format does not need to distinguish the two conditions, as  St = ω

st r t × st



 +v

 0 . st

(2.58)

In the above equation, it is easy to see that the instantaneous screw is a sixdimensional vector. According to the linear superposition property of velocities, two velocities generated by the same rigid body at a given pose can be linearly composited. This means that two instantaneous screws can be directly added together to get the composition screw, as St,ab = St,a + St,b ,

(2.59)

2.3 Instantaneous Screw

41

where St,ab is used to denote the composition instantaneous screw of two screws St,a and St,b , which are in the form of Eq. (2.58) as  St,a = ωa

st,a r t,a × st,a



 + va

0 st,a

 ,

(2.60)

and  St,b = ωb

st,b r t,b × st,b



 + vb

 0 . st,b

(2.61)

The composition instantaneous screw in Eq. (2.59) can be rewritten into the standard format in Eq. (2.58) as  St,ab = ωab

st,ab r t,ab × st,ab



 + vab

0 st,ab

 ,

(2.62)

where ωa st,a + ωb st,b , st,ab =  ωa st,a + ωb st,b  r t,ab =

ωa r t,a × st,a + ωb r t,b × st,b + va st,a + vb st,b   × st,ab ± kst,ab , k ∈ R, ωa st,a + ωb st,b    ωab = ωa st,a + ωb st,b , T  vab = ωa r t,a × st,a + ωb r t,b × st,b + va st,a + vb st,b st,ab .

Similar to finite screws, two instantaneous screws and their composition screw form a triangle, too. This triangle is a vector triangle, as shown in Fig. 2.4. Any two vectors in the triangle can be linearly added to get the third one. Fig. 2.4 Screw triangle of instantaneous screws

42

2 Finite and Instantaneous Screw Theory

The relationships between the three screws in Fig. 2.4 can be expressed by Eq. (2.59) together with the following two equations, −St,a + St,ab = St,b ,

(2.63)

  St,ab + −St,b = St,a .

(2.64)

and

2.3.2 Algebraic Structure of Instantaneous Screw When the instantaneous motion axis, the magnitude of linear velocity and the magnitude of angular velocity are arbitrarily selected, the following instantaneous screw expression defines the entire set of instantaneous screws as 

st St = ω r t × st





 0 +v , st , r t ∈ R3 , ω, t ∈ R, st

(2.65)

which is a vector space because of its linear property as shown in Eqs. (2.59)–(2.62). The instantaneous screw set in Eq. (2.65) should form a Lie algebra which is isomorphic with the Lie algebra of SE(3), i.e., se(3), since it is a mathematical tool to describe instantaneous motions. In order to verify this conjecture, it will be proved whether this instantaneous screw set satisfies the properties of Lie algebra under the screw product, i.e., the closure, the distributivity, the inverse commutativity, and the cyclosymmetric annihilation. (a) The closure For any two elements St,a and St,b in the instantaneous screw set, their screw product can be computed as SCt,ab = St,a × St,b           st,a 0 st,b 0 + va × ωb + vb = ωa r t,a × st,a r t,b × st,b st,a st,b   ωa st,a × ωb st,b   , = ωa st,a × ωb r t,b × st,b + vb st,b + ωa r t,a × st,a + va st,a × ωb st,b (2.66) where SCt,ab is used to denote the screw product of St,a and St,b , which indicates a screw along the common perpendicular of the two screws. SCt,ab can be rewritten into the standard form of the instantaneous screw in Eq. (2.58) as

2.3 Instantaneous Screw

43

 SCt,ab

=

C ωab

sCt,ab C r t,ab × sCt,ab



 +

C vab

0 sCt,ab

 ,

(2.67)

where

r Ct,ab =

ωa st,a

ωa st,a × ωb st,b , sCt,ab =  ωa st,a × ωb st,b      × ωb r t,b × st,b + vb st,b + ωa r t,a × st,a + va st,a × ωb st,b   ωa st,a × ωb st,b 

× sCt,ab ± ksCt,ab ,

k ∈ R,   C ωab = ωa st,a × ωb st,b ,

   T   C vab = ωa st,a × ωb r t,b × st,b + vb st,b + ωa r t,a × st,a + va st,a × ωb st,b sCt,ab . Thus, the screw product is a closure operation with respect to the elements in the instantaneous screw set. This means that the set is closed under the screw product. (b) The distributivity If the screw product satisfies the distributive law when operating on instantaneous screws, Eqs. (2.68)–(2.70) should all hold,  St,a + St,b × St,c = St,a × St,c + St,b × St,c ,

(2.68)

  St,a × St,b + St,c = St,a × St,b + St,a × St,c ,

(2.69)

      k St,a × St,b = St,a × k St,b = k St,a × St,b .

(2.70)



Taking Eq. (2.68) for example, its left side can be expanded through computing the screw product twice, as 

 St,a + St,b × St,c          st,a 0 st,b 0 + va + ωb + vb = ωa r t,a × st,a r t,b × st,b st,a st,b      st,c 0 + vc × ωc r t,c × st,c st,c   ⎛ ⎞ ωa st,a +   ωb st,b × ωc st,c  ⎠. =⎝ (2.71) ωa st,a + ωb st,b × ωc r t,c × st,c + vc st,c  + ωa r t,a × st,a + va st,a + ωb r t,b × st,b + vb st,b × ωc st,c

44

2 Finite and Instantaneous Screw Theory

The above equation can be rewritten as 

 St,a + St,b × St,c ⎞ ⎛ ωa st,a × ωc st,c +   ωb st,b × ωcst,c   ⎠ = ⎝  ωa st,a × ωc r t,c × st,c  + vc st,c +ωb st,b × ωc r t,c × st,c + vc st,c  + ωa r t,a × st,a + va st,a × ωc st,c + ωb r t,b × st,b + vb st,b × ωc st,c   ωa st,a × ωc st,c   = ωa st,a × ωc r t,c × st,c + vc st,c + ωa r t,a × st,a + va st,a × ωc st,c   ωb st,b × ωc st,c   + , ωb st,b × ωc r t,c × st,c + vc st,c + ωb r t,b × st,b + vb st,b × ωc st,c (2.72)

which equals to the right side of Eq. (2.68). In this way, Eq. (2.68) is verified to be true. Using the same manner, we can verify that Eqs. (2.69)–(2.70) also hold. Hence, it can be concluded that the screw product operating on the instantaneous set obeys the distributive law. (c) The inverse commutativity Change the sequence of the two elements St,a and St,b in the screw product of Eq. (2.66). Another screw along their common perpendicular can be obtained as SCt,ba = St,b × St,a    ωb st,b × ωa st,a   . =  ωb r t,b × st,b + vb st,b × ωa st,a + ωb st,b × ωa r t,a × st,a + va st,a (2.73) Comparing the above equation with Eq. (2.66), it is easy to see that SCt,ba = −SCt,ab .

(2.74)

Thus, it is verified that the screw product satisfies the inverse commutative law when it operates on the instantaneous screw set. (d) The cyclosymmetric annihilation The cyclosymmetric annihilation formula of the screw product is given as       St,a × St,b × St,c + St,b × St,c × St,a + St,c × St,a × St,b = 0.

(2.75)

2.3 Instantaneous Screw

45

This formula can be proved by expanding the three ternary-computations using the algorithms of the screw product. Firstly, we expand the first ternary-computation in Eq. (2.75) as   St,a × St,b × St,c      st,a 0 + va = ωa r t,a × st,a st,a           st,b 0 st,c 0 × ωb + vb × ωc + vc r t,b × st,b r t,c × st,c st,b st,c   ⎛ ⎞ ωa st,a × ωb st,b × ωc st,c   ⎟  ⎜ ⎜ ωa st,a × ωb r t,b × st,b + vb st,b × ωc st,c ⎟ ⎟ (2.76) =⎜ ⎜ + ω s × ω s × ω r × s + v s  ⎟ a t,a b t,b c t,c t,c c t,c ⎝ ⎠     + ωa r t,a × st,a + va st,a × ωb st,b × ωc st,c In the same manner, the second and third ternary-computations in Eq. (2.75) are expanded as   St,b × St,c × St,a ⎞ ⎛   ω   b st,b × ωc st,c × ωa st,a ⎟ ⎜ ωb st,b × ωc r t,c × st,c + vc st,c × ωa st,a  ⎟,   =⎜ ⎝ + ωb st,b × ωc st,c × ωa r t,a × st,a + va st,a ⎠     + ωb r t,b × st,b + vb st,b × ωc st,c × ωa st,a   St,c × St,a × St,b ⎞ ⎛   ω    c st,c × ωa st,a × ωb st,b ⎜ ωc st,c × ωa r t,a × st,a + va st,a × ωb st,b ⎟  ⎟   =⎜ ⎝ + ωc st,c × ωa st,a × ωb r t,b × st,b + vb st,b ⎠.     + ωc r t,c × st,c + vc st,c × ωa st,a × ωb st,b

(2.77)

(2.78)

When the right sides of Eqs. (2.76)–(2.78) are added together, based on the principle of three-vector cross-product expansion, the resultant expression has four annihilation sub-expressions in the form of a × (b × c) + b × (c × a) + c × (a × b) = 0,

(2.79)

where a, b and c are three arbitrary three-dimensional vectors. The upper part of the resultant expression has one annihilation sub-expression as     ωa st,a × ωb st,b × ωc st,c +ωb st,b × ωc st,c × ωa st,a   +ωc st,c × ωa st,a × ωb st,b = 0.

(2.80)

46

2 Finite and Instantaneous Screw Theory

Its lower part has three annihilation sub-expressions as follows,      ωa st,a × ωb r t,b × st,b + vb st,b × ωc st,c + ωb r t,b × st,b + vb st,b      × ωc st,c × ωa st,a + ωc st,c × ωa st,a × ωb r t,b × st,b + vb st,b = 0, (2.81)    ωa st,a × ωb st,b × ωc r t,c × st,c + vc st,c + ωb st,b    × ωc r t,c × st,c + vc st,c × ωa st,a     + ωc r t,c × st,c + vc st,c × ωa st,a × ωb st,b = 0, (2.82)     ωa r t,a × st,a + va st,a × ωb st,b × ωc st,c + ωb st,b    × ωc st,c × ωa r t,a × st,a + va st,a + ωc st,c    (2.83) × ωa r t,a × st,a + va st,a × ωb st,b = 0. In this way, both the upper and lower parts of the resultant expression are proved to be null. Hence, the cyclosymmetric annihilation formula of the screw product holds. From the above analysis, the algebraic structure of the instantaneous screw set in Eq. (2.65) is strictly proved to be a Lie algebra, because the elements in this set under the screw product satisfy the closure, the distributivity, the inverse commutativity, and the cyclosymmetric annihilation [1]. In this manner, it is named as the instantaneous screw Lie algebra. The relationship between this Lie algebra and the finite screw Lie group will be investigated in Sect. 2.2.4.

2.3.3 Twist and Wrench with Reciprocal Product In the above sections, we discussed the instantaneous screw in six-dimensional vector form, which is used to describe the instantaneous motion of a rigid body based upon Mozzi’s theorem. Besides, there exists another kind of instantaneous screw describing the exerted force on a rigid body. In order to distinguish the two kinds of instantaneous screws, the former one is called twist, and the latter one is called wrench. According to Eqs. (2.57)–(2.58), when one or none of ω and v equals zero, a unit twist has one of the following three forms: (1) A linear vector with zero pitch that describes an angular instantaneous motion,

Sˆ t,r =



 st,r . r t,r × st,r

(2.84)

2.3 Instantaneous Screw

47

(2) A couple with an infinite pitch that describes a linear instantaneous motion, Sˆ t,t =



 0 . st,t

(2.85)

(3) A general twist with non-zero and finite pitch that describes a helical instantaneous motion combining a unit angular part with a linear part along the same directions, Sˆ t,h =



st,h r t,h × st,h



 + pt

0 st,h

 .

(2.86)

In Eqs. (2.84)–(2.86), Sˆ t , st and r t with specific subscripts denote a unit twist, the unit direction vector and position vector of the corresponding twist axis. As the counterpart of twist, wrench describes the force on a rigid body. The Poinsot’s theorem gives that the resultant force of several exerted forces on a rigid body can be equivalently regarded as a force along an axis together with a torque about the same axis. In this way, the six-dimensional Plücker axis coordinates are employed to express the axis of the resultant force as,  T L w = (r w × sw )T sTw ,

(2.87)

where L w denotes the Plücker coordinates of the axis of the resultant force, sw (|sw | = 1) is the unit direction vector of the axis, and r w is the axis’ position vector pointing from the original point O of the fixed reference frame to an arbitrary point on the axis. According to Poinsot’s theorem, an exerted force of a rigid body can be expressed by a wrench that is the combination of the Plücker coordinates of its axis, the intensity and a pitch. The pitch is defined as the ratio between the magnitude of the torque about the axis and the magnitude of the force along that axis, which is expressed as pw =

τ , f

(2.88)

where pw denotes the pitch of the exerted force (wrench), τ and f are the magnitudes of the torque about the wrench axis and the force along that axis. Accordingly, the intensity of wrench is defined as  aw =

f f = 0 , τ f =0

(2.89)

48

2 Finite and Instantaneous Screw Theory

where aw is the intensity of wrench, which is given in two situations including general force ( f = 0) as well as pure torque ( f = 0). Using the Plücker axis coordinates, the pitch, and the intensity in Eqs. (2.87)– (2.89), the standard format of the wrench is given as ⎞ ⎧ ⎛ τ ⎪ r s × s + ⎪ w w w ⎪ ⎪ f ⎠ f = 0 ⎨ f⎝ sw , Sw =   ⎪ ⎪ s ⎪ ⎪ f =0 τ w ⎩ 0

(2.90)

where Sw denotes wrench. The standard format can be further rewritten into a general format, which does not need to distinguish the two situations, as  Sw = f

r w × sw sw





 sw +τ . 0

(2.91)

Similar to instantaneous motions, the forces exerted on the same rigid body at a given pose can be linearly composited. This means that wrenches are also sixdimensional vectors, and two wrenches can be directly added together to get the composition wrench, as Sw,ab = Sw,a + Sw,b ,

(2.92)

where Sw,ab is used to denote the composition wrench of two wrenches Sw,a and Sw,b . Sw,a and Sw,b are in the form of Eq. (2.91) as  Sw,a = f a

r w,a × sw,a sw,a



 + τa

 sw,a , 0

(2.93)

 sw,b . 0

(2.94)

and  Sw,b = f b

r w,b × sw,b sw,b



 + τb

It can be easily seen that the entire set of wrenches is also a vector space that is closure under additional operation. Similar to the three kinds of unit twists, when one or none of f and τ in Eqs. (2.90) or (2.91) equals zero, a unit wrench has one of the following three forms: (1) A linear vector with zero pitch that describes a unit pure force, Sˆ w, f =



r w, f × sw, f sw, f

 .

(2.95)

2.3 Instantaneous Screw

49

(2) A couple with an infinite pitch that describes a unit torque, Sˆ w,c =



 sw,c . 0

(2.96)

(3) A general wrench with nonzero and finite pitch that describes a unit hybrid force combining a unit pure force part with a torque part along the same directions, Sˆ w,h =



r w,h × sw,h sw,h



 + pw

 sw,h . 0

(2.97)

In Eqs. (2.95)–(2.97), Sˆ w , sw and r w with specific subscripts denote a unit wrench, the unit direction vector and position vector of the corresponding wrench axis. Consider that the multiplication of velocity and force achieves the power. In the screw theory, the reciprocal product of twist and wrench is defined to express the power done by or given to a rigid body. The reciprocal product is expressed as STt Sw

  = ω

st r t × st





0 +v st

T      r w × sw s f +τ w sw 0

= (ωst )T ( f r w × sw + τ sw ) + (ωr t × st + vst )T ( f sw ).

(2.98)

When a reciprocal product of a twist and a wrench equals to zero, STt Sw = 0,

(2.99)

they are reciprocal to each other. There are totally nine combinations of unit twists and unit wrenches, and the geometric conditions for their reciprocal relationships are listed as follows: T (1) Sˆ t,r Sˆ w, f = 0:

   T sTt,r r w, f × sw, f + r t,r × st,r sw, f = 0 T    ⇒ r w, f − r t,r sw, f × st,r = 0

(2.100)

which means that are parallel  the twist and the wrench are coplanar.  Their directions   sw, f = ±st,r , or their axes have intersecting points r w, f = r t,r .

50

2 Finite and Instantaneous Screw Theory

T (2) Sˆ t,r Sˆ w,c = 0:

T  sTt,r sw,c + r t,r × st,r 0 = 0 ⇒ sTt,r sw,c = 0,

(2.101)

which means that the twist and the wrench are perpendicular to each other, i.e., they have vertical directions. T (3) Sˆ t,r Sˆ w,h = 0:    T sTt,r r w,h × sw,h + pw sw,h + r t,r × st,r sw,h = 0 T    ⇒ r w,h − r t,r sw,h × st,r + pw sTt,r sw,h = 0,

(2.102)

which means that the twist and the wrench are coplanar, and they have vertical directions, i.e., their axes are intersecting and perpendicular to each other. T (4) Sˆ t,t Sˆ w, f = 0:

  0T r w, f × sw, f + sTt,t sw, f = 0 ⇒ sTt,t sw, f = 0,

(2.103)

which is similar to Eq. (2.101), and means that the twist and the wrench have vertical directions. T (5) Sˆ t,t Sˆ w,c = 0: 0T sw,c + sTt,t 0 = 0,

(2.104)

which shows that any linear velocity and any torque are reciprocal to each other. T (6) Sˆ t,t Sˆ w,h = 0:

  0T r w,h × sw,h + pw sw,h + sTt,t sw,h = 0 ⇒ sTt,t sw,h = 0,

(2.105)

which means that the twist and the wrench have vertical directions. T (7) Sˆ t,h Sˆ w, f = 0:    T sTt,h r w, f × sw, f + r t,h × st,h + pt st,h sw, f = 0 T    ⇒ r w, f − r t,h sw, f × st,h + pt sTt,h sw, f = 0,

(2.106)

which is similar to Eq. (2.102), and means that the twist and the wrench have intersecting and perpendicular axes.

2.3 Instantaneous Screw

51

T (8) Sˆ t,h Sˆ w,c = 0:

T  sTt,h sw,c + r t,h × st,h + pt st,h 0 = 0 ⇒ sTt,h sw,c = 0,

(2.107)

which is similar to Eq. (2.105), and means that the twist and the wrench have vertical directions. T (9) Sˆ t,h Sˆ w,h = 0:

   T sTt,h r w,h × sw,h + pw sw,h + r t,h × st,h + pt st,h sw,h = 0 T    ⇒ r w,h − r t,h sw,h × st,h + ( pt + pw )sTt,h sw,h = 0,

(2.108)

which shows that the twist and the wrench have intersecting and perpendicular axes. The linear combinations of several twists (wrenches) form a twist (wrench) space, which is spanned by several twists (wrenches). These twists (wrenches) are called base twists (wrenches) of the twist (wrench) space. The twist space describes all the feasible velocities of a rigid body at a given pose, while the wrench space describes all the forces exerted on the rigid body at that pose. When each base twist of the twist space is reciprocal to all the base wrenches of the wrench space, the screw spaces are reciprocal to each other. In this situation, the wrench space describes all the constraint forces that restrict the rigid body motion.

2.3.4 Classification of Twist Spaces and Wrench Spaces For an f-dimensional ( f ≤ 6) twist space, it is spanned by f base screws. Each screw is a unit twist in one of the three forms as shown in Eqs. (2.84)–(2.86). Suppose that among the f base screws, the number of linear vectors is n r , the number of couples is n t , and the number of general twists is f − n r − n t , the twist space can be expressed as St =

nr  i=1

ωr,i Sˆ t,r,i +

n r +n t i=nr +1

vt,i Sˆ t,t,i +

f 

ωh,i Sˆ t,h,i , ωr,i , vt,i , ωh,i ∈ R,

i=nr +n t +1

(2.109) which is the general form of twist spaces. A twist space can be classified according to the numbers of linear vectors, couples, and general twists contained in its base screws, as well as the geometric relations among the base screws. The total number of the base screws equals to its dimension. In this way, the different kinds of twist spaces with one to five dimensions will be classified and illustrated in detail.

52

2 Finite and Instantaneous Screw Theory

(a) One-dimensional twist spaces A one-dimensional twist space has only one base screw. It can be spanned by either a linear vector, or a couple, or a general twist. If the twist space is spanned by a linear vector, it can be expressed as St,1−1 = ωr,1 Sˆ t,r,1 , ωr,1 ∈ R.

(2.110)

Similarly, if the twist space is spanned by a couple, it can be expressed as St,1−2 = vt,1 Sˆ t,t,1 , vt,1 ∈ R.

(2.111)

In a similar way, if the twist space is spanned by a general screw, it can be expressed as St,1−3 = ωh,1 Sˆ t,h,1 , ωh,1 ∈ R.

(2.112)

Here, St,1−i (i = 1, 2, 3) are used to denote the one-dimensional twist spaces. (b) Two-dimensional twist spaces There are two base screws in a two-dimensional twist space, which can be spanned by one of the following pairs of unit twists (1) two linear vectors with different directions; (2) two linear vectors with the same direction but different axes, or a linear vector and a couple with the perpendicular directions; (3) two couples with different directions; (4) two general vectors with different directions; (5) two general vectors with the same directions but different axes, or a general vector and a couple with different directions (specific conditions between their directions); (6) two general vectors with the same axes but different pitches, or a general vector and a couple with the same directions, or a linear vector and a couple with the same directions, or a general vector and a linear vector with the same axes; (7) a linear vector and a couple with non-perpendicular and different directions; (8) a general screw and a linear vector with different directions; (9) a general screw and a linear vector with the same directions but different axes, or a general screw and a couple with different directions (specific conditions between their directions). Here, the opposite unit directions are regarded in the same directions. The expressions of these two-dimensional twist spaces can be, respectively, obtained by spanning the corresponding pairs of base screws. For the first and second two-dimensional twist spaces that are spanned by two linear vectors, their expressions are

2.3 Instantaneous Screw

53

St,2−1/2 = ωr,1 Sˆ t,r,1 + ωr,2 Sˆ t,r,2 , ωr,1 , ωr,2 ∈ R

(2.113)

where Sˆ t,r,1 and Sˆ t,r,2 have the formats of Eq. (2.84), as Sˆ t,r,1 =



   st,r,1 st,r,2 , Sˆ t,r,2 = . r t,r,1 × st,r,1 r t,r,2 × st,r,2

(2.114)

Thus, for the first twist space that has two linear vectors with different directions as base screws, its expression can be obtained as  St,2−1 = ωr,1

st,r,1 r t,r,1 × st,r,1



 + ωr,2

 st,r,2 , st,r,1 × st,r,2 = 0, r t,r,2 × st,r,2 ωr,1 , ωr,2 ∈ R. (2.115)

When the two linear vectors with different directions are in the same plane, i.e., they have intersecting axes, the twist space spanned by them consists of all the linear vectors in that plane passing through their intersection point O  . In this situation, the base screws of the twist space can be arbitrarily selected as any two linear vectors in the plane whose axes pass through O  , as  St,2−2 = ωr,1



 + ωr,2

st,r,1 r o × st,r,1



 + ωr,2



st,r,2  r o × st,r,2



 st,r,2 , r o × st,r,2     T   sT t,r,1 s t,r,1 × s t,r,2 = 0, s t,r,2 s t,r,1 × s t,r,2 = 0, ωr,1 , ωr,2 ∈ R, (2.116)

 = ωr,1



st,r,1  r o × st,r,1

where r o is the position vector of the intersection point O . For the second twist space whose base screws are two linear vectors with the same directions, its expression is 

   st,r,1 st,r,1 + ωr,2 St,2−2 = ωr,1 r t,r,1 × st,r,1 r t,r,2 × st,r,1     st,r,1 = ωr,1 + ωr,2 r t,r,1 × st,r,1 ⎛ ⎞ 0    ⎜  ⎟ + ωr,2  r t,r,2 − r t,r,1 × st,r,1 ⎝ r t,r,2 − r t,r,1 × st,r,1 ⎠,     r t,r,2 − r t,r,1 × st,r,1  ωr,1 , ωr,2 ∈ R,

(2.117)

which can be regarded to be spanned by a linear vector and a couple with the   perpendicular directions, because r t,r,2 − r t,r,1 × st,r,1 is perpendicular to st,r,1 . For the third twist space, it is spanned by two couples and can be expressed as

54

2 Finite and Instantaneous Screw Theory

St,2−3 = vt,1 Sˆ t,t,1 + vt,2 Sˆ t,t,2 , vt,1 , vt,2 ∈ R,

(2.118)

where Sˆ t,t,1 and Sˆ t,t,2 have the formats of Eq. (2.85), as Sˆ t,t,1 =





0 st,t,1

, Sˆ t,t,2 =



0 st,t,2

 .

(2.119)

It is obvious that the directions of the two couples are different because the base screws should be different couples. The two base couples can be replaced by any two couples that are planar with them, 

0





0



  + vt,2 s Tt,t,1 st,t,1 × st,t,2 = 0  st,t,1 st,t,2      T  v , v ∈ R s t,t,2 st,t,1 × st,t,2 = 0, t,1 t,2 0 0   + vt,2  , = vt,1  s t,t,1 s t,t,2 (2.120)

St,2−3 = vt,1

For the fourth, the fifth, and the sixth twist spaces, each space has two general screws as its base screws. Thus, their expressions are St,2−4/5/6 = ωh,1 Sˆ t,h,1 + ωh,2 Sˆ t,h,2 , ωh,1 , ωh,2 ∈ R,

(2.121)

where Sˆ t,h,1 and Sˆ t,h,2 have the formats of Eq. (2.86), as Sˆ t,h,1 = Sˆ t,h,2 =

 

st,h,1 r t,h,1 × st,h,1 st,h,2 r t,h,2 × st,h,2



 + pt,1



 + pt,2

0 st,h,1 0 st,h,2

 ,  .

(2.122)

In this way, for the fourth twist space, it’s base screws are two general vectors with different directions and can be expressed as  St,2−4 = ωh,1  + ωh,2

st,h,1 r t,h,1 × st,h,1 st,h,2 r t,h,2 × st,h,2



 + ωh,1 pt,1



 + ωh,2 pt,2

0 st,h,1 0 st,h,2

  , sR,1 × sR,2 = 0, ωh,1 , ωh,2 ∈ R. (2.123)

For the fifth one, its two base screws are general vectors with the same directions but different axes,

2.3 Instantaneous Screw

55

 St,2−5 = ωh,1 

st,h,1 r t,h,1 × st,h,1



 + ωh,1 pt,1





0



st,h,1

 st,h,1 0 + ωh,2 pt,2 r t,h,2 × st,h,1 st,h,1       0 st,h,1 + pt,1 = ωh,1 + ωh,2 r t,h,1 × st,h,1 st,h,1 ⎞ ⎛ 0 + ωh,2 |a|⎝ a ⎠, ωh,1 , ωh,2 ∈ R, |a|

+ ωh,2

(2.124)

where     a = r t,h,2 − r t,h,1 × st,h,1 + pt,2 − pt,1 st,h,1 , and Eq. (2.124) can be regarded to be spanned by a general vector and a couple with the specific conditions between their directions. For the sixth one, its two base screws are general vectors with the same axes but different pitches, and its expression can be written in the following four ways,  St,2−6 = ωh,1  + ωh,2

st,h,1 r t,h,1 × st,h,1 st,h,1 r t,h,1 × st,h,1



 + ωh,1 pt,1



 + ωh,2 pt,2

0 st,h,1 0 st,h,1

  , ωh,1 , ωh,2 ∈ R, (2.125)

   0 st,h,1 + pt,1 St,2−6 r t,h,1 × st,h,1 st,h,1       0 , ωh,1 + ωh,2 , ωh,2 pt,2 − pt,1 ∈ R, (2.126) + ωh,2 pt,2 − pt,1 st,h,1     st,h,1 St,2−6 = ωh,1 + ωh,2 r t,h,1 × st,h,1     0 , ωh,1 + ωh,2 , ωh,1 pt,1 + ωh,2 pt,2 ∈ R, + ωh,1 pt,1 + ωh,2 pt,2 st,h,1 (2.127)   St,2 - 6 = ωh,1 pt,1 + ωh,2 pt,2     st,h,1 0 + st,h,1  pt,2 ∈ R,  r t,h,1 × st,h,1    ωh,1 pt,1 + ωh,2 + ωh,1 1 − pt,1 + ωh,2 1 − pt,2 ωh,1 1 − pt,1 + ωh,2 1 − pt,2 ∈ R   st,h,1 , r t,h,1 × st,h,1 (2.128)    = ωh,1 + ωh,2

56

2 Finite and Instantaneous Screw Theory

which can be regarded to be spanned by a general vector and a couple with the same directions, or a linear vector and a couple with the same directions, or a general vector and a linear vector with the same axes. For the seventh twist space, the base screws are a linear vector and a couple with non-perpendicular and different directions. The space is expressed as  St,2−7 =ωr,1

st,r,1 r t,r,1 × st,r,1



 + vt,2

0 st,t,2

 , st,r,1 × st,t,2 = 0, sTt,r,1 st,t,2 = 0,

ωr,1 , vt,2 ∈ R.

(2.129)

For the eighth and the ninth twist spaces, the base screws are a general screw and a linear vector, respectively. Thus, their expressions are  St,2−8/9 = ωh,1

st,h,1 r t,h,1 × st,h,1



 + ωh,1 pt,1



0 st,h,1

 + ωr,2

ωh,1 , ωr,2 ∈ R.

 st,r,2 , r t,r,2 × st,r,2 (2.130)

For the eighth twist space, the directions of its two base screws are different, and the expression for the space is  St,2−8 = ωh,1

st,h,1 r t,h,1 × st,h,1



 + ωh,1 pt,1

0 st,h,1



 + ωr,2

st,h,1 × st,r,2 = 0, ωh,1 , ωr,2 ∈ R.

 st,r,2 , r t,r,2 × st,r,2 (2.131)

For the ninth one, the directions of its two base screws are the same, but their axes are different. Thus, the expression for the space is 

St,2−9

     st,h,1 0 st,h,1 + ωh,1 pt,1 + ωr,2 = ωh,1 r t,h,1 × st,h,1 r t,r,2 × st,h,1 st,h,1 ⎞ ⎛     0   0 st,h,1 = ωh,1 + ωr,2 + pt,1 + ωr,2 |b|⎝ b ⎠ r t,h,1 × st,h,1 st,h,1 |b|   r t,h,1 − r t,r,2 × st,h,1 = 0, ωh,1 , ωr,2 ∈ R, (2.132)

where   b = r t,r,2 − r t,h,2 × st,h,1 − pt,1 st,h,1 , and Eq. (2.132) can be regarded to be spanned by a general vector and a couple with the specific conditions between their directions. Here, St,2−i (i = 1, . . . , 9) are used to denote the nine kinds of two-dimensional twist spaces.

2.3 Instantaneous Screw

57

Based on the above analysis, all the nine kinds of two-dimensional twist spaces are enumerated and expressed. The one-dimensional and two-dimensional twist spaces discussed can be directly used to constitute the spaces with more than two dimensions. (c) Twist spaces with more than two dimensions The twist spaces with more than two dimensions can be spanned by the union of several one-dimensional and two-dimensional twist spaces. Generally, we can conclude the constructions of three-dimensional, four-dimensional, and five-dimensional twist spaces as follows: (1) a three-dimensional twist space is spanned by the union of a one-dimensional space and a two-dimensional space; (2) a four-dimensional twist space is spanned by the union of two two-dimensional spaces; (3) a five-dimensional twist space is spanned by the union of a one-dimensional space and two two-dimensional spaces. The equivalent rewriting of any pair of base screws of a twist space can be referred to the equivalence relationships shown above. Here, we give an example to show the constructions of these twist spaces. A threedimensional twist space spanned by three couples can be spanned by the union of St,1−2 in Eq. (2.111) and St,2−3 in Eq. (2.120), as St,3 =St,1−2 + St,2−3 =vt,t,1 Sˆ t,t,1 + vt,t,2 Sˆ t,t,2 + vt,t,3 Sˆ t,t,3 , vt,1 , vt,2 , vt,3 ∈ R,

(2.133)

where the three couples have the formats of Eq. (2.85), as Sˆ t,t,1 =



     0 0 ˆ ˆ , St,t,2 = , St,t,3 = , st,t,1 st,t,2 st,t,3 0

(2.134)

with the condition that their unit directions st,t,1 , st,t,2 , and st,t,3 are linearly independent. In a similar manner, all the different kinds of three-dimensional, four-dimensional, and five-dimensional twist spaces, and their equivalent expressions can be enumerated. Here, we could not list all the details due to space limitations. The wrench spaces can be classified according to the classification of twist spaces. On one hand, similar to those of twist spaces, the base screws of wrench spaces are linear vectors, couples, and general screws. On the other hand, for any wrench space, a twist space can be found reciprocal to the wrench space. In this way, the wrench spaces with no more than three dimensions can be classified in the same way as the classification of twist spaces with no more than three dimensions, and the wrench spaces with more than three dimensions can be classified based upon their reciprocal twist spaces. Hence, it can be concluded that the wrench spaces can be

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2 Finite and Instantaneous Screw Theory

Table 2.1 Classification of twist and wrench spaces

Classification of twist spaces

Equivalent classification of wrench spaces

One-dimensional (Five-dimensional)

One-dimensional (Five-dimensional)

Two-dimensional (Four-dimensional)

Two-dimensional (Four-dimensional)

Three-dimensional

Three-dimensional

classified according to the classification of one-dimensional, two-dimensional, and three-dimensional twist spaces, as shown in Table 2.1. Based upon the above analysis and the properties of reciprocal screw spaces, only the classification of one-dimensional, two-dimensional, and three-dimensional screw spaces is needed. The classification of four-dimensional and five-dimensional spaces can, respectively, refer to that of two-dimensional and one-dimensional spaces.

2.4 Differential Mapping Between the Screws After the algebraic structures of the finite screw Lie group and instantaneous screw Lie algebra are proved, the relationship between them will be revealed in this section by investigating the differential property of the screw triangle product.

2.4.1 One-DoF Motion For a rigid body realizing a general one-DoF motion, it moves from its initial pose to other poses in a continuous path. During the motion process, the finite motion of the rigid body with respect to the initial pose continuously varies. Thus, the Plücker coordinates of the finite motion axis, the corresponding rotational angle and translational distance can be regarded as functions of the same parameter. Denoting the parameter as x, the expression of finite screw in Eq. (2.9) can be rewritten as     θ (x) 0 s f (x) + t(x) . S f (x) = 2 tan r f (x) × s f (x) s f (x) 2

(2.135)

The above equation expresses the varying displacement of the rigid body with respect to its initial pose when it moves to different poses. Thus, if the parameter x is supposed to denote time, the expression of the velocity that the rigid body obtains at each pose can be solved through differentiating this displacement expression, as

2.4 Differential Mapping Between the Screws

59

  ˙ θ(x) s f (x) θ (x) r f (x) × s f (x) cos2 2   θ (x) s˙ f (x) + 2 tan r˙ f (x) × s f (x) + r f (x) × s˙ f (x) 2     0 0 + t(x) . + t˙(x) s˙ f (x) s f (x)

S˙ f (x) =

(2.136)

At the initial pose, the parameter x equals zero, and the rigid body achieves no displacement. This means that both the rotational angle and translational distance are zero at that pose, i.e., θ (0) = 0, t(0) = 0.

(2.137)

Substituting Eq. (2.137) into Eq. (2.136), the differential of the one-DoF finite motion at the initial pose is derived as ˙ S˙ f (0) = θ(0)



s f (0) r f (0) × s f (0)





 0 + t˙(0) . s f (0)

(2.138)

According to the form of instantaneous screw (twist) in Eq. (2.58), it can be rewritten as     s f (0) 0 ˙S f (0) = St = ω +v . (2.139) r f (0) × s f (0) s f (0) In summary, one-DoF finite motion of a rigid body can be expressed by a oneparameter finite screw expression, which denotes a one-dimensional finite screw sub-set. The time differential of the finite screw sub-set at the initial pose is a oneDoF instantaneous screw sub-space, because the relationship between the parameter x and time can be arbitrarily set. The instantaneous screws in the sub-space have the same instantaneous motion axis, which is coincident with the finite motion axis at the initial pose. This instantaneous screw sub-space expresses all the feasible instantaneous motions that the rigid body can obtain at its initial pose. In short, the differential of a one-DoF finite screw sub-set is proved to be a one-DoF instantaneous screw sub-space. Hence, there exists a differential mapping between the two kinds of screws [1], as  S˙ f θ=0, = St ,

(2.140)

t=0

where the axes of the screws are coincident at the initial pose,  L t = L f θ=0, . t=0

(2.141)

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2 Finite and Instantaneous Screw Theory

2.4.2 Multi-DoF Motion In order to extend the situation of differential mapping between screws from one-DoF motion to multi-DoF motion, the differential property of the screw triangle product will be investigated. Rewrite the screw triangle product of two finite screws as S f,ab = S f,a  S f,b =

F , G

(2.142)

where F = S f,a + S f,b +

  S f,b × S f,a 0 , −H tb s f,a + ta s f,b 2

G = 1 − H sTf,a s f,b , H = tan

θb θa tan . 2 2

Here, S f,a and S f,b with different varying parameters denote two one-DoF finite screw sub-sets, and S f,ab is a two-DoF finite screw sub-set realized by a rigid body. Thus, the differential of Eq. (2.142) can be solved as ˙ F G˙ F S˙ f,ab = − 2, G G

(2.143)

where ˙ = S˙ f,a + S˙ f,b + 1 S˙ f,b × S f,a + 1 S f,b × S˙ f,a F 2 2     0 0 −H , − H˙ t˙b s f,a + t˙a s f,b tb s f,a + ta s f,b G˙ = − H˙ sTf,a s f,b , H˙ =

θa θ˙a θb θ˙b   tan + tan  . θa θb 2 2 2 2 2 cos 2 cos 2 2

Similar with the one-DoF situation, θa , ta , θb , tb equal to zero at the initial pose of the rigid body, which leads to H = 0, F = 0, G = 1,

(2.144)

˙ = S˙ f,a + S˙ f,b , G˙ = 0. H˙ = 0, F

(2.145)

2.4 Differential Mapping Between the Screws

61

Hence, the differential of S f,ab at the initial pose is derived as  S˙ f,ab θi =0,

ti =0 i=a,b

  = S˙ f,a θa =0, + S˙ f,b θb =0, , ta =0

(2.146)

tb =0

which means that the differential of a two-DoF finite screw sub-set is a two-DoF instantaneous screw sub-space. This conclusion can be further extended to multi-DoF motion in the same manner, as  S˙ f,abc··· θi =0,

ti =0 i=a,b,c,···

   = S˙ f,a θa =0, + S˙ f,b θb =0, + S˙ f,c θc =0, + · · · ta =0

tb =0

tc =0

= St,a + St,b + St,c + · · · .

(2.147)

From Eq. (2.147), it is clear to see that the differential of a finite screw set is a twist space. In other words, the tangent space of any curve in the finite screw Lie group is the sub-space of the instantaneous screw Lie algebra. Thus, the instantaneous screw Lie algebra is the corresponding Lie algebra of the finite screw Lie group.

2.5 Discussion on the Algebraic Structures of FIS The formats, algebraic structures, and computational properties of finite and instantaneous screws discussed above constitute the FIS theory. As a description method of rigid body motion, it has close relationships with other methods, like matrix Lie group and Lie algebra [19], dual quaternions [12]. The instantaneous screw has the similar linear format with the element of matrix Lie algebra and pure dual quaternion,  ω˜st ωr t × st + vst , ξ= 0 0

(2.148)

d = ωst + ε(ωr t × st + vst ),

(2.149)



where s˜ t is the skew-symmetric matrix of st . For this reason, the instantaneous screw Lie algebra, matrix Lie algebra and pure dual quaternions for instantaneous motion description are all linear vector spaces, and their algebraic structures are isomorphic to each other [20]. Thus, only the mathematical tools for finite motion description will be compared in this section. Six-dimensional Special Euclidean group SE(3) is the most well-known mathematical tool to describe rigid body finite motions. Any element of SE(3) consists of a rotational matrix of three-dimensional Special Orthogonal group SO(3) and a three-dimensional translational vector, as m = (R, t), R ∈ SO(3), t ∈ R3 ,

(2.150)

62

2 Finite and Instantaneous Screw Theory

where R and t denote the rotational matrix and translational vector, respectively. Their expressions involving the three basic elements of finite motion are given as  2 R = E 3 + sin θ s˜ f + (1 − cos θ ) s˜ f ,

(2.151)

  t = (E 3 − R) r f − sTf r f s f + t s f ,

(2.152)

where E 3 is a three-order unit matrix. Element of SE(3) is commonly represented by two other mathematical tools, i.e., transformation matrix and dual quaternion. The former has the following formats as 

 R t , 0 1

(2.153)

 R ˜tR , 0 R

(2.154)

T = R + ε ˜tR .

(2.155)

T= or  T= or

The format of the latter is shown in Eqs. (2.3) and (2.6). The entire set of transformation matrices is known as a matrix Lie group. It is an isomorphism of SE(3) because there exists a one-to-one mapping between its elements and those of SE(3). The same situation happens to the entire set of dual quaternions with positive rotational angles. Those dual quaternions also form an isomorphism of SE(3). Consider the finite screw Lie group in Eq. (2.39), the mapping between finite screws and elements of SE(3) is bijective, which shows that this Lie group is an isomorphism of SE(3), too. Among these three mathematical tools for finite motion description, it is easy to see that finite screw has the simplest and most concise format because it has only six items and clearly describes the three basic elements of finite motion. However, the transformation matrix has at least twelve items, and the dual quaternion has eight. The differences among finite screw Lie group, matrix Lie group, and dual quaternions rise from their different algebraic structures. In order to distinguish these three mathematical tools clearly, and to explain why finite screw is a better choice for describing rigid body finite motion, the relationships among them and the Special Euclidean group SE(3) will be further discussed. Any transformation matrix in the matrix Lie group can be represented by a 4×4 real matrix, a 6×6 real matrix, or a 3×3 dual matrix, etc. The entire set of each kind of these matrices has the same inner closure and associative properties with SE(3). Hence, the matrix Lie group forms a homomorphism of SE(3). What’s more, it is

2.5 Discussion on the Algebraic Structures of FIS

63

an isomorphism of SE(3) because of the bijective mapping between it and SE(3). Furthermore, the matrix Lie group is a representation of SE(3), because the matrices play as linear transformations acting on the six-dimensional vector space. A similar situation happens to dual quaternion. Half part of the entire set of dual quaternions with positive rotational angles is also an isomorphism and a representation of SE(3). In this way, the entire set of dual quaternions is a double cover of SE(3). With linear transformation formats, the transformation matrices can be composited by multiplication. So can the dual quaternions for the same reason. However, for finite motion composition, matrix and dual quaternion multiplications are hardly able to directly give out the composited Chasles’ axis with the corresponding rotational angle and translational distance of the resultant motion, because these multiplications maintain the linear transformation formats and cannot extract the three basic elements of Chasles’ motion. Different from the transformation matrix and dual quaternion, finite screw is invented to break the linear transformation format of finite motion description, which can be regarded as a general extension of Gibbs vector. Finite screw does not act on any vector space, so it cannot transform any coordinates of geometric point or line. It is a mathematical tool purely for finite motion description, and it can express the basic elements of Chasles’ motion in a straightforward manner. The composition algorithm of finite screws, i.e., the screw triangle product, maintains the screw format, which directly leads to the expressions of basic elements of the resultant Chasles’ motion. Although the entire set of finite screws under the screw triangle product has the same inner closure and associative properties with SE(3), it has no sense to be the representation of SE(3). In other words, it only forms an isomorphism of SE(3). Any element of SE(3) is a combination of rotational matrix and translational vector, and it is a homogeneous transformation of the coordinates of points. In this way, all representations of SE(3) cannot break the inherent linear transformation formats. Hence, only finite screw under the screw triangle product can express and composite rigid body finite motions in a non-redundant and direct manner. In this section, the algebraic structures of FIS are discussed in detail and compared with matrix and dual quaternion representations of SE(3) and se(3). It is shown that FIS is the simplest and most concise mathematical tool for finite and instantaneous motion description and composition.

2.6 Conclusion This chapter provides the mathematical background of this book. FIS theory including the basic concepts, expressions, operations, and properties of the finite screw and instantaneous screw is introduced. It is a powerful mathematical tool in description and modeling of rigid body motions, and in type synthesis and performance analysis of robotic mechanisms. The detailed usage of the FIS theory will be discussed in the following chapters. The main points of this chapter are listed below for the readers’ convenience.

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2 Finite and Instantaneous Screw Theory

(1) Finite screw and instantaneous screw have the forms of six-dimensional quasivector and vector, respectively. And there exists differential mapping between them, which closely relates these two kinds of screws and connects type synthesis with performance analysis and design of robotic mechanisms. (2) The entire set of finite screws is a Lie group under the screw triangle product, which is named as the finite screw Lie group. It is an isomorphism of SE(3), but not a representation of it. The entire set of instantaneous screws is a Lie algebra under the screw product, which is named as the instantaneous screw Lie algebra. It is the corresponding Lie algebra of the finite screw Lie group. (3) Compared with the matrix and dual quaternion representations of SE(3), the finite screw Lie group is the simplest and most concise mathematical tool for finite motion description and composition.

References 1. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 2. Sun T, Yang SF, Huang T et al (2018) A finite and instantaneous screw based approach for topology design and kinematic analysis of 5-axis parallel kinematic machines. Chin J Mech Eng 31:44 3. Sun T, Yang SF (2019) An approach to formulate the Hessian matrix for dynamic control of parallel robots. IEEE-ASME Trans Mechatron 24(1):271–281 4. Sun T, Song YM, Gao H et al (2015) Topology synthesis of a 1-translational and 3-rotational parallel manipulator with an articulated traveling plate. J Mech Rob Trans ASME 7(3):031015 5. Yang SF, Sun T, Huang T et al (2016) A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech Mach Theory 104:405–419 6. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 7. Sun T, Song YM, Li YG et al (2010) Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. J Mech Rob Trans ASME 2(3):031009 8. Sun T, Song YM, Dong G et al (2012) Optimal design of a parallel mechanism with three rotational degrees of freedom. Rob Comput Integr Manuf 28(4):500–508 9. Sun T, Liang D, Song YM (2018) Singular-perturbation-based nonlinear hybrid control of redundant parallel robot. IEEE Trans Industr Electron 65(4):3326–3336 10. Sun T, Lian BB, Song YM et al (2019) Elastodynamic optimization of a 5-DoF parallel kinematic machine considering parameter uncertainty. IEEE ASME Trans Mechatron 24(1):315–325 11. Ball RS (1900) A treatise on the theory of screws. Cambridge University Press, Cambridge 12. McCarthy JM, Soh GS (2011) Geometric design of linkages, 2nd edn. Springer, New York 13. Hunt KH, Parkin IA (1995) Finite displacements of points, planes, and lines via screw theory. Mech Mach Theory 30(2):177–192 14. Huang C, Chen CM (1995) The linear representation of the screw triangle—a unification of finite and infinitesimal kinematics. J Mech Des Trans ASME 117(4):554–560 15. Hervé JM (1999) The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mech Mach Theory 34(5):719–730 16. Ceccarelli M (2000) Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion. Mech Mach Theory 35(6):761–770

References

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17. Tsai LW (1999) Robot analysis: the mechanics of serial and parallel manipulators. Wiley, New York 18. Huang Z, Li QC, Ding HF (2013) Theory of parallel mechanisms. Springer, Berlin 19. Meng J, Liu GF, Li ZX (2007) A geometric theory for analysis and synthesis of sub-6 DoF parallel manipulators. IEEE Trans Rob 23(4):625–649 20. Murray RM, Li ZX, Sastry SS (1994) A mathematical introduction to robotic manipulation. CRC Press, Boca Raton

Chapter 3

Topology and Performance Modeling of Robotic Mechanism

3.1 Introduction Topology of a robotic mechanism [1], describing the arrangement of joints including number, sequence, type, and axis (or direction), denotes the basic mechanical structure of the robotic mechanism [2]. Topology determines motion capability thus directly affects the kinematic, stiffness, and dynamic performance of the robotic mechanism [3–5]. As the complete feature of the mechanism is aimed to be analyzed, topology model is expected to be formulated in the finite motion level. Whereas the performance models mainly concern the instantaneous motions and the exerted forces, thus they are built in the instantaneous motion level. The topology and performance models are, respectively, used for type synthesis [6–9] and performance analysis [10–14], leading to the topology and performance design of robotic mechanisms. It has long been a desire that an integrated design method considering both topology and performance could be proposed, which requires unifying the finite motion based topology model and instantaneous motion based performance model under the same mathematical framework. By employing FIS theory given in Chap. 2, a general and unified method of topology and performance modeling and analysis of robotic mechanisms is presented in this chapter. The underlying relationship of topology and performance models of the robotic mechanism is analyzed based on the finite and instantaneous motions described by FIS theory. The finite motions of joints, open-loop, and closed-loop mechanisms are expressed by finite screws. It leads to the topology models of joints and mechanisms which can also be regarded as their displacement models. By applying the differential mapping between finite and instantaneous screws [15], the instantaneous motion model of joints and mechanisms are directly obtained. They correspond to the velocity, acceleration, and jerk models that are closely linked to the kinematic, stiffness, and dynamic performances [16, 17]. Examples of formulating topology and performance models of typical open-loop and closed-loop mechanisms are given. Finally, an integrated framework for type synthesis and performance analysis of © Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_3

67

68

3 Topology and Performance Modeling of Robotic Mechanism

robotic mechanisms is proposed. The details on the methodology of type synthesis and performance analysis of robotic mechanisms will be discussed in the following chapters.

3.2 FIS Based Topology and Performance Modeling Suppose the robotic mechanism is a rigid body system. It starts from one rigid body connecting to another rigid body by a joint. If n joints are used to connect n + 1 rigid bodies serially with the 1st rigid body as the fixed base, the rigid body system forms an open-loop mechanism consisting of n joints. The (n + 1)th body is called its end-effector. If two open-loop mechanisms having ni + 1 (i = 1, 2) rigid bodies, respectively, the ni th joints share the same end-effector and the 1st rigid body in each open-loop mechanism is fixed, the rigid body system forms a closed-loop mechanism regarding the two open-loop mechanisms as its two limbs. Their shared end-effector is its moving platform. The evolution from a single rigid body to openloop and closed-loop mechanisms is shown in Fig. 3.1. Furthermore, the open-loop mechanism and closed-loop mechanism can be regarded as subsystems. By connecting the subsystems in serial, parallel, or hybrid structures, the rigid body systems form the robotic mechanisms with complex structures, such as hybrid mechanisms and multi-closed-loop mechanisms.

Fig. 3.1 The evolution of a rigid body to an open-loop mechanism and a closed-loop mechanism

3.2 FIS Based Topology and Performance Modeling

69

Topology of the robotic mechanism describes the types and structures of the above mentioned rigid body systems [18]. As the joint is the basic element determining the motion capability of the robotic mechanism, topology of a mechanism involves the types, sequences, directions, and positions of all the joints that constitute the mechanism. The directions and positions of the joints are measured at the initial pose of the mechanism for the convenience of formulating the topology of the mechanism reflecting its full-cycle motion characteristics. For an open-loop mechanism, its topology contains (a) the types of all the joints, i.e., one-DoF revolute (R) joint, prismatic (P) joint, helical (H) joint, two-DoF universal (U) joint, cylindrical (C) joint, and threeDoF spherical (S) joint, the two-DoF and three-DoF joints are equivalent to two and three one-DoF joints, respectively; (b) the sequences of the joints, where the joints are numbered from the fixed base to the end-effector in ascending order; (c) the directions of the joints, which can be expressed by their unit direction vectors; (d) the positions of the joints, among which the positions of P joints do not need to be concerned, they have no influence on the kinematic properties of the mechanism. For a closed-loop mechanism, besides the topology of its each limb as an openloop mechanism, the topology of the mechanism also contains the geometrical relationships among all its limbs. For a more complicated mechanism, the topology of each subsystem is included, which is the topology of open-loop mechanism, closed-loop mechanism, or both, and the relationships among the subsystems are contained. Topology determines the robotic mechanism’s motion which is divided into finite and instantaneous motions. In this book, finite motion and instantaneous motion models of robotic mechanisms are formulated in the form of screws. Analyzing the finite motion model and instantaneous motion model, respectively, built by finite screw and instantaneous screw, it is found that the finite screw based finite motion model contains all the topology information on the joints, because (a) the finite motion axes of finite screws can describe the directions and positions of different kinds of joints, such as one-DoF R joint, P joint, H joint, two-DoF U joint, C joint, and three-DoF S joint; (b) the relations between the finite screw of a mechanism and those of all its joints reflect the sequences of the joints in the mechanism. However, the instantaneous motion model contains the types, directions, and positions of the joints but not their sequence in the robotic mechanism. Since the finite screw based finite motion model reflects all the topology information of a robotic mechanism, it can serve as topology model of the mechanism. In the meantime, the finite motion generated by a mechanism can describe its displacement. Hence, the topology model based on finite screw can describe both topology and displacement of a mechanism in an algebraic manner.

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3 Topology and Performance Modeling of Robotic Mechanism

As has mentioned, instantaneous motion is the infinitesimal displacement of a rigid body at a given pose [19]. For a robotic mechanism, if the first-order differential of displacement model is considered, the instantaneous motion model serves served as the velocity model [20]. Similarly, considering the high-order differential of the displacement model, the acceleration model and jerk model of the robotic mechanism can be obtained [17]. The velocity, acceleration, and jerk reflected by the instantaneous motion models are the prerequisites for the motion and force analysis of the robotic mechanism, which are also called performance models. Thus, the instantaneous motion models are applied to performance modeling and analysis of the robotic mechanism. It has been proved in Chap. 2 that there exists a differential mapping between finite and instantaneous screws. The velocity model in the form of an instantaneous screw can be directly formulated through differentiating the corresponding finite screw based topology model [16, 17], as St = S˙ f .

(3.1)

Considering the higher-order differentials of finite screws as the differentials of instantaneous screws, Eqs. (3.2) and (3.3) are obtained [17], Sa = S˙ t ,

(3.2)

Sj = S˙ a ,

(3.3)

where Sa and Sj are vectors to express acceleration and jerk in forms of screw as well. Since the differentials of velocity lead to acceleration and jerk, the differential of instantaneous screws can be used for acceleration and jerk modeling. The velocity, acceleration, and jerk models are essential for performance modeling and analysis, which are formulated through differentiating the finite screw based topology model. The relations of topology model and performance models based on FIS theory are shown in Fig. 3.2. Select any pose of a mechanism as its initial pose. The velocity model of the mechanism at that pose is the differential of its topology model. In this way, the velocity model at each pose is obtained. Further, differentiating the velocity model leads to the acceleration and jerk models of the mechanism. Using the differential mappings between topology and performance models of mechanisms, these models are integrated into the framework of FIS theory and formulated in a simple and straightforward manner [16, 17]. Thus, the topology characteristics and performance properties of mechanisms are closely related.

3.3 FIS Based Finite and Instantaneous Motion Modeling

71

Fig. 3.2 Topology and performance models based on FIS theory

3.3 FIS Based Finite and Instantaneous Motion Modeling In this section, the finite motion and instantaneous motion models of robotic mechanisms are algebraically formulated using the FIS theory. They represent the topology and performance models, respectively. In this way, a FIS theory based unified method of topology and performance modeling of mechanisms is proposed. The establishment of topology model starts from describing the finite motions of joints by finite screws. Then, the finite motion of the end-effector of open-loop mechanism can be computed by the screw triangle product. Next, the finite motion of the moving platform of closed-loop mechanism is obtained using the intersection algorithm of finite screws. After the finite motion model of the robotic mechanism is formulated, the instantaneous motion models can be directly built by the differentiation of finite motion. Hence, the performance models are connected to the topology model by the differential relationship between finite and instantaneous screws. This corresponds to the fact that the differentiations of displacement model are the velocity, acceleration, and jerk models of the robotic mechanism.

3.3.1 The Finite Motion Modeling As discussed in Chap. 2, the finite motion of a rigid body can be described by a finite screw. The screw directly reflects the elements of finite motion. As finite screws can be composited by the screw triangle product in Eq. (2.24), the motions generated by rigid body systems can be described by screws as well. The finite motion generated by a rigid body system is the composition and intersection result of the motions between the rigid bodies linked to the fixed body and to the end body, and the motions between any two of the connected rigid bodies. Since each motion can be described by a finite screw in form of Eq. (2.9), the finite motion of the end body in the rigid body system can be expressed by the nonlinear

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3 Topology and Performance Modeling of Robotic Mechanism

computation result of several finite screws, which is also a finite screw according to the closure property of the finite screw Lie group. The finite motion of the robotic mechanism is generated by the joints in it. For each one-DoF joint, its finite motion can be expressed by a finite screw having the joint axis as the finite motion axis together with the rotational angle/translational distance. For each multi-DoF joint, the screw of the joint is the composition of several screws which are generated by the one-DoF joints constituting the joint. The finite motion of an open-loop or a closed-loop mechanism is defined as the output motion of its end body, which is the computation result of all its joint motions. The computation process depends on the structure of the mechanism. (1) The output motion of end-effector for an open-loop mechanism is the composition of the motions of all its joints from the one connected to the fixed body to the one connected to the end body; (2) The output motion of a moving platform for a closed-loop mechanism is the intersection of the motions of its limbs, and the motion of each limb is the composition of the motions of all the joints in the limb. For mechanisms with complex structures, such as hybrid mechanisms and multiclosed-loop mechanisms, each individual part of them is an open-loop mechanism or a closed-loop mechanism, thus, their finite motions can be computed by composition and intersection of the motions of the parts contained in them. The basic rules of motion computation of these mechanisms are listed as follows: (1) If two parts in a mechanism are connected to form a serial structure, i.e., the fixed body in one part is fixed to the end body in another part, the total output motion of the two parts is the composition of their generated motions; (2) If two parts in a mechanism are connected to form a parallel structure, i.e., the two parts share the same end body, their total output motion is the intersection of the motions generated by them. Based upon the above analysis, the formulation of the finite motion model follows the topology layout from joint to open-loop mechanism then closed-loop mechanism. The finite motion of more complicated mechanism can be further obtained by the composition and/or intersection of open-loop mechanisms and/or closed-loop mechanisms. Therefore, the finite motion models of joint, open-loop mechanism and closed-loop mechanism are introduced in sequence in this section. The finite motions of the other types of mechanisms follow similar composition and intersection algorithms.

3.3.1.1

Joint

As the topology models of the joints are the basic elements of the motions of robotic mechanisms. The motion models of different types of joints are firstly discussed. In this section, the joints are assumed to be connected to the fixed base. The commonly used joints are R joint, P joint, H joint, U joint, C joint, and S joint.

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73

Fig. 3.3 R joint

For an R joint shown in Fig. 3.3, its topology model is formulated by using finite screw to describe its finite motion as   θ s Sf = 2 tan , (3.4) 2 r×s where s (|s| = 1) is the unit direction vector of the R joint axis, r is its position vector pointing from the origin of the fixed reference frame to an arbitrary point on the axis, and θ is the rotational angle measured from the initial pose of the joint. In the similar manner, for a P joint shown in Fig. 3.4, its topology model is formulated as   0 Sf = t , (3.5) s where s (|s| = 1) is the unit direction vector of the P joint, and t is the translational distance measured from its initial pose. For an H joint shown in Fig. 3.5, its topology model is formulated as Fig. 3.4 P joint

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3 Topology and Performance Modeling of Robotic Mechanism

Fig. 3.5 H joint

Sf = 2 tan

    θ s 0 + hθ , s 2 r×s

(3.6)

where s (|s| = 1) is the unit direction vector of the H joint axis, r is its position vector pointing from the origin of the fixed reference frame to an arbitrary point on the axis, θ is the rotational angle measured from the initial pose of the joint, and h is the pitch (the ratio between the translational distance and rotational angle) of the H joint. For a U joint shown in Fig. 3.6, it can be regarded as the combination of two R joints having the intersecting axes. Its topology model is the composition of the models of the two R joints. Using screw triangle product, the topology model of U joint is formulated Sf = 2 tan

    θ1 θ2 s2 s1 2 tan , 2 rO × s2 2 rO × s1

(3.7)

where s1 and s2 are the unit direction vectors of the two R joint axes, rO is the position vector pointing from the origin of the fixed reference frame to the center O of the U Fig. 3.6 U joint

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75

Fig. 3.7 C joint

joint, θ1 and θ2 are the rotational angles of the two R joints measured from the initial pose of the U joint. For a C joint shown in Fig. 3.7, it is the combination of one R joint and one P joint. The two joints have the same directions. The topology model of the C joint is the composition of the models of the R joint and the P joint, which is formulated as     θ2 0 s2 t1 , Sf = 2 tan r × s s 2 2 2 1

(3.8)

where s1 and s2 are the unit direction vectors of the two joints, s1 = s2 , r2 is the position vector of the R joint, t1 is the translational distance of the P joint, and θ2 is the rotational angle of the R joint measured from the initial pose of the C joint. The three-DoF S joint shown in Fig. 3.8 is the combination of three R joints having the intersecting axes. Its topology model is the composition of the models of the three R joints. The topology model of the S joint is formulated as Fig. 3.8 S joint

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Table 3.1 Topology models of joints Type

Direction

Position

R

s

r

P

s

N/A

H

s

r

U

s1 , s2

rO

C

s1 , s2

r2

S

s1 , s2 , s3

rO

Model Sf = 2 tan

Sf = t

θ 2





s r×s

  0 s

θ Sf = 2 tan 2





s

  0

+ hθ r×s s     θ2 θ1 s2 s1 Sf = 2 tan 2 tan 2 rO × s2 2 rO × s1 θ2 Sf = 2 tan 2 Sf = θ3 2 tan 2





s2 r2 × s2

s3 rO × s3



 t1



θ2 2 tan 2

0



s1 

s2 rO × s2



θ1 2 tan 2





s1 rO × s1

      θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan , Sf = 2 tan 2 rO × s3 2 rO × s2 2 rO × s1

(3.9)

where s1 , s2 , and s3 are the unit direction vectors of the three R joint axes decomposing from the S joint, rO is the position vector pointing from the origin of the fixed reference frame to the center O of the S joint, θ1 , θ2 and θ3 are the rotational angles of the three R joints measured from the initial pose of the S joint. The topology models of the joints are summarized in Table 3.1.

