Control of Underactuated Manipulators: Design and Optimization 9819908892, 9789819908899

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Jundong Wu · Pan Zhang · Qingxin Meng · Yawu Wang

Control of Underactuated Manipulators Design and Optimization

Control of Underactuated Manipulators

Jundong Wu · Pan Zhang · Qingxin Meng · Yawu Wang

Control of Underactuated Manipulators Design and Optimization

Jundong Wu School of Automation China University of Geosciences Wuhan, Hubei, China

Pan Zhang School of Automation China University of Geosciences Wuhan, Hubei, China

Qingxin Meng School of Automation China University of Geosciences Wuhan, Hubei, China

Yawu Wang School of Automation China University of Geosciences Wuhan, Hubei, China

ISBN 978-981-99-0889-9 ISBN 978-981-99-0890-5 (eBook) https://doi.org/10.1007/978-981-99-0890-5 Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. © Science Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Underactuated mechanical systems are a kind of nonlinear system whose number of control inputs is less than the number of degrees of freedom. Nowadays, underactuated mechanical systems have been widely used in industrial production and daily life. Studying the control of underactuated mechanical systems has great significance in improving production efficiency, reducing system weight, etc. Moreover, it enables a fully-actuated mechanical system to continue its operation even if some of the actuators encounter failure. This guarantees the reliability of the system. As a typical representative of underactuated mechanical systems, underactuated manipulators have attracted worldwide attention. Generally speaking, a manipulator becomes underactuated mainly due to the occurrence of an actuator failure, which turns some joints of the manipulator into passive ones. Another situation is when joints or links of the manipulator have flexibility, which also brings underactuated characteristics to the system. In most cases, an underactuated manipulator has nonholonomic constraints, which means that some of its generalized coordinates have no corresponding actuators. Meanwhile, the dynamic model of the underactuated manipulator is a nonlinear model which cannot be strictly linearized. These two factors bring great difficulties to the control of underactuated manipulators. Nowadays, with the development of advanced control methods and artificial intelligence, control methods based on intelligent optimization have provided new possibilities in achieving the control performance, which is difficult to achieve using traditional control methods. In this monograph, we summarized our work and experience in the design and optimization for the control of underactuated manipulators over the last decade with the hope to provide a useful reference for researchers in the related field, and to help graduate students to gain a more thorough understanding of the subject. We also hope that this monograph can attract more researchers to devote themselves to the research of underactuated mechanical system control based on intelligent optimization algorithms. The monograph consists of six chapters. Chapter 1 mainly introduces the basic concepts and application background of the underactuated manipulators and the current research status of the underactuated manipulators. State of art of the subject v

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is also summarized, including characteristic analysis, control method design, etc. Chapter 2 focuses on the modeling of the underactuated manipulators, in which dynamic models of the planar underactuated manipulator, the vertical underactuated manipulator, the multi-link underactuated manipulator, the flexible manipulator, and other underactuated manipulator systems are established, and the characteristics of the underactuated manipulators are analyzed. Chapter 3 addresses control problems of the vertical underactuated manipulators. Effective control methods are proposed to achieve the swing-up control and balance control of such manipulators. Chapter 4 mainly discusses how to use the motion-state constraints on angles and angular velocities to achieve stable control of the planar manipulators with a passive first joint. Segmented control method and optimization-based continuous control method are proposed. Chapter 5 describes the control of the planar underactuated manipulators with a passive non-first joint, and some optimization-based methods are used to design the controller. Chapter 6 focuses on control problems of the flexible manipulator, in which manipulators with flexible joints or flexible links are studied, and some effective stable control and vibration suppression methods are proposed. The monograph is under the support of the Natural Science Foundation of China under Grant 60674044, 61074112, 61374106, 61773353, the 111 project under Grant B17040, and the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010. We would also like to express our gratitude for the support of scholars both in domestic and abroad. We would like to thank Prof. Xuzhi Lai, Prof. Min Wu, Prof. Weihua Cao, Prof. Yong He, Prof. Xin Chen, Prof. Luefeng Chen, Prof. Xiongbo Wan, and Prof. Huafeng Ding of China University of Geosciences, Prof. Chun-Yi Su of Concordia University, Prof. Jinhua She and Prof. Yasuhiro Ohyama of Tokyo University of Technology, and Prof. Simon X. Yang of University of Guelph, for patiently helping us in improving this monograph. We would also like to thank Prof. Ancai Zhang of Linyi University, Assoc. Prof. Changzhong Pan, and Dr. Peiyin Xiong of Hunan University of Science and Technology, Mr. Wenjun Ye of Concordia University, for their efforts and work on this monograph. Last but not least, we very much appreciate our colleagues for their work in typesetting and sorting this monograph, including Dr. Zixin Huang of Wuhan Institute of Technology, Dr. Lejun Wang of Chongqing University of Posts and Telecommunications, Ms. Fengjiao Zhu and Messrs. Zhen Zhang, Yibiao Luo, Shengqiang Cao, Yang Sheng, Junqing Cao, Ke Zeng of Central South University, Mses. Ze Yan, Huiqing Yang, Siyu Chen, and Messrs. Haoqiang Chen, and Dong Liu of China University of Geosciences. Wuhan, China September 2022

Jundong Wu Pan Zhang Qingxin Meng Yawu Wang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Underactuated Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Control Method Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Energy Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Backstepping Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Partial Feedback Linearization Control . . . . . . . . . . . . . . . . . . 1.3.5 Approximate Linearization Control . . . . . . . . . . . . . . . . . . . . . 1.3.6 Model Reduction Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Intelligent and Optimization Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Neural Network Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Optimization-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 6 7 8 8 10 11 11 12 13 14 16

2 Modeling and Characteristics Analysis of Underactuated Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Dynamic Modeling of Typical Underactuated Manipulators . . . . . . . 2.1.1 The Inertia Wheel Pendulum Manipulator . . . . . . . . . . . . . . . 2.1.2 Two-Link Underactuated Manipulator . . . . . . . . . . . . . . . . . . . 2.1.3 Multi-link Underactuated Manipulator . . . . . . . . . . . . . . . . . . 2.1.4 Flexible Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Characteristics Analysis of Underactuated Mechanical System . . . . 2.2.1 Dynamics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Controllability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 26 30 39 52 53 55 57 57

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3 Control of Vertical Underactuated Manipulator . . . . . . . . . . . . . . . . . . . 3.1 A Lyapunov-Based Unified Control Strategy . . . . . . . . . . . . . . . . . . . 3.1.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Control Strategy for the Swing-Up Area . . . . . . . . . . . . . . . . . 3.1.3 Control Strategy for the Attractive Area . . . . . . . . . . . . . . . . . 3.1.4 Local and Global Stability Analysis . . . . . . . . . . . . . . . . . . . . . 3.1.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A Rewinding Approach-Based Control Strategy . . . . . . . . . . . . . . . . 3.2.1 Modeling and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Trajectory Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stable Control of Three-Link Underactuated Manipulators . . . . . . . 3.3.1 Model and Division of Motion Space . . . . . . . . . . . . . . . . . . . 3.3.2 Swing-Up Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Singularity Avoidance in Swing-Up Controller . . . . . . . . . . . 3.3.4 Balance Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stable Control of n-Link Underactuated Manipulators . . . . . . . . . . . 3.4.1 Dynamic Model and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Controller Design in Stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Controller Design in Stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 63 65 66 70 74 74 77 79 81 82 84 86 88 93 94 96 96 101 104 109 111 113

4 Control of Planar Underactuated Manipulator with a Passive First Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Motion-State Constraint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Motion-State Constraint on Angular Velocities . . . . . . . . . . . 4.1.3 Motion-State Constraint on Angles . . . . . . . . . . . . . . . . . . . . . 4.2 Motion-State Constraint-Based Control of Planar Acrobot . . . . . . . . 4.2.1 Motion Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Motion Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stable Control of Planar Three-Link PAA Manipulator . . . . . . . . . . . 4.3.1 Motion Strategy of Angles to Target Values . . . . . . . . . . . . . . 4.3.2 PSO-Based Target Angle Optimization . . . . . . . . . . . . . . . . . . 4.3.3 Target Angle-Based Controllers Design . . . . . . . . . . . . . . . . . 4.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 119 119 120 124 125 126 127 129 130 131 133 134 138

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4.4 Two-Stage Control of Planar n-Link (n > 3) PAn−1 Manipulator . . . 4.4.1 Controllers Design of Stage 1 for Model Reduction . . . . . . . 4.4.2 Controllers Design of Stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 GA-PSO-Based Target Angle Optimization . . . . . . . . . . . . . . 4.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Intelligent Optimization-Based Continuous Control Method . . . . . . 4.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Continuous Controller Design for Active Joints . . . . . . . . . . . 4.5.3 Optimization of Design Parameters and Target Angles . . . . . 4.5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Nonlinear MPC-Based Robust Control Method . . . . . . . . . . . . . . . . . 4.6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Control Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Nonlinear Model Predictive Control-Based Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Fast Terminal Sliding Mode Controller Design . . . . . . . . . . . 4.6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Online Iterative Correction-Based Robust Control Method . . . . . . . . 4.7.1 Uncertain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Uncertain Planar Virtual PAA System . . . . . . . . . . . . . . . . . . . 4.7.3 Controller Design for Model Reduction . . . . . . . . . . . . . . . . . 4.7.4 Control of Planar Virtual PAA System . . . . . . . . . . . . . . . . . . 4.7.5 Online Iterative Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Control of Planar Underactuated Manipulator with an Active First Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fourier Transformation-Based Control Method for Planar Pendubot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Dynamic Model of Planar Pendubot . . . . . . . . . . . . . . . . . . . . 5.1.2 Controllers Design with Disturbance Observer . . . . . . . . . . . 5.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Energy Attenuation Approach-Based Control Method for Planar APAA Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Target Angles Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Stable Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Chained Form-Based Control Method for Planar AAPA Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Controllers Design for the Manipulator with Reduced Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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140 141 143 144 146 147 148 149 150 152 155 155 156 157 159 165 167 167 169 170 174 175 176 179 180 183 183 184 185 189 192 193 197 202 204 204 211 215

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5.4 A General Position Control Method for Planar Underactuated Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Dynamic Model and Control Scheme . . . . . . . . . . . . . . . . . . . 5.4.2 Bi-Directional Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Trajectory Tracking Controllers Design . . . . . . . . . . . . . . . . . 5.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Control of Flexible Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Stable Control for Flexible-Joint Manipulator . . . . . . . . . . . . . . . . . . . 6.1.1 EID Based Control Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Trajectory Tracking Control for Flexible-Joint Manipulator . . . . . . . 6.2.1 Uncertain System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Stable Control for Flexible-Link Manipulator . . . . . . . . . . . . . . . . . . . 6.3.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 FGA-Based Online Optimization . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Position Control with Zero Residual Vibration for Flexible-Link Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Tracking Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Position Control for Flexible-Joint Manipulator with a Passive Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Property Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Energy-Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Position Control for Flexible-Link Manipulator with a Passive Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Dynamic Coupling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Target Angles Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Controller Design for the Active Joint . . . . . . . . . . . . . . . . . . . 6.6.4 Parameters Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

220 221 224 228 230 234 234 237 237 238 244 245 245 246 249 250 251 254 256 261 261 263 268 270 271 272 273 275 277 277 279 281 283 284 287 288

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

About the Authors

Jundong Wu was born in Hunan, China, in 1992. He received a B.S. degree in microelectronics from the School of Electronics Engineering and Computer Science, Peking University, Beijing, China, in 2014, a M.S. degree in electrical engineering from the School of Engineering and Applied Sciences, Harvard University, Cambridge, the United States, in 2017, and a Ph.D. degree in mechanical engineering from the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, Canada, in 2022. In 2022, he joined the School of Automation, China University of Geosciences as a Professor. His current research interests include underactuated robot control, soft robot modeling and control, and nonlinear system control, of which he has over 20 paper publications. He currently owns 3 research projects, funded by the National Natural Science Foundation of China and the China Postdoctoral Science Fund. He was also elected into the Talent Plan of Hubei Province.

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Pan Zhang received a B.S. and a Ph.D. degree in engineering from China University of Geosciences, Wuhan, China, in 2015 and 2020, respectively. From 2017 to 2020, she was a Research Intern with the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada. In 2020, she joined the School of Automation, China University of Geosciences, where she is currently an Associate Professor. Her current research interests include underactuated robot control, nonlinear system control, and intelligent control.

Qingxin Meng received a B.S. degree in engineering from the Hebei University of Technology, Tianjin, China, in 2016, and a Ph.D. degree in engineering from the China University of Geosciences, Wuhan, China, in 2021. From 2019 to 2021, he was a Research Intern with the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada. In 2022, he joined the School of Automation, China University of Geosciences, where he is currently a Professor. His current research interests include flexible robot control, vibration control, and nonlinear system control. Yawu Wang received a B.S. and M.S. degrees in engineering from the Hubei University of Technology, Wuhan, China, in 2011 and 2015, respectively, and a Ph.D. degree in engineering from the China University of Geosciences, Wuhan, China, in 2018. He was a joint doctoral student with the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada, from 2017 to 2018. In 2019, he joined the School of Automation, China University of Geosciences, where he is currently a Professor. His current research interests include underactuated robot control, soft robot modeling and control, and intelligent control.

Acronyms, Notations, and Symbols

AAAP AAP AAPA ACO Acrobot APA APAA BPNN CNN DEA DOF EID FGA FMP FTSM GA IDA-PBC IRP LIS LSTMNN LQR MPC NSLF ODE OIC PAA PAAA PDE

A four-link manipulator with a fourth passive joint A three-link manipulator with a third passive joint A four-link manipulator with a third passive joint Ant colony optimization A two-link manipulator with a single actuator at the second joint A three-link manipulator with a second passive joint A four-link manipulator with a second passive joint Back Propagation neural network Convolution neural network Differential evolution algorithm Degree of freedom Equivalent-input-disturbance Fuzzy-Genetic Algorithm Forward-middle-position Fast terminal sliding mode Genetic algorithm Interconnection and damping assignment passivity-based control Initial-rest-position Largest invariant set Long-Short Term Memory neural network Linear quadratic regulation Model predictive control Nonsmooth Lyapunov function Ordinary differential equation Online iterative correction A three-link manipulator with a first passive joint A four-link manipulator with a first passive joint Partial differential equation

xiii

xiv

Pendubot PFUM PPA PRJP PSLFLM PSO RBFNN RMP RV SA SCPS SLFJM SMC SMVS STLC TDE TLRF TORA TRP UM UMS USLFJM UTGR VUM WCLF R R+ Z Z+ Rn Rn×m PT P −1 def(P) rank(P) E(P) P > 0(P  0) In 0m×n diag{· · · } mod{x,  y} max f (x)

Acronyms, Notations, and Symbols

A two-link manipulator with a single actuator at the first joint Planar four-link underactuated manipulator A three-link manipulator with a single actuator at the third joint Planar rigid-joint Pendubot Planar single-link flexible-link manipulator Particle swarm optimization Radial basis function neural network Reverse-middle-position Residual vibration Simulated annealing Spring-couping Pendubot system Single-link flexible-joint maniputor Sliding mode controller Sliding mode variable structure Small time local controllability Time delay estimation Two-link rigid-flexible Translational oscillator with rotational actuator Target-rest-position Underactuated manipulator Underactuated mechanical system Uncertain single-link flexible-joint maniputor Underactuated three-link gymnast robot Vertical underactuated manipulator Weak-control Lyapunov function Field of real numbers Field of positive real numbers Field of integer Field of positive integer n-dimensional real vectors n×m-dimensional real matrices Transposed matrix of P Inverse matrix of P Determinant of matrix P Rank of matrix P Eigenvalues of matrix P P is a positive definite (semi-definite) matrix n-unit matrix n×m-null matrix Diagonal matrix Remainder of x divided by y, and the symbol is the same as y Maximum value of function f (x)

Acronyms, Notations, and Symbols

  min f (x) Lf h(x)  f ,g adkf g(x) α2 σmax () G∞

xv

Minimum value of function f (x) Lie derivative of function h(x) along vector field f (x) Lie bracket of vector field g(x) with respect to vector field f (x) k(k  1)-lie bracket of vector field g(x) with respect to vector field f (x) √ 2-norm of vector α, and α2 = α T α Maximum singular value function   H∞ norm of function G(x), and G∞ = sup σmax G(jw) 0ω∞

Chapter 1

Introduction

Manipulators are one of the most typical type of mechanical systems in industrial scenes. As a class of manipulators, underactuated manipulators have the advantages of increasing the flexibility of the system, reducing costs and energy consumption, etc. Nowadays, with the aggravation of the energy shortage, the research on such systems has attracted more and more attention and has gradually become a new research hotspot. This chapter will provide a comprehensive review for the current state of the underactuated manipulators, including their categories in applications, research significance, system characteristics, and corresponding control methods. The aim is to help readers have a deeper understanding of the development history of such systems.

1.1 Underactuated Manipulators Mechanical systems are the tools designed by humans to achieve specific production demands. At present, the development level of mechanical systems is closely related to the level of economy and society, and it has become an important indicator to evaluate the development degree of social production, science and technology [1]. For most mechanical systems, each degree of freedom (DOF) corresponds to an actuator. That is, the number of control inputs is equal to that of DOF, and such systems are called fully-actuated mechanical systems. However, in actual applications, a fully-actuated system may be changed due to external environment or human factors, turning it into an underactuated mechanical system (UMS). Obviously, the UMS is a class of nonlinear systems, whose number of control inputs is less than that of DOF [2–6]. In this case, the dimension of the system control input is less than that of the system configuration space [7]. Reasons why a mechanical system becomes an underactuated one are usually as follows:

© Science Press 2023 J. Wu et al., Control of Underactuated Manipulators, https://doi.org/10.1007/978-981-99-0890-5_1

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1. The system has some motion constraints that make it an underactuated system or some motions of the system itself lack actuation, such as mobile robots [8], spacecrafts [9], helicopters [10], etc. 2. In certain environments (such as microgravity environments), in order to decrease energy consumption and reduce system costs, some mechanical systems are designed to be compact and flexible, such as spatial flexible link manipulator [11]. In addition, from the perspective of human-machine interaction safety, in order to avoid rigid collision, some structures of mechanical systems are often designed with flexible materials, such as flexible joint manipulators [12]. 3. The system becomes an underactuated system when one or more actuators of the fully-actuated mechanical system are damaged, or the actuators have a long reaction time lag. 4. Since many physical systems in real life are underactuated, studying the control of such systems is of great importance. Thus, the researchers artificially develop the simplified underactuated system in the laboratory based on the actual physical system and use this as a platform to verify control algorithms, such as inverted pendulum [13, 14], wheel pendulum [15], Translational Oscillator with Rotational Actuator (TORA) [16], beam-and-ball system [17], Furuta pendulum [18], etc. Manipulators are a kind of mechanical system composed of links and joints. which are widely used in industrial production. For a manipulator, when one or more joint actuators on the manipulator are damaged, it will become an underactuated manipulator. Furthermore, when the joints or links of the manipulator have flexible characteristics, it also has underactuated characteristics. Hence, the underactuated manipulators are a typical representative of UMS, which makes the control of the underactuated manipulators an important topic in the field of UMS control. The research on the control theory of the underactuated manipulators, including system analysis and control design, originated in the 1990s. In the past three decades, with the development of control theory and the needs of engineering practice, especially the development of robotics and aerospace technology, this research has received increasing attention from scholars all over the world [19–22], and relative control strategies have been thoroughly studied and have been applied in various practical scenarios. So far, scholars have carried out various studies on the characteristic analysis and control method design of the underactuated manipulators. The efforts have expanded the research to a large extent in connection with modern computational techniques and advanced mathematical methodology. Nowadays, the control theory of underactuated manipulators has been greatly developed with the help of intelligent optimization technology. Some intelligent control methods have been proposed and improved in recent years to solve the challenging problems in the control of underactuated manipulators, which are usually difficult to be solved by classic control methods. In the following sections, a comprehensive review on the state of the art of the underactuated manipulators is presented.

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1.2 Characteristic Analysis A typical way of categorizing underactuated manipulators is to consider whether the system is affected by gravity, based on which the underactuated manipulators can be divided into two main categories: one is the vertical underactuated manipulators, which move in a vertical plane, such as the vertical two-link underactuated manipulator (the Acrobot/Pendubot) [23, 24]; the other is the planar underactuated manipulator, which moves in a horizontal plane, such as the planar two-link underactuated manipulator (the planar Acrobot/Pendubot) [25, 26], and the space flexible manipulator [27]. In order to study the control of underactuated manipulators, scholars first made a lot of exploration on the characteristics of the systems. In [28], the integrability of the constraints existing in the underactuated manipulators was discussed in detail, and the necessary and sufficient conditions for the second-order nonholonomic, firstorder nonholonomic, and holonomic constraints were given. According to this result, most of the underactuated manipulators are second-order nonholonomic systems. In [29–31], the coupling relationship among all links of the underactuated manipulators was discussed, and the metric for the degree of coupling was given. In [32], the authors studied the nonlinear characteristics of a planar two-link underactuated manipulator with a second passive joint (i.e., the planar Pendubot) and obtained the conclusion that, by using Poincar mapping, the system appears to be chaotic phenomenon when the control input is a small periodic signal. In [33], a performance metric called Actuability for the underactuated manipulators was introduced, which provided guidance for the construction of the actual control system. As a precondition for the study of the control methods, the controllability of underactuated manipulators is an important focus of the topic. The system controllability refers to the nature of the system moving from one state to another state under the appropriate input. So far, the controllability analysis of linear systems is relatively mature. However, the research progress on the controllability of nonlinear systems is still preliminary, which is usually limited to the research on local controllability. As a special kind of nonlinear system, the controllability analysis of the underactuated manipulators is mainly focused on the local controllability, including local linear controllability analysis and local nonlinear controllability analysis. The main idea of local linear controllability analysis is to convert a nonlinear underactuated manipulator into a linear system around the equilibrium point and then use the controllability criterion of the linear system to judge the controllability of this nonlinear underactuated manipulator in this small region. The conversion methods are mainly divided into two types. One is the linear approximation method [34], which performs linear approximation directly at the equilibrium point to obtain a linear system. Another is the pseudo-linearization method [35], which first approximates the underactuated manipulator to a nonlinear system with feedback linearization around the equilibrium point and then uses the corresponding tool to obtain a linear system. Based on these two methods and the linear system controllability criterion, it can be proved that the vertical underactuated manipulators with one passive joint are locally linear controllable, but the

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planar underactuated manipulators are not. Further, analysis of the local nonlinear controllability is mainly based on the STLC (Small Time Local Controllability) of the affine nonlinear system [36], which has been a topic that was widely studied in recent years. For example, in [37], the authors discussed the STLC of various planar two-link underactuated manipulators and pointed out that the equilibrium point of the system satisfying the STLC was at most one-dimensional submanifold; in [38], the authors presented a few controllability properties of a planar three-link underactuated manipulator with different actuator configurations and indicated that the planar three-link manipulator with a passive first joint did not satisfy the sufficient conditions for STLC, while other configurations with a passive second or third joint would do. Reference [39] generalized this conclusion to a planar n-link (n > 3) underactuated manipulator with a passive joint, and presented a simple sufficiency criterion for the system being STLC. In addition to discussing the STLC of the underactuated manipulators, scholars have explored other forms of controllability of such systems. For example, [40, 41] studied the local motion controllability of the system, and [42] studied the vibration controllability of the system. The research on various controllability properties not only deepens people’s understanding of the underactuated manipulators, but also provides necessary theoretical guidance for the design of motion trajectories and the development of control methods. Scholars have also carried out a variety of researches on the oscillation characteristics of underactuated manipulators in recent years. In [43], based on the constraint among all states of the planar underactuated manipulator, an equation about the underactuated states was obtained by taking the actuated states as the periodic function, and then the periodic oscillation of the system was discussed by using the average method. In [44], the periodic motion characteristics of the Acrobot around the zero equilibrium point were explored by introducing a virtual constraint between the actuated states and the underactuated states. This virtual constraint method is further developed later. For example, in [45], using this method, the orbital stabilization problem of the Pendubot was discussed, and the conditions for generating stable limit cycle orbits were given. In [46], the authors analyzed the property of the Pendubot, which showed that, under high-frequency excitation of the actuated first link, various bifurcation phenomena would emerge in the unactuated link, based on which the motion control of the system was realized without using the underactuated states. In [47], the inertia wheel pendulum manipulator was studied. By limiting the control input of the system, some nonlinear phenomena were analyzed in detail, including balance point bifurcation and homoclinic ring bifurcation.

1.3 Control Method Design On the basis of the characteristics analysis, extensive researches on the control problems of underactuated manipulators have been carried out. Since the mid-1990s, many effective control strategies have been proposed according to the different

1.3 Control Method Design

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characteristics of different underactuated manipulators [48–51], including energy based control, sliding mode control, backstepping control, partial feedback linearization control, linear approximation control, model reduction control, etc.

1.3.1 Energy Based Control Generally speaking, the energy based control strategy is essentially a generalized state feedback strategy, whose main idea is to change the energy of the underactuated manipulators by means of state feedback, so the closed-loop system has the energy of the desired stable states [52–54]. This control strategy consists of two types of methods. The first one is the Controlled Lagrangian method [55, 56], which is an energy reconstruction control strategy that analyzes the expression of the desired kinetic or potential energy from the view of the Lie group and transforms the solution problem of the state feedback law into that of a set of partial differential equations. The design process of such methods is obscure, and it does not consider the situation where the kinetic energy and potential energy of the system are reconstructed at the same time. The second method is the IDA-PBC (Interconnection and Damping Assignment Passivity Based Control) method [57–60], which can be considered as a generalization of the Controlled Lagrangian method. It is suitable for the case in which the kinetic energy and the potential energy are constructed respectively, and also suitable for the case of the reconstructing of both energies at the same time. Like the control Lagrangian method, the IDA-PBC method also transforms the solution problem of the state feedback law into that of a set of partial differential equations, but its design process is simpler, and its control law is designed as the sum of energy reconstruction and input damping, so that makes the physical meaning of using this method more obvious. In recent years, the IDA-PBC method has been greatly developed. Plenty of discussions and researches are carried out [61–63] to improve its theoretical methods and the applicable ranges, gradually making the method a complete theoretical system. Although the above two methods are theoretically rigorous, they have to solve a set of partial differential equations to obtain the state feedback control law. The solution is not difficult for some systems with simple nonlinear characteristics, but it becomes very difficult when the nonlinear characteristics become slightly more complicated, and sometimes it is necessary to select appropriate free variables for the solution, which limits the applications of the two methods to some extent. Based on the above two methods, combined with the motion characteristics of the underactuated manipulator, scholars proposed several energy based control strategies for underactuated manipulators. Typical examples are the swing-up controllers of vertical underactuated manipulators [64, 65]. In these strategies, the expression of the system energy and the desired system energy required for the straight-up equilibrium point are directly constructed with a Lyapunov function. Then, the control law of the system is obtained by solving the inequality, in which the derivative of the Lyapunov function is less than zero. The obtained control law ensures that the energy

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of the vertical underactuated manipulator gradually increases and reaches the energy required for the target equilibrium point, achieving the swing-up control of these underactuated manipulators. In recent years, these energy based control strategies have been extended to the control of planar underactuated manipulators [66] and flexible manipulators [67].

1.3.2 Sliding Mode Control As an important part of modern control theory, the sliding mode variable structure (SMVS) control method has become an effective and commonly used method to solve the control problems of nonlinear systems. Its basic control principle is to design the switching hyperplane according to the dynamic characteristics expected by the system and make the system states converge from outside the superplane to the switching hyperplane by the sliding mode controller (SMC). Once the system reaches the switching hyperplane, the control will ensure that the system reaches the system origin along the switching hyperplane to achieve the control objective. Since the characteristics and parameters of the control system only rely on the designed switching hyperplane and are unaffected by external disturbances, the control method has strong robustness under external disturbances, parameter perturbation, and unmodeled dynamics. In addition, this control algorithm is simple with a fast response speed, which makes it a widely-used strategy in robot control [68–70]. With the rising requirements on the anti-interference ability and robust performance of the underactuated manipulators, the SMVS control method has eventually been introduced to the control of the system. However, since decoupling between various state variables of the underactuated manipulators cannot be achieved, the traditional SMVS control method cannot be directly applied to such systems. To solve this problem, tremendous efforts have been made, and some corresponding research results have been obtained. Orlov et al. adopted the SMC to realize the swing of the system near the target point by designing a critical state to make the control switch back and forth indefinitely, which overcomes the problem of uncertainty and external disturbances to some extent [71]. Some scholars have also proposed improved SMVS control methods based on traditional SMVS [72, 73], in which two typical representatives are the two-layer SMVS control method [74, 75] and the cascaded SMVS control method [76, 77]. In the former method, sliding surfaces are designed for each subsystem of the underactuated manipulators, based on which the sliding surface for the whole system is designed. With this strategy, the control law is found to make the sliding surface of each subsystem close to zero, so that the state variables of the whole system approach zero. As for the latter method, the first sliding surface is designed based on the first subsystem, and the second sliding surface is designed based on the first sliding surface and the second state variables, the other sliding surfaces are designed in a similar manner, until the final sliding surface is obtained. Since this final sliding surface contains the information of all state variables

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of the system, when it approaches zero, all other sliding mode surfaces should also approach zero, thereby making the state variables of the system approach zero. Compared with other control methods, the most prominent features of the SMVS control method are its strong robustness, and that it can directly deal with systems with complex nonlinearities. However, since this control method needs to be switched between different switching surfaces, it will inevitably cause chattering problems, which increases energy consumption. Therefore, overcoming such shortcomings and realizing better control performance with the SMVS control method are the hotspots of current research.

1.3.3 Backstepping Control In the backstepping method, the Lyapunov function is constructed forwardly, stepby-step. In each step, the state variation, the adaptive adjustment function of the uncertain parameter and the stabilization function of the virtual control system are considered together to gradually amend the algorithm to design the stable controllers. The Lyapunov function corresponding to the backstepping method is composed of the Lyapunov function of the first subsystem and the square term of the state variables of the second subsystem. Using a statically compensated virtual control, the first subsystem can achieve stable control through the virtual control of the latter subsystem. Thus, in the controller design, the focus is to construct a suitable virtual controller to meet the stable control requirements of the recursive subsystem. The backstepping method is quite effective in implementing the tracking and stabilization control of a simple nonlinear system [78–81], but it has not been applied to complex systems like the UMSs for a long time ever since it was proposed. It was not until the late 1990s that it was applied to the tracking control of UMSs such as ships [82, 83]. Since then, scholars have made great efforts to extend the backstepping method to underactuated manipulator control, where the representative one is the backstepping method based on coordinate transformation proposed by Olfati-Saber et al. [84, 85]. Consider that the underactuated manipulator has the features of complex nonlinearity and strong coupling, leading to the difficulty of designing a stable controller, and the method introduced a differential homeomorphic coordinate transformation to convert the original system to a relatively simple nonlinear system, and used the traditional nonlinear backstepping method [86] to asymptotically stabilize the new system. The asymptotic stability of the original system at the target point is guaranteed by the homeomorphism of the coordinate transformation. Utilizing this control strategy, [87] achieved the asymptotic stability of the Acrobot. Although this control strategy based on the coordinate transformation successfully extends the backstepping method to underactuated manipulator control, there also exists some problems. For example, the nonlinearity of the transformed system is still complex. Furthermore, it is difficult to obtain the explicit expression of the virtual input of the transformed system. Even if the explicit expression can be obtained, the expression is also complicated, resulting in a computational burden. In response to

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these problems, based on the traditional backstepping method, numerous improved control methods have emerged, including the sliding surface-based backstepping method [88], the adaptive backstepping method [89], the improved backstepping method [90], etc.

1.3.4 Partial Feedback Linearization Control Feedback linearization is a control method that turns a nonlinear system into a linear system in a new coordinate by applying the state feedback and non-singular coordinate transformation to the nonlinear system. Generally speaking, according to the applicable range of non-singular coordinate transformation, the feedback linearization method can be subdivided into global feedback linearization and local feedback linearization. For the former method, the coordinate transformation does not have singular points in the global range, while the latter only guarantees that the coordinate transformation does not appear singular in a small range. Currently, studies on the analysis and design of the systems are relatively mature, as a result, the feedback linearization provides an excellent solution to the control problems of the nonlinear systems. In recent years, this simple and effective nonlinear control method has been widely applied [91–94]. For an affine nonlinear system, [95] proposed the sufficient conditions for feedback linearization of the system. According to these conditions, the underactuated manipulator is usually unable to achieve full-state feedback linearization. To solve this problem, a partial feedback linearization method was proposed in [96] to achieve the stable control of the Acrobot. Furthermore, in [97], the partial feedback linearization method was extended to the control of an underactuated biped robot. Generally speaking, the partial feedback linearization method provides an effective strategy to solve the control problem of complex nonlinear systems like the underactuated manipulators. In recent years, such a method has been widely extended to the control of other UMSs, and achieved fair control performance [98–100]. However, the partial feedback linearization method also has shortcomings. For example, it is sometimes difficult to obtain the desired trajectory tracked by the linearized states. Meanwhile, this method only ensures that a part of the states of the UMS can be controlled, while other states are kept on a degraded manifold, and the completion of the control objective relies on the structure of the manifold.

1.3.5 Approximate Linearization Control Due to the strong nonlinearity of the underactuated manipulators, traditional linear control methods are difficult to achieve its stable control. Some scholars began to study the approximate linearization control method, which approximately linearizes the nonlinear model of the underactuated manipulator into a linear model, to realize

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the control of the system by using linear control methods [101, 102]. Based on characteristic analysis for the underactuated manipulator, it has been discovered that the vertical underactuated manipulator can be approximated to a linear system only in a small region around the origin, and this linear system is controllable [103–105]. Based on this characteristic, some scholars proposed the zoning control strategy to implement the approximate linearization control for the vertical underactuated manipulator [106–108]. The zoning control strategy is often utilized for vertical underactuated manipulators, whose control objective is to control the manipulator to swing-up from the straight-down position to the straight-up position. To achieve this, the motion space of the vertical underactuated manipulators is divided into the swing-up area and attraction area. The attraction area is a small region around the straight-up position, where the system model can be approximately linearized into a linear model, and the other areas are the swing-up area. Meanwhile, the rest of the regions corresponds to the swing-up area. During the control process, the controller of the swing-up area first works to drive the manipulator close to the attraction area. Once the system reaches the attraction area, the controller is switched into a controller that works to stabilize the system to the control objective. This is the general process of the zoning control. For the swing-up area, controllers can be designed based on strategies presented in previous subsections, which include energy-based controllers, sliding mode controllers, backstepping control controllers, etc. For the attraction area, since the dynamic model of the vertical underactuated manipulator is usually linearized, a commonly used controller is the linear quadratic optimal controller [109–111]. Linear quadratic optimal control is an important part of modern control theory, which has been widely used in the control system. This control theory mainly discusses how to make the system operate in a way that makes a given performance index reach the best value. If the dynamics of a system can be represented by a set of linear differential equations, its performance index function is a quadratic function, which is called the quadratic optimal indicator. Hence, the optimal control problem of this system can be called the linear quadratic problem, and the solution to this problem is the linear quadratic regulator (LQR) [112]. For this optimization problem, many computational softwares (such as MATLAB) have provided related toolboxes, which make it easy to find the optimal solution that satisfies the quadratic optimal indicator. In [113], an LQR controller was used to achieve the balance control for a vertical Acrobot in the attraction area. In [114], a three-link underactuated manipulator with a single actuator at the first joint was balanced at its straight-up position by using the LQR controller. Generally speaking, the approximate linearization method is an effective strategy for the vertical underactuated manipulator, in the situation where the system successfully reaches its attraction area. However, for planar underactuated manipulators, there is no controllable approximate linear models near the target position [115]. Therefore, the approximate linearization control method has its limitations in its utilization.

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1.3.6 Model Reduction Control For the underactuated manipulator, the increase in the number of the system links will significantly increase its nonlinear characteristics, which makes the control much more challenging [116]. To address this challenge, scholars proposed the model reduction control strategy. For this strategy, the general idea is to reduce the system’s DOF by making multiple connected links move as a whole with a fixed posture. The control procedure is usually divided into various steps, and in different steps, different model reduction strategies are applied [117]. The purpose of applying the model reduction is different for the vertical underactuated manipulator and the planar underactuated manipulator. For the vertical underactuated manipulator, the model reduction can effectively increase the range of the attraction area and make it easier to control the system to reach the attraction area. In [118], an n-DOF vertical underactuated manipulator was taken as the research object, and a model reduction control method based on torque-coupled was proposed to realize the control objective of this system. In [64], the model reduction control method was proposed to address the motion control problem of an n-link vertical underactuated manipulator with any one of its joints being passive. The nlink systems considered in this paper were reduced to a virtual Acrobot or a virtual Pendubot according to the position of the passive joint. For the planar underactuated manipulator, the model reduction is utilized mainly to introduce a constraint on the angles of the system, which only happened for the two-link planar underactuated manipulator with a passive first joint (i.e., the planar Acrobot). The constraint on the links’ angles of the planar Acrobot strictly limits the angle variation relationship between the two links of the system, which means that the stability angle of the passive link can be calculated according to the stability angle of the active link. The angle constraint of the planar Acrobot makes it possible to realize the angle control of its two links simultaneously. To use this angle constraint of the planar Acrobot in the control of the planar multi-link underactuated manipulators, [119] reduced the planar three-link underactuated manipulator with a passive first joint to two virtual planar Acrobot, and achieved stabilization of the three links at their target angles through a two-stage control. This control strategy was further extended into a position-posture control strategy for the planar four-link underactuated manipulator with a passive first joint [120]. Furthermore, [121] provided a control strategy in which the angle constraint of the planar Acrobot was utilized for the control of a planar multi-link underactuated manipulator whose passive link is not the first link. In this study, a planar four-link underactuated manipulator with a passive third joint was considered as a demonstration of the proposed control strategy, and was successively reduced to a planar virtual three-link manipulator with a passive third joint and a virtual planar Acrobot. The control strategy was divided into three steps, and four links of the system were finally successfully stabilized at their respective target angles. To sum up, the model reduction control usually consists of multiple control stages during the control process, which is sometimes time-consuming. Meantime, this

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control strategy becomes even more complicated when the number of control stages increases. Therefore, researchers still tend to seek other advanced control strategies to realize the control objective of the multi-link underactuated manipulators with only one control stage.

1.4 Intelligent and Optimization Control With the rapid development of computer technology and artificial intelligence, more and more scholars have introduced the logic, reasoning, or heuristic knowledge to the traditional automatic control theory. A set of intelligent control theories and techniques for complex systems has been established by combining automatic control, artificial intelligence, and some related disciplines in system science. That is to say, such intelligent control methods are based on a series of disciplines, such as automatic control theory, computer science, artificial intelligence, systems engineering, etc. In recent years, intelligent control methods have been proven to have outstanding performances in solving nonlinear system problems and to have unprecedented their advantages in self-learning, self-adaption, robustness, and other aspects. As a result, intelligent control methods are widely applied to the control of various nonlinear systems. So far, a variety of intelligent control methods has been thoroughly researched and has been utilized in underactuated manipulator control, this includes fuzzy control, neural network control, optimization-based control, etc.

1.4.1 Fuzzy Control For the control of underactuated manipulators, the most typical intelligent control method is fuzzy control, which is an intelligent control method based on fuzzy set theory, fuzzy language variables, and fuzzy logic reasoning. In real life, the reasoning and decision-making processes of human beings are often in uncertain, imprecise, and fuzzy ways, which depend on the experience and understanding of the causal laws of things. However, the automatic operation of a mechanical system often follows a set of fixed binary logic [122]. This mismatch between human beings and machines makes it impossible for automatic control to complete a specific task like an experienced engineer, which sometimes makes automatic control unable to achieve the performance of manual control. Fuzzy control is an effective solution to the above problem, and it is a control method that can imitate the fuzzy reasoning and decision-making process of human beings. Firstly, the fuzzy control method compiles the experience of operators or experts into fuzzy rules. Then, the method fuzzifies the real-time state information of the system measured by the sensor and takes the fuzzified state information as the input of fuzzy rules. Based on the established fuzzy rules, the output of the fuzzy controller is obtained by fuzzy reasoning and applied to the system through

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1 Introduction

the actuator. Arguably, fuzzy control is the embodiment of human experience and wisdom on machines. Therefore, the fuzzy controller often does not rely on the precise model of the controlled object, while the fuzzy rules designed based on the expert experience can usually satisfy the requirement of controller robustness. In the control of the underactuated manipulators, the fuzzy control is mainly used to combine with the energy based control method to achieve the swing-up control objective of the vertical underactuated manipulators [123]. Compared with the traditional energy based control method, the addition of fuzzy control enables the controller to coordinate state changes between the energy and posture of the system and ensures the smooth switching between the swing-up controller and the balance controller in the zoning control strategy [124, 125]. Furthermore, the fuzzy control is also used to coordinate the switching of controllers according to different control stages to improve the control performance. For example, in [126], a fuzzy PI + PD control strategy was proposed for the swing-up control of a Pendubot, and in [127], the fuzzy control was combined with the genetic algorithm to optimize a parameter of an energy-based controller online, so as to quickly realize the stable control of a planar single-link flexible manipulator. Moreover, the fuzzy system has the function of universal approximation [128, 129], so some fuzzy control methods were developed to estimate the uncertainties in the control of some underactuated manipulators. In [130], an adaptive fuzzy hierarchical sliding-mode control method was introduced to achieve the trajectory tracking control of an uncertain underactuated mobile inverted pendulum system. In [131], an adaptive fuzzy control method was proposed for a single-link flexible manipulator to handle its input dead-zones.

1.4.2 Neural Network Control Neural network control is an intelligent control method inspired by the physiological structure of the human brain. It realizes the intelligent behavior of machines by artificially simulating the working mechanism of the human brain [132]. Scientists use mathematical methods to abstract and simulate the structure and function of the human brain neural network, including designing mathematical models of artificial neurons and connecting various neurons through different topological structures to form different neural networks. In most cases, the neural networks have three layers of structure: input layer, hidden layer, and output layer. The input layer is the first layer of the neural network. It receives the input signal of the neural network and passes it to the next layer without performing any operation. The hidden layer may contain multiple hierarchical structures, which are mainly responsible for the operation of the neural network. In the hidden layer, the artificial neurons are connected with each other in a variety of topological structures. The input signals of the neural network are calculated and transmitted among different artificial neurons and hierarchical structures. Finally, the last layer of the hidden layer transmits the calculated values to the output layer. The output layer is the last layer of the neural network, which is used to receive and output the values from the last layer of the hidden layer.

1.4 Intelligent and Optimization Control

13

According to the connection mode of the artificial neurons, the neural networks can be divided into feedforward neural networks and feedback neural networks [133]. The feedforward neural networks utilize a unidirectional multi-layer topology, in which each layer contains several neurons, and the artificial neurons in each layer only receive inputs from the previous layer. Meanwhile, there is no feedback structure between the neurons. The commonly used feedforward neural networks include the Back Propagation neural network (BPNN), the Radial Basis Function neural network (RBFNN) and the convolution neural network (CNN). As for the feedback neural networks, there is a feedback structure from its output layer to its input layer. That is, each artificial neuron simultaneously feedbacks its own output signal to other artificial neurons as an input signal. This neural network is a feedback dynamic system that requires a period of time to achieve stability. The Hopfield neural network and the Long-Short Term Memory neural network (LSTMNN) are the commonly used feedback neural networks. The application of neural network control for underactuated manipulators is mainly based on the universal approximation characteristic of the neural networks that the neural networks can approximate any complexity functions with any accuracy through learning and training. There are two ways to learn and train, one is offline and the other is online. In [134], the BPNN was used to describe the coupling relationship among links of a planar four-link unactuated manipulator. In this study, a large number of experimental data on the coupling among links are collected and used to train the established BPNN offline. In [134], the RBFNN was used to implement a reinforcement learning scheme, which is used to adjust the parameters of the adaptive PID controller online to achieve the tracking control of a two-link underactuated manipulator. In addition, in the control studies of some underactuated manipulators, such as the three-link planar underactuated manipulator [135], the n-link planar manipulator [136], the two-link rigid-flexible manipulator [137], etc. The neural networks are also often used to approach the uncertainties of the system online. In these studies, the RBFNN was combined with the sliding mode control method, and the approximation error of the RBFNN about the system uncertainties was overcome by the sliding mode control.

1.4.3 Optimization-Based Control The idea of optimization is to obtain the most suitable solution to an optimization problem based on optimization algorithms, which is an important subject in the fields of control science and engineering. Such technique often appears in the process of choosing parameters with the purpose of satisfying the requirements of optimal motion trajectory, realizing better control performance, minimizing energy consumption, etc. Intelligent optimization is a type of optimization strategy based on intelligent algorithms, which often borrow ideas from intelligent phenomena of biotic populations in the natural world. These algorithms have simple logic, relatively simple processes

14

1 Introduction

and simple implementations, and they can usually get satisfactory solutions within acceptable time spans. Different intelligent optimization methods can also be further combined to deal with optimization problems that cannot be solved using conventional optimization techniques. Common intelligent optimization algorithms include genetic algorithm (GA), particle swarm optimization (PSO) algorithm, differential evolution algorithm (DEA), ant colony optimization (ACO) algorithm, etc. Among these intelligent optimization methods, the PSO algorithm has superiorities in its easy implementation, fast convergence, simplicity and the ability to perform parallel computing, So far, the PSO algorithm has been applied in various scenarios for underactuated manipulator control. E.g., [138] presented a PSO-based motion planning and control method, where the PSO was used in the nonlinear model predictive control to obtain the optimized trajectories of the joints. In addition, the PSO algorithm was also used to calculate the target angle values of the joints [139], to solve the parameters of the joint motion trajectories [140], and to obtain the optimized trajectories which reduce the amplitude of residual vibration of the flexible manipulator [141]. The GA was first presented by Professor John H. Holland in a monograph entitled Adaptation in Natural and Artificial Systems in 1975 [142], which was based on the phenomena in evolutionary biology, such as inheritance, mutation, natural selection and hybridization. For a tree-link underactuated manipulator, [143] presented a GA-based switching control method, where the GA was used to optimize the gain parameters of the switching control. Reference [144] presented a GA-based tip position and vibration suppression control method for a planar two-link rigid-flexible underactuated manipulator, where the GA is used to optimize the parameters of the designed controller. The GA is also a good choice to calculate the target angle values of the joints [145, 146].

1.5 Organization of This Book In summary, the analysis and control design of the UMS are very meaningful and challenging. On the one hand, the research on such systems can help humans more deeply understand the characteristics of the system, so that it can better serve production and life. On the other hand, the research can provide some valuable experience for other systems that existed in nature, so that the ability to transform and utilize nature is continuously improving with important scientific significance and practical value. The chapters of the book and their contents are arranged as follows: This chapter explains the characteristics of the underactuated manipulator, and the classical and intelligent control methods used in the underactuated manipulators. Chapter 2 mainly discusses the dynamic modeling of the underactuated manipulators based on the Lagrangian method and further analyses their characteristics.

1.5 Organization of This Book

15

Chapter 3 mainly discusses the vertical underactuated manipulators and presents some control methods. Firstly, a Lyapunov-based unified control strategy is proposed for the Acrobot and Pendubot, where the motion area is divided into the swing-up area and attractive area and the controllers are designed separately for each area. In order to overcome the sudden change of torque when switching controllers, a rewinding approach-based control strategy is proposed. In addition, a partition control method is applied to the three-link gymnast robot, and a reduced-order control method is proposed for a n-link underactuated manipulator with any one of its joints being passive. Chapter 4 mainly discusses the planar n-link (n  2) underactuated manipulators with a passive first joint and presents some control methods. The motion-state constraints of such a manipulator are firstly analyzed and obtained. Based on the motion-state constraint on angles, this chapter gives a stable control method to realize the position control of the planar two-link underactuated manipulator (i.e., planar Acrobot). Then, based on the motion-state constraint on angular velocities, two segment control methods are presented for the planar n-link (n  3) underactuated manipulators. With the help of the intelligent optimization algorithm, a continuous control method is further presented for this kind of manipulators. Finally, two robust control methods are designed for the manipulators with parametric perturbations and external disturbances. Chapter 5 mainly discusses the control of the planar underactuated manipulators with a passive non-first joint. Firstly, a control strategy based on Fourier transformation and intelligent optimization is proposed for the planar two-link underactuated manipulator (i.e., planar Pendubot). For a planar four-link one with a second passive joint, an energy attenuation approach-based control method is proposed. Furthermore, an effective model reduction method combined with DEA is presented for the four-link underactuated manipulator with a third passive joint. Finally, a general position control strategy on the basis of the DEA and bi-directional motion planning is designed for the planar underactuated manipulators with a passive non-first joint. Chapter 6 mainly focuses on the control of the flexible manipulators. Firstly, the flexible-joint manipulator is taken as the control object, and an equivalent-inputdisturbance (EID) based control approach is proposed to achieve the stable control and trajectory tracking control of this system. Then, the control problem of the flexible-link manipulator is analyzed. An online optimization approach and an optimization-based motion planning approach are proposed to achieve the stable control and position control with zero residual vibration for the flexible-link manipulator. Finally, an energy control method based on parameter optimization is proposed to handle the position control of an underactuated flexible-link manipulator with a passive joint.