3.3.1.2

Open-loop Mechanism

Without loss of generality, an open-loop mechanism consisting of n one-DoF joints is considered. The joints are numbered from the one connected to the fixed base to the one connected to the end-effector in ascending order. Based upon the evolution from joints to open-loop mechanism, the finite motion of the open-loop mechanism is the composition of all the motions generated by its joints. Hence, the finite motion of the mechanism is formulated by composting the finite screws of the n joints by using the screw triangle product as Sf = Sf ,n  · · · Sf ,1 ,

(3.10)

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where Sf denotes the finite motion of the open-loop mechanism, and Sf ,k (k = 1, . . . , n) denotes the finite motion generated by the kth joint in the mechanism with sk (and rk ) as its unit direction vector (and position vector), θk or tk as its rotational angle or translational distance, and hk as the pitch of the corresponding H joint. Each Sf ,k is in the form of Eq. (2.9). According to the closure of finite screw Lie group, Sf can be always rewritten into the form of Eq. (2.9). Equation (3.10) is the finite motion model of the open-loop mechanism in finite screw form. Due to the algebraic algorithms of the screw triangle product, the model is an analytical expression and can reflect all the finite motion characteristics of the mechanism.

3.3.1.3

Closed-loop Mechanism

A closed-loop mechanism is assembled by two open-loop mechanisms sharing the same end-effector, which are its two limbs. We consider that each limb consists of ni (i = 1, 2) one-DoF joints. According to the finite motion model of open-loop mechanism in Eq. (3.10), the model of each limb in the closed-loop mechanism can be formulated as Sf ,i = Sf ,i,ni  · · · Sf ,i,1 , i = 1, 2,

(3.11)

where Sf ,i denotes the finite motion generated by the ith limb, Sf ,i,k (k = 1, . . . , ni ) is the finite motion of the kth joint in the ith limb, si,k (and ri,k ) is its unit direction vector (and position vector), θi,k or ti,k is its rotational angle or translational distance, and hi,k is the pitch of the corresponding H joint. In the same way, all Sf ,i,k and Sf ,i are in the form of Eq. (2.9). Based upon the evolution from the open-loop mechanism to closed-loop mechanism, the finite motion of the closed-loop mechanism is the intersection of the motions of its two limbs. Thus, the finite motion model of the closed-loop mechanism is formulated as Sf = Sf ,1 ∩ Sf ,2 ,

(3.12)

where Sf denotes the finite motion of the closed-loop mechanism, which is also in the form of Eq. (2.9). As both the finite motions of the limbs (open-loop mechanisms) are expressed analytically, the motion of the closed-loop can be obtained through algebraic derivations. According to the evolution from a single rigid body to a joint, then an open mechanism and finally a closed-loop mechanism, the finite motion models of joints and mechanisms are formulated through expressing their motions by finite screws.

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3.3.2 The Instantaneous Motion Modeling A differential mapping between the finite screw and instantaneous screw, shown in Eqs. (2.140) and (2.147), has been strictly proved in Chap. 2. Using this mapping, the instantaneous motion models of joints, open-loop, and closed-loop mechanisms in forms of instantaneous screw can be directly obtained through differentiating the corresponding finite motion models formulated in Sect. 3.3.1. The finite motion model of a mechanism can also be regarded as its displacement model because it describes the rotations and translations generated by the mechanism. As discussed above, the finite motion model of the robotic mechanism is formulated in form of finite screw. The model can be regarded as a function of the finite motion parameters of all the one-DoF joints constituting the mechanism, Sf = ff (q),

(3.13)

where Sf denotes finite motion (displacement) of the mechanism, q denotes the vector constituted by the finite motion parameters of all the joints in the mechanism, i.e., rotational angles of all the R joints and the H joints, and translational distances of all the P joints. By using the differential mapping between finite screw and instantaneous screw revealed in Chap. 2, the velocity model of the mechanism can be directly formulated through differentiating its finite motion model in Eq. (3.13), as  St = S˙ f q=0 = ft (˙q),

(3.14)

where St is an instantaneous screw in the form of Eq. (2.58) denoting the mechanism’s velocity. Furthermore, solving the higher-order differentiations of the finite motion model leads to the acceleration and jerk models of the mechanism, Sa = S˙ t = fa (¨q),

(3.15)

Sj = S˙ a ... = fj ( q ),

(3.16)

where Sa and Sj , respectively, denote acceleration and jerk of the mechanism. Equations (3.14)–(3.16) are velocity, acceleration, and jerk models of the mechanism. They are known as the instantaneous motion models of the mechanism, which are in the forms of an instantaneous screw. As shown in Eqs. (3.14)–(3.16), the instantaneous motion models of the mechanism can be regarded as functions of the different order instantaneous motion parameters of all the joints. In a similar manner, much higher-order instantaneous motion models can be formulated. We only

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79

consider the models with no more than three orders in this book, i.e., velocity, acceleration, and jerk models. The differential mappings between the finite motion and instantaneous motion models match the fact that velocity, acceleration, and jerk are different order differentials of displacement. We will discuss the general method and procedures of solving the velocity, acceleration, and jerk models of joint, open-loop mechanism, and closed-loop mechanism through differentiating the corresponding topology (displacement) models in the following sections.

3.3.2.1

Joint

For the R joint, its velocity model is formulated through differentiating the topology model of it in Eq. (3.4) at the initial pose. The obtained velocity model is  St = S˙ f θ=0

  s  =  θ  r × s  cos2 2 θ=0   s =ω r×s θ˙



= ωSˆ t ,

(3.17)

where ω is the angular velocity of the R joint, and Sˆ t is the instantaneous motion axis of the joint, which is the unit instantaneous screw with zero pitch. When the R joint moves away from its initial pose, its instantaneous motion axis is unchanged. Thus, its velocity model is constant at any pose, as shown in Fig. 3.9.

Fig. 3.9 The R joint from initial pose to arbitrary pose

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3 Topology and Performance Modeling of Robotic Mechanism

Hence, the acceleration and jerk models of the joint can be obtained through differentiating the velocity model in Eq. (3.17) as Sa = S˙ t = ω˙ Sˆ t = α Sˆ t ,

(3.18)

Sj = S˙ a = α˙ Sˆ t = β Sˆ t ,

(3.19)

where α and β are angular acceleration and jerk of the R joint. The velocity model of the P joint is formulated as  St = S˙ f t=0   0  = ˙t s t=0   0 =v s = vSˆ t ,

(3.20)

where v is the linear velocity of the P joint, and Sˆ t denotes the instantaneous motion direction of the joint, which is the unit instantaneous screw with infinite pitch. The direction of the P joint’s velocity is unchanged, when it moves away from the initial pose. As shown in Fig. 3.10, the velocity model of the P joint is constant at any pose.

Fig. 3.10 The P joint from initial pose to arbitrary pose

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81

Hence, the P joint’s acceleration and jerk models are obtained by differentiating Eq. (3.20) as Sa = S˙ t = ˙t Sˆ t = aSˆ t ,

(3.21)

Sj = S˙ a = a˙ Sˆ t = j Sˆ t ,

(3.22)

where a and j are linear acceleration and jerk of the P joint. For the H joint, the velocity model is formulated as  St = S˙ f θ=0 ⎛ ⎞      s 0 ⎟ ⎜ θ˙ =⎝ + hθ˙ ⎠ θ  r × s s  cos2 2 θ=0   s =ω r × s + hs = ωSˆ t ,

(3.23)

where ω is the angular velocity of the H joint, and Sˆ t is the unit instantaneous screw with h pitch. When the H joint moves away from its initial pose, its instantaneous motion axis is unchanged. As shown in Fig. 3.11, its velocity model is constant at any pose.

Fig. 3.11 The H joint from initial pose to arbitrary pose

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3 Topology and Performance Modeling of Robotic Mechanism

Thus, the acceleration and jerk models of the H joint are obtained as Sa = S˙ t = ω˙ Sˆ t = α Sˆ t ,

(3.24)

Sj = S˙ a = α˙ Sˆ t = β Sˆ t ,

(3.25)

where α and β are angular acceleration and jerk of the H joint. For the U joint, its velocity model is derived through differentiating the corresponding topology model. The obtained velocity model is  St = S˙ f θ1 =0, θ =0   2  θ1 θ2 = ω2 Sˆ t,2 2 tan Sˆ t,1 + 2 tan Sˆ t,2 ω1 Sˆ t,1  θ1 =0, 2 2 θ2 =0

= ω2 Sˆ t,2 + ω1 Sˆ t,1 ,

(3.26)

where ω1 and ω2 are the angular velocities of the two R joints that constitute the U joint, Sˆ t,1 and Sˆ t,2 denote the instantaneous motion axes of the two R joints at the initial pose of the U joint. When the U joint moves away from its initial pose, the instantaneous motion axis of the R1 joint, which is connected to the fixed base, is unchanged. However, the instantaneous motion axis of the R2 joint is changed by the rotation of R1 . As shown in Fig. 3.12, when the rotational angle of the R1 joint is θ1 , the instantaneous motion axis of the R2 joint is changed to

Fig. 3.12 The U joint from initial pose to arbitrary pose

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83

 s2 rO × s2   s 2 → Sˆ  t,2 = rO × s 2    0 s˜1 s2 . = exp θ1 rO × s2 r˜O s˜1 − s˜1 r˜O s˜1

Sˆ t,2 =



(3.27)

The velocity model of the U joint at any pose is 

St = ω2 Sˆ t,2 + ω1 Sˆ t,1 .

(3.28)

Hence, the U joint’s acceleration and jerk models are obtained by differentiating Eq. (3.28) as Sa = S˙ t 

= ω˙ 2 Sˆ t,2 + ω2 θ˙1

 0 ˆ s˜1 S + ω˙ 1 Sˆ t,1 r˜O s˜1 − s˜1 r˜O s˜1 t,2





= α2 Sˆ t,2 + α1 Sˆ t,1 + ω1 Sˆ t,1 × ω2 Sˆ t,2 ,

(3.29)

Sj = S˙ a   = α˙ 2 Sˆ t,2 + ω1 Sˆ t,1 × α2 Sˆ t,2 + α˙ 1 Sˆ t,1      + α1 Sˆ t,1 × ω2 Sˆ t,2 + ω1 Sˆ t,1 × α2 Sˆ t,2 + ω1 Sˆ t,1 × ω2 Sˆ t,2   = β2 Sˆ t,2 + β1 Sˆ t,1 + 2ω1 Sˆ t,1 × α2 Sˆ t,2     + α1 Sˆ t,1 × ω2 Sˆ t,2 + ω1 Sˆ t,1 × ω1 Sˆ t,1 × ω2 Sˆ t,2 ,

(3.30)

where α1 , α2 , β1 , and β2 are angular accelerations and jerks of the two R joints. For the C joint, its velocity model is the differential of the topology model as  St = S˙ f t1 =0, θ =0   2  θ2 = ω2 Sˆ t,2 t1 Sˆ t,1 + 2 tan Sˆ t,2 v1 Sˆ t,1  t1 =0, 2 θ2 =0

= ω2 Sˆ t,2 + v1 Sˆ t,1 ,

(3.31)

where v1 and ω2 are the linear velocity of the P joint and the angular velocity of the R joint, Sˆ t,1 and Sˆ t,2 denote the instantaneous motion direction of the P joint and the instantaneous motion axis of the R joint. When the C joint moves away from its initial pose, Sˆ t,1 and Sˆ t,2 are unchanged. Thus, its velocity model is constant at any pose, as shown in Fig. 3.13.

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3 Topology and Performance Modeling of Robotic Mechanism

Fig. 3.13 The C joint from initial pose to arbitrary pose

The acceleration and jerk models of the C joint are obtained through differentiating the velocity model as Sa = S˙ t = ω˙ 2 Sˆ t,2 + v˙ 1 Sˆ t,1 = α2 Sˆ t,2 + a1 Sˆ t,1 ,

(3.32)

Sj = S˙ a = α˙ 2 Sˆ t,2 + a˙ 1 Sˆ t,1 = β2 Sˆ t,2 + j1 Sˆ t,1 ,

(3.33)

where a1 and j1 are linear acceleration and jerk of the P joint, α2 and β2 are angular acceleration and jerk of the R joint. The velocity model of the S joint is formulated by differentiating its topology model as  St = Sf θ1 =0,

θ2 =0, θ3 =0,

⎞ θ2 θ1  ω3 Sˆ t,3 2 tan Sˆ t,2 2 tan Sˆ t,1   ⎟ ⎜ 2 2 ⎟ ⎜ θ3 ˆ θ  ⎟ ⎜ = ⎜ +2 tan St,3 ω2 Sˆ t,2 2 tan 1 Sˆ t,1 ⎟  ⎟ ⎜ 2 2 ⎠ ⎝ θ3 ˆ θ2 ˆ  +2 tan St,3 2 tan St,2 ω1 Sˆ t,1 θ1 =0, 2 2 θ2 =0, ⎛

θ3 =0,

= ω3 Sˆ t,3 + ω2 Sˆ t,2 + ω1 Sˆ t,1 ,

(3.34)

where ω1 , ω2 , and ω3 are the angular velocities of the three R joints, Sˆ t,1 , Sˆ t,2 and Sˆ t,3 denote the instantaneous motion axes of the three R joints.

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85

Fig. 3.14 The S joint from initial pose to arbitrary pose

When the S joint moves away from its initial pose, the instantaneous motion axis of the R1 joint is unchanged, but the instantaneous motion axes of the R2 and R3 joints are changed. However, as shown in Fig. 3.14, no matter which pose the S joint arrives, the three-DoF instantaneous rotations of the joint can be decomposed to three one-DoF rotations about the directions of s1 , s2 , and s3 . It should be noted that the angular velocities are no longer ω1 , ω2 , and ω3 , but ω1 , ω2 , and ω3 . The velocity model of the S joint at any pose is St = ω3 Sˆ t,3 + ω2 Sˆ t,2 + ω1 Sˆ t,1 .

(3.35)

Thus, the acceleration and jerk models of the S joint are obtained through differentiating the velocity model as Sa = S˙ t = ω˙ 3 Sˆ t,3 + ω˙ 2 Sˆ t,2 + ω˙ 1 Sˆ t,1 = α3 Sˆ t,3 + α2 Sˆ t,2 + α1 Sˆ t,1 ,

(3.36)

Sj = S˙ a = α˙ 3 Sˆ t,3 + α˙ 2 Sˆ t,2 + α˙ 1 Sˆ t,1 = β3 Sˆ t,3 + β2 Sˆ t,2 + β1 Sˆ t,1 ,

(3.37)

where α1 , α2 , α3 , and β1 , β2 , β3 are angular accelerations and jerks of the three R joints, respectively. The instantaneous motion models of all joints are summarized in Table 3.2.

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Table 3.2 Performance models of joints Type R P H U

Velocity St = ωSˆ t

Acceleration Sa = α Sˆ t

Jerk

St = v Sˆ t St = ωSˆ t

Sa = aSˆ t Sa = α Sˆ t

St = ω2 Sˆ  t,2 + ω1 Sˆ t,1

Sa = α2 Sˆ  t,2 + α1 Sˆ t,1

Sj = β Sˆ t Sj = j Sˆ t Sj = β Sˆ t Sj = β2 Sˆ  t,2 + β1 Sˆ t,1

+ ω1 Sˆ t,1 × ω2 Sˆ  t,2

+ 2ω1 Sˆ t,1 × α2 Sˆ  t,2 + α1 Sˆ t,1 × ω2 Sˆ  t,2   + ω1 Sˆ t,1 × ω1 Sˆ t,1 × ω2 Sˆ  t,2

C

St = ω2 Sˆ t,2 + v1 Sˆ t,1

S

S t 3=

Sa = α2 Sˆ t,2 + a1 Sˆ t,1 Sa =

3

 ˆ k=1 αk St,k

 ˆ k=1 ωk St,k

3.3.2.2

Sj = β2 Sˆ t,2 + j1 Sˆ t,1 Sj =

3

ˆ k=1 βk St,k

Open-loop Mechanism

For an open-loop mechanism discussed in Sect. 3.3.1.2, by solving the differential of the finite motion model in Eq. (3.10) at the initial pose of the mechanism, its velocity model is obtained as,  St = S˙ f θk =0 t =0, k=1,...,n   k ˙ = Sf ,n  · · · Sf ,1 + · · · + Sf ,n  · · · S˙ f ,1 θk =0 tk =0,   = S˙ f ,n + · · · + S˙ f ,1 θk =0 tk =0,

k=1,...,n

k=1,...,n

= St,n + · · · + St,1 n  = St,k ,

(3.38)

k=1

where St denotes the velocity of the open-loop mechanism, St,k (k = 1, . . . , n) denotes the velocity generated by the kth joint of the mechanism with sk (and rk ) as its unit direction vector (and position vector), and ωk or vk as its angular velocity or linear velocity. St,k is the differential of Sf ,k at the initial pose,  St,k = S˙ f ,k θk =0 or tk =0 ,

(3.39)

which is obtained by using the differential mapping between the finite and instantaneous models of each joint. As Eq. (3.38) is a linear expression, it can be rewritten into the following matrix form as

3.3 FIS Based Finite and Instantaneous Motion Modeling

87

  St = Sˆ t,1 · · · Sˆ t,n q˙ = J q˙ ,

(3.40)

where Sˆ t,k (k = 1, . . . , n) denotes the instantaneous motion axis (associated with pitch) or instantaneous motion direction of the kth joint in the mechanism at its initial pose, J is the Jacobian matrix of the open-loop mechanism, and q˙ is the vector containing all the angular and linear velocities of all the joints in the mechanism. The Jacobian mapping follows the kinematic principle that the velocity of an open-loop mechanism is the linear addition of the velocities of all its joints. Through Eq. (3.38), only the velocity model at the initial pose can be obtained. When the mechanism moves away from its initial pose, all the instantaneous motion axes and instantaneous motion directions of the joints are all changed except those of the 1st joint connected to the fixed base. Considering the axes of finite motions cover the instantaneous axes at the initial pose, the instantaneous motion axis or instantaneous motion direction of the kth joint is changed by the motions of the 1st to (k − 1)th joints. When the mechanism moves to another pose, the topology model should be formulated in order to obtain the velocity model at that pose. The axis or direction of each joint at that pose is computed as ⎛  Sˆ t,k

=⎝

k−1 

⎞    ˜ˆ exp qj St,j ⎠Sˆ t,k , k = 2, . . . , n, Sˆ t,1 = Sˆ t,1 ,

(3.41)

j=1

where Sˆ  t,k and Sˆ t,k denote the axis (associated with pitch) or direction of the kth joint at that pose and at the initial pose of the mechanism, respectively, qj is the rotational angle or translational distance of the jth joint from the initial pose to that pose, as  qj =

θj R/H joint , j = 1, . . . , n, tj P joint

(3.42)

˜ and Sˆ t,j is the 6 × 6 matrix form of the axis or direction of the jth joint, as

˜ Sˆ t,j

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

 s˜k 0 R joint r˜k s˜k − s˜k r˜k s˜k

 0 0 P joint , j = 1, . . . , n. = ⎪ s˜k 0 ⎪ 

⎪ ⎪ ⎪ 0 s˜k ⎪ ⎪ H joint ⎩ r˜k s˜k − s˜k r˜k + hk s˜k s˜k

(3.43)

Hence, the topology model of the open-loop mechanism with respect to the new pose can be formulated as Sf = Sf ,n  · · · Sf ,1 ,

(3.44)

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3 Topology and Performance Modeling of Robotic Mechanism

where Sf and Sf ,k (k = 1, . . . , n) are the finite motions of the mechanism and its kth joint measured from the new pose,

Sf ,k =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

θk  R joint Sˆ 2 t,k  ˆ tk St,k P joint , k = 1, . . . , n,    θ  0 0 + θk − 2 tan k hk Sˆ H joint E3 0 t,k 2 (3.45) 2 tan

⎪ ⎪ θ  ⎪ ⎪ ⎩ 2 tan k Sˆ t,k 2

where θk and tk are the rotational angle and translational distance of the kth joint measured from the new pose. Similar to Eqs. (3.38) and (3.40), the velocity model and Jacobian matrix of the open-loop mechanism at the new pose is obtained by solving the differential of the finite motion model in Eq. (3.44) equation at that pose, as   St = S˙ f θk =0

t  =0,

k=1,··· ,n

  k   = S˙ f ,n + · · · + S˙ f ,1 θk =0 =



 Sˆ t,1

   · · · Sˆ t,n q˙

tk =0,

k=1,...,n

= J  q˙  ,

(3.46)

where St and J  denote the velocity and Jacobian matrix of the mechanism at that pose, q˙  is the vector containing the angular and linear velocities of all the joints at that pose, as      T  q˙  = θ˙1 ˙t1 · · · θ˙ n t˙ n θk =0

tk =0,

.

(3.47)

k=1,...,n

It is found that when all qj (j = 1, . . . , n) in Eqs. (3.43) and (3.46) equal to zero, the new pose is coincident with the initial pose. So the velocity models in Eqs. (3.46) and (3.40) can be merged together into St = J  q˙ ,

(3.48)

where St and q˙ can describe the velocity of the open-loop mechanism and the vector of joint velocity at any pose, and J  is the Jacobian matrix constituted by the unit parametric twists of all joints at any pose, which contain qj (j = 1, . . . , n) as the parameters. In this way, the acceleration and jerk models of the mechanism are derived through differentiating its velocity model in Eq. (3.48), as

3.3 FIS Based Finite and Instantaneous Motion Modeling

89

Sa = S˙ t = J  q¨ + J˙ q˙ = J  q¨ + q˙ T H q˙ ,

(3.49)

and Sj = S˙ a ... = J  q + J˙ q¨ + J¨ q˙ + J˙ q¨ ... = J  q + 2˙qT H q¨ + J¨ q˙ ,

(3.50)

where Sa and Sj denote acceleration and jerk of the open-loop mechanism, H is the Hessian matrix of the mechanism. In the same manner, the instantaneous motion model of an open-loop mechanism can be formulated by differentiating its finite motion model. Substituting the values of joint parameters qj (j = 1, . . . , n) which are determined by the pose of the mechanism in St , Sa , and Sj leads to the velocity, acceleration, and jerk models at that pose.

3.3.2.3

Closed-loop Mechanism

Similar with open-loop mechanism, for the closed-loop mechanism discussed in Sect. 3.3.1.3, its instantaneous motion model can be obtained through solving the differential of its finite motion model in Eq. (3.12) at the initial pose,  St = S˙ f  θi,k =0, i=1,2, ti,k =0 k=1,...,ni   = S˙ f ,1 ∩ S˙ f ,2  θi,k =0,

i=1,2, ti,k =0 k=1,...,ni

= St,1 ∩ St,2 = J q˙ ,  St,i = S˙ f ,i  =

ni  k=1 

St,i,k

= Sˆ t,i,1 = J i q˙ i

(3.51)

θi,k = 0 ti,k = 0, k = 1, . . . , ni , i = 1, 2,

(3.52)

 · · · Sˆ t,i,ni q˙ i

where St and St,i (i = 1, 2) are the velocities of the closed-loop mechanism and its ith limb at the initial pose, J is the Jacobian matrix of the mechanism, which is constituted by all the unit base screws of the instantaneous screw system St , q˙ is the vector containing all the angular and linear velocities corresponding to these base

90

3 Topology and Performance Modeling of Robotic Mechanism

screws, St,i,k (k = 1, . . . , ni ) denotes the velocity of the kth joint in the ith limb, Sˆ t,i,k is its unit instantaneous motion axis with si,k and ri,k as its unit direction vector and position vector, and ωi,k or vi,k as its angular velocity or linear velocity, J i is the Jacobian matrix of the ith limb, and q˙ i denotes the vector that contains all the angular and linear velocities of all the joints in the limb. When the closed-loop mechanism moves away from the initial pose, all the velocity models of its limbs change in a similar way with open-loop mechanism. The topology model of the mechanism with respect to the new pose is S f = S f ,1 ∩ S f ,2 ,

(3.53)

S f ,i = S f ,i,ni  · · · S f ,i,1 , i = 1, 2,

(3.54)

where S f , S f ,i , and S f ,i,k (k = 1, . . . , ni ) are the finite motions of the mechanism, its ith limb, and the kth joint in the limb measured from the new pose, S f ,i,k is computed by Eqs. (3.55)–(3.58), as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

 θi,k

 ˆ R joint S t,i,k Sˆ  t,i,k ti,k 2     θi,k θi,k   hi,k P joint , k = 1, . . . , ni , − 2 tan Sˆ t,i,k + θi,k S f ,i,k = 2 tan 2 2 ⎪ ⎪

 ⎪ ⎪ 0 0 ˆ ⎪ ⎪ H joint S t,i,k ⎩ E3 0 (3.55) ⎛ ⎞   k−1  ˜ Sˆ  t,i,k = ⎝ exp qi,j Sˆ t,i,j ⎠Sˆ t,i,k , k = 2, . . . , ni , Sˆ  t,i,1 = Sˆ t,i,1 (3.56)

2 tan

j=1



θi,j R/H joint , j = 1, . . . , ni ti,j P joint ⎧

 ⎪ s˜i,k 0 ⎪ ⎪ R joint ⎪ ⎪ r˜i,k s˜i,k − s˜i,k r˜i,k s˜i,k ⎪ ⎪

 ⎨ 0 0 P joint , j = 1, . . . , ni , = ⎪ ˜ s i,k 0 ⎪ 

⎪ ⎪ ⎪ 0 s˜i,k ⎪ ⎪ H joint ⎩ r˜i,k s˜i,k − s˜i,k r˜i,k + hi,k s˜i,k s˜i,k qi,j =

˜ Sˆ t,i,j

(3.57)

(3.58)

where the denotations of the symbols in Eqs. (3.55)–(3.58) can be referred to those in Eqs. (3.41)–(3.43) and (3.45). Similar with Eqs. (3.51)–(3.52), the velocity models and Jacobian matrices of the closed-loop mechanism and its limbs at the new pose are obtained by solving the differential of the finite motion models in Eqs. (3.53)–(3.54), as

3.3 FIS Based Finite and Instantaneous Motion Modeling

91

  S t = S˙  f  θi,k =0,

i=1,2,  =0 k=1,...,ni ti,k

= S t,1 ∩ S t,2 = J  q˙  ,

(3.59)

  S t,i = S˙  f ,i θi,k =0  =0, k=1,...,n ti,k i   , i = 1, 2,  = Sˆ t,i,1 · · · Sˆ  t,i,ni q˙  i

(3.60)

= J  i q˙  i where S t and S t,i (i = 1, 2) are the velocities of the closed-loop mechanism and its ith limb at the new pose, J  is the Jacobian matrix at that pose, q˙  is the vector containing all the angular and linear velocities corresponding to the unit base screws in J  . The denotations of the symbols in Eq. (3.60) can be referred to those in Eq. (3.46), and       T  ˙ti,1 · · · θ˙ i,ni t˙ i,ni , i = 1, 2. (3.61) q˙  i = θ˙i,1  θi,k =0,  =0 ti,k

k=1,...,ni

When all qi,j (i = 1, 2, j = 1, . . . , ni ), in Eqs. (3.56), (3.59) and (3.60) equal to zero, the new pose is coincident with the initial pose. Hence, Eqs. (3.59) and (3.51) can be combined as St = J  q˙ ,

(3.62)

where St and q˙ can describe the velocity of the closed-loop mechanism and the vector of initiative velocities of its joints at any pose, and J  is the Jacobian matrix at any pose, which contains the parameters in q. Thus, the acceleration and jerk models of the closed-loop mechanism are derived through differentiating its velocity model, as Sa = S˙ t = J  q¨ + J˙ q˙ = J  q¨ + q˙ T H q˙ ,

(3.63)

and Sj = Sa ... = J  q + J˙ q¨ + J¨ q˙ + J˙ q¨ ... = J  q + 2˙qT H q¨ + J¨ q˙ ,

(3.64)

where Sa and Sj denote acceleration and jerk of the closed-loop mechanism, and H is the Hessian matrix of the mechanism. Substituting the values of parameters in q

92

3 Topology and Performance Modeling of Robotic Mechanism

that are determined by the pose of the mechanism into those in St , Sa , and Sj leads to the velocity, acceleration, and jerk models at that pose. The finite and instantaneous motion models of complex mechanisms can be formulated regarding open-loop and closed-loop mechanisms as their parts. In this way, the finite motion models of these mechanisms are obtained by using the composition and intersection of the models of open-loop and closed-loop mechanisms, and their instantaneous models can be directly solved as the differential of the corresponding finite motion models.

3.4 Example 3.4.1 Typical Open-loop Mechanism The general topology and performance models of open-loop mechanisms have been formulated in Sects. 3.3.1.2 and 3.3.2.2, and so do the models of closed-loop mechanisms in Sects. 3.3.1.3 and 3.3.2.3. The models of some commonly used open-loop mechanisms will be discussed accordingly in this section. (a) PPRH mechanism For example, a four-DoF open-loop mechanism consisting of two P joints, one R joint, and one H joint is shown in Fig. 3.15. The mechanism is denoted as P1 P2 R3 H4 . Based upon Eq. (3.10), the topology model of this mechanism is formulated in finite screw form as Fig. 3.15 A P1 P2 R3 H4 mechanism

3.4 Example

93

    θ4 0 s4 + h4 θ4 Sf = 2 tan s4 2 r4 × s4       θ3 0 0 s3 t2 t1 , 2 tan s2 s1 2 r3 × s3 

(3.65)

where the denotations of the symbols in Eq. (3.65) can be referred to those in Eqs. (3.4)–(3.6). It is clear that this equation contains all the types, sequences, directions, and positions of all the joints in the P1 P2 R3 H4 open-loop mechanism. According to the discussions in Sect. 3.2, the performance models of open-loop mechanisms can be directly obtained through solving the multi-order differentials of the corresponding topology models. Hence, the velocity model of the P1 P2 R3 H4 mechanism is formulated by differentiating its topology model in Eq. (3.65) with respect to the initial pose as  St = S˙ f t1 =0,t2 =0, θ3 =0,θ4 =0         s4 s3 0 0 + ω3 + v2 + v1 = ω4 r4 × s4 + h4 s4 r3 × s3 s2 s1 ⎛ ⎞ v1 .  ⎜ v2 ⎟ ⎟ = Sˆ t,1 Sˆ t,2 Sˆ t,3 Sˆ t,4 ⎜ ⎝ ω3 ⎠ ω4

(3.66)

= J q˙ When the mechanism moves to other poses, based upon the analysis in Eqs. (3.41)– (3.48), its velocity model changes to   St = Sˆ  t,1 Sˆ  t,2 Sˆ  t,3 Sˆ  t,4 q˙ = J  q˙

,

(3.67)

where Sˆ  t,1 = Sˆ t,1 ,  ˆS t,2 = exp t1 0 s˜1  0 Sˆ  t,3 = exp t1 s˜1  ˆS t,4 = exp t1 0 s˜1

0 0

 Sˆ t,2 ,

 0 exp t2 s˜2   0 0 exp t2 s˜2 0 0 0



0 0 0 0

 Sˆ t,3 , 

 exp θ3

0 s˜3 r˜3 s˜3 − s˜3 r˜3 s˜3

 Sˆ t,4 .

In this way, the acceleration and jerk models of the P1 P2 R3 H4 mechanism are obtained as the differentiations of Eq. (3.67), as

94

3 Topology and Performance Modeling of Robotic Mechanism

Sa = S˙ t = J  q¨ + q˙ T H q˙ ,

(3.68)

and Sj = S˙ a ... = J  q + 2˙qT H q¨ + J¨ q˙ .

(3.69)

For an n-DoF open-loop mechanism, it consists of n one-DoF joints. Each joint is one of R, P, H joints. Thus, the total number of structures of n-DoF open-loop mechanisms is 3n without considering the geometric relationships among the directions and positions of the joints. Here, we enumerate the topology models of some commonly used open-loop mechanisms for the readers’ convenience. Following the same method and procedures, the velocity, acceleration, jerk models, and the involved Jacobian and Hessian matrices of these mechanisms and any other open-loop mechanisms can all be formulated in a simple and straightforward manner. (b) PU mechanism A PU mechanism is shown in Fig. 3.16, which can be regarded as a P1 R2 R3 mechanism, its topology model is formulated as       θ2 θ3 0 s3 s2 2 tan t1 , Sf = 2 tan s1 2 rO × s3 2 rO × s2

(3.70)

where the denotations of the symbols can be referred to the topology models of P and U joints in Eqs. (3.5) and (3.7). (c) UP mechanism As the kinematic inverse of the PU mechanism, a UP mechanism is shown in Fig. 3.17, which can be regarded as a R1 R2 P3 mechanism. Similar with the PU mechanism, its Fig. 3.16 PU mechanism

3.4 Example

95

Fig. 3.17 UP mechanism

topology model is formulated as  S f = t3

     θ2 θ1 0 s2 s1 2 tan 2 tan , s3 2 rO × s2 2 rO × s1

(3.71)

where the denotations of the symbols can also be referred to the topology models of U and P joints. It is easy to see that the topology models of PU and UP mechanisms have the same three factors in the screw triangle product, but they have the opposite sequences. They are called kinematic inverse of each other. (d) UPU mechanism As shown in Fig. 3.18, a UPU mechanism has two U joints and one P joint. It can be regarded as a R1 R2 P3 R4 R5 mechanism of which the topology model is formulated as       θ4 θ5 0 s5 s4 2 tan t3 Sf = 2 tan s3 2 rO2 × s5 2 rO2 × s4     θ1 θ2 s2 s1 2 tan , (3.72) 2 tan 2 rO1 × s2 2 rO1 × s1 where the denotations of the symbols can be referred to the topology models of U and P joints. The kinematic inverse of UPU mechanism is itself. (e) PRS mechanism A PRS mechanism is shown in Fig. 3.19, which can be regarded as a P1 R2 R3 R4 R5 mechanism. Its topology model is formulated as

96

3 Topology and Performance Modeling of Robotic Mechanism

Fig. 3.18 UPU mechanism

Fig. 3.19 PRS mechanism

    θ4 θ5 s5 s4 2 tan Sf = 2 tan 2 rO × s5 2 rO × s4       θ2 θ3 0 s3 s2 2 tan t1 , 2 tan s1 2 rO × s3 2 r2 × s2

(3.73)

where the denotations of the symbols can be referred to the topology models of P, R and S joints in Eqs. (3.5), (3.4) and (3.9).

3.4 Example

97

Fig. 3.20 RPS mechanism

(f) RPS mechanism As shown in Fig. 3.20, the RPS mechanism can be regarded as a R1 P2 R3 R4 R5 mechanism. Similar with the PRS mechanism, its topology model is formulated as     θ4 θ5 s5 s4 Sf = 2 tan 2 tan 2 rO × s5 2 rO × s4       θ1 θ3 0 s3 s1 t2 2 tan , (3.74) 2 tan s2 2 rO × s3 2 r1 × s1 where the denotations of the symbols can also be referred to the topology models of R, P and S joints.

3.4.2 Typical Closed-loop Mechanism Having the topology models of different open-loop mechanisms at hand, the topology models of limbs in closed-loop mechanisms can be obtained. According to Eq. (3.12), the topology model of a closed-loop mechanism is the intersection of its limbs and can thus be formulated. Here, we give two examples of closed-loop mechanisms and another two examples of multiple closed-loop limbs.

98

3 Topology and Performance Modeling of Robotic Mechanism

Fig. 3.21 RRR-PPR closed-loop mechanism

(a) RRR-PPR mechanism As shown in Fig. 3.21, the RRR-PPR closed-loop mechanism is also denoted as R1 R1 R1 -P2 P3 R2 . The link between the joints R1 and P2 is the fixed base, and the link between the joints R1 and R2 is the moving platform. Using the topology models of R and P joints, the topology models of the two limbs are formulated by Eq. (3.11) as Sf ,1 = 2 tan

      θ1,2 θ1,1 θ1,3 s1 s1 s1 2 tan 2 tan , 2 r1,3 × s1 2 r1,2 × s1 2 r1,1 × s1 (3.75)       θ2 0 0 s2 Sf ,2 = 2 tan t3 t2 . (3.76) s3 s2 2 r2 × s2

where the denotations of the symbols in Eqs. (3.75) and (3.76) can be referred to those in Eq. (3.11). The relationships among the joint directions are sT2 s1 = 0, sT3 s1 = 0, s2 × s3 = 0.

(3.77)

According to Eq. (3.12), the topology model of the mechanism is computed as Sf = Sf ,1 ∩ Sf ,2     0 0 t3 . = t2 s2 s3

(3.78)

From the topology model formulated, it is easy to see that the closed-loop mechanism has two DoFs, and P2 and P3 can be selected as the actuation joints of the mechanism.

3.4 Example

99

Using the method based on FIS theory, the velocity model of the mechanism is obtained as  St = S˙ f t2 =0,t3 =0, θ1,i =0,θ2 =0, i=1,2,3   = S˙ f ,1 ∩ S˙ f ,2 t2 =0,t3 =0, θ1,i =0,θ2 =0,

i=1,2,3

          0 0 0 0  t3 + t2 v3 = v2 t2 =0,t3 =0, s2 s3 s2 s3 θ1,i =0,θ2 =0,     0 0 + v3 = v2 s2 s3  

v2 0 0 = v3 s2 s3

i=1,2,3

.

(3.79)

= J q˙ When the mechanism moves from its initial pose to other poses, its velocity model is unchanged. Thus, through differentiating Eq. (3.79), the acceleration and jerk models of the mechanism are Sa = S˙ t = J q¨ ,

(3.80)

Sj = S˙ a ... = J q.

(3.81)

and

(b) RRRR Bennett closed-loop mechanism As shown in Fig. 3.22, the RRRR closed-loop mechanism, which is called Bennett mechanism, has four R joints. The mechanism is also denoted as R1 R2 -R4 R3 . The link between the joints R1 and R4 is the fixed base, and the link between the joints R2 and R3 is the moving platform. Using the topology models of R joints, the topology models of the two limbs are formulated as     θ1 θ2 s2 s1 2 tan , (3.82) Sf ,1 = 2 tan 2 r2 × s2 2 r1 × s1     θ4 θ3 s3 s4 Sf ,2 = 2 tan 2 tan , (3.83) 2 r3 × s3 2 r4 × s4 where the denotations of the symbols in Eqs. (3.82) and (3.83) can be referred to those in Eq. (3.11). The relationships among the direction and position vectors of

100

3 Topology and Performance Modeling of Robotic Mechanism

Fig. 3.22 RRRR Bennett closed-loop mechanism

the joints are         cos−1 sT1 s4 = cos−1 sT2 s3 = α, cos−1 sT1 s2 = cos−1 sT3 s4 = β, |r4 − r1 | = |r2 − r3 | = a, |r2 − r1 | = |r3 − r4 | = b, sin α sin β = . (3.84) a b The topology model of the mechanism is computed by using Eq. (3.12), as Sf = Sf ,1 ∩ Sf ,2 α+β     sin θ1 θ1 s2 s1 2 2 tan . =2 tan β −α 2 r2 × s2 2 r1 × s1 sin 2

(3.85)

Thus, Bennett mechanism has only one rotational DoF, and R1 can be selected as the actuation joint. Using the method based on FIS theory, the velocity model of the mechanism is obtained as  St = S˙ f  θi = 0, i = 1, . . . , 4   = S˙ f ,1 ∩ S˙ f ,2  θi = 0, i = 1, . . . , 4 ⎞ ⎛ α+β     sin s2 s1 ⎟ ⎜ 2 + =⎝ (3.86) ⎠ω1 β − α r2 × s2 r1 × s1 sin 2

3.4 Example

101

In Eq. (3.86), the base twist of the Bennett mechanism is not rewritten into a unit twist for simplicity. The velocity model in Eq. (3.86) is changed when the mechanism moves from its initial pose to other poses. It is changed into ⎞ α+β     s1 exp(θ1 s˜1 )s2 ⎟ ⎜ 2 + St = ⎝ ⎠ω1 . β − α (r1 + exp(θ1 s˜1 )(r2 − r1 )) × exp(θ1 s˜1 )s2 r1 × s1 sin 2 (3.87) ⎛

sin

Thus, through differentiating Eq. (3.87), the acceleration and jerk models of the mechanism are Sa = S˙ t

⎞ α+β ⎛ ω1 s˜1 exp(θ1 s˜1 )s2 2 ⎝ = (ω1 s˜1 exp(θ1 s˜1 )(r2 − r1 ) × exp(θ1 s˜1 )s2 ⎠ω1 β −α +(r1 + exp(θ1 s˜1 )(r2 − r1 )) × (ω1 s˜1 exp(θ1 s˜1 )s2 )) sin 2 ⎞ ⎛ α+β     sin s1 exp(θ1 s˜1 )s2 ⎟ ⎜ 2 + +⎝ ⎠α1 , β − α (r1 + exp(θ1 s˜1 )(r2 − r1 )) × exp(θ1 s˜1 )s2 r1 × s1 sin 2 (3.88) sin

and α+β 2 Sj = S˙ a = β −α sin 2 ⎞ ⎛ α1 s˜1 exp(θ1 s˜1 )s2 + (ω1 s˜1 )2 exp(θ1 s˜1 )s2   ⎟ ⎜ ⎟ ⎜ α1 s˜1 exp(θ1 s˜1 )(r2 − r1 ) + (ω1 s˜1 )2 exp(θ1 s˜1 )(r2 − r1 ) ⎟ ⎜ ⎟ ×⎜ ⎜ × exp(θ1 s˜1 )s2 + 2ω1 s˜1 exp(θ1 s˜1 )(r2 − r1 ) × (ω1 s˜1 exp(θ1 s˜1 )s2 ) ⎟ω1 ⎟ ⎜ + (r1 + exp(θ1 s˜1 )(r2 − r1 )) ⎠ ⎝   sin

× α1 s˜1 exp(θ1 s˜1 )s2 + (ω1 s˜1 )2 exp(θ1 s˜1 )s2

⎞ α+β ⎛ ω1 s˜1 exp(θ1 s˜1 )s2 2 ⎝ +2 (ω1 s˜1 exp(θ1 s˜1 )(r2 − r1 ) × exp(θ1 s˜1 )s2 ⎠α1 β −α +(r1 + exp(θ1 s˜1 )(r2 − r1 )) × (ω1 s˜1 exp(θ1 s˜1 )s2 )) sin 2 ⎞ ⎛ α+β     sin s1 exp(θ1 s˜1 )s2 ⎟ ⎜ 2 + +⎝ ⎠β1 . β − α (r1 + exp(θ1 s˜1 )(r2 − r1 )) × exp(θ1 s˜1 )s2 r1 × s1 sin 2 (3.89) sin

102

3 Topology and Performance Modeling of Robotic Mechanism

As the Myard and Goldberg mechanisms are the combinations of two Bennett mechanisms, the models obtained here can be directly used in the formulation of the models of these mechanisms as well as the complex assembly of several Bennett mechanisms. (c) 4PPRH mechanism 4PPRH mechanism is composed of four four-DoF limbs, as shown in Fig. 3.23. Each limb consists of two P joints, one R joint, and one H joint. The four limbs can be regarded as open-loop mechanisms, and their topology models are formulated by referring to Eq. (3.65) as      θi,4 0 si,4 + hi,4 θi,4 Sf ,i = 2 tan si,4 2 ri,4 × si,4       θi,3 0 0 si,3 ti,2 ti,1 , i = 1, . . . , 4 2 tan si,2 si,1 2 ri,3 × si,3

(3.90)

where the denotations of ti,1 , ti,2 , θi,3 , θi,4 , and hi,4 are similar to those in Eq. (3.65). The geometrical relationships among the joints of the four limbs are: (1) the directions of the second P joints and the H joints in all limbs are parallel to each other, s1,2 = s2,2 = s3,2 = s4,2 = s1,4 = s2,4 = s3,4 = s4,4 ;

(3.91)

(2) the directions of the R joints in all limbs are parallel to each other, s1,3 = s2,3 = s3,3 = s4,3 ; (3) The translational planes of all limbs have common vertical as s1,3 , Fig. 3.23 4PPRH mechanism

(3.92)

3.4 Example

103

s1,1 × s1,2 = s2,1 × s2,2 = s3,1 × s3,2 = s4,1 × s4,2 = s1,3 .

(3.93)

The topology model of the mechanism can be formulated as the intersection of the models of the two closed-loop mechanisms contained in it. According to the analysis in Sect. 3.3.1.3 and Eq. (3.65), the topology model of the 4PPRH mechanism is the intersection of the models of two 2PPRH closed-loop mechanisms, as     Sf = Sf ,1 ∩ Sf ,2 ∩ Sf ,3 ∩ Sf ,4 ,

(3.94)

which can be computed using the geometrical relationships through algebraic derivations. Finally, the topology mode of the 4PPRH mechanism is       θ3,3 0 0 s3,3 t2,2 t1,1 . Sf = 2 tan s2,2 s1,1 2 r3,3 × s3,3

(3.95)

It is obvious that the parallel mechanism has three DoFs. P1,1 , P2,2 , and R3,3 can be selected as the actuation joints of the mechanism. Its velocity model is  St = S˙ f ti,1 =0,ti,2 =0, θi,3 =0,θi,4 =0, i=1,...,4   ˙ = Sf ,1 ∩ S˙ f ,2 ∩ S˙ f ,3 ∩ S˙ f ,4 ti,1 =0,ti,2 =0, θi,3 =0,θi,4 =0

i=1,...,4

    ⎞  s3,3 0 0  ω t t 3,3 2,2 1,1 ⎜ ⎟ r3,3 × s3,3 s2,2 s1,1 ⎟ ⎜ ⎜     ⎟   ⎜ θ3,3 0 0 ⎟ s3,3 ⎜ ⎟ v2,2 t1,1 = ⎜ +2 tan  ⎟ r × s s s 2 3,3 3,3 2,2 1,1 ⎟ ⎜ ⎜     ⎟  ⎝ θ3,3 0 0 ⎠ s3,3 +2 tan t2,2 v1,1 ti,1 =0,ti,2 =0, s2,2 s1,1 2 r3,3 × s3,3 θi,3 =0,θi,4 =0       s3,3 0 0 + v2,2 + v1,1 = ω3,3 r3,3 × s3,3 s2,2 s1,1 ⎞ ⎛  v1,1

0 0 s3,3 ⎝ v2,2 ⎠ = s1,1 s2,2 r3,3 × s3,3 ω3,3 ⎛



i=1,...,4

= J q˙ .

(3.96)

When the mechanism moves to other poses, its velocity model changes to

St =

0 s1,1

= J  q˙ .

⎞ ⎛  v1,1 0  s3,3 ⎝ v2,2 ⎠  s2,2 r3,3 + t1,1 s1,1 + t2,2 s2,2 × s3,3 ω3,3 (3.97)

104

3 Topology and Performance Modeling of Robotic Mechanism

In this way, its acceleration model and jerk models are obtained as Sa = S˙ t = J  q¨ + q˙ T H q˙

⎞ ⎛  a1,1 0  s3,3 ⎝ a2,2 ⎠  = s1,1 s2,2 r3,3 + t1,1 s1,1 + t2,2 s2,2 × s3,3 α3,3 ⎞ ⎛

 v1,1 00 0 ⎝ v2,2 ⎠  + 0 0 v1,1 s1,1 + v2,2 s2,2 × s3,3 ω3,3

0

(3.98)

and ... Sj = J  q + 2˙qT H q¨ + J¨ q˙

⎞ ⎛  j1,1 0  s3,3 ⎝ j2,2 ⎠  = s1,1 s2,2 r3,3 + t1,1 s1,1 + t2,2 s2,2 × s3,3 β3,3 ⎞ ⎛

 a1,1 00 0 ⎝ a2,2 ⎠  +2 0 0 v1,1 s1,1 + v2,2 s2,2 × s3,3 α3,3 ⎞ ⎛

 v1,1 00 0 ⎝ v2,2 ⎠  + 0 0 j1,1 s1,1 + j2,2 s2,2 × s3,3 ω3,3

0

(3.99)

(d) 2UPR-SPR mechanism 2UPR-SPR mechanism, which is known as Exechon robot, is composed of two four-DoF limbs and one five-DoF limb, as shown in Fig. 3.24. Each limb can be regarded as an open-loop mechanism. The topology model of the UPR limb and SPR limb are formulated according to Eq. (3.11) as     θi,4 si,4 0 ti,3 si,3 2 ri,4 × si,4     θi,1 θi,2 si,2 si,1 2 tan , i = 1, 3, (3.100) 2 tan 2 ri,20 × si,2 2 ri,1 × si,1       θi,3 θi,5 0 si,5 si,3 ti,4 2 tan = 2 tan × si,5 × si,3 si,4 2 ri,5 2 ri,3     , i = 2, θ θi,2 si,2 si,1 i,1 2 tan 2 tan 2 ri,2 × si,2 2 ri,1 × si,1 (3.101)

Sf ,i = 2 tan

Sf ,i

3.4 Example

105

Fig. 3.24 2UPR-SPR mechanism

where the denotations can be referred to those in Eq. (3.11). The geometric relations among the joints of the limbs are (1) The first axes of U joints in two UPR limbs should be colinear, which are parallel with the axis of R joint in SPR limb at the initial pose, s1,1 = s3,1 = s2,5 , r1,1 × s1,1 = r3,1 × s3,1 .

(3.102)

(2) The directions of R joints in RPU limbs are parallel with each other, which are also parallel with the second axes of U joints, s1,2 = s3,2 = s1,4 = s3,4 .

(3.103)

Based upon Sect. 3.3.1.3 and Eq. (3.12), the Exechon robot’s topology model can be described as, Sf = Sf ,1 ∩ Sf ,2 ∩ Sf ,3 .

(3.104)

As illustrated in Fig. 3.24, the three rotational axes of the S joint in SPR limb are considered parallel with x-, y-, z- axis, respectively. Finally, Eq. (3.104) could be further solved as,  S f = tz

       θ1,1 θ2,2 0 s1,1 exp θ1,1 s˜1,1 s2,2 2 tan 2 tan . sz 2 r1,1 × s1,1 2 r2,2 × exp θ1,1 s˜1,1 s2,2 (3.105)

106

3 Topology and Performance Modeling of Robotic Mechanism

where sz is the unit vector along the z axis of the fixed reference frame, i.e., sz =  T 0 0 1 , t z is the translational distance of the mechanism along sz measured from its initial pose. It indicates from Eq. (3.105) that the Exechon robot has one translational and two rotational motions. The velocity model of Exechon robot at initial pose can be derived by differentiating Eq. (3.105) as, St = ˙tz

=



0 sz



+ θ˙1,1



s1,1 r1,1 × s1,1

s2,2 0 s1,1 sz r1,1 × s1,1 r2,2 × s2,2



+ θ˙2,2



⎞ ⎛  v4 ⎝ ω1,1 ⎠

s2,2 r2,2 × s2,2



ω2,2

= J q˙ ,

(3.106)

where v4 is the linear velocity of the moving platform. When the mechanism moves to a new pose, its velocity model changes to St = ˙t4

=



0 s4



+ θ˙1,1

0 s1,1 s4 r1,1 × s1,1

   exp θ1,1 s˜1,1 s2,2 r2,2 × exp θ1,1 s˜1,1 s2,2 ⎞ ⎛    v4 exp θ1,1 s˜1,1 s2,2 ⎝ ω1,1 ⎠ r2,2 × exp θ1,1 s˜1,1 s2,2 ω2,2



s1,1 r1,1 × s1,1



+ θ˙2,2



= J  q˙ ,

(3.107)

where J  is the Jacobian matrix of the parallel mechanism when the mechanism moves to any poses, which denotes the linear twist space of the moving platform,       T  exp θ1,1 s˜1,1 s2,2 × exp θ2,2 exp θ1,1 s˜1,1 s˜2,2 exp θ1,1 s˜1,1 s1,1  s4 =     T   .    exp θ1,1 s˜1,1 s2,2 × exp θ2,2 exp θ1,1 s˜1,1 s˜2,2 exp θ1,1 s˜1,1 s1,1  By differentiating the velocity model of the mechanism expressed in Eq. (3.107), the acceleration model can be formulated as,

Sa =

s2,2 0 s1,1 s4 r1,1 × s1,1 r2,2 × s2,2

⎞ ⎛  a4 ⎝ α1,1 ⎠ α2,2

⎞ ⎛  v4 0 0 α1,1 sz ⎝ ω1,1 ⎠. + α1,1 s2,2 − α2,2 s1,1 0 r2,2 × α1,1 sz ω2,2

(3.108)

3.4 Example

107

It would turn to be the jerk model by taking further differential operation of Eq. (3.108), as

Sj =

s2,2 0 s1,1 sz rA1 × s1,1 r2,2 × s2,2

+2

+

⎤ ⎡  j5 ⎣ β1,1 ⎦ β2,2

0 0 α1,1 sz α1,1 s2,2 − α2,2 s1,1 0 r2,2 × α1,1 sz

0 0 β1,1 sz β1,1 s2,2 − β2,2 s1,1 0 r2,2 × β1,1 sz

⎞ ⎛  a4 ⎝ α1,1 ⎠

α2,2 ⎤ ⎡  v4 ⎣ ω1,1 ⎦.

(3.109)

ω2,2

The above examples show the procedures of formulating the topology and performance models of closed-loop mechanisms. Following the same manner, topology model of any mechanism can be formulated. The performance models and the involved Jacobian and Hessian matrices can all be obtained using this simple and straightforward method by differentiating the topology model.

3.5 Integrated Framework for Type Synthesis and Performance Analysis Different topological structure leads to different motion capability of the robotic mechanism. In the analysis and design of the robotic mechanism, the topological structure design, i.e., type synthesis, is firstly implemented to make the mechanism realize certain mobility. For this purpose, the adopted topology model should (1) contain the complete topology information to uniquely determine the mechanism, and (2) describe the full-cycle motion characteristics to guarantee the mobility of the mechanism. The finite screw based topology model covers the above-mentioned requirements thus can be used for type synthesis of the mechanism, which will be introduced in detail from Chaps. 4–6 in this book. On the whole, the topology model based on finite screw is in form of analytical expression and the robotic mechanism can be synthesized in an algebraic manner. The instantaneous screw based performance model reflects different order differential kinematics of a mechanism. For the first-order differentiation of the displacement model, the resulted instantaneous motion is called twist described by instantaneous screw, as has mentioned in Chap. 2. The exerted force to the mechanism can also be described by instantaneous screw, which is called wrench. The twist and wrench models can be directly applied to analyze and evaluate the kinematic features of the robotic mechanism. Specifically, if the elastic features of the bodies are considered, the deformation twist is included in the twist model of the mechanism.

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3 Topology and Performance Modeling of Robotic Mechanism

As a result, the stiffness analysis and evaluation of the mechanism can be implemented. For the second-order of the displacement model, the instantaneous motion contains both velocity and acceleration of the mechanism, from which the dynamic performance model can be formulated. The details of the performance analysis and modeling will be introduced from Chaps. 7–9 in this book. With the performance models available at hand, the optimal design of the robotic mechanism (illustrated in Chap. 10) can be implemented. Both the topology model and performance model have effects on the design of the robotic mechanism. It has long been a desire to unify the type synthesis and performance analysis under the same framework. Based on the FIS theory, the topology model of the robotic mechanism is formulated by the finite screw, and the performance analysis can be implemented in both finite and instantaneous screw expressions. The topology and performance of the robotic mechanism are closely related by the differential mapping between the finite screw and instantaneous screw. Therefore, type synthesis and performance analysis of the robotic mechanisms are integrated into the framework of FIS theory. The procedures of the proposed method based upon the FIS theory for unified topology and performance modeling are shown in Fig. 3.25. As discussed in Chap. 2, the FIS theory is the simplest and most concise mathematical tool for motion description. Additionally, the FIS theory can consistently express the motions of mechanisms in an algebraic manner, resulting in analytical expressions for topology and performance models of different kinds of mechanisms. In this way, the process of type synthesis and performance analysis of mechanisms is extremely simplified, and the obtained results are guaranteed to be correct. Since the performance models of mechanisms are directly obtained by the differentiation of the corresponding topology models, the topology and performance of the robotic mechanism can be unified in the FIS based framework.

Fig. 3.25 Integrated framework for type synthesis and performance analysis of robotic mechanisms

3.6 Conclusion

109

3.6 Conclusion Based on the FIS theory, this chapter presents a unified method of topology and performance modeling of robotic mechanisms. The algebraic and closely related topology and performance models are formulated, which lays a theoretical foundation of type synthesis and performance analysis of mechanisms. The main key points of this chapter are listed below for the readers’ convenience. (1) Topology models of joints, open-loop mechanisms, and closed-loop mechanisms, which contain all the topology information of mechanisms, are formulated in an algebraic manner through describing their finite motions in the forms of finite screws. (2) The topology model can be used as a displacement model. The performance models are closely linked to the finite and instantaneous motion of the robotic mechanism. Specifically, instantaneous motion models involving the velocity, acceleration, and jerk integrate different order kinematics of mechanisms under the FIS theory, which are essential in the kinematic, stiffness, and dynamic performance analysis. (3) By using the differential mapping between the finite screw and instantaneous screw, the underlying relationships between topology structures and performance properties of robotic mechanisms are revealed. An integrated framework for the topology and performance modeling of robotic mechanisms by using FIS theory is proposed. (4) The proposed method of topology and performance modeling is general and suitable for any mechanism, leading to a solid foundation of type synthesis and performance analysis of robotic mechanisms.

References 1. Sun T, Song YM, Gao H et al (2015) Topology synthesis of a 1-translational and 3-rotational parallel manipulator with an articulated traveling plate. J Mech Robot Trans ASME 7(3):031015(9 pages) 2. Huang Z, Li QC (2002) General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. Int J Robot Res 21(2):131–145 3. Gosselin CM (2002) Stiffness mapping for parallel manipulators. IEEE Trans Robot Autom 6(3):377–382 4. Merlet JP (1993) Direct kinematics of parallel manipulators. IEEE Trans Robot Autom 9(6):842–846 5. Gallardo J, Rico JM, Frisoli A et al (2003) Dynamics of parallel manipulators by means of screw theory. Mech Mach Theory 38(11):1113–1131 6. Qi Y, Sun T, Song YM et al (2015) Topology synthesis of three-legged spherical parallel manipulators employing Lie group theory. Proc Inst Mech Eng Part C J Mech Eng Sci 229(10):1873–1886 7. Yang SF, Sun T, Huang T et al (2016) A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech Mach Theory 104:405–419

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8. Yang SF, Sun T, Huang T et al (2017) Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mech Mach Theory 109:220–230 9. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 10. Sun T, Song YM, Li YG et al (2010) Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. J Mech Robot Trans ASME 2(3):031009(8 pages) 11. Sun T, Song YM, Dong G et al (2012) Optimal design of a parallel mechanism with three rotational degrees of freedom. Robot Comput Integr Manuf 28(4):500–508 12. Lian BB, Sun T, Song YM et al (2015) Stiffness analysis and experiment of a novel 5-DoF parallel kinematic machine considering gravitational effects. Int J Mach Tools Manuf 95:82–96 13. Sun T, Liang D, Song YM Singular-perturbation-based nonlinear hybrid control of redundant parallel robot. IEEE Trans Ind Electron 65(4):3326–3336 14. Sun T, Lian BB, Song YM et al (2019) Elasto-dynamic optimization of a 5-DoF parallel kinematic machine considering parameter uncertainty. IEEE-ASME Trans Mechatron 24(1):315–325 15. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 16. Sun T, Yang SF, Huang T et al (2018) A finite and instantaneous screw based approach for topology design and kinematic analysis of 5-axis parallel kinematic machines. Chin J Mech Eng 31(2):66–75 17. Sun T, Yang SF (2019) An approach to formulate the Hessian matrix for dynamic control of parallel robots. IEEE-ASME Trans Mechatron 24(1):271–281 18. Kong XW, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Berlin, Heidelberg 19. Hunt KH (1978) Kinematic geometry of mechanisms. Clarendon Press, Oxford 20. Angeles J (2014) Fundamentals of robotic mechanical systems: Theory, Methods, and Algorithms, 4th edn. Springer, New York

Chapter 4

Type Synthesis Method and Procedure of Robotic Mechanism

4.1 Introduction Topology of a robotic mechanism is also called its type [1], which denotes the mechanism’s basic structure including number and sequence of all joints in the mechanism, as well as the classifications and axes/directions of the joints. The type determines the motions and performances of the mechanism. Thus, type design is the prerequisite step in the innovation design of robotic mechanisms [2–7]. In the design of a robotic mechanism, the primary objective is making the mechanism have desired mobility that is usually defined by the number and types of DoFs. In particular, mobility of the mechanism concerns the continuous set of poses of its end effector or moving platform, which is named as motion pattern in this book. There are numerous mechanisms that can realize the same motion pattern. Therefore, obtaining all the feasible mechanical structures generating the expected motion pattern is the first step in the design of the mechanism, which is called type synthesis. The two main issues involved in type synthesis of robotic mechanisms are: (1) How to define, distinguish, and express different motion patterns through using analytical expressions; (2) How to obtain the whole set of feasible mechanical structures having the expected motion pattern in an algebraic manner. With the topology model formulated in Chap. 3, these two issues will be investigated in depth by using a finite screw [8] as the mathematical tool. In this chapter, a motion pattern is defined as the analytical set of finite screws, on which basis, the general procedures of type synthesis method for different kinds of mechanisms, including open-loop, closed-loop, and other complex mechanisms, are presented. Following the type synthesis method, the commonly used motion patterns are firstly described by finite screws. Then, the key topics in terms of limb synthesis, mechanism assembly, and actuation arrangements are investigated detailedly. Finally, type synthesis of the mechanisms with invariable and variable rotation axes are introduced and briefly discussed, which will be illustrated in detail in Chaps. 5 and 6. © Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_4

111

112

4 Type Synthesis Method and Procedure of Robotic Mechanism

This chapter gives the generic procedures for type synthesis of robotic mechanisms and proposes a finite screw method [9–12] based upon analytical expressions and algebraic derivations. The proposed method is a general one, which is suitable for any mechanism.

4.2 General Procedure of Finite Screw Based Type Synthesis The objective of type synthesis is obtaining all the feasible mechanical structures that generate the expected motion pattern. A motion pattern [2] is defined as a continuous set of poses, which describes the desired motions (the number and types of DoFs) at the end effector or moving platform of a mechanism. For example, a five-DoF motion pattern may mean that the end effector or moving platform can realize three-DoF translations and two-DoF rotations around two given directions. The mechanical structure includes the number and sequence of all joints in the mechanism as well as the classification, position and/or direction of each joint. Hence, the input of type synthesis is an expected motion pattern, and the output is the whole set of feasible mechanisms [13–19]. The fundamental issue about a type synthesis method is the way in which the motion pattern is defined, i.e., the mathematical tool that is used. In order to describe the full motion characteristics of the mechanism, finite motion expression is preferred to be employed in motion pattern description instead of instantaneous motion one [20], because the finite motion model involves all the topology information of the mechanism, as discussed in Chap. 3. Among the mathematical tools that can be used to express finite motions of mechanisms, the finite screw has the simplest and most concise format [8], as strictly proved in Chap. 2. Through expressing the desired pose as a finite screw, the motion pattern can be directly defined as the continuous set of finite screws, leading to the precise description of the full-cycle motion characteristics of mechanisms. Since finite screw has clear algebraic structure with analytical composition and intersection algorithms, the feasible structures which generate the expected motion pattern can be rigorously synthesized without relying on the design experience. After describing the expected motion pattern by a set of finite screws, the next step is determining which kind of mechanisms is desired to be synthesized. According to the analysis in Chap. 3, robotic mechanisms can be classified into three categories. (1) Open-loop mechanism, which has a serial structure and consists of several oneDoF joints; (2) The closed-loop mechanism includes single closed-loop mechanism and multiclosed-loop mechanism. It is composed of two or more limbs in parallel structure. Each limb consists of several one-DoF joints; (3) Hybrid mechanism, which also has a serial structure and consists of several parts. Each part is an open-loop mechanism or a closed-loop mechanism.

4.2 General Procedure of Finite Screw Based Type Synthesis

113

Fig. 4.1 The procedure for type synthesis of robotic mechanisms

With the known category of the mechanisms, the problem turns into the type synthesis of the specific kind of mechanisms. The general procedures for the type synthesis of robotic mechanisms are summarized in Fig. 4.1, from which the type synthesis of open-loop mechanisms, closed-loop mechanisms, and hybrid mechanisms are illustrated as follows: The finite screw based type synthesis of open-loop mechanisms is divided into three steps. Step 1: Rewrite the expected motion of the end-effector into a composition expression of the finite motions of several one-DoF joints; Step 2: Get the finite motion of each joint by decomposing the mechanism motion expression, which corresponds to a one-DoF joint. The sequence of the joints in the mechanism is the same as the sequence of their motions in the mechanism motion expression; Step 3: Obtain the type, direction, and position of each one-DoF joint from the finite motion expression of joint, leading to the mechanical structure of a feasible mechanism. Because the composition expressions of different joint motions in Step 2 can be formed in different ways, the whole set of feasible open-loop mechanisms can be synthesized. The synthesis process of open-loop mechanisms is called limb synthesis, which is the foundation for the synthesis of the other kinds of mechanisms.

114

4 Type Synthesis Method and Procedure of Robotic Mechanism

A closed-loop mechanism can be treated as a combination of several open-loop mechanisms. The motion of a closed-loop mechanism can be rewritten as the intersection of different limbs’ motions. For each limb motion, it can be decomposed into different joints’ motions. In this way, the whole set of feasible closed-loop mechanisms can be synthesized. The synthesis procedure of closed-loop mechanisms is summarized as Step 1: Rewrite the mechanism motion expression into the intersection of several limbs’ motion expressions; Step 2: Extract the motion of each limb from the mechanism motion. All the limbs should satisfy the assembly conditions to construct the limbs and form the desired mechanism. The assembly conditions are guaranteed by the intersection expression in Step 1; Step 3: Regard the motion of each limb as the motion pattern of open-loop mechanisms, and do the synthesis procedures of open-loop mechanisms to obtain all the feasible limb structures. A hybrid mechanism has several open-loop or/and closed-loop parts connected in serial or parallel structure. The motion of a hybrid mechanism can be rewritten as the composition of different parts’ motions. In the meantime, some parts’ motions can be treated as the motion patterns of either open-loop or closed-loop mechanisms. Using the synthesis procedures of open-loop and closed-loop mechanisms, all the feasible hybrid mechanisms can be synthesized. The synthesis procedures of hybrid mechanisms contain the following three steps. Step 1: Rewrite the mechanism motion expression into the composition of several parts’ motions; Step 2: Obtain the motion of each part by decomposing the mechanism motion expression, which corresponds to an open-loop or a closed-loop mechanism. The sequence of the parts in the mechanism is the same as the sequence of their motions in the mechanism motion expression; Step 3: Regard each part’s motion as the motion pattern of open-loop or closed-loop mechanisms. For each open-loop part, do the synthesis procedures of open-loop mechanisms. For each closed-loop part, do the synthesis procedures of closed-loop mechanisms. By using the above procedures, all the robotic mechanisms that generate the same motion pattern can be synthesized when the expected motion pattern is given. These mechanisms are all called motion generators of the pattern. It is found from the procedures that there are three important issues in the type synthesis. (1) Motion pattern, describes the motion characteristics of the mechanism. The expected motion pattern, including number and types of DoFs, is expressed by the set of finite screws; (2) Limb synthesis, synthesizes all the feasible structures of open-loop mechanisms when the limb motion expression is determined. Each feasible limb structure involves the types, positions and/or directions of all the joints in it.

4.2 General Procedure of Finite Screw Based Type Synthesis

115

Limb synthesis is implemented based upon the composition algorithms of finite screws; (3) Assembly condition, makes sure the mechanism constituted by the synthesized limbs can realize the expected motion pattern. It provides the geometrical relationships of the limbs in a closed-loop mechanism. When the limb structures are given, the assembly conditions among them and the proper actuation arrangements are derived according to the intersection algorithms of finite screws. The commonly used motion patterns, limb synthesis, assembly conditions, and actuation arrangements will be deeply discussed and investigated in the following three sections.

4.3 The Commonly Used Motion Pattern Using the topology models of robotic mechanisms formulated in Chap. 3, the commonly used motion patterns can be directly defined and expressed by finite screws [8].

4.3.1 One-DoF Motion Pattern (1) One-DoF translation along fixed direction This motion is a translation along a fixed direction, as shown in Fig. 4.2, which is expressed as S f,t = t

Fig. 4.2 One-DoF translation along a fixed direction

  0 , t ∈ R, s

(4.1)

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.3 One-DoF translation along a circle

where s denotes the unit vector of the translational direction, and t denotes the translational distance. (2) One-DoF translation along a circle Unlike the first translational motion pattern, this type of translation has variable direction. As shown in Fig. 4.3, it translates along a circle, and can be expressed as  S f,ct =

 0 , θ ∈ R, (exp(θ s˜ ) − E 3 )r

(4.2)

where s is the unit vector perpendicular to the translation circle, and r is the vector along the radius of the circle at the initial pose, θ is the angle that the translational arc corresponds to. (3) One-DoF rotation around fixed axis This kind of motion is a rotation that has fixed axis. It means that both the direction and position of the rotation axis are fixed, as shown in Fig. 4.4. The expression for this motion pattern is Fig. 4.4 One-DoF rotation around a fixed axis

4.3 The Commonly Used Motion Pattern

117

Fig. 4.5 One-DoF helical motion around a fixed axis

S f,r

θ = 2 tan 2



 s , θ ∈ R, r×s

(4.3)

where s and r denote the unit direction vector and position vector of the rotation axis, respectively, and θ denotes the rotational angle. (4) One-DoF helical motion around fixed axis As shown in Fig. 4.5, this one-DoF helical motion rotates around an axis having fixed direction and position, while translating along the same axis with fixed pitch. Its expression is S f,h

θ = 2 tan 2



s r×s



  0 + hθ , θ ∈ R, s

(4.4)

where s and r are the unit direction vector and position vector of the rotation axis, respectively, θ is the rotational angle, and h is the pitch which denotes the fixed ratio between the translational distance and rotational angle. (5) One-DoF bifurcate rotation around two fixed axes This motion is a special kind of one-DoF motion pattern. It has two fixed rotation axes that can be switched to each other at the initial pose. Fig. 4.6 One-DoF bifurcate rotation around two fixed axes

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4 Type Synthesis Method and Procedure of Robotic Mechanism

As shown in Fig. 4.6, the one-DoF bifurcate rotation is the intersection of two similar two-DoF rotations. Both two-DoF rotations have the same two rotation axes, but the sequences of the axes are opposite. In this way, this motion pattern is expressed as      θ1 θ2 s2 s1 2 tan S f,br = 2 tan 2 r × s2 2 r × s1      θ2 θ1 s1 s2 2 tan ∩ 2 tan 2 r × s1 2 r × s2     θ2 θ1 s1 s2 ∪ 2 tan , θ1 , θ2 ∈ R, (4.5) = 2 tan 2 r × s1 2 r × s2 where s1 and s2 are the unit direction vectors of the two rotation axes, r is their position vector, θ1 and θ2 are the rotational angles about the two axes. (6) One-DoF rotation with variable axis This motion pattern is a one-DoF rotation. When moving away from the initial pose, the rotation axis varies with the current pose. This special one-DoF rotation can be regarded as the intersection of a couple of two-DoF rotations, as shown in Fig. 4.7. There are totally four one-DoF rotations with the same fixed position vector involved in this motion pattern. Hence, this motion pattern is expressed as      θ1 θ2 s2 s1 2 tan S f,vr = 2 tan 2 r × s2 2 r × s1      θ3 θ4 s4 s3 2 tan ∩ 2 tan 2 r × s4 2 r × s3   θ s = 2 tan , 2 r×s s and tan

θ can be computed as 2

Fig. 4.7 One-DoF rotation with a variable axis

(4.6)

4.3 The Commonly Used Motion Pattern



sT s1 − cos ϕ1 sT s2 =

tan



119

sT (s1 × s2 ) sin2 ϕ1   2 sT s2 − cos ϕ1 sT s1 − sT (s1 × s2 ) cos ϕ1

sT s3 − cos ϕ2 sT s4



sT (s3 × s4 ) sin2 ϕ2 ,   2 sT s4 − cos ϕ2 sT s3 − sT (s3 × s4 ) cos ϕ2

(4.7)

sT (s1 × s2 ) sin2 ϕ1 θ = , (4.8)    2 2 sT s1 − cos ϕ1 sT s2 sT s2 − cos ϕ1 sT s1 − sT (s1 × s2 ) cos ϕ1

    where ϕ1 = cos−1 sT1 s2 , ϕ2 = cos−1 sT3 s4 . s1 , s2 , s3 , s4 denote the unit direction vectors of the four rotation axes, r denotes their position vector, and θ1 , θ2 , θ3 , θ4 (θi ∈ R, i = 1, . . . , 4) denote the rotational angles about these four axes. s is the unit direction vector of the variable rotation axis of this motion pattern, and θ is the rotational angle about this axis. The six common one-DoF motion patterns have been defined and expressed by finite screws and summarized in Table 4.1. Having these expressions at hand, the multi-DoF motion patterns can be expressed as the composition of them. Table 4.1 One-DoF motion patterns No.

Motion pattern

1

Translation

2

3

Circle translation

Expression

S f,t = t

5

6

s 

(exp(θ s˜ ) − E 3 )r   θ s = 2 tan 2 r×s

Rotation

Helical motion

S f,h = 2 tan

Bifurcate rotation

S f,br

Variable rotation

S f,vr

tan



0

S f,ct =

S f,r 4

  0

θ 2



s

 + hθ

  0

r×s s     θ1 θ2 s1 s2 = 2 tan ∪ 2 tan 2 r × s1 2 r × s2   θ s = 2 tan , 2 r×s

sT (s1 × s2 ) sin2 ϕ1 θ =     2 T T T 2 s s1 − cos ϕ1 s s2 s s2 − cos ϕ1 sT s1 − sT (s1 × s2 ) cos ϕ1 = 

sT s

3

− cos ϕ2

sT s

 4

sT (s3 × s4 ) sin2 ϕ2   2 T T 4 − cos ϕ2 s s 3 − s (s 3 × s 4 ) cos ϕ2

sT s

120

4 Type Synthesis Method and Procedure of Robotic Mechanism

4.3.2 Multi-DoF Motion Pattern Multi-DoF motion patterns can be obtained by compositing the one-DoF motion patterns selected from Table 4.1. Here, we only list some motion patterns that can be generated by commonly used lower-mobility robotic mechanisms due to the space limitation. (1) Two-DoF and three-DoF translations The two-DoF translations are the composition of two one-DoF translations along two fixed directions, as shown in Fig. 4.8. Its expression is the composition of two expressions that have the format of Eq. (4.1), S f,2t = S f,t,2 S f,t,1     0 0 + t1 , = t2 s2 s1

(4.9)

where the basic motion generator of the motion pattern is shown in the same figure. When three translations are independent, for example, three translations along three different fixed directions, the composition of them leads to the three-DoF translations. As shown in Fig. 4.9, the three-DoF translations mean arbitrary translational motions in the three-dimensional space. Its expression can be obtained as S f,3t = S f,t,3 S f,t,2 S f,t,1       0 0 0 + t2 + t1 . = t3 s3 s2 s1 The three-DoF translations can also be rewritten in the following ways,

Fig. 4.8 Two-DoF translations along two fixed directions

(4.10)

4.3 The Commonly Used Motion Pattern

121

Fig. 4.9 Three-DoF translations

S f,3t = S f,ct,3 S f,t,2 S f,t,1       0 0 0 = + t2 + t1 , s3 × (s2 × s1 ) = 0, s2 s1 (exp(θ3 s˜ 3 ) − E 3 )r 3 (4.11) S f,3t = S f,ct,3 S f,ct,2 S f,t,1     0 0 + = s˜ ) − E 3 )r 3 (exp(θ2 s˜ 2 ) − E 3 )r 2 (exp(θ  3 3   0 2 , sT2 s1 + |s3 × s2 |2 = 0, + t1 s1

(4.12)

S f,3t = S f,ct,3 S f,ct,2 S f,ct,1     0 0 + = (exp(θ2 s˜ 2 ) − E 3 )r 2 (exp(θ3 s˜ 3 ) − E 3 )r 3   0 , |s2 × s1 |2 + |s3 × s2 |2 = 0. + (exp(θ1 s˜ 1 ) − E 3 )r 1

(4.13)

(2) Two-DoF and three-DoF rotations As shown in Figs. 4.10 and 4.11, there are two kinds of commonly used two-DoF rotational motion patterns. Each is composited by two one-DoF rotations around different fixed axes. For the first kind of two-DoF rotational motion patterns, the composited rotations have intersection axes. The two-DoF rotational motion pattern is obtained as

122

4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.10 Two-DoF rotations with intersection axes

Fig. 4.11 Two-DoF rotations with skew axes

S f,2r (1) = S f,r,2 S f,r,1     θ1 θ2 s2 s1 2 tan = 2 tan 2 r × s2 2 r × s1 ⎛ ⎞ θ2 θ2 θ1 θ1 tan s2 + tan s1 + tan tan s1 × s2 ⎜ 2 2 2 ⎟  2 2⎝ ⎠ θ2 θ2 θ1 θ1 r × tan s2 + tan s1 + tan tan s1 × s2 2 2 2 2 = . θ2 T θ1 1 − tan tan s1 s2 2 2

(4.14)

For the second kind of two-DoF rotational motion patterns, the two one-DoF rotations have skew axes. The composition result is

4.3 The Commonly Used Motion Pattern

123

S f,2r (2) = S f,r,2 S f,r,1     θ1 θ2 s2 s1 2 tan = 2 tan 2 r 2 × s2 2 r 1 × s1 ⎞ ⎛ θ2 θ1 θ1 θ2 θ2 s s tan s r + tan + tan × s tan × s tan 2 1 1 2 2 2 ⎟ ⎜ 2 2 2 2 2 2⎝ ⎠ θ1 θ1 θ2 + tan r 1 × s1 + tan tan (s1 × (r 2 × s2 ) + r 1 × s1 × s2 ) 2 2 2 = . θ2 θ1 1 − tan tan sT1 s2 2 2 (4.15) Similarly, there are five kinds of three-DoF rotational motion patterns that are composited by three one-DoF rotations around fixed axes, as shown in Figs. 4.12, 4.13, and 4.14. They, respectively, have: (1) three one-DoF rotations with intersection axes, (2) three one-DoF rotations with skew axes, (3) the first two one-DoF rotations with intersection axes, and the third one having skew axis with the first two, (4) the last two one-DoF rotations with intersection axes, and the first one having skew axis with the last two,

Fig. 4.12 Three-DoF rotations with intersection axes

Fig. 4.13 Three-DoF rotations with skew axes

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.14 Three-DoF rotations with two intersection axes and a skew axis

(5) the first and the last one-DoF rotations with intersection axes, and the second one having skew axis with them. Thus, their motion expressions are the compositions of the ones of three one-DoF rotations, as S f,3r (1) = S f,r,3 S f,r,2 S f,r,1       θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan = 2 tan 2 r × s 2 r × s2 2 r × s1 3 θ s 3 = 2 tan , s ∈ R , |s| = 1, 2 r×s

(4.16)

S f,3r (2) = S f,r,3 S f,r,2 S f,r,1       θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan , (4.17) = 2 tan 2 r 3 × s3 2 r 2 × s2 2 r 1 × s1 S f,3r (3) = S f,r,3 S f,r,2 S f,r,1       θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan , = 2 tan 2 r 23 × s3 2 r 23 × s2 2 r 1 × s1 (4.18)

4.3 The Commonly Used Motion Pattern

125

S f,3r (4) = S f,r,3 S f,r,2 S f,r,1       θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan , = 2 tan 2 r 3 × s3 2 r 12 × s2 2 r 12 × s1 (4.19) S f,3r (5) = S f,r,3 S f,r,2 S f,r,1       θ2 θ1 θ3 s3 s2 s1 2 tan 2 tan , = 2 tan 2 r 13 × s3 2 r 2 × s2 2 r 13 × s1 (4.20) where r 23 is the common position vector of the first two one-DoF rotations, r 12 is the common position vector of the last two one-DoF rotations, and r 13 is the common position vector of the first and the last one-DoF rotations. Figure 4.14d shows the motion generator of Eq. (4.18). (3) Three-DoF motions with two translations and one rotation There are two kinds of three-DoF motion patterns with two translations and one rotation. One is that the rotation direction is perpendicular to both of the two translations, and the other is that the rotation direction is not perpendicular to the translational plane, as shown in Figs. 4.15 and 4.16. The expressions for these two kinds of motion patterns are obtained by compositing the expressions of two translations and one rotation, S f,2t1r (1) = S f,t,3 S f,t,2 S f,r,1        θ1 0 0 s1 = t3 + t2 2 tan s3 s2 2 r 1 × s1     θ1 s1 0 + 2 tan = t3 s 3 + t2 s 2 2 r 1 × s1   θ1 0 + tan 2 s1 × (t3 s3 + t2 s2 )   θ1 s1 , = 2 tan 2 r × s1

Fig. 4.15 Planar motion

(4.21)

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.16 Three-DoF motion pattern with two translations and one rotation

where s2 × s3 = 0, s1 × (s2 × s3 ) = 0. It is called the planar motion, because it contains all the translation in a plane and all the rotations whose axes are perpendicular to the plane; S f,2t1r (2) = S f,t,3 S f,t,2 S f,r,1        θ1 0 0 s1 + t2 2 tan = t3 s3 ⎛ s2 2 r1 × ⎞s 1 s1  θ1  ⎠ (t3 s3 + t2 s2 ) = 2 tan ⎝ r1 − × s1 2 2   0 + , s1 × (s2 × s3 ) = 0. t 3 s 3 + t2 s 2

(4.22)

More kinds of motion patterns will be obtained when we replace the one-DoF rotation by a one-DoF helical motion. (4) Three-DoF motions with one translation and two rotations Considering the relationships between the directions and positions of the two rotations, three kinds of motion patterns, each of which contains one translation having fixed direction and two rotations, are widely used, as (1) both the two rotations have fixed rotation axes, and their axes are intersection ones, (2) both the two rotations have fixed rotation axes, and their axes are skew ones, (3) one rotation has fixed rotation axis, however, the axis of the other has fixed position but varying direction, and the two axes are skew ones. These three motion patterns are illustrated in Figs. 4.17, 4.18, and 4.19. Among them, the third one is known as the motion of Exechon mechanism.

4.3 The Commonly Used Motion Pattern

Fig. 4.17 Three-DoF motions with intersection rotation axes

Fig. 4.18 Three-DoF motions with skew rotation axes Fig. 4.19 Motion of Exechon mechanism

127

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Composite the expressions of one translation and different couples of two rotations, these three kinds of motions are expressed as S f,1t2r (1) = S f,t,3 S f,r,2 S f,r,1       θ2 θ1 0 s2 s1 2 tan 2 tan = t3 s3 2 r × s2 2 r × s1 ⎛ ⎞ θ2 θ2 θ1 θ1 tan s2 + tan s1 + tan tan s1 × s2 ⎜ 2 2 2 ⎟  2 2⎝ ⎠ θ2 θ2 θ1 θ1   r × tan s2 + tan s1 + tan tan s1 × s2 2 2 2 2 0  = t3 , θ2 T θ1 s3 1 − tan tan s1 s2 2 2 (4.23) S f,1t2r (2) = S f,t,3 S f,r,2 S f,r,1       θ2 θ1 0 s2 s1 = t3 2 tan 2 tan 2 r 2 × s2 2 r 1 × s1 s3 ⎛ ⎞ θ2 θ1 θ1 θ2 θ2 tan s2 + tan s1 + tan tan s1 × s2 tan r 2 × s2 ⎜ ⎟ 2 2 2 2 2 ⎜ ⎟ θ1 θ1 ⎜ ⎟ 2⎜ + tan r 1 × s1 + tan ⎟ 2 2 ⎝ ⎠ θ2   tan (s1 × (r 2 × s2 ) + r 1 × s1 × s2 ) 0 2 = t3 ,  θ2 θ1 s3 tan sT 1 − tan s 2 2 2 1

(4.24)

S f,1t2r (1) = S f,t,3 S f,r,2 S f,1       θ2 θ1 0 s2 exp(θ2 s˜ 2 )s1 2 tan 2 tan , (4.25) = t3 s3 2 r 2 × s2 2 r 1 × exp(θ2 s˜ 2 )s1 where the third factor in Eq. (4.25) is a rotation with fixed position vector but varying direction vector. Its direction vector varies with the rotational angle of the second factor. The motion in Eq. (4.25) is the motion of Exechon mechanism and Exechonlike mechanisms. (5) Four-DoF motions with three translations and one rotation This motion pattern is the composition of three translations and a one-DoF rotation, as shown in Fig. 4.20. Hence, the corresponding motion expression can be obtained as

4.3 The Commonly Used Motion Pattern

129

Fig. 4.20 Schoenfilies motion

S f,3t1r = S f,3t S f,r,1 = S f,t,4 S f,t,3 S f,t,2 S f,r,1          θ1 0 0 0 s1 + t3 + t2 2 tan = t4 s4 s3 s2 2 r 1 × s1    θ1 s1 0 + 2 tan = , t4 s 4 + t 3 s 3 + t 2 s 2 2 r 1× s1  θ1 0 + tan (t4 s4 +t3 s3+ t2 s2 ) 2 s1 ×  θ1 0 s1 + t1 , r ∈ R3 = 2 tan s1 2 r × s1

(4.26)

which is called Schoenfilies motion. It should be noted that replacing the one-DoF rotation by a one-DoF helical motion has no influence on the Schoenfilies motion. If we replace the one-DoF rotation with fixed axis to a bifurcate rotation or a variable rotation, two new kinds of motions patterns will be obtained. (6) Four-DoF motions with one translation and three rotations having intersecting axes According to the above-mentioned analysis about three-DoF rotations and one-DoF translation, there are about ten and even more kinds of motion patterns if changing the directions and positions of different joints. So, we just take the general situation into account because of the space limitation. The composition motion of one translation and three rotations having intersecting axes is shown in Fig. 4.21, which can be denoted as

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.21 Four-DoF motions with one translation and three rotations having intersecting axes

S f,1t3r = S f,t,4 S f,r,3 S f,r,2 S f,r,1         θ3 θ2 θ1 0 s3 s2 s1 2 tan 2 tan 2 tan = t4 s4 2 r × s3 2 r × s2 2 r × s1     θ 0 s 2 tan = t4 . (4.27) s4 2 r×s (7) Four-DoF motions with two translations and two rotations The motion pattern of two translations along fixed axes with two rotations about intersection or skew axes can be obtained as the composition of four one-DoF motions, as shown in Fig. 4.22. The corresponding expressions can be obtained as S f,2t2r (1) = S f,t,4 S f,t,3 S f,r,2 S f,r,1          θ2 θ1 0 0 s2 s1 = t4 + t3 2 tan 2 tan 2 r × s2 2 r × s1 s4 s3 ⎛ ⎞ θ2 θ2 θ1 θ1 tan s1 × s2 tan s2 + tan s1 + tan ⎜ 2 2 2 ⎟  2 2⎝ ⎠ θ2 θ2 θ1 θ1   tan s1 × s2 r × tan s2 + tan s1 + tan 2 2 2 2 0 = ,  θ2 T θ1 t3 s3 + t4 s4 tan s1 s2 1 − tan 2 2

(4.28)

4.3 The Commonly Used Motion Pattern

131

Fig. 4.22 Four-DoF motions with two translations and two rotations

S f,2t2r (2) = S f,t,4  S f,t,3  S f,r,2  S f,r,1          θ2 θ1 s2 s1 0 0 + t3  2 tan  2 tan = t4 2 r 2 × s2 2 r 1 × s1 s4 s3 ⎛ ⎞ θ2 θ1 θ1 θ2 tan s2 + tan s1 + tan tan s1 × s2 ⎜ ⎟ 2 2 2 2 2⎝ ⎠ θ2 θ1 θ1 θ2   tan r 2 × s2 + tan r 1 × s1 + tan tan (s1 × (r 2 × s2 ) + r 1 × s1 × s2 ) 0 2 2 2 2 =  θ1 θ2 t3 s 3 + t 4 s 4 1 − tan tan sT1 s2 2 2

(4.29)

(8) Five-DoF motions with three translations and two rotations The composition of three translations and two rotations is called double-Schoenfilies motion, as shown in Fig. 4.23. It can also be regarded as the composition of two Schoenfilies motions. The positions of the two rotation axes can be arbitrarily located. The motion expression is obtained as

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4 Type Synthesis Method and Procedure of Robotic Mechanism

Fig. 4.23 Double-Schoenfilies motion

S f,3t2r = S f,3t S f,2r (1) = S f,t,5 S f,t,4 S f,t,3 S f,r,2 S f,r,1          θ2 0 0 0 s2 + t4 + t3 2 tan = t5 s5 s4 s3 2 r 2 × s2   θ1 s1 2 tan 2 r 1 × s1    θ1 0 s1 + t1 = 2 tan r × s1 s1 2      θ2 0 s2 + t2 , r, r  ∈ R3 ,  2 tan s2 2 r  × s2

(4.30)

(9) Five-DoF motions with two translations and three rotations Here, we only take three rotations with skew axes into account, as shown in Fig. 4.24. The composition of two translations and three rotations has many different kinds. Other kinds could be evolved from this motion. The expression for this kind of motion pattern is obtained as S f,2t3r = S f,2t S f,3r = S f,t,5 S f,t,4 S f,r,3 S f,r,2 S f,r,1       θ3 θ2 s3 s2 0 2 tan 2 tan = t 4 s 4 + t5 s 5 2 r 3 × s3 2 r 2 × s2   θ1 s1 . (4.31) 2 tan 2 r 1 × s1 Employing finite screw, the different kinds of motion patterns have been precisely defined, resulting in analytical motion expressions. We use “precisely defined” here for the following reasons.

4.3 The Commonly Used Motion Pattern

133

Fig. 4.24 Five-DoF motions with two translations and three rotations

(1) All the characteristics of the motions involved in a motion pattern are analytically described, such as the number of motions, the type, direction, and position of each motion; 2) Motion patterns having the same number and basic types of motions, for example, the different kinds of three-DoF rotations, can be clearly distinguished.

4.4 Limb Synthesis With the motion pattern available at hand, the motion of each joint in the open-loop mechanism can be directly obtained, as mentioned in Sect. 4.2. Type synthesis of open-loop mechanisms can be regarded as limb synthesis of closed-loop mechanisms, which is the core part in the synthesis of closed-loop mechanisms, as well as synthesis of hybrid mechanisms. Limb synthesis leads to all the feasible limbs of the closedloop mechanisms. Since the motion of closed-loop mechanism is the intersection of its limb motions, the objective of limb synthesis is synthesizing all the limbs which generate the motions that contain the mechanism motion. In other words, the limb motions should have the mechanism motion pattern as sub-sets. Thus, the process of limb synthesis is summarized as follows. Step 1: Find out all the expressions of limb motions that contain the expected motion pattern of closed-loop mechanisms as sub-sets. The equivalent expressions are regarded as one expression. Step 2: Obtain the basic limb structure that generates each limb motion expression, which consists of only one-DoF joints. Step 3: Derive the equivalent expressions of each limb motion expression to synthesize all the feasible limb structures. Unlike the other methods that depend on the experience of designers, the finite screw method obtains limb structures by derivations of analytical expressions. Hence, it is an algebraic method.

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4 Type Synthesis Method and Procedure of Robotic Mechanism

4.4.1 Standard Limb Structure The expected motion pattern of closed-loop mechanisms can be denoted by S f . The simplest way to get the limb motions containing its motion pattern is adding oneparameter finite screws to its screw triangle product. The one-parameter finite screws are the screws generated by R, P, and H joints, as

S f,J

  θ s , S f,J = 2 tan 2 r×s   0 S f,J = t , s     θ s 0 = 2 tan + hθ . s 2 r×s

(4.32) (4.33) (4.34)

The meanings of the symbols in Eqs. (4.32)–(4.34) can be referred to the explanations in Chap. 3. If S f can be written as the screw triangle product of several one-parameter screws, the limb motions can be obtained by directly adding zero, one, or more one-parameter screws into the product as S f,L = S f  S f,J1  · · · S f,Jm , m ∈ N,   

(4.35)

m

where S f,J j ( j = 1, . . . , m) is generated by one of R, P, H joints. In other words, S f,J j is in form of one of Eqs. (4.32)–(4.34). If S f cannot be written as the screw triangle product of several one-parameter screws, it should be rewritten into the intersection of several multi-parameter screws, as S f = S f,1 ∩ S f,2 ∩ · · · ∩ S f,l .

(4.36)

Hence, the limb motion can be obtained by adding zero, or one, or more oneparameter screw expression as, S f,L = S f,i  S f,J1  · · · S f,Jm , i = 1, 2, . . . , l, m ∈ N.   

(4.37)

m

Following these manners, some basic limb motion expressions can be formulated. According to the corresponding relationships between the one-parameter finite screws and the one-DoF joints generating them, the basic limb structures which generate these motion expressions can be synthesized. They are called standard limb structures.

4.4 Limb Synthesis

135

Based on the above analysis, the standard limb structures have the following characteristics (1) They consist of only one-DoF joints; (2) Their motion expressions are derived by adding one-parameter screws at the end of the screw triangle product of the expected motion pattern or the motions containing the motion pattern.

4.4.2 Derivative Limb Structure Having the standard limb structures available at hand, more limb structures can be derived from them, which are called derivative limb structures. The derivative limb structures are classified into two categories. Each limb structure in the first category generates the same motion with one of the standard limb structures, as Sdf,L = S f,L ,

(4.38)

where S f,L denotes the motion generated by a standard limb, and Sdf,L denotes the motion generated by a derivative limb of that standard limb. Each limb structure in the second category generates different motion with any standard limb structure, but it still generates the motion that contains the expected motion pattern of the closed-loop mechanisms, as Sdf,L = S f,L , and Sdf,L ⊇ S f .

(4.39)

When a standard limb structure is obtained, there are two ways to get more limb structures which are candidates of derivative limb structures. (1) Arbitrarily switch the sequence of joints in it; (2) Adjust the types of some adjacent joints without changing the limb motion, and redo (1). The motion equivalence between different groups of joints can be revealed by using the computation algorithms of the screw triangle product. The algorithms will also be used to verify if a candidate belongs to one of the two categories of derivative limb structures. In this way, all the feasible limb structures can be obtained through analytical derivations.

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4 Type Synthesis Method and Procedure of Robotic Mechanism

4.4.3 Composition Algorithms Among Joint Motions If the expected motion pattern is expressed by a finite screw, a set of standard limb structures can be obtained following steps 1 and 2. By equivalently rewriting the motion expressions of the standard structures in different ways, the whole set of feasible limbs can be derived. When rewriting the limb motion, the composition of joint motions based on the screw triangle product is involved. In order to provide reference to derive all the equivalent motion expressions, the composition algorithms are introduced as follows: (1) Generating translations by rotations with parallel axes A circle translation can be generated by two rotations with parallel axes. The circle plane is perpendicular to the two axes. The radius of the circle is the distance between the axes, as shown in Fig. 4.25. The algorithm for this motion equivalence is given as     θ1,1 θ1,2 s1 s1 2 tan 2 tan 2 r 1,2 × s1 2 r 1,1 × s1     θ1,1 + θ1,2 0  s1   . (4.40) = 2 tan r 1,2 × s1 exp(θ1,1 s˜ 1 ) − E 3 (r 1,2 − r 1,1 ) 2 Similarly, two circle translations in the same plane can be generated by three rotations with parallel axes. As shown in Fig. 4.26, the circle plane is perpendicular to the three axes, and the radii of the two circles are the distances between the adjacent axes,

Fig. 4.25 Generating a circle translation by two parallel rotations

4.4 Limb Synthesis

137

Fig. 4.26 Generating two circle translations by three parallel rotations

     θ1,2 θ1,1 s1 s1 s1 2 tan 2 tan 2 2 r 1,3 × s1 r 1,2 × s1 r 1,1 × s1     θ1,1 + θ1,2 + θ1,3 0 s1   = 2 tan    2 r 1,3 × s1 exp (θ1,1 + θ1,2 )˜s1 − E 3 (r 1,3 − r 1,2 )   0  (4.41)   exp(θ1,1 s˜ 1 ) − E 3 (r 1,2 − r 1,1 )

θ1,3 2 tan 2



Combined with Eqs. (4.11)–(4.13), the computation algorithms in Eqs. (4.40)– (4.41) are very useful in rewriting the translational factors into some rotational factors with parallel axes, leading to some equivalent expressions of the standard limbs. (2) Changing positions of rotations in planar, Schoenfilies, double-Schoenfilies motions By using the screw triangle product in Eq. (2.24), it is easy to verify that the positions of rotations in planar, Schoenfilies and double-Schoenfilies motions can be arbitrarily adjusted without changing the motion patterns. (a) For planar motion The following three expressions are equivalent,      θ1 s1 0 0 t2 2 tan , = t3 s3 s2 2 r 1 × s1       θ1 0 0 s1 2 tan t2 , = t3 s3 s2 2 r 1 × s1       θ1 0 0 s1 t3 t2 . = 2 tan s3 s2 2 r 1 × s1 

S f,2t1r (1) S f,2t1r (1) S f,2t1r (1)

(4.42) (4.43) (4.44)

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4 Type Synthesis Method and Procedure of Robotic Mechanism

(b) For Schoenfilies motion The following four expressions are equivalent, 

S f,3t1r S f,3t1r S f,3t1r S f,3t1r

       θ1 0 0 0 s1 t3 t2 2 tan , = t4 s4 s3 s2 2 r 1 × s1         θ1 0 0 0 s1 t3 2 tan t2 , = t4 s4 s3 s2 2 r 1 × s1         θ1 0 0 0 s1 2 tan t3 t2 , = t4 s4 s3 s2 2 r 1 × s1         θ1 0 0 0 s1 t4 t3 t2 . = 2 tan s4 s3 s2 2 r 1 × s1

(4.45) (4.46) (4.47) (4.48)

(c) For double-Schoenfilies motion The following ten expressions are equivalent, 

         θ2 θ1 0 0 0 s2 s1 t4 t3 2 tan 2 tan , s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.49)           θ2 θ1 0 0 0 s2 s1 t4 2 tan t3 2 tan , = t5 s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.50)           θ2 θ1 0 0 0 s2 s1 2 tan t4 t3 2 tan , = t5 s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.51)           θ1 θ2 0 0 0 s2 s1 t5 t4 t3 2 tan , = 2 tan s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.52)           θ2 θ1 0 0 0 s2 s1 t4 2 tan 2 tan t3 , = t5 s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.53)           θ2 θ1 0 0 0 s2 s1 2 tan t4 2 tan t3 , = t5 r × s r × s s5 s s 2 2 2 2 4 1 1 3 (4.54)           θ1 θ2 s2 s1 0 0 0 t5 t4 2 tan t3 , = 2 tan s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.55)

S f,3t2r = t5

S f,3t2r

S f,3t2r

S f,3t2r

S f,3t2r

S f,3t2r

S f,3t2r

4.4 Limb Synthesis

139



         θ2 θ1 0 0 0 s2 s1 2 tan 2 tan t4 t3 , s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.56)           θ1 θ2 0 0 0 s2 s1 t5 2 tan t4 t3 , = 2 tan s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.57)           θ1 θ2 0 0 0 s2 s1 2 tan t5 t4 t3 . = 2 tan s5 s4 s3 2 r 2 × s2 2 r 1 × s1 (4.58)

S f,3t2r = t5

S f,3t2r

S f,3t2r

These computation algorithms are used to obtain the derivative limb structures from the standard structures having planar, Schoenfilies, or double-Schoenfilies motions. (3) Switching the positions of one-DoF motions Since the screw triangle product does not obey the commutative law, the positions of one-DoF joint motions in a limb motion expression cannot be directly exchanged. The following algorithms should be complied while switching the positions of one-DoF motions. (a) Switch two translations  t2

       0 0 0 0 t1 = t1 t2 . s2 s1 s1 s2

(4.59)

(b) Switch one translation and one rotation        θ1 θ1 s1 s1 0 0 t2 2 tan = 2 tan t2 , s2 exp(θ1 s˜ 1 )s2 2 r 1 × s1 2 r 1 × s1 (4.60)         θ2 θ2 0 0 s2 s2 2 tan t1 = t1 2 tan . s1 s1 2 r 2 × s2 2 (r 2 + t1 s1 ) × s2 (4.61) 

(c) Switch two rotations    θ1 s2 s1 2 tan 2 r 1 × s1 r 2 × s2     θ1 θ2 s1 exp(θ1 s˜ 1 )s2 = 2 tan . 2 tan 2 r 1 × s1 2 (r 1 + exp(θ1 s˜ 1 )(r 2 − r 1 )) × (exp(θ1 s˜ 1 )s2 )

θ2 2 tan 2



(4.62)

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4 Type Synthesis Method and Procedure of Robotic Mechanism

The computation algorithms in Eqs. (4.59)–(4.62) are beneficial for verifying the motion equivalence between the limbs with the same number and types of joints but the different sequences.

4.4.4 Equivalent Groups of Joints Here, we list some groups of adjacent joints that have equivalent motions with other groups. The subscripts of joints denote their directions. These equivalent groups are very useful in limb synthesis. (1) Equivalent groups with P1 P2 Ra In P1 P2 Ra , the directions of the two P joints are perpendicular to that of the R joint. Its motion expression is θa S f = 2 tan 2



     0 0 sa t2 t1 , r a × sa s2 s1

s1 × s2 = 0, sa × (s1 × s2 ) = 0.

(4.63)

By using the composition algorithms among joint motions in Sect. 4.4.3, it is found that this motion expression is equivalent to those generated by P1 Ra P2 , Ra P1 P2 , P1 Ra Ra , Ra P1 Ra , Ra Ra P1 , and Ra Ra Ra . For instance, the equivalence between the motions of P1 P2 Ra and P1 Ra Ra is proved as follows: The motion generated by P1 Ra Ra is expressed as θa,2 S f = 2 tan 2



     θa,1 0 sa sa 2 tan t1 , r a,2 × sa s1 2 r a,1 × sa

(4.64)

which can be rewritten by using Eq. (4.40), as θa,1 + θa,2 S f = 2 tan 2



     0 0 sa    . t1 r a,2 × sa exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 ) s1

(4.65) According to the algorithms of the screw triangle product, Eqs. (4.63) and (4.65) can be, respectively, computed as ⎞ ⎛   sa  θa ⎝  0 ⎠ s + t s t S f = 2 tan , + 1 1 2 2 ra + × sa t 1 s 1 + t2 s 2 2 2

(4.66)

4.4 Limb Synthesis

141

⎛

⎞ sa       θa,1 + θa,2 ⎜ ⎟ exp θa,1 s˜ a − E 3 r a,2 − r a,1 + t1 s1 S f = 2 tan ⎝ ⎠ r a,2 + × sa 2 2   0 . (4.67) +  exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 ) + t1 s1 

Both the value range of θa in   Eq. (4.66) and that of θa,1 + θa,2 in Eq. (4.67) are R. Meanwhile, when  r a,2 − r a,1  → ∞, the value ranges of the vector constituted by the last three items of Eq. (4.66) and the vector constituted by those of Eq. (4.67) are both the set of all the three-dimensional vectors perpendicular to sa . In this way, the motions generated by limb P1 P2 Ra and limb P1 Ra Ra are equivalent to each other. (2) Equivalent groups with P1 P2 P3 Ra The three P joint directions in P1 P2 P3 Ra are arbitrary but independent, and the direction of P1 joint is not perpendicular to the R joint direction. Its motion is expressed by θa S f = 2 tan 2



       0 0 0 sa T t3 t2 t1 , sT 1 (s 2 × s 3 )  = 0, s 1 sa  = 0. r a × sa s3 s2 s1

(4.68) This motion expression is easily proved to be equivalent to those generated by P1 P2 Ra P3 , P1 Ra P2 P3 , Ra P1 P2 P3 , P1 P2 Ra Ra , P1 Ra P2 Ra , Ra P1 P2 Ra , P1 Ra Ra P2 , Ra P1 Ra P2 , Ra Ra P1 P2 , P1 Ra Ra Ra , Ra P1 Ra Ra , Ra Ra P1 Ra , Ra Ra Ra P1 , and others containing H joints. For instance, the equivalence between the motions of P1 P2 P3 Ra and P1 P2 Ra Ra is proved as follows: The motion generated by P1 P2 Ra Ra is expressed as     θa,1 θa,2 sa sa 2 tan 2 r a,2 × sa 2 r a,1 × sa     0 0 t1 , t2 s2 s1

S f = 2 tan

(4.69)

which can be rewritten by using Eq. (4.40), as     θa,1 + θa,2 0  sa    r a,2 × sa exp(θa,1 s˜ a ) − E 3 r a,2 − r a,1 2     0 0 t1 . (4.70) t2 s2 s1

S f = 2 tan

Equations (4.68) and (4.70) can be respectively computed as

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4 Type Synthesis Method and Procedure of Robotic Mechanism

⎞ ⎛   sa  θa ⎝  0 ⎠ t1 s 1 + t2 s 2 + t3 s 3 , S f = 2 tan + × sa ra + t 1 s 1 + t2 s 2 + t3 s 3 2 2 (4.71)

⎞ ⎛ s      a  θa,1 + θa,2 ⎜  ⎟ exp θa,1 s˜ a − E 3 r a,2 − r a,1 + t1 s1 + t2 s2 S f = 2 tan ⎠ ⎝ r a,2 + × sa 2 2   0    . +   (4.72) exp θa,1 s˜ a − E 3 r a,2 − r a,1 + t1 s1 + t2 s2

The value rangeof θa in Eq.(4.71) and that of θa,1 +θa,2 in Eq. (4.72) are the same. Meanwhile, when  r a,2 − r a,1  → ∞, the value ranges of the vector constituted by the last three items of Eq. (4.71) and the vector constituted by those of Eq. (4.72) are both R3 . Hence, the motions generated by P1 P2 P3 Ra and P1 P2 Ra Ra are equivalent to each other. (3) Equivalent groups with P1 P2 P3 Ra Rb In P1 P2 P3 Ra Rb , the three P joint directions are arbitrary but independent. The direction of P1 joint is not perpendicular to either of the two R joint directions. Its motion expression is S f = 2 tan

θb 2



         θa sb sa 0 0 0 2 tan t3 t2 t1 , r b × sb s3 s2 s1 2 r a × sa (4.73)

where sT1 (s2 × s3 ) = 0, sT1 sa = 0, sT1 sb = 0. By proving the equivalence among different finite screws, this motion expression is verified as equivalence of those generated by P1 P2 Ra P3 Rb , P1 Ra P2 P3 Rb , Ra P1 P2 P3 Rb , P1 P2 Ra Rb P3 , P1 Ra P2 Rb P3 , P1 Ra Rb P2 P3 , Ra P1 P2 Rb P3 , Ra P1 Rb P 2 P3 , Ra Rb P1 P2 P3 , P1 P2 Ra Ra Rb , P1 Ra P2 Ra Rb , P1 Ra Ra P2 Rb , P1 Ra Ra Rb P2 , Ra P1 P2 Ra Rb , Ra P1 Ra P2 Rb , Ra Ra P1 P2 Rb , Ra P1 Ra Rb P2 , Ra Ra P1 Rb P2 , Ra Ra Rb P1 P2 , P1 P2 Ra Rb Rb , P1 Ra P2 Rb Rb , P1 Ra Rb P2 Rb , P1 Ra Rb Rb P2 , Ra P1 P2 Rb Rb , Ra P1 Rb P2 Rb , Ra Rb P1 P2 Rb , Ra P1 Rb Rb P2 , Ra Rb P1 Rb P2 , Ra Rb Rb P1 P2 , P1 Ra Ra Rb Rb , Ra P1 Ra Rb Rb , Ra Ra P1 Rb Rb , Ra Ra Rb P1 Rb , Ra Ra Rb Rb P1 , P1 Ra Ra Ra Rb , Ra P1 Ra Ra Rb , Ra Ra P1 Ra Rb , Ra Ra Ra P1 Rb , Ra Ra Ra Rb P1 , P1 Ra Rb Rb Rb , Ra P1 Rb Rb Rb , Ra Rb P1 Rb Rb , Ra Rb Rb P1 Rb , Ra Rb Rb Rb P1 , Ra Ra Ra Rb Rb , Ra Ra Rb Rb Rb , and others containing H joints. For instance, the motion equivalence between P1 P2 P3 Ra Rb and P1 Ra Ra Rb Rb is proved as follows. The motion generated by P1 Ra Ra Rb Rb is expressed as      θb,1 θa,2 sb sb sa 2 tan 2 tan r b,2 × sb 2 r b,1 × sb 2 r a,2 × sa     θa,1 0 sa t1 (4.74) 2 tan s1 2 r a,1 × sa

θb,2 S f =2 tan 2



4.4 Limb Synthesis

143

which can be rewritten by using Eqs. (4.40) and (4.60), as       θa,1 + θa,2 θb,1 + θb,2 0 sb sa 2 tan t1 S f = 2 tan r b,2 × sb r a,2 × sa s1 2 2   0       exp θa,1 + θa,2 s˜ a exp(θb,1 s˜ b ) − E 3 r b,2 − r b,1   0      . (4.75)  exp θa,1 s˜ a − E 3 r a,2 − r a,1 Equations (4.73) and (4.75) can be, respectively, computed as ⎛

⎞ θa θb θb θa tan sa + tan sb + tan tan sb × sa ⎜ ⎟ 2 2 2  2 ⎜ θa θb ⎟ θb θa ⎜ ⎟ tan r a × sa + tan r b × sb + tan tan ⎜ ⎟ 2 2 2 2 ⎜ ⎟ 2⎜ t 1 s 1 + t2 s 2 + t3 s 3 ⎟ ⎜ (sb × (r a × sa ) + (r b × sb ) × sa ) + ⎟ ⎜ 2  ⎟  ⎝ ⎠ θb θa θb θa , × tan sa + tan sb + tan tan sb × sa 2 2 2 2 Sf = θb T θa 1 − tan tan sa sb 2 2   0 + t 1 s 1 + t2 s 2 + t3 s 3 ⎛     θa θb θa θb tan sa + (tan sb + tan tan sb × sa ⎜ ⎜⎛ 2 2 2 2 ⎜     ⎜ θ θ ⎜ ⎝tan a r a,2 × sa + tan b r b,2 × sb + tan θ a tan θ b ⎜ 2 2 2 2 2⎜ ⎜ ⎜ s × r × s  + r × s  × s  a b,2 b a ⎜ b ⎛a,2 ⎞⎞ ⎜     ⎜ t θ θ θ θ ⎝ + × ⎝tan a s + tan b s + tan a tan b s × s ⎠⎠ a b b a 2 2 2 2 2 Sf =   θa θb T tan sa sb 1 − tan 2 2   0 + , t where 



θ a = θa,1 + θa,2 , θ b = θb,1 + θb,2 ,     t = exp θ a s˜ a exp(θb,1 s˜ b ) − E 3 r b,2 − r b,1   + exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 ) + t1 s1 . 



(4.76)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.77)

144

4 Type Synthesis Method and Procedure of Robotic Mechanism 



The value ranges  of θa , θb in  Eq. (4.76)and those of θ a , θ b in Eq. (4.77) are all R. Meanwhile, when  r a,2 − r a,1  → ∞ and  r b,2 − r b,1  → ∞, the value ranges of the vector constituted by the last three items of Eq. (4.76) and the vector constituted by those of Eq. (4.77) are both R3 . Thus, the motion equivalence between P1 P2 P3 Ra Rb and P1 Ra Ra Rb Rb is proved. (4) Equivalent groups with Ra Rb Rc The three R joint directions in Ra Rb Rc are arbitrary, and any two of them are not parallel. Their axes are intersecting at a common point O. The motion expression of Ra Rb Rc is S f = 2 tan

θc 2



     θb θa sc sb sa 2 tan 2 tan , r O × sc 2 r O × sb 2 r O × sa

(4.78)

where sa × sb = 0, sa × sc = 0, sb × sc = 0. The sequence of the three R joints can be arbitrarily switched without changing the generated motion. It means that Ra Rc Rb , Rc Ra Rb , Rc Rb Ra , Rb Ra Rc , Rb Rc Ra all have the same motion with Ra Rb Rc . This motion is equivalent to the motion generated by any three R joints having their axes un-parallel and intersecting at point O, such as Rb Rc Rd , which has motion expression as S f = 2 tan

θd 2



     θc θb sd sc sb 2 tan 2 tan , r O × sd 2 r O × sc 2 r O × sb

(4.79)

where sb × sc = 0, sb × sd = 0, sc × sd = 0. The motions generated by Ra Rb Rc , Rb Rc Rd , and their equivalences can all be computed as θ S f = 2 tan 2



 s , θ ∈ R, s ∈ R3 and |s| = 1. rO × s

(4.80)

The equivalent groups of joints discussed in this section are very useful in obtaining the derivative limb structures. Their detailed usage will be shown in the next two chapters while synthesizing some typical robotic mechanisms. In most cases, we only take the limb structures consisting of one-DoF joints into account. However, more limb structures will be obtained by replacing two/three adjacent one-DoF joints by a two/three-DoF joint. For instance, one R joint and its adjacent P joint with the same direction can be replaced by a C joint; two adjacent R joints having intersecting axes can be replaced by a U joint; three adjacent R joints having intersecting axes can be replaced by a S joint. Because the motion expressions generated by the limbs before and after the replacement are the same, they are regarded as the same limb structures. The replacing between one-DoF joints and two/three-DoF joints are listed in Table 4.2.

4.5 Assembly Condition and Actuation Arrangement Table 4.2 One-DoF joints to two/three-DoF joints

Adjacent one-DoF joints

145 Two/three-DoF joint

Ra Pa , Pa Ra

Ca

Ra Rb (axes are intersecting at point O)

U (center O)

Ra Rb Rc (axes are intersecting at point O)

S (center O)

4.5 Assembly Condition and Actuation Arrangement After the available limbs are obtained, the assembly conditions and actuation arrangements are determined by the geometrical relationships among the limbs. Here, the intersection algorithms among limb motions will be involved.

4.5.1 Assembly Condition When the expected motion pattern of the closed-loop mechanisms is given, the whole set of their feasible limb structures (the standard ones and the derivative ones) can be obtained based on the methods introduced in the above sections. Two or more limbs will be assembled to obtain a desired closed-loop mechanism, which are required to satisfy the assembly conditions that the intersection of limb motions is the expected motion pattern, as S f,1 ∩ · · · ∩ S f,i · · · ∩ S f,l = S f ,

(4.81)

where S f,i is the motion generated by the ith (i = 1, 2, . . . , l) limb. According to the limb synthesis, each limb motion contains the expected motion pattern, S f,i ⊇ S f .

(4.82)

It should be noted that the assembly conditions are the geometric relations among the links of limbs, not the relations among the joint parameters. The conditions contain the following two kinds of geometric relations: (1) The relations among the directions and positions of different joints in a limb; (2) The relations among the directions and positions of joints in different limbs. The assembly conditions of a closed-loop mechanism are determined by the selected limbs. Selecting different limbs to synthesize closed-loop mechanisms leads to different assembly conditions.

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4 Type Synthesis Method and Procedure of Robotic Mechanism

4.5.2 Non-redundant Actuation Arrangement After a closed-loop mechanism with the expected motion pattern is synthesized, actuations will be assigned to the mechanism and acted as the inputs. Some joints of the mechanism should be arranged as the actuation joints. The actuation arrangement is proper by confirming the moving platform of the mechanism is fixed when all the assigned actuation joints are locked, i.e., the actuation joints do not generate motions. For a single closed-loop mechanism within which each limb has actuation joints, if the actuation joints are locked, the motion of the ith limb is changed from S f,i to − S f,i by eliminating the screws generated by the actuation joints. If the actuation arrangement for this mechanism is appropriate, Eq. (4.83) should be satisfied, S f,2 = ∅. − S f,1 ∩ −

(4.83)

For an n-DoF closed-loop mechanism, n joints should be selected as the actuation joints in order to realize non-redundant actuation arrangement. There may be different actuation arrangements for a mechanism. It is believed that the actuation joints would better be P joints than R joints, and they should be near the fixed base and be away from the moving platform. Some scholars also believe that the simpler the mapping between the actuation joint parameters and the moving platform motion is, the better the selected actuation arrangement is.

4.5.3 Intersection Algorithms Among Limb Motion It is found in Eqs. (4.81) and (4.83) that the assembly conditions and actuation arrangements are determined by the intersection algorithms among limb motions. Hence, the assembly conditions among the limbs can be obtained through checking if Eq. (4.84) has solutions. These equations are formulated by the finite screw expressions of the limb motions and the expected motion pattern, as S f,1 = · · · = S f,i = · · · = S f,l = S f .

(4.84)

Furthermore, the actuation arrangements can be verified by checking that Eq. (4.85) has no solutions under the assembly conditions, S f,i = · · · = − S f,l . − S f,1 = · · · = −

(4.85)

In summary, the intersection algorithms among limb motions should guarantee that Eq. (4.84) has solutions in the range values of all the joint parameters under the assembly conditions, and that Eq. (4.85) has no solutions if the actuation arrangements are appropriate. The derivations to obtain the assembly conditions and verify the actuation arrangements by using the intersection algorithms depend on the specific mechanisms. Thus,

4.5 Assembly Condition and Actuation Arrangement

147

the general idea is given here. The details will be presented while synthesizing some typical mechanisms in the next two chapters.

4.6 Type Synthesis of Robotic Mechanism From the point of number and types of DoFs, it is found that type synthesis of threeDoF translational mechanisms and three-DoF rotational mechanisms, i.e., mechanisms with pure translations or pure rotations, have been investigated via different mathematical tools. The motion patterns of such mechanisms contain only translation/rotation along/about each axis, thus the limb synthesis is easy to be carried out and the assembly conditions are easy to be guaranteed. However, type synthesis of mechanisms with one-DoF and two-DoF rotations is seldom found, because the instantaneous rotation axes of these mechanisms are not the same as their finite rotation axes and the motion descriptions are difficult for the conventional mathematical tools. Specifically, mechanisms with one-DoF rotation and two-DoF rotations can be divided into mechanisms with invariable and variable rotation axes. The term “invariable” or “variable” indicates that the mechanism has fixed or unfixed rotation axes. On one hand, type synthesis of mechanisms with invariable rotation axes can be found in some literature where the motion characteristics are not summarized as fixed rotation axes, for instance, mechanisms with Schoenfiles motion, planar motion, double-Schoenfiles motion, and Tricept motion. Although type synthesis of these mechanisms has been done, motion features that invariable rotation axes indicate invariable directions of finite motions cannot be revealed or explained due to the limitation of the adopted mathematical tools. On the other hand, type synthesis of mechanism with variable rotation axes has long been a challenge despite some promising mechanisms, such as Exechon and Z3 mechanisms, were invented. It hinders not only the possibility of inventing more mechanisms with similar motion capabilities but also the comprehensive understanding of the motion characteristics of a mechanism with known topology that leads to variable rotation axes. In this chapter, a general and rigorous type synthesis method based on the finite screw is proposed. The presented method can be applied to synthesize any mechanism, including mechanisms with pure translations and rotations, mechanisms with invariable rotation axes, and mechanisms with variable rotation axes. This is because (1) the finite motion based description can describe the full-cycle motion characteristic of the mechanism; (2) the finite screw based motion pattern contains all the information on the DoFs of the mechanism, including the number and types, and (3) the mechanisms with expected motion pattern can be synthesized by analytical expressions and algebraic derivations of finite screws. The type synthesis of mechanisms with invariable rotation axes and mechanisms with variable rotation axes will be illustrated in detail in Chaps. 5 and 6. By applying the finite screw, the classification of mechanisms with invariable rotation axes will be given in an analytical manner, and the mechanisms have invariable rotation directions

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4 Type Synthesis Method and Procedure of Robotic Mechanism

in finite motion level will be proved. In addition, the motion patterns of mechanisms with variable rotation axes will be explicitly expressed. The synthesis problems of these mechanisms will be solved and many novel mechanisms will be invented.