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109. L. Weerakoon, N. Chopra, Swing up control of a soft inverted pendulum with revolute base, in Proceedings of the 2021 60th IEEE Conference on Decision and Control (IEEE, 2021), pp. 685–690 110. G. Turrisi, M. Capotondi, C. Gaz et al., On-line learning for planning and control of underactuated robots with uncertain dynamics. IEEE Robot. Autom. Lett. 7(1), 358–365 (2021) 111. X. Xin, K. Makino, S. Izumi et al., Anti-swing control of the Pendubot using damper and spring with positive or negative stiffness. Int. J. Robust Nonlinear Control 31(9), 4227–4246 (2021) 112. P.O. Scokaert, J.B. Rawlings, Constrained linear quadratic regulation. IEEE Trans. Autom. Control 43(8), 1163–1169 (1998) 113. L. Wang, S. Chen, P. Zhang et al., A simple control strategy based on trajectory planning for vertical acrobot 10(12), 308 (2021) 114. Y. Liu, X. Xin, Global motion analysis of energy-based control for 3-link planar robot with a single actuator at the first joint. Nonlinear Dyn. 88(3), 1749–1768 (2017) 115. T. Chen, B. Goodwine, Controllability and accessibility results for n-link horizontal planar manipulators with one unactuated joint. Automatica 125, 109480 (2021) 116. K. Izumi, K. Watanabe, K. Ichida et al., The design of fuzzy energy regions optimized by GA for a switching control of multi-link underactuated manipulators, in Proceedings of the 2007 International Conference on Control, Automation and Systems (IEEE, 2007), pp. 24–29 117. Z. Huang, X. Lai, P. Zhang et al., Virtual model reduction-based control strategy of planar three-link underactuated manipulator with middle passive joint. Int. J. Control, Autom. Syst. 19(1), 29–39 (2021) 118. A. Zhang, X. Lai, M. Wu et al., Nonlinear stabilizing control for a class of underactuated mechanical systems with multi degree of freedoms. Nonlinear Dyn. 89(3), 2241–2253 (2017) 119. X. Lai, Y. Wang, M. Wu et al., Stable control strategy for planar three-link underactuated mechanical system. IEEE/ASME Trans. Mechatron. 21(3), 1345–1356 (2016) 120. X. Lai, P. Zhang, Y. Wang et al., Position-posture control of a planar four-link underactuated manipulator based on genetic algorithm. IEEE Trans. Ind. Electron. 64(6), 4781–4791 (2017) 121. D. Liu, X. Lai, Y. Wang et al., Position control for planar four-link underactuated manipulator with a passive third joint. ISA Trans. 87, 46–54 (2019) 122. D. Driankov, H. Hellendoorn, M. Reinfrank, An Introduction to Fuzzy Control (Springer Science & Business Media, 2013) 123. K. Ichida, K. Izumi, K. Watanabe et al., Control of three-link underactuated manipulators using a switching method of fuzzy energy regions. IEEE Trans. Syst., Man, Cybern. Part B (Cybern.) 39(2), 389–398 (2009) 124. X. Q. Ma, C. Y. Su, A new fuzzy approach for swing up control of Pendubot, in Proceedings of the 2002 American Control Conference (IEEE, 2002), pp. 1001–1006 125. K. Ichida, K. Watanabe, K. Izumi et al., Fuzzy switching control of underactuated manipulators with approximated switching regions, in Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 586–591 126. E. Sanchez, V. Flores, Real-time underactuated robot swing-up via fuzzy PI+PD control. J. Intell. Fuzzy Syst. 17(1), 1–13 (2006) 127. Q. Meng, X. Lai, Y. Wang et al., A fast stable control strategy based on system energy for a planar single-link flexible manipulator. Nonlinear Dyn. 94, 615–626 (2018) 128. R.R. Yager, V. Kreinovich, Universal approximation theorem for uninorm-based fuzzy systems modeling. Fuzzy Sets Syst. 140(2), 331–339 (2003) 129. L. Wang, J.M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal leastsquares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992) 130. C.-L. Hwang, C.-C. Chiang, Y.-W. Yeh, Adaptive fuzzy hierarchical sliding-mode control for the trajectory tracking of uncertain underactuated nonlinear dynamic systems. IEEE Trans. Fuzzy Syst. 22(2), 286–299 (2014) 131. C. Zhang, T. Yang, N. Sun et al., An adaptive fuzzy control method of single-link flexible manipulators with input dead-zones. Int. J. Fuzzy Syst. 22, 2521–2533 (2020)

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132. F.W. Lewis, S. Jagannathan, A. Yesildirak, Neural Network Control of Robot Manipulators and Non-linear Systems (Taylor & Francis, 1999) 133. J. Anderson, An Introduction to Neural Networks (The MIT Press, 1995) 134. J. Wu, J. She, Y. Wang et al., Position and posture control of planar four-link underactuated manipulator based on neural network model. IEEE Trans. Ind. Electron. 67(6), 4721–4728 (2019) 135. P. Zhang, X. Lai, Y. Wang et al., Motion planning and adaptive neural sliding mode tracking control for positioning of uncertain planar underactuated manipulator. Neurocomputing 334, 197–205 (2019) 136. Y. Wang, X. Lai, P. Zhang et al., Adaptive robust control for planar n-link underactuated manipulator based on radial basis function neural network and online iterative correction method. J. Frankl. Inst. 355, 8373–8391 (2018) 137. X. Zhou, H. Wang, Y. Tian, Adaptive boundary iterative learning vibration control using disturbance observers for a rigid-flexible manipulator system with distributed disturbances and input constraints. J. Vib. Control 28(11–12), 1324–1340 (2022) 138. P. Zhang, X. Lai, Y. Wang et al., PSO-based nonlinear model predictive planning and discretetime sliding tracking control for uncertain planar underactuated manipulators. Int. J. Syst. Sci. 53(10), 2075–2089 (2022) 139. X. Lai, Y. Wang, M. Wu et al., Stable control strategy for planar three-link underactuated mechanical system. IEEE/ASME Trans. Mechatron. 21(3), 1345–1356 (2016) 140. P. Zhang, X. Lai, Y. Wang et al., Chaos-PSO-based motion planning and accurate tracking for position-posture control of a planar underactuated manipulator with disturbance. Int. J. Control, Autom. Syst. 19, 3511–3521 (2021) 141. E. Khanmirza, K. Daneshjou, A. Ravandi, Underactuated flexible aerial manipulators: a new framework for optimal trajectory planning under constraints induced by complex dynamics. J. Intell. Robot. Syst. 92, 599–613 (2018) 142. H.H. John, Adaptation in Natural and Artificial Systems (The MIT Press, 1975) 143. K. Ichida, S. Katayama, K. Watanabe, Design parameter setting by GA with switching method for three-DOF underactuated manipulators. J. Inst. Ind. Appl. Eng. 3(4), 178–182 (2015) 144. Q. Meng, X. Lai, Z. Yan et al., Tip position control and vibration suppression of a planar two-link rigid-flexible underactuated manipulator. IEEE Trans. Cybern. 52(7), 6771–6783 (2022) 145. P. Xiong, X. Lai, M. Wu, Position and posture control for a class of second-order nonholonomic underactuated mechanical system. IMA J. Math. Control Inf. 35(2), 523–533 (2018) 146. X. Lai, P. Zhang, Y. Wang et al., Position and posture control for a class of second-order nonholonomic underactuated mechanical system. IEEE Trans. Ind. Electron. 64(6), 4781– 4791 (2017)

Chapter 2

Modeling and Characteristics Analysis of Underactuated Manipulators

Before discussing the control of an underactuated manipulator, it is necessary to construct a mathematical model for the system [1–3]. For an underactuated manipulator, different modeling methods lead to different system descriptions resulting in huge differences in system analysis and control design. Therefore, it is very important to choose an appropriate modeling method. At present, the Lagrangian method is a typical modeling method in the field of engineering machinery, which is popular among engineers and technicians due to its concise modeling process [4–6]. This chapter mainly discusses how this method is applied to establish the dynamic models of the underactuated manipulators and to analyze their characteristics.

2.1 Dynamic Modeling of Typical Underactuated Manipulators Let q = [q1 , q2 , . . . , qn ]T be the generalized coordinate of an n-DOFs manipulator, and define the Lagrange function of the system as L (q, q) ˙ = K (q, q) ˙ − P (q) ,

(2.1)

where K (q, q) ˙ and P (q) are the kinetic energy and potential energy, respectively. According to the Lagrangian method, the dynamics of the manipulator satisfy the following Lagrange dynamics equations ˙ ∂ L (q, q) ˙ d ∂ L (q, q) − = τi , i = 1, 2, . . . , n, dt ∂ q˙i ∂qi

© Science Press 2023 J. Wu et al., Control of Underactuated Manipulators, https://doi.org/10.1007/978-981-99-0890-5_2

(2.2)

23

24

2 Modeling and Characteristics Analysis of Underactuated Manipulators

where τi is the control input of the ith dynamics equation. For an underactuated manipulator, because the number of its control inputs is less than that of its DOFs, the control inputs in (2.2) usually satisfy τr 1 = 0, τr 2 = 0, . . . , τr m = 0,

(2.3)

where r1 , r2 , . . . , rm ∈ { 1, 2, . . . , n} are the positions of the dynamics equations that the system has no control input, and 1  m < n. The above describes the basic theory of Lagrangian modeling, and it is applied to modeling for several typical underactuated manipulators.

2.1.1 The Inertia Wheel Pendulum Manipulator The wheel pendulum manipulator , which is illustrated in Fig. 2.1, is consisted of a link and a rotating wheel at the end. The link is underactuated, and only one actuator is installed on the wheel to make the system move in a vertical plane. Obviously, it is a very typical 2-DOF underactuated system. In Fig. 2.1, q1 is the rotation angle of the link, m 1 is the mass of the link, L 1 is the length of the link, L c1 is the distance between the passive joint and the center of mass of the link, J1 is the moment of inertia of the link around its center of mass, q2 is the rotation angle of the wheel, m 2 is the mass of the wheel, J2 is the moment of inertia of the wheel, g is the gravity acceleration, and τ2 is the torque applied to the wheel. Define the generalized coordinates of the system as q = [q1 , q2 ]T , and it is easy to obtain the position coordinates of the center of mass of the link and the wheel, X 1 and X 2 , which are

Fig. 2.1 Wheel pendulum manipulator

2.1 Dynamic Modeling of Typical Underactuated Manipulators

X 1 = [L c1 sin q1 , L c1 cosq1 ] , X 2 = [L 1 sin q1 , L 1 cosq1 ] .

25

(2.4)

The time derivative of which can be calculated as X˙ 1 = [L c1 q˙1 cos q1 , −L c1 q˙1 sinq1 ] , X˙ 2 = [L 1 q˙1 cos q1 , −L 1 q˙1 sinq1 ] .

(2.5)

˙ and K 2 (q, q), ˙ Thus, the kinetic energy of the link and the wheel is defined as K 1 (q, q) respectively, ⎧ 1  2 ⎪ ⎪ ˙ = m 1  X˙ 1  + ⎪ K 1 (q, q) ⎨ 2 ⎪ ⎪ 1  2 ⎪ ⎩ K 2 (q, q) ˙ = m 2  X˙ 2  + 2

 1 1 J1 q˙12 = m 1 L 2c1 + J1 q˙12 , 2 2

1 1 1 J1 (q˙1 + q˙2 )2 = m 2 L 21 q˙12 + J2 (q˙1 + q˙2 )2 . 2 2 2 (2.6) From (2.6), the total kinetic energy of the wheel pendulum manipulator becomes K (q, q) ˙ = K 1 (q, q) ˙ + K 2 (q, q) ˙ = J2 q˙1 q˙2 +

 1 1 J2 q˙22 + m 1 L 2c1 + m 2 L 21 + J1 + J2 q˙12 . 2 2

(2.7)

If the plane with x-axis is taken as the zero potential energy surface, then the gravitational potential energy of the system can be obtained as P (q) = (m 1 L c1 + m 2 L 1 ) g cos q1 .

(2.8)

Combining (2.1) , (2.7) and (2.8) gets L (q, q) ˙ =

1 1 m 11 q˙12 + m 12 q˙1 q˙2 + m 22 q˙22 − M0 cos q1 , 2 2

where m 11 = m 1 L 2c1 + m 2 L 21 + J1 + J2 , m 12 = m 21 = m 22 = J2 , and (m 1 L c1 + m 2 L 1 ) g. From (2.9), the following equations can be obtained

(2.9) M0 =

⎧ ∂ L (q, q) ˙ ∂ L (q, q) ˙ ⎪ = m 11 q˙1 + m 12 q˙2 , = M0 sin q1 , ⎪ ⎪ ⎨ ∂ q˙1 ∂q1 ⎪ ⎪ ∂ L (q, q) ˙ ∂ L (q, q) ˙ ⎪ ⎩ = m 21 q˙1 + m 22 q˙2 , = 0. ∂ q˙2 ∂q2

(2.10)

Finally, combine (2.2) and (2.10), and the dynamic model of the wheel pendulum manipulator is given by

26

2 Modeling and Characteristics Analysis of Underactuated Manipulators

⎡ ⎣

m 11 m 12 m 21 m 22

⎤⎡ ⎦⎣

q¨1 q¨2

⎤

⎡

⎦+⎣

−M0 sin q1

⎤

⎡

⎦=⎣

0

0 τ2

⎤ ⎦.

(2.11)

2.1.2 Two-Link Underactuated Manipulator Underactuated manipulators can be divided into two categories according to different motion planes. One is the vertical underactuated manipulator, which moves in a vertical plane [7–10]. The other one is the planar underactuated manipulator, which moves in a horizontal plane [11–14]. In this section, the dynamic model of a two-link fully-actuated vertical manipulator is first built, then the dynamic models of two-link underactuated manipulators are introduced. The structure diagram of the two-link fully-actuated manipulator is depicted in Fig. 2.2. In Fig. 2.2, in which qi (i=1,2), m i and L i are the angle, mass and length, respectively, of the ith link, L ci is the distance between the ith joint and the center of mass of the ith link, Ji is the moment of inertia of the ith link around its center of mass, τi is the applied torque of the ith joint, and g is the gravity acceleration. For this system, the generalized coordinates of the system is q = [q1 , q2 ]T . From Fig. 2.2, the position coordinates of center of the mass of the first link and the second link, X 1 and X 2 , respectively, are ⎧ ⎨ X 1 = [L c1 sin q1 , L c1 cosq1 ] , ⎩

(2.12) X 2 = [L 1 sin q1 + L c2 sin(q1 + q2 ), L 1 cosq1 + L c2 cos(q1 + q2 )] .

Then, one has

Fig. 2.2 Fully-actuated two-link manipulator

2.1 Dynamic Modeling of Typical Underactuated Manipulators

⎧ X˙ 1 = [L c1 q˙1 cos q1 , −L c1 q˙1 sin q1 ] , ⎪ ⎪ ⎪ ⎪ ⎨ X˙ 2 = [L 1 q˙1 cos q1 + L c2 (q˙1 + q˙2 ) cos(q1 + q2 ) ⎪ ⎪ ⎪ ⎪ ⎩ − L 1 q˙1 sin q1 − L c2 (q˙1 + q˙2 )sin(q1 + q2 )].

27

(2.13)

Hence, the kinetic energy of the first link, K 1 (q, q), ˙ is ˙ = K 1 (q, q)

2 1  1 1  m 2  X˙ 1  + J1 q˙12 = m 1 L 2c1 + J1 q˙12 , 2 2 2

(2.14)

˙ is and the kinetic energy of the second link, K 2 (q, q), ˙ = K 2 (q, q) =

2 1 1  m 2  X˙ 2  + J2 (q˙1 + q˙2 )2 2 2  1 m 2 L 21 + m 2 L 2c2 + J2 + 2m 2 L 1 L 2 cos q2 q˙12 2

   1 m 2 L 2c2 + J2 q˙22 . + m 2 L 2c2 + J2 + m 2 L 1 L c2 cos q2 q˙1 q˙2 + 2 (2.15) From (2.14) and (2.15), the total kinetic energy of the manipulator is obtained as ˙ + K 2 (q, q) ˙ = K (q, q) ˙ = K 1 (q, q)

1 1 m 11 q˙12 + m 12 q˙1 q2 + m 22 q˙22 , 2 2

(2.16)

where m 11 = a1 + a2 + 2a3 cos q2 , m 12 = a2 + a3 cos q2 , m 22 = a2 , (2.17) a1 = m 1 L 2c1 + m 2 L 21 + J1 , a2 = m 2 L 2c2 + J2 , a3 = m 2 L 1 L c2 . If the plane with x-axis is taken as the zero potential energy surface, the potential energies of the first link and the second link of the manipulator, P1 (q) and P2 (q), respectively, are P1 (q) = m 1 gL c1 cos q1 , P2 (q) = m 2 g (L 1 cos q1 + L c2 cos (q1 + q2 )) . (2.18) Therefore, the total potential energy of the manipulator is P (q) = P1 (q) + P2 (q) = a4 cos q1 + a5 cos (q1 + q2 ) , where a4 = (m 1 L c1 + m 2 L 1 ) g and a5 = m 2 L c2 g. Combining (2.1), (2.16) with (2.19) yields

(2.19)

28

2 Modeling and Characteristics Analysis of Underactuated Manipulators

L (q, q) ˙ =

1 1 m 11 q˙12 + m 12 q˙1 q˙2 + m 22 q˙22 − a4 cos q1 − a5 cos (q1 + q2 ) . (2.20) 2 2

From (2.20), the following equations can be obtained ∂ L (q, q) ˙ ∂ q˙1 ∂ L (q, q) ˙ ∂q1 ∂ L (q, q) ˙ ∂ q˙2 ∂ L (q, q) ˙ ∂q2

= m 11 q˙1 + m 12 q˙2 ,

(2.21a)

= a4 sin q1 + a5 sin (q1 + q2 ) ,

(2.21b)

= m 12 q˙1 + m 22 q˙2 ,

(2.21c)

= −a3 (q˙1 + q˙2 ) q˙2 sin q1 + a5 sin (q1 + q2 ) .

(2.21d)

Finally, from (2.2), the dynamic equation of the manipulator is given by ⎡ ⎣

m 11 m 12

⎤⎡ ⎦⎣

m 12 m 22

q¨1 q¨2

⎤

⎡

⎦+⎣

c11 c12 c21 0

⎤⎡ ⎦⎣

q˙1 q˙2

⎤

⎡

⎦+⎣

g1 g2

⎤

⎡

⎦=⎣

τ1 τ2

⎤ ⎦,

(2.22)

where c11 = −a3 q˙2 sin q2 , c12 = −a3 (q˙1 + q˙2 ) sin q2 , c21 = a3 q˙1 sin q2 , g1 = −a4 sin q1 − a5 sin (q1 + q2 ), g2 = −a5 sin (q1 + q2 ).

(2.23)

For the two-link fully-actuated vertical manipulator, if there is no torque applied to its first joint, that is, τ1 = 0, it becomes the underactuated Acrobot system as shown in Fig. 2.3. Correspondingly, if there is no torque applied to its second joint, that is, τ2 = 0, which leads to the underactuated Pendubot system as shown in Fig. 2.4. These 2-DOF underactuated manipulators are the most discussed systems by scholars, and their dynamic equations can be obtained by substituting τ1 = 0 or τ2 = 0 into (2.22).

Fig. 2.3 Underactuated Acrobot system

2.1 Dynamic Modeling of Typical Underactuated Manipulators

29

Fig. 2.4 Underactuated Pendubot system

Fig. 2.5 Planar two-link manipulator

In addition, assuming that the two-link manipulator is not subjected to gravity, that is, g = 0, the manipulator will move in a horizontal plane. The structural sketch of a planar two-link fully-actuated manipulator is illustrated in Fig. 2.5, and its dynamic equation is obtained by substituting g = 0 into (2.22) as ⎡ ⎣

m 11 m 12 m 12 m 22

⎤⎡ ⎦⎣

q¨1 q¨2

⎤

⎡

⎦+⎣

c11 c12 c21 0

⎤⎡ ⎦⎣

q˙1 q˙2

⎤

⎡

⎦=⎣

τ1 τ2

⎤ ⎦.

(2.24)

Similar to the case of the two-link vertical manipulator, when there is no torque applied to the first joint of the planar fully-actuated manipulator, it becomes a planar underactuated Acrobot system as shown in Fig. 2.6. When there is no torque applied to the second joint, it becomes a planar underactuated Pendubot system as shown in Fig. 2.7. The dynamic equations of these two planar underactuated manipulators can be obtained by substituting τ1 = 0 or τ2 = 0 into (2.24). Remark 2.1 The control of the planar underactuated manipulator is a recognized problem that has not been solved very well in the field of engineering control. The main reason is that the linear approximate model at any equilibrium point of the planar underactuated manipulator is not controllable. However, for the spring-coupled

30

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Fig. 2.6 Planar underactuated Acrobot system

Fig. 2.7 Planar underactuated Pendubot system

planar Pendubot, the difficulty of its control is reduced since installing a spring at the underactuated joint transforms the planar Pendubot system into a controllable system. In addition, because the price of spring is much lower than that of actuators and the spring elasticity will not disappear when the system is in outer space, this spring-coupled manipulator has good application prospects in the fields of industrial manufacturing, medical and health care, and outer space exploration.

2.1.3 Multi-link Underactuated Manipulator In this section, the modeling of multi-link underactuated manipulators is discussed. Since the three-link underactuated manipulator is the simplest multi-link underactuated manipulator [15–18], its dynamic model is first built. The structure of a three-link fully-actuated vertical manipulator is depicted in Fig. 2.8, where qi (i=1, 2, 3), m i and L i are the angle, mass and length, respectively, of the ith link, L ci is the distance

2.1 Dynamic Modeling of Typical Underactuated Manipulators

31

Fig. 2.8 Fully-actuated three-link manipulator

between the ith joint and the center of mass of the ith link, Ji is the moment of inertia of the ith link around its center of mass, τi is the applied torque of the ith joint, and g is the gravity acceleration. The generalized coordinates of this three-link system is q = [q1 , q2 , q3 ]T . From Fig. 2.8, the position coordinate, X i , of the ith link is obtained as X 1 = [L c1 sin q1 , L c1 cos q1 ] ,

(2.25a)

X 2 = [L 1 sin q1 + L c2 sin (q1 + q2 ) , L 1 cos q1 + L c2 cos (q1 + q2 )] , X 3 = [L 1 sin q1 + L 2 sin (q1 + q2 ) + L c3 sin (q1 + q2 + q3 ) , L 1 cos q1 + L 2 cos (q1 + q2 ) + L c3 cos (q1 + q2 + q3 )] .

(2.25b) (2.25c)

From (2.25), one obtains X˙ 1 = [L c1 q˙1 cos q1 , −L c1 q˙2 sin q1 ] , X˙ 2 = [L 1 q˙1 cos q1 + L c2 (q˙1 + q˙2 ) cos (q1 + q2 ) −L 1 q˙1 sin q1 − L c2 (q˙1 + q˙2 ) sin (q1 + q2 )] , X˙ 3 = [L 1 q˙1 cos q1 + L 2 (q˙1 + q˙2 ) cos (q1 + q2 ) + L c3 (q˙1 + q˙2 + q˙3 ) cos (q1 + q2 + q3 ) , − L 1 q˙1 sin q1 − L 2 (q˙1 + q˙2 ) sin (q1 + q2 ) −L c3 (q˙1 + q˙2 + q˙3 ) sin (q1 + q2 + q3 )] .

(2.26a) (2.26b) (2.26c)

32

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Therefore, the kinetic energy of the first link, K 1 (q, q), ˙ is ˙ = K 1 (q, q)

2 1  1 1  m 1 L 2c1 + J1 q˙12 , m 2  X˙ 1  + J1 q˙12 = 2 2 2

(2.27)

˙ is the kinetic energy of the second link, K 2 (q, q), ˙ = K 2 (q, q) =

2 1 1  m 2  X˙ 2  + J2 (q˙1 + q˙2 )2 2 2  1 m 2 L 21 + m 2 L 2c2 + J2 + 2m 2 L 1 L 2 cos q2 q˙12 2

   1 m 2 L 2c2 + J2 q˙22 , + m 2 L 2c2 + J2 + m 2 L 1 L c2 cos q2 q˙1 q˙2 + 2 (2.28) and the kinetic energy of the third link, K 3 (q, q), ˙ is 2 1 1  m 3  X˙ 3  + J3 (q˙1 + q˙2 + q˙3 )2 2 2  1  2 m 3 L 1 + L 22 + L 2c3 + 2L 1 L 2 cos q2 + 2L 2 L c3 cos q3 q˙12 = 2   1 1  2 m 3 L 2 + L 2c3 + 2L 2 L c3 cos q3 + J3 q˙22 + m 3 J3 q˙12 + 2 2

   + m 3 L 2c3 + L 2 L c3 cos q3 + L 1 L c3 cos (q2 + q3 ) + J3 q˙1 q˙3 1 + m 3 2L 1 L c3 cos (q2 + q3 ) q˙12

2    + m 3 L 2c3 + L 2 L c3 cos q3 + J3 q˙1 q˙3 

 + m 3 L 22 + L 2c3 + L 1 L 2 cos q2 + 2L 2 L c3 cos q3 q˙1 q˙2  1 m 3 L 2c3 +J3 q˙32 +m 3 J3 q˙1 q˙2 +m 3 L 1 L c3 q˙1 q˙2 cos (q2 +q3 ) . + 2 (2.29) Then, the total kinetic energy of the manipulator is ˙ = K 3 (q, q)

K (q, q) ˙ = K 1 (q, q) ˙ + K 2 (q, q) ˙ + K 3 (q, q) ˙ =

1 1 1 m 11 q˙12 + m 22 q˙22 + m 33 q˙32 + m 12 q˙1 q˙2 + m 13 q˙1 q˙3 + m 23 q˙2 q˙3 , 2 2 2 (2.30)

where m 11 = a1 + a2 + a4 + 2a5 cos (q2 + q3 ) + 2a3 cos q2 + 2a6 cos q3 , m 12 = m 21 = a2 + a4 + a3 cos q2 + a5 cos (q2 + q3 ) + 2a6 cos q3 ,

(2.31a) (2.31b)

m 13 = m 31 = a4 + a5 cos (q2 + q3 ) + a6 cos q3 ,

(2.31c)

2.1 Dynamic Modeling of Typical Underactuated Manipulators

33

m 22 = a2 + a4 + 2a6 cos q3 ,

(2.31d)

m 23 = m 32 = a4 + a6 cos q3 , m 33 = a4 ,

(2.31e) (2.31f)

a1 = m 1 L 2c1 + J1 + (m 2 + m 3 ) L 21 , a2 = J2 + m 2 L 2c2 + m 3 L 22 , a3 = (m 2 L c2 + m 3 L 2 ) L 1 , a4 = J3 + a5 = m 3 L c3 L 1 , a6 = m 3 L c3 L 2 .

m 3 L 2c3 ,

(2.31g) (2.31h) (2.31i)

If the plane with x-axis is taken as the zero potential energy surface, the potential energy Pi (q) (i = 1, 2, 3) of ith link of the manipulator is ⎧ ⎨ P1 (q) = m 1 gL c1 cos q1 , P2 (q) = m 2 g [L 1 cos q1 + L c2 cos (q1 + q2 )] , ⎩

P3 (q) = m 3 g [L 1 cos q1 + L 2 cos (q1 + q2 ) + L c3 cos (q1 + q2 + q3 )] . (2.32) From (2.32), the total potential energy of the manipulator can be obtained as P (q) = P1 (q) + P2 (q) + P3 (q) (2.33) = β1 cos q1 + β2 cos (q1 + q2 ) + β3 cos (q1 + q2 + q3 ) , where β1 = (m 1 L c1 + m 2 L c2 + m 3 L c3 ) g, β2 = (m 2 L c2 + m 3 L 2 ) g, β3 = m 3 L c3 g. Combining (2.1), (1.39) and (1.42) yields L (q, q) ˙ =

1 1 1 m 11 q˙12 + m 22 q˙22 + m 33 (q) q˙32 + m 12 q˙1 q˙2 2 2 2 + m 13 q˙1 q˙3 + m 23 q˙2 q˙3 − β1 cos q1

(2.34)

− β2 cos (q1 + q2 ) − β3 cos (q1 + q2 + q3 ) . Further, the following equations can be obtained ∂ L (q, q) ˙ ∂ q˙1 ∂ L (q, q) ˙ ∂q1 ∂ L (q, q) ˙ ∂ q˙2 ∂ L (q, q) ˙ ∂q2

= m 11 q˙1 + m 12 q˙2 + m 13 q˙3 ,

(2.35a)

= β1 sin q1 + β2 sin (q1 + q2 ) + β3 sin (q1 + q2 + q3 ),

(2.35b)

= m 12 q˙2 + m 22 q˙2 + m 13 q˙3 ,

(2.35c)

= −a3 (q˙1 + q˙2 ) q˙1 sin q2 − a5 (q˙1 + q˙2 + q˙3 ) q˙1 sin (q2 + q3 ) (2.35d) + β2 sin (q1 + q2 ) + β3 sin (q1 + q2 + q3 ),

∂ L (q, q) ˙ = m 13 q˙1 + m 23 q˙2 + m 33 q˙3 , ∂ q˙3

(2.35e)

34

2 Modeling and Characteristics Analysis of Underactuated Manipulators

∂ L (q, q) ˙ = −a5 (q˙1 + q˙2 + q˙3 ) q˙1 sin (q2 + q3 ) + β3 sin (q1 + q2 + q3 ) (2.35f) ∂q3   −a6 q22 + 2q˙1 q˙2 + q˙1 q˙3 + q˙2 q˙3 sin q3 . Based on (2.2), the dynamic equation of the vertical three-link fully-actuated manipulator is given by ⎡

m 11 m 12 m 13

⎤⎡

q¨1

⎤

⎡

c11 c12 c13

⎤⎡

q˙1

⎤

⎡

g1

⎤

⎡

τ1

⎤

⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ m 21 m 22 m 23 ⎥ ⎢ q¨2 ⎥ + ⎢ c21 c22 c23 ⎥ ⎢ q˙2 ⎥ + ⎢ g2 ⎥ = ⎢ τ2 ⎥ , ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ m 31 m 32 m 33 q¨3 c31 c32 c33 q˙3 g3 τ3

(2.36)

where c11 = −a5 (q˙2 + q˙3 ) sin (q2 + q3 ) − a3 q˙2 sin q2 − a6 q˙3 sin q3 , c12 = −a3 (q˙2 +q˙3 ) sin q2 −a6 q˙2 sin q3 −a5 (q˙1 +q˙2 +q˙3 ) sin (q2 +q3 ), c13 = − (q˙1 + q˙2 + q˙3 ) (a5 sin (q2 + q3 ) + a6 sin q3 ), c21 = a5 q˙1 sin (q1 + q2 ) + a3 q˙1 sin q2 − a6 q˙3 sin q3 , c22 = −a6 q˙3 sin q3 , c23 = −a6 (q˙1 + q˙2 + q˙3 ) sin q3 ,

(2.37d) (2.37e)

c31 = a5 q˙1 sin (q2 + q3 ) + a6 (q˙2 + q˙3 ) sin q3 , c32 = a6 (q˙2 + q˙3 ) sin q3 , c33 = 0,

(2.37f) (2.37g)

g1 = −β1 sin q1 − β2 sin (q1 + q2 ) − β3 sin (q1 + q2 + q3 ), g2 = −β2 sin (q1 + q2 ) − β3 sin (q1 + q2 + q3 ),

(2.37h) (2.37i)

g3 = −β3 sin (q1 + q2 + q3 ).

(2.37j)

(2.37a) (2.37b) (2.37c)

For the fully-actuated manipulator depicted in Fig. 2.8, it will become an underactuated manipulator when its one joint is passive joint. For example, when τ1 = 0, it becomes the Passive-Active-Active (PAA) underactuated manipulator as shown in Fig. 2.9, and the corresponding dynamic equation can be obtained by substituting τ1 = 0 into (2.36). Similarly, when τ2 = 0 or τ3 = 0, it becomes the Active-PassiveActive (APA) or Active-Active-Passive (AAP) underactuated manipulator respectively. Since the structure diagrams of the AAP and APA underactuated manipulators are similar to that of the PAA underactuated manipulator, their structure diagrams will not be given here. On the basis of the PAA underactuatred manipulator, let τ2 = 0, then, the fully-actuated manipulator becomes the Passive-Passive-Active (PPA) underactuated manipulator, and its structure diagram is shown in Fig. 2.10. The dynamic equation of this manipulator can be obtained by combining τ1 = τ2 = 0 with (2.36). In addition, substituting τ1 = 0, g = 0 into (2.36) yields

2.1 Dynamic Modeling of Typical Underactuated Manipulators

35

Fig. 2.9 PAA underactuated manipulator

Fig. 2.10 PPA underactuated manipulator

⎡

m 11 m 12 m 13

⎤⎡

q¨1

⎤

⎡

c11 c12 c13

⎤⎡

q˙1

⎤

⎡

0

⎤

⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ m 21 m 22 m 23 ⎥⎢ q¨2 ⎥ + ⎢ c21 c22 c23 ⎥⎢ q˙2 ⎥ = ⎢ τ2 ⎥ , ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ m 31 m 32 m 33 q¨3 c31 c32 c33 q˙3 τ3

(2.38)

36

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Fig. 2.11 Planar PAA underactuated manipulator

Fig. 2.12 Fully-actutaed four-link manipulator

which is the dynamic model of a planar PAA underactuated manipulator as shown in Fig. 2.11. The dynamic equations have been given for different types of three-link underactuated manipulator. Next, the case with four-link underactuated manipulator is discussed. Figure 2.12 shows the four-link fully-actuated vertical manipulator. Because the modeling process of this manipulator is similar to that of the threelink manipulator, the complicated derivation process is omitted, and the dynamic equation is given directly as

2.1 Dynamic Modeling of Typical Underactuated Manipulators

⎡

m 11 m 12 m 13 m 14

⎤⎡

q¨1

⎤ ⎡

c11 c12 c13 c14

⎤⎡

q˙1

⎤ ⎡

37

g1

⎤ ⎡

τ1

⎤

⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ m 21 m 22 m 23 m 24 ⎥⎢ q¨2 ⎥ ⎢ c21 c22 c23 c24 ⎥⎢ q˙2 ⎥ ⎢ g2 ⎥ ⎢ τ2 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥+⎢ ⎥=⎢ ⎥ , (2.39) ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ m 31 m 32 m 33 m 34 ⎥⎢ q¨3 ⎥ ⎢ c31 c32 c33 c34 ⎥⎢ q˙3 ⎥ ⎢ g3 ⎥ ⎢ τ3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ m 41 m 42 m 43 m 44 q¨4 c41 c42 c43 c44 q˙4 g4 τ4 where m 11 = a1 + a2 + a4 + 2a3 cos q2 + 2a3 cos (q2 + q3 ) + 2a6 cos q3 + 2a7 cos (q2 + q3 + q4 ) + 2a8 cos (q3 + q4 ) + 2a9 cos q4 + a10 , m 12 = m 21 = a2 + a4 + a3 cos q2 + a3 cos (q2 + q3 ) + 2a6 cos q3 + a7 cos (q2 + q3 + q4 ) + 2a9 cos q4 + a10 , m 13 = m 31 = a4 + a3 cos (q2 + q3 ) + a6 cos q3 + a7 cos (q2 + q3 + q4 ) m 14 m 24

+ a8 cos (q3 + q4 ) + +2a9 cos q4 + a10 , = m 41 = a7 cos (q2 + q3 + q4 ) + 2a8 cos (q3 + q4 ) + a9 cos q4 + a10 , = m 42 = a4 + a6 cos q3 + a8 cos (q3 + q4 ) + 2a9 cos q4 + a10 ,

m 33 = a4 + 2a9 cos q4 + a10 , m 34 = m 43 = a4 + a6 cos q3 + a8 cos (q3 + q4 ) + 2a9 cos q4 + a10 , m 44 = a10 , are the elements of a positive-definite symmetric matrix M (q) and ⎡ ⎤ ⎡ ⎤⎡ ⎤ q˙1 c11 c12 c13 c14 H1 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ H2 ⎥ ⎢ c21 c22 c23 c24 ⎥ ⎢ q˙2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥⎢ ⎥ H (q, q) ˙ =⎢ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ H3 ⎥ ⎢ c31 c32 c33 c34 ⎥ ⎢ q˙3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ H4 c41 c42 c43 c44 q˙4 is the matrix of Coriolis force and centrifugal force, where

38

2 Modeling and Characteristics Analysis of Underactuated Manipulators

H1 = −a3 (2q˙1 +q˙2 ) q˙2 sin q2 − a6 (2q˙1 +2q˙2 + q˙3 ) q˙3 sin q3 − a5 (2q˙1 +q˙2 + q˙3 ) (q˙2 +q˙3 ) sin (q2 + q3 ) − a7 (2q˙1 +q˙2 + q˙3 + q˙4 ) (q˙2 +q˙3 + q˙4 ) sin (q2 + q3 + q4 ) − a8 (2q˙1 +q˙2 + q˙3 + q˙4 ) (q˙3 + q˙4 ) sin (q3 + q4 ) − a9 (2q˙1 +q˙2 + 2q˙3 + q˙4 ) q˙4 sin q4 , H2 = −a3 q˙12 sin q2 + a5 q12 sin (q2 + q3 ) − a6 (2q˙1 +2q˙2 + q˙3 ) q˙3 sin q3 + a7 q12 sin (q2 + q3 + q4 ) − a9 (2q˙1 +q˙2 + 2q˙3 + q˙4 ) q˙4 sin q4 − a8 (2q˙1 +q˙2 + q˙3 + q˙4 ) (q˙3 + q˙4 ) sin (q3 + q4 ) , H3 = −a3 q˙12 sin (q2 + q3 ) + a6 (q˙1 +q˙2 )2 sin q3 + a7 q12 sin (q2 + q3 + q4 ) + a8 (q˙1 +q˙2 )2 sin (q˙3 + q˙4 ) − a9 (2q˙1 +q˙2 + 2q˙3 + q˙4 ) q˙4 sin q4 , H4 = a7 q12 sin (q2 + q3 + q4 ) + a8 (q˙1 +q˙2 )2 sin (q˙3 + q˙4 ) + a9 (q˙1 +q˙2 + 2q˙3 )2 sin q4 , and G (q) = [g1 , g2 , g3 , g4 ]T is the matrix of gravity, where g1 = −β1 sin q1 − β2 sin (q1 + q2 ) − β3 sin (q1 + q2 + q3 ) − β4 sin (q1 + q2 + q3 + q4 ) , g2 = −β2 sin (q1 + q2 ) − β3 sin (q1 + q2 + q3 ) − β4 sin (q1 + q2 + q3 + q4 ) , g3 = −β3 sin (q1 + q2 + q3 ) − β4 sin (q1 + q2 + q3 + q4 ) , g4 = −β4 sin (q1 + q2 + q3 + q4 ) , The vector q = [q1 , q2 , q3 , q4 ]T in (2.39) is the generalized coordinates of this fourlink system, and the physical meanings of m i , L i , L ci , Ji , τi (i = 1, 2, 3, 4) are consistent with that of three-link manipulator, and a1 = J1 + m 1 L 2c1 + (m 2 + m 3 + m 4 ) L 21 , a2 = J2 + m 2 L 2c2 + (m 3 + m 4 ) L 22 , a3 = (m 2 L c2 + m 3 L 2 + m 4 L 2 ) L 1 , a4 = J3 L 2 L c3 + m 4 L 1 L 3 , a5 = m 4 L 1 L c4 , 2 a6 = m 4 L 2 L c4 , a7 = m 4 L 1 L c4 , a8 = m 3 L 2 L c4 , a9 = m 4 L 1 L 3 , a10 = J4 + m 4 lc4 ,

β1 = (m 1 L c1 + m 2 L 1 + m 3 L 1 + m 4 L 1 ) g, β2 = (m 2 L c2 + m 3 L 2 + m 4 L2) g, β3 = (m 3 L c3 + m 4 L 3 ) g, β4 = m 4 L c4 g, are the structural parameters. The dynamic equation of the four-link underactuated manipulator can be obtained based on (2.39). For example, the dynamic equation of the Passive-Active-ActiveActive (PAAA) underactuated manipulator as shown in Fig. 2.13 can be obtained by substituting τ1 = 0 into (2.39), and similarly, the dynamic equation of the planar PAAA underactuated manipulator can be obtained by substituting τ1 = 0 and g = 0 into (2.39).

2.1 Dynamic Modeling of Typical Underactuated Manipulators

39

Fig. 2.13 Planar PAAA Underactuated manipulator

2.1.4 Flexible Manipulator The modeling of rigid-joint rigid-link manipulators is introduced in previous subsections. Additionally, in some practical situations, in order to make the whole operation safer and have lower energy consumption, people may choose lightweight materials or slender links for the joints or links in manipulators. These kinds of manipulators are called flexible manipulators. Modeling for flexible manipulators is slightly different due to the flexible elements (flexible joints or flexible links). In this subsection, the modeling methods for the flexible joint manipulators and the flexible link manipulators are introduced, respectively. When an active joint of the manipulator is flexible, the active link corresponding to this joint is connected with the joint motor through a spring [19–21]. Define q f as the link angle and θ f as the motor angle, where f is the position of the flexible joint. Thus, the spring is relaxed as q f = θ f , and the elastic potential energy of this flexible joint can be expressed as P f (q) =

1 k f (q f − θ f )2 , 2

(2.40)

where k f is the stiffness coefficient of the flexible joint. As an example, the structure of a vertical single-link flexible-joint manipulator is shown in Fig. 2.14, which contains a flexible active joint and a rigid link [22– 25]. As depicted in Fig. 2.15, the flexible joint manipulator with a single link has two DOFs but only one control input from the motor, which means this system has underactuated characteristics. In Figs. 2.14 and 2.15, q1 , θ1 , m 1 , k1 , L c1 , τ1 , and g are the link angle, motor angle, link mass, flexible joint stiffness, distance from flexible joint to COM of the link, motor torque and gravity acceleration. I and J are moments of inertia relative to the link shaft and motor shaft, respectively. The flexible joint is limited to a standard linear elastic spring.

40

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Fig. 2.14 Single-link manipulator with flexible joint

Fig. 2.15 Flexible joint space model

Let q = [q1 , θ1 ]T be the generalized coordinates of this system. Thus, the kinetic energy, K (q), ˙ of the flexible joint manipulator is K (q) ˙ =

1 2 1 2 I q˙ + J θ˙ , 2 1 2 1

(2.41)

and the potential energy, P(q), of the system is P(q) =

1 k1 (q1 − θ1 )2 + m 1 gL c1 (1 − cos q1 ). 2

(2.42)

The Lagrangian function of this manipulator is K (q) ˙ − L(q, q) ˙ = P(q). By using the Euler-Lagrange modeling equations, the dynamic model of the flexible joint manipulator is obtained as M q¨ + G = τ, (2.43) where τ = [0, τ1 ]T ,    I 0 k1 (q1 − θ1 ) + m 1 gL c1 sin q1 , M= , G= −k1 (q1 − θ1 ) 0 J 

(2.44)

are inertial matrix and the matrix of gravity and elastic force, respectively. When a passive joint of the manipulator is flexible, the passive link corresponding to this joint is connected with the previous link through a spring. Generally speaking,

2.1 Dynamic Modeling of Typical Underactuated Manipulators

41

Fig. 2.16 Spring-couping Pendubot system

when the spring is relaxed, q f = 0, where f denotes the position of the flexible joint. In this way, the elastic potential energy of the system at this flexible joint can be expressed as 1 (2.45) P f (q) = k f (q f )2 . 2 As an example, the structure of a spring-coupled planar Pendubot is shown in Fig. 2.16. This system is equivalent to the manipulator in Fig. 2.7 whose second joint has an external elastic force. Combining (2.1), (2.16) with (2.45), it is not difficult to obtain the dynamic equation of the spring-coupled planar Pendubot as ⎡ ⎣

m 11 m 12 m 12 m 22

⎤⎡ ⎦⎣

q¨1 q¨2

⎤

⎡

⎦+⎣

c11 c12

⎤⎡ ⎦⎣

c21 0

q˙1 q˙2

⎤

⎡

⎦+⎣

⎤

0

⎡

⎦=⎣

k 2 q2

τ1

⎤ ⎦.

(2.46)

0

where k2 > 0 is the elastic coefficient of spring and the elements in (2.46) are given in (2.17) and (2.23). Clearly, if k2 = 0, the system (2.46) is equivalent to the underactuated Pendubot system given in Fig. 2.7. Similarly, the dynamic equation of a spring-coupled planar PAA manipulator can be obtained by referring to (2.38), whose structure is shown in Fig. 2.17. For this system, its first joint is flexible and passive, and the dynamic equation of this system is expressed as ⎡

m 11 m 12 m 13

⎤⎡

q¨1

⎤

⎡

c11 c12 c13

⎤⎡

q˙1

⎤

⎡

k 1 q1

⎤

⎡

0

⎤

⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ m 21 m 22 m 23 ⎥ ⎢ q¨2 ⎥ + ⎢ c21 c22 c23 ⎥ ⎢ q˙2 ⎥ + ⎢ 0 ⎥ = ⎢ τ2 ⎥ . (2.47) ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎣ ⎦ ⎣ ⎦ m 31 m 32 m 33 q¨3 c31 c32 c33 q˙3 τ3 0

42

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Fig. 2.17 Planar Spring-PAA underactuated manipulator

where the elements in (2.47) are given in (2.31) and (2.37). When one link of the manipulator is a flexible link, the flexible link will produce elastic deformation in the process of movement [26–28]. Define ω(x, t) as the elastic deflection of one point on the flexible link at position x and at time t. Obviously, the motion state of the flexible link cannot be expressed only by the lumped state at its center of mass position like the rigid link, which makes the modeling process of the flexible link manipulator different from that of the rigid link manipulator. In fact, the flexible link manipulator is a distributed parameter system whose dynamics should be expressed by a partial differential equation (PDE) dynamic model [29– 31]. However, the PDE model brings difficulties to the characteristic analysis and controller design of the flexible link manipulator [32]. Hence, in this subsection, the assumed mode method, which is the most widely used modeling method for the flexible link manipulator [33, 34], will be introduced to establish the approximate discrete ordinary differential equation (ODE) dynamic model of the flexible link manipulator. As the most simplified flexible link manipulator, in this subsection, a planar singlelink flexible-link manipulators (PSLFLM) is introduced as an example to express the modeling progress of the flexible link manipulators [35]. The structure model of this manipulator is shown in Fig. 2.18, and it contains one rigid joint and one flexible link. Since the link of the manipulator moves in the horizontal plane, the gravity influence on the manipulator is ignored. Meantime, the flexible link of the manipulator is assumed to have a slender structure, so it can be regarded as an Euler-Bernoulli beam. X OY and X  OY  are the inertia coordinate system and rotating coordinate system for the system, respectively. Mt is the payload mass of the manipulator. For the flexible link, q1 is its rotation angle, L 1 is its length, ρ A is its mass per unit length, and E I is its flexural rigidity. For the joint motor, Ih is its inertia and τ1 is its torque. ω(x, t) is the elastic deflection of one point on the flexible link relative to the rotating coordinate X  OY  at position x and time t, where x ∈ [0, L 1 ] and t ∈ [0, +∞).

2.1 Dynamic Modeling of Typical Underactuated Manipulators

43

Fig. 2.18 Physical structure of the planar single-link flexible-link manipulators

Based on the theory of the Euler-Bernoulli beam, the following vibration equation [36] and the boundary conditions [37] can be obtained ⎧ E I ∂ 4 ω(x, t) ∂ 2 ω(x, t) ⎪ ⎪ ⎪ + = −x q, ¨ ⎪ 4 ⎪ ∂t 2 ⎪ ⎨ ρ A ∂x ω(0, t) = ω (0, t) = ω (L 1 , t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Mt ω (L 1 , t) + ρ Aω (L 1 , t) = 0.

(2.48)

According to the assumed mode method [38–41], the elastic deflection ω(x, t) can be approximated as ω(x, t) =

∞ 

φi (x) pi (t),

(2.49)

i=1

where φi (x) and pi (t) are the ith vibration modal function and ith vibration modal coordinates, respectively [42]. According to [43], the vibration modal function φi (x) can be expressed as ⎧ φ (x) = ⎪ ⎪ ⎨ i

1 Hi

[cosh(λi x) − cos(λi x) − ai (sinh(λi x) − sin(λi x))], (2.50)

cosh(λi L 1 ) + cos(λi L 1 ) ⎪ ⎪ ⎩ ai = . sinh(λi L 1 ) + sin(λi l L 1 ) If Mt is equal to zero, Hi will be equal to

√

L 1 , otherwise, it will be equal to



   21 ρ A 1 + cosh(λi L 1 ) cos(λi L 1 ) 2 , Hi = L 1 + sinh(λi L 1 ) sin(λi L 1 ) Mt λi2

(2.51)

44

2 Modeling and Characteristics Analysis of Underactuated Manipulators

where λi can be obtained by solving the following characteristic equation Mt λi [sinh(λi L 1 ) cos(λi L 1 ) − cosh(λi L 1 ) sin(λi L 1 )] ρA

(2.52)

+cosh(λi L 1 ) cos(λi L 1 ) = −1. and it is the ith positive solution. There are orthogonalities among the vibration modal functions, which can be expressed as ⎧  L1 ⎪ ⎨ρ A 0 φi (x)φ j (x)dx+Mt φi (L 1 )φ j (L 1 )=0, ⎪ ⎩

ρA

 L1 0

i = j, (2.53)

φi (x)φ j (x)dx+Mt φi (L 1 )φ j (L 1 )=ρ A, i = j,

and ⎧ L 1   ⎪ ⎨ E I 0 φi (x)φ j (x)dx = 0, ⎪ ⎩

EI

 L1 0

i = j, (2.54)

φi (x)φ j (x)dx = ρ Awi2 , i = j,

where wi is the ith natural frequency of vibration for the PSLFLM. The system with infinite assumed modes is difficult to carry out system characteristic analysis and controller design [44, 45]. Because the low-order modes play a leading role in the vibration of the system, while the influence of the high-order modes on the vibration of the system is so small that it can be ignored, only the first n modes are considered. Then, (2.49) is rewritten as ω(x, t) =

n 

φi (x) pi (t),

(2.55)

i=1

where n is the number of the assumed modes, and the model accuracy of the system will increase with the increase of the number n. Unlike the rigid link manipulator, the flexible link manipulator has m × n + j

generalized coordinates, where m is the number of flexible links of the manipulator and j is the is the number of joints of the manipulator. Thus, for the PSLFLM shown in Fig. 2.18, it has n + 1 generalized coordinates. Define q = [q1 , p1 , p2 , . . . , pn ]T as the generalized coordinates vector for the PSLFLM, where p = [ p1 , p2 , . . . , pn ] is the first n assumed modes coordinates of the system. Let R to be the position vector of an arbitrary point on the flexible link, which is shown in Fig. 2.18. Then, the position vector R can be written as

2.1 Dynamic Modeling of Typical Underactuated Manipulators

 R=

   X (x, t) x cos q1 − ω(x, t) sin q1 . = x sin q1 + ω(x, t) cos q1 Y (x, t)

45

(2.56)

Based on (2.56), the kinetic energy of the system can be expressed as T =

1 2



L1

  ρ A X˙ (x, t)2 + Y˙ (x, t)2 dx

0

  1 1 + Mt X˙ (L 1 , t)2 + Y˙ (L 1 , t)2 + Ih q˙12 . 2 2 The elastic potential energy of the system can be written as  L1 1 2 [ω(x, t) ] dx. D = EI 2 0

(2.57)

(2.58)

Set the Euler-Lagrange function of the system as L = T − D.