4.7 Conclusion Based upon the topology model formulated in Chap. 3, the type synthesis method and procedures of robotic mechanisms are discussed in this chapter. The proposed method is a general and rigorous one because it is based on the finite screw that is regarded as the simplest and most convenient mathematical tool to be used in type synthesis. The key problems, including motion pattern description, limb synthesis, and assembly conditions are investigated in depth. In the next two chapters, type synthesis of some typical mechanisms will be illustrated by using the proposed method. The main key points of this chapter are listed below for the readers’ convenience. (1) The general method and procedures for type synthesis of robotic mechanisms including open-loop, closed-loop, and hybrid mechanisms, are presented. (2) Different kinds of motion patterns are precisely defined in an algebraic manner, resulting in analytical expressions in form of finite screws. (3) Given the expected motion pattern, the standard and derivative limb structures can be synthesized by using the composition algorithms among joint motions. (4) The assembly conditions and actuation arrangements of mechanisms are discussed by investigating the intersection algorithms among limb motions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Huang Z, Li QC, Ding HF (2013) Theory of parallel mechanisms. Springer, Dordrecht Kong XW, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Berlin Gogu G (2008) Structural synthesis of parallel robots. Springer, Dordrecht Gao F, Yang JL (2013) Topology synthesis for parallel robotic mechanisms. Warsaw university of technology Press, Warsaw Liu XJ, Wang JS (2014) Parallel kinematics: type, kinematics, and optimal design. Springer, Berlin Yang TL, Liu AX, Shen HP et al (2018) Topology design of robot mechanisms. Springer, Singapore Li QC, Hervé JM, Ye W (2020) Geometric method for type synthesis of parallel manipulators. Springer, Singapore Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 Yang SF (2017) Type synthesis of parallel mechanisms based upon finite screw theory. Dissertation, Tianjin University Yang SF, Sun T, Huang T et al (2016) A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech Mach Theory 104:405–419

References

149

11. Yang SF, Sun T, Huang T (2017) Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mech Mach Theory 109:220–230 12. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 13. Huang Z, Li QC (2002) General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators. Int J Robot Res 21(2):131–145 14. Fang YF, Tai LW (2002) Structure synthesis of a class of 4-DoF and 5-DoF parallel manipulators with identical limb structures. Int J Robot Res 21(9):799–810 15. Huang Z, Li QC (2003) Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method. Int J Robot Res 22(1):59–79 16. Fang YF, Tai LW (2004) Structure synthesis of a class of 3-DOF rotational parallel manipulators. IEEE Trans Robot Autom 20(1):117–121 17. Kong XW, Gosselin CM (2004) Type synthesis of three-degree-of-freedom spherical parallel manipulators. Int J Robot Res 23(3):237–245 18. Li QC, Huang Z, Hervé JM (2004) Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements. IEEE Trans Robot Autom 20(2):173–180 19. Kong XW, Gosselin CM (2004) Type synthesis of 3T1R 4-DOF parallel manipulators based on screw theory. IEEE Trans Robot Autom 20(2):181–190 20. Li QC, Hervé JM (2014) Type synthesis of 3-DOF RPR-equivalent parallel mechanisms. IEEE Trans Rob 30(6):1333–1343

Chapter 5

Type Synthesis of Mechanisms with Invariable Rotation Axes

5.1 Introduction Mechanisms with invariable and variable rotation axes [1–4] are of interest in this book because the type synthesis of these mechanisms has not been thoroughly investigated due to the limitation of the adopted mathematical tools. Based upon the finite screw based type synthesis method proposed in Chap. 4, the type synthesis of mechanisms with invariable rotation axes is implemented in this chapter and the type synthesis of mechanisms in the other category will be carried out in Chap. 6. In this chapter, firstly, the motion characteristics of the mechanisms with invariable rotation axes are discussed and classified. Strict derivations are given to demonstrate why the rotation axes of this category of mechanisms are “invariable”. The two subcategories, i.e., the mechanisms with one invariable rotation and the mechanisms with two invariable rotation axes, are, respectively, discussed. Secondly, the synthesis examples of the mechanisms having one invariable rotation axis are presented. The open-loop mechanisms with SCARA (Schoenfiles) motion and the ones with planar motion are taken as two typical examples. Finally, the synthesis of the single closedloop mechanisms with double-Schoenfiles motion and the closed-loop mechanisms with Tricept motion are shown as the examples for the type synthesis of mechanisms with two invariable rotation axes. The detailed utilization of the finite screw-based type synthesis method is given. All the synthesis procedures are presented with analytical expressions and algebraic derivations. Through using finite screw [5], all the mechanisms are synthesized in finite motion level, which avoids the problem that is caused by the disparity between the finite and instantaneous motion characteristics.

© Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_5

151

152

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

5.2 Mechanism with Invariable Rotation Axes The mechanisms with invariable rotation axes have one or two-DoF rotations. The rotations generated by these mechanisms have invariable rotation directions. However, they may have fixed or unfixed rotation positions. In other words, the phase “invariable rotation axes” only denotes that the mechanisms have fixed rotation directions. Motion patterns of mechanisms with invariable rotation axes can be described as sets of finite screws [5]. In the expression of a motion pattern, the one or two rotation factors have invariable directions that are denoted by unit direction vectors. An invariable (fixed) unit direction vector means that the motion parameters, i.e., the translational distances and rotational angles, have no influence on the direction vector. The mechanisms with invariable rotation axes can be further divided into two subcategories: (1) mechanisms with one invariable rotation axis and (2) mechanisms with two invariable rotation axes.

5.2.1 Mechanism with One Invariable Rotation Axis For the mechanisms with one invariable rotation axis, there are totally nine kinds of motion patterns listed in Table 5.1. Some of them have been expressed in Sect. 4.3. It can be seen that a mechanism that generates the motion pattern shown in Table 5.1 has invariable rotation direction denoted as s1 . For the R joint motion and the C joint motion, the corresponding rotation axes have fixed position vectors as r 1 . For the planar motion and the Schoenfiles motion, the position vectors of the rotation axes can be arbitrarily selected, i.e., r ∈ R3 . For the un-co-axial 1T1R motion, the position vector is ⎧ t2 s 2 ⎪ θ1 = 0 ⎪ ⎨ r1 − 2 t t 2 2 r= r − s − (s × s1 ) θ1 = 0 . 1 2 ⎪ θ1 2 ⎪ 2 ⎩ 2 tan 2

(5.1)

The expression of un-co-axial 1T1R motion can be rewritten as ⎛ S f,1t1r = 2 tan

⎛

θ1 ⎜ ⎜ t2 ⎜⎜ 2 ⎝ ⎝ r 1 − s2 − 2

s1

⎞

⎞

⎟ ⎟ t2 sT2 s1 ⎟ ⎟. (s2 × s1 )⎠ × s1 + s1 ⎠ θ1 θ1 2 tan 2 tan 2 2 (5.2) t2

5.2 Mechanism with Invariable Rotation Axes

153

Table 5.1 The motion patterns with one invariable rotation axis Motion pattern

Expression θ1 = 2 tan 2

1R: R joint motion

S f,r

1T1R: C joint motion

S f,1t1r (1)

1T1R: un-co-axial 1T1R motion

S f,1t1r (2)



s1

, θ1 ∈ R r 1 × s1

θ1 s1 0 = 2 tan + t1 , θ1 , t 1 ∈ R 2 r 1 × s1 s1 ⎞ ⎛ s1  θ1 ⎝  ⎠ +t2 0 , = 2 tan t2 2 r 1 − s2 × s1 s2 2

θ1 , t2 ∈ R 1R1T: un-co-axial 1R1T motion

S f,1r 1t

⎛ ⎞ s2  θ2 ⎝  ⎠ + t1 0 , = 2 tan t1 2 r 2 + s1 × s2 s1 2

t1 , θ2 ∈ R 2T1R: planar motion

2T1R: un-planar 2T1R motion

S f,2t1r (2)

⎞ ⎛ s1 θ1 ⎝   ⎠+ = 2 tan t2 t3 2 r 1 − s2 − s3 × s1 2 2

0

t2 s 2 + t3 s 3

1T1R1T: un-planar 1T1R1T motion

3T1R: SCARA (Schoenfiles) motion

S f,2t1r (1)



1R2T: un-planar 1R2T motion



θ1 = 2 tan 2

s1 r × s1

, θ1 ∈ R, r ∈ R3

, θ1 , t 2 , t 3 ∈ R

⎞ ⎛ s3 θ3 ⎝   ⎠+ S f,1r 2t = 2 tan t1 t2 2 r 3 + s1 + s2 × s3 2 2

0 , t1 , t2 , θ1 ∈ R t1 s 1 + t2 s 2 ⎞ ⎛ s2 θ2 ⎝   ⎠+ S f,1t1r 1t = 2 tan t1 t3 2 r 3 − s1 + s3 × s2 2 2

0 , t1 , t2 , θ1 ∈ R t1 s 1 + t3 s 3

θ1 0 s1 S f,3t1r = 2 tan + t1 , θ1 , t1 ∈ R, 2 r × s1 s1 r ∈ R3

154

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

The expression of the un-co-axial 1R1T motion is similar to the un-co-axial 1T1R motion. For the un-planar 2T1R motion, the position vector is changeable and expressed as ⎧ t2 ⎪ ⎪ ⎨ r 1 − 2 s2 − r = r − t2 s − 1 2 ⎪ ⎪ 2 ⎩

t3 s3 2 t3 s3 − 2

θ1 = 0 t2 θ1 2 tan 2

(s2 × s1 ) −

t3 θ1 2 tan 2

(s3 × s1 ) θ1 = 0 . (5.3)

The expression of the un-planar 2T1R motion can be rewritten as ⎛

S f,2t1r

⎞ s1 θ1 ⎜ t2 sT s1 + t3 sT3 s1 ⎟ = 2 tan ⎜ r × s1 + 2 s1 ⎟ ⎝ ⎠. 2 θ1 2 tan 2

(5.4)

Similar situations happen to the un-planar 1R2T motion and the un-planar 1T1R1T motion.

5.2.2 Mechanism with Two Invariable Rotation Axes The mechanism with two invariable rotation axes has sixteen different types. For two-DoF rotations, the most basic expression with invariable directions and fixed positions is S f,2r (1) = 2 tan

θ2 2



   θ1 s2 s1 2 tan . r 1 × s2 2 r 1 × s1

(5.5)

By using the computation algorithm in Eq. (4.62), Eq. (5.5) can be rewritten as S f,2r (1) = 2 tan

θ1 2



   θ2 s1 exp(θ1 s˜ 1 )s2 2 tan . r 1 × s1 2 r 1 × (exp(θ1 s˜ 1 )s2 )

(5.6)

Equations (5.5) and (5.6) are equivalent to each other. Hence, even though the rotation direction of the second factor in Eq. (5.6) is dependent with the motion parameter of the first factor, Eq. (5.6) can be rewritten into Eq. (5.5) in which the rotation directions of the two factors are independent with any motion parameter. This means that when we say a mechanism has two invariable rotation axes, its motion pattern may be written into an expression with variable rotation axis. However, the expression can be always equivalently rewritten into another expression with invariable rotation axes. The physical meanings of the above two equations are interpreted as follows. For a U joint with two rotation axes whose directions are s1 and s2 (the rotation axis with s1 direction is connected to the fixed base), Eq. (5.5) will be obtained if the rotation axis with s2 direction rotates first, and then the one with s1 rotates, because the rotation

5.2 Mechanism with Invariable Rotation Axes

155

about s2 has no influence on s1 . If the rotation axis with s1 direction rotates an angle θ1 first, the direction of the other axis will be changed from s2 to exp(θ1 s˜ 1 )s2 , and then exp(θ1 s˜ 1 )s2 will be the direction of that axis, leading to Eq. (5.6). All the basic expressions for the motion patterns of the 16 kinds of mechanisms with two invariable rotation axes are shown in Tables 5.2 and 5.3. Any Mechanism with a shown motion pattern has two invariable rotation directions. Table 5.2 The two- and three-DoF motion patterns with two invariable rotation axes Motion pattern

Expression

2R: U joint motion

S f,2r (1) = 2 tan

θ2 2

2R: skew 2R motion

S f,2r (2) = 2 tan

θ2 2

1T2R: CR motion or 1T2R motion without co-axial motion

2R1T: RC motion or 2R1T motion without co-axial motion

S f,1t2r (1)

s2

2 tan

r 1 × s2



s2 r 2 × s2

θ2 = 2 tan 2



2 tan

θ1 2

θ1 2

s2 r 2 × s2

+ t2



s1

, θ1 , θ2 ∈ R

r 1 × s1

s1 r 1 × s1

0

, θ1 , θ2 ∈ R



θ1 2 tan 2

s2



s1 r 1 × s1

,

θ1 , θ2 , t2 ∈ R ⎛ S f,1t2r (2)

⎞ ⎛ ⎞

s2  θ2 ⎝  θ1 0 ⎠ s1 ⎝ ⎠ = 2 tan + t3 2 tan , t3 2 2 r 1 × s1 r 2 − s3 × s2 s3 2

θ1 , θ2 , t3 ∈ R S f,2r 1t (1)

θ2 = 2 tan 2



s2 r 2 × s2



θ1  2 tan 2



s1 r 1 × s1

+ t1

0 s1

,

θ1 , t1 , θ2 ∈ R S f,2r 1t (2)

θ3 = 2 tan 2



s3 r 3 × s3

⎞ ⎛ ⎞ s2  θ  2 ⎠ + t 1 0 ⎠, ⎝2 tan ⎝ t1 2 r 2 + s1 × s2 s1 2

⎛

t1 , θ2 , θ3 ∈ R ⎛

S f,1r 1t1r 1R1T1R: 1R1T1R motion without co-axial motion



⎞ ⎛ ⎞

s3  θ θ1 0 ⎠ s1  3 ⎝ ⎠ ⎝ = 2 tan 2 tan + t2 t2 2 2 r 1 × s1 r 3 + s2 × s3 s2 2 , ⎛ ⎞

⎛ ⎞ s 1 θ3 θ1  s3  ⎠ + t2 0 ⎠ ⎝2 tan ⎝ = 2 tan t2 2 r 3 × s3 2 r 1 − s2 × s1 s2 2

θ1 , t2 , θ3 ∈ R

1T2R1T: 1T2R1T motion

2R2T: 2R2T motion with or without planar motion

2T2R: 2T2R motion with or without planar motion

Motion pattern

θ4 = 2 tan 2

S f,2r 2t (1)

S f,2r 2t (2)

r × s2

s2

θ1 2 tan 2

r 1 × s1

s1 , θ1 , θ2 ∈ R, r ∈ R3



θ3 2 tan 2

r × s3

s3 , θ3 , θ4 ∈ R, r ∈ R3

s4

⎛ ⎞

⎛

⎞ s3 θ3 ⎝  0  ⎠, ⎠ ⎝  2 tan + t1 t2 2 r 3 + s1 + s2 × s3 t1 s 1 + t2 s 2 r 4 × s4 2 2

r 4 × s4

s4

t1 , θ2 , θ3 , t4 ∈ R (continued)

⎛ ⎛ ⎞ ⎞

⎞ ⎛

⎞ s3  s2  θ θ 0 0   3 2 ⎠+ ⎠+ ⎠⎝2 tan ⎝ ⎠, = ⎝2 tan ⎝ t4 t1 2 2 r 3 − s4 × s3 r 2 + s1 × s2 t4 s 4 t1 s 1 2 2

t1 , t2 , θ3 , θ4 ∈ R ⎛

θ4 = 2 tan 2

S f,1t2r 1t



⎛ ⎞

⎞

s2 θ θ s 0   2 1 1 ⎠2 tan ⎠+ = ⎝2 tan ⎝ , t3 t4 2 2 r 1 × s1 r 2 − s3 − s4 × s2 t3 s 3 + t4 s 4 2 2 ⎛

θ1 , θ2 , t3 , t4 ∈ R

S f,2t2r (2)

S f,2t2r (1)

θ2 = 2 tan 2

Expression

Table 5.3 The four- and five-DoF motion patterns with two invariable rotation axes

156 5 Type Synthesis of Mechanisms with Invariable Rotation Axes

1T1R1T1R: 1T1R1T1R motion without planar motion

1R2T1R: 1R2T1R without planar motion

Motion pattern

Table 5.3 (continued)

θ1 , t2 , θ3 , t4 ∈ R

S f,1t1r 1t1r

⎛

(continued)

⎛ ⎞

⎞

s3 θ θ s 0   3 1 1 ⎠+ ⎠2 tan = ⎝2 tan ⎝ t4 t2 2 2 r 1 × s1 r 3 − s4 + s2 × s3 t2 s 2 + t4 s 4 2 2 ⎛ ⎞ ⎞ ⎞ ⎛ ⎛ ⎛

⎞ s3  s1  θ θ 0 0   3 1 ⎠+ ⎠+ ⎠⎝2 tan ⎝ ⎠, = ⎝2 tan ⎝ t4 t2 2 2 r 3 − s4 × s3 r 1 − s2 × s1 t4 s 4 t2 s 2 2 2

t1 , θ2 , θ3 , t4 ∈ R

S f,1r 2t1r

⎛

⎛ ⎛ ⎞ ⎞

⎞ ⎛

⎞ s4  s1  θ θ 0 0   4 1 ⎠+ ⎠+ ⎠⎝2 tan ⎝ ⎠ = ⎝2 tan ⎝ t3 t2 2 2 r 4 + s3 × s4 r 1 − s2 × s1 t3 s 3 t2 s 2 2 2 ⎞ ⎛

⎛

⎞ s1 θ4 θ 0 s4   1 ⎠+ ⎠ = 2 tan ⎝2 tan ⎝ t2 t3 2 r 4 × s4 2 r 1 − s2 − s3 × s1 t2 s 2 + t3 s 3 2 2 ⎛ ⎞ ⎛

⎞

s4 θ1 θ4 ⎝  s1 0  ⎠ ⎠ ⎝ , = 2 tan + 2 tan t2 t3 2 2 r 1 × s1 r 4 + s2 + s3 × s4 t2 s 2 + t3 s 3 2 2

Expression

5.2 Mechanism with Invariable Rotation Axes 157

3T2R: double-Schoenfiles motion

1R1T1R1T: 1R1T1R1T motion without planar motion

Motion pattern

Table 5.3 (continued)

S f,3t2r

θ2 = 2 tan 2

s2

s4

r  × s2



+ t2



⎛

s2

0

θ1  2 tan 2

r × s1

s1

+ t1

s1

0

, θ1 , t1 , θ2 , t2 ∈ R, r, r  ∈ R3



⎛ ⎞

⎞ s2 θ 0   2 ⎠+ ⎠ ⎝2 tan ⎝ t1 t3 2 r 2 + s1 − s3 × s2 r 4 × s4 t1 s 1 + t3 s 3 2 2 ⎛ ⎞ ⎞ ⎛ ⎛

⎞ ⎛

⎞ s4  s2  θ θ 0 0   4 2 ⎠+ ⎠+ ⎠⎝2 tan ⎝ ⎠, = ⎝2 tan ⎝ t3 t1 2 2 r 4 + s3 × s3 r 2 + s1 × s2 t3 s 3 t1 s 1 2 2

θ4 = 2 tan 2

t1 , θ2 , t3 , θ4 ∈ R

S f,1r 1t1r 1t

Expression

158 5 Type Synthesis of Mechanisms with Invariable Rotation Axes

5.2 Mechanism with Invariable Rotation Axes

159

The mechanisms with one or two invariable rotation axes have wide applications in various industrial areas. Thus, their synthesis problems are important issues in the research of mechanisms and robotics. In the following two sections, some typical examples will be given to show the detailed procedures for the synthesis of these mechanisms.

5.3 Examples with One Invariable Rotation Axis Among the mechanisms generating the motion patterns with one invariable rotation axis shown in Table 5.1, the ones having 2T1R planar motion [6–8] and the ones having 3T1R Schoenfiles motion [9–13] are the most popular and the most commonly used. In this section, open-loop mechanisms with Schoenfiles motion will be synthesized first, and then the ones with planar motion. Even though these mechanisms have been discussed and synthesized by other methods, the synthesis procedures presented by the finite screw method are quite different, because all the procedures are strictly based upon analytical expressions and algebraic derivations.

5.3.1 Open-loop Mechanisms with Schoenfiles Motion By using the method proposed in Sect. 4.4, the open-loop mechanisms with 3T1R Schoenfiles motion are synthesized as follows. (1) The standard structure The 3T1R expected motion pattern of this kind of mechanisms is expressed by S f,3t1r

θ1 = 2 tan 2



s1 r × s1



 + t1

 0 . s1

(5.7)

Without loss of generality, sa is used to denote the rotation direction, and s1 , s2 , and s3 are applied to respectively denote the three translation directions. Following this manner, we can rewrite the above equation into  S f,3t1r = t3

       θa 0 0 0 sa t2 t1 2 tan . s3 s2 s1 2 r a × sa

(5.8)

According to the relationship between the screw factors in the expression and the joints in the mechanical structure, the structure that generates the finite screw in Eq. (5.8) is obtained as Ra P1 P2 P3 shown in Fig. 5.1, which is regarded as the standard structure.

160

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Fig. 5.1 Ra P1 P2 P3 open-loop mechanism

(2) The derivative structures As discussed in Sect. 4.4.4, the structure Ra P1 P2 P3 is equivalent to P1 P2 P3 Ra . Thus, all the equivalent structures of P1 P2 P3 Ra are equivalent to Ra P1 P2 P3 , which are all derivative structures generating the Schoenfiles motion, as listed in Table 5.4. The geometric conditions among the joint directions of the structures in Table 5.4 are sT1 (s2 × s3 ) = 0,

(5.9)

Table 5.4 Standard and derivative structures generating the Schoenfiles motion No.

Structure

No.

Structure

No.

Structure

No.

Structure

1

Ra P1 P2 P3

2

P1 Ra P2 P3

3

P1 P2 Ra P3

4

P1 P2 P3 Ra

5

Ra Ra P1 P2

6

P1 Ra Ra P2

7

P1 P2 Ra Ra

8

Ra P1 Ra P2

9

Ra P1 P2 Ra

10

P1 Ra P2 Ra

11

Ra Ra Ra P1

12

Ra Ra P1 Ra

13

Ra P1 Ra Ra

14

P1 Ra Ra Ra

15

Ha P1 P2 P3

16

P1 Ha P2 P3

17

P1 P2 Ha P3

18

P1 P2 P3 Ha

19

Ha Ra P1 P2

20

Ra Ha P1 P2

21

Ha Ha P1 P2

22

P1 Ha Ra P2

23

P1 Ra Ha P2

24

P1 Ha Ha P2

25

P1 P2 Ha Ra

26

P1 P2 Ra Ha

27

P1 P2 Ha Ha

28

Ha P1 Ra P2

29

Ra P1 Ha P2

30

Ha P1 Ha P2

31

Ha P1 P2 Ra

32

Ra P1 P2 Ha

33

Ha P1 P2 Ha

34

P1 Ha P2 Ra

35

P1 Ra P2 Ha

36

P1 Ha P2 Ha

37

Ha Ra Ra P1

38

Ra Ha Ra P1

39

Ra Ra Ha P1

40

Ha Ha Ra P1

41

Ha Ra Ha P1

42

Ra Ha Ha P1

43

Ha Ha Ha P1

44

Ha Ra P1 Ra

45

Ra Ha P1 Ra

46

Ra Ra P1 Ha

47

Ha Ha P1 Ra

48

Ha Ra P1 Ha

49

Ra Ha P1 Ha

50

Ha Ha P1 Ha

51

Ha P1 Ra Ra

52

Ra P1 Ha Ra

53

Ra P1 Ra Ha

54

Ha P1 Ha Ra

55

Ha P1 Ra Ha

56

Ra P1 Ha Ha

57

Ha P1 Ha Ha

58

P1 Ha Ra Ra

59

P1 Ra Ha Ra

60

P1 Ra Ra Ha

61

P1 Ha Ha Ra

62

P1 Ha Ra Ha

63

P1 Ra Ha Ha

64

P1 Ha Ha Ha

5.3 Examples with One Invariable Rotation Axis

161

which indicates that the directions of P1 , P2 , and P3 joints are not coplanar. Their translations are independent with each other, and sT1 sa = 0,

(5.10)

which indicates that the direction of P1 joint is not perpendicular to that of Ra joint, and the translations generated by a P1 joint is independent with the translations generated by parallel Ra joints. Through using the computation algorithms of the screw triangle product, the motion expression of any open-loop mechanism in Table 5.4 can be proved to be equivalent to Eq. (5.8). Here, we take Ha P1 P2 P3 , Ha Ha P1 P2 , and Ha Ha Ha P1 as examples to show the derivations. Example 1: Ha P1 P2 P3 The motion expression of Ha P1 P2 P3 is formulated by describing the motions of its joints as finite screws as  S f = t3

          θa 0 0 0 0 sa t2 t1  2 tan + h a θa . (5.11) s3 s2 s1 sa 2 r a × sa

The above equation can be rewritten as  S f = t1

         θa 0 0 0 0 sa t2 t3 h a θa 2 tan . s1 s2 s3 sa 2 r a × sa

(5.12)

Through using the algorithms of the screw triangle product, the motion expressions of Ra P1 P2 P3 and Ha P1 P2 P3 are, respectively, computed as    θa sa 0 2 tan = t1 s 1 + t2 s 2 + t3 s 3 2 r a × sa     θa 0 sa + = 2 tan t1 s 1 + t2 s 2 + t3 s 3 2 r a × sa   θa 0 + tan 2 sa × (t1 s1 + t2 s2 + t3 s3 ) ⎞ ⎛   sa  θa ⎝  0 ⎠ t1 s 1 + t2 s 2 + t3 s 3 , + = 2 tan ra − × sa t 1 s 1 + t2 s 2 + t3 s 3 2 2 (5.13) 

S f,3t1r

and

162

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

   θa sa 0 2 tan Sf = t1 s1 + t2 s2 + t3 s3 + h a θa sa 2 r a × sa ⎞ ⎛ s a  θa  ⎠ t1 s1 + t2 s2 + t3 s3 + h a θa sa = 2 tan ⎝ ra − × sa 2 2   0 + . t1 s1 + t2 s2 + t3 s3 + h a θa sa 

(5.14)

Compare Eq. (5.14) with Eq. (5.13). Both the value ranges of the vector constituted by the last three items of S f , and the vector constituted by those of S f,3t1r are all R3 . Therefore, it is strictly proved that Ha P1 P2 P3 generates equivalent motion with Ra P1 P2 P3 . Example 2: Ha Ha P1 P2 The motion generated by Ha Ha P1 P2 is obtained as 

        θa,2 0 0 0 sa t1  2 tan + h a,2 θa,2 s2 s1 sa 2 r a,2 × sa      θa,1 sa 0  2 tan + h a,1 θa,1 , (5.15) sa 2 r a,1 × sa

S f = t2

which can be derived by rewriting and reordering the screw factors in the equation by using the computation algorithms of the screw triangle product as 

       0 0 0 0 t2 h a,1 θa,1 h a,2 θa,2 S f = t1 s1 s2 sa sa     θa,1 + θa,2 sa  0     2 tan r a,2 × sa exp θa,1 s˜ a − E 3 (r a,2 − r a,1 ) 2         0 0 0 0 t2 h a,1 θa,1 h a,2 θa,2 = t1 s1 s2 sa sa    0    exp −(θa,1 + θa,2 )˜sa exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 )   θa,1 + θa,2 sa . (5.16) 2 tan r a,2 × sa 2 Equation (5.16) is computed as

5.3 Examples with One Invariable Rotation Axis

163

⎞ 0    ⎠ S f = ⎝ t1 s1 + t2 s2 + h a,1 θa,1 + h a,2 θa,2 sa         + exp − θa,1 + θa,2 s˜ a exp θa,1 s˜ a − E 3 r a,2 − r a,1   θa,1 + θa,2 sa 2 tan r a,2 × sa 2 ⎞ ⎛   sa  θa,1 + θa,2 ⎝  ⎠+ 0 , t = 2 tan (5.17) r a,2 − × sa t 2 2 ⎛

where   t = t1 s1 + t2 s2 + h a,1 θa,1 + h a,2 θa,2 sa    + exp −(θa,1 + θa,2 )˜sa exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 ). The value range of θa,1 + θa,2  in Eq. (5.17) and that of θa in Eq. (5.13) are both R. Meanwhile, when  r a,2 − r a,1  → ∞, both the value ranges of the vector constituted by the last three items of S f and the vector constituted by those of S f,3t1r are R3 . Hence, the motions generated by Ha Ha P1 P2 and Ra P1 P2 P3 are equivalent to each other. Example 3: Ha Ha Ha P1 Similarly, the motion of Ha Ha Ha P1 is expressed by 

      θa,3 0 0 sa  2 tan + h a,3 θa,3 S f = t1 s1 sa 2 r a,3 × sa      θa,2 0 sa + h a,2 θa,2  2 tan sa 2 r a,2 × sa      θa,1 0 sa + h a,1 θa,1 ,  2 tan sa 2 r a,1 × sa

(5.18)

which can be rewritten as         0 0 0 0 S f = t1 h a,1 θa,1 h a,2 θa,2 h a,3 θa,3 s1 sa sa sa   0           exp − θa,1 + θa,2 + θa,3 s˜ a exp θa,1 + θa,2 s˜ a − E 3 r a,3 − r a,2     0        exp − θa,1 + θa,2 + θa,3 s˜ a exp θa,1 s˜ a − E 3 r a,2 − r a,1   θa,1 + θa,2 + θa,3 sa . (5.19) 2 tan r a,3 × sa 2

164

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Equation (5.19) is computed by using the algorithms of the screw triangle product as ⎞ ⎛    sa  θa ⎝  0 ⎠ t S f = 2 tan + , × sa r a,3 − t 2 2

(5.20)

where 

θ a = θa,1 + θa,2 + θa,3 ,   t = t1 s1 + h a,1 θa,1 + h a,2 θa,2 + h a,3 θa,3 sa         exp (θa,1 +θa,2 )˜sa − E 3 r a,3 − r a,2 + exp −θ a s˜ a      + exp θa,1 s˜ a − E 3 r a,2 − r a,1 . By comparing Eq. (5.20) with Eq. (5.13), it is easy to see that the value ranges of 

θ a and θa are the same. Additionally, both the value ranges of the vector constituted constituted by those of S f,3t1r are R3 , by the last three items of S f and the vector       when r a,2 − r a,1 → ∞, r a,3 − r a,2 → ∞. In this way, Ha Ha Ha P1 is proved to generate the equivalent motion with Ra P1 P2 P3 . In the same manner, all the 64 open-loop mechanisms in Table 5.4 can be proved to generate the expected motion pattern S f,3t1r . They are all motion generators of the 3T1R Schoenfiles motion with one invariable rotation axis. More equivalent structures of them can be synthesized by utilizing the replacements between oneDoF joints and two-, three-DoF joints. We only list the structures that consist of one-DoF joints due to the space limitation. Figure 5.2 shows some typical ones among them.

5.3.2 Open-loop Mechanisms with Planar Motion Similar to the open-loop mechanisms with 3T1R Schoenfiles motion, the open-loop mechanisms with 2T1R planar motion are synthesized as follows: (1) The standard structure The 2T1R planar motion with one invariable rotation axis is expressed by S f,2t1r (1) = 2 tan

θ1 2



 s1 . r × s1

(5.21)

Without loss of generality, sa is used to replace s1 in Eq. (5.21) to denote the rotation direction, s1 and s2 are used to, respectively, denote the two translation directions. The geometrical relationship between them is

5.3 Examples with One Invariable Rotation Axis

165

Fig. 5.2 Typical open-loop mechanisms with 3T1R Schoenfiles motion

sa × (s1 × s2 ) = 0,

(5.22)

which indicates that both s1 and s2 are perpendicular to sa . In this way, Eq. (5.21) can be rewritten as  S f,2t1r (1) = t2

     θa 0 0 sa t1 2 tan . s2 s1 2 r a × sa

(5.23)

As the sequence among the two translational factors and one rotational factor in Eq. (5.23) has no influence on this motion pattern, they can be arbitrarily adjusted. Thus, we use Eq. (5.24) as the standard expression of S f,2t1r (1) , because it is convenience to the derivation process of the derivative structures, S f,2t1r (1)

θa = 2 tan 2



     sa 0 0 t2 t1 . r a × sa s2 s1

(5.24)

The three factors in the above equation correspond to an R joint with direction sa , and two P joints with directions s1 and s2 , respectively. Hence, the standard

166

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Fig. 5.3 P1 P2 Ra open-loop mechanism

mechanical structure that generates the motion expression in Eq. (5.24) is P1 P2 Ra , as shown in Fig. 5.3. (2) The derivative structures All the structures that are equivalent to P1 P2 Ra are listed in Table 5.5. The equivalence between the motions of P1 Ra Ra and P1 P2 Ra has been proved in Sect. 4.4.4. Here, we give the proof on the equivalence between the motions of Ra Ra Ra and P1 P2 Ra . The motion expression of Ra Ra Ra is formulated as θa,3 S f = 2 tan 2



     θa,2 θa,1 sa sa sa 2 tan 2 tan . r a,3 × sa 2 r a,2 × sa 2 r a,1 × sa (5.25)

By using the computation algorithms of the screw triangle product, Eq. (5.25) is rewritten as   θa,1 + θa,2 + θa,3 sa S f = 2 tan r a,3 × sa 2    0      exp θa,1 + θa,2 s˜ a − E 3 r a,3 − r a,2   0    , (5.26)    exp θa,1 s˜ a − E 3 r a,2 − r a,1 Table 5.5 Standard and derivative structures generating the planar motion No.

Structure

No.

Structure

No.

Structure

1

Ra P1 P2

2

P1 Ra P2

3

P1 P2 Ra

4

P1 Ra Ra

5

Ra P1 Ra

6

Ra Ra P1

7

Ra Ra Ra

5.3 Examples with One Invariable Rotation Axis

167

which can be further written into    0 sa  r a,3 × sa t ⎞ ⎛    sa  θa ⎝  ⎠+ 0 , t = 2 tan r a,3 + × sa t 2 2 

S f = 2 tan

θa 2



(5.27)

where 

θ a = θa,1 + θa,2 + θa,3 ,        t = exp (θa,1 + θa,2 )˜sa − E 3 r a,3 − r a,2 + exp(θa,1 s˜ a ) − E 3 (r a,2 − r a,1 ). Through comparing Eqs. (5.27) and (4.66), it is easy to see that the value range of θa      in Eq.  (4.66) and that of θ a in Eq. (5.27) are both R. Meanwhile, when r a,2 − r a,1 → ∞,  r a,3 − r a,2  → ∞, the value ranges of the vector constituted by the last three items of Eq. (5.27) and the vector constituted by those of Eq. (4.66) are both the set of all the three-dimensional vectors perpendicular to sa . Hence, the motion generated by Ra Ra Ra is proved to be equivalent to those generated by P1 P2 Ra . The equivalence between any of the derivative structures in Table 5.5 and P1 P2 Ra can be proved by following the similar manner. Two of the derivative structures are shown in Fig. 5.4.

Fig. 5.4 Typical open-loop mechanisms with 2T1R planar motion

168

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

5.4 Examples with Two Invariable Rotation Axes In Sect. 5.3, the open-loop mechanisms having Schoenfiles motion and the ones having planar motion are synthesized. More complicated synthesis examples will be shown in this section which is focused on the closed-loop mechanisms with two invariable rotation axes. Among the motion patterns with two invariable rotation axes in Tables 5.2 and 5.3, the 3T2R double-Schoenfiles motion [14–16] and the special 1T2R UP motion [17– 19] are widely used. The single closed-loop mechanisms with double-Schoenfiles motion and the closed-loop mechanisms with UP motion will be synthesized by following the procedures in Sects. 5.4.1 and 5.4.2.

5.4.1 Single Closed-loop Mechanisms with Double-Schoenfiles Motion The 3T2R double-Schoenfiles motion is expressed as           θ1 θ2 0 0 s2 s1 + t  2 tan + t . S f,3t2r = 2 tan 2 1 s2 s1 2 r  × s2 2 r × s1 (5.28) Here, we use sa and sb to denote the two rotation directions, and use s1 , s2 , and s3 to respectively denote the three translation directions. After the replacement of these symbols, Eq. (5.28) can be rewritten as 

       θb 0 0 0 sb t2 t1 2 tan s3 s2 s1 2 r b × sb   θa sa 2 tan , 2 r a × sa

S f,3t2r = t3

(5.29)

which can be further written into           θa θb 0 0 0 sb sa S f,3t2r = 2 tan 2 tan t3 t2 t1 . s3 s2 s1 2 r b × sb 2 r a × sa (5.30) Equation (5.30) is regarded as the expected motion pattern of the mechanisms with double-Schoenfiles. (1) The standard limb structure The six-DoF limbs are not considered here. Thus, for five-DoF closed-loop mechanisms, the limb motion should be the same with the expected motion pattern,

5.4 Examples with Two Invariable Rotation Axes

169

Fig. 5.5 P1 P2 P3 Ra Rb limb

as S f,L = 2 tan

θb 2



         θa 0 0 0 sb sa 2 tan t3 t2 t1 . r b × sb s3 s2 s1 2 r a × sa (5.31)

In this way, the standard limb structure is obtained as P1 P2 P3 Ra Rb , as shown in Fig. 5.5. (2) The derivative limb structures As discussed in Sect. 4.4.4, there are many limbs that can generate the equivalent motions as P1 P2 P3 Ra Rb . They are all derivative limb structures of the standard structure P1 P2 P3 Ra Rb , which are listed in Sect. 4.4.4. Besides them, more structures containing H joints are also derivative structures of P1 P2 P3 Ra Rb . All the feasible limb structures of the mechanisms with double-Schoenfiles can be synthesized using the limb synthesis method proposed in Sect. 4.4. The limbs are listed in Table 5.6. Any derivative limb in Table 5.6 can be analytically proved to generate the equivalent motion with P1 P2 P3 Ra Rb by using the computation algorithms of the screw triangle product. Here, we take P1 P2 P3 Ha Rb , P1 Ha Ha Hb Hb , and Ha Ha Hb Hb Hb as examples to show the detailed derivations. Example 1: P1 P2 P3 Ha Rb Through employing finite screw to express the motion generated by each joint, the motion of P1 P2 P3 Ha Rb is formulated as θb S f = 2 tan 2



      θa 0 sb sa  2 tan + h a θa r b × sb sa 2 r a × sa       0 0 0 t2 t1 . t3 s3 s2 s1

(5.32)

By using the computation algorithms in Sect. 4.4.3, Eq. (5.32) can be rewritten and computed as

170

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Table 5.6 Standard and derivative limb structures of mechanisms with double-Schoenfiles motion No.

Structure

1

P1 P2 P3 Ra Rb

No.

5

P1 P2 Ra Rb P3

6

P1 Ra P2 Rb P3

7

P1 Ra Rb P2 P3

8

Ra P1 P2 Rb P3

9

Ra P1 Rb P2 P3

10

Ra Rb P1 P2 P3

11

P1 P2 Ra Ra Rb

12

P1 Ra P2 Ra Rb

13

P1 Ra Ra P2 Rb

14

P1 Ra Ra Rb P2

15

Ra P1 P2 Ra Rb

16

Ra P1 Ra P2 Rb

17

Ra Ra P1 P2 Rb

18

Ra P1 Ra Rb P2

19

Ra Ra P1 Rb P2

20

Ra Ra Rb P1 P2

21

P1 P2 Ra Rb Rb

22

P1 Ra P2 Rb Rb

23

P1 Ra Rb P2 Rb

24

P1 Ra Rb Rb P2

25

Ra P1 P2 Rb Rb

26

Ra P1 Rb P2 Rb

27

Ra Rb P1 P2 Rb

28

Ra P1 Rb Rb P2

29

Ra Rb P1 Rb P2

30

Ra Rb Rb P1 P2

31

P1 Ra Ra Rb Rb

32

Ra P1 Ra Rb Rb

33

Ra Ra P1 Rb Rb

34

Ra Ra Rb P1 Rb

35

Ra Ra Rb Rb P1

36

P1 Ra Ra Ra Rb

37

Ra P1 Ra Ra Rb

38

Ra Ra P1 Ra Rb

39

Ra Ra Ra P1 Rb

40

Ra Ra Ra Rb P1

41

P1 Ra Rb Rb Rb

42

Ra P1 Rb Rb Rb

43

Ra Rb P1 Rb Rb

44

Ra Rb Rb P1 Rb

45

Ra Rb Rb Rb P1

46

Ra Ra Ra Rb Rb

47

Ra Ra Rb Rb Rb

48

P1 P2 P3 Ha Rb

49

P1 P2 P3 Ra Hb

50

P1 P2 P3 Ha Hb

51

P1 P2 Ha P3 Rb

52

P1 P2 Ra P3 Hb

53

P1 P2 Ha P3 Hb

54

P1 Ha P2 P3 Rb

55

P1 Ra P2 P3 Hb

56

P1 Ha P2 P3 Hb

57

Ha P1 P2 P3 Rb

58

Ra P1 P2 P3 Hb

59

Ha P1 P2 P3 Hb

60

P1 P2 Ha Rb P3

61

P1 P2 Ra Hb P3

62

P1 P2 Ha Hb P3

63

P1 Ha P2 Rb P3

64

P1 Ra P2 Hb P3

65

P1 Ha P2 Hb P3

66

P1 Ha Rb P2 P3

67

P1 Ra Hb P2 P3

68

P1 Ha Hb P2 P3

69

Ha P1 P2 Rb P3

70

Ra P1 P2 Hb P3

71

Ha P1 P2 Hb P3

72

Ha P1 Rb P2 P3

73

Ra P1 Hb P2 P3

74

Ha P1 Hb P2 P3

75

Ha Rb P1 P2 P3

76

Ra Hb P1 P2 P3

77

Ha Hb P1 P2 P3

78

P1 P2 Ha Ra Rb

79

P1 P2 Ra Ha Rb

80

P1 P2 Ra Ra Hb

81

P1 P2 Ha Ha Rb

82

P1 P2 Ha Ra Hb

83

P1 P2 Ra Ha Hb

84

P1 P2 Ha Ha Hb

85

P1 Ha P2 Ra Rb

86

P1 Ra P2 Ha Rb

87

P1 Ra P2 Ra Hb

88

P1 Ha P2 Ha Rb

89

P1 Ha P2 Ra Hb

90

P1 Ra P2 Ha Hb

91

P1 Ha P2 Ha Hb

92

P1 Ha Ra P2 Rb

93

P1 Ra Ha P2 Rb

94

P1 Ra Ra P2 Hb

95

P1 Ha Ha P2 Rb

96

P1 Ha Ra P2 Hb

97

P1 Ra Ha P2 Hb

98

P1 Ha Ha P2 Hb

99

P1 Ha Ra Rb P2

100

P1 Ra Ha Rb P2

101

P1 Ra Ra Hb P2

102

P1 Ha Ha Rb P2

103

P1 Ha Ra Hb P2

104

P1 Ra Ha Hb P2

105

P1 Ha Ha Hb P2

106

Ha P1 P2 Ra Rb

107

Ra P1 P2 Ha Rb

108

Ra P1 P2 Ra Hb

109

Ha P1 P2 Ha Rb

110

Ha P1 P2 Ra Hb

111

Ra P1 P2 Ha Hb

112

Ha P1 P2 Ha Hb

113

Ha P1 Ra P2 Rb

114

Ra P1 Ha P2 Rb

115

Ra P1 Ra P2 Hb

116

Ha P1 Ha P2 Rb

117

Ha P1 Ha P2 Rb

118

Ra P1 Ha P2 Hb

119

Ha P1 Ha P2 Hb

120

Ha Ra P1 P2 Rb

121

Ra Ha P1 P2 Rb

122

Ra Ra P1 P2 Hb

123

Ha Ha P1 P2 Rb

124

Ha Ra P1 P2 Hb

125

Ra Ha P1 P2 Hb

126

Ha Ha P1 P2 Hb

127

Ha P1 Ra Rb P2

128

Ra P1 Ha Rb P2

129

Ra P1 Ra Hb P2

130

Ha P1 Ha Rb P2

131

Ha P1 Ra Hb P2

132

Ra P1 Ha Hb P2

133

Ha P1 Ha Hb P2

134

Ha Ra P1 Rb P2

135

Ra Ha P1 Rb P2

136

Ra Ra P1 Hb P2

137

Ha Ha P1 Rb P2

138

Ha Ra P1 Hb P2

139

Ra Ha P1 Hb P2

140

Ha Ha P1 Hb P2

141

Ha Ra Rb P1 P2

142

Ra Ha Rb P1 P2

143

Ra Ra Hb P1 P2

144

Ha Ha Rb P1 P2

2

Structure P1 P2 Ra P3 Rb

No. 3

Structure P1 Ra P2 P3 Rb

No. 4

Structure Ra P1 P2 P3 Rb

(continued)

5.4 Examples with Two Invariable Rotation Axes

171

Table 5.6 (continued) No.

Structure

No.

Structure

No.

Structure

No.

Structure

145

Ha Ra Hb P1 P2

146

Ra Ha Hb P1 P2

147

Ha Ha Hb P1 P2

148

P1 P2 Ha Rb Rb

149

P1 P2 Ra Hb Rb

150

P1 P2 Ra Rb Hb

151

P1 P2 Ha Hb Rb

152

P1 P2 Ha Rb Hb

153

P1 P2 Ra Hb Hb

154

P1 P2 Ha Hb Hb

155

P1 Ha P2 Rb Rb

156

P1 Ra P2 Hb Rb

157

P1 Ra P2 Rb Hb

158

P1 Ha P2 Hb Rb

159

P1 Ha P2 Rb Hb

160

P1 Ra P2 Hb Hb

161

P1 Ha P2 Hb Hb

162

P1 Ha Rb P2 Rb

163

P1 Ra Hb P2 Rb

164

P1 Ra Rb P2 Hb

165

P1 Ha Hb P2 Rb

166

P1 Ha Rb P2 Hb

167

P1 Ra Hb P2 Hb

168

P1 Ha Hb P2 Hb

169

P1 Ha Rb Rb P2

170

P1 Ra Hb Rb P2

171

P1 Ra Rb Hb P2

172

P1 Ha Hb Rb P2

173

P1 Ha Rb Hb P2

174

P1 Ra Hb Hb P2

175

P1 Ha Hb Hb P2

176

Ha P1 P2 Rb Rb

177

Ra P1 P2 Hb Rb

178

Ra P1 P2 Rb Hb

179

Ha P1 P2 Hb Rb

180

Ha P1 P2 Rb Hb

181

Ra P1 P2 Hb Hb

182

Ha P1 P2 Hb Hb

183

Ha P1 Rb P2 Rb

184

Ra P1 Hb P2 Rb

185

Ra P1 Rb P2 Hb

186

Ha P1 Hb P2 Rb

187

Ha P1 Rb P2 Hb

188

Ra P1 Hb P2 Hb

189

Ha P1 Hb P2 Hb

190

Ha Rb P1 P2 Rb

191

Ra Hb P1 P2 Rb

192

Ra Rb P1 P2 Hb

193

Ha Hb P1 P2 Rb

194

Ha Rb P1 P2 Hb

195

Ra Hb P1 P2 Hb

196

Ha Hb P1 P2 Hb

197

Ha P1 Rb Rb P2

198

Ra P1 Hb Rb P2

199

Ra P1 Rb Hb P2

200

Ha P1 Hb Rb P2

201

Ha P1 Rb Hb P2

202

Ra P1 Hb Hb P2

203

Ha P1 Hb Hb P2

204

Ha Rb P1 Rb P2

205

Ra Hb P1 Rb P2

206

Ra Rb P1 Hb P2

207

Ha Hb P1 Rb P2

208

Ha Rb P1 Hb P2

209

Ra Hb P1 Hb P2

210

Ha Hb P1 Hb P2

211

Ha Rb Rb P1 P2

212

Ra Hb Rb P1 P2

213

Ra Rb Hb P1 P2

214

Ha Hb Rb P1 P2

215

Ha Rb Hb P1 P2

216

Ra Hb Hb P1 P2

217

Ha Hb Hb P1 P2

218

P1 Ha Ra Rb Rb

219

P1 Ra Ha Rb Rb

220

P1 Ra Ra Hb Rb

221

P1 Ra Ra Rb Hb

222

P1 Ha Ha Rb Rb

223

P1 Ha Ra Hb Rb

224

P1 Ha Ra Rb Hb

225

P1 Ra Ha Hb Rb

226

P1 Ra Ha Rb Hb

227

P1 Ra Ra Hb Hb

228

P1 Ha Ha Hb Rb

229

P1 Ha Ha Rb Hb

230

P1 Ha Ra Hb Hb

231

P1 Ra Ha Hb Hb

232

P1 Ha Ha Hb Hb

233

Ha P1 Ra Rb Rb

234

Ra P1 Ha Rb Rb

235

Ra P1 Ra Hb Rb

236

Ra P1 Ra Rb Hb

237

Ha P1 Ha Rb Rb

238

Ha P1 Ra Hb Rb

239

Ha P1 Ra Rb Hb

240

Ra P1 Ha Hb Rb

241

Ra P1 Ha Rb Hb

242

Ra P1 Ra Hb Hb

243

Ha P1 Ha Hb Rb

244

Ha P1 Ha Rb Hb

245

Ha P1 Ra Hb Hb

246

Ra P1 Ha Hb Hb

247

Ha P1 Ha Hb Hb

248

Ha Ra P1 Rb Rb

249

Ra Ha P1 Rb Rb

250

Ra Ra P1 Hb Rb

251

Ra Ra P1 Rb Hb

252

Ha Ha P1 Rb Rb

253

Ha Ra P1 Hb Rb

254

Ha Ra P1 Rb Hb

255

Ra Ha P1 Hb Rb

256

Ra Ha P1 Rb Hb

257

Ra Ra P1 Hb Hb

258

Ha Ha P1 Hb Rb

259

Ha Ha P1 Rb Hb

260

Ha Ra P1 Hb Hb

261

Ra Ha P1 Hb Hb

262

Ha Ha P1 Hb Hb

263

Ha Ra Rb P1 Rb

264

Ra Ha Rb P1 Rb

265

Ra Ra Hb P1 Rb

266

Ra Ra Rb P1 Hb

267

Ha Ha Rb P1 Rb

268

Ha Ra Hb P1 Rb

269

Ha Ra Rb P1 Hb

270

Ra Ha Hb P1 Rb

271

Ra Ha Rb P1 Hb

272

Ra Ra Hb P1 Hb

273

Ha Ha Hb P1 Rb

274

Ha Ha Rb P1 Hb

275

Ha Ra Hb P1 Hb

276

Ra Ha Hb P1 Hb

277

Ha Ha Hb P1 Hb

278

Ha Ra Rb Rb P1

279

Ra Ha Rb Rb P1

280

Ra Ra Hb Rb P1

281

Ra Ra Rb Hb P1

282

Ha Ha Rb Rb P1

283

Ha Ra Hb Rb P1

284

Ha Ra Rb Hb P1

285

Ra Ha Hb Rb P1

286

Ra Ha Rb Hb P1

287

Ra Ra Hb Hb P1

288

Ha Ha Hb Rb P1 (continued)

172

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Table 5.6 (continued) No.

Structure

No.

Structure

No.

Structure

No.

Structure

289

Ha Ha Rb Hb P1

290

Ha Ra Hb Hb P1

291

Ra Ha Hb Hb P1

292

Ha Ha Hb Hb P1

293

P1 Ha Ra Ra Rb

294

P1 Ra Ha Ra Rb

295

P1 Ra Ra Ha Rb

296

P1 Ra Ra Ra Hb

297

P1 Ha Ha Ra Rb

298

P1 Ha Ra Ha Rb

299

P1 Ha Ra Ra Hb

300

P1 Ra Ha Ha Rb

301

P1 Ra Ha Ra Hb

302

P1 Ra Ra Ha Hb

303

P1 Ha Ha Ha Rb

304

P1 Ha Ha Ra Hb

305

P1 Ha Ra Ha Hb

306

P1 Ra Ha Ha Hb

307

P1 Ha Ha Ha Hb

308

Ha P1 Ra Ra Rb

309

Ra P1 Ha Ra Rb

310

Ra P1 Ra Ha Rb

311

Ra P1 Ra Ra Hb

312

Ha P1 Ha Ra Rb

313

Ha P1 Ra Ha Rb

314

Ha P1 Ra Ra Hb

315

Ra P1 Ha Ha Rb

316

Ra P1 Ha Ra Hb

317

Ra P1 Ra Ha Hb

318

Ha P1 Ha Ha Rb

319

Ha P1 Ha Ra Hb

320

Ha P1 Ra Ha Hb

321

Ra P1 Ha Ha Hb

322

Ha P1 Ha Ha Hb

323

Ha Ra P1 Ra Rb

324

Ra Ha P1 Ra Rb

325

Ra Ra P1 Ha Rb

326

Ra Ra P1 Ra Hb

327

Ha Ha P1 Ra Rb

328

Ha Ra P1 Ha Rb

329

Ha Ra P1 Ra Hb

330

Ra Ha P1 Ha Rb

331

Ra Ha P1 Ra Hb

332

Ra Ra P1 Ha Hb

333

Ha Ha P1 Ha Rb

334

Ha Ha P1 Ra Hb

335

Ha Ra P1 Ha Hb

336

Ra Ha P1 Ha Hb

337

Ha Ha P1 Ha Hb

338

Ha Ra Ra P1 Rb

339

Ra Ha Ra P1 Rb

340

Ra Ra Ha P1 Rb

341

Ra Ra Ra P1 Hb

342

Ha Ha Ra P1 Rb

343

Ha Ra Ha P1 Rb

344

Ha Ra Ra P1 Hb

345

Ra Ha Ha P1 Rb

346

Ra Ha Ra P1 Hb

347

Ra Ra Ha P1 Hb

348

Ha Ha Ha P1 Rb

349

Ha Ha Ra P1 Hb

350

Ha Ra Ha P1 Hb

351

Ra Ha Ha P1 Hb

352

Ha Ha Ha P1 Hb

353

Ha Ra Ra Rb P1

354

Ra Ha Ra Rb P1

355

Ra Ra Ha Rb P1

356

Ra Ra Ra Hb P1

357

Ha Ha Ra Rb P1

358

Ha Ra Ha Rb P1

359

Ha Ra Ra Hb P1

360

Ra Ha Ha Rb P1

361

Ra Ha Ra Hb P1

362

Ra Ra Ha Hb P1

363

Ha Ha Ha Rb P1

364

Ha Ha Ra Hb P1

365

Ha Ra Ha Hb P1

366

Ra Ha Ha Hb P1

367

Ha Ha Ha Hb P1

368

P1 Ha Rb Rb Rb

369

P1 Ra Hb Rb Rb

370

P1 Ra Rb Hb Rb

371

P1 Ra Rb Rb Hb

372

P1 Ha Hb Rb Rb

373

P1 Ha Rb Hb Rb

374

P1 Ha Rb Rb Hb

375

P1 Ra Hb Hb Rb

376

P1 Ra Hb Rb Hb

377

P1 Ra Rb Hb Hb

378

P1 Ha Hb Hb Rb

379

P1 Ha Hb Rb Hb

380

P1 Ha Rb Hb Hb

381

P1 Ra Hb Hb Hb

382

P1 Ha Hb Hb Hb

383

Ha P1 Rb Rb Rb

384

Ra P1 Hb Rb Rb

385

Ra P1 Rb Hb Rb

386

Ra P1 Rb Rb Hb

387

Ha P1 Hb Rb Rb

388

Ha P1 Rb Hb Rb

389

Ha P1 Rb Rb Hb

390

Ra P1 Hb Hb Rb

391

Ra P1 Hb Rb Hb

392

Ra P1 Rb Hb Hb

393

Ha P1 Hb Hb Rb

394

Ha P1 Hb Rb Hb

395

Ha P1 Rb Hb Hb

396

Ra P1 Hb Hb Hb

397

Ha P1 Hb Hb Hb

398

Ha Rb P1 Rb Rb

399

Ra Hb P1 Rb Rb

400

Ra Rb P1 Hb Rb

401

Ra Rb P1 Rb Hb

402

Ha Hb P1 Rb Rb

403

Ha Rb P1 Hb Rb

404

Ha Rb P1 Rb Hb

405

Ra Hb P1 Hb Rb

406

Ra Hb P1 Rb Hb

407

Ra Rb P1 Hb Hb

408

Ha Hb P1 Hb Rb

409

Ha Hb P1 Rb Hb

410

Ha Rb P1 Hb Hb

411

Ra Hb P1 Hb Hb

412

Ha Hb P1 Hb Hb

413

Ha Rb Rb P1 Rb

414

Ra Hb Rb P1 Rb

415

Ra Rb Hb P1 Rb

416

Ra Rb Rb P1 Hb

417

Ha Hb Rb P1 Rb

418

Ha Rb Hb P1 Rb

419

Ha Rb Rb P1 Hb

420

Ra Hb Hb P1 Rb

421

Ra Hb Rb P1 Hb

422

Ra Rb Hb P1 Hb

423

Ha Hb Hb P1 Rb

424

Ha Hb Rb P1 Hb

425

Ha Rb Hb P1 Hb

426

Ra Hb Hb P1 Hb

427

Ha Hb Hb P1 Hb

428

Ha Rb Rb Rb P1

429

Ra Hb Rb Rb P1

430

Ra Rb Hb Rb P1

431

Ra Rb Rb Hb P1

432

Ha Hb Rb Rb P1 (continued)

5.4 Examples with Two Invariable Rotation Axes

173

Table 5.6 (continued) No.

Structure

No.

Structure

No.

Structure

No.

Structure

433

Ha Rb Hb Rb P1

434

Ha Rb Rb Hb P1

435

Ra Hb Hb Rb P1

436

Ra Hb Rb Hb P1

437

Ra Rb Hb Hb P1

438

Ha Hb Hb Rb P1

439

Ha Hb Rb Hb P1

440

Ha Hb Rb Hb P1

441

Ra Hb Hb Hb P1

442

Ha Hb Hb Hb P1

443

Ha Ra Ra Rb Rb

444

Ra Ha Ra Rb Rb

445

Ra Ra Ha Rb Rb

446

Ra Ra Ra Hb Rb

447

Ra Ra Ra Rb Hb

448

Ha Ha Ra Rb Rb

449

Ha Ra Ha Rb Rb

450

Ha Ra Ra Hb Rb

451

Ha Ra Ra Rb Hb

452

Ra Ha Ha Rb Rb

453

Ra Ha Ra Hb Rb

454

Ra Ha Ra Rb Hb

455

Ra Ra Ha Hb Rb

456

Ra Ra Ha Rb Hb

457

Ra Ra Ra Hb Hb

458

Ha Ha Ha Rb Rb

459

Ha Ha Ra Hb Rb

460

Ha Ha Ra Rb Hb

461

Ha Ra Ha Hb Rb

462

Ha Ra Ha Rb Hb

463

Ha Ra Ra Hb Hb

464

Ra Ha Ha Hb Rb

465

Ra Ha Ha Rb Hb

466

Ra Ha Ra Hb Hb

467

Ra Ra Ha Hb Hb

468

Ha Ha Ha Hb Rb

469

Ha Ha Ha Rb Hb

470

Ha Ha Ra Hb Hb

471

Ha Ra Ha Hb Hb

472

Ra Ha Ha Hb Hb

473

Ha Ha Ha Hb Hb

474

Ha Ra Rb Rb Rb

475

Ra Ha Rb Rb Rb

476

Ra Ra Hb Rb Rb

477

Ra Ra Rb Hb Rb

478

Ra Ra Rb Rb Hb

479

Ha Ha Rb Rb Rb

480

Ha Ra Hb Rb Rb

481

Ha Ra Rb Hb Rb

482

Ha Ra Rb Rb Hb

483

Ra Ha Hb Rb Rb

484

Ra Ha Rb Hb Rb

485

Ra Ha Rb Rb Hb

486

Ra Ra Hb Hb Rb

487

Ra Ra Hb Rb Hb

488

Ra Ra Rb Hb Hb

489

Ha Ha Hb Rb Rb

490

Ha Ha Rb Hb Rb

491

Ha Ha Rb Rb Hb

492

Ha Ra Hb Hb Rb

493

Ha Ra Hb Rb Hb

494

Ha Ra Rb Hb Hb

495

Ra Ha Hb Hb Rb

496

Ra Ha Hb Rb Hb

497

Ra Ha Rb Hb Hb

498

Ra Ra Hb Hb Hb

499

Ha Ha Hb Hb Rb

500

Ha Ha Hb Rb Hb

501

Ha Ha Rb Hb Hb

502

Ha Ra Hb Hb Hb

503

Ra Ha Hb Hb Hb

504

Ha Ha Hb Hb Hb

S f = 2 tan ⎛

θb 2





θa sb sa 0 0 0 0 2 tan h a θa t1 t2 t3 2 r a × sa sa s1 s2 s3 r b × sb

⎞ θb θa θb θa tan sa × sb tan sa + tan sb + tan ⎜ ⎟ 2 2 2 2 ⎜ ⎟ ⎜ tan θa r × s + tan θb r × s + tan θa tan θb (s × (r × s ) + (r × s ) × s ) ⎟ a a b b a b b a a b ⎟ 2⎜ 2 2 2 2 ⎜  ⎟  ⎝ ⎠ θb θa h a θa sa + t1 s1 + t2 s2 + t3 s3 θa θb + × tan sa + tan sb + tan tan sa × sb 2 2 2 2 2 = θa θb 1 − tan tan saT sb 2 2

0 . + (5.33) h a θa sa + t1 s1 + t2 s2 + t3 s3

Compare Eq. (5.32) with Eq. (4.76), both the value ranges of the vector constituted by the last three items of Eq. (5.32), and the vector constituted by those of Eq. (4.76) are R3 . Hence, it is strictly proved that P1 P2 P3 Ha Rb generates equivalent motion with P1 P2 P3 Ra Rb . Example 2: P1 Ha Ha Hb Hb The motion of P1 Ha Ha Hb Hb is expressed as

174

5 Type Synthesis of Mechanisms with Invariable Rotation Axes







θb,1 sb sb 0 0 + h b,2 θb,2  2 tan + h b,1 θb,1 2 sb sa r b,2 × sb r b,1 × sb



θa,2 0 sa  2 tan + h a,2 θa,2 2 sa r a,2 × sa



θa,1 0 0 sa  2 tan (5.34) + h a,1 θa,1 t1 , 2 sa s1 r a,1 × sa

S f = 2 tan

θb,2 2



which can be rewritten and computed as



θb,1 θa,2 θa,1 sb sb sa sa 2 tan 2 tan 2 tan 2 2 2 r b,2 × sb r b,1 × sb r a,2 × sa r a,1 × sa

    0 0 0     h a,1 θa,1 + h a,2 θa,2 t1  h b,1 θb,1 + h b,2 θb,2 exp θa,1 + θa,2 s˜ a sb sa s1

θa,1 + θa,2 θb,1 + θb,2 sb sa 2 tan = 2 tan 2 2 r b,2 × sb r a,2 × sa



0 0               exp θa,1 + θa,2 s˜ a exp θb,1 s˜ b − E 3 r b,2 − r b,1 exp θa,1 s˜ a − E 3 r a,2 − r a,1

  0   0 0     h a,1 θa,1 + h a,2 θa,2 t1  h b,1 θb,1 + h b,2 θb,2 exp θa,1 + θa,2 s˜ a sb sa s1 ⎛ ⎞     θa θb θa θb ⎜ ⎟ tan sa + tan sb + tan tan sa × sb ⎜ ⎛ ⎟ 2 2 2 2 ⎜ ⎟    ⎜ ⎟ ⎜ ⎝tan θ a r × s + tan θ a tan θ b  s ×  r × s  +  r × s  × s  ⎟ a,2 a a b,2 b a,2 a b ⎟ 2⎜ 2 2 2 ⎜ ⎟ ⎜ ⎛ ⎞⎞ ⎟ ⎜ ⎟      ⎜ ⎟ ⎝ + tan θ b r b,2 × sb + t × ⎝tan θ a sa + tan θ b sb + tan θ a tan θ b sa × sb ⎠⎠ ⎠ 2 2 2 2 2 2 0 + , =   t θa θb T tan s sb 1 − tan 2 2 a

S f = 2 tan

θb,2 2



(5.35)

where 



θ a = θa,1 + θa,2 , θ b = θb,1 + θb,2 ,            t = exp θ a s˜ a exp θb,1 s˜ b − E 3 r b,2 − r b,1 + h b,1 θb,1 + h b,2 θb,2 sb        + exp θa,1 s˜ a − E 3 r a,2 − r a,1 + h a,1 θa,1 + h a,2 θa,2 sa + t1 s1 . 



Both the value  ranges of θ a and θ b in Eq. (5.35), and  those of θa and θb in Eq. (4.76) are R. When  r a,2 − r a,1  → ∞ and  r b,2 − r b,1  → ∞, the value ranges of the vector constituted by the last three items of Eq. (5.35) and the vector constituted by those of Eq. (4.76) are R3 . In this way, the motions of P1 Ha Ha Hb Hb is proved to be equivalent with P1 P2 P3 Ra Rb .

5.4 Examples with Two Invariable Rotation Axes

175

Example 3: Ha Ha Hb Hb Hb The motion expression of Ha Ha Hb Hb Hb is obtained as S f = 2 tan

θb,3 2

sa r b,3 × sa

+ h b,3 θb,3

0 sa

 2 tan

θb,2 2

sb r b,2 × sb

+ h b,2 θb,2

0 sb









θb,1 θa,2 sb sa 0 0  2 tan + h b,1 θb,1  2 tan + h a,2 θa,2 2 2 r b,1 × sb r a,2 × sa sb sa



θa,1 0 sa + h a,1 θa,1 . (5.36)  2 tan 2 sa r a,1 × sa

Rewriting and reordering the rotational factors and translational factors in the screw triangle product of Eq. (5.36), and then computing the resultant screw, leads to ⎛

⎞     θa θb θa θb ⎜ ⎟ sa + tan sb + tan tan sa × sb tan ⎜ ⎛ ⎟ 2 2 2 2 ⎜ ⎟    ⎜ ⎟       θ θ θ a a b ⎜ ⎝tan ⎟ + r × s × s + tan × r × s × s r tan s a,2 a a b,3 b a,2 a b ⎜ ⎟ 2⎜ 2 2 2 ⎟ ⎜ ⎟ ⎞ ⎞ ⎛ ⎜ ⎟      ⎜ ⎟ t θ θ θ θ θ b a b a b ⎝ + tan ⎠ ⎠ ⎠ ⎝ r b,3 × sb + × tan sa + tan sb + tan tan sa × sb 2 2 2 2 2 2 Sf =



1 − tan

0 + , t



θa θb T tan s sb 2 2 a

(5.37)

where 



θ a = θa,1 + θa,2 , θ b = θb,1 + θb,2 + θb,3 ,

              t = exp θ a s˜ a exp θb,1 s˜ b − E 3 r b,2 − r b,1 + exp θb,1 +θb,2 s˜ b − E 3 r b,3 − r b,2           + exp θa,1 s˜ a − E 3 r a,2 − r a,1 + h b,1 θb,1 + h b,2 θb,2 + h b,3 θb,3 exp θ a s˜ a sb   + h a,1 θa,1 + h a,2 θa,2 sa .

      When  r a,2 − r a,1  → ∞,  r a,3 − r a,2  → ∞, and  r b,2 − r b,1  → ∞, for the similar reason with Example 2, the motion equivalence between Ha Ha Hb Hb Hb and P1 P2 P3 Ra Rb is proved. In this way, all limb structures listed in Table 5.6 can be proved to be feasible limbs of mechanisms with double-Schoenfiles motion. According to the replacements between one-DoF and two-, three-DoF joints, more structures can be synthesized.

176

5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Fig. 5.6 Typical derivative limb structures with 3T2R double-Schoenfiles motion

Here, we don’t list the structures containing two- or three-DoF joints due to the space limitation. Some typical derivative limbs are shown in Fig. 5.6. (3) Assembly condition In a single closed-loop mechanism, there are two limbs. According to the above analysis, all the feasible limb structures for the mechanisms with double-Schoenfiles motion generate the same double-Schoenfiles motion. Hence, only the simple assembly conditions should be satisfied for any two selected limbs to constitute an expected single closed-loop mechanism: the two rotation directions of one limb must be assembled to have the same directions with the corresponding rotation directions in the other limb, as s1,a = s2,a , s1,b = s2,b .

(5.38)

5.4 Examples with Two Invariable Rotation Axes

177

Under these conditions, Eq. (5.39) always holds to guarantee the right assembly between any two feasible limbs in Table 5.6, S f,1 ∩ S f,2 = S f,3t2r ,

(5.39)

where S f,1 = S f,3t2r ,

S f,2 = S f,3t2r .

(4) Actuation arrangement In order to realize non-redundant actuation arrangements of the five-DoF 3T2R mechanisms with double-Schoenfiles motion, five joints in a mechanism should be selected as actuation joints. Based on the principles discussed in Sect. 4.6.2, the specific criteria to select actuation joints for this kind of single closed-loop mechanisms are listed. (1) At least one Ra (Ha ) and one Rb (Hb ) should be actuation joints for the two-DoF rotations; (2) In a mechanism, the P joints, the dyads, and triads of parallel R and/or H joints except for one Ra (Ha ) and/or one Rb (Hb ) are used to generate translations. Among these joints, the actuation joints for the three-DoF translations should be the three joints in one limb, or two joints in one limb and one joint in the other limb, which guarantee that in the remaining ones in these joints, there are no parallel P joints, nor dyad of R and/or H joints in a limb whose axis direction is perpendicular to two P joints in the other limb, nor two dyads of R and/or H joints with the some directions and common perpendiculars in two limbs. Here, a dyad of parallel R and/or H joints refers to a group of two parallel R and/or H joints in the same limb; a triad of parallel R and/or H joints refers to a group of three parallel R and/or H joints in the same limb. By following these criteria, the recommended actuation arrangements are (1) The numbers of actuation joints in two limbs are two and three, respectively; (2) In one limb, the two actuation joints are selected as one Ra (Ha ) and one Rb (Hb ) that are nearest to the fixed base. In the other limb, the three actuations are selected as the ones except for one Ra (Ha ) and one Rb (Hb ) which are nearest to the moving platform. For example, for a mechanism composed by two Ha Ha Hb Hb Hb limbs, which is denoted as 2Ha Ha Hb Hb Hb or Ha Ha Hb Hb Hb -Ha Ha Hb Hb Hb , the recommended actuation joints are the first Ha and Hb in the first limb, and the first Ha and the first two Hb in the second limb, as Ha Ha Hb Hb Hb -Ha Ha Hb Hb Hb . By using the assembly conditions and actuation arrangements, any single closedloop mechanism with double-Schoenfiles motion can be synthesized by combining two feasible limbs. Some mechanisms with symmetric structures are shown in Fig. 5.7.

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5 Type Synthesis of Mechanisms with Invariable Rotation Axes

Fig. 5.7 Typical symmetric single closed-loop mechanisms with 3T2R double-Schoenfiles motion

5.4.2 Closed-loop Mechanisms with Tricept Motion The 1T2R Tricept motion belongs to the 1T2R motion without co-axial motion. The two rotations in it have common position vector. Thus, it can be named as UP motion, which is expressed as ⎞ ⎛ ⎞     s2   θ θ1 s1 0 2 ⎠ ⎝ ⎝ ⎠ t3 . + t3 = 2 tan 2 tan r 1 − s3 × s2 s3 2 2 r 1 × s1 2 (5.40) ⎛

S f,1t2r (2)

If we denote the translation direction as s1 , and the two rotation directions as sa and sb , the above equation can be rewritten into  S f,1t2r (2) = t1

     θb θa 0 sb sa 2 tan 2 tan . s1 2 r a × sb 2 r a × sa

(5.41)

5.4 Examples with Two Invariable Rotation Axes

179

Fig. 5.8 Tricept mechanism

The well-known parallel mechanism with the UP motion is the Tricept mechanism whose structure is 3UPS-UP, as shown in Fig. 5.8. The mechanism has three six-DoF UPS limbs and a three-DoF UP limb. Because there is no equivalent structure of UP limb, in order to obtain more parallel mechanisms which have similar structures with the Tricept mechanism, we will try to find the 3XXX-UP mechanisms that have three five-DoF limbs which contain the UP motion in Eq. (5.41). Compared with the Tricept mechanism, these mechanisms with fewer joints generate the same motion pattern with it. Thus, they will have similar practical applications, and can bring cost-savings to the constructions of this kind of mechanisms. (1) Standard limb structure In order to have the similar symmetric structures with Tricept mechanism, the synthesized structures should be 3XXX-UP. If XXX is a four-DoF limb, its motion pattern should be obtained by adding one rotational or translational factor into Eq. (3.41). The obtained motion pattern will be one of the following seven expressions as      θb 0 sc sb t1 2 tan r c × sc s1 2 r a × sb   θa sa , 2 tan 2 r a × sa       θc θb 0 sc sb 2 tan 2 tan = t1 s1 2 r c × sc 2 r a × sb   θa sa , 2 tan 2 r a × sa       θb θc 0 sb sc 2 tan 2 tan = t1 s1 2 r a × sb 2 r c × sc   θa sa , 2 tan 2 r a × sa

S f,1r 1t2r = 2 tan

S f,1t3r (1)

S f,1t3r (2)

θc 2



(5.42)

(5.43)

(5.44)

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5 Type Synthesis of Mechanisms with Invariable Rotation Axes



     θb θa 0 sb sa 2 tan 2 tan s1 2 r a × sb 2 r a × sa   θc sc , (5.45) 2 tan 2 r c × sc         θb θa 0 0 sb sa S f,2t2r = t2 t1 2 tan 2 tan , (5.46) s2 s1 2 r a × sb 2 r a × sa         θb θa 0 0 sb sa S f,1t1r 1t1r = t1 2 tan t2 2 tan , s1 s2 2 r a × sb 2 r a × sa (5.47)         θb θa 0 0 sb sa 2 tan 2 tan t2 . (5.48) S f,1t2r 1t = t1 s1 s2 2 r a × sb 2 r a × sa S f,1t3r (3) = t1

In the limbs that generate Eqs. (5.42)–(5.45), the positions of the rotations about sa and sb are fixed. All the rotation axes with the same directions in the three limbs should be coincident. This is impossible in a closed-loop mechanism with symmetric structure. In the limbs that generate Eqs. (5.46)–(5.48), at least one of the positions of the rotations is fixed. All the rotation axes with this rotation direction in the three limbs should be coincident. This is also impossible. Hence, four-DoF limbs are not feasible. If XXX is a five-DoF limb, its motion pattern should be obtained by adding two rotational factors, or one rotational and one translational factors, or two translational factors into Eq. (5.41). For a similar reason with that of the four-DoF limbs, the five-DoF limbs whose motion pattern is obtained by adding two rotational factors, or one rotational and one translational factor into Eq. (5.41) are not feasible. Therefore, the motion pattern of the five-DoF limbs could only be S f,3t2r which is the same with Eq. (5.30). This means that the standard limb structure for XXX is also P1 P2 P3 Ra Rb , as shown in Fig. 5.5. (2) Derivative limb structures Since the standard limb structure for XXX is P1 P2 P3 Ra Rb , all it’s derivative limb structures are those listed in Table 5.6. (3) Assembly conditions The three XXX limbs are numbered as the 1st to the 3rd limbs, and the UP limb is numbered as the 4th limb. The assembly conditions among the four limbs are: the same rotation direction of each limb must be assembled as colinear, s1,a = s2,a = s3,a = s4,a , s1,b = s2,b = s3,b = s4,b .

(5.49)

In this way, the motion of the synthesized 3XXX-UP mechanism is obtained as, S f,1 ∩ S f,2 ∩ S f,3 ∩ S f,4 = S f,1t2r (2) , which is the same as the Tricept mechanism.

(5.50)

5.4 Examples with Two Invariable Rotation Axes

181

Fig. 5.9 Typical symmetric mechanisms with Tricept motion

(4) Actuation arrangements Similar to the Tricept mechanism, the actuation joints of a symmetric three-DoF 3XXX-UP mechanism are selected as three P/R joints in the three XXX limbs. Each XXX limb has the 3rd joint in it as an actuation joint. The UP limb is un-actuated limb. When the actuation joints are locked, the mechanism’s moving platform cannot move, as S f,2 ∩ − S f,3 ∩ S f,4 = ∅. − S f,1 ∩ −

(5.51)

Unlike the Tricept, the synthesized mechanisms are over-constrained ones, among which two typical ones are shown in Fig. 5.9.

5.5 Conclusion Using the type synthesis method and procedures of robotic mechanisms proposed in Chap. 4, the mechanisms with invariable rotation axes are synthesized in this chapter. The detailed utilization of the finite screw based type synthesis method is shown. All the synthesis procedures are strictly presented with analytical expressions and algebraic derivations. The main key points of this chapter are listed below for the readers’ convenience. (1) Mechanisms with invariable rotation axes indicate the rotation directions of the mechanism are fixed. Using finite motion descriptions, the motion characteristics of the mechanism with invariable rotation axes are discussed. (2) The motion patterns with invariable rotation axes are classified into two subcategories, i.e., the ones with one invariable rotation and the ones with two invariable rotation axes.

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5 Type Synthesis of Mechanisms with Invariable Rotation Axes

(3) The open-loop mechanisms with Schoenfiles motion and the ones with planar motion are synthesized, leading to novel structures for open-loop mechanisms and limbs of closed-loop mechanisms. (4) The single closed-loop mechanisms with double-Schoenfiles motion and the systematic closed-loop mechanisms with Tricept motion are synthesized.

References 1. Kong XW, Gosselin CM (2007) Type synthesis of parallel mechanisms. Springer, Berlin 2. Yang SF, Sun T, Huang T (2017) Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mech Mach Theory 109:220–230 3. Qi Y, Sun T, Song YM (2017) Type synthesis of parallel tracking mechanism with varied axes by modeling its finite motions algebraically. ASME J Mech Robot 9(5):054504 (6 pages) 4. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 5. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 6. Li QC, Huang Z (2003) A family of symmetrical lower-mobility parallel mechanisms with spherical and parallel subchains. J Robot Syst 20(6):297–305 7. Bonev IA, Zlatanov D, Gosselin CM (2003) Singularity analysis of 3-DOF planar parallel mechanisms via screw theory. ASME J Mech Des 125(3):573–581 8. Kong XW, Gosselin CM (2005) Type synthesis of 3-DOF PPR-equivalent parallel manipulators based on screw theory and the concept of virtual chain. ASME J Mech Des 127(6):1113–1121 9. Kong XW, Gosselin CM (2004) Type synthesis of 3T1R 4-DOF parallel manipulators based on screw theory. IEEE Trans Robot Autom 20(2):181–190 10. Carricato M (2005) Fully isotropic four-degrees-of-freedom parallel mechanisms for Schoenflies motion. Int J Robot Res 24(5):397–414 11. Lee CC, Hervé JM (2009) Type synthesis of primitive Schoenflies-motion generators. Mech Mach Theory 44(10):1980–1997 12. Lee CC, Hervé JM (2009) On some applications of primitive Schönflies-motion generators. Mech Mach Theory 44(12):2153–2163 13. Lee CC, Hervé JM (2011) Isoconstrained parallel generators of Schoenflies motion. ASME J Mech Robot 3(2):021006 (10 pages) 14. Lee CC, Hervé JM (2006) Translational parallel manipulators with doubly planar limbs. Mech Mach Theory 41(4):433–455 15. Li QC, Hervé JM (2009) Parallel mechanisms with bifurcation of Schoenflies motion. IEEE Trans Rob 25(1):158–164 16. Lee CC, Hervé JM (2010) Generators of the product of two Schoenflies motion groups. Eur J Mech-A/Solids 29(1):97–108 17. Siciliano B (1999) The Tricept robot: inverse kinematics, manipulability analysis and closedloop direct kinematics algorithm. Robotica 17(4):437–445 18. Zhang D, Gosselin CM (2002) Kinetostatic analysis and design optimization of the Tricept machine tool family. ASME J Manuf Sci Eng 124(3):725–733 19. Joshi S, Tsai LW (2003) A comparison study of two 3-DOF parallel manipulators: one with three and the other with four supporting legs. IEEE Trans Robot Autom 19(2):200–209

Chapter 6

Type Synthesis of Mechanism with Variable Rotation Axes

6.1 Introduction Type synthesis of mechanisms with variable rotation axes has long been a challenge in the mechanism research community. Compared with the mechanisms with invariable rotation axes synthesized in Chap. 5, the motion patterns of mechanisms with variable rotation axes are more complex [1–4], leading to more complicated limb synthesis and assembly conditions. There have been robotic mechanisms that belong to this type of mechanisms being well appreciated in the industry, such as Exechon [1] and Z3 [2] mechanisms. The variable rotation axes enlarge the workspace of these mechanisms and bring in excellent performances. Hence, addressing the type synthesis of mechanisms with variable rotation axes, on one hand, provides novel and promising topology structures to the industry, on the other hand, obtains a thorough understanding of the motion characteristics of the mechanisms and offers guidance to the following performance analysis. In this chapter, typical mechanisms with variable rotation axes are synthesized by using the method and procedures proposed in Chap. 4. Firstly, the mechanisms generating the motion patterns with variable rotation axes can be classified into three subcategories, i.e., the ones with one variable rotation axis, the ones with one invariable and one variable rotation axes, and the ones with two variable rotation axes. Discussions on the characteristics of the motions with variable rotation axes are given based on the finite screw expressions. Then, type synthesis of mechanisms in these three subcategories is carried out in sequence. For the first subcategory, the single closed-loop mechanisms with 1R1T motion and the closed-loop mechanisms with 3T1R motion are taken as examples. For the second subcategory, the synthesis of open-loop mechanisms with 3T2R motion and the closed-loop mechanisms with Exechon motion are presented. For the third subcategory, the synthesis procedures of the single closed-loop mechanisms with 2R motion and the closed-loop mechanisms with Z3 motion are provided as examples.

© Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_6

183

184

6 Type Synthesis of Mechanism with Variable Rotation Axes

Since the motion patterns with variable rotation axes can be analytically expressed by finite screws [5], mechanisms generating such motions can be synthesized rigorously. The examples listed in this chapter can be synthesized by the proposed method in finite motion level. Hence, the advantages of the finite screw method type synthesis of robotic mechanisms are further illustrated.

6.2 Mechanism with Variable Rotation Axes Variable rotation axes indicate that the rotation directions are different with the change of motion parameters. Mechanism with variable rotation axes has unfixed rotation directions, but its rotation positions can be fixed or unfixed. Mechanisms with variable rotation axes can be classified into three subcategories, i.e., the ones with one variable rotation axis, the ones with one invariable and one variable rotation axes, and the ones with two variable rotation axes. Most of the mechanisms with variable rotation axes could solely be generated by closed-loop mechanisms. However, some patterns in the second subcategory (motion patterns with one invariable and one variable rotation axes) can be generated by both openand closed-loop mechanisms. The motion pattern in each subcategory will be analytically expressed by finite screw [5] in the following sections, where the features of variable rotation axes will be explained.

6.2.1 Mechanism with One Variable Rotation Axis To the authors’ knowledge, for a mechanism with one variable rotation axis, the rotation can only be the one-DoF rotation with variable direction as expressed by Eqs. (4.6)–(4.8). This rotation is the output motion of a general spherical 4R closedloop mechanism [6–8], as      θa θb sb sa 2 tan S f,vr = 2 tan 2 r a × sb 2 r a × sa      θc θd sd sc 2 tan ∩ 2 tan 2 r a × sd 2 r a × sc   θvr svr , =2 tan 2 r a × svr where svr and θvr can be computed from

(6.1)

6.2 Mechanism with Variable Rotation Axes

185

sTvr (sa × sb ) sin2 ϕab   2 sTvr sa − cos ϕ1 sTvr sb sTvr sb − cos ϕ1 sTvr sa − sTvr (sa × sb ) cos ϕab , sTvr (sc × sd ) sin2 ϕcd =    2 sTvr sc − cos ϕ2 sTvr sd sTvr sd − cos ϕ2 sTvr sc − sTvr (sc × sd ) cos ϕcd     ϕab = cos−1 saT sb , ϕcd = cos−1 sTc sd , 

tan



θvr sTvr (sa × sb ) sin2 ϕab = .    2 2 sTvr sa − cos ϕ1 sTvr sb sTvr sb − cos ϕ1 sTvr sa − sTvr (sa × sb ) cos ϕab

More motion patterns with one variable rotation axis can be obtained by adding 1–3 translational factors into Eq. (6.1), leading to the six motion patterns as follows 

   θvr 0 svr 2 tan , s1 2 r a × svr     θvr 0 svr S f,1vr 1t = 2 tan t1 , s1 2 r a × svr       θvr svr 0 0 S f,2t1vr = t2 t1 2 tan , s2 s1 2 r a × svr       θvr 0 0 svr S f,1vr 2t = 2 tan t2 t1 , s2 s1 2 r a × svr       θvr 0 0 svr S f,1t1vr 1t = t2 2 tan t1 , r × s s2 s 2 a vr 1         θvr 0 0 0 svr S f,3t1vr = t3 t2 t1 2 tan . s3 s2 s1 2 r a × svr S f,1t1vr = t1

(6.2) (6.3) (6.4) (6.5) (6.6) (6.7)

Because the S f,vr can only be generated by a spherical 4R closed-loop mechanism, the motion generators of the patterns in Eqs. (6.1)–(6.7) can only be closed-loop mechanisms.

6.2.2 Mechanism with One Invariable and One Variable Rotation Axes It is known from Chap. 5 that the open-loop mechanisms with two parallel Rb joints having no Rb joint between them can still generate rotations with two invariable rotation direction. Based on this, if we add one or several Ra joints between two parallel Rb joints, the mechanism will generate the motions with one invariable and one variable rotation axes. The same motions sometimes can also be generated by a closed-loop mechanism. Therefore, mechanism with one invariable and one variable rotation axes is divided into two types. One is formed by open-loop or closed-loop mechanisms, the other is formed only by closed-loop mechanisms.

186

6 Type Synthesis of Mechanism with Variable Rotation Axes

In the first type that can be open-loop mechanism or closed-loop mechanism, for the three-DoF motions, including one invariable rotation axis, one variable rotation axis and one translation, the only expression is S f,1r 1t1vr

θb = 2 tan 2



       θa ˜ s s 0 sb exp θb,1 b a    2 tan , (6.8) r b × sb t b⊥ 2 r a × exp θb,1 s˜ b sa

where t b⊥ denotes a translation vector that is perpendicular to sb . For four-DoF motions, there are five different cases, whose expressions are 

   θb 0 sb 2 tan S f,1t1r 1t1vr = t1 s1 2 r b × sb       θa 0 exp θb,1  s˜ b sa  2 tan , t b⊥ 2 r a × exp θb,1 s˜ b sa       θb 0 0  sb   t1 S f,1r 2t1vr = 2 tan t b⊥ exp θb,1 s˜ b s1 2 r b × sb     θa exp θb,1  s˜ b sa , 2 tan 2 r a × exp θb,1 s˜ b sa     θb 0 sb S f,1r 1t1vr 1t (1) = 2 tan  t b⊥ 2 r b × sb       θa s˜ b sa 0 exp θb,1  t1 , 2 tan s1 2 r a × exp θb,1 s˜ b sa     θb 0 sb S f,1r 1t1vr 1t (2) = 2 tan  t b⊥ 2 r b × sb       θa exp θb,1  s˜ b sa  0  t1 , 2 tan exp θb,1 s˜ b s1 2 r a × exp θb,1 s˜ b sa     θb 0 sb S f,1r 1t1vr 1t (3) = 2 tan  t b⊥ 2 r b × sb       θa ˜ s s 0 exp θb,1 b a      , 2 tan exp θb,1 s˜ b t a⊥ 2 r a × exp θb,1 s˜ b sa

(6.9)

(6.10)

(6.11)

(6.12)

(6.13)

where t a⊥ denotes a translational vector that is perpendicular to sa . For five-DoF motions, the motion expression is 

     0 0 0 t2 t1 s3 s2 s1       θa θb sb exp θb,1  s˜ b sa 2 tan . 2 tan 2 r b × sb 2 r a × exp θb,1 s˜ b sa

S f,3t1r 1vr = t3

(6.14)

6.2 Mechanism with Variable Rotation Axes

187

The motion generator of Eq. (6.8) is Rb Ra Rb , and the generators of Eqs. (6.9)–(6.13) are, respectively, Rb Ra Rb P1 , Rb Ra P1 Rb , P1 Rb Ra Rb , Rb P1 Ra Rb , and Rb Ra Ra Rb . The motion generators of Eq. (6.14) will be synthesized in detail in Sect. 6.4.1. In the second type that is formed by closed-loop mechanisms, there is only one motion pattern that is the Exechon motion generated by Exechon closed-loop mechanism. The motion pattern is expressed as  S f,1t1r 1vr = tz

     θy θx 0 sx exp(θx s˜ x )s y 2 tan 2 tan . sz 2 r x × sx 2 r y × exp(θx s˜ x )s y (6.15)

It can also be written as     θy 0 sy     S f,1t1r 1vr = tz 2 tan exp(−θx s˜ x ) r y − r x + r x × s y sz 2   θx sx . (6.16) 2 tan 2 r x × sx Since the position of an invariable rotation can only be changed by the translational parameters, Eqs. (6.15)–(6.16) show that there is one invariable and one variable rotation axes in the motion pattern. The motion generators of S f,1t1r 1vr can be synthesized as the single closed-loop and closed-loop mechanisms that have the same motions with Exechon mechanism.

6.2.3 Mechanism with Two Variable Rotation Axes There are many different cases in the third subcategory, i.e., mechanisms with two variable rotation axes. Besides the two variable rotations, these mechanisms have zero, one, two, or three translations. Here, we list two typical motion patterns from these mechanisms. One is a 2R pattern, and the other is 1T2R pattern. The 2R pattern is generated by P1 S-Ca Uab single closed-loop mechanism, as      θb,1 θc sc sb 2 tan S f,2vr = 2 tan 2 r a,1 × sc 2 r a,1 × sb     θa,1 0 sa t1 2 tan s1 2 r a,1 × sa      θb,2 θa,3 sa sb 2 tan ∩ 2 tan 2 r a,3 × sa 2 r a,3 × sb     θa,2 0 sa ta . 2 tan sa 2 r a,2 × sa

(6.17)

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6 Type Synthesis of Mechanism with Variable Rotation Axes

The three-DoF 1T2R pattern is generated by Z3 closed-loop mechanism, as ⎛

S f,1t2vr



   ⎞ θi,c,2 0 si,c ⎜ ti,z s z 2 tan 2 ⎟ r i,c,1 × si,c ⎜ ⎟ ⎜ ⎟   3 ⎜ ⎟ si,c ⎜ 2 tan θi,c,1 ⎟, = ⎜ ⎟ r i,c,2 × si,c 2 ⎟ i=1 ⎜ ⎜  ⎟   ⎝ ⎠ θi,a θi,b si,b si,a 2 tan 2 tan r × s r × s 2 2 i,b i,b i,a i,a sTz si,c = 0, i = 1, 2, 3,

(6.18)

where √ sT1,c s2,c

=

sT2,c s3,c

=

sT3,c s1,c

=−

3 . 2

The synthesis procedures of the mechanisms having these motion patterns will be presented in Sect. 6.5 in detail.

6.3 Example with One Variable Rotation Axis As has been mentioned, mechanisms with one variable rotation axis have seven motion patterns in total. Among these mechanisms, the ones having 1R1T motion and ones having 3T1R motion [3] are the most typical mechanisms which will be synthesized in these sections.

6.3.1 Single Closed-loop Mechanism with 1R1T Motion Single closed-loop mechanisms with 1R1T motion are synthesized by using the method proposed in Sects. 4.4 and 4.5. (1) Standard limb structure The expected motion pattern for this kind of mechanisms is expressed in Eq. (6.3), which can be rewritten into        θa θb 0 sb sa S f,1vr 1t = 2 tan 2 tan t1 s1 2 r a × sb 2 r a × sa        θd θc 0 sd sc ∩ 2 tan 2 tan t1 . (6.19) s1 2 r a × sd 2 r a × sc

6.3 Example with One Variable Rotation Axis

189

In order to obtain symmetric structures of the single closed-loop mechanisms, the standard limb motions are formulated in the following ways. (1) For three-DoF limbs, the motion expression is S1f,L

θb = 2 tan 2



     θa 0 sb sa 2 tan t1 . r a × sb s1 2 r a × sa

(6.20)

(2) For four-DoF limbs, the motion expression is S2f,L

θb = 2 tan 2



       θa 0 0 sb sa 2 tan t2 t1 . (6.21) r a × sb s2 s1 2 r a × sa

The two standard limb structures corresponding to Eqs. (6.20)–(6.21) are P1 Ra Rb and P1 P2 Ra Rb , respectively. (2) Derivative limb structure (1) P1 Ra Rb Through arbitrarily adjusting the sequence of P1 joint in P1 Ra Rb , the structures Ra P1 Rb and Ra Rb P1 can be obtained. But these two structures generate different motions with P1 Ra Rb ; they are not derivative structures of P1 Ra Rb . Thus, P1 Ra Rb is the only three-DoF limb structure for this kind of mechanisms, as shown in Fig. 6.1. (2) P1 P2 Ra Rb Two structures, i.e., P1 Ra P2 Rb and P1 Ra Rb P2 , can be obtained by arbitrarily adjusting the sequence of P2 joint in P1 P2 Ra Rb . Although the motions generated by the two structures are different from P1 P2 Ra Rb , their motions still contain S f,1vr 1t as subsets, S f = S2f,L , Fig. 6.1 P1 Ra Rb limb

(6.22)

190

6 Type Synthesis of Mechanism with Variable Rotation Axes

and S f ⊇ S f,1vr 1t .

(6.23)

Hence, these two structures are derivative limb structures of P1 P2 Ra Rb . If the directions of P1 and P2 are perpendicular to Ra , as (s1 × s2 ) × sa = 0,

(6.24)

six more derivative limb structures can be synthesized based upon the equivalent structures of P1 P2 Ra in Table 5.5. Table 6.1 lists all the feasible four-DoF limb structures, and Fig. 6.2 shows two typical derivative ones of the limbs in the above table. (3) Assembly condition The systematical structures of the single closed-loop mechanisms are considered. The assembly condition of the mechanism with two three-DoF limbs (P1 Ra Rb and Table 6.1 Standard and derivative limb structures of mechanisms with 1R1T motion No.

Structure

No.

Structure

No.

Structure

1

P1 P2 Ra Rb

2

P1 Ra P2 Rb (s1 × s2 ) × sa = 0

3

P1 Ra Rb P2

4

P1 Ra P2 Rb (s1 × s2 ) × sa = 0

5

Ra P1 P2 Rb

6

P1 Ra Ra Rb

7

Ra P1 Ra Rb

8

Ra Ra P1 Rb

9

Ra Ra Ra Rb

Fig. 6.2 Typical derivative limb structures of mechanisms with 1R1T motion

6.3 Example with One Variable Rotation Axis

191

P1 Rc Rd ) is that the axes of the four R joints intersect at a common point at the initial pose of the mechanism. The assembly conditions of mechanisms with two identical four-DoF limbs are (1) One Ra joint, the Rb joint in one limb, and one Rc joint, the Rd joint in the other limb should intersect at a common point at the initial pose of mechanism; (2) The translational planes of two limbs should not be parallel to each other, as 

   s1,1 × s1,2 × s2,1 × s2,2 = 0.

(6.25)

For example, the assembly conditions of the mechanism constructed by P1,1 P1,2 Ra Rb and P2,1 P2,2 Rc Rd limbs can be  2  2 s1,1 = s2,1 , sT1,2 s1,1 + sT2,2 s2,1 = 0.

(6.26)

(4) Actuation arrangement The two actuation joints should be selected according to the following criteria. (1) The actuation joint for the rotation should be one of the four R joints that intersect at the common point; (2) The actuation joint for the translation should be one of the P joints, or one of an R joint in the dyad/triad of parallel R joints. For example, P1,1 and Rc can be selected as the actuation joints of the mechanism P1,1 P1,2 Ra Rb -P2,1 P2,2 Rc Rd and P1,1 P1,2 Ra Rb -P2,1 P2,2 Rc Rd . By following the above assembly conditions and actuation arrangements, all the single closed-loop mechanisms with 1R1T motion can be synthesized, among which two typical ones are shown in Fig. 6.3.

Fig. 6.3 Typical single closed-loop mechanisms with 1R1T motion

192

6 Type Synthesis of Mechanism with Variable Rotation Axes

6.3.2 Closed-loop Mechanism with 3T1R Motion Similar to the single closed-loop mechanisms with 1R1T motion, the closed-loop mechanisms with 3T1R motion are synthesized as follows. (1) Standard limb structure Rewriting this 3T1R expected motion pattern expressed in Eq. (6.7) results in

 







θa 0 0 0 sb sa 2 tan t3 t2 t1 2 r a × sa r a × sb s3 s2 s1  







θd θc 0 0 0 sd sc ∩ 2 tan 2 tan t3 t2 t1 . 2 r a × sd 2 r a × sc s3 s2 s1

θb S f,3t1vr = 2 tan 2



(6.27) Hence, the standard limb motion is expressed as S f,L

θb = 2 tan 2



         θa 0 0 0 sb sa 2 tan t3 t2 t1 , r a × sb s3 s2 s1 2 r a × sa (6.28)

the standard limb structure corresponding to the motion expression is P1 P2 P3 Ra Rb . (2) Derivative limb structure Based upon the analysis and derivations in Sect. 5.4.1, all the equivalent structures have been synthesized as listed in Table 5.6, which can serve as derivative limb structures of the closed-loop mechanisms with 3T1R motion. (3) Assembly condition In a 3T1R closed-loop mechanism having variable rotation axis, there are four limbs. The four limbs can be separated into two groups. The limbs in each group have the same limb structures and they are placed oppositely. In the first group, the rotation direction of R and H joints are sa and sb , while in the second group the rotation direction of R and H joints are sc and sd . The intersection of the motions generated by the four limbs should be S f,3t1vr . According to Eq. (6.1), the variable rotation direction is computed by solving Eq. (6.29), as

6.3 Example with One Variable Rotation Axis

193

sTvr (sa × sb ) sin2 ϕab   2 sTvr sa − cos ϕ1 sTvr sb sTvr sb − cos ϕ1 sTvr sa − sTvr (sa × sb ) cos ϕab , (6.29) sTvr (sc × sd ) sin2 ϕcd =    2 sTvr sc − cos ϕ2 sTvr sd sTvr sd − cos ϕ2 sTvr sc − sTvr (sc × sd ) cos ϕcd 



where svr denotes the variable rotation direction of S f,3t1vr . Equation (6.29) is the axode equation of svr . The svr cannot be a constant direction or an arbitrary direction. Hence, the assembly conditions of the four limbs are summarized as (1) sa = sc and sb = sd ; (2) one of the following three conditions holds (a) sa = sd and sb = sc , (b) sa = sd and saT sb = sTc sd , (c) sb = sc and saT sb = sTc sd . By following these assembly conditions, any symmetrical 3T1R closed-loop mechanism having a variable rotation axis can be synthesized. Some typical ones are given in Fig. 6.4, within which the U joints are the replacements of two adjacent Ra and Rb joints, and/or two adjacent Rc and Rd joints. (4) Actuation arrangement According to Sect. 4.5.2, when the actuation joints are locked, the moving platform of the 3T1R closed-loop mechanism with variable rotation axis should satisfy − S f,1 ∩ − S f,2 ∩ − S f,3 ∩ − S f,4 = ∅.

(6.30)

One joint in each limb can be selected as an actuation joint since the mechanism is with symmetrical structure. The actuation arrangements are summarized as follows. (1) The three actuation joints generating the three-DoF translational motions should be one of the following cases, (a) three unparallel P joints, (b) two unparallel P joints, and an R or H joint, (c) one P joint and two R and H joints, (d) three R and H joints. It should be noted that the actuation R and H joints should belong to a dyad or triad of parallel R and H joints. (2) The actuation joint generating the one-DoF rotational motion can be arbitrarily selected. (3) For the three-DoF translational motions, the actuation R and H joints in the opposite limbs should have different sequences in the limbs.

194

6 Type Synthesis of Mechanism with Variable Rotation Axes

Fig. 6.4 Typical symmetrical 3T1R closed-loop mechanisms with one variable rotation axis

6.4 Example with One Invariable and One Variable Rotation Axes

195

6.4 Example with One Invariable and One Variable Rotation Axes For the mechanisms with one invariable and one variable rotation axes, the open-loop mechanisms with 3T2R motion [4] will be synthesized first, and then the closed-loop mechanisms with Exechon motion [1, 9–12].

6.4.1 Open-loop Mechanism with 3T2R Motion The basic motion expression for the open-loop mechanisms with this 3T2R motion is shown in Eq. (6.14). All the open-loop structures generating the 3T2R motion having one invariable and one variable rotation axes will be listed since they do not have equivalent motions. (1) P1 P2 Rb Ra Rb and its similar structures In Eq. (6.14), θa , θb , t1 , t2 , and t3 are the five independent parameters. This means that, in that equation θb,1 is not an independent parameter, which depends on the five independent parameters. The relationship between θb,1 and θa , θb , t1 , t2 , and t3 is determined by the specific structure of the open-loop mechanisms that generates the 3T2R motion. The specific values of the two position vectors of the rotations, r b and r a , have no influences on S f,3t1r 1vr because of the three-DoF translations. Hence, r b and r a can be arbitrarily selected, i.e., r b ∈ R3 and r a ∈ R3 . The simplest limb structure generating 3T2R motion with one invariable and one variable rotation axes is P1 P2 Rb Ra Rb . Its motion expression is       θa θb,1 θb,2 sb sa sb 2 tan 2 tan 2 r b,2 × sb 2 r a × sa 2 r b,1 × sb     0 0 (6.31) t2  t1  , s2 s1

S f = 2 tan

  where s1 × s2 × sb = 0. Equation (6.31) can be equivalently rewritten by using the properties of the screw triangle product, and the resultant expression is obtained as

196

6 Type Synthesis of Mechanism with Variable Rotation Axes

  θb,1 + θb,2 sb r b,2 × sb 2     θa exp θb,1 s˜ b s a      2 tan r b,1 + exp θb,1 s˜ b r a − r b,1 × exp θb,1 s˜ b sa 2       0         ,  exp θa exp θb,1 s˜ b sa × exp θb,1 s˜ b − E 3 r b,2 − r b,1 +t2 s2 + t1 s1 (6.32)

S f = 2 tan

where [s×] is an equivalent expression of s˜ . The following substitutions of symbols can be made in the above equation without changing the value range of S f , as θb,1 + θb,2 ∈ R → θb ∈ R,           exp θa exp θb,1 s˜ b sa × exp θb,1 s˜ b − E 3 r b,2 − r b,1 +t2 s2 +t1 s1 ∈ R3 → t3 s3 +t2 s2 +t1 s1 ∈ R3 ,     r b,2 → ∀r b ∈ R3 , r b,1 + exp θb,1 s˜ b r a − r b,1 → ∀r a ∈ R3 . In this way, Eq. (6.32) is proved to have the same form as Eq. (6.14). The relationship between θb,1 and θa , t1 , t2 , t3 of P1 P2 Rb Ra Rb can be derived from Eq. (6.33) by using Euler’s formula to rewrite exp θb,1 s˜ b ,       T      exp θa exp θb,1 s˜ b sa × exp θb,1 s˜ b − E 3 r b,2 − r b,1 s1 × s2   = (t3 s3 +t2 s2 +t1 s1 )T s1 × s2 . (6.33) The open-loop structures generating the 3T2R motion that are similar to P1 P2 Rb Ra Rb are listed in Table 6.2 (containing P1 P2 Rb Ra Rb ). For each open-loop mechanism, the corresponding equation contains three scalar equations with three dependent parameters, i.e., θb,1 , t1 , and t2 , to be solved. These three parameters can be, respectively, solved as the expressions of the other four/five independent parameters, θa , (θb ), t1 , t2 , and t3 . Hence, the relationship between the dependent parameter θb,1 and these independent parameters can be derived. (2) P1 Rb Ra Ra Rb and its similar structures P1 Rb Ra Ra Rb is also a motion generator of the 3T2R motion. The motion expression of this open-loop mechanism is

6.4 Example with One Invariable and One Variable Rotation Axes

197

Table 6.2 The open-loop structures generating the 3T2R motion that are derived from P1 P2 Rb Ra Rb Equation for deriving the relationship between θb,1 and θa , θb , t1 , t2 , and t3   1 P1 P2 Rb Ra Rb A (B 1 − E 3 ) r b,2 − r b,1 + t2 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1   2 P1 Rb P2 Ra Rb A (B 1 − E 3 ) r b,2 − r b,1 + t2 B 1 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1   3 P1 Rb Ra P2 Rb A (B 1 − E 3 ) r b,2 − r b,1 + t2 B 1 As2 + t1 s1 = t3 s3 + t2 s2 + t1 s1   4 P1 Rb Ra Rb P2 A (B 1 − E 3 ) r b,2 − r b,1 + t2 B 1 AB 2 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1     5 Rb P1 P2 Ra Rb A (B 1 − E 3 ) r b,2 − r b,1 + B 1 t2 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1   6 Rb P1 Ra P2 Rb A (B 1 − E 3 ) r b,2 − r b,1 + t2 B 1 As2 + t1 B 1 s1 = t3 s3 + t2 s2 + t1 s1   7 Rb P1 Ra Rb P2 A (B 1 − E 3 ) r b,2 − r b,1 + t2 B 1 AB 2 s2 + t1 B 1 s1 = t3 s3 + t2 s2 + t1 s1     8 Rb Ra P1 P2 Rb A (B 1 − E 3 ) r b,2 − r b,1 + B 1 A t2 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1   9 Rb Ra P1 Rb P2 A (B 1 − E 3 ) r b,2 − r b,1 +t2 B 1 As2 +t1 B 1 AB 2 s1 = t3 s3 +t2 s2 +t1 s1     10 Rb Ra Rb P1 P2 A (B 1 − E 3 ) r b,2 − r b,1 + B 1 AB 2 t2 s2 + t1 s1 = t3 s3 + t2 s2 + t1 s1           Noted s1 × s2 × sb = 0, A = exp θa exp θb,1 s˜ b sa × , A = exp(θa s˜ a ), B 1 = exp θb,1 s˜ b ,    B 2 = exp θb − θb,1 s˜ b . No.

Structure

   θa,2 sb sa 2 tan r b,2 × sb 2 r a,2 × sa       θb,1 θa,1 sa sb 0 2 tan t1  , 2 tan s1 2 r a,1 × sa 2 r b,1 × sb

θb,2 S f = 2 tan 2



(6.34)

which can be equivalently rewritten as  sb r b,2 × sb     θa,1 + θa,2 ˜ s s exp θ b,1 b a       2 tan r b,1 + exp θb,1 s˜ b r a,2 − r b,1 × exp θb,1 s˜ b sa 2 ⎛ ⎞     0        ⎝ exp θa,1 + θa,2 exp θb,1 s˜ b sa × exp θb,1 s˜ b − E 3 r b,2 − r b,1 ⎠        + exp θb,1 s˜ b exp θa,1 s˜ a − E 3 r a,2 − r a,1 +t1 s1 (6.35)

S f = 2 tan

θb,1 + θb,2 2



Without changing the value range of the above equation, the following substitutions of symbols can be made θb,1 + θb,2 ∈ R → θb ∈ R, θa,1 + θa,2 ∈ R → θa ∈ R,

          exp θa exp θb,1 s˜ b sa × exp θb,1 s˜ b − E 3 r b,2 − r b,1       + exp θb,1 s˜ b exp θa,1 s˜ a − E 3 r a,2 − r a,1 +t1 s1 ∈ R3 → t3 s3 +t2 s2 +t1 s1 ∈ R3 ,

198

6 Type Synthesis of Mechanism with Variable Rotation Axes

    r b,2 → ∀r b ∈ R3 , r b,1 + exp θb,1 s˜ b r a,2 − r b,1 → ∀r a ∈ R3 . There are more open-loop structures that are similar to P1 Rb Ra Ra Rb and generate 3T2R motion, as listed in Table 6.3. For each structure in Table 6.3, the corresponding equation leads to the relationship between θb,1 and θa , (θb ,) t1 , t2 , and t3 . (3) P1 Rb Ra Rb Rb , P1 Rb Rb Ra Rb , and their similar structures, Rb Ra Ra Rb Rb , Rb Rb Ra Ra Rb , and Rb Ra Ra Ra Rb In a similar manner, the P1 Rb Ra Rb Rb , P1 Rb Rb Ra Rb , Rb Ra Ra Rb Rb , Rb Rb Ra Ra Rb , and Rb Ra Ra Ra Rb open-loop mechanisms can be proved to generate the 3T2R motion with one invariable and one variable rotation axes. More structures generating this kind of motion can be derived from P1 Rb Ra Rb Rb and P1 Rb Rb Ra Rb . Due to space limitations, the derivations are not listed here. All these open-loop structures are shown in Table 6.4. Table 6.3 The open-loop structures generating the 3T2R motion that are derived from P1 Rb Ra Ra Rb Equation for deriving the relationship between θb,1 and θa , θb , t1 , t2 , and t3 1 P1 Rb Ra Ra Rb A (B − E ) r − r  + B ( A − E ) r − r  + t  s = 1 3 b,2 b,1 1 1 3 a,2 a,1 1 1 t3 s 3 + t 2 s 2 + t 1 s 1 2 Rb P1 Ra Ra Rb A (B − E ) r − r  + B ( A − E ) r − r  + t  B s = 1 3 b,2 b,1 1 1 3 a,2 a,1 1 1 1 t3 s 3 + t 2 s 2 + t 1 s 1 3 Rb Ra P1 Ra Rb A (B − E ) r − r  + B ( A − E ) r − r  + t  B A s = 1 3 b,2 b,1 1 1 3 a,2 a,1 1 1 1 1 t3 s 3 + t 2 s 2 + t 1 s 1 4 Rb Ra Ra P1 Rb A (B − E ) r − r  + B ( A − E ) r − r  + 1 3 b,2 b,1 1 1 3 a,2 a,1 t1 B 1 A1 A2 s1 = t3 s3 + t2 s2 + t1 s1 5 Rb Ra Ra Rb P1 A (B − E ) r − r  + B ( A − E ) r − r  + 1 3 b,2 b,1 1 1 3 a,2 a,1 t1 B 1 A1 A2 B 2 s1 = t3 s3 + t2 s2 + t1 s1           Noted A = exp θa exp θb,1 s˜ b sa × , A1 = exp θa,1 s˜ a , B 1 = exp θb,1 s˜ b , B 2 =    exp θb − θb,1 s˜ b . No.

Structure

Table 6.4 Other open-loop structures generating the 3T2R motion No.

Structure

No.

Structure

No.

Structure

1

P1 Rb Ra Rb Rb

2

Rb P1 Ra Rb Rb

6

P1 Rb Rb Ra Rb

11

Rb Ra Ra Rb Rb

7

Rb P1 Rb Ra Rb

12

3

Rb Ra P1 Rb Rb

Rb Rb Ra Ra Rb

8

Rb Rb P1 Ra Rb

13

Rb Ra Ra Ra Rb

4

Rb Ra Rb P1 Rb

9

Rb Rb Ra P1 Rb

5

Rb Ra Rb Rb P1

10

Rb Rb Ra Rb P1

6.4 Example with One Invariable and One Variable Rotation Axes

199

Fig. 6.5 Typical 3T2R open-loop mechanisms with one invariable and one variable rotation axes

All the 28 open-loop motion generators of the 3T2R motion with one invariable and one variable rotation axes are synthesized through the above analysis and derivations. The typical ones among them are shown in Fig. 6.5. These mechanisms can be used as limbs in the synthesis of the closed-loop mechanism with 3T2R motion.

6.4.2 Closed-loop Mechanism with Exechon Motion The 1T2R closed-loop mechanisms that have the same motions with the Exechon mechanism are synthesized in this section.

200

6 Type Synthesis of Mechanism with Variable Rotation Axes

(1) Standard limb structure Rewrite the Exechon motion shown in Eq. (6.15) into the intersection of several motions, leading to S f,1t1r 1vr = S1f,L ∩ S2f,L ,

(6.36)

where      θy sy 0 0 t1 2 tan = t2 s2 s1 2 r y × sy   θx sx 2 tan 2 r x × sx       θy 0 0 sy     tz 2 tan = tx exp(−θx s˜ x ) r y − r x + r x × s y sx sz 2   θx sx 2 tan 2 r x × sx         θy θx 0 0 sx exp(θx s˜ x )s y tz 2 tan 2 tan , = tx sx sz 2 r x × sx 2 r y × exp(θx s˜ x )s y (6.37) 

S1f,L

(s1 × s2 ) × s y = 0, 

       θy θz 0 0 sz sy t3 2 tan 2 tan s4 s3 2 r y × sz 2 r y × sy   θx sx 2 tan 2 r y × sx         θy θx 0 0 sx sy tz 2 tan 2 tan = ty sy sz 2 r y × sx 2 r y × sy   θz sz 2 tan 2 r y × sz         θy θx 0 0 sx exp(θx s˜ x )s y tz 2 tan 2 tan = ty sy sz 2 r x × sx 2 r y × exp(θx s˜ x )s y   θz sz , (6.38) 2 tan 2 r y × sz

S2f,L = t4

(s3 × s4 ) × s x = 0. where S1f,L and S2f,L are the simplest motions with minimum numbers of screw factors that contain S f,1t1r 1vr as subsets. Hence, the four- and five-DoF standard limb structures are Rx Ry P1 P2 and Rx Ry Rz P3 P4 .

6.4 Example with One Invariable and One Variable Rotation Axes

201

Because the expected closed-loop mechanisms have three DoFs, there should be three limbs in each mechanism with one actuation in each limb. Therefore, two fourDoF limbs generating S1f,L and one five-DoF limb generating S2f,L will be selected to constitute a 1T2R closed-loop mechanism with Exechon motion. The synthesis of the derivative limb structures is presented as follows. (2) Derivative limb structure (1) Four-DoF derivative limb structure According to Table 5.5, under the condition (s1 × s2 ) × s y = 0, the seven openloop structures are equivalent to each other, i.e., Ry P1 P2 , P1 Ry P2 , P1 P2 Ry , P1 Ry Ry , Ry P1 Ry , Ry Ry P1, and Ry Ry Ry . By replacing the “Ry P1 P2 ” in the standard structure Rx Ry P1 P2 with its equivalent structures, six four-DoF derivative structures are synthesized as shown in Table 6.5. The adjacent R joints with intersecting axes are replaced by U joints. Two typical ones in Table 6.5 are shown in Fig. 6.6. Table 6.5 Four-DoF standard and derivative limb structures of closed-loop mechanisms with Exechon motion No.

Structure

No.

Structure

No.

Structure

1

UP1 P2

2

Rx P1 Ry P2

3

Rx P1 P2 Ry

4

Rx P1 Ry Ry

5

UP1 Ry

6

URy P1

7

URy Ry

Fig. 6.6 Typical four-DoF limb structures of closed-loop mechanisms with Exechon motion

202

6 Type Synthesis of Mechanism with Variable Rotation Axes

(2) Five-DoF derivative limb structure Equation (6.38) can be rewritten as 

S2f,L

       θx θz 0 0 sx sz t3 2 tan 2 tan = t4 s4 s3 2 r y × sx 2 r y × sz   θy sy . (6.39) 2 tan 2 r y × sy

It indicates that the five-DoF standard limb structure can also be denoted as Ry Rz Rx P3 P4 . Similar to the synthesis of four-DoF limbs, under the condition (s3 × s4 )×s x = 0, the structures Rx P3 P4 , P3 Rx P4 , P3 P4 Rx , P3 Rx Rx , Rx P3 Rx , Rx Rx P3, and Rx Rx Rx are equivalent. By replacing the “Rx P3 P4 ” in the standard structure Ry Rz Rx P3 P4 with its equivalent structures, six five-DoF derivative structures can be synthesized. More open-loop structures can be obtained by replacing the adjacent Ry and Rz joints with U joints or replacing the adjacent Ry , Rz , and Rx joints with S joint. The obtained limb structures are listed in Table 6.6 and the typical ones are shown in Fig. 6.7. Table 6.6 Five-DoF standard and derivative limb structures of closed-loop mechanisms with Exechon motion No.

Structure

No.

Structure

No.

Structure

1

SP3 P4

2

UP3 Rx P4

3

UP3 P4 Rx

4

UP3 Rx Rx

5

SP3 Rx

6

SRx P3

7

SRx Rx

Fig. 6.7 Typical five-DoF limb structures of closed-loop mechanisms with Exechon motion

6.4 Example with One Invariable and One Variable Rotation Axes

203

(3) Assembly condition The limb synthesis procedures show that the arbitrarily selected two four-DoF limbs and one five-DoF limb lead to the following expression, S1f,L ∩ S1f,L ∩ S2f,L = S f,1t1r 1vr ,

(6.40)

which indicates that the only assembly condition is that the direction of each joint corresponds to its subscript correctly. (4) Actuation arrangement The actuation joint can be selected from each limb. Following the principle that the actuation joints should be near the fixed base, the criteria for the actuation arrangements are (1) For a limb with three joints such as UP1 Ry or SP3 Rx , the middle one-DoF joint is selected as the actuation joint. (2) For a limb with four joints such as Rx P1 Ry P2 or UP3 Rx P4 , the second joint which is countered from the one connected to the fixed base is selected as the actuation one. Based upon the above analysis, novel 1T2R closed-loop mechanisms having the same motions as the Exechon mechanism can be synthesized. The Exechon mechanism actuated by P joints is shown in Fig. 6.8a, and a novel mechanism with the same motions that are actuated by R joints is shown in Fig. 6.8b.

Fig. 6.8 Mechanisms with Exechon motion

204

6 Type Synthesis of Mechanism with Variable Rotation Axes

6.5 Example with Two Variable Rotation Axes The motions with two variable rotation axes are similar to the Exechon motion, as (1) The motion can only be generated by closed-loop mechanisms. (2) There is only one way to rewrite the motion into the intersection of several motions. Therefore, the synthesis problem of the mechanisms is degenerated into limb synthesis. The assembly conditions are naturally ensured by the intersection of limbs. In this section, the 2R single closed-loop mechanisms having the same motions with P1 S-Ca Uab will be synthesized first, and then the closed-loop mechanisms having the same motions with Z3 mechanism [2, 13–16].

6.5.1 Single Closed-loop Mechanisms with 2R Motion The 2R single closed-loop mechanisms generating equivalent motion as P1 S-Ca Uab are synthesized in the following way. (1) Standard structure Equation (6.17) is the simplified way to rewrite the motion into the intersection of several motions. Thus, the standard limb motions are       θc θb,1 θa,1 sc sb sa 2 tan 2 tan 2 r a,1 × sc 2 r a,1 × sb 2 r a,1 × sa   0 , (6.41) t1 s1

S1f,L = 2 tan

and       θb,2 θa,2 θa,3 sa sb sa 2 tan 2 tan 2 r a,3 × sa 2 r a,3 × sb 2 r a,2 × sa   0 , (6.42) ta sa

S2f,L = 2 tan

which correspond to the standard limb structures P1 S and Pa Ra Rb Ra , respectively. (2) Derivative structure There is no equivalent structure to P1 S. The derivative structure of Pa Ra Rb Ra is Ra Pa Rb Ra which is another way of denoting Ca Uba .

6.5 Example with Two Variable Rotation Axes

205

Fig. 6.9 Two single closed-loop mechanisms having 2R motion with two variable axes

(3) Assembly condition Two types of single closed-loop mechanisms with 2R motion are obtained, i.e., P1 S-Pa Ra Rb Ra and P1 S-Ra Pa Rb Ra . The assembly conditions for these mechanisms are directly determined by Eq. (6.17), as (1) The directions of P1 and Pa should not be parallel; (2) The axes of the R joints in limbs Pa Ra Rb Ra and Ra Pa Rb Ra should not pass through the center of the S joint. (4) Actuation arrangement Because the mechanisms are two-DoF ones, one actuation joint is arranged in each limb. The proper arrangements can be P1 S-Pa Ra Rb Ra and P1 S-Ra Pa Rb Ra . According to the assembly conditions and actuation arrangements, the two mechanisms with 2R motion are shown in Fig. 6.9.

6.5.2 Closed-loop Mechanisms with Z3 Motion (1) Standard structure To the authors’ knowledge, Eq. (6.18) is the only way for rewriting the Z3 motion into the intersection of several motions. Hence, the standard limb motion is obtained as       θc,2 θc,1 sc sc 0 2 tan 2 tan S f,L = tz sz 2 r c,1 × sc 2 r c,2 × sc     θa θb sb sa 2 tan , sTz sc = 0, (6.43) 2 tan 2 r b × sb 2 r a × sa whose corresponding standard limb structure is Pz Rc S.

206

6 Type Synthesis of Mechanism with Variable Rotation Axes

(2) Derivative structure The standard limb structure can also be denoted as Pz Rc Rc Rb Ra . Under the condition sTz sc = 0, the six equivalent structures of “Pz Rc Rc ” can be synthesized based upon Table 5.5, as Pz P1 Rc , Pz Rc P1 , P1 Pz Rc , Rc Pz Rc , Rc Rc Pz , and Rc Rc Rc , where the direction of P1 is s1 = s z × sc . By using these equivalent structures to replace “Pz Rc Rc ” in the standard structure, six derivative structures can be obtained accordingly, as listed in Table 6.7. In the table, the adjacent two R joints are replaced by a U joint, and the adjacent three R joints are replaced by an S joint. Two typical structures are shown in Fig. 6.10. (3) Assembly conditions By using three identical limbs with any selected structure in Table 6.7 to constitute a symmetric closed-loop mechanism with Z3 motion, the assembly conditions are directly determined by Eq. (6.18) as 2 1) The angle between any two si,c is π. 3 2) The directions of the Pz joints in all limbs are parallel to each other.

Table 6.7 Standard and derivative limb structures of closed-loop mechanisms with Z3 motion No.

Structure

No.

Structure

No.

Structure

1

Pz Rc S

2

Pz P1 S

3

Pz Rc P1 U

4

P1 Pz S

5

Rc Pz S

6

Rc Rc Pz U

7

Rc Rc S

Fig. 6.10 Typical limb structures of closed-loop mechanisms with Z3 motion

6.5 Example with Two Variable Rotation Axes

207

Fig. 6.11 Two typical closed-loop mechanisms with Z3 motion

(4) Actuation arrangement The actuations of the mechanism with Z3 motion that can be determined by arranging one actuation joint in each limb. The criteria for the actuation arrangements are as follows. (1) For a limb with three joints such as Pz Rc S, the middle one-DoF joint is selected as the actuation joint. (2) For a limb with four joint such as Pz Rc P1 U, the second joint which is countered from the one connected fixed base is selected as the actuation one. Through the analysis and derivations in this section, closed-loop mechanisms with Z3 motion can be synthesized. The 3Pz Rc S Z3 mechanism and the 3Rc Pz S mechanism are shown in Fig. 6.11 as two typical examples.

6.6 Conclusion Motions with variable rotation axes are difficult to be expressed by the existing mathematical tools, which hinder the synthesis and analysis of the mechanisms with these motion patterns. By applying the finite screw-based method proposed in Chap. 4, the motions with variable rotation axes are analytically described, and the mechanisms are synthesized in the finite motion level for the first time. The main key points of this chapter are listed below for the readers’ convenience. (1) The variable characteristics of the mechanisms’ rotations are investigated, leading to the underlying differences between the rotations with variable axes and the ones with invariable axes.