(2.59)

Based on (2.59), the dynamic model of the system is obtained as M (q) q¨ + C(q, q) ˙ q˙ + K q = τ,

(2.60)

where matrix M is a positive-definite symmetric matrix, matrix C is expressed as C(q, q) ˙ q˙ =

1 ∂[q˙ T ∂ M(q)q] ˙ dM(q) q˙ − , dt 2 ∂q

(2.61)

which is the combination of the Coriolis and centrifugal forces, and matrix K is the elasticity matrix of the system. τ = [τ1 , 0, 0, . . . , 0]T is the input torques matrix for the system. Matrix M can be expressed as ⎡ M(q) = ⎣

Mqq Mq p MqTp

⎤ ⎦ ∈ R(n+1)×(n+1) ,

(2.62)

M pp

where Mqq = m 11

 n   1 3 2 2 = Ih + ρ AL 1 + Mt L 1 + ρ A pi , 3 i=1

(2.63a)

46

2 Modeling and Characteristics Analysis of Underactuated Manipulators

⎡ ⎢ ⎢ MqTp = ⎢ ⎣ ⎡

⎤

⎡

⎤ σ1 ⎥ ⎢ σ2 ⎥ ⎥ ⎢ ⎥ ⎥ = ⎢ .. ⎥ ∈ Rn×1 , ⎦ ⎣ . ⎦

m 21 m 31 .. .

σn

m (n+1)1 m 22 m 32 .. .

⎢ ⎢ M pp = ⎢ ⎣

m 23 m 33 .. .

··· ···

m 2(n+1) m 3(n+1) .. .

(2.63b) ⎤ ⎥ ⎥ ⎥ ⎦

(2.63c)

m (n+1)2 m (n+1)3 · · · m (n+1)(n+1) ⎤ η11 η12 · · · η1n ⎢ η21 η22 · · · η2n ⎥ ⎢ ⎥ n×n =⎢ . . .. ⎥ ∈ R , ⎣ .. .. ⎦ . ⎡

ηn1 ηn2 · · · ηnn

in which 

L1

σi = ρ A

 x · φi (x)dx + Mt L 1 φi (L 1 ), ηi j =

0

0, i = j ρ A, i = j

(2.64)

where i, j = 1, 2, . . . , n. The form of matrix K can be specifically expressed as K = diag [0, k1 , k2 , . . . , kn ] ,

(2.65)

where ki = ρ Awi2 (i = 1, 2, . . . , n). For the flexible manipulator containing two or more links, its dynamic model can be derived by combining the modeling processes of the multi-link rigid manipulator and the single-link flexible manipulator. Here, the modeling process of a planar two-link rigid-flexible (TLRF) manipulator is given. The structure diagram of the planar TLRF manipulator is shown in Fig. 2.19. The first link of this manipulator is rigid while the second one is flexible. Because this system moves in the horizontal plane, the gravity of the system is neglected. The flexible link of the system is assumed to be a clamped-free cantilever EulerBernoulli beam with a tip payload. Because the longitudinal stiffness of the flexible link is much greater than that of bending, the longitudinal deformation of the flexible link can be neglected for this system. As shown in Fig. 2.19, (xt , yt ) is the coordinate of the tip position. X OY is the base coordinate system of the manipulator. X 1 OY1 is the rotating coordinate system of the rigid link while X 2 O2 Y2 is the rotating coordinate system of the flexible link. qi and L i are the ith link angle and ith link length of the manipulator, respectively (i = 1, 2). For the rigid link of this manipulator, m 1 is its mass, L c1 is the distance from its centroid to the first joint, and J1 is its moment of inertia. For the flexible link of the manipulator, ρ, A are its material density and section area, E I is its flexural rigidity and Ih is its moment of inertia. ω(x, t) is used to represent the

2.1 Dynamic Modeling of Typical Underactuated Manipulators

47

Fig. 2.19 The planar two-link rigid-flexible manipulator

elastic deflection of the flexible link at position x and at time t. Moreover, Mh is the mass of the second joint, Mt is the mass of the payload, and  input   τi is the = ∂∗ ∂ x and torque applied to the ith joint. For the convenience of description, ∗  (˙∗) = ∂∗ ∂t are used. According to the kinematics analysis, for one point on the flexible link, its position vector can be expressed as follows ⎡

⎤ L 1 cos q1 +x cos(q1 +q2 )−ω(x, t) sin(q1 +q2 ) ⎦, R =⎣ L 1 sin q1 +x sin(q1 +q2 )+ω(x, t) cos(q1 +q2 )

(2.66)

where x is x-coordinate of this point relative to X 2 O2 Y2 . Clearly, for the tip payload of this manipulator, its position vector is Rt = R|x=L 2 . By using the assumed mode method, the elastic deflection ω(x, t) can be approximated as n  φi (x) pi (t), (2.67) ω(x, t) = i=1

where n denotes the number of the selected flexible modes, φi (x) denotes the ith mode shape function, and pi (t) denotes the ith generalized coordinates. Since the flexible link of the planar TLRF manipulator is also a clamped-free cantilever EulerBernoulli beam, the form of its ith mode shape function is the same as that of the PSLFLM, which is given in (2.50)–(2.54). For the planar TLRF manipulator, its total kinetic energy can be expressed as T =

1 1 (m 1 L 2c1 + J1 )q˙12 + [Mh L 21 q˙12 + Ih (q˙1 + q˙2 )2 ] 2 2  L2 1 ˙ + 1 Mt R˙ tT R˙ t . R˙ T Rdx + ρA 2 2 0

(2.68)

48

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Furthermore, its total potential energy can be express as 1 D = EI 2



L2

[ω(x, t) ] dx. 2

(2.69)

0

The function L = T − D is chosen as the Lagrange function. By using the Lagrangian modeling method, the dynamic model of the system can be obtained as    ⎧  d ∂L ∂L ⎪ ⎪ − = τi , i = 1, 2, ⎪ ⎪ ∂qi ⎨ dt ∂ q˙i     ⎪ ⎪ d ∂L ∂L ⎪ ⎪ ⎩ − = 0, j = 1, 2, . . . , n. dt ∂ p˙ j ∂ pj

(2.70)

Define q = [q1 , q2 , p1 , p2 , . . . , pn ]T as the generalized coordinates vector for the TLRF manipulator. Substituting Eqs. (2.68) and (2.69) into (2.70), it has M(q)q¨ + C(q, q) ˙ q˙ + K q = τ,

(2.71)

where M(q) is the inertia matrix of the planar TLRF manipulator, and it is a positive definite symmetric matrix, which can be expressed as ⎡ M(q) = ⎣

Mqq Mq p MqTp

⎤ ⎦ ∈ R(n+2)×(n+2) ,

(2.72)

M pp

where Mqq ∈ R2×2 and can be expressed as  Mqq =

 m 11 m 12 . m 21 m 22

(2.73)

The elements in the matrix Mqq are (2.74a) m 11 = J1 + Ih + m 1 L 2c1 + L 21 Mh + Mt (L 21 + L 22 + 2L 1 L 2 cos q2 )    n  n   1 3 L 2 +L 21 L 2 +L 1 L 22 cos q2 + pi2 −2L 1 ψi pi sin q2 , +ρ A 3 i=1 i=1  n   ψi pi sin q2 (2.74b) m 12 = m 21 = Ih + Mt (L 22 + L 1 L 2 cos q2 ) − L 1  +ρ A

1 3 1 L + L 1 L 22 cos q2 + 3 2 2

n  i=1

 pi2 ,

i=1

2.1 Dynamic Modeling of Typical Underactuated Manipulators

49



m 22 = Ih +

Mt L 22

 n 1 3  2 L + + ρA p . 3 2 i=1 i

(2.74c)

where ψi (i = 1, 2, . . . , n) is the structural parameter and is expressed as 

L2

ψi = ρ A

φi (x)dx + Mt φi (L 2 ).

(2.75)

0

For the MqTp ∈ Rn×2 in (2.72), it has ⎡ ⎢ ⎢ MqTp = ⎢ ⎣

m 31 m 41 .. .

⎤

⎡

σ1 +ψ1 L 1 cos q2 ⎥ ⎢σ2 +ψ2 L 1 cos q2 ⎥ ⎢ ⎥=⎢ .. ⎦ ⎣ .

m 32 m 42 .. .

σn +ψn L 1 cos q2

m (n+2)1 m (n+2)2

⎤ σ1 σ2 ⎥ ⎥ .. ⎥ , .⎦

(2.76)

σn

where σi (i = 1, 2, . . . , n) is the structural parameter and is expressed as 

L2

σi = ρ A

xφi (x)dx + Mt L 2 φi (L 2 ).

(2.77)

0

Furthermore, M pp ∈ Rn×n in (2.72) is expressed as ⎡ ⎢ ⎢ M pp = ⎢ ⎣

m 33 m 43 .. .

m 34 m 44 .. .

··· ···

m 3(n+2) m 4(n+2) .. .

m (n+2)3 m (n+2)4 · · · m (n+2)(n+2)

⎤

⎡

η11 ⎥ ⎢ η21 ⎥ ⎢ ⎥ = ⎢ .. ⎦ ⎣ .

η12 · · · η22 · · · .. .

⎤ η1n η2n ⎥ ⎥ .. ⎥, . ⎦

(2.78)

ηn1 ηn2 · · · ηnn

where ηi j (i, j = 1, 2, . . . , n) is the structural parameter and is expressed as  ηi j =

0, i = j, ρ A, i = j.

(2.79)

Define matrix H (q, q) ˙ as the matrix caused by the Coriolis and centrifugal forces. Then, based on (2.71), H (q, q) ˙ can be expressed as H (q, q) ˙ = C(q, q) ˙ q˙ =

1 ∂[q˙ T M(q)q] ˙ dM(q) q˙ − . dt 2 ∂q

(2.80)

The matrix K in (2.71) is the matrix of system elasticity forces, which is a diagonal matrix of order n + 2 and can be written as K = diag [0, 0, k1 , k2 , . . . , kn ] ,

(2.81)

50

2 Modeling and Characteristics Analysis of Underactuated Manipulators

Fig. 2.20 The planar two-link rigid-flexible underactuated manipulator with a passive first joint

where ki = ρ Awi2 (i = 1, 2, . . . , n), and wi is the ith natural frequency of vibration for the TLRF manipulator. The vector τ in (2.71) is the control inputs vector, which is expressed as τ = [τ1 , τ2 , 0, 0, . . . , 0]T ∈ Rn+2 .

(2.82)

When a joint motor of the flexible link manipulator completely loses its efficiency, this joint will become a passive joint. Next, a planar TLRF underactuated manipulator with a passive first joint will be taken as an example to introduce how to obtain the dynamic model of the underactuated flexible link manipulator based on the modeling experience of the underactuated manipulator and the flexible link manipulator. The structure diagram of the planar TLRF underactuated manipulator with a passive first joint is shown in Fig. 2.20. Compared with the manipulator shown in Fig. 2.19, the manipulator shown in Fig. 2.20 does not have a motor in the first joint to provide control torque for the system, which means that τ1 ≡ 0 in the control process. Substituting τ1 ≡ 0 into (2.71)–(2.82), the dynamic model of this planar TLRF underactuated manipulator is obtained. The following is the dynamic model of the planar TLRF underactuated manipulator with a passive first joint considering the first two assumed modes. M(q)q¨ + C(q, q) ˙ q˙ + K q = τ,

(2.83)

where q = [q1 , q2 , p1 , p2 ]T is the generalized coordinates vector for the system. M(q) is the inertia matrix of the planar TLRF underactuated manipulator with a passive first joint, and it is a positive definite symmetric matrix, which can be expressed as

2.1 Dynamic Modeling of Typical Underactuated Manipulators

⎡

m 11 ⎢ m 21 ⎢ M(q) = ⎣ m 31 m 41

m 12 m 22 m 32 m 42

m 13 m 23 m 33 m 43

⎤ m 14 m 24 ⎥ ⎥, m 34 ⎦ m 44

51

(2.84)

where m 11 = J1 + m 1 L 2c1 + L 21 (Mh + Mt + ρ AL 2 ) + a1 + a2 + a3 cos q2 +

m 12 m 13

+ 2a6 p1 p2 − 2L 1 (a7 p1 + a8 p2 ) sin q2 , 1 = m 21 = a1 + a2 + a3 cos q2 + a4 p12 + a5 p22 2 + 2a6 p1 p2 − L 1 (a7 p1 + a8 p2 ) sin q2 , = m 31 = a9 + L 1 a7 cos q2 , a4 p12

+

(2.85a)

a5 p22

(2.85b)

(2.85c)

m 14 = m 41 = a10 + L 1 a8 cos q2 ,

(2.85d)

m 22 = a1 + a2 +

(2.85e)

a4 p12

+

a5 p22

+ 2a6 p1 p2 ,

m 23 = m 32 = a9 , m 24 = m 42 = a10 , m 33 = a4 , m 34 = m 43 = a6 , m 44 = a5 .

(2.85f) (2.85g)

The structural parameters are 1 ρ AL 32 , a3 = L 1 L 2 (2Mt + ρ AL 2 ), 3 a4 = η11 , a5 = η22 , a6 = η12 , a7 = ψ1 ,

(2.86b)

a8 = ψ2 , a9 = σ1 , a10 = σ2 ,

(2.86c)

a1 = Mt L 22 + Ih , a2 =

(2.86a)

where ψi , σi , and ηi j (i, j = 1, 2) can be obtained from (2.75), (2.77), and (2.79). Define matrix H (q, q) ˙ as the matrix caused by the Coriolis and centrifugal forces, which can be expressed as H (q, q) ˙ = C(q, q) ˙ q˙ =

1 ∂[q˙ T M(q)q] ˙ dM(q) q˙ − . dt 2 ∂q

(2.87)

Matrix K is the matrix of system elasticity forces, which is expressed as ⎡

0 ⎢0 K =⎢ ⎣0 0

0 0 0 0

0 0 k1 0

⎤ 0 0⎥ ⎥, 0⎦ k2

(2.88)

where k1 and k2 are the same as (2.81), and τ = [0, τ2 , 0, 0]T is the vector of the system control inputs.

52

2 Modeling and Characteristics Analysis of Underactuated Manipulators

2.2 Characteristics Analysis of Underactuated Mechanical System In this section, the dynamic model of a general underactuated mechanical system is built based on the Lagrange method. Characteristics of the system are discussed based on the proposed dynamic model. Suppose that the underactuated system discussed has n-DOF (n  2) including m (1  m < n) underactuated DOF. As mentioned in above section, if L (q, q) ˙ = K (q, q) ˙ − P (q) is taken as the Lagrangian function of the system, then, the dynamic equation of the system is given by ˙ ∂ L (q, q) ˙ d ∂ L (q, q) − = τ, dt ∂ q˙ ∂q

(2.89)

where q = [q1 , q2 , ..., qn ]T ∈ Rn , τ = [τ1 , τ2 , . . . , τn ]T ∈ Rn , and τr 1 = τr 2 = · · · τr m = 0, where r1 , r2 , . . . rm ∈ {1, 2, . . . , n} are the positions of the m underactuated DOF. From the modeling process in the previous section, it can be obtained that there ˙ exists a symmetric positive definite matrix M (q) so that K (q, q) ˙ = q˙ T M (q) q/2. Since ˙ d d ∂ L (q, q) dM (q) = q, ˙ ˙ = M (q) q¨ + [M (q) q] dt ∂ q˙ dt dt  ∂ P (q) ∂ L (q, q) ˙ 1 ∂ T ∂ P (q) = q˙ M (q) q˙ − = W (q, q) ˙ q˙ − , ∂q 2 ∂q ∂q ∂q where

⎡1 ⎢2 ⎢ ⎢ ⎢ W (q, q) ˙ =⎢ ⎢ ⎢ ⎢ ⎣1 2

q˙ T

q˙ T

∂ M (q) ⎤ ∂q1 ⎥ ⎥ ⎥ ⎥ .. ⎥. . ⎥ ⎥ ⎥ ∂ M (q) ⎦

(2.90a) (2.90b)

(2.91)

∂qn

It is not difficult to find that (2.89) is equivalent to  M (q) q¨ +

 dM (q) ∂ P (q) − W (q, q) ˙ q˙ + = τ. dt ∂q

(2.92)

2.2 Characteristics Analysis of Underactuated Mechanical System

53

2.2.1 Dynamics Analysis By carrying out the dynamics analysis of the underactuated manipulators, the rela tionship among the variables in (2.92) is analysed. Let M (q) = Mi j (q) n×n , and then ⎡

⎡

∂ M11 ⎢ ⎢ ∂q1 ⎢ ⎢ ⎢ ⎢ ∂ M11 ⎢ ⎢ ⎢ ⎢ M˙ (q) =⎢q˙ T ⎢ ∂q2 ⎢ ⎢ . ⎢ ⎢ .. ⎢ ⎢ ⎣ ⎣ ∂ M11 ∂qn ⎡

⎡

⎤ ⎡ ∂ Mn1 ∂ M12 ∂ M1n ··· ⎢ ∂q1 ∂q1 ∂q1 ⎥ ⎥ ⎢ ⎢ ∂ Mn1 ∂ M12 ∂ M1n ⎥ ⎥ ⎢ ··· ⎥ ∂q2 ∂q2 ⎥ · · · q˙ T ⎢ ⎢ ∂q2 ⎢ . .. .. ⎥ ⎢ .. . . ⎥ ⎥ ⎢ ⎣ ∂ Mn1 ∂ M12 ∂ M1n ⎦ ··· ∂qn ∂qn ∂qn

∂ M11 ⎢ ⎢ ∂q1 ⎢ ⎢ ⎢ ⎢ ∂ M11 ⎢ q˙ T⎢ ⎢ ⎢ W˙ (q, q) ˙ =⎢ ⎢ ∂q2 ⎢ 2⎢ . ⎢ ⎢ .. ⎢ ⎢ ⎣ ⎣ ∂ M11 ∂qn

∂ M12 ∂ M1n ··· ∂q1 ∂q1 ∂ M12 ∂ M1n ··· ∂q2 ∂q2 .. .. . . ∂ M12 ∂ M1n ··· ∂qn ∂qn

⎤

⎡

⎤⎤T ∂ Mn2 ∂ Mnn ··· ⎥ ∂q1 ∂q1 ⎥ ⎥⎥ ⎥ ∂ Mn2 ∂ Mnn ⎥⎥ ⎥ ··· ⎥ ∂q2 ∂q2 ⎥ ⎥⎥ , ⎥ .. .. ⎥⎥ . . ⎥⎥ ⎥ ∂ Mn2 ∂ Mnn ⎦⎦ ··· ∂qn ∂qn

∂ Mn1 ⎥ ⎢ ∂q1 ⎥ ⎢ ⎥ ⎢ ⎥ q˙ T ⎢ ∂ Mn1 ⎥ ⎢ ⎥· · · ⎢ ∂q2 ⎥ 2 ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎦ ⎣ ∂ Mn1 ∂qn

∂ Mn2 ∂ Mnn ··· ∂q1 ∂q1 ∂ Mn2 ∂ Mnn ··· ∂q2 ∂q2 .. .. . . ∂ Mn2 ∂ Mnn ··· ∂qn ∂qn

⎤⎤T ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ . ⎥⎥ ⎥⎥ ⎥⎥ ⎦⎦

If introducing a matrix as follows n 

 C (q, q) ˙ = Ci j (q, q) λi jk q˙k , i, j = 1, 2, . . . , n, (2.93a) ˙ n×n , Ci j (q, q) ˙ =

λi jk = λik j

1 = 2



k=1

 ∂ Mi j (q) ∂ Mik (q) ∂ M jk (q) , k = 1, 2, . . . , n, + − ∂qk ∂q j ∂qi (2.93b)

then, it is easy to verify 

 dM (q) ˙ q. ˙ − W (q, q) ˙ q˙ = C (q, q) dt

In general, λi jk in (2.77) is Christoffel symbol. In addition, since

(2.94)

54

2 Modeling and Characteristics Analysis of Underactuated Manipulators

dMi j (q)  ∂ Mi j (q) = q˙k , dt ∂qk k=1  n  1  ∂ Mi j (q) ∂ Mik (q) ∂ M jk (q) q˙k . Ci j (q, q) + − ˙ = 2 k=1 ∂qk ∂q j ∂qi n

then

 n   ∂ M jk (q) ∂ Mik (q) dMi j (q) q˙k . − 2Ci j (q, q) − ˙ = dt ∂qi ∂q j k=1

(2.95a) (2.95b)

(2.96)

Further, it can be obtained that ⎧ 0, i = j, ⎪ ⎪ ⎨ dMi j (q)   − 2Ci j (q, q) ˙ = dM ji (q) ⎪ dt ⎪ − , i = j, − 2C q) ˙ (q, ⎩ ji dt

(2.97)

which means that M˙ (q) − 2C (q, q) ˙ is an antisymmetric matrix, that is  ξ

T

 dM (q) − 2C (q, q) ˙ ξ = 0, ∀ξ ∈ Rn×1 . dt

(2.98)

Based on the relationship among variables in (2.92), and combining (2.94) with ˙ + P (q) is (2.98), the derivation of the system total energy E (q, q) ˙ = q˙ T M (q) q/2 obtained as 1 1 ∂ P (q) dE (q, q) ˙ 1 dM (q) = q¨ T M (q) q˙ + q˙ T q˙ + q˙ T M (q) q¨ + q˙ T dt 2 2 dt 2 ∂q   ∂ P (q) 1 dM (q) + q˙ T q˙ = q˙ T M (q) q¨ + ∂q 2 dt     1 dM (q) dM (q) − W (q, q) ˙ q˙ + q˙ T q˙ = q˙ T τ + dt 2 dt 1 dM (q) = q˙ T [τ − C (q, q) q˙ ˙ q] ˙ + q˙ T 2 dt   dM (q) 1 = q˙ T τ + q˙ T − 2C (q, q) ˙ q˙ 2 dt n−m  = q˙ T τ = q˙ηk τηk ,

(2.99)

k=1

where ηk (k = 1, 2, . . . , n − m) is the position of n − m DOF in sequence, that is, ηk ∈ {1, 2, . . . , n} \ {r1 , r2 , . . . , rm } and η1 < η2 < · · · < ηn−m .

2.2 Characteristics Analysis of Underactuated Mechanical System

55

2.2.2 Controllability Analysis In this subsection, the controllability of the system at its open-loop equilibrium point is analysed. From (2.92) and (2.94), it has ˙ q˙ + G (q)] + M −1 (q) τ, q¨ = −M −1 (q) [C (q, q)

(2.100)

T T

where G (q) = ∂ P (q) /∂q. If let x = x1T , x2T = q T , q˙ T , then, the state equation of the system is (2.101) x˙ = f (x) + g (x) τα , T

where τα = τη1 , τη2 , . . . , τηn−m , and ⎡ f (x) = ⎣

−M

−1

g (x) = ⎣

0n×(n−m) M

n×(n−m)

, B˜ i j =

⎦

˙ q˙ + G (q)] (q) [C (q, q) ⎡

  B˜ = B˜ i j

⎤

q˙

−1

(q) B˜

,

q=x1 ,q=x ˙ 2

⎤ ⎦

,

q=x1 ,q=x ˙ 2

⎧ ⎨ 1, i = ηk , j = k, k = 1, 2, . . . , n − m ⎩

.

0, otherwise

Let τα = 0 in (2.101), then, the open-loop equilibrium point set of the system as ⎧ ⎨

⎫ ⎬  ε= x ∗ = ⎣ ⎦ x1∗ = q ∗ ∈ Rn , x2∗ = 0n×1 , G x1∗ = 0 . ⎩ ⎭ x2∗ ⎡

x1∗

⎤





(2.102)

    because C x1∗ , 0 = 0, and G x1∗ = 0, for x = x ∗ , it can be obtained that ∂ M −1 (x1 ) (C (x1 , x2 ) x2 + G (x)) ∂G (x1 ) = M −1 (x1 ) , ∂ x1 ∂ x1 ∂ M −1 (x1 ) (C (x1 , x2 ) x2 + G (x1 )) = 0. ∂ x2 It is easy to obtain the linear approximation model of the system (2.101) at x = x ∗ , given by (2.103) x˙ = Ax + Bτα , where

56

2 Modeling and Characteristics Analysis of Underactuated Manipulators

⎡ A=⎣

0n×n In

⎤

⎡

⎦, B = ⎣

0n×(n−m)

⎤ ⎦,

(2.104a)

Aˆ 0n×n Bˆ   ∂G (x1 ) ˜ |x=x ∗ , Bˆ = M −1 x1∗ B. Aˆ = −M −1 (x1 ) ∂ x1

(2.104b)

From (2.101), it has ⎡ AB = ⎣

⎡

⎤

Bˆ

⎦ , A2 B = ⎣

0n×(n−m)

⎤

⎡

⎦ , . . . , A2n−1 B = − ⎣

⎤

Aˆ n−1

⎦. 0n×(n−m) (2.105)

 When rank B, AB, . . . , A2n−1 B = 2n, the linear approximation model of the system (2.101) around the equilibrium point is completely controllable. In particular, when the system (2.101) is not affected by potential energy, that is, G (q) = 0 , from (2.102), it can be obtained that the open-loop equilibrium point of the system T

(2.101) is x ∗ = q ∗T , 01×n , where q ∗ ∈ Rn×1 is any column vector. Moreover, from (2.103) and (2.105) , it has 0n×(n−m)

Aˆ Bˆ

⎡



rank B, AB, . . . , A2n−1 B = rank ⎣

0n×(n−m)

bˆ

Bˆ

0n×(n−m)

⎤ ⎦ = 2 (n − m) , (2.106)

which means, the system (2.103) has 2m uncontrollable modes, that is, (A, B) is not completely controllable. In summary, the dynamic equation of an underactuated mechanical system with (n − m) control inputs can be expressed as

or

˜ α, M (q) q¨ + C (q, q) ˙ q˙ + G (q) = Bτ

(2.107)

˜ α, M (q) q¨ + H (q, q) ˙ + G (q) = Bτ

(2.108)

˙ ˙ ∈ where M (q) ∈ Rn×n is the symmetric matrix, H (q, . . . q) = C (q, q)(C (q, q) of the Coriolis and centrifugal forces, G ∈ Rn×n , q˙ ∈ Rn ) is the combination (q) 

Rn×1 is potential energy, τα = τη1 , τη2 , . . . , τηn−m is the vector of applied torques, η1 < η2 < · · · < ηn−m ∈ {1, 2, . . . , n} and 

B˜ = B˜ i j

 n×(n−m)

, B˜ i j =

⎧ ⎨ 1, i = ηk , j = k, k = 1, 2, . . . , n − m ⎩

0, otherwise

For the system (2.107), the following properties are true.

.

References

57

Property 2.1 M (q) is a positive definite symmetric matrix, and G (q) = where P (q) is potential energy of the system.

 Property 2.2 If let C (q, q) ˙ = Ci j (q, q) ˙ n×n , M˙ (q) − 2C (q, q) ˙ is an antisymmetric matrix.

Ci j (q, q) ˙ =

n #

∂ P (q) , ∂q

λi jk q˙k , then,

k=1

Property 2.3 If let E (q, q) ˙ be the total energy of the system, then, E˙ (q, q) ˙ = n−m # ˜ α= q˙ηk τηk . q˙ T Bτ k=1 ∗ Property 2.4  T  The open-loop equilibrium point of the system (2.107) is x = ∗ q , 01×n , and the sufficient and necessary conditions for the complete control  lability of the linear approximate model at x ∗ is rank B, AB, . . . , A2n−1 B = 2n, where q ∗ satisfies G (q ∗ ) = 0, and the definition of A and B are shown in (2.104). Particularly, when G (q) = 0, q ∗ ∈ Rn is any column vector, and the linear approximate model of the system (2.107) is not completely controllable at x ∗ .

2.3 Conclusions This Chapter mainly introduces basic concepts in the control of underactuated manipulators. First of all, the dynamic model of different types of underactuated manipulators is established based on the Lagrangian modeling method, in which the dynamic equation corresponding to the passive joint can clearly reflect the underactuated characteristics of the system. Meanwhile, the strategy to establish the dynamic model of the flexible-joint manipulators and the flexible-link manipulators by referring to the modeling process of the rigid underactuated manipulators is introduced. Based on the established dynamic model of the underactuated manipulators, the characteristics of such systems are analyzed including the relationship among the system variables and the controllability of the systems. Finally, four properties of the underactuated manipulators are summarized in this chapter. The contents of this Chapter lays a theoretical foundation for the design of subsequent control strategies for underactuated manipulators.

References 1. A. Jain, G. Rodriguez, An analysis of the kinematics and dynamics of underactuated manipulators. IEEE Trans. Robot. Autom. 9(4), 411–422 (1993) 2. N. Ma, X. Dong, D. Axinte, Modeling and experimental validation of a compliant underactuated parallel kinematic manipulator. IEEE/ASME Trans. Mechatron. 25(3), 1409–1421 (2020) 3. Y. Liu, Y. Xu, M. Bergerman, Cooperation control of multiple manipulators with passive joints. IEEE Trans. Robot. Autom. 15(2), 258–267 (1999)

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2 Modeling and Characteristics Analysis of Underactuated Manipulators

4. Z. Li, J. Zhang, Y. Yang, Motion control of mobile under-actuated manipulators by implicit function using support vector machines. IET Control. Theory & Appl. 4(11), 2356–2368 (2010) 5. X. Lai, Y. Wang, M. Wu et al., Stable control strategy for planar three-link underactuated mechanical system. IEEE/ASME Trans. Mechatron. 21(3), 1345–1356 (2016) 6. J. Wu, W. Ye, Y. Wang et al., A general position control method for planar underactuated manipulators with second-order nonholonomic constraints. IEEE Trans. Cybern. 51(9), 4733– 4742 (2021) 7. X. Lai, J. She, S.X. Yang et al., Comprehensive unified control strategy for underactuated twolink manipulators. IEEE Trans. Syst. Man Cybern. Part B (Cybern). 39(2), 389–398 (2009) 8. L. Wang, X. Lai, P. Zhang, et al., A unified and simple control strategy for a class of n-link vertical underactuated manipulator. ISA Trans. 128(A), 198–207 (2022) 9. X. Xin, M. Kaneda, Swing-up control for a 3-dof gymnastic robot with passive first joint: design and analysis. IEEE Trans. Robot. 23(6), 1277–1285 (2007) 10. X. Xin, Y. Liu, Reduced-order stable controllers for two-link underactuated planar robots. Autom. 49(7), 2176–2183 (2013) 11. J. Wu, Y. Wang, W. Ye et al., Control strategy based on fourier transformation and intelligent optimization for planar pendubot. Inf. Sci. 491, 279–288 (2019) 12. H. Arai, K. Tanie, N. Shiroma, Nonholonomic control of a three-dof planar underactuated manipulator. IEEE Trans. Robot. Autom. 14(5), 681–695 (1998) 13. Y. Wang, H. Yang, P. Zhang, Iterative convergence control method for planar underactuated manipulator based on support vector regression model. Nonlinear Dyn. 102(4), 2711–2724 (2020) 14. A.D. Luca, R. Mattone, G. Oriolo, Stabilization of an underactuated planar 2r manipulator. Int. J. Robust Nonlinear Control. 10(4), 181–198 (2000) 15. P. Zhang, X. Lai, Y. Wang et al., Effective position-posture control strategy based on switching control for planar three-link underactuated mechanical system. Int. J. Syst. Sci. 48(10), 2202– 2211 (2017) 16. X. Lai, C. Pan, M. Wu et al., Control of an underactuated three-link passive–active–active manipulator based on three stages and stability analysis. J. Dyn. Syst. Meas. Control. 137(2), 021007 (2015) 17. P. Zhang, X. Lai, Y. Wang et al., Effective position-posture control strategy based on switching control for planar three-link underactuated mechanical system. Int. J. Syst. Sci. 48(10), 2202– 2211 (2017) 18. P. Zhang, X. Lai, Y. Wang et al., PSO-based nonlinear model predictive planning and discretetime sliding tracking control for uncertain planar underactuated manipulators. Int. J. Syst. Sci. 53(10), 2075–2089 (2022) 19. W. He, Z. Yan, Y. Sun et al., Neural-learning-based control for a constrained robotic manipulator with flexible joints. IEEE Trans. Neural Netw. Learn. Syst. 29(12), 5993–6003 (2018) 20. S. Dian, Y. Hu, T. Zhao et al., Adaptive backstepping control for flexible-joint manipulator using interval type-2 fuzzy neural network approximator. Nonlinear Dyn. 97(2), 1567–1580 (2019) 21. Z. Yan, X. Lai, Q. Meng et al., Tracking control of single-link flexible-joint manipulator with unmodeled dynamics and dead zone. Int. J. Robust Nonlinear Control. 31(4), 1270–1287 (2021) 22. Z. Yan, X. Lai, Q. Meng et al., A novel robust control method for motion control of uncertain single-link flexible-joint manipulator. IEEE Trans. Syst. Man Cybern. Syst. 51(3), 1671–1678 (2021) 23. J.T. Agee, S. Kizir, Z. Bingul, Intelligent proportional-integral (ipi) control of a single link flexible joint manipulator. J. Vib. Control. 21(11), 2273–2288 (2015) 24. J.P. Singh, K. Lochan, N.V. Kuznetsov et al., Coexistence of single-and multi-scroll chaotic orbits in a single-link flexible joint robot manipulator with stable spiral and index-4 spiral repellor types of equilibria. Nonlinear Dyn. 90(2), 1277–1299 (2017) 25. A.-C. Huang, Y.-C. Chen, Adaptive sliding control for single-link flexible-joint robot with mismatched uncertainties. IEEE Trans. Control. Syst. Technol. 12(5), 770–775 (2004)

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26. A.A. Shabana, Z. Bai, Actuation and motion control of flexible robots: small deformation problem. J. Mech. Robot. 14(1) 27. P. Sarkhel, N.B. Hui, N. Banerjee, Dynamic formulation of a two link flexible manipulator and its comparison analysis with a knuckle joint cantilever. Int. J. Veh. Inf. Commun. Syst. 7(1), 82–99 (2022) 28. Y. Yang, J. Shi, Z. Liu et al., Vibration and position tracking control for a flexible timoshenko robot arm with disturbance rejection mechanism. Assem. Autom. 42(2), 248–257 (2022) 29. W. He, X. He, M. Zou et al., PDE model-based boundary control design for a flexible robotic manipulator with input backlash. IEEE Trans. Control. Syst. Technol. 27(2), 790–797 (2019) 30. X. Zhang, W. Xu, S.S. Nair et al., PDE modeling and control of a flexible two-link manipulator. IEEE Trans. Control. Syst. Technol. 13(2), 301–312 (2005) 31. Z. Liu, J. Liu, W. He, Adaptive boundary control of a flexible manipulator with input saturation. Int. J. Control. 89(6), 1191–1202 (2016) 32. H. Gao, W. He, C. Zhou et al., Neural network control of a two-link flexible robotic manipulator using assumed mode method. IEEE Trans. Ind. Inform. 15(2), 755–765 (2019) 33. K. Lochan, B.K. Roy, B. Subudhi, A review on two-link flexible manipulators. Annu. Rev. Control. 42, 346–367 (2016) 34. A. Abe, Trajectory planning for residual vibration suppression of a two-link rigid-flexible manipulator considering large deformation. Mech. Mach. Theory. 44(9), 1627–1639 (2009) 35. Q. Meng, X. Lai, Y. Wang et al., A fast stable control strategy based on system energy for a planar single-link flexible manipulator. Nonlinear Dyn. 94(1), 615–626 (2018) 36. L. Tian, C. Collins, Adaptive neuro-fuzzy control of a flexible manipulator. Nonlinear Dyn. 15(10), 1305–1320 (2005) 37. Z. Luo, Direct strain feedback control of flexible robot arms: new theoretical and experimental results. IEEE Trans. Autom. Control. 38(11), 1610–1622 (1993) 38. Z. Zhao, S. Wen, F. Li et al., Free vibration analysis of multi-span timoshenko beams using the assumed mode method. Arch. Appl. Mech. 88(7), 1213–1228 (2018) 39. A. Ward, M. Towers, H. Barker, Dynamic analysis of helical springs by the assumed mode method. J. Sound Vib. 112(2), 305–320 (1987) 40. C. Guoping, H. Jiazhen, Assumed mode method of a rotating flexible beam. Acta Mech. Sin. 37(1), 48–56 (2005) 41. J. Xie, J. Liu, J. Chen et al., Blade damage monitoring method base on frequency domain statistical index of shaft’s random vibration. Mech. Syst. Signal Process. 165, 108351 (2022) 42. G. Cai, C.W. Lim, Active control of a flexible hub-beam system using optimal tracking control method. Int. J. Mech. Sci. 48(10), 1150–1162 (2006) 43. H. Gao, W. He, C. Zhou et al., Neural network control of a two-link flexible robotic manipulator using assumed mode method. IEEE Trans. Ind. Inform. 15(2), 755–765 (2019) 44. J. Martins, M.A. Botto, J.S.D. Costa, Modeling of flexible beams for robotic manipulators. Multibody Syst. Dyn. 7(1), 79–100 (2002) 45. J. Ju, W. Li, Y. Wang et al., Two-time scale virtual sensor design for vibration observation of a translational flexible-link manipulator based on singular perturbation and differential games. Sens. 16(11), 1804 (2016)

Chapter 3

Control of Vertical Underactuated Manipulator

Vertical underactuated manipulators (VUMs) [1–6] moving in a gravitational environment are a class of nonlinear systems, whose control inputs are less than their degrees of freedom [7, 8]. Such systems have the advantages of saving energy [9], reducing cost and weight [10], enhancing the flexibility of the system [11], etc. However, the absence of some actuators poses a great challenge to the control of VUMs [12–14]. For the VUM, its control objective is usually to control its end-point to swing from the straight-down equilibrium position to the straight-up equilibrium position, and stabilize it [15–17]. Due to the nonholonomic characteristics [18–20] of the VUM, it is hard to realize the control objective through a smooth controller [21–24]. However, the approximate linear model of the VUM near the straight-up equilibrium point is controllable [25–28], so the whole space of the system is generally partitioned into a swing-up area and an attraction area [29–31], and the control objective can be achieved by adopting different control strategies for the two areas [32–34]. Though the partition control method is effective, it also has shortcomings, such as torque mutation when switching controllers and singularity in the control process. Hence, some control methods without partitioning are also applied to the control of VUM. In this chapter, novel control strategies for the VUMs are presented. These control strategies are designed from different aspects to provide an overview of typical methods on this topic nowadays.

3.1 A Lyapunov-Based Unified Control Strategy In this section, a Lyapunov-based unified control strategy [35] is proposed for the twolink VUM to achieve its control objective. In this strategy, the motion area is divided into the swing-up area and attractive area. For the swing-up area, a weak-control Lyapunov-based swing-up controller is designed to increase the system energy to reach the potential energy at the straight-up equilibrium point, and also to make the © Science Press 2023 J. Wu et al., Control of Underactuated Manipulators, https://doi.org/10.1007/978-981-99-0890-5_3

61

62

3 Control of Vertical Underactuated Manipulator

state of the actuated link approach its target value. Here, a time-varying parameter is introduced into the weak-control Lyapunov function (WCLF) to avoid the singularity. For the attractive area, a non-smooth Lyapunov-based optimal controller is employed to guarantee the stability of the system. Combined with the WCLF and non-smooth Lyapunov function (NSLF) , the global stability of the system is ensured. Simulation results show the validity of the strategy on the two-link VUM.

3.1.1 Dynamic Model Models of the Acrobot and Pendubot have been illustrated in Figs. 2.4 and 2.5. By employing the Euler–Lagrange method, the dynamic equation of the two-link manipulator is M(q)q¨ + H (q, q) ˙ + G(q) = τ, (3.1) where M(q), H (q, q) ˙ and G(q) are the inertia matrix, combination of Coriolis force and centrifugal force and gravity matrix respectively. Their expressions are shown in (2.22). τ = [τ1 , τ2 ]T is the torque, where τ1 = 0 when the system is Acrobot, and τ2 = 0 when the system is Pendubot. Assuming that jth link is the active link ( j = 1 or 2), and x = [x1 , x2 , x3 , x4 ]T = [q1 , q2 , q˙1 , q˙2 ]T , the state space of the dynamic equation (3.1) of two-link VUM is

where



f 1 (x) f 2 (x)



x˙ = f a (x) + ba (x)τ j ,

(3.2)

f a (x) = [x3 , x4 , f 1 (x), f 2 (x)]T , ba (x) = [0, 0, b1 (x), b2 (x)]T

(3.3)

= M −1 (x) [−H (x) − G(x)] ,



b1 (x) b2 (x)



  1 −m 21 (x) , j = 1, det[M(x)]  m 11 (x) = 1 m (x) ⎪ 12 ⎪ , j = 2, ⎩ det[M(x)] m 11 (x) ⎧ ⎪ ⎪ ⎨

(3.4)

and det[M(x)] is the determinant of the square matrix M. The whole motion space is defined as Σ, and the attractive area Σ2 is defined as 

     x  

ε3 x3 x1 + x2  1  



  ε1 , mod mod   ε2 , ε4 x4  ε5 , 2π 2π

(3.5)

which satisfies |E(x) − E 0 |  ε E ,

(3.6)

where mod(x/y) is the remainder of x divided by y, E 0 is the system potential energy at the straight-up position, x is the Euclidean norm of the vector x, ε1 , ε2 > 0, ε3 , ε4 , ε5  0 and ε E > 0.

3.1 A Lyapunov-Based Unified Control Strategy

63

Then, the rest area is defined as the swing-up area Σ1 = Σ − Σ2 . Combined with the energy and posture, the proposed strategy can easily move the VUM from the swing-up area into the attractive area, and it can make the system energy approach E 0 during the swing-up process, while making the state of the active link approach its target state as well.

3.1.2 Control Strategy for the Swing-Up Area Taking the Acrobot as an example, the design process of the swing-up controller will be illustrated in this subsection. A Lyapunov-based swing-up controller will be designed to quickly control the system to go from the swing-up area into the attractive area, in which the Lyapunov function V1 (x) is constructed as: V1 (x) =

 1 α1 E x2 + α2 x22 + β(x)x42 + Δ, 2 E x = E(x) − E 0 ,

(3.7) (3.8)

where α1 , α2 , Δ > 0, β(x) is a time-varying design parameter which satisfies β(x) > 0, and E(x) is given by Sect. 2.2. The time derivative of E(x) is ˙ E(x) = x4 τ2 .

(3.9)

Combined with (3.9), the time derivative of V1 (x) is ˙ V˙1 (x) = [α2 x2 + β(x) f 2 (x) + 0.5β(x)x 4 ]x 4 + [α1 E x + β(x)b2 (x)] x 4 τ2 . (3.10) When α1 E x + β(x)b2 (x) = 0, the swing-up controller is designed to be τ2 =

˙ −α2 x2 − β(x) f 2 (x) − 0.5β(x)x 4 − γ x4 , α1 E x + β(x)b2 (x)

(3.11)

where γ > 0. Clearly, (3.11) ensures that V˙1 (x) = −γ x42 < 0, x4 ∈ R \ {0}.

(3.12)

However, when α1 E x + β(x)b2 (x) = 0, the denominator of τ2 becomes zero, resulting in singularity. In order to avoid the singularity, the denominator of τ2 needs to be further discussed. Since α1 > 0 and −2E 0  E x < 0 before the energy reaches E 0 , α1 E x < 0. In addition, b2 (x) is

64

3 Control of Vertical Underactuated Manipulator

b2 (x) =

m 11 (x) . m 11 (x)m 22 (x) − m 212 (x)

(3.13)

Substituting the parameters into (3.13) yields b2 (x) =

θ1 + θ2 + 2θ3 cos x2 . θ1 θ2 − θ32 cos2 x2

(3.14)

Simple calculations yield the minimum b2m =

θ1 + θ2 − 2θ3 , b2m > 0. θ1 θ2 − θ32

(3.15)

θ1 + θ2 + 2θ3 . θ1 θ2 − θ32

(3.16)

and the maximum b2M = So,

0 < b2m  b2 (x)  b2M .

(3.17)

In (3.11), since β(x)b2 (x) > 0, if the design parameters α1 and β(x) are not chosen properly, singularity may occur when α1 E x < 0. To avoid this, the parameter β(x) is chosen as α1 |E x | . (3.18) β(x) > b2m When the initial condition is the downward equilibrium position, b2 (x)|t=0 is b2M . So, a large constant β satisfying (3.18) will make the denominator of τ2 very large, and in turn, it would make τ2 very small. This would prevent a quick response. Thus, β(x) is chosen to be η , (3.19) β(x) = b2 (x) where η > 2α1 E 0 > 0. Consequently, α1 E x + β(x)b2 (x) > 0,

(3.20)

which ensures that no singularity occurs. β(x) is a function of b2 (x), which makes the denominator of (3.11) change in a suitable range during the swing-up process. In this way, the system has a quick response at the beginning of the swing-up motion. Although the above method avoids the singularity and produces a better transient response than previously reported ones, there is still room for improvement. Substituting (3.11) into (3.9) yields ˙ E(x) = ς1 (x) + γ ς2 (x),

(3.21)

3.1 A Lyapunov-Based Unified Control Strategy

65

where ˙ −α2 x2 − β(x) f 2 (x) − 0.5x4 β(x) x4 , α1 E x + β(x)b2 (x) −γ x42 . ς2 (x) = α1 E x + β(x)b2 (x) ς1 (x) =

(3.22) (3.23)

The value and sign of ς1 (x) change with the system state, the parameter β(x), and ˙ its derivative β(x). Likewise, the value of ς2 (x) changes with the system state, β(x) and γ . From (3.20), (3.23), and γ > 0, it is clear that ς2 (x) < 0. Hence, when ς1 (x) < 0, the swing-up controller (3.11) with (3.19) reduces the energy regardless of the value of γ . To slow down the rate of decrease in the energy, γ should be as small as possible. On the other hand, when ς1 (x) > 0, a small γ leads to a small ς2 (x). This rapidly increases the system energy, and may cause it to greatly exceed E 0 . So, an unsuitable choice of γ can cause the undesirable oscillations in the control torque. Consequently, it is difficult for the swing-up controller (3.11) to produce a satisfactory response when γ is constant. In order to improve the performance, γ is chosen as the following nonlinear function of the system state, rather than a constant: γ (x) = γ0 E ε (x),

(3.24)

where γ0 > 0, and E ε (x) is defined to be E ε (x) = E(x) + E 0 + ε,

(3.25)

where ε > 0. A small ε is taken to ensure that E ε (x) > 0, so making γ (x) increase monotonically with energy and preventing abrupt changes in the system energy. In addition, if the value of γ0 in (3.24) is smaller, it will lead to a larger energy overshoot, and if its value is larger, the time required for the energy to reach E 0 will be longer. The choice (3.19) avoids singularities in the swing-up controller (3.11) for the VUM and also ensures that the Lyapunov function V1 (x) converges to zero. Next, it is proved that the system energy converges to E 0 , and the angle and angular velocity of the actuated link both converge to zero. In this way, the manipulator can easily move into the attractive area.

3.1.3 Control Strategy for the Attractive Area The swing-up controller (3.11) guarantees that the manipulator moves into the attractive area Σ2 from the swing-up area. Then, the following linear approximate model is calculated at the straight-up position x˙ = Ax + Bτ2 .

(3.26)

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3 Control of Vertical Underactuated Manipulator

The balance controller is designed based on the optimal control method. The second Lyapunov function V2 (x) is constructed as V2 (x) = x T P x.

(3.27)

where P = P T > 0 is a solution of the Riccati equation AT P + P A − P B R −1 B T P + Q = 0,

(3.28)

in which Q  0, R > 0, and (Q 1/2 , A) is observable. Thus, the optimal controller is designed as C2 : τ2 = −F x, F = R −1 B T P.

(3.29)

which guarantees that V˙2 (x) < 0, ∀x ∈ R4 \ {0} in the attractive area, so ensuring the convergence of the motion of the VUM to the straight-up position.