208

6 Type Synthesis of Mechanism with Variable Rotation Axes

(2) The motion patterns with variable rotation axes are classified into three subcategories, i.e., the ones with one variable rotation, the ones with one invariable and one variable rotations, and the ones with two variable rotations. (3) The 1R1T single closed-loop mechanisms and the 3T1R closed-loop mechanisms with one-DoF spherical rotations are synthesized. (4) The challenges on the synthesis of the 3T2R open-loop mechanisms, the closedloop mechanisms with Exechon motion and Z3 motion, and the 2R single closed-loop mechanisms are addressed, which provides solid foundations for the innovative design of robots with variable rotations.

References 1. Sun T, Yang SF (2019) An approach to formulate the Hessian matrix for dynamic control of parallel robots. IEEE-ASME Trans Mechatron 24(1):271–281 2. Sun T, Huo XM (2018) Type synthesis of 1T2R parallel mechanisms with parasitic motions. Mech Mach Theory 128:412–428 3. Yang SF, Sun T, Huang T (2017) Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mech Mach Theory 109:220–230 4. Yang SF, Li YM (2019) Different kinds of 3T2R serial kinematic chains and their applications in synthesis of parallel mechanisms. Mech Mach Theory. In press 5. Sun T, Yang SF, Huang T et al (2017) A way of relating instantaneous and finite screws based on the screw triangle product. Mech Mach Theory 108:75–82 6. Ruth DA, McCarthy JM (1999) The design of spherical 4R linkages for four specified orientations. Mech Mach Theory 34(5):677–692 7. McCarthy JM, Bodduluri RM (2000) Avoiding singular configurations in finite position synthesis of spherical 4R linkages. Mech Mach Theory 35(3):451–462 8. Mullineux G (2011) Atlas of spherical four-bar mechanisms. Mech Mach Theory 46(11):1811– 1823 9. Bi ZM, Jin Y (2011) Kinematic modeling of Exechon parallel kinematic machine. Robot Comput-Integr Manuf 27(1):186–193 10. Bi ZM (2014) Kinetostatic modeling of Exechon parallel kinematic machine for stiffness analysis. Int J Adv Manuf Technol 71(1–4):325–335 11. Zhang J, Zhao YQ, Jin Y (2016) Kinetostatic-model-based stiffness analysis of Exechon PKM. Robot Comput-Integr Manuf 37:208–220 12. Hu B (2016) Kinematically identical manipulators for the Exechon parallel manipulator and their comparison study. Mech Mach Theory 103:117–137 13. Huang Z, Tao WS, Fang YF (1996) Study on the kinematic characteristics of 3 DOF in-parallel actuated platform mechanisms. Mech Mach Theory 31(8):999–1007 14. Huang Z, Wang J, Fang YF (2002) Analysis of instantaneous motions of deficient-rank 3-RPS parallel manipulators. Mech Mach Theory 37(2):229–240 15. Huynh P, Hervé JM (2005) Equivalent kinematic chains of three degree-of-freedom tripod mechanisms with planar-spherical bonds. ASME J Mech Des 127(1):95–102 16. Li QC, Chen Z, Chen QH et al (2011) Parasitic motion comparison of 3-PRS parallel mechanism with different limb arrangements. Robot Comput-Integr Manuf 27(2):389–396

Chapter 7

Kinematic Modeling and Analysis of Robotic Mechanism

7.1 Introduction Robot kinematics, the study of the motions, is the prerequisite for statics, dynamics, accuracy analysis and design of robotic mechanism [1–3]. Kinematics of robotic mechanism concerns the motion transmissions from the input to the output. It is divided into two main topics, forward and inverse kinematics [4, 5]. Forward kinematics addresses the problems of finding the pose of end-effector or moving platform when the joint actuations are known [6, 7]. On the contrary, inverse kinematics is to compute the actuated joint parameters from a given pose of the end-effector or moving platform [8, 9]. Both forward and inverse kinematics involve the displacement model of the robotic mechanism [10]. Different from the existing methods, the displacement model is obtained by the finite screw base topology model, from which the forward and inverse kinematics are solved [11, 12]. Based on the displacement model, the reachable range of the end-effector or moving platform, i.e., the workspace of the robotic mechanism can be determined considering mechanism constraints [13]. Through the differentiation of the finite screw base displacement model, the instantaneous screw based velocity model of robotic mechanism is further established. The instantaneous screw can be applied to describe the twist or wrench of the robotic mechanism [14–17]. The former corresponds to the instantaneous motion whereas the latter is the exerted force. By employing the reciprocal property of the twist and wrench [18, 19], the generalized Jacobian matrix is obtained. Hence, the velocity model between actuated joint parameter and the instantaneous motion of the end-effector or the moving platform can be formulated. This chapter presents the displacement and velocity modeling methods of the robotic mechanism, concerning mainly the relations between input actuations and output motions. Specially, workspace of the robotic mechanism is analyzed on the basis of the displacement model. In addition, the generalized Jacobian matrix is built considering the twist and wrench mappings. According to the differential mapping between finite and instantaneous screws, differentiating the finite motion model © Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_7

209

210

7 Kinematic Modeling and Analysis of Robotic Mechanism

results in the instantaneous motion model at the initial pose. Hence, the velocity model at a new pose can be obtained by the differentiation of the displacement model regarding the new pose as a new initial pose. For description simplicity, the symbols describing instantaneous motions at initial pose are adopted to describe the instantaneous motions at any pose. The presented work lays a solid foundation for the modeling of stiffness and dynamic performances.

7.2 Displacement Modeling Displacement model of the robotic mechanism represents the kinematic equations from the actuated joint parameters to the mechanism poses. Based upon the analysis in Chap. 3, the topology model of robotic mechanism serves as its displacement model. Hence, the formulation of kinematic equations can be referred to the topology models.

7.2.1 Forward Kinematics Modeling 7.2.1.1

Open-loop Mechanism

The kinematic equation of an open-loop mechanism is formulated by the composition of all its joint motions. For a non-redundant open-loop mechanism, all the joints are actuation joints. Consider an open-loop mechanism consisting of n one-DoF joints. The kinematic equation is formulated by utilizing the topology model as S f = S f,n  · · · S f,1 ,

(7.1)

where S f denotes the displacement of the end-effector, S f, j ( j = 1, . . . , n) denotes the displacement generated by the jth joint. The joints connecting to the fixed base and the end-effector are named as the 1st and the nth joint, respectively. All the other joints are named in ascending order. The finite motion of the end-effector can be regarded as the motion contributed from the nth joint to the 1st joint in descending order. That is to say, the finite motion of the end-effector is obtained by the accumulation of the nth joint, then the (n − 1)th joint, and so forth. As the finite motion denotes the displacement from the initial pose to the current pose, the displacement of the open-loop mechanism can be calculated by the motion composition of each joint whose axis is described at the initial pose. The finite motion of the end-effector can also be regarded as the motion composition from the 1st joint to the nth joint, i.e., first the 1st joint, then the 2nd joint, and so on. Suppose the n-DoF open-loop mechanism is composed of n R joints as shown in Fig. 7.1. When the 1st R joint rotates, the axis of the jth R joint is changed from

7.2 Displacement Modeling

211

Fig. 7.1 Composition of motions in an open-loop mechanism from the first joint to the nth joint

s j to s1j ( j = 2, . . . , n). It indicates that the motion of the previous joint has effects on the motions of the latter joints. Hence, kinematic equation of an n-DoF open-loop mechanism by the motion composition from the 1st joint to the nth joint is expressed as n−1 S f = S f,1 S1f,2  · · · Sn−2 f,n−1 S f,n ,

(7.2)

j−1

where S f, j denotes the finite motion of the jth joint resulted from the previous j − 1 joints, and

j−1

S f, j =

j−1

Herein, s j

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

θj tan 2



⎪ ⎪ ⎪  ⎪ j−1 ⎪ ⎪ θ sj j ⎪ ⎪ ⎪ ⎩ 2 tan 2 r j−1 × s j−1 j j j−1

and r j

j−1

sj j−1 j−1 r j × sj   0 tj j−1 sj 



+ h jθj

R joint



0 j−1 sj



P joint .

(7.3)

H joint

can be computed by the following steps:

(1) If the 1st joint is an R joint, s1j and r 1j are computed as  s1j = exp(θ1 s˜ 1 )s j , r 1j = r 1 + exp(θ1 s˜ 1 ) r j − r 1 .

(7.4)

If the 1st joint is a P joint, s1j and r 1j are computed as s1j = s j , r 1j = r j + t1 s1 . If the 1st joint is an H joint, s1j and r 1j are computed as

(7.5)

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7 Kinematic Modeling and Analysis of Robotic Mechanism

 s1j = exp(θ1 s˜ 1 )s j , and r 1j = r 1 + exp(θ1 s˜ 1 ) r j − r 1 + h 1 θ1 s1 ,

(7.6)



   s1j 0 where or denotes the axis or direction of the jth joint after 1 1 1 s r j × sj j the movement of the 1st joint. (2) If the kth (k = 2, . . . , j − 1) joint is an R joint, skj and r kj are computed as  





k−1 k−1 k−1 k−1 k−1 k ˜ skj = exp θk s˜ k−1 s r . s , and r = r + exp θ − r k k k j j k j k (7.7) If the kth joint is a P joint, skj and r kj are computed as k−1 k skj = sk−1 + tk sk−1 j , and r j = r j k .

(7.8)

If the kth joint is an H joint, skj and r kj are computed as 

k skj = exp θk s˜ k−1 sk−1 k j , and r j 



r k−1 + h k θk sk−1 = r k−1 + exp θk s˜ k−1 − r k−1 k k j k k ,

(7.9)



   skj 0 where or is the axis or direction of the jth joint after the k k skj r j × sj movement of the first k joints. Referring to Chap. 2, S f remains unchanged if the positions of the one-DoF motions in the screw triangle product are switched. As a result, it can be proved that Eq. (7.2) is equivalent to Eq. (7.1), indicating that the kinematic equation of an open-loop mechanism can be formulated in the form of either of these two equations. Equation (7.1) is commonly found in the robotic kinematic analysis as its formulation. However, Eq. (7.2) and its variants are very helpful in identifying the specific motion patterns of some mechanisms. If the joints do not move one after one, the same principle applied in Eq. (7.2) is applicable. That is, the motion of the jth joint will affect axes of the ( j + 1)th joint to the nth joint, but will not affect the axes of the 1st to the ( j − 1)th joint. Suppose the mth joint moves at the beginning of the mechanism motion and the uth joint moves later (m < u). c1 , c2 , . . . are numbers of joints between m and u. The kinematic equation of the open-loop mechanism can be formulated as 1 ,c2 ,... ··· , S f = S f,m  · · · Scf,u

(7.10)

7.2 Displacement Modeling

213

1 ,c2 ,... where Scf,u denotes the motion generated by the uth joint whose joint axis is 1 ,c2 ,... affected by the motions of the c1 th, c2 th, … joints. Scf,u can be computed in the

j−1

similar way as S f, j . (1) If the c1 th joint is an R joint, scu1 and r cu1 are computed as    scu1 = exp θc1 s˜ c1 su , r cu1 = r c1 + exp θc1 s˜ c1 r u − r c1 .

(7.11)

If the c1 th joint is a P joint, scu1 and r cu1 are computed as scu1 = su , r cu1 = r u + tc1 sc1 .

(7.12)

If the c1 th joint is an H joint, scu1 and r cu1 are computed as    scu1 = exp θc1 s˜ c1 su , r cu1 = r c1 + exp θc1 s˜ c1 r u − r c1 + h c1 θc1 sc1 ,

(7.13)

0 scu1 or is the axis or direction of the uth joint after the where r cu1 × scu1 scu1 movement of the c1 th joint. (2) If the c2 th joint is an R joint, scu1 ,c2 and r cu1 ,c2 are computed as

   scu1 ,c2 = exp θc2 s˜ cc12 scu1 , r cu1 ,c2 = r cc12 + exp θc2 s˜ cc12 r cu1 − r cc12 .

(7.14)

If the c2 th joint is a P joint, scu1 ,c2 and r cu1 ,c2 are computed as scu1 ,c2 = scu1 , r cu1 ,c2 = r cu1 + tc2 r cc12 .

(7.15)

If the c2 th joint is an H joint, scu1 ,c2 and r cu1 ,c2 are computed as    scu1 ,c2 = exp θc2 s˜ cc12 scu1 , r cu1 ,c2 = r cc12 + exp θc2 s˜ cc12 r cu1 − r cc12 + h c2 θc2 scc12 , (7.16)



0 scu1 ,c2 or is the axis or direction of the uth joint after r cu1 ,c2 × scu1 ,c2 scu1 ,c2 the movement of the c1 th and c2 th joints.

where

Consequently, suc1 ,c2 ,... and r uc1 ,c2 ,... can be computed following the same manner. Hence, the kinematic equation shown in Eq. (7.10) can be computed.

7.2.1.2

Closed-loop Mechanism

Suppose a closed-loop mechanism with l limbs is considered. The ith limb consists of n i one-DoF joints, as shown in Fig. 7.2. The kinematic equation of the ith limb can be formulated in the similar way as Eq. (7.1),

214

7 Kinematic Modeling and Analysis of Robotic Mechanism

Fig. 7.2 The ith limb with n i joints in closed-loop mechanism

S f,i = S f,i,ni  · · · S f,i,1 ,

(7.17)

where S f,i denotes the finite motion of the end-effector of the ith limb (the moving platform of the closed-loop mechanism), S f,i,k (k = 1, . . . , n i ) denotes the finite motion generated by the kth joint in the ith limb. The forward kinematics of the closed-loop mechanism is computing the finite motion of the moving platform by the parameters of actuated joints. In common cases, there is one actuated joint in a limb. The parameter of this actuated joint is the known variable. The parameters of the remaining n i − 1 joints in the ith limb are determined by the known actuation parameter of the ith limb and the actuation parameters of the other limbs. Define S¯ f to denote the finite motion having unknown motion parameter. If the ai th joint in the ith limb is selected as the actuated joint, the kinematic equation of the ith limb can be expressed as S¯ f,i = S¯ f,i,ni  · · · S f,i,ai  · · ·  S¯ f,i,1 ,

(7.18)

where S¯ f,i denotes the finite motion of the ith limb. S f,i,ai denotes the finite motion of the actuated joint in ith limb. S¯ f,i,k (k = 1, . . . , ai − 1, ai + 1, . . . , n i ) denotes the finite motion of kth non-actuated joint whose joint parameter is unknown. S¯ f,i,k is expressed as

S¯ f,i,k

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

θ¯i,k si,k tan R joint s 2 r i,k ×

i,k 0 P joint , = t¯i,k ⎪ s i,k ⎪



⎪ ⎪ ⎪ θ¯i,k 0 si,k ⎪ ⎪ + h i,k θ¯i,k H joint ⎩ tan r × s s 2 i,k i,k i,k

(7.19)

where θ¯i,k and t¯i,k are the unknown parameters of the non-actuation joints. The finite motion of the limbs is the same since they are connected to the same moving platform. Hence, the following equation can be formulated:

7.2 Displacement Modeling

215

S¯ f,1 = · · · = S¯ f,i = · · · = S¯ f,l .

(7.20)

The unknown parameters of the non-actuated joints in the ith limb are first solved, which can be expressed as the function of all the actuated joint parameters as  p¯ i,k = f p1,a1 , . . . , pl,al ,

(7.21)

where p¯ i,k denotes the non-actuation parameter of the kth joint in the ith limb. pi,ai denotes the actuation parameter, and  p¯ i,k =

θ¯i,k R or H joint , k = 1, . . . , ai − 1, ai + 1, . . . , n i , P joint t¯i,k  θ R or H joint pi,ai = i,ai . P joint ti,ai

(7.22) (7.23)

With the parameters of both actuated and non-actuated joint, the finite motion of the moving platform can be solved by the finite motion of the ith limb as shown in Eq. (7.17). Following this procedure, there are two ways of addressing the forward kinematics problem of the closed-loop mechanism. One way is solving the simultaneous equations, and the other way is computing the finite motion by step motion. (a) Solving the Simultaneous Equations Equations (7.18)–(7.20) constitute simultaneous equations. There are l actuated joints in the closed-loop mechanism. Their actuation parameters are known in the forward kinematic problem. The parameters of the non-actuation joints are the unknown  parameters. Hence, there are li=1 (n i − 1) unknown parameters. Assume that each limb is a lower mobility limb with no more than five one-DoF joints. The number of unknown parameters is no more than 4l. Equations (7.18)–(7.20) can be written into 6 × (l − 1) scalar equations, in which l is no less than two because a closed-loop mechanism has at least two limbs. As a result, the number of scalar equations is more than the number of unknown parameters. The unknown parameters can be computed by solving the simultaneous equations. Finally, the current pose of the closed-loop mechanism can be obtained by Eq. (7.17). Although formulating the forward kinematics model of a closed-loop mechanism by solving the simultaneous equation is feasible, it is computationally expensive. Especially when a closed-loop mechanism with complicated motion patterns is involved, the equation solving process can be extremely difficult. (b) Computing the Pose by Step Motions The finite motion of the closed-loop mechanism can be regarded as the composition of the motion generated by all the actuated joints. The actuated joints move one by one from the one in the 1st limb to the one in the lth limb. These actuation joints are named as the 1st actuated joint to the lth actuated joint. When the actuation joint in

216

7 Kinematic Modeling and Analysis of Robotic Mechanism

the 1st limb moves, the finite motion of the moving platform generated by the joint motion can be obtained by Eq. (7.18) as S¯ f,1 = S¯ f,1,n 1  · · · S f,1,a1  · · ·  S¯ f,1,1 .

(7.24)

After the movement of the 1st actuated joint, the axes and/or directions of the joints in the other limbs will be changed. When the 2nd actuated joint moves, the finite motion of the moving platform generated by the joint motion is 1 1 1 S¯ f,2 = S¯ f,2,n 2  · · · S1f,2,a2  · · ·  S¯ f,2,1 ,

(7.25)

where S¯ f,2 is the finite motion of moving platform generated by the 2nd actuated joint after the movement of the 1st actuated joint. S1f,2,a2 is the finite motion generated by the 2nd actuated joint after the movement of the 1st actuated joint. S1f,2,a2 is expressed as ⎧

θ2,a2 s12,a2 ⎪ ⎪ ⎪ tan R joint ⎪ ⎪ r 1 × s1 2 ⎪ ⎪ 2,a2 2,a2 ⎨ 0 P joint . t2,a2 1 S1f,2,a2 = (7.26) ⎪ s ⎪ 2,a2



⎪ ⎪ ⎪ θ2,a2 s12,a2 0 ⎪ ⎪ + h 2,a2 θ2,a2 1 H joint ⎩ tan 1 1 s r × s 2 2,a2 2,a2 2,a2 1

1 S¯ f,2,k (k = 1, . . . , a2 − 1, a2 + 1, . . . , n 2 ) denotes the finite motion generated by the kth non-actuation joint in the 2nd limb after the movement of the 1st actuated joint. 1 S¯ f,2,k is expressed as

1 S¯ f,2,k

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

s12,k R joint r 1 × s1 2,k 2,k 0 P joint , t¯2,k 1 = ⎪ s ⎪ 2,k



⎪ ⎪ ⎪ θ¯2,k s12,k 0 ⎪ ⎪ + h 2,k θ¯2,k 1 H joint ⎩ tan 1 s2,k 2 r 2,k × s12,k tan

θ¯2,k 2



(7.27)

where θ¯2,k and t¯2,k are the non-actuation parameters of the second limb, determined by the first two actuation parameters. The relationship between these non-actuation parameters and the first two actuation parameters can be written as  p¯ 2,k = f 2,k p1,a1 , p2,a2 .

(7.28)

s12,k and r 12,k (k = 1, . . . , n 2 ) are the unit direction vector and position vector of the kth joint in the 2nd limb, after the movement of the 1st actuation joint.

7.2 Displacement Modeling

217

Similarly, before the jth actuation joint moves, the axes and/or directions of the joints in this limb have been changed by the movements of the previous j −1 actuation joints. The finite motion of the moving platform generated by the jth actuation joint is j−1 j−1 j−1 j−1 S¯ f,i = S¯ f,i,n j  · · · S f,i,a j  · · ·  S¯ f,i,1 ,

(7.29)

where S¯ f,i is the finite motion of the moving platform generated by the jth actuation j−1 joint after the movement of the previous j − 1 actuation joints. S f,i,a j is the pose generated by the jth actuation joint after the movement of the previous j −1 actuation j−1 joints. S f,i,a j is expressed as j−1

j−1

S f,i,a j =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

θi,ai tan 2



j−1



si,ai j−1 j−1 r ×s i,ai  i,ai 0 ti,ai j−1 si,ai 

⎪ ⎪ ⎪  ⎪ j−1 ⎪ ⎪ θi,ai si,a j ⎪ ⎪ tan ⎪ j−1 j−1 ⎩ 2 r i,ai × si,ai

R joint



+ h i,ai θi,ai

 0 j−1

si,ai

P joint ,

(7.30)

H joint

j−1  S¯ f,i,k k = 1, . . . , a j − 1, a j + 1, . . . , n j denotes the pose generated by the kth nonactuation joint in the ith limb after the movement of the previous j − 1 actuation j−1 joints, S¯ f,i,k is expressed as

j−1 S¯ f,i,k =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

θ¯i,k tan 2



j−1



si,k j−1 j−1 r ×s i,k i,k 0 t¯i,k j−1 si,k

⎪ ⎪  ⎪ ⎪ j−1 ⎪ θ¯ j,k ⎪ si,k ⎪ ⎪ tan ⎩ j−1 j−1 2 r i,k × si,k

+ h i,k θ¯i,k

R joint P joint ,

0 j−1

si,k

(7.31)

H joint

where θ¯i,k and t¯i,k are the non-actuation parameters of the ith limb, which are deterj−1 j−1 mined by the previous j − 1 actuation parameters. si,k and r i,k (k = 1, . . . , n i ) are the unit direction vector and position vector of the kth joint in the ith limb, after the movement of the first j − 1 actuation joints. The relationship between the nonactuation parameters and the previous j − 1 actuation parameters can be written as  p¯ i,k = f i,k p1,a1 , . . . , p j−1,a j−1 .

(7.32)

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7 Kinematic Modeling and Analysis of Robotic Mechanism

In summary, the motion of the moving platform from its initial pose to the current pose can be divided into step motions. Hence, the motion of the moving platform is obtained by moving one actuation joint at a time, as shown in the above analysis. In this way, the closed-loop mechanism can be regarded as an open-loop mechanism, and its motion is the composition of the motions generated by all the l actuation joints. Thus, the kinematic equation of the closed-loop mechanism is formulated as j−1 l−1 S f = S¯ f,1  · · ·  S¯ f,i  · · ·  S¯ f,l .

(7.33)

In the second way to solve forward kinematics problem of closed-loop mechaj−1 j−1 nisms, θ¯i,k and t¯i,k (i = 1, . . . , l, k = 1, . . . , n i ), si,k and r i,k (i = 2, . . . , l, k = 1, . . . , n i ) are the unknown parameters. They can be obtained by analyzing the geometrical conditions between joints and limbs. The computation of these parameters has to follow the motion sequence of the actuation joints. It indicates that θ¯i,k and j−1 j−1 t¯i,k can be computed only after si,k and r i,k have been obtained.

7.2.2 Inverse Kinematics Modeling The inverse kinematics is to solve the actuation joint parameters given the pose of the end-effector or the moving platform of the robotic mechanism. It is the inverse process of forward kinematics.

7.2.2.1

Open-loop Mechanism

Consider an open-loop mechanism consisting of n one-DoF joints that are all actuated joints. The kinematic equation of the open-loop mechanism can be formulated as Eq. (7.1). Using pa j to denote the actuation joint parameters, the kinematic equation can be rewritten as  S f = f pa1 , . . . , pan ,

(7.34)

which shows the mapping between the mechanism pose and joint parameters. The pose of the end-effector can be expressed as S f = 2 tan

θ 2



s r×s

+t

0 . s

(7.35)

The mechanism pose is the function of the joint parameters. Hence, the elements in Eq. (7.35), including the rotational angle, translational distance unit vector and position vector of the motion axis, are the functions of these parameters. Suppose there are m rotational parameters, θ1 , …, θm , in the open-loop mechanism, Eq. (7.35)

7.2 Displacement Modeling

219

can be divided as the following four functions: s = f s (θ1 , . . . , θm ),

(7.36)

θ = f θ (θ1 , . . . , θm ), 2  r = fr pa1 , . . . , pan ,

(7.38)

 t = f t pa1 , . . . , pan .

(7.39)

tan

(7.37)

We can also get another function by separating Eq. (7.35) as θ r × s + ts 2   = 2 f θ (θ1 , . . . , θm ) · fr pa1 , . . . , pan × f s (θ1 , . . . , θm )  + f t pa1 , . . . , pan · f s (θ1 , . . . , θm ).  = f p pa1 , . . . , pan

2 tan

(7.40)

Equations (7.36)–(7.40) reveal the algebraic mappings between the joint parameters and the finite motion of the end-effector. Using these mappings, all the joint parameters for the open-loop mechanism at the given pose can be analytically solved through algebraic derivations. From the above analysis, the general procedures for inverse kinematics are summarized as follows: Step 1: Derive the composition of poses generated by all joints using screw triangle product. Rewrite the kinematic equation as shown in Eq. (7.34); Step 2: Build the mappings between the rotational parameters of joints and the given orientation, as shown in Eqs. (7.36)–(7.37); Step 3: Build the mapping between all the joint parameters and the last three items of the given pose, as shown in Eqs. (7.38)–(7.39); Step 4: Solve the formulated Eqs. (7.36)–(7.39) through vector and polynomial analysis. Obtain the analytical solutions of all joint parameters.

7.2.2.2

Closed-loop Mechanism

The closed-loop mechanism can be regarded as several open-loop mechanisms sharing the same end-effector. Hence, the inverse kinematics of the closed-loop mechanism can be divided into several inverse kinematics problems of open-loop mechanisms having the same given pose. The kinematic equation of ith limb can be rewritten in a similar manner as Eq. (7.34)  S f,i = f pi,1 , pi,2 , . . . , pi,ai . . . , pi,ni ,

(7.41)

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7 Kinematic Modeling and Analysis of Robotic Mechanism

The motion from the initial pose to the current pose of moving platform can be expressed as S f,i

θi = 2 tan 2



si r i × si

+ ti

0 . si

(7.42)

Similar to the open-loop mechanism, the function in Eq. (7.42) can be divided as the following four functions:  si = f s θi,1 , . . . , θi,m ,  θi = f θ θi,1 , . . . , θi,m , 2  r i = fr pi,1 , . . . , pi,ni ,

tan

 ti = f t pi,1 , . . . , pi,ni .

(7.43) (7.44) (7.45) (7.46)

By separating Eq. (7.42), we have θi r i × si + ti si 2    = 2 f θ θi,1 , . . . , θi,m · fr pi,1 , . . . , pi,ni × f s θi,1 , . . . , θi,m   + f t pi,1 , . . . , pi,ni · f s θi,1 , . . . , θi,m .  = f p pi,1 , . . . , pi,ni

2 tan

(7.47)

By Eqs. (7.43)–(7.47), all the joint parameters of each limb with the given pose can be analytically solved through algebraic derivations. From the above analysis, the general procedure for inverse kinematics of closed-loop mechanism is summarized as follows: Step 1: Derive the composition of poses generated by all limbs, rewrite the kinematic equation of ith limb as shown in Eq. (7.41); Step 2: Build the mappings between the rotational parameters of joints and the given orientation, as shown in Eqs. (7.43)–(7.44); Step 3: Build the mapping between all the joint parameters and the last three items of the given pose, as shown in Eqs. (7.45)–(7.46); Step 4: Solve the formulated Eqs. (7.43)–(7.46) through vector and polynomial analysis. Obtain the analytical solutions of all joint parameters.

7.3 Workspace Analysis

221

7.3 Workspace Analysis The kinematic equation is the mapping from the actuation parameters to the current pose of the mechanism. For an n-DoF nonredundant open-loop or closed-loop mechanism, it has n actuation joints, and this mapping is formulated as  S f = f pa1 , . . . , pan .

(7.48)

If the range of each actuation parameter is given, the whole set of S f can be computed, which  is denoted by {S f }. Because the DoFs of the mechanism is no more than six, S f denotes the sub-six-dimensional workspace of the mechanism. The pose of a mechanism is generated by the motions of its actuation joints. In this book, we only consider the mechanisms with one-DoF actuation joints. If an actuation joint is an R joint or an H joint, the value range of its parameter is within [0, 2π]. If an actuation joint is a P joint, the value range of its parameter is the field of real number R. Considering that the geometric constraints of robotic mechanisms such as position limitation  of motors and structure interference, the set of reachable poses is the subset of S f . If the number of R and H actuation joints is m and the number of P actuation joints is n − m, the set of reachable poses is computed as 

Sf

r

 =

   pak ∈ [αk , βk ], k = 1, . . . , m f pa1 , . . . , pan  ,  pak ∈ [ak , bk ], k = m + 1, . . . , n 

(7.49)

 r where S f denotes the set of reachable poses of the robotic mechanism, 

Sf

r

  ⊆ Sf .

(7.50)

αk and βk are the lower and upper boundaries for rotational angle of an R or H joint, boundaries for translational distance of a P joint. ak and bk are the lower  and  upper r As the subset of S f , S f is also a sub-six-dimensional workspace which is a subset of the finite screw Lie group discussed in Chap. 2. When the value ranges of all the actuation joints are given,the whole set of reachr able poses can be computed by Eq. (7.49). The resultant set S f contains various finite screws that describe different poses of the end-effector or moving platform of the mechanism measured from its initial pose. A pose of the mechanism can be divided into two parts, i.e., orientation and position. Hence, the set of reachable poses can be divided into the set of reachable orientations and the set of reachable positions, which means that sub-six-dimensional workspace are separated into orientation space and position space. In order to clearly show the characteristics of a mechanism’s workspace, we will discuss the orientation space and position space separately.

222

7 Kinematic Modeling and Analysis of Robotic Mechanism

7.3.1 Sub-three-dimensional Orientation Space The orientation of a mechanism can be described by a Gibson form consisting of a three-dimensional vector and an angle as θ g = 2 tan s, 2

(7.51)

where g describes the orientation in Gibson form, s is a unit vector that expresses the rotation direction from the initial orientation to the current orientation, θ is the rotational angle which is measured from the initial orientation. When a mechanism undergoes two orientations successively, both of these orientations can be expressed in Gibson form as g a = 2 tan

θa θb sa , g b = 2 tan sb . 2 2

(7.52)

The resultant orientation of the mechanism is the composition of the above two orientations as

g ab =

2 tan

θb θa θb θa sa + 2 tan sb + 2 tan tan sb × sa 2 2 2 2 , θb T θa 1 − tan tan sb sa 2 2

(7.53)

where g ab denotes the current orientation of the mechanism. The composition of several Gibson forms has closure property. Thus, g ab can be rewritten into Gibson form as θab sab , (7.54) g ab = 2 tan 2 where θab

⎞ ⎛   tan θa sa + tan θb sb + tan θa tan θb sb × sa  ⎟ ⎜ 2 2 2 2 ⎟, = 2 arctan⎜ ⎠ ⎝ θb T θa 1 − tan tan sb sa 2 2

and

θa θb sa + tan sb + tan 2 2 =  θ θ a tan sa + tan b sb + tan  2 2 tan

sab

θb θa tan sb × sa 2 2 .  θb θa tan sb × sa  2 2

(7.55)

(7.56)

The composition algorithm of Gibson forms is similar to the screw triangle product. (1) The composition algorithm of Gibson forms is the degeneration of the screw triangle product into pure rotation level. (2) The screw triangle product can be

7.3 Workspace Analysis

223

regarded as the extension of the composition algorithm of Gibson forms from angles, vectors to dual angles, dual vectors by using the “transference principle”. The Gibson form can be obtained by extracting the first three items from the finite screw in quasi-vector form. Hence, there exists a linear mapping between finite screw and Gibson form. The mapping is built as



  θ θ s 0 +t 2 tan s = E 3 03×3 2 tan , s 2 2 r×s ⇒ g = T SG S f

(7.57)

  where T SG = E 3 03×3 is the transformation matrix from a finite screw to the corresponding Gibson form. The orientation space of a robotic can be expressed by the set of Gibson   mechanism forms, which can be denoted as g M . It is the subset of the three-dimensional whole set of Gibson forms,      θ  g M ⊆ {g}, {g} = 2 tan sθ ∈ [0, 2π], s ∈ R3×1 , |s| = 1 . (7.58) 2   Therefore, g M is a sub-three-dimensional orientation space. Using the transformation matrix between finite screw and Gibson form, the mapping between a mechanism’s sub-six-dimensional workspace and the sub-threedimensional orientation space is formulated as 

  r g M = T SG S f .

(7.59)

The orientation space of a mechanism contains all the orientations it can realize. It can be used to profile the mechanism’s workspace in terms of orientations regardless of positions. From the expressions of Gibson form, it is found that the dimensions of the mechanism have little effects on the orientation space. A small mechanism may have the same or more flexible orientation space than a large one.

7.3.2 Sub-three-dimensional Position Space A three-dimensional position vector can be used to describe the position of a mechanism with respect to its initial pose. It is constituted by a distance value and a unit vector, d = d s,

(7.60)

where d is the magnitude of the distance vector, and s(|s| = 1) is its direction vector. Consider the position space of an open-loop mechanism consisting of n-DoF joints. As usual, the n joints are numbered from the one connected to the fixed base to the one connected to the end-effector, and they move from the nth one to the 1st

224

7 Kinematic Modeling and Analysis of Robotic Mechanism

one. When the nth joint moves, the position vector  of the center of the end-effector before and after the movement of this joint are r r 0 and r 1 , respectively. In a similar manner, the position vector before and after the movement of the jth ( j = n, . . . , 1) joint are r n− j and r n− j+1 , respectively. If the jth joint is an R joint, the distance vector d j generated by the joint is obtained as    d j = exp θ j s˜ j − E 3 r j−1 − r j ,

(7.61)

r j can thus be computed as r j =r j−1 + d j    =r j−1 + exp θ j s˜ j − E 3 r j−1 − r j .

(7.62)

If the jth joint is a P joint, the distance vector d j is obtained as d j = tj sj,

(7.63)

r j can be computed as r j =r j−1 + d j =r j−1 + t j s j .

(7.64)

If the jth joint is an H joint, the distance vector d j is obtained as    d j = exp θ j s˜ j − E 3 r j−1 − r j + h j θ j s j ,

(7.65)

r j can be computed as r j =r j−1 + d j .    =r j−1 + exp θ j s˜ j − E 3 r − r j + h j θ j s j

(7.66)

In this way, d j generated by each one of the n joints can be obtained. Hence, the total position of the mechanism generated by its n joints is computed as d=

n 

d j,

(7.67)

j=1

where d denotes the current position of the mechanism after the movements of all its joints. It should be noted that d j and d can all be rewritten into the standard form of distance vectors in Eq. (7.60). Unlike g M , there is no explicit mapping between d and S f . However, d can also be regarded as a function of actuation joint parameters. For an n-DoF open-loop

7.3 Workspace Analysis

225

mechanism having n one-DoF joints, the function is expressed as  d = f pa1 , . . . , pan .

(7.68)

Hence, when the value ranges of the joint parameters are given, the position space of open-loop mechanism can be computed in the following way:  {d} =

   pak ∈ [αk , βk ], k = 1, . . . , m f pa1 , . . . , pan  ,  pak ∈ [ak , bk ], k = m + 1, . . . , n 

(7.69)

the denotations of αk , βk , ak , and bk can be referred to Eq. (7.49). Consider an n-DoF closed-loop mechanism with l limbs. Each limb has n i (i = 1, . . . , l) one-DoF joints. The position space of the ith limb can be obtained using Eq. (7.69) as    {d i } = f pi,1 , . . . , pi,ni .

(7.70)

Because all the limbs share the same moving platform, the position space of the mechanism should be the intersection of those of its limbs, as {d} = {d 1 } ∩ · · · ∩ {d l },

(7.71)

which can be rewritten as the value range of the function of all its n actuation joint parameters as    {d} = f pa1 , . . . , pal .

(7.72)

From the expression of d, the position space is determined by the dimension of links. For the robotic mechanism with the same topology, the mechanism with larger link dimensions, the larger position space it will get.

7.3.3 Workspace Regardless of Initial Pose As discussed above, the pose, i.e., orientation and position of an open-loop or closedloop mechanism, are r  from its initial pose. Therefore, different initial pose  measured leads to different S f , g M and {d}. Sometimes, the feature of the workspace is analyzed no matter where the initial pose is. Hence, the workspace regardless of initial pose is investigated in this section. Suppose a fixed reference frame and a moving reference frame are given to a robotic mechanism. The moving reference frame, sometimes called tool frame, is attached to the center of the end-effector or moving platform, as shown in Fig. 7.3.

226

7 Kinematic Modeling and Analysis of Robotic Mechanism

Fig. 7.3 The reference frame connected to the mechanism’s end-effector or moving platform

The initial pose can be described by a finite screw denoting the motion transformation between the fixed reference frame and the moving reference frame. The finite screw describing the initial pose of the mechanism can be written as SIPf = 2 tan

θ IP 2



sIP IP r × sIP

+ t IP

0 , sIP

(7.73)

 where sIP ; r IP × sIP is the axis of Chasles’ motion between the fixed reference frame and the reference frame connected to the mechanism’s end-effector or moving platform at its initial pose. sIP and r IP are its unit vector and position vector. θ IP and t IP are the corresponding rotational angle and translational distance. The workspace regardless of initial pose can be obtained by compositing the finite screw in Eq. (7.73) and the workspace of the mechanism as 

  r S f = SIPf S f .

(7.74)

By using the linear mapping between the workspace and the orientation space of the mechanism, the orientation space regardless of initial pose is obtained as 

   r g  M = T SG SIPf S f .

(7.75)

In order to obtain the position space regardless of initial pose, the position vector of the center of the mechanism’s end-effector or moving platform at the initial pose is required. It can be computed by using the elements of SIPf , as  



r = exp θ IP s˜ IP − E 3 r IP + t IP sIP .

(7.76)

7.3 Workspace Analysis

227

The position space regardless of initial pose can be obtained by adding this position vector to each distance vector in {d}, as   d = {r + d}.

(7.77)

7.4 Velocity Analysis After addressing the displacement model of the robotic mechanism in the previous sections, the velocity model is considered in this section. Velocity model of the robotic mechanism has been discussed in Chap. 3, where the mappings between the joint parameters and the velocity of end-effector or the moving platform are revealed. In this chapter, the motion mapping between the inputs and the outputs of the mechanism is considered. The motion and constraint of the mechanism, expressed by the twist and wrench, will be introduced to compute the Jacobian matrix between the actuation joint parameters and the output velocity. It is worth mentioning that the velocity model denotes the instantaneous motion model of the mechanism at any pose, which is obtained by differentiating the finite motion model regarding the new pose as the new initial pose.

7.4.1 Jacobian Matrix of Open-loop Mechanism By differentiating the displacement model of the open-loop mechanism, the velocity model is computed as St =

n 

q˙ j Sˆ t, j .

(7.78)

j=1

T  where St = ωT vT denotes the velocity of the end-effector. q˙ j is the velocity of the jth joint, and Sˆ t, j is the unit twist of the jth joint at current pose. As mentioned in Sect. 2.3, the wrench space of an open-loop mechanism can be classified based on the reciprocal twist space. Hence, the wrench acting on the jth joint is computed by the reciprocal twist of the rest joints. Suppose there are n joints in the open-loop mechanism. If n < 6, one can find (6 − n) wrenches reciprocal to all the joint twist. The computation process is referred to Eqs. (2.100)–(2.108). Name the wrench exerting on the joint as actuation wrench and the wrench reciprocal to all the joint twists as constraint wrench. The wrenches of an open-loop mechanism are given as

228

7 Kinematic Modeling and Analysis of Robotic Mechanism

Sw = Swa + Swc =

n 

f a, ja Sˆ wa, ja +

ja=1

6−n 

f c, jc Sˆ wc, jc ,

(7.79)

jc=1

where Swa and Swc are the actuation and constraint wrenches of the open-loop mechanism. Sˆ wa, ja and Sˆ wc, jc stand for the unit actuation wrench screw and unit constraint wrench screw. f a, ja and f c, jc represent the intensity of actuation and constraint wrench. In order to obtain the Jacobian matrix denoting the mapping relations between actuation joint parameters and the velocity of end-effector, first multiply Sˆ wa, ja to both side of Eq. (7.78). Since the Sˆ wa, ja is reciprocal to the twist of the other joints except for the jath actuated joint, we have T T Sˆ wa, ja St = q˙ ja Sˆ wa, ja Sˆ t, ja , ja = 1, 2 . . . , n.

(7.80)

As the Sˆ wc, jc is reciprocal to all the joint twist in the open-loop mechanism, multiplying Sˆ wc, jc to both sides of Eq. (7.78) yields, T Sˆ wc, jc St = 0,

jc = 1, 2 . . . , 6 − n.

(7.81)

Rewrite Eqs. (7.80)–(7.81) into a matrix form as ˙ J w St = J q q,

(7.82)

 T  q˙ a J wa , q˙ = Jw = , q˙ a = q˙1 · · · q˙n , J wc 0 "T ! J wa = Sˆ wa,1 Sˆ wa,2 · · · Sˆ wa,n , "T ! J wc = Sˆ wc,1 Sˆ wc,2 · · · Sˆ wc,6−n , ⎤ ⎡ T Sˆ wa,1 Sˆ t,1 ··· ··· ··· ··· ⎥ ⎢ T ⎢ ··· Sˆ wa,2 Sˆ t,2 · · · ··· ···⎥ ⎥ ⎢ ⎢ .. .. .. ⎥ .. . Jq = ⎢ . . . . ⎥ ⎥ ⎢ ⎥ ⎢ T ⎣ ··· ··· · · · Sˆ wa,n Sˆ t,n · · · ⎦ ··· ··· ··· ··· 0 

If J w is full rank, Eq. (7.82) can be further written as ˙ = J q, ˙ St = J −1 w Jqq

(7.83)

where J is the Jacobian matrix denoting the mapping between the actuation joint parameters and the velocity of the end-effector.

7.4 Velocity Analysis

229

Equation (7.83) can also be expressed as q˙ =G St , ) *T Sˆ wa,1 Sˆ wa,n Sˆ wa,2 ··· T G= . T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2 Sˆ wa,n Sˆ t,n

(7.84)

7.4.2 Jacobian Matrix of Closed-loop Mechanism Since all the limbs share the same moving platform, the velocity of the moving platform can be computed by the twist of each limb as St,i =

ni 

" ! St,i,k = Sˆ t,i,1 · · · Sˆ t,i,ni q˙ i ,

(7.85)

k=1

where St,i,k (k = 1, . . . , n i ) denotes the velocity of the kth joint in the ith limb. Sˆ t,i,k is its unit instantaneous motion axis with si,k (and r i,k ) as its unit direction vector (and position vector). q˙ i denotes the vector that contains all the angular and linear velocities of all the joints in the limb. Similar to the open-loop mechanism, the wrenches of the ith limb is computed by the reciprocal twist space. The wrench space of the closed-loop mechanism is formed by the actuation wrenches from the actuated joints and the constraint wrenches of the limbs. Suppose there are n i joints in ith limb and gi of them are actuated. The wrench space of the closed-loop mechanism can be expressed as Sw = Swa + Swc =

gi l   i=1 ka=1

f a,i,ka Sˆ wa,i,ka +

l 6−n  i

f c,i,kc Sˆ wc,i,kc ,

(7.86)

i=1 kc=1

where Swa and Swc are the actuation and constraint wrench of the closed-loop mechanism. Sˆ wa,i,ka and Sˆ wc,i,kc stand for the unit actuation wrench screw and unit constraint wrench screw. f a,i,ka and f c,i,kc denote the intensity of the actuation and constraint wrench screws. Taking inner products of the unit wrench screws Sˆ wa,i,ka and Sˆ wc,i,kc to both sides of Eq. (7.85), respectively, we have T T Sˆ wa,i,ka St = qa,i,ka Sˆ wa,i,ka Sˆ t,i,ka , ka = 1, 2 . . . , gi , i = 1, 2 . . . l,

(7.87)

230

7 Kinematic Modeling and Analysis of Robotic Mechanism T Sˆ wc,i,kc St = 0, kc = 1, 2 . . . , 6 − n i .

(7.88)

Rewrite Eqs. (7.87)–(7.88) into a matrix form as ˙ J w St = J q q,  Jw =

(7.89)

  J qa q˙ a J wa , Jq , q˙ = , J wc 0 0

"T ! J wa = Sˆ wa,1,1 · · · Sˆ wa,i,gi · · · Sˆ wa,l,gl , , J wc

⎡

⎤ J wc,1 ⎢ J wc,2 ⎥ ⎢ ⎥ = ⎢ . ⎥, ⎣ .. ⎦ J wc,l

"T

!

J wc,i = Sˆ wc,i,1 Sˆ wc,i,2 · · · Sˆ wc,i,6−ni , ⎤ ⎡ T Sˆ wa,1,1 Sˆ t,1,1 · · · ··· ··· ··· ⎥ ⎢ .. .. .. .. .. ⎥ ⎢ . ⎥ ⎢ . . . . ⎥ ⎢ T ⎥. ˆ ˆ J qa = ⎢ ··· ··· · · · Swa,i,gi St,i,gi · · · ⎥ ⎢ ⎥ ⎢ . . . . .. ⎥ ⎢ .. .. .. . . . ⎦ ⎣ T ··· ··· ··· · · · Sˆ wa,l,gl Sˆ t,l,gl If J w is full rank, Eq. (7.89) can be written as St =

˙ J −1 w Jqq

  q˙ a , St = J a q˙ a , = J q˙ = J a J c 0 

(7.90)

where J is the Jacobian matrix denoting the mapping between the actuation joint parameters and the velocity of the moving platform. Equation (7.90) can also be expressed as q˙ a = G a St ,

(7.91)

where ) Ga =

Sˆ wa,1,1 T Sˆ wa,1,1 Sˆ t,1,1

···

Sˆ wa,l,gl T Sˆ wa,l,gl Sˆ t,l,gl

*T .

7.5 Example

231

7.5 Example 7.5.1 Typical Open-loop Mechanism 7.5.1.1

6R Mechanism

(1) Forward kinematics As shown in Fig. 7.4, all the joints of 6R mechanism are actuations. The kinematic equation is formulated as S f = S f,6  · · · S f,1 ,

(7.92)

where S f denotes the current pose of the end-effector of 6R mechanism, S f, j ( j = 1, . . . , 6) denotes the pose of the open-loop mechanism generated by the jth joint. According to the finite motion description of R joint, the kinematic equation is given as

s6 2 tan r 6 × s6

θ3 s3 2 tan 2 tan 2 r 3 × s3

S f = 2 tan

θ6 2



θ4 s5 s4 2 tan  r 5 × s5 2 r 4 × s4

θ2 θ1 s2 s1 2 tan , 2 r 2 × s2 2 r 1 × s1 θ5 2



(7.93)

herein, θ1 to θ6 are the rotational angles of the actuation joint that are known !  T "T parameters in the forward kinematics. sTj r j × s j is the axis of the ith joint ( j = 1, . . . , 6) at the initial pose, which is also known. The finite screw of the end-effector can thus be computed. Fig. 7.4 6R mechanism

232

7 Kinematic Modeling and Analysis of Robotic Mechanism

(2) Inverse kinematics The current pose of the 6R mechanism is known. It can be expressed in form of finite screw as

θ s 0 S f = 2 tan +t . (7.94) r × s s 2 The current pose is the function of the joint parameters. Hence, the following equations can be formulated referring to Eqs. (7.36)–(7.39) as s = f s (θ1 , . . . , θ6 ), tan

θ = f θ (θ1 , . . . , θ6 ), r = fr (θ1 , . . . , θ6 ), t = f t (θ1 , . . . , θ6 ). 2 (7.95)

The analytical solution of θ1 to θ6 can be obtained through the vector and polynomial analysis. (3) Workspace analysis The forward kinematics is applied to analyze the workspace of the 6R mechanism. Rotational ranges of the actuation joints are the main constraints that determine the boundary of the orientation workspace. The reachable workspace can be computed by 

Sf

r

     = S f  θ j ∈ θ Lj , θ Uj , j = 1, . . . , 6 ,

(7.96)

where θ Lj , θ Uj are the lower and upper rotational limit of the jth R joint. The orientation workspace is expressed as the set of the Gibson form referring to Eq. (7.59) as 

  r g M = T SG S f,M .

(7.97)

Now consider the position workspace. The position vector for the center of endeffector at initial pose is known as r 0 . The distance vector d j generated by the jth joint is computed as    d j = exp θ j s˜ j − E 3 r j−1 − r j , j = 1, . . . , 6.

(7.98)

Hence, the position of the end-effector generated by all the joints is expressed as Eq. (7.67)

7.5 Example

233

d=

6 

d j.

(7.99)

j=1

d is regarded as the function of actuation joint parameters. Hence, we have d = f (θ1 , . . . , θ6 ).

(7.100)

The position workspace can be computed by      {d} = f (θ1 , . . . , θ6 )θ j ∈ θ Lj , θ Uj , j = 1, . . . , 6 .

(7.101)

(4) Jacobian matrix between actuation joints and the velocity of end-effector The velocity model of the 6R open-loop mechanism is computed referring to Eq. (7.78) as St =

6 

θ˙ j Sˆ t, j =

j=1

6 

θ˙ j



j=1

sj , r j × sj

(7.102)

where St is the twist of end-effector. Sˆ t, j and θ˙ j are the unit twist screw and intensity of each actuation joint. The actuation wrench of the joints constitutes the wrench space of the 6R mechanism. The wrench is expressed according to Eq. (7.79) as Sw =

6 

f a, ja Sˆ wa, ja ,

(7.103)

ja=1

where Sˆ wa, ja is the unit actuation wrench screw and f a, ja is the intensity. Taking inner product of the actuation wrenches and the twist of the 6R mechanism results in St = J θ˙ or θ˙ = G St ,

(7.104)

where J is the Jacobian matrix of 6R mechanism between the actuation joints and the velocity of the end-effector, and ) G = J −1 =

Sˆ wa,1

Sˆ wa,2

T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2

···

Sˆ wa,6 T Sˆ wa,6 Sˆ t,6

*T .

(7.105)

234

7 Kinematic Modeling and Analysis of Robotic Mechanism

Fig. 7.5 SPR mechanism

7.5.1.2

SPR Mechanism

(1) Forward kinematics The SPR mechanism is realized by the R1 R2 R3 P4 R5 mechanism whose joints are all actuated as shown in Fig. 7.5. The forward kinematics of the R1 R2 R3 P4 R5 mechanism is computed by the topology model as S f = S f,5  · · · S f,1 ,

(7.106)

where S f denotes the current pose of the end-effector of SPR mechanism, S f, j ( j = 1, . . . , 5) denotes the pose of the open-loop mechanism generated by the jth joint. According to the finite motion description for R and P joints, the forward kinematics of SPR mechanism is formulated as



θ3 θ5 0 s5 s3 t4 2 tan  S f = 2 tan s4 2 r 5 × s5 2 r 3 × s3

θ2 θ1 s2 s1 2 tan 2 tan (7.107) 2 r 2 × s2 2 r 1 × s1 where θ1 , θ2 , and θ3 are the rotational angle of S joint, i.e., rotational angles of the first equivalent R joints, with respect to the initial states. t4 is the translational distance !  T "T of P joint. θ5 is the rotational angle of the last R joint. sTj r j × s j is the axis  T T of the jth ( j = 1, 2, 3, 5) R joint. 0 s4 is the direction of the P joint.

7.5 Example

235

The axis and direction of the R and P joints at the initial pose are known. The rotational angles and translation distance are also known in forward kinematics. Hence, the kinematic equation in Eq. (7.107) can be solved. (2) Inverse kinematics The current pose of the SPR mechanism is known. It can be expressed in form of finite screw as

θ s 0 S f = 2 tan +t . (7.108) s 2 r×s Equation (7.108) can be denoted as the function of the joint parameters. According to Eqs. (7.36)–(7.39), the following equations are obtained. θ = f θ (θ1 , θ2 , θ3 , θ5 ), 2

(7.109)

r = fr (θ1 , θ2 , θ3 , t4 , θ5 ), t = f t (θ1 , θ2 , θ3 , t4 , θ5 ).

(7.110)

s = f s (θ1 , θ2 , θ3 , θ5 ), tan

The analytical solution of θ1 to θ5 can be obtained through the vector and polynomial analysis in Eqs. (7.109) and (7.110). (3) Workspace analysis Rotational angles of the R joints and traveling distance of the P joint are the boundary determining the workspace of the SPR mechanism. The reachable workspace is computed by 

Sf

r

 =

   t4 ∈ t L , t U  4 4    , Sf  θ j ∈ θ Lj , θ Uj , j = 1, 2, 3, 5

(7.111)

where θ Lj and θ Uj are the lower and upper rotational limit of the jth R joint. t4L and t4U are the lower and upper limits for translational distances of the P joint. The orientation workspace is obtained by the set of Gibson form as 

  r g M = T SG S f .

(7.112)

The distance vector d j generated by the jth R joint is referred to Eq. (7.61) as    d j = exp θ j s˜ j − E 3 r j−1 − r j , j = 1, 2, 3, 5.

(7.113)

The distance vector d 4 generated by the P joint is computed as d 4 = t4 s 4 .

(7.114)

236

7 Kinematic Modeling and Analysis of Robotic Mechanism

Therefore, the position of the end-effector generated by all the joints is expressed as d=

5 

d j.

(7.115)

j=1

The position workspace is finally computed by  {d} =

   t4 ∈ t L , t U  4 4    f (θ1 , θ2 , θ3 , t4 , θ5 ) .  θ j ∈ θ Lj , θ Uj , j = 1, 2, 3, 5

(7.116)

(4) Jacobian matrix between actuation joints and the velocity of end-effector The velocity model of the SPR mechanism is expressed according to Eq. (7.78) as St = Sˆ t, j = θ˙ j



5 

q˙i Sˆ t,i ,

(7.117)

i=1

sj 0 ˆ ˙ , St,4 = t4 , q˙ j = θ˙ j , j = 1, 2, 3, 5, q˙4 = t˙4 , r j × sj s4

where St is the twist of end-effector of SPR mechanism, Sˆ t,i and qi stand for the unit twist screw and intensity of each actuation joint. There are five actuation wrenches corresponding to the actuation joints. According to the reciprocal property of twist and wrench, there is one constraint wrench reciprocal to all the twists within the SPR mechanism. Hence, the wrench space is expressed as Sw =

5 

f a, ja Sˆ wa, ja + f c Sˆ wc ,

(7.118)

ja=1

where Sˆ wa, ja and f a, ja stand for the unit actuation wrench screw and its intensity. Sˆ wc and f c are the unit constraint wrench and its intensity. Taking Sˆ wa, ja and Sˆ wc to both sides of Eq. (7.117) results in St = J q˙ or q˙ = G St ,

(7.119)

where J is the Jacobian matrix denoting the mapping between the actuation joint parameters and the velocity of end-effector, and ) G=

Sˆ wa,1

Sˆ wa,2

T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2

···

Sˆ wa,5 T Sˆ wa,5 Sˆ t,5

*T .

7.5 Example

237

Fig. 7.6 Schematic diagram of Exechon mechanism

7.5.2 Typical Closed-loop Mechanism (1) Forward kinematics As shown in Fig. 7.6, the topology of the Exechon mechanism is 2UPR-SPR. The UPR limbs are named as the 1st and the 3rd limbs, and the SPR limb is named as the 2nd limb. The kinematic equations of the ith limb is expressed as



θi,2 θi,4 0 si,4 si,2 ti,3 2 tan  si,3 2 r i,4 × si,4 2 r i,2 × si,2

θi,1 si,1 2 tan , i = 1, 3, (7.120) 2 r i,1 × si,1



θi,3 θi,5 0 si,5 si,3 ti,4 2 tan  =2 tan si,4 2 r i,5 × si,5 2 r i,3 × si,3

θi,2 θi,1 si,2 si,1 2 tan 2 tan , i = 2, (7.121) 2 r i,2 × si,2 2 r i,1 × si,1

S f,i = 2 tan

S f,i

where S f,i denotes the current pose of the end-effector of the ith limb. θi,k (k = 1, 2, 4 when i = 1, 3, k = 1, 2, 3, 5 when i = 2) is the rotational angle of the kth joint !  T "T T is the axis of the kth joint in the ith limb. in the ith limb. si,k r i,k × si,k  T T ti,3 (i = 1, 3) and ti,4 (i = 2) are the translational distances. 0 si,3 (i = 1, 3) and  T T 0 si,4 (i = 2) are the unit screws of the P joints.

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7 Kinematic Modeling and Analysis of Robotic Mechanism

Since all the limbs share the same moving platform, the kinematic equation of Exechon mechanism can be formulated as S f = S f,1 = S f,2 = S f,3 .

(7.122)

It can be written into simultaneous equations. There are three passive joints in the UPR limb and four passive joints in SPR limb. In total, there are 10 unknown parameters. The number of scalar equations resulted from Eq. (7.122) is 12. Therefore, the unknown joint parameters can be computed by solving the equations. Finally, the finite motion of the Exechon mechanism is obtained by Eq. (7.120) or Eq. (7.121). (2) Inverse kinematics The UPR limbs and the SPR limb share the same moving platform. The inverse kinematics problem of the Exechon mechanism can be divided into three inverse kinematics problems of limbs. The kinematic equation of the ith limb can be written as  S f,i = f θi,1 , θi,2 , ti,3 , θi,4 , i = 1, 3,

(7.123)

 S f,i = f θi,1 , θi,2 , θi,3 , ti,4 , θi,5 , i = 2.

(7.124)

The current pose of the moving platform is known in the inverse kinematics problem. The finite motion of the moving platform from the initial pose to the current pose can be expressed by the finite screw as S f,i

θi = 2 tan 2



si r i × si

+ ti

0 . si

(7.125)

Substituting Eq. (7.123) into Eq. (7.125) yields   θi si = f s θi,1 , θi,2 , θi,4 , tan = f θ θi,1 , θi,2 , θi,4 , 2   r i = fr θi,1 , θi,2 , ti,3 , θi,4 , ti = f t θi,1 , θi,2 , ti,3 , θi,4 .

(7.126) (7.127)

Similarly, substituting Eq. (7.124) into Eq. (7.125) leads to   θi si = f s θi,1 , θi,2 , θi,3 , θi,5 , tan = f θ θi,1 , θi,2 , θi,3 , θi,5 , 2   r i = fr θi,1 , θi,2 , θi,3 , ti,4 , θi,5 , ti = f t θi,1 , θi,2 , θi,3 , ti,4 , θi,5 .

(7.128) (7.129)

Solve the Eqs. (7.126)–(7.129) through vector and polynomial analysis. The analytical solutions of all joint parameters are finally obtained.

7.5 Example

239

(3) Workspace analysis The position workspace of the Exechon mechanism is considered. The position vector of the center of moving platform before the movement of the kth R joint in the ith limb is known as r 0 . The distance vector generated by the kth R joint in the kth limb is expressed as    d i.k = exp θi,k s˜ i,k − E 3 r k−1 − r i,k .

(7.130)

Similarly, the distance vector generated by the P joint in the ith limb is given as d i,3 = ti,3 si,3 , i = 1, 3 or d i,4 = ti,4 si,4 , i = 2.

(7.131)

The position of the ith limb generated by all the joints is computed as di =

4 

d i,k , i = 1, 3 or d i =

k=1

5 

d i,k , i = 2.

(7.132)

k=1

d i can be regarded as a function of joint parameters. Hence, the position workspace of the ith limb can be obtained as     L U , ti,3   ti,3 ∈ ti,3 {d i } = f θi,1 , θi,2 , ti,3 , θi,4  , i = 1, 3, (7.133)  L U  θi,k ∈ θi,k , θi,k , k = 1, 2, 4     L U , ti,4   ti,4 ∈ ti,4 {d i } = f θi,1 , θi,2 , θi,3 , ti,4 , θi,5  , i = 2,  L U  θi,k ∈ θi,k , θi,k , k = 1, 2, 3, 5 (7.134) L where θi,k and θ U are the lower and upper rotational limit of the kth R joint in the ith  L i,k U  U L limb. ti,3 ti,4 and ti.3 ti,4 are the lower and upper translational limits of the P joints. Finally, the position workspace of the Exechon mechanism is computed by the intersection of the position space of each limb as

{d} = {d 1 } ∩ {d 2 } ∩ {d 3 }.

(7.135)

(4) Jacobian matrix between actuation joints and the velocity of moving platform The velocity of the moving platform at current pose can be computed by the twist of each limb as St,i =

4  k=1

q˙i,k Sˆ t,i,k , i = 1, 3,

(7.136)

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7 Kinematic Modeling and Analysis of Robotic Mechanism

St,i =

5 

q˙i,k Sˆ t,i,k , i = 2,

(7.137)

k=1

where Sˆ t,i,k is the unit instantaneous motion axis of the kth joint in the ith limb with si,k (and r i,k ) as its unit direction vector (and position vector). q˙i,k is its intensity. The P joint in each limb is selected as the actuated joint. For the UPR limb, there is one actuation wrench reciprocal to the R joints in the limb, and there are two constraint wrenches reciprocal to all the joints. Hence, the wrench of the UPR limbs is expressed as Sw,i

= f a,i,3 Sˆ wa,i,3 +

2 

f c,i,kc Sˆ wc,i,kc , i = 1, 3,

(7.138)

kc=1

where Sˆ wa,i,3 and Sˆ wc,i,kc stand for the unit actuation wrench of the actuated joint and constraint wrench of the UPR limb. f a,i,3 and f c,i,kc are the intensity of actuation and constraint wrenches, and Sˆ wa,i,3 =





r i,3 × si,3 r i,2 × si,2 si,2 × si,1 ˆ ˆ , Swc,i,1 = , Swc,i,2 = . si,3 si,2 0

For the SPR limb, there is one actuation wrench reciprocal to the R joints in the limb, and there is one constraint wrench reciprocal to all the joints. Hence, the wrench of the SPR limb is expressed as Sw,i = f a,i,4 Sˆ wa,i,4 + f c,i,1 Sˆ wc,i,1 , i = 2,

(7.139)

where Sˆ wa,i,4 and Sˆ wc,i,1 stand for the unit actuation wrench and constraint wrench of the SPR limb. f a,i,4 and f c,i,1 are the intensity of actuation and constraint wrenches, and

r i,4 × si,4 r i,1 × si,1 , Sˆ wc,i,1 = . Sˆ wa,i,4 = si,4 si,1 Noting that there are five constraint wrenches offered by the limbs, among which three wrenches are independent, indicating that the Exechon mechanism is overconstrained. Take inner product of the unit actuation wrench and independent constraint wrench to both side of the twist of the ith limb, and write into matrix form, resulting in ˙ J w St = J q q,    T  q˙ a J wa J qa , Jq = , q˙ a = t˙1,3 t˙2,4 t˙3,3 , Jw = , q˙ = J wc 0 03×1

(7.140)

7.5 Example

241

"T "T ! ! J wa = Sˆ wa,1,3 Sˆ wa,2,4 Sˆ wa,3,3 , J wc = Sˆ wc,1,1 Sˆ wc,2,1 Sˆ wc,3,1 , ⎡ T ⎤ Sˆ wa,1,3 Sˆ t,1,3 ⎢ ⎥ T ⎢ ⎥ Sˆ wa,2,4 Sˆ t,2,4 ⎢ ⎥. J q =⎢ T ⎥ ⎣ Sˆ wa,3,3 Sˆ t,3,3 ⎦ 0

7.6 Conclusion Kinematics of the robotic mechanism concerning the input and output motions are addressed. The forward and inverse kinematics are solved based on the finite screw descriptions. On this basis, the workspace of the robotic mechanism is analyzed, which is divided into sub-three-dimensional orientation and position workspace. Velocity model of the robotic mechanism is formulated and the Jacobian matrix denoting the velocity mapping between actuation inputs and velocity output is especially concerned. For the reader’s convenience, the key points of this chapter are represented as follows: (1) Through building the mapping between the actuated joints and the finite motion of open-loop mechanism or closed-loop mechanism, forward kinematics and inverse kinematics are solved in an analytical manner; (2) Using the kinematic equation, sub-six-dimensional workspace is defined as the set of finite screws. On this basis, the sub-three-dimensional orientation and position spaces are defined as set of Gibson forms and set of distance vectors; (3) Based on the differential map between finite and instantaneous screw, the velocity model of the robotic mechanism is formulated. The difference of the instantaneous motion model in this chapter and the previous chapter is that the Jacobian matrix between input and output instantaneous motions are concerned; (4) Typical examples of open-loop mechanisms (6R, SPR) and closed-loop mechanism (Exechon) are given to demonstrate the procedure of kinematic modeling and analysis. The presented kinematic modeling and analysis method is a generic one which reveals the motion transmissibility from the input to the output. It is the foundation for the following performance analysis.

242

7 Kinematic Modeling and Analysis of Robotic Mechanism

References 1. Gao Z, Zhang D, Ge Y (2010) Design optimization of a spatial six degree-of-freedom parallel manipulator based on artificial intelligence approaches. Robot Comput Integr Manuf 26:180– 189 2. Merlet JP (2006) Parallel robots, 2nd edn. Springer, Netherlands 3. Zhang D (2010) Parallel robotic machine tool. Springer, New York 4. Liu XJ, Wang JS (2014) Parallel kinematics. Springer, Berlin, Heidelberg 5. Angeles J (2014) Fundamentals of robotic mechanical systems: theory, methods, and algorithms, 4th edn. Springer, New York 6. Paul RP, Stevenson CN (1983) Kinematics of robot wrists. Int J Robot Res 2:31–38 7. Song YM, Gao H, Sun T et al (2014) Kinematic analysis and optimal design of a novel 1T3R parallel manipulator with an articulated travelling plate. Robot Comput Integr Manuf 30(5):508–551 8. Huo XM, Sun T, Song YM (2017) A geometric algebra approach to determine motion/constraint, mobility and singularity of parallel mechanism. Mech Mach Theory 116:273–293 9. Song YM, Lian BB, Sun T et al (2014) A novel five-degree-of-freedom parallel manipulator and its kinematic optimization. ASME Trans J Mech Robot 6(4):410081–410089 10. Sun T, Song YM, Dong G et al (2012) Optimal design of a parallel mechanism with three rotational degrees of freedom. Robot Comput Integr Manuf 28(4):500–508 11. Sun T, Yang SF, Huang T, Dai JS (2018) A finite and instantaneous screw based approach for topology design and kinematic analysis of 5-axis parallel kinematic machines. Chin J Mech Eng 31:44 12. Angeles J, Park FC (2008) Performance evaluation and design criteria. In: Siciliano B, Khatib O (eds) Springer handbook of robotics. Springer, Berlin, Heidelberg 13. Sun T, Song YM, Li YG et al (2010) Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. ASME Trans J Mech Robot 2(3):310091–310098 14. Mccarthy JM, Soh GS (2011) Geometric design of linkages. Springer, New York 15. Xie FG, Liu XJ, Wang JS (2012) A 3-DOF parallel manufacturing module and its kinematic optimization. Robot Comput Integr Manuf 28:334–343 16. Tsai MJ, Lee HW (1994) Generalized evaluation for the transmission performance of mechanisms. Mech Mach Theory 29:607–618 17. Sun T (2012) Performance evaluation index framework of lower mobility parallel manipulators. Dissertation, Tianjin University 18. Ball RS (1990) A treatise on the theory of screws. Cambridge University Press, London 19. Gosselin CM, Angeles J (1991) A global performance index for the kinematic optimization of robotic manipulators. J Mech Des 113:220–226

Chapter 8

Static Modeling and Analysis of Robotic Mechanism

8.1 Introduction Stiffness is a measure of the resistance offered by an elastic body to deformation in response to an applied force [1–4]. The static stiffness of the robotic mechanism denotes the elastic deformation when the robot remains stationary and is under static wrenches [5, 6]. In robotic applications such as assembling large components in aviation or aerospace, the assembling motion is relatively slow. External wrenches from the assembled components are applied to the robot. In such quasi-static or static application scenarios, static stiffness becomes an essential criterion in evaluating the performance capability of the robot [7–11]. Hence, stiffness modeling is of significance to the comprehensive analysis of the robotic mechanism. Stiffness modeling is the process of obtaining the stiffness of the whole mechanism from the stiffness or compliance of the components or limbs. No matter the obtained stiffness model is for performance analysis of a known robotic mechanism or is being adopted for constructing design objectives of an unknown robotic mechanism, the stiffness model is required to (1) be able to compute mechanism stiffness at any configuration within workspace and (2) contain the complete and accurate deformation of the whole mechanism. The existing stiffness modeling methods are either numerical or analytical. The numerical method [12–15] based on finite element analysis (FEA) software can precisely describe mechanism stiffness but is applicable at one configuration at a time, thus is computationally expensive. The analytical method [16–20] is efficient in analyzing mechanism stiffness within workspace. However, it adopts a standard element to resemble the mechanism, affecting the accuracy of the stiffness description. Combining the merits of the numerical and analytical methods, a semi-analytical stiffness modeling method is proposed in this chapter. The complete deformation of components is taken into account and the overall stiffness performance within workspace can be comprehensively analyzed.

© Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_8

243

244

8 Static Modeling and Analysis of Robotic Mechanism

Based on the kinematic analysis of the robotic mechanism in Chap. 7, the twist and wrench analysis of the open-loop mechanism and closed-loop mechanism are carried out, in which the deformation of the elastic body is included. Then, the m-DoF virtual spring is defined to represent the elastic features of the components in robotic mechanisms. On this basis, the stiffness modeling method of open-loop mechanism and closed-loop mechanism are presented with the aid of virtual work principle. Finally, the 6R, SPR open-loop mechanism, and Exechon closed-loop mechanism are taken as examples to illustrate stiffness modeling process.

8.2 Twist and Wrench Analysis The twist and wrench mapping models in Chap. 7 focus on the relations of rigid body motions and actuations/constraint forces of the robotic mechanisms. Since the stiffness of the robotic mechanism is considered in this chapter, all the components are regarded as elastic bodies. The relation of deflection twist and actuation/constraint wrench of the robotic mechanisms is required to be analyzed for formulating the stiffness models.

8.2.1 Open-loop Mechanism For an open-loop mechanism, the static stiffness is obtained from the external wrench and the deformation twist of the end-effector when the actuated joints are locked. The twist analysis of the open-loop mechanism is firstly analyzed. When the actuated joints are locked, the twist of the end-effector under the external wrenches is contributed by the deformation twist of all the joints and components as St = Ste ,

(8.1)

where Ste denotes the deformation twist of the open-loop mechanism. The deformation of the end-effector is contributed by the serially connected elastic components. Suppose there are N elastic components in the open-loop mechanism. The deformation twist can be formulated by the deformation twist of components as Ste =

N 

Ste, j ,

j=1

where Ste, j is the deformation twist of the jth elastic component. Hence, the twist of the open-loop mechanism can be expressed as

(8.2)

8.2 Twist and Wrench Analysis

245 N 

St =

Ste, j .

(8.3)

j=1

Assume that the open-loop mechanism has n DoFs. Referring to Chap. 7, the wrench model of the n-DoF open-loop mechanism is given by Sw = Swa + Swc =

n 

f a, ja Sˆ wa, ja +

ja=1

6−n 

f c, jc Sˆ wc, jc ,

(8.4)

jc=1

T  where Sw = τ TE f TE , f E , and τ E represent external wrench force and moment. Sˆ wa, ja and f a, ja stand for the unit actuation wrench and the corresponding intensity. Sˆ wc, jc and f c, jc denote the unit constraint wrench and the intensity. Equation (8.4) can be further written as  f   a , Sw = Sˆ wa,1 · · · Sˆ wa,n Sˆ wc,1 · · · Sˆ wc,6−n fc

(8.5)

where f a and f c denote the reaction forces from the end-effector, and fa =



f a,1 · · · f a,n

T

, fc =



f c,1 · · · f c,6−n

T

.

Referring to Eq. (8.1), it is found that the external wrench does work on the Ste . The following equations are obtained. T T Sˆ wa St = Sˆ wa Ste = ρ ta ,

(8.6)

T T Sˆ wc St = Sˆ wc Ste = ρ tc ,

(8.7)

where ρ ta and ρ tc denote the deformation resulted from f a and f c , respectively. Equations (8.6)–(8.7) can also be written into matrix form as T T Sˆ w St = Sˆ w Ste = ρ t .

(8.8)

8.2.2 Closed-loop Mechanism An n-DoF closed-loop mechanism with l limbs is considered. Since all the limbs share the same moving platform, the twist of the moving platform is the same as the twist of each limb. The components within each limb are considered as elastic

246

8 Static Modeling and Analysis of Robotic Mechanism

bodies. When the actuation joints are locked, the deformation twist of the moving platform can be formulated as St,i , i = 1, . . . , l, St = St,i = Ste,i + −

(8.9)

St,i is the twist of the ith where Ste,i denote the deformation twist of the ith limb. − limb after the actuated joints are locked. Referring to Eq. (8.2), the deformation twist of elastic components of the ith limb is given as Ste,i =

Ni 

Ste,i,k ,

(8.10)

k=1

where Ni denotes the number of elastic bodies in the ith limb. Similarly, the − St,i is formulated by the linear combination of the unit twist of the passive joints as n i −gi

− St,i =



δqi,k − Sˆ t,i,k ,

(8.11)

k=1

where n i and gi are the total number of joints and the number of actuation joints in the ith limb. − Sˆ t,i,k denotes the unit twist of the kth passive joint. δqi,k is the corresponding amplitude. Hence, the deformation twist of the ith limb is given as St,i =

Ni  k=1

n i −gi

Ste,i,k +



δqi,k − Sˆ t,i,k .

(8.12)

k=1

From the free body analysis on the moving platform, the external wrench of the closed-loop mechanism is the sum of the wrenches provided by each limb. The limb wrench can be divided into actuation and constraint wrenches. Hence, the wrench model of the closed-loop mechanism is expressed as Sw =

l  i=1

Sw,i =

g l i   i=1

ka=1

f a,i,ka Sˆ wa,i,ka +

6−n i

f c,i,kc Sˆ wc,i,kc ,

(8.13)

kc=1

 T where Sw = τ TE f TE , τ E and f E represent external moment and external force. Sˆ wa,i,ka and f a,i,ka denote the kath actuation wrench and its intensity of the ith limb. Sˆ wc,i,kc and f c,i,kc are the kcth constraint wrench and intensity of the ith limb. It is known from Chap. 7 that the limb wrench of the closed-loop mechanism depends

8.2 Twist and Wrench Analysis

247

on the topology model of the mechanism and the actuation methods. For the nonredundantly actuated and non-over-constrained closed-loop mechanism, the actuation and constraint wrenches form the base of the wrench space. Hence, the wrenches are usually written into matrix form and applied to the formulation of twist mapping model, as has been discussed in Chap. 7. For the redundantly actuated or the over-constraint closed-loop mechanism, the linear independent wrenches are usually selected to form the matrix. However, the elastic features of the limbs under all the wrenches are considered in the stiffness analysis. Therefore, the redundant actuation wrench or constraint wrench should also be included. According to the reciprocal property between twist and wrench, the actuation and constraint wrenches do not do work on the twist of passive joints but do work on the deformation twists of components. For the ith limb (i = 1, 2, . . . , l), the following relations are obtained. St,i = 0, STwc,i − St,i = 0, STwa,i −

(8.14)

T T T Sˆ wa,i St = Sˆ wa,i St,i = Sˆ wa,i Ste,i = ρ ta,i ,

(8.15)

T T T Sˆ wc,i St = Sˆ wc,i St,i = Sˆ wc,i Ste,i = ρ tc,i .

(8.16)

where ρ ta,i and ρ tc,i denote the deformation resulted from f a,i and f c,i in the ith limb.

8.3 Stiffness Modeling With the twist and wrench mapping model available at hand, the stiffness model of the robotic mechanism can be formulated from component to limb and finally the whole mechanism by Hooke’s law and virtual work principle. Hence, the stiffness modeling process is divided into two steps: (1) from the component to the limb, formulate the deformation twist of the open-loop mechanism or the deformation twist of the ith limb of closed-loop mechanism and (2) from the limb to the mechanism, apply Hooke’s law and virtual work principle to obtain the stiffness model of the whole mechanism. For the convenience of formulating the deformation twist, the m-DoF virtual spring is proposed for representing the elastic features of components and joints.

8.3.1 m-DoF Virtual Spring It has been widely recognized that the elastic body can be regarded as a rigid body and a virtual spring [11, 12]. The equivalent rigid body keeps the shape and size while the virtual spring denotes the elastic features of the elastic body. When an external wrench

248

8 Static Modeling and Analysis of Robotic Mechanism

is applied to the elastic body, the equivalent rigid body has rigid body motion and the elastic deformation is generated by the virtual spring. Assume that each component of the open-loop mechanism or closed-loop mechanism is represented by a virtual spring. The open-loop mechanism or limb of closed-loop mechanism is replaced by the serially connected rigid components and virtual springs. The formulation of deformation twist relies on the virtual springs. According to Hooke’s law, the stiffness of the virtual spring is computed by the division of external wrench and the resulted deformation twist when the equivalent rigid body is subjected to certain constraints. The inverse of stiffness is the compliance of the virtual spring. Described in R3 space, the dimension of the stiffness/compliance matrix should be 6 × 6 since the dimension of wrench is 6. The stiffness/compliance matrix of a virtual spring is referred to the material mechanics and obtained either by FEA software or analytical computation. Taking a cantilever beam as an example, a body fixed frame is attached to the end with fixed constraints as in Fig. 8.1. The compliance matrix of the cantilever beam in frame O − x yz is computed by the deformation twist under the applied wrench as follows. ⎡ ⎤ δϕx (τx ) δϕx (τ y ) δϕx ( f z ) ··· ⎢ τx τy fz ⎥ ⎢ ⎥ ⎢ δϕ y (τx ) δϕ y (τ y ) δϕ y ( f z ) ⎥ ⎢ ⎥ ··· St ⎢ τy fz ⎥ = ⎢ τx (8.17) C= ⎥ , ⎢ ⎥ Sw .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ δpz (τx ) δpz (τ y ) δpz ( f z ) ⎦ ··· τx τy fz 6×6  T where Sw = τx τ y τz f x f y f z . Each column corresponds to the six-dimensional deformation under the applied wrench. Take the first column as an example, the applied wrench is the torque about x-direction τx . Hence, δϕx (τx ), δϕ y (τx ), and δϕz (τx ) represent the angular deformation about the x-axis, y-axis, and z-axis under τx . δpx (τx ), δp y (τx ) and δpz (τx ) denote the linear deformation along the x-axis, y-axis, and z-axis under τx , respectively. The analytical method is to compute the deformations based on material mechanics. According to the material property and the geometric dimensions, the compliance matrix of the cantilever beam can be formulated analytically as

Fig. 8.1 A cantilever beam and its equivalent elastic model

8.3 Stiffness Modeling

249

⎤

⎡

L ⎢ G Ip ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ L L ⎥ ⎥ − E Iy 2E I y ⎥ ⎥ ⎥ L L2 ⎥ ⎥ E Iz 2E Iz ⎥, ⎥ L ⎥ ⎥ EA ⎥ 2 3 ⎥ L L ⎥ ⎥ 2E Iz 3E Iz ⎥ 2 3 ⎦ L L − 2E I y 3E I y 2

(8.18)

where L is the link length. A is the cross-sectional area. I y and Iz are the crosssectional motion of inertia. I p is the polar motion of inertia. E and G are the Young’s and Coulomb’s modules. The numerical method is applying the unit wrench to the output point, i.e., the free end of the beam, and directly measuring the deformations. As the applied force or moment is unit, the compliance matrix can be constructed by the measured deformation as ⎤ ⎡ c11 c12 · · · c16 ⎢ c21 c22 · · · c26 ⎥ ⎥ ⎢ (8.19) C=⎢ . . . ⎥, ⎣ .. .. · · · .. ⎦ c61 c62 · · · c66 where ci, j (i = 1, . . . , 6, j = 1, . . . , 6) denotes corresponding element in Eq. (8.17) when the applied force or torque is unit. The inverse of the compliance matrix is the stiffness matrix. The 6 × 6 stiffness matrix indicates that the virtual spring of the cantilever beam has six DoFs. However, there might be redundant DoF in the virtual spring if the cantilever beam is connected with a passive joint in the open-loop mechanism or the limb of closedloop mechanism. For instance, if the cantilever beam links to the other components by an R joint whose rotational axis is along the z-axis, the angular stiffness about the z-axis of the virtual spring are compensated by the R joint. Considering the effects of the connected passive joints, the virtual spring of components should be m-DoF (1 ≤ m ≤ 6), where m = 6 − f p and f p is the DoF of the connected passive joint.

8.3.2 Open-loop Mechanism Without loss of generality, an open-loop mechanism with N elastic components is considered. The open-loop mechanism can be described by the m-DoF virtual springs as shown in Fig. 8.2. The compliance matrix of the m-DoF virtual spring in the component reference frame has been discussed. From the Hooke’s law, the jth component has the following relation,

250

8 Static Modeling and Analysis of Robotic Mechanism

Fig. 8.2 The elastic model of the open-loop mechanism

Ste, p, j = C p, j Sw, p, j ,

(8.20)

where Ste, p, j is the resulted deformation twist of the jth component in the component reference frame. C p, j is the compliance matrix of the virtual spring in the component reference frame. Sw, p, j is the wrench acting on the jth component and described in the component reference frame. Referring to Eq. (8.3), the deformation twist of each component is accumulated to the deformation twist of the open-loop mechanism. The jth component deformation twist should be transferred to the fixed reference frame as Ste, j = T j Ste, p, j , j = 1, 2, . . . , N ,

(8.21)

where T j is the adjoint transformation matrix, and  Tj =

 Rj 0 , r˜ j R j R j

(8.22)

herein R j is the rotation matrix of component reference frame with respect to the fixed reference frame. r j is the position vector of the origin of component reference frame and r˜ j denotes its skew-symmetric matrix. Applying the adjoint transformation matrix, the wrench of component follows as Sw, p, j = T Tj Sw, j .

(8.23)

According to virtual work principle, we have STw, j Ste, j = STw, p, j Ste, p, j .

(8.24)

Substitute Eqs. (8.20)–(8.23) into Eq. (8.24), the compliance matrix of component described in the fixed reference frame can be expressed as Ste, j = C j Sw, j , C j = T j C p, j T Tj .

(8.25)

T T Multiply Sˆ wa, j and Sˆ wc, j on the both sides of Eq. (8.25), respectively, leading to

8.3 Stiffness Modeling

251 T T Sˆ wa, j C j Swa, j = Sˆ wa, j Ste, j = ρ ta ,

(8.26)

T T Sˆ wc, j C j Swc, j = Sˆ wc, j Ste, j = ρ tc .

(8.27)

Equations (8.26)–(8.27) can be further written as T T Sˆ w, j C j Sˆ w, j f j = Sˆ w, j Ste, j = ρ t ,

(8.28)

where       Sˆ w, j = Sˆ wa, j Sˆ wc, j , f j = f a, j f c, j , ρ t = ρ ta ρ tc . Considering all the virtual springs in the open-loop mechanism, Eq. (8.28) turns into T T Sˆ w C L Sˆ w f = Sˆ w Ste = ρ t ,

(8.29)

 where C L = Nj=1 C j . From Hooke’s law, Eq. (8.29) can be written as −1  T ρt . f = Sˆ w C L Sˆ w

(8.30)

The virtual work principle is formulated as STt Sw = ρ Tt f .

(8.31)

Applying the twist and wrench mapping model of the open-loop mechanism, Eq. (8.31) can be expressed as  T −1 T Sˆ w Ste . Sw = Sˆ w Sˆ w C L Sˆ w

(8.32)

The stiffness or compliance model of the components in the open-loop mechanism is directly obtained from Eq. (8.32) as  T −1 T Sˆ w . K L = Sˆ w Sˆ w C L Sˆ w

(8.33)

The elastic deformation of the fixed based and the end-effector can be considered by the deformation superposition principle. The stiffness model of the open-loop mechanism is finally obtained as  −1 , K = C −1 = C B + K −1 L + CP

(8.34)

252

8 Static Modeling and Analysis of Robotic Mechanism

where C B = T B C B T TB , C P = T P C P T TP . C B , C P are the compliance matrices of the fixed base and the end-effector in the component reference frame. T B and T P are the adjoint transformation matrices,  TB =

   E3 0 R 0 ,TP = , r˜ O E 3 r˜ R R

(8.35)

herein r O is the position vector of the center of the fixed base. R is the rotation matrix of moving reference frame with respect to the fixed reference frame. r is the position vector of the origin of moving reference frame.

8.3.3 Closed-loop Mechanism The elastic model of the closed-loop mechanism is as shown in Fig. 8.3. The stiffness modeling of the closed-loop mechanism can be divided into three levels: component, limb, and mechanism. The free body analysis of the moving platform indicates that the external wrench equals the actuation wrench and constraint wrench provided by the limbs. From the action and reaction force principle, the wrench applied to each limb is the actuation wrench and/or constraint wrench of the limb, which is similar to the applied wrench of open-loop mechanism. Therefore, from the components to the limb, the stiffness modeling process is the same as that of open-loop mechanism. From the limb to the mechanism, the virtual work equation of the moving platform is formulated and the stiffness model of the closed-loop mechanism can be obtained. The compliance matrices of the components in the component reference frames are obtained either by FEA software or analytical computation. They are converted to the compliance matrices in the fixed reference frame and added to formulate the limb

Fig. 8.3 The elastic model of the closed-loop mechanism

8.3 Stiffness Modeling

253

compliance model. Therefore, the compliance model of the ith limb is expressed as C L ,i =

Ni 

T T i,k C i,k T i,k , i = 1, 2, . . . , l,

(8.36)

k=1

where C i,k represents the compliance matrix of kth component in the ith limb. T i,k is the adjoint transformation matrix, and  T i,k =

 0 Ri,k , r˜ i,k Ri,k Ri,k

(8.37)

herein Ri,k is the rotation matrix of component reference frame with respect to the fixed reference frame. r i,k is the position vector of the origin of component reference frame. Referring to the stiffness modeling of the open-loop mechanism, the deformation twist of the ith limb and the wrench has the following relationship,  T −1 T Sˆ w,i Ste,i , Sw,i = Sˆ w,i Sˆ w,i C L ,i Sˆ w,i

(8.38)

Therefore, the stiffness model of the ith limb can be obtained as  T −1 T Sˆ w,i . K L ,i = Sˆ w,i Sˆ w,i C L ,i Sˆ w,i

(8.39)

The virtual work equation of the moving platform is formulated as STt,E Sw,E =

l 

STt,i Sw,i .

(8.40)

i=1

where Sw,E and St,E denote the external wrench acting on the moving platform and the resulted twist. Substitute Eqs. (8.39)–(8.40), the stiffness model of all the limbs is obtained as KL =

l 

K L ,i .

(8.41)

i=1

The closed-loop mechanism can be viewed as a serial linkage in the order of the fixed base, the limbs, and the moving platform. Considering the elastic deformation of the fixed base and the moving platform, the overall stiffness model of the closed-loop mechanism is formulated as −1  , K = C −1 = C B + K −1 L + CP

(8.42)

where C B = T B C B T TB , C P = T P C P T TP . C B and C P are the compliance matrices of the fixed base and the moving platform in the component reference frame. T B and

254

8 Static Modeling and Analysis of Robotic Mechanism

T P are the adjoint transformation matrices,  TB =

   E3 0 R 0 ,TP = , r˜ O E 3 r˜ R R

(8.43)

herein r O is the position vector of the center of fixed base. R is the rotation matrix of moving reference frame with respect to the fixed reference frame. r is the position vector of the end reference point.

8.4 Example 8.4.1 Typical Open-loop Mechanism 8.4.1.1

6R Mechanism

The topology model of 6R mechanism is shown in Fig. 7.4. There are five components connected by six R joints. The base link is attached to the 1st R joint and the endeffector is connected with the 6th R joint. Each R joint is actuated by a servo motor, respectively, indicating that there is no passive joint within 6R mechanism. Therefore, six-DoF virtual spring is applied to describe the elastic feature of the components. The elastic model of 6R mechanism is as shown in Fig. 8.4. According to Eqs. (8.2)–(8.3) for the twist and wrench of an open-loop mechanism, the twist and wrench models of 6R mechanism can be formulated as St = Ste =

5 

Ste, j , j = 1, 2, . . . , 5,

(8.44)

j=1

Sw = Swa =

6 

f a, ja Sˆ wa, ja .

(8.45)

ja=1

Referring to Eq. (8.25), the Hooke’s law is expressed in fixed reference frame as Ste, j = C j Sw, j ,

Fig. 8.4 The elastic model of 6R mechanism

(8.46)

8.4 Example

255

C j = T j C p, j T Tj ,

(8.47)

where C p, j denotes the compliance of component in the component reference frame. It is computed by FEA software or analytical computation. T j is the adjoint transformation matrix, and  Tj =

 Rj 0 , r˜ j R j R j

(8.48)

herein R j is the rotation matrix of component reference frame with respect to the fixed reference frame. r j denotes the position vector of the origin of component reference frame. T There are only actuation wrenches in 6R mechanism. Multiply Sˆ wa, j to both sides of Eq. (8.46) yields T T Sˆ wa, j C j Sˆ wa, j f a, j = Sˆ wa, j Ste, j = ρ ta, j , j = 1, 2, . . . , 5,

(8.49)

Considering all the virtual springs of the components, Eq. (8.49) can be rewritten as T T Sˆ w C L Sˆ w f w = Sˆ w Ste = ρ t ,

(8.50)

 where C L = 5j=1 C j . The virtual work principle for the 6R mechanism is formulated as STt Sw = ρ Tt f .

(8.51)

Applying the twist and wrench mapping model of the 6R mechanism, Eq. (8.32) can be expressed as  T −1 T Sˆ w Ste . Sw = Sˆ w Sˆ w C L Sˆ w

(8.52)

The stiffness or compliance model of the components in the open-loop mechanism is directly obtained from Eq. (8.52) as  T −1 T Sˆ w . K L = Sˆ w Sˆ w C L Sˆ w

(8.53)

The elastic deformations of the fixed based and the end-effector are considered. According to the deformation superposition principle, the stiffness model of 6R mechanism is derived as −1  , K = C −1 = C B + K −1 L + CP

(8.54)

256

8 Static Modeling and Analysis of Robotic Mechanism

where C B = T B C B T TB , C P = T P C P T TP . C B , C P are the compliance matrices of the fixed base and the end-effector in the component reference frame. T B and T P are the adjoint transformation matrices,  TB =

   E3 0 R 0 ,TE = , r˜ O E 3 r˜ R R

herein r O is the position vector of the center of the fixed base. R is the rotation matrix of moving reference frame with respect to the fixed reference frame. r is the position vector of the end reference point.

8.4.1.2

SPR Mechanism

The topology model of the SPR mechanism is shown in Fig. 7.5. The S joint is replaced with three equivalent R joints. Hence, the SPR mechanism has four components connected by five actuated one-DoF joints. The elastic model of the SPR mechanism is as shown in Fig. 8.5. The twist and wrench models of SPR mechanism can be formulated as St = Ste =

4 

Ste, j ,

(8.55)

j=1

Sw = Swa + Swc =

5 

f a, ja Sˆ wa, ja + f c Sˆ wc, jc .

(8.56)

ja=1

The Hooke’s law of SPR mechanism can be expressed in fixed reference frame as Ste, j = C j Sw, j ,

(8.57)

C j = T j C p, j T Tj ,

(8.58)

where C p, j denotes the compliance matrix of the jth component in component reference frame. It is obtained by FEA software or analytical computation. T j is the adjoint transformation matrix, and

Fig. 8.5 The elastic model of SPR mechanism

8.4 Example

257

 Tj =

 Rj 0 , r˜ j R j R j

(8.59)

herein R j is the rotation matrix of component reference frame with respect to the fixed reference frame. r j denotes the position vector of the origin of component reference frame to the origin of fixed reference frame. Multiplying the unit actuation wrench or constraint wrench to both sides of Eq. (8.57) results in T T Sˆ w C L Sˆ w f w = Sˆ w Ste = ρ t ,

(8.60)

 where C L = 4j=1 C j . The virtual work principle for SPR mechanism is formulated as STt Sw = ρ Tt f .

(8.61)

Substituting the twist and wrench mapping model of the SPR mechanism to Eq. (8.61) yields  T −1 T Sˆ w Ste . Sw = Sˆ w Sˆ w C L Sˆ w

(8.62)

Referring to Hooke’s law, the stiffness model of the components in the open-loop mechanism is directly obtained from Eq. (8.62) as  T −1 T Sˆ w . K L = Sˆ w Sˆ w C L Sˆ w

(8.63)

Considering the elastic deformation of the fixed based and the end-effector, the stiffness model of SPR mechanism is derived according to the deformation superposition principle as −1  , K = C −1 = C B + K −1 L + CP

(8.64)

where C B = T B C B T TB , C P = T P C P T TP . C B , C P are the compliance matrices of the fixed base and the end-effector in the component reference frame. T B and T P are the adjoint transformation matrices,  TB =

   E3 0 R 0 ,TP = , r˜ O E 3 r˜ R R

(8.65)

herein r O is the position vector of fixed base. R is the rotation matrix of moving reference frame with respect to the fixed reference frame. r is the position vector of the end reference point.

258

8 Static Modeling and Analysis of Robotic Mechanism

8.4.2 Typical Closed-loop Mechanism The topology model of the Exechon mechanism is as shown in Fig. 7.6. The P joint at each limb is regarded as the actuation joint while the rest are passive joints. Hence, the deformations of the components connected to the P joints are denoted by six-DoF virtual spring. The deformation of components linked by f p -DoF passive joints are denoted by m-DoF virtual springs (m = 6 − f p ). The elastic model of the Exechon closed-loop mechanism is as shown in Fig. 8.6. Considering both deformation and the rigid body motion of the passive joints, the twist of the ith limb is formulated as St,i , i = 1, 2, 3, St = St,i = Ste,i + −

(8.66)

St,i denote the deformation twist and rigid body motion twist of the where Ste,i and − ith limb. The deformation twist of both joints and components are considered. The twist of each limb can be further expressed as St,i =

2 

Ste,i,k +

k=1

St,i =

2 

3 

δqi,k − Sˆ t,i,k , i = 1, 3,

(8.67)

k=1

Ste,i,k +

k=1

4 

δqi,k − Sˆ t,i,k , i = 2.

(8.68)

k=1

From the free body analysis of the moving platform, the external wrench on the moving platform equals to the wrenches provided by each limb. Sw =

3 

Sw,i .

(8.69)

i=1

The limb wrenches can be divided into actuation and constraint wrenches according to Sect. 7.5.2 as

Fig. 8.6 The elastic model of Exechon mechanism

8.4 Example

259

Sw,i = f a,i,ka Sˆ wa,i,ka +

2 

f c,i,kc Sˆ wc,i,kc , i = 1, 3,

(8.70)

kc=1

Sw,i = f a,i,ka Sˆ wa,i,ka + f c,i,kc Sˆ wc,i,kc , i = 2,

(8.71)

where Sˆ wa,i,ka and f a,i,ka denote the actuation wrench and the intensity of ith limb. Sˆ wc,i,kc and f c,i,kc are the constraint wrench and intensity of the ith limb. For the SPR limb, the stiffness model is referred to Eq. (8.63) as  T −1 T Sˆ w,2 , K L ,2 = Sˆ w,2 Sˆ w,2 C L ,2 Sˆ w,2

(8.72)

 where C L ,2 = 2k=1 C 2,k . The S joint is decomposed by three nonlinear R joints and the compliance matrix of each one-DoF joint is included. For the UPR limbs, the Hooke’s law for each component expressed in fixed reference frame is expressed as Ste,i,k = C i,k Sw,i,k , i = 1, 3,

(8.73)

T C i,k = T i,k C i, p,k T i,k , i = 1, 3,

(8.74)

where C i, p,k denotes the compliance matrix of component in the component reference frame obtained by FEA software or analytical computation. T i,k is the adjoint transformation matrix, and  T i,k =

 0 Ri,k , r˜ i,k Ri,k Ri,k

herein Ri,k is the rotation matrix of component reference frame with respect to the fixed reference frame. r i,k denotes the position vector of the origin of component reference frame. Multiplying the unit wrench of the UPR limb to the virtual work equation of each components results in T T Sˆ w,i,k C i,k Sˆ w,i,k f w,i,k = Sˆ w,i,k Ste,i,k = ρ t,i,k , i = 1, 3.

(8.75)

Considering all the virtual springs of the UPR limb, Eq. (8.75) turns into T T Sˆ w,i C L ,i Sˆ w,i f w,i = Sˆ w,i Ste,i = ρ t,i , i = 1, 3,

where C L ,i = included.

2 k=1

(8.76)

C i,k . Herein the compliance matrix of the one-DoF joint is also

260

8 Static Modeling and Analysis of Robotic Mechanism

The virtual work principle for the UPR limb is formulated as STt,i Sw,i = ρ Tt,i f w,i , i = 1, 3.

(8.77)

Substitute Eqs. (8.76)–(8.77), the following equation can be obtained.  T −1 T Sˆ w,i Ste,i , i = 1, 3. Sw,i = Sˆ w,i Sˆ w,i C L ,i Sˆ w,i

(8.78)

The stiffness or compliance model of the components in the open-loop mechanism is directly obtained from Eq. (8.78) as  T −1 T Sˆ w,i , i = 1, 3. K L ,i = Sˆ w,i Sˆ w,i C L ,i Sˆ w,i

(8.79)

With the stiffness models of each limb is available at hand, the stiffness model of the whole mechanism can be formulated with the aid of virtual work principle. The virtual work equation of the moving platform is formulated as STt,E Sw,E =

3 

STt,i Sw,i .

(8.80)

i=1

Substitute Eqs. (8.78)–(8.80), the stiffness model of all the limbs is obtained as KL =

3 

K L ,i , i = 1, 2, 3.

(8.81)

i=1

The closed-loop mechanism can be viewed as a serial linkage in the order of the fixed base, the limbs and the moving platform. Considering the elastic deformation of the fixed base and the moving platform, the overall stiffness model of the Exechon mechanism is formulated as −1  , K = C −1 = C B + K −1 L + CP

(8.82)

where C B = T B C B T TB , C P = T P C P T TP . C B and C P are the compliance matrices of the fixed base and the moving platform in the component reference frame. T B and T P are the adjoint transformation matrices,  TB =

   E3 0 R 0 ,TP = , r˜ O E 3 r˜ R R

(8.83)

herein r O is the position vector of the center of fixed base. R is the rotation matrix of moving reference frame with respect to the fixed reference frame. r is the position vector of the end reference point.

8.5 Conclusion

261

8.5 Conclusion This chapter presents a generic stiffness modeling method by the FIS theory. The deformation of the elastic body is regarded as an instantaneous motion resulted from external wrenches. Hence, deformation twist of components is included in the twist and wrench analysis of robotic mechanisms. An m-DoF virtual spring is defined to denote the elastic features of the components considering the effects of the connecting passive joints. Based on the twist and wrench analysis, the m-DoF virtual spring, and the virtual work principle, the stiffness model of any robotic mechanism can be formulated. The presented stiffness modeling method considers the complete deformations of the components and reflects the accurate deformation transformations of robotic mechanisms. For the reader’s convenience, the key points of this chapter are represented as follows: (1) Applying FIS theory, the twist and wrench analysis between joint space and operated space of open-loop and closed-loop mechanisms are analyzed. The deformations of elastic body are considered as an instantaneous motion and included in the twist analysis of robotic mechanisms. (2) The virtual spring is applied to denote the elastic features of components whose compliance matrix can be obtained either by FEA software or by analytical computation. Specifically, an m-DoF virtual spring is proposed considering the effects of connecting passive joints. (3) Based on the twist and wrench analysis, the m-DoF virtual spring, the virtual work, and Hooke’s law, the generic stiffness modeling method for open-loop and closed-loop mechanism is obtained. It combines the accuracy of the numerical method and the completeness of analytical method. The obtained stiffness model can be applied to the analysis of static performance of robotic mechanisms and the formulation of static performance objectives in the robotic design. (4) The stiffness modeling of 6R, SPR open-loop mechanisms, and the Exechon closed-loop mechanism are given to illustrate the presented stiffness modeling method.

References 1. Gosselin CM (1990) Stiffness mapping for parallel manipulators. IEEE Trans Robot Autom 6(3):377–382 2. Klimchik A, Chablat D, Pashkevich A (2014) Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings. Mech Mach Theory 79:1–28 3. Portman VT (2011) Stiffness evaluation of machines and robots: minimum collinear stiffness value approach. J Mech Robot 3:011015-1–011015-9 4. Dai JS, Ding XL (2006) Compliance analysis of a three-legged rigidly-connected platform device. J Mech Des 128:755–764

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5. Lian BB, Sun T, Song YM (2017) Stiffness modeling, analysis and evaluation of a 5 degree of freedom hybrid manipulator for friction stir welding. Proc Inst Mech Eng Part C J Mech Eng Sci 231(23):4441–4456 6. El-Khasawneh BS, Ferreira PM (1999) Computation of stiffness bounds for parallel link manipulators. Int J Mach Tools Manuf 39:321–342 7. Ceccarelli M, Carbone G (2002) A stiffness analysis for CaPaMan (Cassino parallel manipulator). Mech Mach Theory 37:427–439 8. Li YM, Xu QS (2008) Stiffness analysis for a 3-PUU parallel kinematic machine. Mech Mach Theory 43:186–200 9. Wu J, Wang JS, Wang LP (2009) Study on the stiffness of a 5-DOF hybrid machine tool with actuation redundancy. Mech Mach Theory 44:289–305 10. Pashkevich A, Klimchik A, Chablat D (2011) Enhanced stiffness modeling of manipulators with passive joints. Mech Mach Theory 46:662–679 11. Lian BB (2017) Methodology of multi-objective optimization for a five degree-of-freedom parallel manipulator. Dissertation, Tianjin University 12. Lian BB, Sun T, Song YM (2016) Stiffness analysis of a 2-DoF over-constrained RPM with an articulated traveling platform. Mech Mach Theory 96:165–178 13. Pashkevich A, Chablat D, Wenger P (2009) Stiffness analysis of overconstrained parallel manipulators. Mech Mach Theory 44:966–982 14. Majou F, Gosselin CM, Wenger P (2007) Parametric stiffness analysis of the orthoglide. Mech Mach Theory 42:296–311 15. Shneor Y, Portman VT (2010) Stiffness of 5-axis machines with serial, parallel, and hybrid kinematics: evaluation and comparison. CIRP Ann Manuf Technol 59:409–412 16. Lian BB, Sun T, Song YM (2015) Stiffness analysis and experiment of a novel 5-DoF parallel kinematic machine considering gravitational effects. Int J Mach Tools Manuf 95:82–96 17. Lian BB, Sun T, Song YM (2016) Passive and active gravity compensation of horizontallymounted 3-RPS parallel kinematic machine. Mech Mach Theory 190–201 18. Klimchik A, Pashkevich A, Chablat D (2013) CAD-based approach for identification of elastostatic parameters of robotic manipulators. Finite Elem Anal Des 75:19–30 19. Liu H, Huang T, Chetwynd DG et al (2017) Stiffness modeling of parallel mechanisms at limb and joint/link levels. IEEE Trans Rob 33(3):734–741 20. Wang M, Liu H, Huang T et al (2015) Compliance analysis of a 3-SPR parallel mechanism with consideration of gravity. Mech Mach Theory 84:99–112

Chapter 9

Dynamic Modeling and Analysis of Robotic Mechanism

9.1 Introduction Dynamics, associated with velocity, acceleration, and force, is a key performance for the robotic mechanisms working with high speed or under heavy load [1–3]. Dynamic modeling is the determination of the relation between motions and forces of the robotic mechanism [4, 5]. In general, there are two types of dynamic models [6]. One is direct dynamics that determines the mechanism motion being given the actuated joint forces [7, 8]. The other one is inverse dynamics that determines the actuated joint forces being given the trajectory, velocity, and acceleration of the endeffector or moving platform, which is usually applied in control development or optimal design [9–16]. This chapter presents the inverse dynamic modeling method based on the FIS theory. The previous chapters have analyzed kinematic and stiffness performances based on the differential mapping from topology to velocity models, from which the Jacobian matrix can be adopted to compute the velocities of all the components. In this chapter, differentiation of velocity model is further implemented by the FIS theory and the Hessian matrix is obtained. Based on the velocities and the Hessian matrix, acceleration models among actuated joints, components, and end-effector/moving platform are formulated in a straightforward manner. Hence, the forces including actuated forces, inertia forces, and gravity can be expressed. With the velocities, accelerations, and forces of joints, components and end-effector/moving platform available at hand, a dynamic model of robotic mechanism is derived according to the virtual work principle. The proposed dynamic modeling method based on FIS theory has the following merits. (1) Acceleration model is obtained by the differentiation of twist model that is the differentiation of the topology model. The modeling process is explicit. (2) The underlying relationships between velocity, force, acceleration, and the topology of a robotic mechanism are revealed, which completes the integrated design framework involving topology and performance models.

© Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_9

263

264

9 Dynamic Modeling and Analysis of Robotic Mechanism

In this chapter, velocity models of all the components are firstly obtained based on the previous study in Chap. 7. Then, FIS theory based acceleration models are formulated. Next, the wrenches of components and end-effector/moving platform are analyzed. Finally, dynamic models of open-loop and closed-loop mechanisms are obtained. Two open-loop and one closed-loop robotic mechanisms are taken as examples to illustrate the dynamic modeling method in detail. It is worth mentioning that the velocities and accelerations are the ones at any pose. They are obtained by the differentiations of topology model described from the concerned pose.

9.2 Velocity Modeling Velocity model between the actuated joints and the end-effector/moving platform of open-loop mechanism/closed-loop mechanism has been formulated in Chap. 7. Based on this, the velocities of all the components are analyzed in this section for the dynamic modeling. Herein, center mass is assumed to be the same as the geometric center of the component.

9.2.1 Open-loop Mechanism The velocities at the center of the end-effector and all the components are calculated in this section. Firstly, the velocity at the center of the end-effector can be obtained as p

St = T p St ,

(9.1)

 E3 0 is the transformation matrix between the velocity of endwhere T p = −˜rp E3 effector described at point O and that described at point P. rp is the position vector from point O to P. Component is the rigid body connecting by two kinematic joints. Without loss of generality, the jth (j = 1, 2, . . . , n − 1) component in the open-loop mechanism is defined as the linkage connecting the jth and (j + 1)th joints. To this end, the velocity of each component in an open-loop mechanism could be developed by three steps. 

Step 1: Solve the velocities of the 1st to the jth joints; Step 2: Calculate the velocity of the jth component at connecting point with the jth joint by the linear combination of the results from step 1; Step 3: Obtain the velocity of the jth component at the center by multiplying the transformation matrix.

9.2 Velocity Modeling

265

The detail process is explained in the following. It is noted that there are no passive joints in an open-loop robotic mechanism. Referred to Eq. (7.84), the mapping between velocities of joints and that of endeffector is formulated as q˙ = GSt ,

(9.2)

where q˙ is the vector including the velocities of all the n one-DoF joints. G is derived from the Jacobian matrix J as 

G = G1 G2 · · · Gn

T

 =

Sˆ wa,1

Sˆ wa,2

T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2

···

T

Sˆ wa,n T Sˆ wa,n Sˆ t,n

.

herein, Sˆ t,j (j = 1, 2, . . . , n) is the unit twist generated by the jth joint. Sˆ wa,j is the actuation wrench corresponding to the jth joint. Referred to Eq. (9.2), the velocities of j joints can be measured by taking out j rows from G as q˙ j = Gbj St ,

(9.3)

T  where q˙ j denotes a j-dimension vector. Gbj = G1 G2 · · · Gj is a block of G, which is composed by G1 , G2 , . . . , Gj . Velocity of the jth component is computed by the addition of velocities from the 1st to the jth joint. Considering also the transformation matrix, the velocity at the center of the jth component can be obtained as Sct,j = T c,j

j

q˙ k Sˆ t,k ,

(9.4)

k=1

 E3 0 is where denotes the velocity of the jth component center. T c,j = −˜rc,j E3 the transformation matrix between the velocity obtained by point O and that by the center of the jth component. rc,j is the position vector from point O to the center of the jth component. Applying Eqs. (9.3)–(9.4), the velocity at the component center is written as 

Sct,j

Sct,j = T c,j J bj Gbj St = T c,j J bj q˙ j ,

(9.5)

  where J bj = Sˆ t,1 Sˆ t,2 · · · Sˆ t,j is a 6 × j block, composed by j unit twists of joints.

266

9 Dynamic Modeling and Analysis of Robotic Mechanism

9.2.2 Closed-loop Mechanism The velocity mapping between the moving platform and actuated joints has been analyzed in Chap. 7. In this section, the relationship between the velocity of the moving platform, all the joints including passive and actuated joints and components are investigated by four steps. Step 1: Describe the velocity of the moving platform at center point P; Step 2: Solve the velocities of all joints in the ith limb (i = 1, 2, . . . , l); Step 3: Collect the velocities of k adjacent joints and calculate the velocity of the kth component in ith limb; Step 4: Repeat step 2–3 until the velocities of all the components are obtained. The velocity at the center of the moving platform in closed-loop mechanism could also be solved in the same way as Eq. (9.1). Similar to Eq. (9.2), the velocities of all the joints in the ith limb in a closed-loop mechanism could be obtained as q˙ i = Gi St ,

(9.6)

where q˙ i is the vector containing the velocities of all the joints of the ith limb, Gi could be described as  Gi =

Sˆ wa,i,1

Sˆ wa,i,2

T T Sˆ wa,i,1 Sˆ t,1,i Sˆ wa,i,2 Sˆ t,i,2

···

Sˆ wa,i,ni T Sˆ wa,i,ni Sˆ t,i,ni

T .

From Eq. (7.90), St is the velocity of the moving platform, St = J a q˙ a ,

(9.7)

where q˙ a denotes the vector including the velocities of actuated joints. Referring to Eq. (9.5), the velocity of the kth component in the ith limb is the accumulation of k joints and could be described in task space as Sct,i,k = T c,i,k

k

q˙ i,j Sˆ t,i,j = T c,i,k J bi,k Gbi,k St ,

j=1

where    E3 0 , J bi,k = Sˆ t,i,1 Sˆ t,i,2 · · · Sˆ t,i,k , −˜rc,i,k E3 T  = Gi,1 Gi,2 · · · Gi,k . 

T c,i,k = Gbi,k

Substituting Eq. (9.7) into Eq. (9.8), leads to an expression in joint space,

(9.8)

9.2 Velocity Modeling

267

Sct,i,k = T c,i,k J bi,k Gbi,k J a q˙ a .

(9.9)

9.3 Acceleration Modeling In this chapter, accelerations of all the components in open-loop and closed-loop robotic mechanisms are constructed by differential mapping of the velocity models. Firstly, the acceleration models of joints and end-effector/moving platform for robotic mechanism are developed, respectively. Then the accelerations of components are solved by Hessian matrix. The acceleration models of open-loop mechanism could be utilized in the process of closed-loop mechanism.

9.3.1 Open-loop Mechanism The acceleration model of the open-loop mechanism can be described as Sa = J q¨ + q˙ T H q˙ ,

(9.10)

where H is defined as the Hessian matrix. From Eq. (9.10), the accelerations of joints could be solved as  q¨ = G Sa − q˙ T H q˙ ,

(9.11)

where G has the same form with that in Eq. (9.2). Sa is known as the acceleration of end-effector at point O, which could be described at the center point P as Spa = T p Sa ,

(9.12)

where Spa represents the acceleration at the center of the end-effector. T p is referred to Eq. (9.1). The acceleration of the jth component of the open-loop mechanism could be expressed in task space as

Sca,j

= T c,j

J bj Gbj

   b T T T b Sa − St G HGSt + Gj St H j×j Gj St ,

(9.13)

where H j×j is the first j × j matrix block of H. Substituting Eq. (9.10) into Eq. (9.13), the acceleration of the jth component center could take the form in joint space as   Sca,j = T c,j J bj q¨ j + q˙ Tj H j×j q˙ j .

(9.14)

268

9 Dynamic Modeling and Analysis of Robotic Mechanism

9.3.2 Closed-loop Mechanism The acceleration model of closed-loop mechanism can be described as  ˙ a q˙ a , Sa = J a q¨ a + J −1 w h q

(9.15)

 where J w is referred to Eq. (7.89), h q˙ a is the matrix constituted by the reciprocal product of the unit wrenches and the Coriolis term of each limb acceleration, i.e.,  ⎤ T T Sˆ wa,1,1 G1 J a q˙ a H 1 G1 J a ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ T ⎥   T ⎢ˆ ⎥ ⎢ Swa,1,g1 G1 J a q˙ a H 1 G1 J a ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎥  ⎢ . ⎢   ⎥. h q˙ a = ⎢ T T ⎥ ⎢ Sˆ wa,l,gl Gl J a q˙ a H l Gl J a ⎥ ⎢ T ⎥   T ⎢ ˆ ⎥ ⎢ Swc,1,1 G1 J a q˙ a H 1 G1 J a ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦    T T Sˆ wc,l,gl Gl J a q˙ a H l Gl J a ⎡

Herein, H i is the Hessian matrix of the ith limb. The acceleration at the center point P of moving platform is solved as Spa = T p Sa ,

(9.16)

where T p is referred to that in Eq. (9.1). Equation (9.15) shows the mapping between the acceleration of actuations and moving platform in a closed-loop mechanism, which could be rewritten as   ˙ a q˙ a . q¨ a = Ga Sa − J −1 w h q

(9.17)

Similarly, the acceleration of the ith limb in a closed-loop mechanism could be derived from Eq. (9.11) as  q¨ i = Gi Sa − qTi H i qi ,

(9.18)

where Gi has the same format with that in Eq. (9.6). Equation (9.18) can also be expressed as  q¨ i = Gi Sa − (Gi J a q˙ a )T H i Gi J a q˙ a .

(9.19)

For the kth component in the ith limb, the acceleration could be formulated as

9.3 Acceleration Modeling

269

 Sca,i,k = T c,i,k J bi,k q¨ i,k + q˙ Ti,k H i,k×k q˙ i,k .

(9.20)

Based on Eqs. (9.19)–(9.20), acceleration of kth component in the ith limb is written in task and joint space, respectively, as     T Sca,i,k = T c,i,k J bi,k Gbi,k Sa − (Gi St )T H i Gi St + Gbi,k St H i,k×k Gbi,k St ,

(9.21)

   T   ˙ a q˙ a − Gi J a q˙ a H i Gi J a q˙ a Sca,i,k = T c,i,k J bi,k Gbi,k J a q¨ a + J −1 w h q   T + Gbi,k J a q˙ a H i,k×k Gbi,k J a q˙ a . (9.22)

9.4 Dynamic Modeling Having the velocities and accelerations of all the components at hand, dynamic modeling based on virtual work principle is presented. In the following, the wrenches at the moving platform, components, and joints in a robotic mechanism are analyzed. Then the equations based on virtual work principle are formulated. Finally, dynamic equations of open-loop mechanism and closed-loop mechanism described in joint space and task space are constructed.

9.4.1 Wrench Analysis 9.4.1.1

Open-loop Mechanism

The wrenches of an open-loop mechanism include actuation, inertia, gravity, and external wrenches. The actuation wrench of the open-loop mechanism has been analyzed in Chap. 7, which is denoted by Swa,j . As shown in Fig. 9.1, the external wrench is applied to the center of end-effector. Hence, the wrench of the end-effector is formulated as E Sw = SIw + SG w + Sw ,

(9.23)

E where SIw , SG w , and Sw denote inertia, gravity, and external wrenches, respectively. According to the D’Alembert principle, the gravity and inertia wrenches at the center of moving platform are described as

 SG w =

 0 , mg

(9.24)

270

9 Dynamic Modeling and Analysis of Robotic Mechanism

Fig. 9.1 Forces applied to the end-effector and components in an open-loop mechanism

 SIw =

 I ω˙ + ω × Iω p p = mS˙ t + V mSt , m˙v

(9.25)

 T where g = 0 0 g denotes the gravity acceleration vector,  m=

     I 0 ω˜ 0 0 ω˜ p p , S˙ t = Spa + ΩSt , V = , Ω= . 0 mE3 0 v˜ 00

herein, I is the inertia matrix of the end-effector. S˙ t represents the acceleration state at the center of end-effector. v, ω, and m are the linear, angular velocities and the mass of the end-effector. Besides describing by the motions of the end-effector, Eq. (9.25) could be mapped into joint space as p

   SIw = m T p J q¨ + q˙ T H q˙ + ΩT p J q˙ + V mT p J q˙ .

(9.26)

According to the virtual work principle, internal wrenches among the components are not considered. As shown in Fig. 9.1, the wrenches of the jth component in the open-loop mechanism can be formulated as Scw,j = SIw,j + SG w,j ,

(9.27)

where SIw,j and SG w,j denote the inertia and gravity wrenches of the jth component.  0 , = mj g   I j ω˙ j + ωj × I j ωj c = mj S˙ t,j + V j mj Sct,j , = mj v˙ j 

SG w,j

SIw,j

(9.28) (9.29)

9.4 Dynamic Modeling

271

where  mj =

     Ij 0 ω˜ j 0 0 ω˜ j c , S˙ t,j = Sca,j + Ω j Sct,j , V j = , Ωj = . 0 0 0 mj E3 0 v˜ j

where S˙ t,j represents the acceleration state at the center of jth component, of which vj and ωj denote the linear and angular velocities, respectively. I j is the inertia matrix and mj denotes the mass of jth component. Equation (9.29) can be further written in task space and joint space, respectively, as

⎛   ⎞   b T b b T T b T J G S + G G − S G HGS S H S c,j a j×j j j t j t j t t ⎠ SIw,j = mj ⎝ b b + Ω j T c,j J j Gj St c

+ V j mj T c,j J bj Gbj St ,     SIw,j = mj T c,j J bj q¨ j + q˙ Tj H j×j q˙ j + Ω j T c,j J bj q˙ j + V j mj T c,j J bj q˙ j ,

9.4.1.2

(9.30) (9.31)

Closed-loop Mechanism

For the moving platform in a closed-loop mechanism as shown in Fig. 9.2, the wrenches at the center can be expressed as E Sw = SIw + SG w + Sw

(9.32)

E where SIw , SG w , and Sw denote inertial, gravity and external wrenches of moving platform.

Fig. 9.2 Forces at moving platform and components in a closed-loop mechanism

272

9 Dynamic Modeling and Analysis of Robotic Mechanism

From Eqs. (9.28)–(9.29), the gravity and inertia wrenches of the moving platform are given as 

 0 = mg   I ω˙ + ω × Iω p p = mS˙ t + V mSt , SIw = m˙v SG w

(9.33) (9.34)

where      I 0 ω˜ 0 0 ω˜ p p p ˙ , St = Sa + ΩSt , V = m= , Ω= , 0 mE3 0 v˜ 00 

herein, the meanings of I, v, ω, and m refer to Eqs. (9.24)–(9.25). The inertial wrench could be rewritten into joint space as    ˙ a q˙ a + ΩT p J a q˙ a + V mT p J a q˙ a . SIw = m T p J a q¨ a + J −1 w h q

(9.35)

In a similar manner, the resultant force at the center of a component could be obtained as Scw,i,k = SIw,i,k + SG w,i,k ,

(9.36)

where SIw,i,k and SG w,i,k are the inertia and gravity forces of the kth component in the ith limb, respectively. The gravity and inertia wrenches of the kth component in the ith limb are in the similar form as Eqs. (9.28)–(9.29),  SG w,i,k  SIw,i,k =

=

I i,k ω˙ i,k + ωi,k × I i,k ωi,k mi,k v˙ i,k



0

 (9.37)

mi,k g = mi,k S˙ t,i,k + V i,k mi,k Sct,i,k c

(9.38)

where  I i,k 0 c , S˙ t,i,k = Sca,i,k + Ω i,k Sct,i,k , = 0 mi,k E3     ω˜ i,k 0 0 ω˜ i,k , Ω i,k = , = 0 0 0 v˜ i,k 

mi,k V i,k

herein, the meanings of I i.k , mi.k , vi,k , and ωi,k could be referred to Eqs. (9.28)–(9.29). From Sect. 9.3.2, Sct,i,k and Sca,i,k represent the velocity and acceleration state at the center of the kth component in the ith limb, respectively.

9.4 Dynamic Modeling

273

Equation (9.38) can be further expressed as 

 ⎞ J bi,k Gbi,k Sa − (Gi St )T H i Gi St + ⎜ T c,i,k  b T ⎟ c = mi,k ⎝ ⎠ + V i,k mi,k St,i,k , Gi,k St H i,k×k Gbi,k St + Ω i,k Sct,i,k (9.39) ⎛

SIw,i,k

SIw,i,k can also be described in joint space in form

SIw,i,k

 ⎞ T   ˙ ˙ ˙ ˙ q q q h q − G J H G J J bi,k Gbi,k J a q¨ a + J −1 i a i i a a a a a w ⎜ T c,i,k ⎟ T  = mi,k ⎝ ⎠ + Gbi,k J a q˙ a H i,k×k Gbi,k J a q˙ a c + Ω i,k St,i,k ⎛



+ V i,k mi,k Sct,i,k .

(9.40)

9.4.2 Dynamic Modeling The virtual work principle states that if a mechanism is in equilibrium under the effect of external actions, the work produced by the corresponding forces with any virtual velocity must be null [17]. Hence, the work done by the inertia forces, the gravities, the actuation forces, and the external force of the robotic mechanism equals zero.

9.4.2.1

Open-loop Mechanism

For the open-loop mechanism, the virtual work equation could be written as n−1  T    

p T I T E ˙ T f = 0, Sct,j SIw,j + SG Sw + SG w,j + St w + St Sw + q

(9.41)

j=1

T  where f = f1 · · · fn denotes the actuation force/torque vector of the n actuation joints. Referring to Eqs. (9.1)–(9.2), (9.41) could be expanded as ⎞ ⎛ n−1  T  

 T I G E T ⎠ T c,j J bj Gbj SIw,j + SG STt ⎝ = 0. w,j + T p Sw + Sw + Sw + G f j=1

(9.42)

274

9 Dynamic Modeling and Analysis of Robotic Mechanism

Hence, the actuation force/torque can be derived as ⎛

⎞ n−1  T  

 T I G E⎠ f = −J T ⎝ T c,j J bj Gbj SIw,j + SG w,j + T p Sw + Sw + Sw .