3.1.4 Local and Global Stability Analysis For the two-link VUM, it is difficult to design a smooth controller to achieve its control objective, so a switching control strategy is used, and multiple Lyapunov functions are employed to guarantee the global stability of the switching control system. Note that β(x) in the Lyapunov function V1 (x) in (3.7) is time-varying. Although V1 (x) decreases monotonically, it cannot guarantee that x(t) converges to zero. So, in this subsection, the concept of WCLF [36, 37] is proposed to further analyze the stability of system in the swing-up area. Definition 3.1 (WCLF) Let the function V (x) : R4 → R be continuously differentiable, and positive definite for x = 0. It is said to be a WCLF for (3.2) if for each x ∈ R4 \ {0}, there exists τ2 such that L Fa (x,τ2 ) V (x) =

∂ V (x) Fa (x, τ2 )  0. ∂x

(3.30)

According to Definition 3.1, under the swing-up controller (3.11), V1 (x) in (3.7) is a WCLF. Substituting (3.11) into (3.2) yields x˙ = Fc (x).

(3.31)

Then, the concept of an invariant set will be introduced to analyze the stability of (3.31) in the swing-up area.

3.1 A Lyapunov-Based Unified Control Strategy

67

Definition 3.2 (Invariant set) A set Ω is said to be an invariant set with respect to (3.31) if x(0) ∈ Ω ⇒ x(t) ∈ Ω, ∀t  0. Next, LaSalle’s invariance principle is used to examine the stability. Lemma principle) Let Ω be an invariant set of (3.31),  3.1 (LaSalle’s invariance  Ψ = x(t) ∈ Ω|V˙1 (x) = 0 , and M be the largest invariant set (LIS) contained in Ψ . Then, every solution x(t) starting in Ψ converges to M as t → ∞. The set Ψ is the solution of (3.2) for V˙1 (x) = 0, or equivalently, x4 = 0, which means that both V1 (x) and x2 are constant. From (3.9), the energy E(x) is also constant. Therefore, the difference E x , defined in (3.8) must be constant. Two situations are considered: E x = 0 and E x = 0. If E x = 0 and x4 = 0, then (3.2) and (3.11) yield x2 = 0. Hence, (x2 , x4 ) = (0, 0) and E(x) = E 0 . In this case, the passive link travels in a periodic circle orbit, which can be derived from (3.8) to be Pendubot θ2 x32 = 2θ5 g(1 − cos x1 ),

(3.32)

Acrobot (θ1 +θ2 +2θ3 )x32 = 2(θ4 +θ5 )g(1−cos x1 ).

(3.33)

The situation E x = 0 cannot be the steady state, for which V˙1 (x) = 0 and x˙2 = 0. So, the following reductio ad absurdum are used to demonstrate it. First, x3 = 0 is shown. For an Acrobot, it is known from (3.9) that the energy E(x) is constant. Thus, the expression (3.8) can be simplified to S1 x32 + T1 sin x1 + U1 cos x1 = W1 ,

(3.34)

where S1 , T1 , U1 , and W1 are constants. Since x˙2 = 0, from (3.2) and (3.11), one can obtain α2 x2 + β(x) f 2 (x) f 2 (x) =− . τ2 = − (3.35) b2 (x) α1 E x + β(x)b2 (x) From (3.35), one can obtain τ2 = −

α2 x2 f 2 (x) =− = constant. b2 (x) α1 E x

(3.36)

Therefore, substituting the parameters into (3.36) yields S2 x32 + T2 sin x1 + U2 cos x1 = W2 ,

(3.37)

where S2 , T2 , U2 , and W2 are constants. Combining with (3.34) and (3.37) yields T3 sin x1 + U3 cos x1 = W3 ,

(3.38)

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3 Control of Vertical Underactuated Manipulator

where T3 , U3 , and W3 are constants. From (3.38), the variable x1 must be a constant. Thus, x3 = 0. Now, consider a Pendubot. The method that produces x3 = 0 from V˙1 (x) = 0 is quite different from that for an Acrobot. Since x1 is not a constant, the sine and cosine of x1 are not constant in m 22 (x), m 12 (x), m 21 (x), h 1 (x), or h 2 (x). Since x4 = 0, one can obtain from (3.2) that f 1 (x) x˙3 f 2 (x) = − . b2 (x) b1 (x) b1 (x)

(3.39)

And x4 = 0 gives h 1 (x) = 0. Equations (3.4) and (3.39) thus yield m 12 (x)g1 (x) − m 11 (x)h 2 (x) − m 11 (x)g2 (x) m 11 (x) x˙3 −m 22 (x)g1 (x) + m 21 (x)h 2 (x) + m 21 (x)g2 (x) − . = −m 21 (x) b1 (x)

(3.40)

The above expression is simplified to x˙3 = −

g1 (x) , m 11 (x)

(3.41)

that is, x˙3 = δ1 sin(x1 + x2 ),

(3.42)

where δ1 = θ5 g/θ2 . According to (3.42), x3 = δ2 cos(x1 + x2 ) + δ3 ,

(3.43)

where δ2 , δ3 and E x are constants. Thus, the expression (3.8) is simplified to S˜1 x32 + T˜1 cos(x1 + x2 ) = W˜ 1 ,

(3.44)

where S˜1 , T˜1 , and W˜ 1 are constants. Substituting (3.43) into (3.44) yields cos2 (x1 + x2 ) + M1 cos(x1 + x2 ) + M2 = 0,

(3.45)

where M1 and M2 are constants. Therefore, cos(x1 + x2 ) is constant. Since x2 is a constant, x3 = 0. Next, it is shown that the condition x3 = x4 = 0 leads to a contradiction. For both a Pendubot and an Acrobot, when x3 = 0 and x4 = 0, the state equation (3.2) is written as f 1 (x) + b1 (x)τ2 = 0, f 2 (x) + b2 (x)τ2 = 0.

(3.46) (3.47)

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69

Since x3 = 0 and x4 = 0, h 1 (x) = 0 and h 2 (x) = 0. From (3.4), (3.46), and (3.47), it is clear that −m 22 (x)g1 (x) + m 12 (x)g2 (x) m 12 (x) =− . (3.48) m 21 (x)g1 (x) − m 11 (x)g2 (x) m 11 (x) From (3.48), the following equation can be obtained det[M(x)] = 0,

(3.49)

which contradicts the characteristics of the parameters of the manipulators. Hence, the solution xδ , for which E(xδ ) is constant and not equal to zero under the condition V˙1 (x) = 0, cannot be a steady state in Ψ . Therefore, according to Lemma 3.1, the LIS of the system (3.31) is given by (3.32) for a Pendubot and (3.33) for an Acrobot with (x2 , x4 ) = (0, 0). The following stability theorem for the swing-up area summarizes the above results. Theorem 3.1 Let xδ = M be the LIS of the system (3.31). V1 (x) is a WCLF, is a compact closed and bounded set that contains all the initial states of the system (3.31), and Ψ is a set with states in and V˙1 (x) = 0. The controller (3.11) based on the first Lyapunov function (3.7) makes the system (3.2) move toward the invariant set M. Assume that the controller is switched from the swing-up controller (3.11) to the balance controller (3.29) at time tα . Define tα− as the nearest moment before tα , which corresponds to the last moment of the swing-up process. The stability of the control system for 0  t < tα− is guaranteed by Theorem 3.1. And the stability of the control system for t  tα− is guaranteed by an NSLF, as discussed below. Assumption 3.1 Σ j is the jth subspace and C j is the jth controller for Σ j ( j = 1, 2). Assumption 3.2 There exists a Lyapunov function, J j (x), for each pair (C j , Σ j ) ( j = 1, 2). Assumption 3.3 Σ1 and Σ2 cover the motion space for t  tα− , where (C1 , Σ1 ) is only used at t = tα− and (C2 , Σ2 ) is used when t > tα− . Definition 3.3 (NSLF) An NSLF is a Lyapunov function, J (x), given by J (x) =

2 

J j (x), tα−  t,

(3.50)

j=1

that satisfies

J [x(t2 )] < J [x(t1 )], tα−  t1 < t2 ,

(3.51)

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3 Control of Vertical Underactuated Manipulator

where

J1 (x) = V1 (xtα− ), t = tα− .

(3.52)

J2 (x) = V2 (x), t  tα .

(3.53)

According to the stability analysis for switched control systems [38–40], the stability of the control system for t  tα− is guaranteed by the following theorem. Theorem 3.2 If J (x) defined in (3.50) is used for the system when t  tα− , then controller based on the NSLF guarantees the stability of the two-link VUM for t  tα− . Δ in (3.7) is selected as Δ = max[J2 (x)] = max[V2 (x)],

(3.54)

so ensuring J1 (x) > J2 (x) in Definition 3.3. Note that V2 (x)  max[V2 (x)] =

4 4  

| Pi j xi max x j max |,

(3.55)

i=1 j=1

where, for i, j = 1, 2, 3, 4, Pi j is the ith row and jth column element of the positivedefinite symmetrical matrix P, and xi max (or x j max ) is the maximum of the state variable xi (or x j ) when t  tα . From (2.28), (2.31), (2.32), (3.5), and (3.6), one can obtain x1 max = ε1 , x2 max = ε1 + ε2 and xi max = 4ε E /m (i−2)(i−2) (i = 3, 4). Therefore, J (x) in (3.50) decreases monotonically starting at t = tα− , which avoids a shock in the switching surface. The controller (3.29) guarantees that a two-link VUM converges to the straight-up position. Thus, the stability of the control system for 0  t < tα− and t  tα− is guaranteed by Theorems 1 and 2, respectively. In particular, since the NSLF J (x) decreases for t > 0, the stability is guaranteed in the whole motion space. This is summarized in the following theorem. Theorem 3.3 A switched control system for a two-link VUM that employs the WCLF V1 (x) when 0  t < tα− and the NSLF J (x) when t  tα− is globally stable.

3.1.5 Simulation Results The parameters for the attractive area for both the two-link VUM are chosen to be ε1 = ε2 =

π , ε3 = ε4 = 10−3 , ε5 = 103 , ε E = 1.0 J. 6

and their parameters are shown in Table 3.1.

(3.56)

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71

Table 3.1 Parameters of the Two-link VUM used in simulations m 1 /kg I1 /(kg · m2 ) L 1 /m L g1 /m m 2 /kg I2 /(kg · m2 ) Pendubot 0.39 Acrobot 1.00

4.23 × 10−2 8.33 × 10−2

0.20 1.00

0.10 0.50

1.59 1.00

7.11 × 10−2 0.33

L 2 /m

L g2 /m

0.08 2.00

0.04 1.00

Remark 3.1 ε1 and ε2 are two important parameters that define the attractive area. In this area, the linear approximate model (3.26) is obtained by using the approximations sin(x1 ) ≈ x1 and sin(x1 + x2 ) ≈ x1 + x2 in the nonlinear model (3.2). Since sin(π/4) = 0.7071 and π/4 = 0.7854, roughly speaking, sin ξ ≈ ξ is true for 0  |ξ |  π/4. So, ε1 and ε2 should be less than π/4. While a suitable choice of ε1 and ε2 ensures the convergence of the motion when the balance controller (3.29) is used in the attractive area, it is difficult to capture the manipulators in the attractive area if the two parameters are very small. In addition, (3.29) cannot ensure the convergence of the motion in the attractive area if ε1 and ε2 are very large because the linear approximate model does not describe the motion of the VUM precisely in that case. (1) Case for Pendubot The parameters in (3.11), (3.19), (3.24), and (3.25) are chosen as: α1 = 0.5, α2 = 10, η = 15, γ0 = 0.13, ε = 0.5, and E 0 = 13.35 J. Q = I4 and R = 0.5 are selected for the design of the state-feedback controller, and the gain F = [−109.544, −109.697, −14.311, −18.558] ,

(3.57)

is employed when the system enters the attractive area. The initial state is x(0) = [0, π, 0, 0]T . Figure 3.1 shows the simulation results for the Pendubot. The controller is switched at t = 1.400 s. By this time, the swing-up controller increase the energy and stretch out the links. At t = 1.400 s, the switching condition for entering the attractive area is satisfied, and the balance controller takes over. The controller successfully makes the system converge to the straight-up position smoothly and quickly. These results show the validity of the described control strategy. (2) Case for Acrobot The parameters in (3.11), (3.19), (3.24), and (3.25) are chosen as α1 = 0.5, α2 = 30, η = 25, γ0 = 1.6, ε = 0.5, and E 0 = 24.5 J. And Q and R are same as the Case for Pendubot. The gain by using a linear quadratic regulator is obtained: F = [−260.559, −104.448, −112.604, −52.944] .

(3.58)

The initial state is x(0) = [π, 0, 0, 0]T . Figure 3.2 shows the simulation results for the Acrobot. The controller is switched at t = 7.664 s. The two controllers properly control the Acrobot, and make it smoothly converge to the straight-up position.

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3 Control of Vertical Underactuated Manipulator

Fig. 3.1 Overall control results for the Pendubot

A comparison of the results for the two-link VUM shows that even though the same control strategy for both and judiciously adjust the control parameters are used, the results are quite different. In particular, the time required for the energy to reach E 0 is much longer for the Acrobot than for the Pendubot. This shows that motion control is much more difficult for an Acrobot than for a Pendubot. Figures 3.1 and 3.2 demonstrate the effectiveness of the proposed unified motion control method of a Pendubot and an Acrobot, which employs a WCLF and an NSLF. Unlike most existing methods, the proposed swing-up controller adjusts not only the energy but also the posture of the manipulator, which ensures that the actuated link stretches out in a natural way while the energy increases. So, in contrast to the existing methods, the presented strategy makes it easy to capture the manipulator in the attractive area and easy to keep the motion smooth when the controller is switched. In addition, the presented method does not need repeated adjustment, simplifies the controller design and improves the efficiency of the design process. The choice of the positive, time-varying design parameter in V1 is very important. It adjusts the denominator of the swing-up controller within a suitable range during the whole swing-up process, thereby ensuring that the torque has a suitable value at the beginning of the swing-up motion. In this way, the VUM has a fast response. Meantime, the singularities are avoided in swing-up controller. For the parameters in Table 3.1, trial and error show that β(x) = 70 is a possible choice. However, the simulation results (shown in Fig. 3.3) show that it takes about 180 s for the angle of the second link to settle at zero. In contrast, the time required for the energy to reach E 0 in the proposed strategy is only about 10 s.

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73

Fig. 3.2 Overall control results for an Acrobot

Fig. 3.3 Simulation results for an Acrobot that use a large constant β(x) (=70) to avoid a singularity

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3 Control of Vertical Underactuated Manipulator

Moreover, the proposed strategy is a unified control strategy for the two-link VUM. Because different underactuated manipulators have completely different characteristics, most of the existing methods are only suitable for the control of an underactuated manipulator with a specified configuration. In addition, since the stability in the control of VUM is a challenging issue, many methods mainly focus on the stability of the swing-up area, which is quite complicated. In this section, the global stability of the two-link VUM is ensured by using WCLF and NSLF, and its control performance is also better than the existing control methods.

3.2 A Rewinding Approach-Based Control Strategy In the Sect. 3.1, a comprehensive unified switching control method has been designed to achieve the control objective of the underactuated two-link manipulators. However, in this method, the controller needs to be switched, and multiple controllers need to be designed. The torque may change abruptly when the controller is switched, which leads to instability of the system. Based on the above consideration, this section presents a method of finding a time-optimal trajectory for an Acrobot from the straight-down position all the way to the straight-up position, and achieve the control objective by a single controller.

3.2.1 Modeling and Analysis From (3.1), the dynamic equation of the Acrobot is ˙ q˙ + G(q) = [0, τ2 ]T , M(q2 )q¨ + H (q, q)

(3.59)

Let x = [x1 , x2 , x3 , x4 ]T = [q1 , q2 , q˙1 , q˙2 ]T . Then, the dynamic equation (3.59) can be rewritten in a state-space form, which is x˙ = f (x) + g(x)τ2 ,

(3.60)

where ⎤ ⎡ ⎤ 0 x3 ⎥ ⎢ ⎢ x4 0  ⎥ ⎥ , g(x) = ⎢ ⎥. f (x) = ⎢ ⎣ −1 ⎣ −M −1 (q2 )H1 (q, q) ˙ ⎦ 0 ⎦ M (q2 ) ˙ −M −1 (q2 )H2 (q, q) 1 ⎡

(3.61)

∗ = [0, 0, 0, 0]T , and the It is not difficult to verify that both the straight-up position xsu ∗ T straight-down position xsd = [π, 0, 0, 0] , are equilibrium points of the open-loop ∗ , one has system (3.60). Since K (q, q) ˙  0 and P(q) is minimum at xsd

3.2 A Rewinding Approach-Based Control Strategy

75

 ∗ E(x)  E sd := E(x)x=x ∗ = −(a4 + a5 ),

(3.62)

sd

where E(x) = K (q, q) ˙ + P(q) is the total energy of the Acrobot. So, V (x) = ∗ is a non-negative storage function of the Acrobot system. Moreover, E(x) − E sd since   1 dM(q2 ) ∂ P(q) , i = 1, 2, (3.63) q˙ T − C(q, q) ˙ q˙ = 0, G i (q) = 2 dt ∂qi it follows from (3.59) that dV (x) dE(x) = = x4 τ2 . dt dt

(3.64)

which means that the Acrobot system is a lossless passive mechanical system with the output being y = x4 . In fact, the term x4 τ2 is precisely the power supplied to the system. ∗ . If an artificial frictional torque, Note that the total energy E(x), is minimum at xsd τ2 = −μx4 ,

(3.65)

is added to the second joint, where μ > 0 is the artificial friction coefficient, then, ∗ . from the standpoint of the physics, the Acrobot will asymptotically converge to xsd The following lemma provides a rigorous theoretical proof of the statement. Lemma 3.2 Combining (3.60) and (3.65) yields the closed-loop system x˙ = f (x) − μx4 g(x) = F(x),

(3.66)

for which two statements hold: ∗ ∗ (1) The system has four equilibrium points: xe1 = [0, π, 0, 0]T , xe2 = [π, π, 0, 0]T , ∗ ∗ ∗ xsu , and xsd . Among them, only xsd is stable. ∗ as t → +∞ for any initial condition, (2) The system asymptotically converges to xsd ∗ ∗ ∗ x(0), other than xe1 , xe2 , or xsu .

Proof First, Statement (1) is proved. Letting F(x) = 0 yields x3 = x4 = 0 and G(q) = 0. That gives (3.67) sin x1 = 0, sin(x1 + x2 ) = 0. Since x1 and x2 are cyclic variables in S1 , one can obtain x1 , x2 = 0 or π (mod 2π). ∗ ∗ ∗ ∗ , xe2 , xsu , and xsd are the only four equilibrium points of (3.66). Thus, xe1

(3.68)

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3 Control of Vertical Underactuated Manipulator

To determine the stability of the system (3.66) at the equilibrium points, it is linearized around each equilibrium point and one can find that the four Jacobian matrices have the same form ⎤ ⎡ 0 0 1 0 ⎢ 0 0 0 1 ⎥ ⎥ (3.69) J =⎢ ⎣ J31 J32 0 J34 ⎦ . J41 J42 0 J44 The Routh-Hurwitz criterion tells us that the matrix J is stable if and only if ⎧ Λ1 ⎪ ⎪ ⎨ Λ2 Λ3 ⎪ ⎪ ⎩ Λ4

= J44 < 0, = J34 J41 + J42 J44 > 0, = J31 J42   − J32 J41 > 0,   = J41 × J32 J44 − J42 J34 J44 + J31 J44 − J34 J41 J34 > 0. (3.70) As shown in [28], ai > 0 (i = 1, 2, . . . , 5), a1 a2 − a32 > 0, a2 a4 > a3 a5 , and a3 a4  ∗ , a1 a5 . It is not difficult to verify that, for the equilibrium point xsu ∗ Λ2 (xsu )

  μ (a1 + a3 )2 a5 + (a2 + a3 )2 a4 =− < 0,  2 a1 a2 − a32

(3.71)

∗ ∗ for the equilibrium points xe1 and xe2 , ∗ ∗ ) = Λ3 (xe2 )=− Λ3 (xe1

a4 a5 < 0, a1 a2 − a32

(3.72)

∗ , and for the equilibrium point xsd

⎧ −μ(a1 + a2 + 2a3 ) ∗ ⎪ ⎪ < 0, ⎪ Λ1 (xsd ) = ⎪ a1 a2 − a32 ⎨ ∗ ∗ ∗ ∗ ) = −Λ2 (xsu ) > 0, Λ3 (xsd ) = −Λ3 (xe1 ) > 0, . Λ2 (xsd 2  ⎪ ⎪ a − a a + a a − a a ) μ(a ⎪ 3 4 1 5 2 4 3 5 ∗ ⎪ ⎩ Λ4 (xsd )= > 0. a1 a2 − a32

(3.73)

∗ ∗ ∗ Therefore, the equilibrium points xe1 , xe2 , and xsu are unstable, and the equilibrium ∗ point xsd is stable. Now, Statement (2) is proved. The storage function V (x) is chosen to be the Lyapunov function of (3.66). Combining (3.64) and (3.65) yields V˙ = −μx42  0. This means that V (x) decreases monotonically as t > 0. However, it does not guarantee that V (x) converges to zero because V (x) remains unchanged when x4 = 0. So, the convergence of V (x) needs to be further analyzed.

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77

Letting V˙ ≡ 0 yields x4 ≡ 0. From (3.64) and (3.66), E(x) = E¯ and x2 = x¯2 are constant. Combining (2.28) and (2.31), and E(x) = E¯ yields W1 x32 + W2 cos x1 + W3 sin x1 = W4 ,

(3.74)

where Wi (i = 1, 2, 3, 4) are constants. In addition, the fourth entry in (3.66) and x4 = 0 give (3.75) S1 x32 + S2 cos x1 + S3 sin x1 = S4 , where Si (i = 1, 2, 3, 4) are constants. Combining (3.74) and (3.75) yields T1 cos x1 + T2 sin x1 = T3 ,

(3.76)

where Ti (i = 1, 2, 3) are constants. This means that x1 = x¯1 is a constant. Thus, ¯ = 0. So, x¯ is an x3 = 0. Let x¯ = [x¯1 , x¯2 , 0, 0]T , it follows from (3.66) that F(x) equilibrium point of the system (3.66). From LaSalle’s invariance theorem, the system ∗ ∗ ∗ ∗ ∗ (3.66) converges to one of the equilibrium points  x∗e1 , x∗e2 , x∗su, or xsd . Since xsd is the only stable equilibrium point and since x(0) ∈ / xe1 , xe2 , xsu , (3.66) asymptotically ∗ as t → +∞. converges to xsd

3.2.2 Motion Planning ∗ For an Acrobot, an upward trajectory is defined to be a trajectory from xsd to xε , ∗ which is a position very close to xsu . In addition, a downward trajectory is defined to be the opposite. Furthermore, a stabilizing trajectory is defined to be a trajectory ∗ . The upward, downward and stabilizing trajectories are denoted by from xε to xsu xu (t), xd (t) and xs (t), respectively. This subsection concerns the motion planning of ∗ ∗ and xsu (Fig. 3.4). an Acrobot between xsd Let the initial position of the Acrobot be xε = [ε, 0, 0, 0]T , where ε > 0 is a very ∗ . From Statement 2 of small number. Clearly, xε is in a small neighborhood of xsu Lemma 3.1, the downward trajectory and downward torque are

⎧ ⎨



 qd (t) , xd (0) = xε , t ∈ [0, td ], q˙d (t) ⎩ τ2d (t) = −μq˙2d (t), t ∈ [0, td ], xd (t) =

(3.77)

where td > 0 is the response time required for the state of the Acrobot to enter the ∗ , steady-state set around xsd 

|min (q1 , 2π − q1 ) − π| < 0.05π, |q˙1 | < 0.05, |min (q2 , 2π − q2 )| < 0.05, |q˙2 | < 0.05,

and remain there when t > td .

(3.78)

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3 Control of Vertical Underactuated Manipulator

Fig. 3.4 Process of constructing desired trajectory xw (t)

Based on the downward trajectory and downward torque, the idea of rewinding is used to construct the upward trajectory and the corresponding torque:  qd (td −t) , τ2u (t) = τ2d (td −t), t ∈ [0, td ]. xu (t) = −q˙d (td −t) 

(3.79)

The idea comes from the process of first watching a video tape and then watching it in reverse as it rewinds [41]. Clearly, xu (t) is a rewinding trajectory of xd (t) when t ∈ [0, td ]. Note that the finial position of the upward trajectory xu (td ) = xε = ∗ , and that the upward torque τ2u (t) equals zero at td . [ε, 0, 0, 0]T , is very close to xsu ∗ A period tΔ is introduced and the reference stabilizing trajectory xs (t) is set to be xsu ∗ during the interval t ∈ (td , td + tΔ ] in order to drive the state from xu (td ) to xsu . So, the definitions of the whole trajectory of the Acrobot and the corresponding control input are  xw (t) =

 xu (t), t ∈ [0, td ], τ (t), t ∈ [0, td ], τ2w (t) = 2u xs (t), t ∈ (td , td + tΔ ], 0, t ∈ (td , td + tΔ ],

(3.80)

Thus, xw (t) is the trajectory of the Acrobot from the point [qdT (td ), −q˙dT (td )]T , which ∗ ∗ , to xsu . Since C(q, −q) ˙ = −C(q, q) ˙ and q¨u (t) = q¨d (td − t), and is very close to xsd since xd (t) and τ2d (t), xu (t) and τ2u (t) satisfy the dynamic equation. It thus follows from (3.80) that xw (t) = [q1w (t), q2w (t), q˙1w (t), q˙2w (t)]T and τ2w (t) satisfy M(q2w )q¨w + C(qw , q˙w )q˙w + G(qw ) = [0, τ2w ]T .