(9.43)

j=1

Substituting Eqs. (9.24)–(9.25), (9.28), and (9.30) into Eq. (9.43), the dynamic model in task space is obtained as  f = −J T M 1 Sa + C r,1 St + FG,1 + SEw ,

(9.44)

where n−1  T

T c,j J bj Gbj mj T c,j J bj Gbj + T Tp mT p , M1 = j=1

FG,1 = T Tp SG w +

n−1  T

T c,j J bj Gbj SG w,j , j=1

C r,1 = T Tp mΩT p + T Tp V mT p +

n−1  T    T  

T c,j J bj Gbj mj T c,j −J bj Gbj STt GT HG + Gbj St H j×j Gbj j=1

  + Ω j T c,j J bj Gbj + V j mj T c,j J bj Gbj Considering Eqs. (9.24), (9.26), (9.28), and (9.31) yields the dynamic model in joint space,  f = −J T M 2 q¨ + C r,2 q˙ + FG,2 + SEw , where M2 =

n−1  T

  T c,j J bj Gbj mj T c,j J bj Λ + T Tp mT p J, Λ = Ej×j 0 j×n , j=1

FG,2 = T Tp SG w +

n−1  T

T c,j J bj Gbj SG w,j , j=1

 T Tp mT p q˙ T H + mΩT C r,2 =   p J + V mT p J   T m T q˙ T H + m Ω T J b n−1  j c,j j j×j j j c,j j b b T c,j J j Gj + Λ . +V j mj T c,j J bj j=1

(9.45)

9.4 Dynamic Modeling

275

In this way, the dynamic equation of open-loop mechanism is formulated. The formulated dynamic model in Eq. (9.45) allows directly computing the actuation forces/torques required for controlling motion of the robot.

9.4.2.2

Closed-loop Mechanism

In closed-loop mechanism, the dynamic equation could be constructed based on virtual work as l n

i −1

 c T  I  p T  I T E ˙ Ta f a = 0. St,i,k Sw,i,k + SG Sw + SG w,i,k + St w + (St ) Sw + q i=1 k=1

(9.46) In this way, Eq. (9.46) could be expanded as  STt

l n

i −1

 T   T I G E T T c,i,k J bi,k Gbi,k SIw,i,k + SG w,i,k + T p Sw + Sw + Sw + Ga f a

 = 0.

i=1 k=1

(9.47) To this end, the dynamic model could be derived as  l n −1  i

   T T I G E T c,i,k J bi,k Gbi,k SIw,i,k + SG f a = −J Ta w,i,k + T p Sw + Sw + Sw . i=1 k=1

(9.48) Substituting Eqs. (9.33)–(9.34), (9.37), and (9.39) into Eq. (9.48) yields the dynamic model in task space,  f a = −J Ta M 1 Sa + C r,1 St + FG,1 + SEw , where M1 =

l

n−1

 T T c,i,k J bi,k Gbi,k mi,k T c,i,k J bi,k Gbi,k + T Tp mT p , i=1 k=1

FG,1 = T Tp SG w +

l n

i −1

 T T c,i,k J bi,k Gbi,k SG w,i,k , i=1 k=1

(9.49)

276

9 Dynamic Modeling and Analysis of Robotic Mechanism

C r,1 =

⎞⎞ ⎛ ⎛   b T T b b b T ⎜ m ⎝ T c,i,k J i,k Gi,k −(Gi St ) H i Gi + Gi,k St H i,k×k Gi,k ⎠ ⎟ i,k ⎟ T c,i,k J bi,k Gbi,k ⎜ ⎝ ⎠ + Ω i,k T c,i,k J bi,k Gbi,k

l n

i −1

i=1 k=1



+ T Tp mΩT p + V mT p

+ V i,k mi,k T c,i,k J bi,k Gbi,k



Substituting Eqs. (9.33), (9.35), (9.37), and (9.40) into Eq. (9.48), the dynamic model in joint space is obtained as  f a = −J Ta M 2 q¨ a + C r,2 q˙ a + FG,2 + SEw ,

(9.50)

where M2 =

l n

i −1



T T c,i,k J bi,k Gbi,k mi,k T c,i,k J bi,k Gbi,k J a + T Tp mT p J a ,

i=1 k=1

FG,2 = T Tp SG w +

l n

i −1

 T T c,i,k J bi,k Gbi,k SG w,i,k , i=1 k=1 ⎛ ⎛

 ⎞⎞⎞ ⎛   T J bi,k Gbi,k J −1 w h q˙ a − Gi J a q˙ a H i Gi J a ⎜ ⎜ ⎠⎟ T T c,i,k ⎝  ⎟⎟ l n

i −1 T ⎜ m ⎜

b J q˙ b J ⎟⎟ ⎜ i,k ⎜ + G H G b b i,k×k i,k a i,k a a ⎝ ⎠⎟ C r,2 = T c,i,k J i,k Gi,k ⎜ ⎟ ⎜ ⎟ b b + Ω i,k T c,i,k J i,k Gi,k J a i=1 k=1 ⎝ ⎠ + V i,k mi,k T c,i,k J bi,k Gbi,k J a      −1 + TT p m T p J w h q˙ a + ΩT p J a + V mT p J a

9.5 Example 9.5.1 Typical Open-loop Mechanism 9.5.1.1

6R Mechanism

Referring to Fig. 7.4, the 6R mechanism is composed of six rotational joints, connecting by a fixed base, five components, and an end-effector. The kinematic mapping

9.5 Example

277

between end-effector and joints has been constructed in Chap. 7. In the following, the velocities and accelerations at the centers of end-effector and components will be solved firstly. The velocity at the center of the end-effector is expressed as p

St = T p St ,

(9.51)

 E3 0 is the transformation matrix between the velocity of end−˜rp E3 effector described at point O and that described at point P. rp is the position vector from point O to P. The velocities of components could be derived as 

where T p =

Sct,j = T c,j J bj Gbj St = T cj J bj q˙ j , j = 1, 2, . . . 5,

(9.52)

  J bj = Sˆ t,1 Sˆ t,2 · · · Sˆ t,j ,

(9.53)

where

 Gbj

=

Sˆ wa,1

Sˆ wa,2

T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2

···

Sˆ wa,j T Sˆ wa,j Sˆ t,j

T .

(9.54)

The acceleration of the end-effector is given as Spa = T p Sa .

(9.55)

The accelerations of components could be formulated as Sca,j

   b T b b T T b = T c,j J j Gj Sa − St G HGSt + Gj St H j×j Gj St , j = 1, 2, . . . 5,

Sca,j

  = T c,j J bj q¨ j + q˙ Tj H j×j q˙ j ,

(9.56) (9.57)

where H denotes the Hessian matrix of 6R open loop mechanism. The resultant force at the center of end-effector of 6R robot could be stated as E Sw = SIw + SG w + Sw ,

(9.58)

E where SIw , SG w , and Sw are six-dimensional vectors, which denote inertia, gravity, and external forces, respectively. According to the D’Alembert principle, the inertia and gravity forces at the center point can be described as

278

9 Dynamic Modeling and Analysis of Robotic Mechanism



 0 = , mg    P SIw = mS˙ t + V mSPt = m T p J q¨ + q˙ T H q˙ + ΩT p J q˙ + V mT p J q˙ , SG w

(9.59) (9.60)

 T where g = 0 0 g denotes the gravity acceleration vector,  m=

     I 0 ω˜ 0 0 ω˜ p p p , St = T p St , S˙ t = Spa + ΩSt , V = ,Ω = , 0 mE3 0 v˜ 00

herein, I is the inertia matrix of the moving platform, respectively. St and S˙ t represent the velocity state and the reduced acceleration state of the moving platform center. v, ω, and m are the linear and angular velocities and the mass of the moving platform. Assuming that gravity force is the only external force exerted at components of 6R mechanism, the resultant force for the center could be obtained in a similar manner as p

p

Scw,j = SIw,j + SG w,j , j = 1, 2, . . . , 5,

(9.61)

where  SG w,j =

 0 . mj g

(9.62)

SIw,j could be mapped into task space and joint space as   SIw,j = m Sca,j + Ω j Sct,j + V j mSct,j

⎛   ⎞   b T b b T T b T c,j J j Gj Sa − St G HGSt + Gj St H j×j Gj St ⎠ = mj ⎝ + Ω j T c,j J bj Gbj St + V j mj T c,j J bj Gbj St ,

(9.63)

    SIw,j = mj T c,j J bj q¨ j + q˙ Tj H j×j q˙ j + Ω j T c,j J bj q˙ j + V j mj T c,j J bj q˙ j , j = 1, 2, . . . , 5 where    ω˜ j 0 0 ω˜ j , Ωj = Vj = . 0 0 0 v˜ j 

(9.64)

9.5 Example

279

The virtual work principle in 6R open-loop mechanism could be constructed as 5 

Sct,j

T     p T I T E ˙ f = 0, SIw,j + SG Sw + SG w,j + St w + St Sw + q

(9.65)

j=1

T  where f = f1 · · · f6 . Eq. (9.63) could be further expressed as ⎞ ⎛ 5  T  

 b b I G T I G E T ⎠ T c,j J j Gj Sw,j + Sw,j + T p Sw + Sw + Sw + G f = 0.

STt ⎝

(9.66)

j=1

When St is valid, dynamic model could be derived as ⎛

⎞ 5  T  

 T I G E⎠ f = −J T ⎝ T c,j J bj Gbj SIw,j + SG w,j + T p Sw + Sw + Sw .

(9.67)

j=1

The dynamic model shown in Eq. (9.67) can be further rewritten into the expressions shown in Eq. (9.44) or Eq. (9.45).

9.5.1.2

SPR Mechanism

The topological layout of the SPR mechanism is as shown in Fig. 7.5. The S joint is replaced by three mutually independent R joints. Hence, there are four components connected by five one-DoF joints. The 4th component links to the end-effector by the R joint. The velocity at the center of the end-effector is expressed as p

St = T p St ,

(9.68)

 E3 0 is the transformation matrix between the velocity of endwhere T p = −˜rp E3 effector described at point O and that described at point P. rp is the position vector from point O to P. Referring to the previous study, velocity of the center of the jth (j = 1, 2, 3, 4) component is derived as 

Sct,j = T cj J bj Gbj St = T cj J bj q˙ j , j = 1, 2, 3, 4

(9.69)

  J bj = Sˆ t,1 Sˆ t,2 · · · Sˆ t,j ,

(9.70)

where

280

9 Dynamic Modeling and Analysis of Robotic Mechanism

 Gbj =

Sˆ wa,1

Sˆ wa,2

T T Sˆ wa,1 Sˆ t,1 Sˆ wa,2 Sˆ t,2

···

Sˆ wa,j

T

T Sˆ wa,j Sˆ t,j

.

(9.71)

The acceleration at the center of the end-effector is given as Spa = T p Sa .

(9.72)

The acceleration at the center of the jth component is computed in task space or joint space, respectively, as

 T   Sca,j = T c,j J bj Gbj Sa − STt GT HGSt + Gbj St H j×j Gbj St , j = 1, 2, 3, 4,

Sca,j

  = T c,j J bj q¨ j + q˙ Tj H j×j q˙ j .

(9.73) (9.74)

The resultant force at the center of end-effector could be stated as E Sw = SIw + SG w + Sw ,

(9.75)

E where SIw , SG w , and Sw are the inertia, gravity, and external forces, respectively. The inertia and gravity forces at the center point of end-effector can be described by



 0 = , mg    P SIw = mS˙ t + V mSPt = m T p J q¨ + q˙ T H q˙ + ΩT p J q˙ + V mT p J q˙ . SG w

(9.76) (9.77)

Assume that gravity force is the only external force exerted at a component composing a robotic mechanism, the resultant force for the center could be obtained in a similar manner as SCw,j = SIw,j + SG w,j , j = 1, 2, 3, 4,

(9.78)

where  SG w,j =

 0 . mj g

SIw,j could be mapped into task space and joint space as

(9.79)

9.5 Example

281

  SIw,j = m Sca,j + Ω j Sct,j + V j mSct,j

⎛   ⎞   b T b b T T b T c,j J j Gj Sa − St G HGSt + Gj St H j×j Gj St ⎠ = mj ⎝ + Ω j T c,j J bj Gbj St + V j mj T c,j J bj Gbj St ,

(9.80)

    SIw,j = mj T c,j J bj q¨ j + q˙ Tj H j×j q˙ j + Ω j T c,j J bj q˙ j + V j mj T c,j J bj q˙ j , j = 1, 2, 3, 4,

(9.81)

where    ω˜ j 0 0 ω˜ j , Ωj = Vj = . 0 0 0 v˜ j 

The virtual work principle of SPR mechanism could be constructed as 4  T    

p T I T E ˙ f = 0, Sct,j SIw,j + SG Sw + SG w,j + St w + St Sw + q

(9.82)

j=1

T  where f = f1 · · · f5 . Hence, the actuation force of the SPR mechanism is computed as ⎛

⎞ 4  T  

 T I G E⎠ f = −J T ⎝ T c,j J bj Gbj SIw,j + SG w,j + T p Sw + Sw + Sw .

(9.83)

j=1

9.5.2 Typical Closed-loop Mechanism Exechon mechanism has two rotational motions and one translational motion, which consists of a fixed base connecting with moving platform by two UPR limbs and one SPR limb, as illustrated in Fig. 7.6. The dynamic model of the Exechon mechanism is as follows. First of all, the velocity state of the moving platform could be obtained as p

St = T p St ,

(9.84) p

where St = J a q˙ a . Herein, J a and q˙ a are referred to Eq. (7.140). St represents the velocity of the center of moving platform. Based upon Eq. (9.6), the mapping between all the joint parameters in the limbs and moving platform could be formulated as

282

9 Dynamic Modeling and Analysis of Robotic Mechanism

q˙ i = Gi St , i = 1, 2, 3,

(9.85)

where Gi is the matrix constituted by all the regularized wrenches in the ith limb,  Gi =  Gi =

Sˆ wa,i,1

Sˆ wa,i,2

Sˆ wa,i,3

Sˆ wa,i,4

T

T T T T Sˆ wa,i,1 Sˆ t,i,1 Sˆ wa,i,2 Sˆ t,i,2 Sˆ wa,i,3 Sˆ t,i,3 Sˆ wa,i,4 Sˆ t,i,4

Sˆ wa,i,1

Sˆ wa,i,2

Sˆ wa,i,3

Sˆ wa,i,4

, i = 1, 3, Sˆ wa,i,5

T

T T T T T Sˆ wa,i,1 Sˆ t,i,1 Sˆ wa,i,2 Sˆ t,i,2 Sˆ wa,i,3 Sˆ t,i,3 Sˆ wa,i,4 Sˆ t,i,4 Sˆ wa,i,5 Sˆ t,i,5

, i = 2,

where Sˆ w,i,k denotes the unit wrench corresponding to the kth one-DoF joint in the ith limb. In this way, the twists of the centers of all the components in the three limbs can be written as SBt i Ai = T Bi Ai J bi,3 Gbi,3 St , i = 1, 3,

(9.86)

SBt i Ai = T Bi Ai J bi,4 Gbi,4 St , i = 2,

(9.87)

where SBt i Ai (i = 1, 2, 3) are the velocities of the centers of the components (links) Bi Ai ,  T   J bi,3 = Sˆ t,i,1 Sˆ t,i,2 Sˆ t,i,3 , Gbi,3 = Gi,1 Gi,2 Gi,3 , i = 1, 3,  T   J bi,4 = Sˆ t,i,1 · · · Sˆ t,i,4 , Gbi,4 = Gi,1 · · · Gi,4 , i = 2. T Bi Ai are the transformation matrices between the velocities of point O and those of these centers, i.e.,  T Bi Ai =

 E3 0 , −˜rBi Ai E3

in which rBi Ai is the position vector from O to the center of the component Bi Ai . In the meanwhile, Eqs. (9.86)–(9.87) could be written in joint space as SBt i Ai = T Bi Ai J bi,3 Gbi,3 J a q˙ a , i = 1, 3,

(9.88)

SBt i Ai = T Bi Ai J bi,4 Gbi,4 J a q˙ a , i = 2.

(9.89)

9.5 Example

283

The acceleration model of Exechon mechanism can be described as  ˙ a q˙ a , Sa = J a q¨ a + J −1 w h q

(9.90)

 where h q˙ a is the matrix constituted by the reciprocal product of the unit wrenches and the Coriolis term of each limb acceleration, i.e.,  ⎤ ⎡ T T Sˆ wa,1,3 G1 J a q˙ a H 1 G1 J a ⎢ T  ⎥ T ⎢ˆ ⎥ ⎢ Swa,2,4 G2 J a q˙ a H 2 G2 J a ⎥ ⎢ T ⎥   T ⎢ ⎥  ⎢ Sˆ wa,3,3 G3 J a q˙ a H 3 G3 J a ⎥ ⎢   ⎥. h q˙ a = ⎢ T T ⎥ ⎢ Sˆ wc,1,1 G1 J a q˙ a H 1 G1 J a ⎥ ⎢ T ⎥   T ⎢ˆ ⎥ ⎢ Swc,2,1 G2 J a q˙ a H 2 G2 J a ⎥ ⎣ T  ⎦ T Sˆ wc,3,1 G3 J a q˙ a H 3 G3 J a Equation (9.90) shows the mapping between the acceleration of actuations and moving platform in a closed-loop mechanism, which could be rewritten as   ˙ a q˙ a . q¨ a = Ga Sa − J −1 w h q

(9.91)

The acceleration at the center point P of moving platform is solved as Spa = T p Sa ,

(9.92)

where T p is referred to that in Eq. (9.1). The accelerate of the ith limb could be formulated by differentiating its twist as Sa,i = J i q¨ i + q˙ Ti H q˙ i , i = 1, 2, 3.

(9.93)

The acceleration mapping between three actuation joints and all the joints in that limb is    T (9.94) q¨ i = Gi Sa − Gi J a q˙ a H i Gi J a q˙ a , i = 1, 2, 3. The accelerations of the centers of all the components in the three limbs can be derived as   SBa i Ai = T Bi Ai J bi,3 Gbi,3 Sa − (Gi St )T H i Gi St  T  + Gbi,3 St H i,3×3 Gbi,3 St , i = 1, 3,

(9.95)

284

9 Dynamic Modeling and Analysis of Robotic Mechanism

    T SBa i Ai = T Bi Ai J bi,4 Gbi,4 Sa − (Gi St )T H i Gi St + Gbi,4 St H i,4×4 Gbi,4 St , i = 2, (9.96) where H i,k×k is the first k × k matrix block of H i . Then in the joint space, we have 

SBa i Ai



  −1 ¨ q J + J h q˙ a q˙ a a  a Tw = T Bi Ai J bi,3 Gbi,3 − Gi J a q˙ a H i Gi J a q˙ a   T + Gbi,3 J a q˙ a H i,3×3 Gbi,3 J a q˙ a , i = 1, 3, 

SBa i Ai

  J a q¨ a + J −1 h q˙ a q˙ a w T  = T Bi Ai − Gi J a q˙ a H i Gi J a q˙ a   T + Gbi,4 J a q˙ a H i,4×4 Gbi,4 J a q˙ a , i = 2.

(9.97)



J bi,4 Gbi,4

(9.98)

The resultant force at the center of moving platform could be stated as E Sw = SIw + SG w + Sw ,

(9.99)

E where SIw , SG w , and Sw are six-dimensional vectors, which denote inertia, gravity, and external forces, respectively. According to the D’Alembert principle, the inertia and gravity forces of the center of moving platform can be described as



SG w

 0 = , mg

P SIw = mS˙ t + V mSPt    ˙ a q˙ a + ΩT p J a q˙ a + V mT p J a q˙ a , = m T p J a q¨ a + J −1 w h q

(9.100)

(9.101)

 T where g = 0 0 g denotes the gravity acceleration vector,      I 0 ω˜ 0 0 ω˜ p p p p ˙ , St = T p St , St = Sa + ΩSt , V = m= , Ω= . 0 mE3 0 v˜ 00 

The resultant force at the center of a component could be obtained as Scw,i,3 = SIw,i,3 + SG w,i,3 , i = 1, 3,

(9.102)

Scw,i,4 = SIw,i,4 + SG w,i,4 , i = 2,

(9.103)

I where SG w,i,ki and Sw,i,ki are the gravity and inertial wrenches of the component in the ith limbs,

9.5 Example

285

 SG w,i,ki

=

0 mi,ki g

 , i = 1, 2, 3, k1 = k3 = 3, k2 = 4.

(9.104)

SIw,i,ki could be mapped into task space and joint space as 

 ⎞ J bi,ki Gbi,ki Sa − (Gi St )T H i Gi St + ⎜ T c,i,ki  b T ⎟ c = mi,ki ⎝ ⎠ + V i,ki mi,k St,i,ki , Gi,ki St H i,ki ×ki Gbi,ki St + Ω i,ki Sct,i,ki (9.105) ⎛

SIw,i,ki

SIw,i,ki

 ⎞ T   ˙ ˙ ˙ ˙ J bi,ki Gbi,ki J a q¨ a + J −1 q h q − G J H G J q q i a i i a a a a a w ⎜ T c,i,ki ⎟ T  = mi,ki ⎝ ⎠ + Gbi,ki J a qa H i,ki ×ki Gbi,ki J a q˙ a + Ω i,k Sct,i,ki ⎛



+ V i,ki mi,ki Sct,i,ki

(9.106)

where Sct,i,ki = T c,i,ki J bi,ki Gbi,ki J a q˙ a = T c,i,ki J bi,ki Gbi,ki St ,     0 ω˜ i,k ω˜ i,ki 0 Ω i,k = . , V i,ki = 0 0 0 v˜ i,ki Based upon the velocities and accelerations of the centers of all the components, the dynamic equation could be constructed based on virtual work as  c T  I  c T  I  c T  I + St,2,4 Sw,2,4 + SG + St,3,3 Sw,3,3 + SG St,1,3 Sw,1,3 + SG w,1,3 w,2,4 w,3,3  p T  T E ˙ Ta f a = 0. + St SIw + SG w + (St ) Sw + q (9.107) In this way, Eq. (9.107) could be expanded as T     b b T I Sw,2,4 + SG T c,1,3 J b1,3 Gb1,3 SIw,1,3 + SG w,1,3 + T c,2,4 J 2,4 G2,4 w,2,4  T   p T  I T E ˙ Ta f a = 0. + T c,3,3 J b3,3 Gb3,3 SIw,3,3 + SG Sw + SG w,3,3 + St w + (St ) Sw + q (9.108)

286

9 Dynamic Modeling and Analysis of Robotic Mechanism

To this end, the dynamic model could be derived as ⎛

T    T  ⎞ b Gb I G b Gb I G S + T S J + S J + S T c,1,3 c,2,4 1,3 1,3 w,1,3 w,1,3 2,4 2,4 w,2,4 w,2,4 ⎟ ⎜ f a = −J Ta ⎝  T   ⎠.  T SI + S G + SE SIw,3,3 + SG + T + T c,3,3 J b3,3 Gb3,3 p w w w w,3,3

(9.109)

9.6 Conclusion In this chapter, dynamic modeling method of open-loop and closed-loop mechanisms is proposed. Based on the FIS theory, the velocities and accelerations of endeffector/moving platform, all the components are computed. The wrenches including actuations, inertia, gravity, and external wrenches are analyzed. On this basis, the virtual work equation is formulated and the dynamic model is obtained. The presented dynamic modeling is straightforward and completes the integrated framework involving type synthesis and performance analysis. For the reader’s convenience, the key points of this chapter are represented as follows: (1) By using the differential mapping between finite instantaneous screws, the velocities of moving platform, components, and joints in open-loop and closed-loop robotic mechanisms are formulated. (2) An FIS based method for solving accelerations of all the components including end-effector/moving platform and components in open-loop and closed-loop mechanisms is proposed in an algebraic manner. (3) All the wrenches which contain inertial and gravity forces, actuations and frictions exerted at moving platform, components, and joints are analyzed and expressed in six-dimensional vector form. (4) Based on the virtual work principle, dynamic modeling for open-loop and closed-loop robotic mechanisms is carried out in both task and joint space.

References 1. Liang D, Song YM, Sun T (2016) Optimum design of a novel redundantly actuated parallel manipulator with multiple actuation modes for high kinematic and dynamic performance. Nonlinear Dyn 83:631–658 2. Wu J, Chen X, Wang L (2014) Dynamic load-carrying capacity of a novel redundantly actuated parallel conveyor. Nonlinear Dyn 78(1):241–250 3. Sun T, Lian BB, Song YM (2019) Elasto-dynamic optimization of a 5-DoF parallel kinematic machine considering parameter uncertainty. IEEE/ASME Trans Mechatron 24(1):315–325

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4. Sun T, Yang SF (2019) An approach to formulate the Hessian matrix for dynamic control of parallel robots. IEEE/ASME Trans Mechatron 24(1):271–281 5. Wang J, Gosselin CM (1998) A new approach for the dynamic analysis of parallel manipulators. Multibody Sys Dyn 2(3):317–334 6. Khalil W, Guegan S (2004) Inverse and direct dynamic modeling of Gough-Stewart robots. IEEE Trans Rob 20(4):754–761 7. Chen GL, Lin ZQ (2009) Forward dynamics analysis of spatial parallel mechanisms based on the Newton-Euler method with generalized coordinates. J Mech Eng 45(7):41–48 8. Miller K (2004) Optimal design and modeling of spatial parallel manipulators. Int J Robot Res 23(2):127–140 9. Yun Y, Li Y (2011) A general dynamics and control model of a class of multi-DOF manipulators for active vibration control. Mech Mach Theory 46(10):1549–1574 10. Liang D, Song YM, Sun T (2017) Nonlinear dynamic modeling and performance analysis of a redundantly actuated parallel manipulator with multiple actuation modes based on FMD theory. Nonlinear Dyn 89(1):391–428 11. Danaei B, Arian A, Tale Masouleh M (2017) Dynamic modeling and base inertial parameters determination of a 2-DOF spherical parallel mechanism. Multibody Sys Dyn 41(4):367–390 12. Shao PJ, Wang Z, Yang SF (2019) Dynamic modeling of a two-DoF rotational parallel robot with changeable rotational axes. Mech Mach Theory 131:318–335 13. Song YM, Dong G, Sun T et al (2016) Elasto-dynamic analysis of a novel 2-DoF rotational parallel mechanism with an articulated travelling platform. Meccanica 51:1547–1557 14. Liang D, Song YM, Sun T (2017) Rigid-flexible coupling dynamic modeling and investigation of a redundantly actuated parallel manipulator with multiple actuation modes. J Sound Vib 403:129–151 15. Tsai LW (2000) Solving the inverse dynamics of a Stewart-Gough manipulator by the principle of virtual work. ASME J Mech Des 122(1):3–9 16. Abdellatif H, Heimann B (2009) Computational efficient inverse dynamics of 6-DOF fully parallel manipulators by using the Lagrangian formalism. Mech Mach Theory 44(1):192–207 17. Gallardo J, Rico JM, Frisoli A (2003) Dynamic of parallel manipulators by means of screw theory. Mech Mach Theory 38(11):1113–1131

Chapter 10

Optimal Design of Robotic Mechanism

10.1 Introduction Optimal design of robotic mechanism is the process of adjusting the structural parameters to achieve optimal performances while keeping them from not violating mechanism constraints [1–3]. This is the step of connecting the mechanism property to the application requirements of the robot. For instance, the robot being applied in manufacturing such as machining and milling is required to be equipped with high stiffness but low weight [4]. Hence, the stiffness and dynamic performances of the robot are considered in the optimal design and regarded to be either objectives or constraints. Corresponding performance models containing the relations between structural parameters and the performances built in the previous chapters are applied. By changing the structural parameters which are called design variables in the optimal design, the optimal performances of the robot meeting the application requirements can be achieved with the aid of optimization algorithm [5–8]. Optimal design of the robotic mechanism mainly deals with the parameters and the concerned performances. On one hand, optimal design provides guidance for building the physical prototype of the robotic mechanism which undergoes component machining and mechanism assembling. Due to the inevitable errors in machining and assembling, the actual parameters of the physical prototype are not exactly the same as the theoretical values given by the optimal design [9, 10]. The differences in the theoretical and actual parameters would result in the differences in the actual performances and the designed performances of the robot. However, the parameter changes, named as parameter uncertainty in this book, caused by the prototype establishment have not been considered in the conventional optimal design procedures. It might lead to unreliable optimal results. On the other hand, the same performances would lead to different designed results if they are set differently as objective or constraints. Taking again the robot to be designed with high stiffness and low weight as an example, both stiffness and mass might be objectives with weighting factors or mass might be objective while stiffness is considered as constraints. These two typical treatments © Springer Nature Singapore Pte Ltd. 2020 T. Sun et al., Finite and Instantaneous Screw Theory in Robotic Mechanism, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-1944-4_10

289

290

10 Optimal Design of Robotic Mechanism

of the performances belong to the single objective optimization problem [11], where the former formulates one objective function by assigning weighting factors [12] to different performances and the latter is to convert the unimportant performances into constraints [13]. The single objective optimization is implemented based on the designers’ experience. However, in most cases, the performances are expected to be treated without any predefined relations hence the unbiased optimal results can be obtained. A multi-objective optimization that simultaneously optimizes the concerned performances is thus considered. This chapter presents the optimal design method of robotic mechanism, mainly addressing the parameter uncertainty and multi-objective optimization problems. Optimal design of the typical open-loop mechanism and closed-loop mechanism are given to illustrate the method. The present work offers reliable parameters having the best compromise among multiple performances to guide the construction of the physical prototype.

10.2 Parameter Uncertainty As has been mentioned, parameter uncertainty refers to the deviations between the actual parameters of the physical prototype and the theoretical parameters offered by the optimal design. Such parameter deviations result from the constructions of the physical prototype, including machining and assembling errors. They can also be caused by the abrasion during operation or deformation when temperature changes. Due to the effects of these practical applications, the parameter deviations are inevitable, leading to the deviations of actual performances and the optimal performances of the robot. If the performance is sensitive to the parameter changes, the actual performances of the physical prototype would largely deviate from the expected values. Hence, the robot fails to be equipped with optimal performances. What’s worse, the parameter deviations might result in the actual parameters violating the constraints, indicating that the engineering requirements could not be met and the physical prototype could not be used. In order to prevent such catastrophe, parameter deviations should be considered in the optimal design. Since the actual parameters are unknown in the design phase, parameter deviations are called parameter uncertainty in this book. The first step of the optimal design concerning parameter uncertainty is to define the objectives and constraints.

10.2.1 Statistical Objective Performance indices are directly applied as objectives in the conventional optimal design. They are formulated based on the performance models built in the previous chapters. The model of the performance index of a robotic mechanism can be expressed as

10.2 Parameter Uncertainty

291

F = f (x),

(10.1)

where F is the performance index, x is the vector of structural parameters that will be treated as the design variable. It is known from Eq. (10.1) that one set of design variables correspond to a certain value of the performance index. Considering the parameter uncertainty, each design variable might have an uncertain deviation that is determined by the actual application. Taking the link length l(l ∈ x) as an example, if the machining error is considered to be the main cause for the parameter deviation, the actual length la  would be the value within the range ln − l, ln + l , where ln is the expected value and l is the machining tolerance of the link. Although the range of the actual parameter can be estimated from the application, the value of the actual parameter is still uncertain, which can be any value within the range. In order to describe the parameter uncertainty, probabilistic simulation method is introduced. Taking l again as an example, N random values are generated by specifying mean value l and standard deviation σl . The mean value l is given by the nominal value ln while standard deviation is assigned by the tolerance. If the sample size N is large enough, there is a big possibility that the minor change of l during construction is captured. In the conventional optimal design, the values of design variables are changed by optimizer to search for the optimal parameters. Considering parameter uncertainty, the value of each design variable visited by optimizer is replaced with the N samples. Therefore, by applying random samples of parameters, design variables become   X = X 1 , X 2 , . . . , X j , . . . , X N , j = 1, 2, . . . , N ,

(10.2)

  X j = x1, j , x2, j , . . . , xi, j , . . . , xn, j , i = 1, 2, . . . , n,

(10.3)

where X is design variables with parameter uncertainty, X j denotes the jth set of random design variables, and xi, j represents the jth random value for design variable xi . Corresponding to the N samples of design variables, there would be N performance indices at each design point according to Eq. (10.1). For instance, l0 ∈ (l L , l U ) is assigned by the optimizer during the optimization. Herein, l L and l U are the lower and upper  bounds for linklength given by design requirements. Replace l0 with l 0 = l0,1 · · · l0,k · · · l0,N . The rest design variables are addressed in the same manner. Performance index of robotic mechanism F0 becomes   F 0 = F0,1 F0,2 · · · F0,k · · · F0,N ,

(10.4)

where F 0 is named as performance index with parameter uncertainty. The statistical features of the performance index can be formulated to address the effects of parameter uncertainty as

292

10 Optimal Design of Robotic Mechanism

  N N 1 

2 1  F0,k − κ0 (F 0 ) , κ0 (F 0 ) = F0,k , σ0 (F 0 ) = N k=1 N k=1

(10.5)

where κ0 denotes the mean value of F 0 , and σ0 is its standard variations. The mean value and standard variation of the performance indices calculated by the N random samples of the design variables are regarded as the objectives of the optimal design considering parameter uncertainty. During the optimization, the design variables leading to the performance with low mean values but high standard variations will not be chosen by optimizer. It indicates that the set of design variables are discarded if parameter uncertainty has great influence on the performance indices. The higher the mean values and lower standard variations, the more reliable and more stable are the future performances of a physical prototype.

10.2.2 Probabilistic Constraint The constraint for the optimal design of robotic mechanism can be classified into geometric and performance constraints. The former belongs to the geometric constraint coming from the geometric features of the robotic mechanism. The latter is the performance constraint interpreted from the target task. In the conventional optimal design, the constraints are expressed as

g(x) ≥ 0 , g(x) = gr (x) − gcL or g(x) = gcU − gr (x)

(10.6)

where gcU , gcL are the upper and lower limits of constraint condition, gr (x) is the model between the constraint condition and the design variables. For example, the lower limit of linear stiffness along x-axis is given as 2 × 106 N/m by end user. Under certain x, gr (x) is calculated as 1.98 × 106 N/m. Then the constraint is violated since g(x) = (1.98 − 2) × 106 < 0. Assume that the values of design variable are given by the optimizer and denoted by x 0 . N random samples of x 0 are generated to consider parameter uncertainty as mentioned in the previous section. Corresponding to these N sets of design variables, N constraint conditions can be calculated referring to Eq. (10.6). The probabilistic constraints can be defined in the form of P(g(X) ≥ 0) ≥ PR ,

(10.7)

where P(g(X) ≥ 0) denotes the probability of the N samples satisfying g(X) ≥ 0. PR ∈ (0, 1) is a predefined value indicating the acceptable probability. Probability computation can be implemented by

10.2 Parameter Uncertainty

293

P(g(X) ≥ 0) =

N 1  I (X), N i=1

(10.8)

where I (X) is an indicator function, I (X) =

1 g(X) ≥ 0 . 0 g(X) < 0

(10.9)

Assume that N = 10,000, PR = 0.95. Equation (10.7) means that the design point x would have little influence on the constrained condition if more than 95% of the 10,000 sets of random design variables satisfy g(X) ≥ 0.

10.3 Multi-objective Optimization 10.3.1 Performance Index Based on the performance model in the previous chapters, performance indices can be defined to be adopted for the formulation of statistical objectives or probabilistic constraints. The performance indices introduced in this section are aimed for clear physical interpretation and unified units.

10.3.1.1

Kinematic Index

At the displacement level, the workspace of the robotic mechanism has been analyzed in Chap. 7, which can be applied as the objective in the optimal design. At the velocity level, the twist and wrench mapping model of the robotic mechanism has also been discussed in Chap. 7. The model reveals the relations of kinematic performance between the joint space and operated space. At the acceleration level, the performance indices will be established by evaluating the dynamic performances. In this chapter, the kinematic indices at the velocity level are concerned. Conventionally, the features of the Jacobian matrix (determinant, condition number, singular number, etc.) are adopted to be the kinematic indices. However, it is recognized that both linear and angular velocities are involved in the Jacobian matrix. The different units might result in an ambiguous assessment of kinematic performance. To address this problem, virtual power transmissibility [14] is proposed. Referring to Chap. 7, the twist mapping model between the joint space and operated space of a robotic mechanism can be given as ˙ J w St = J q q,

(10.10)

294

10 Optimal Design of Robotic Mechanism

where St denotes the twist of end-effector or the twist of moving platform. q˙ is the vector consisting actuation joint parameters. Suppose the matrix J w is full rank, the twist mapping model can be further written as ˙ St = J −1 w J q q,

(10.11)

The columns of J −1 w can be expressed as 

∗ ˆ∗ ∗ ˆ∗ ∗ ˆ∗ . = J −1 S S S ρ ρ · · · ρ w 1 t,1 2 t,2 6 t,6

(10.12)

Taking n-DoF closed-loop mechanism as an example, the twist mapping model can be given as St =

n 

 T  ∗ ∗ Sˆ wa,k,gk Sˆ t,k,gk Sˆ t,k . ρt,k,gk ρt,k

(10.13)

k=1

Referring to Eq. (7.86), the wrench model of the closed-loop mechanism is Sw = Swa + Swc =

gi l  

f a,i,ka Sˆ wa,i,ka +

i=1 ka=1

l 6−n  i

f c,i,kc Sˆ wc,i,kc .

(10.14)

i=1 kc=1

Taking the inner product of Eqs. (10.13) and (10.14) yields T Sˆ w Sˆ t =

n 

 T  T  ∗ ∗ Sˆ wa,k,gk Sˆ t,k Sˆ wa,k,gk Sˆ t,k,gk , f a,k,gk ρt,k,gk ρt,k

(10.15)

k=1 T where Sˆ w Sˆ t is the instantaneous power of the robotic mechanism.

T ∗ ∗ are the intensity of the wrench or twist, Sˆ wa,k,gk Sˆ t,k Since f a,k,gk , ρt,k,gk , and ρt,k T and Sˆ wa,k,gk Sˆ t,k,gk can be applied to assess the power transmitted from the joint space to the operated space. They are named as power transmissibility. If the power transmissibility is close to zero, it indicates that the power offered by the actuation joints cannot transmit to the moving platform and the mechanism might be at the singular pose. The higher the value of the power transmissibility, the better the kinematic performance the robot will obtain. Therefore, the generalized power transmissibility can be defined as ⎧   T   ⎫ T ∗ ˆ  ⎬ ⎪ ⎨  Sˆ wa,k,gk Sˆ t,k   Swa,k,gk Sˆ t,k,gk  ⎪  , T  . (10.16) μk =  T ∗ ˆ  ⎭ ⎪ ⎩  Sˆ wa,k,gk Sˆ t,k   Swa,k,gk Sˆ t,k,gk  ⎪ max

max

10.3 Multi-objective Optimization

295

The generalized power transmissibility varies with the change of configurations. Hence, local and global power transmissibility within workspace can be, respectively, defined by vk = min(μk )

(10.17)

k

 vk =

V

vk d V , v˜ k = V

 V

(vk − v k )2 d V V

(10.18)

where V denotes the volume of workspace. v k denotes the mean value of vk within workspace, and v˜ k denotes its standard variation.

10.3.1.2

Static Stiffness Index

According to mechanics, the virtual work of manipulator will be converted to instantaneous energy regardless of frictions. Virtual work is the product of wrench on the corresponding deformations. That is to say, exerting the same wrench, manipulators with higher stiffness would result in smaller deformation twist. In other words, smaller instantaneous energy means better stiffness performance. Instantaneous energy [15] can be formulated by reciprocal product of deformation twist and wrench. The deformation twist can be divided into three-dimensional instantaneous linear and angular deformation screws. Similarly, the wrench is composed of three-dimensional pure force and moment screws. Instantaneous pure force (moment) screw does not do work on instantaneous angular (linear) deformation twist. Hence, the instantaneous energy can be defined as E = STw Ste =

3  jl =1

E jl +

3 

E ja , jl = ja = 1, 2, 3,

(10.19)

ja =1

where El, jl (E a, ja ) represents the jl th ( ja th) instantaneous energy associated with the instantaneous pure force (moment) screw on the jl th ( ja th) instantaneous linear (angular) deformation screw.     E jl =  STw, jl Ste, jl , E ja =  STw, ja Ste, ja .

(10.20)

Mainly drawing on Hooke’s Law, the relationship between the wrench Sw and deformation twist Ste of robotic mechanism is formulated as Sw = K Ste , Ste = C Sw ,

(10.21)

where K and C are the stiffness and compliance matrices of the robotic mechanism.

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10 Optimal Design of Robotic Mechanism

It is found out that compliance matrix is a Hermite matrix, whose eigenvalue and eigenvector satisfy the following equations 

T Sˆ c,i Sˆ c, j = 1 i = j , C Sˆ c,i = λc,i Sˆ c,i , i = j = 1, 2, . . . 6, T Sˆ c,i Sˆ c, j = 0 i = j

(10.22)

where Sˆ c,i is the eigenvector of compliance matrix, and λc,i denotes the corresponding eigenvalue. If the number of independent eigenvalue is 6, corresponding eigenvectors can be directly applied to Eq. (10.20). If not, the eigenvectors should be formulated by Gram–Schmidt process. The wrench can be uniquely expressed by the basis of the instantaneous payload screw as Sw =

3 

ρl, jl Sˆ l, jl +

jl =1

3 

ρa, ja Sˆ a, ja ,

(10.23)

ja =1

where Sˆ l, jl and Sˆ a, ja denote the base of instantaneous pure force and moment screw, ρl, jl and ρa, ja represent the magnitudes, respectively.    si 0 , i = 1, 2, 3, , Sˆ a,i = 0 si

T

T

T s1 = 1 0 0 , s2 = 0 1 0 , s3 = 0 0 1 .

Sˆ l,i =



(10.24)

The dimension of the space spanned by instantaneous pure force and moment screw is equal to the number of independent eigenvectors of compliance matrix. The base of the wrench space can also be expressed by the linear combination of eigenvectors Sˆ c,i according to linear algebra. Hence, Sˆ l, jl =

6 

δl,i, jl Sˆ c,i , Sˆ a, ja =

i=1

6 

δa,i, ja Sˆ c,i , i = 1, 2, . . . 6,

(10.25)

i=1

where δl,i, jl (δa,i, ja ) represents the linear coefficients of Sˆ c,i , whose value can be calculated as follows 

δl,1, jl · · · δl,6, jl

T

T  = W −1 Sl, jl , δa,1, ja · · · δa,6, ja T

= W −1 Sa, ja , W = Sˆ c,1 · · · Sˆ c,6 .

Substituting Eqs. (10.25)–(10.26) into Eq. (10.23) leads to

(10.26)

10.3 Multi-objective Optimization

Sw =

6 

297

⎛⎛ ⎞ ⎞ ⎛⎛ ⎞ ⎞ 3 6 3    ⎝⎝ ⎝⎝ ρl, jl δl,i, jl ⎠ Sˆ c,i ⎠ + ρa, ja δa,i, ja ⎠ Sˆ c,i ⎠. jl =1

i=1

(10.27)

ja =1

i=1

Once the wrench Sw is determined, corresponding deformation twist Ste can be generated by Eqs. (10.21)–(10.22) and (10.27) as Ste =

6 

⎛⎛ ⎞ ⎞ 3 3   ⎝⎝ ρl, jl δl,i, jl + ρa, ja δa,i, ja ⎠λc,i Sˆ c,i ⎠.

i=1

jl =1

(10.28)

ja =1

Therefore, the instantaneous energy in Eq. (10.19) is rewritten as E = El + E a , El =

6 

⎛⎛ ⎝⎝

3  jl =1

i=1

⎞⎛ ρl, jl δl,i, jl ⎠⎝

3 

ρl, jl δl,i, jl +

jl =1

(10.29) 3 

⎞

⎞

T ρa, ja δa,i, ja ⎠λc,i Sˆ c,i Sˆ c,i ⎠,

ja =1

(10.30) ⎞⎛ ⎞ ⎞ 6 3 3 3     T ⎝⎝ Ea = ρa, ja δa,i, ja ⎠⎝ ρl, jl δl,i, jl + ρa, ja δa,i, ja ⎠λc,i Sˆ c,i Sˆ c,i ⎠, ⎛⎛

i=1

ja =1

jl =1

ja =1

(10.31) where El and E a indicate the instantaneous energy generated by the instantaneous pure force screw and the instantaneous pure moment screw, respectively. Based on Eqs. (10.30)–(10.31), the instantaneous linear/angular stiffness performance index can be formulated by applying the instantaneous pure force/moment screw Sˆ l,i , Sˆ a,i along/about x, y, and z axes as   6  ⎧ "

2  ⎪  ⎪ ηl,x =  ρl,1 δl,i,1 λc,i  ⎪ ⎪ ⎪ i=1 ⎪   6  ⎨ "

2  ηl,y =  ρl,2 δl,i,2 λc,i  , ⎪ ⎪   i=1 ⎪ ⎪ 6 

" 2  ⎪ ⎪ ⎩ ηl,z =  ρl,3 δl,i,3 λc,i  i=1

  6  ⎧ "

2  ⎪  ⎪ ηa,x =  ρa,1 δa,i,1 λc,i  ⎪ ⎪ ⎪ i=1 ⎪   6  ⎨ "

2  ηa,y =  ρa,2 δa,i,2 λc,i  ⎪ ⎪   i=1 ⎪ ⎪ 6 

"  2  ⎪ ⎪ ⎩ ηa,z =  ρa,3 δa,i,3 λc,i 

(10.32)

i=1

where ηl,x , ηl,y , and ηl,z are defined as the instantaneous linear stiffness performance index of the robotic mechanism. Similarly, ηa,x , ηa,y , and ηa,z are known as the instantaneous angular stiffness performance index. The overall stiffness performance within the workspace of the robotic mechanism can be calculated by adding instantaneous linear and angular stiffness performance indices as

298

10 Optimal Design of Robotic Mechanism

 ⎛⎛ ⎞2 ⎞  6  3 3     ⎝⎝ η =  ρl, jl δl,i, jl + ρa, ja δa,i, ja ⎠ λc,i ⎠.  i=1  jl =1 ja =1

(10.33)

Furthermore, the global instantaneous stiffness performance index can be formulated as   2 V (η − η) d V V ηd V , η˜ = , (10.34) η= V V where V describes the volume of the prescribed workspace, η represents the mean value of η over the prescribed workspace, and η˜ is the standard variation.

10.3.1.3

Dynamic Index

Dynamic behavior of a robotic mechanism reflects the relationship of the motions and forces, in which the motions include the acceleration, velocity, and displacement of the mechanism and the forces concern the inertial force, gravity, and external wrenches on the mechanism. Ignoring the external wrench that is controlled by the application scenario of the robot, the inherent dynamic behavior is determined by the terms relating to the acceleration and velocity. Corresponding to the acceleration of the robot, the inertia matrix can be obtained from the dynamic model. It shows the inertia property of the robotic mechanism, including the inertia properties of the components and the moving platform. The condition number of inertia matrix is adopted to access dynamic performance. If the value of condition number is very large, the inertia matrix is close to be singular and the robotic mechanism behaves badly in some region. If the value of the condition number is close to one, the robotic mechanism has similar acceleration capability at any configuration within workspace. Therefore, the inverse of condition number of the inertia matrix is applied to be the performance index of the robotic mechanism as χ = 1/cond(M),

(10.35)

where χ is expected to be close to one. The parameters resulting in the χ close to zero are discarded during optimization. The global indices of the robotic mechanism could be defined as  χ=

V

χdV , χ˜ = V



V (χ

− χ )2 d V V

,

(10.36)

where χ and χ˜ denote the mean value and standard variation within workspace.

10.3 Multi-objective Optimization

299

10.3.2 Response Surface Method Model Based on the performance indices mentioned in the previous sections, sometimes the global performances, such as the overall stiffness performance within workspace, are considered. The formulation of the model between performance indices and structural parameters can be referred to as numerical computation such as discrete searching strategy. For doing so, the workspace is evenly meshed into finite elements. All the nodes of these elements can be described by x, y, z axes of Cartesian coordinates, representing different configurations of the robotic mechanism. The global performance indices can be calculated by the statistical features of the performance at all the nodes. Therefore, the number of nodes affects the accuracy of the computed global performances. This inaccuracy would be further accumulated in the optimization that searches for optimal result by iterative algorithm. In addition, exhaustive calculation at every node within workspace is repeated for each parameter set in the optimization process. This situation would be even worse if we want a large sample size for better estimation of parameter uncertainty at each design variable set given by the optimizer. The enormous iteration and high computational cost would lower the optimization efficiency. To deal with this issue, the response surface method (RSM) model [4] is applied to build an explicit model between performances and design variables. Based on design of experiment (DoE), the core idea of the RSM is to build an approximate model matching the experimental result and verify the model by statistical technique. The establishment of the RSM model is implemented as follows. Step 1: Determine different combinations of parameters by DoE. It can arrange a moderate number of parameter sets that would provide enough information for the formulation of explicit mapping model; Step 2: Calculate performance indices based on the performance models. For the global performance indices, n l nodes are applied to mesh the workspace for computing the global behaviors of the robotic mechanism; Step 3: Formulate the analytical mapping models by RSM. These analytical mapping models are expressed as polynomial surface functions, including linear, quadratic, cubic, and quartic functions; Step 4: Select another m l sets of parameters. Calculate the corresponding performance indices by the performance models and apply them to evaluate the accuracy of RSM models. If the RSMs models are within acceptable accuracy range, they will be selected as the mapping models adopted in the optimization. If not, DoE will be applied again and the establishment of RSM models will be repeated. Mathematically, the linear, quadratic, cubic, and quartic functions for the RSM models can be expressed as

300

10 Optimal Design of Robotic Mechanism

f 1 (x) = a0 +

t 

bi xi , f 2 (x) = a0 +

t 

i=1

bi xi +

i=1

t 

ci xi2 +

t t  

i=1

di j xi x j ,

i=1 i< j

(10.37) f 3 (x) = a0 +

t 

bi xi +

i=1

f 4 (x) = a0 +

t 

bi xi +

i=1

t 

ci xi2 +

i=1 t 

di j xi x j +

i=1 i< j

ci xi2 +

i=1

t t  

t t  

t 

ei xi3 ,

(10.38)

i=1

di j xi x j +

i=1 i< j

t 

ei xi3 +

i=1

t 

f i xi4 ,

i=1

(10.39) where a0 , bi , ci , di j , ei , and f i are calculated regression coefficients. xi x j denotes interactions of any two parameters. xi2 , xi3 , and xi4 represent the second-, third-, and fourth-order nonlinearity. For accuracy assessment of these polynomial surface functions, additional parameter sets are selected and the performance indices are calculated. Four metrics are applied to compute the errors between these calculated results and the polynomial functions [16]. They are relative average absolute error (RAAE), relative maximum absolute error (RMAE), root mean square error (RMSE), and R square (R2 ), expressed as follows ml  " 

RAAE =

q=1 ml "

 yq − yˆq 

    max  yq − yˆ1 , . . . ,  yq − yˆm l  , ml   , RMAE =   "  yq − y   yq − y /m l

q=1

RMSE =

 "  m l y − y 2  q q=1 ml

(10.40)

q=1 ml

"

, R2 = 1 −

q=1 ml "

yq − yˆq



yq − y

2

2

,

(10.41)

q=1

where yq denote the exact value of the response at evaluation parameter set q, yˆq is the calculated value by the surface function, y is the mean value of yq , and m l is the number of additional evaluation parameter sets. In general, a large value of R2 and small values of RAAE, RMAE, and RMSE are preferred. They indicate that the analytical model has high accuracy in the overall design space. But the analytical model can be less accurate in some regions although good global measurements are given by R2 , RAAE, and RMAE. Thus, a smaller RMSE is necessary for focusing on the maximum error in one region. With the consideration of both global accuracy and maximum error from the four matrices, the appropriate analytical model with high accuracy can be achieved. By applying these analytical models, the efficiency of the optimization will be greatly improved.

10.3 Multi-objective Optimization

301

10.3.3 Pareto-based Optimization With the statistical objectives, probabilistic constraints as well as the explicit mapping models between objective/constraints and the design variables available at hand, the multi-objective optimization can be applied to simultaneously optimize several performances without predefined relationships. The multi-objective optimization problem is formulated as min F(X) = { f 1 (X), f 2 (X), . . . , fl (X)} s.t P(gk (X) ≥ 0) ≥ PR , k = 1, 2, . . . , q xL ≤ X ≤ xU

(10.42)

where f 1 (X), f 2 (X), . . . , fl (X) are the statistical objective functions corresponding to l performance indices of the robotic mechanism. P(gk (X) ≥ 0) is the probabilistic constraints considering parameter uncertainty. x U and x L are upper and lower limits of design variables. As has been mentioned, the single optimization method either by assigning weighting factors to multiple performances or converting unimportant performances into constraints requires the subjective operations to construct the single objective function. These subjective operations would impose great influence on the optimizations and lead to different optimal results. Alternatively, Pareto-based method can simultaneously optimize several objectives without any predefined relations among the performances as shown in Eq. (10.42). The optimization result is a cluster of solutions, which is called nondominant solutions. The image of all these solutions is named as Pareto front [2, 9, 10], indicating that none of the performances can be further increased without a decrease of any other remaining performances. Exampled by two-objective optimization whose objectives are global stiffness index and mass of the execution unit of a robot being applied as the machining module, the target is to find out the design variables making both the global stiffness index and the mass of executions as small as possible. The optimization results are shown in Fig. 10.1, in which each point corresponds to a set of design variable values. All the shown points are the sets of design variables satisfying the constraints. Comparing the points on the Pareto front and the points inside the feasible region, the performances of the former behave better than the latter. Taking point I and point II as an example, the mass of the execution unit is the same but the global stiffness index of point II is smaller than the point I. Hence, point I is discarded during optimization since better performance behavior can be obtained. Comparing the point II and point III on the Pareto front, the global stiffness index of point II is smaller than the value of point III, but the mass of execution unit at point II is larger than the value at point III. Both point II and point III are nondominant solutions that can be chosen as the final optimal results. Pareto solutions on the Pareto front are equally good for the optimal design problem because none of them can be better than the others for all objectives. Therefore, Pareto front shows the compromise among multiple performances. The selection of

302

10 Optimal Design of Robotic Mechanism

Fig. 10.1 Two-objective optimization of a robotic mechanism concerning stiffness and mass

final results from the Pareto front is usually left to the designers who choose the final results considering the application conditions. Different solutions might be chosen by different designers even for the same application scenario. To address this problem, a cooperative equilibrium searching method is proposed to find the best compromise among multiple objectives when they are assumed to be equally important. Taking the two-objective optimization again as an example, the cooperative equilibrium searching method is illustrated. If the two objective functions are assumed to correspond to two players, the process to determine the final solution from Pareto front can be regarded as a cooperative game. In the game, each player tries to find his maximum possible benefit considering the shared information from the other player. On the Pareto front, the point with the smallest mass (point IV) has the worst global stiffness performance while the point with the best global stiffness performance (point V) has the largest mass. The ideal solution is the one where both players get their best interests. But it cannot be reached due to the inherent conflict between stiffness and mass of the robotic mechanism. The cooperation between players can be assessed by the gap between existing solutions and the ideal solution. The solution that is the closest to the ideal point is defined as the cooperative equilibrium point, which is selected as the final solution. Mathematically, the cooperative game is played as follows. Dimensionless processing is firstly made to the Pareto solutions. f i (X) =

f i (X) − κ fi , σ fi

(10.43)

where f i (X) denotes the values of performance index on the Pareto front. κ fi and σ fi are the mean values and the standard deviation of the f i (X).

10.3 Multi-objective Optimization

303

Then, the best value of each objective is found on the Pareto front. Ideal optimum is formed by these best values. Distances from Pareto points to this ideal point can be computed as Di =



2  f i (X) − f i,min (X) ,

(10.44)

 where f i,min (X) is the minimal value of the ith performance index, i.e., the best performance on the Pareto front. Therefore, the cooperative equilibrium point is determined as the Pareto point with minimum Di .

10.4 Procedure of Multi-objective Optimization with Parameter Uncertainty Mainly drawing on the parameter uncertainty and the compromise of multiple performances, a procedure of optimal design with parameter uncertainty and multiple objectives is proposed. As shown in Fig. 10.2, the optimal design procedure is divided into three steps. Step 1: Formulation of performance indices and the RSM models Concerning the design requirements, performance indices for the optimal design are determined. For instance, the closed-loop mechanism being used as a machining unit concerns kinematic, stiffness, and dynamic performances and requires the stiffness along the machining directions to be larger than a certain value. Hence, power transmissibility, global stiffness index, stiffness indices along each direction of the global frame, and the inverse of condition number of inertia matrix are formulated based on the kinematic, stiffness, and dynamic models of the closed-loop mechanism. With the models of performance indices and the design variables available at hand, their RSM models are established for improving the efficiency of the optimization. The formulation of RSM models starts from the DoE, with which n groups of design variables are generated within the parameter ranges. The n performance indices can be computed by the models of performance indices. It is worth mentioning that the design variables making the performance be singular will be discarded. With the n sets of design variables and performance indices, the RSM models in Eqs. (10.37)–(10.39) are fitted. The accuracy of the RSM models is evaluated by Eqs. (10.40)–(10.41). Among the first-order to fourth-order functions, the function with the highest accuracy is selected and applied as the analytical model in optimal design. Step 2: Establishment of statistical objectives and probabilistic constraints The performance indices are assigned to be the objectives or constraints. For the closed-loop mechanism being applied as machining unit, the power transmissibility,

304

Fig. 10.2 Multi-objective optimization procedure

10 Optimal Design of Robotic Mechanism

10.4 Procedure of Multi-objective Optimization with Parameter Uncertainty

305

global stiffness index, and the inverse of the condition number of inertia matrix are selected to be the objectives, and the stiffness along the axis of global frame are chosen to be constraint condition. Since the optimizer assigns values to each design variable during optimization, N sets of design variables are randomly generated by regarding the assigned values as mean value and the machining tolerance as the standard variance. For the objectives, N values can be calculated corresponding to each assigned design variables based on the RSM model formulated in Step 1. The mean values and standard variables of these N values are computed, as shown in Eq. (10.5). They are the statistical objectives considering parameter uncertainty. For the constraints, the constraint of these N sets of design variables are calculated based on the corresponding RSM model. The probability of these constrained conditions satisfies the required values that are computed and compared with the predefined value, as shown in Eq. (10.7). The assigned values of design variables are assumed to be safe if the probability is greater than the predefined value. A probabilistic constraint concerning parameter uncertainty is built. Step 3: Implementation of multi-objective optimization and optimal result selection With the statistical objectives and the probabilistic constraints, the optimization model is established as shown in Eq. (10.42). Advanced artificial intelligent optimization algorithm, such as non-dominated sort genetic algorithm and the particle swarm optimization algorithm (PSO) [17], is applied to obtain the Pareto front of the optimization model. The designer can select any solution from the Pareto front. If the best compromise among the performances is considered, the cooperative equilibrium searching method can be adopted to search for the Pareto solution having the minimal distance to the ideal optimum.

10.5 Example 10.5.1 Typical Open-loop Mechanism 10.5.1.1

6R Mechanism

The layout of the 6R mechanism is shown in Fig. 10.3. The centers of the 2nd joint to the 6th joint are denoted by point A to point E. The kinematic, stiffness, and dynamic models have been discussed in the previous chapters. In this chapter, the optimal design of the 6R mechanism is implemented. The design variables are selected as the dimensional and sectional parameters of the components, as shown in Table 10.1. Following the design procedure in Sect. 10.4, the optimal design process is as follows.

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10 Optimal Design of Robotic Mechanism

Fig. 10.3 The schematic diagram of 6R mechanism

Table 10.1 Design variables of the 6R mechanism

Design variables

Range

Unit

The length of the AB (l1 )

(0.50, 0.70)

m

The length of the BC (l2 )

(0.12, 0.18)

m

The length of the C D (l3 )

(0.25, 0.35)

m

The length of the D E (l4 )

(0.25, 0.35)

m

The short side of the cross section 1 (b1 )

(0.12, 0.16)

m

The long side of the cross section 1 (h 1 )

(0.06, 0.10)

m

The diameter of the cross section 2 (d1 )

(0.08, 0.12)

m

Step 1: Formulation of performance indices and the RSM models The 6R mechanism is expected to be used in assembly or welding where high stiffness but the low weight of the execution unit are usually required. Hence, the overall stiffness index shown in Sect. 10.3.1 and the mass of the mechanism are adopted as the objectives. The minimal linear stiffness along each direction is selected to the constraints. The allowable values are given as 1.5 × 107 N/m, 4 × 107 N/m, and 4 × 107 N/m. The RSM models of the objectives and constraints are formulated. 500 sets of design variables are given by the DoE and the responses are calculated by the models of performance indices. Least square fitting is performed to obtain the polynominal functions of the performance indices, as shown in Eqs. (10.37)– (10.39). Accuracy assessments are carried out for each performance index according to Eqs. (10.40)–(10.41). The results are shown in Tables 10.2, 10.3, and 10.4. For

10.5 Example

307

Table 10.2 Accuracy assessment of the RSM models of mass of 6R mechanism κM

RAAE (