(3.81)

To find a time-optimal trajectory xw (t), the downward settling time td needs to be ∗ along xd (t) if the mechanical optimized. Note that the state moves from xε to xsd

3.2 A Rewinding Approach-Based Control Strategy

79

energy of the Acrobot at xε is completely absorbed by the artificial frictional torque, ˙ and the absorptivity is  E˙  = μq˙22 (t). An optimal μ in (3.65) that maximizes | E| needs to be found. In other words, μ is obtained by solving minμ∈(0,+∞) td . Since μ → +∞ means that the artificial friction is very large, it makes the relative speed ˙ is small. On the other hand, μ → 0 means that the |q˙2 (t)|, very small, and thus | E| ˙ for a bounded q˙2 (t). artificial friction is very small, it also gives a small a small | E| Thus, both small and large values of μ result in a very long settling time. So, from the physics, there exists an optimum coefficient μ ∈ (0, +∞), that yields a minimum td and a time-optimal trajectory, xw (t). Remark 3.2 The control torque (3.65) ensures that the mechanical energy E(x), decreases monotonically during the downward motion. So, E(x) must increase monotonically during the upward trajectory because it is the reverse of the downward one. This allows us to achieve the control objective without wasting energy. In addition, the resulting xw (t) and τ2w (t) in (3.80) can be used as guidelines for selecting an actuator as well as for evaluating an Acrobot design.

3.2.3 Trajectory Tracking Control After the desired trajectory xw (t) is obtained, the motion control problem for the Acrobot becomes a tracking control problem. This subsection uses the pole assignment method to design a control law τ2 (t) that forces the Acrobot to track xw (t) for t ∈ [0, td + tΔ ]. Let q = [q1 , q2 ]T = q − qw , τ2e = τ2 − τ2w . Combining (3.59) and (3.81) yields

M(q2w +q2 )q¨ + M(q2w +q2 )− M(q2w ) q¨w + C(qw +q, q˙w



T + q)( ˙ q˙w +q)−C(q ˙ w , q˙w )q˙w + G(qw +q)−G(qw ) = [0, τ2e ] . (3.82) The error dynamics (3.82) are very complicated. In this study, a time-varying linear model for (3.82) along xw (t) is built, after which a time-varying control law τ2e (t) is designed to make the error state, e(t) = [q T (t), q˙ T (t)]T , converge to zero. A simple calculation yields a linear approximate model of the system (3.82) around e(t) = 0 (3.83) e(t) ˙ = A(t)e(t) + B(t)τ2e (t), where  A(t) =

   I2 02×1 02×2 , , B(t) = D −1 (t)E 1 (t) D −1 (t)E 2 (t) D −1 (t)[0, 1]T

 a4 sin q1w + ϕ1 (qw ) ϕ2 (qw , q˙w , q¨w ) + ϕ1 (qw ) , E 1 (t) = ϕ1 (qw ) ϕ3 (qw , q˙w , q¨w ) + ϕ1 (qw )

(3.84)



(3.85)

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3 Control of Vertical Underactuated Manipulator

D(t) = M(q2w ), E 2 (t) = C(q2w , q˙2w ), ϕ1 = a5 cos(q1w +q2w ),

(3.86)

ϕ2 = a3 (2q¨1w + q¨2w ) sin q2w + a3 q˙2w (2q˙1w + q˙2w ) cos q2w ,

(3.87)

  2 cos q2w . ϕ3 = a3 q¨1w sin q2w − q˙1w

(3.88)

Let R0 (t) = B(t), Ri (t) = −A(t)Ri−1 (t) + R˙ i−1 (t) (i = 1, 2, 3). If the condition rank[R0 (t), R1 (t), R2 (t), R3 (t)] = 4, ∀t ∈ [0, td + tΔ ]

(3.89)

holds, then [A(t), B(t)] is controllable while t ∈ [0, td + tΔ ] [42]. For the controllable linear time-varying system (3.83), the control law is designed to be τ2e (t) = −K (t)e(t), K (t) ∈ R1×4 .

(3.90)

Applying Theorem 5.2 in [43] gives us a stability criterion for the close-loop system. Lemma 3.3 The error state e(t) of the closed-loop system, (3.83) and (3.90), exponentially converges to zero if the following conditions hold: ˜ = A(t) − B(t)K (t), λi (i = 1, 2, 3, 4) are constants (1) The eigenvalues of A(t) and all in the left-hand plane. ˙ < β, where P(t) is the matrix of eigenvectors, β is a positive (2) P −1 (t) P(t) ˜ constant that satisfies β < |Re(λ M )|, and λ M is the largest eigenvalue of A(t). From the Lemma 3.3, the tracking controller can be given as follows τ2 (t) = τ2w (t) + τ2e (t),

(3.91)

which makes the Acrobot efficiently track the desired trajectory, xw (t), t ∈ [0, td + tΔ ]. The block diagram of the whole control system in Fig. 3.5 can be obtained by consolidating the above results.

Fig. 3.5 Configuration of tracking control system for an Acrobot

3.2 A Rewinding Approach-Based Control Strategy

81

3.2.4 Simulation Results This subsection presents a numerical example that demonstrates the validity of the method described above. The simulations use the mechanical parameters in Table 3.2. The step size is 0.001 s, and ε in xε is 0.01. For these initial conditions, the relationship between μ and the downward settling time, td , Fig. 3.6 shows that td has the smallest value (4.0 s) when μ = 0.00182. μ is set to this value. In addition, tΔ in (3.80) is chosen to be 2 s. Figure 3.7 shows the simulation results for xw (t) (= [qwT (t), q˙wT (t)]T ), τ2w (t), and E(xw ). Note that xw (t) is almost continuous and that E(xw ) increases monotonically. A simple check shows that the condition (3.89) holds. The eigenvalues of ˜ A(t) in Lemma 3.3 are chosen to be λ1∼4 = −5.5, −8, −10, −15. By using the ˙  5.325 < β = 5.4 (Fig. 3.8) is MATLAB functions eig and norm, P −1 (t) P(t) obtained. Thus, the chosen eigenvalues are suitable. Figure 3.9 shows the simulation ∗ . It is clear that the Acrobot results of the Acrobot under the initial condition xsd ∗ is stabilized at xsu almost precisely along the desired trajectory xw (t). Note that x(t) (= [q T (t), q˙ T (t)]T ) and τ2 (t) exhibit a sudden, small change at t = 4 s. This is due to the small step in q1w (t) at t = 4 s, where the transition from the upward trajectory to the stabilizing trajectory occurs. To determine whether the presented method is practical, simulations are carried out with uncertainties (m 1 and J1 are +10% of their nominal values, and m 2 and J2 are −10% of their nominal values), with white noise in the measurement of q (peak value: 0.1) and q˙ (peak value: 0.3), and with viscous friction torques (first joint: f v1 = −0.0002q˙1 , second joint: f v2 = −0.002q˙2 ). The simulation results Fig. 3.10 show that the motion trajectory of the Acrobot changes a little and that the control torque also changes to robustly stabilize the system. Thus, the presented method is also effective in this case.

Table 3.2 Mechanical parameters of the Acrobot Link i m i /kg L i /m i =1 i =2

0.105 0.080

Fig. 3.6 Relationship between μ and td

0.109 0.215

L ci /m

Ji /(kg · m2 )

0.073 0.1075

1.0396 × 10−4 3.0817 × 10−4

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3 Control of Vertical Underactuated Manipulator

Fig. 3.7 Desired trajectory and control torque Fig. 3.8 Check of Condition (3.3) in Lemma 3.3

3.3 Stable Control of Three-Link Underactuated Manipulators An underactuated three-link gymnast robot (UTGR) [44] is a simple gymnast model with a passive joint located at the first joint. Similarly, the system control objective is also to swing the end-point upward from the straight-down position and stabilize it at the straight-up position. Here, the motion space is divided into two subspaces, which are swing-up area and attractive area, and a controller for each subspace is designed. Since the UTGR is a highly nonlinear system with nonholonomic constraints [45, 46], it is very important to design the swing-up controller to ensure that UTGR enters the attraction area. The focus of this section is to design a swing-up controller without singularities. By introducing virtual coupling between control torques, the problem of avoiding singularity is transformed into the problem of designing reasonable con-

3.3 Stable Control of Three-Link Underactuated Manipulators

83

Fig. 3.9 Tracking control results

Fig. 3.10 Simulation results when there are perturbations in parameters, an external disturbance, and friction acting on the joints

84

3 Control of Vertical Underactuated Manipulator

troller parameters. The swing-up controller designed in this way can ensure that the UTGR enters the attraction area in a natural stretching posture, and make the system stabilize at the target position easily.

3.3.1 Model and Division of Motion Space The model of a UTGR is shown in Fig. 3.11, and the dynamic equation is M(q)q¨ + H (q, q) ˙ q˙ + G(q) = τ,

(3.92)

where q¨ = dq/dt, ˙ τ = [0, τ2 , τ3 ]T , H (q, q) ˙ q˙ + G(q) contains the Coriolis and centrifugal terms, and the gravity term. From (2.97), the following equation can be obtained dE(q, q) ˙ = q˙ T τ = τ2 q˙2 + τ3 q˙3 . dt

(3.93)

Let x = [x1 , x2 , . . . , x5 , x6 ]T = [q T , q˙ T ]T . The state space form of (3.92) is ⎧ x˙1 ⎪ ⎪ ⎨ x˙4 x ˙5 ⎪ ⎪ ⎩ x˙6

= = = =

x4 , x˙2 = x5 , x˙3 = x6 , f 1 (x) + b1 (x)τ2 + c1 (x)τ3 , f 2 (x) + b2 (x)τ2 + c2 (x)τ3 , f 3 (x) + b3 (x)τ2 + c3 (x)τ3 ,

Fig. 3.11 Underactuated three-link gymnast robot (UTGR)

(3.94)

y Active joint Passive joint Center of mass q3

m3 J3

Lc 2

3

Lc 3 L3

L2

m2

q2 J2

2

g

m1 J1

L1

q1

LC1

O

x

3.3 Stable Control of Three-Link Underactuated Manipulators

85

where ⎤ 3  − (H (q, q) ˙ q ˙ − G (q)) i 1 ⎥ ⎢ i=1 1i ⎡ ⎤ ⎥ ⎢ 3 f 1 (x) ⎥ ⎢  ⎣ f 2 (x) ⎦ = M −1 (q) ⎢ − (H2i (q, q) ˙ q˙i − G 2 (q)) ⎥ ⎥ , ⎢ ⎥ ⎢ i=1 f 3 (x) ⎦ ⎣  3 − (H3i (q, q) ˙ q˙i − G 3 (q)) ⎡

(3.95)

i=1

⎡

⎡ ⎤ b1 (x) c1 (x) 0 ⎣ b2 (x) c2 (x) ⎦ = M −1 (q) ⎣ 1 0 b3 (x) c3 (x)

⎤ 0 0⎦. 1

(3.96)

From (3.96), one has  b2 (x) = [M11 (q) M33 (q) − M13 (q) M31 (q)] det [M (q)],  c3 (x) = [M11 (q) M22 (q) − M12 (q) M21 (q)] det [M (q)],  b2 (x)c3 (x) − b3 (x)c2 (x) = M11 (q) det [M (q)],

(3.97) (3.98) (3.99)

where det[M(q)] is the determinant of the matrix M(q). Since M(q) is a positivedefinite matrix, and the terms of M(q) are bounded, one can obtain det[M(q)] > 0, b2 (x) > 0, c3 (x) > 0, and b2 (x)c3 (x) − b3 (x)c2 (x) > 0. These characteristics play an important role in avoiding singularity of the swing-up controller, as discussed in Sect. 3.3.3. The control objective is to control the UTGR to swing up from the straight-down position and finally to stabilize it at the straight-up position (x = 0). The whole motion space Σ is divided into two subspaces: the attractive area Σb and the swingup area Σs , which are shown as follows: ⎧ Ξ (x1 , 2π)  ε1 , ⎪ ⎪ ⎨ Ξ (x1 , 2π) + Ξ (x2 , 2π)  ε2 , Σb : Ξ (x1 , 2π) + Ξ (x2 , 2π) + Ξ (x3 , 2π)  ε3 , ⎪ ⎪ ⎩ |E(x) − E 0 |  ε4 ,

(3.100)

Σs = Σ − Σb ,

(3.101)

where Ξ (y, 2π) = min {mod(y, 2π), 2π − mod(y, 2π)}, mod(y, 2π) is the posi˙ is the total tive remainder of y divided by 2π, εi > 0 (i = 1, 2, 3, 4), E(x) = E(q, q) energy, and E 0 = E(x)|x=0 = β1 + β2 + β3 is the energy of the UTGR at x = 0. So, the attractive area Σb is a small domain around the straight-up position x = 0.

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3 Control of Vertical Underactuated Manipulator

3.3.2 Swing-Up Controller Design In this part, a swing-up controller is designed to make the UTGR approach the attractive area Σb with a naturally stretched-out posture. First, the Lyapunov function is constructed as V (x) =

k1 2 1 2 1 2 k2 2 k3 2 E + x + x + x5 + x6 , 2 x 2 2 2 3 2 2

(3.102)

where E x = E(x) − E 0 , and ki > 0 (i = 1, 2, 3) are design parameters. From (3.93) and (3.94), one can obtain dV (x) = x5 {[k1 E x + k2 b2 (x)] τ2 + x2 + k2 f 2 (x) + k2 c2 (x)τ3 } dt +x6 {[k1 E x + k3 c3 (x)] τ3 + x3 + k3 f 3 (x) + k3 b3 (x)τ2 } . (3.103) A virtual coupling between the control torques τ2 and τ3 is introduced τ3 =

−k3 f 3 (x) − x3 − λ1 x6 − k3 b3 (x)τ2 k1 E x + k3 c3 (x)

(3.104)

where λ1 is a constant which satisfies λ1 > 0. Substituting (3.104) into (3.103) yields dV (x) = x5 [F1 (x)τ2 + F2 (x)] − λ1 x62 , dt

(3.105)

where F1 (x) = k1 E x + k2 b2 (x) −

k2 k3 b3 (x)c2 (x) , k1 E x + k3 c3 (x)

F2 (x) = x2 + k2 f 2 (x) − F3 (x), F3 (x) =

k2 c2 (x) [k3 f 3 (x) + x3 + λ1 x6 ] . k1 E x + k3 c3 (x)

(3.106) (3.107) (3.108)

For a positive constant λ2 , let τ2 =

−F2 (x) − λ2 x5 . F1 (x)

(3.109)

Substituting (3.109) into (3.105) yields dV (x) = −(λ2 x52 + λ1 x62 ) < 0, ∀x5 = 0 or x6 = 0. dt

(3.110)

3.3 Stable Control of Three-Link Underactuated Manipulators

87

In order to obtain the final motion state of the UTGR, the stability of the system needs to be further analyzed based on (3.110). Letting dV (x)/dt ≡ 0 yields x5 = x6 = 0.

(3.111)

From (3.94), (3.111) means that x2 , x3 , and E(x) are all constants x2 = x2∗ , x3 = x3∗ , E(x) = E ∗ .

(3.112)

Now, two different situations are considered. (1) E ∗ = E 0 In this case, (3.104) indicates that x3 = −k3 [ f 3 (x) + b3 (x)τ2 + c3 (x)τ3 ] .

(3.113)

Substituting (3.111) into (3.94) yields f 2 (x) + b2 (x)τ2 + c2 (x)τ3 = 0,

(3.114)

f 3 (x) + b3 (x)τ2 + c3 (x)τ3 = 0.

(3.115)

From which the following expressions for τ2 and τ3 can be obtained τ2 =

c2 (x) f 3 (x) − c3 (x) f 2 (x) , b2 (x)c3 (x) − b3 (x)c2 (x)

(3.116)

τ3 =

b3 (x) f 2 (x) − b2 (x) f 3 (x) . b2 (x)c3 (x) − b3 (x)c2 (x)

(3.117)

Combining (3.113) and (3.115) yields x3 = 0. In addition, (3.109) and (3.116) give k2 [c2 (x) f 3 (x) − c3 (x) f 2 (x)] k2 [c3 (x) f 2 (x) − c2 (x) f 3 (x)] = −x2 − .(3.118) c3 (x) c3 (x) which gives x2 = 0. So, E(x) = E 0 , x2 = x3 = x5 = x6 = 0 in this case. (2) E ∗ = E 0 Substituting (3.111) and (3.112) into (2.20) yields η1 x42 + η2 cos x1 + η3 sin x1 = η4 ,

(3.119)

where ηi (i = 1, 2, 3, 4) are constants. Moreover, (3.104) becomes    τ2  k3 b3 (x) k1 E x + k3 c3 (x) = −k3 f 3 (x) − x3 . τ3

(3.120)

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3 Control of Vertical Underactuated Manipulator

Since both x2 and x3 are constants, M(q) is a constant matrix. Equation (3.96) indicates that b2 (x), b3 (x), c2 (x), and c3 (x) are all constants. Combining (3.95), (3.116), (3.117), and (3.120) yields ν1 x42 + ν2 cos x1 + ν3 sin x1 = ν4 ,

(3.121)

where νi (i = 1, 2, 3, 4) are constants. Equations (3.119) and (3.121) give κ1 cos x1 + κ2 sin x1 = κ3 ,

(3.122)

where κi (i = 1, 2, 3) are constants. Equation (3.122) means that x1 is a constant. From (3.94), it is found that x4 = 0 and f 1 (x) + b1 (x)τ2 + c1 (x)τ3 = 0.

(3.123)

Combining (3.116), (3.117), and (3.123) yields     b2 (x)c3 (x)− b3 (x)c2 (x) f 1 (x) + b3 (x)c1 (x) − b1 (x)c3 (x) f 2 (x)   + b1 (x)c2 (x) − b2 (x)c1 (x) f 3 (x) = 0. (3.124) ˙ q˙ = 0. Substituting (3.95) and (3.96) Since x4 = x5 = x6 = 0, it is clear that C(q, q) into (3.124) yields det(M −1 (q)) = 0, which shows that M(q) is an invertible matrix. invariant set of system (3.94),     From the above discussion, the positive x dV (x)/ dt ≡ 0}, must be a subset of x  E(x) = E 0 , x2 = x3 = x5 = x6 = 0 . From LaSalle’s invariance theorem [47], the following equation can be obtained E(x) → E 0 , x2 → 0, x3 → 0, x5 → 0, x6 → 0. The result is that the UTGR gradually degenerates into a one-link VUM (Fig. 3.12) with energy E 0 in Σs . This guarantees that the UTGR enters the attractive area Σb with a naturally stretched-out posture.

3.3.3 Singularity Avoidance in Swing-Up Controller To avoid singularities in (3.104) and (3.109), the following conditions Λ(x) = k1 E x + k3 c3 (x) = 0, F1 (x) = 0,

(3.125) (3.126)

must be satisfied. Note that Λ(x) is the denominator of the virtual coupling (3.104) and F1 (x) is the denominator of (3.109). A suitable choice of the control parameters ki (i = 1, 2, 3) can ensure (3.125) and (3.126).

3.3 Stable Control of Three-Link Underactuated Manipulators

89

Fig. 3.12 Degenerate three-link gymnast robot

First, k1 and k3 should be chosen such that k3 >

2k1 E 0 , c3m = min [c3 (x)] > 0. c3m

(3.127)

Note that E(x)  −E 0 . This gives E x  −2E 0 . So, from (3.127), Λ(x)  k3 c3m + k1 E x > 2k1 E 0 + k1 E x  0. which avoids the situation Λ(x) = 0. Then, the auxiliary function is introduced Γ (y) = k12 y 2 + k1 φ1 (x)y + k2 k3 φ2 (x),

(3.128)

φ1 (x) = k2 b2 (x) + k3 c3 (x) > 0,

(3.129)

φ2 (x) = b2 (x)c3 (x) − b3 (x)c2 (x) > 0.

(3.130)

where

It is easy to verify that F1 (x) = Γ (E x )/Λ(x). So, F1 (x) = 0 is equivalent to Γ (E x ) = 0. From (3.128), Γ (E x ) > 0 when E x  0. Thus, Γ (E x ) = 0 holds only when E x < 0. Since E x  −2E 0 , the problem of avoiding singularities in the controller (3.109) is equivalent to the statement Γ (E x ) = 0, −2E 0  E x < 0.

(3.131)

90

3 Control of Vertical Underactuated Manipulator

Fig. 3.13 Illustration of (3.131)

y0

Γ ( y)

2E0

O y

For the quadratic function Γ (y), Γ (0) = k2 k3 φ2 (x) > 0 and the axis of symmetry is y0 = −φ1 (x)/2k1 < 0. Figure 3.13 shows that (3.131) holds as long as y0 < −2E 0 and Γ (−2E 0 ) > 0, that is, (3.132) φ1 (x) > 4k1 E 0 , 2k1 E 0 [φ1 (x) − 2k1 E 0 ] . φ2 (x) (3.133) Now, k1 and k2 are chosen to satisfy the following condition 4k12 E 02 − 2k1 E 0 φ1 (x) + k2 k3 φ2 (x) > 0 ⇔ k2 k3 >

k2 >

2k1 E 0 , b2m = min [b2 (x)] > 0. b2m

(3.134)

which gives k2 b2 (x) > 2k1 E 0 .

(3.135)

Based on Eqs. (3.127) and (3.129), (3.132) holds ture. Finally, if the selected ki (i = 1, 2, 3) satisfy k2 k3 > 2k1 E 0 [k3 γ1 + k2 γ2 − 2k1 E 0 γ3 ],

(3.136)

where      c3 (x) b2 (x) 1 > 0, γ2 = max > 0, γ3 = min > 0, γ1 = max φ2 (x) φ2 (x) φ2 (x) (3.137) then it can be proved that (3.133) holds true. From the above statements, the constraints (3.127), (3.134), and (3.136) on ki (i = 1, 2, 3) guarantee that the swing-up controller (3.104) and (3.109) have no singularities. Since E 0 , c3m , and b2m are constants and the constraints (3.127) and (3.134) are known, it is easy to find ki (i = 1, 2, 3). Moreover, note that the left side of (3.136) is k2 k3 , and the right side involves k2 + k3 . Clearly, one can always make (3.136) hold for a constant k1 by increasing k2 or k3 . So, there always exist ki 

3.3 Stable Control of Three-Link Underactuated Manipulators

91

(i = 1, 2, 3) to make the constraints (3.127), (3.134), and (3.136) hold. The above discussion is summarized as a process of parameter selection. Process for choosing ki (i = 1, 2, 3) Step 1: Choose a value for k1 . Step 2: Select k2 and k3 such that (3.127) and (3.134) hold. Step 3: Check if (3.136) holds. If not, increase k2 or k3 until (3.136) is satisfied. Remark 3.3 Since the terms of M(q) are functions of sin q j and cos q j ( j = 2, 3), it follows from (3.97), (3.98), and (3.99) that b2 (x), c3 (x), and φ2 (x) are also functions of sin q j and cos q j ( j = 2, 3). So, the minimum values of φ2 (x)/c3 (x), φ2 (x)/b2 (x), and 1/φ2 (x) for q j ∈ [0, 2π] ( j = 2, 3) can be obtained by using the fmincon function. Then, the periodicity of the sine and cosine functions makes it easy to find the global minimums of φ2 (x)/c3 (x), φ2 (x)/b2 (x), and 1/φ2 (x) for q j ∈ R ( j = 2, 3). This yields values for γ1 , γ2 , and γ3 in (3.137). Remark 3.4 The proposed control strategy can be easily extended to the design of a swing-up controller for an n-link (n > 3) VUM with a passive joint. Assume that the kth joint of the n-link VUM is unactuated (k ∈ {1, 2, . . . , n}). Without loss of generality, it is assumed that k = 1. However, if k = 1, the simple coordinate transformation     q˜1 , q˜2 , . . . , q˜n = qk , q1 , q2 , . . . , qk−1 , qk+1 , . . . , qn , moves qk to the beginning of the position vector, where qi (i = 1, 2, . . . , n) is the angle of the ith link of the manipulator. So, the input vector of the n-link VUM can always be written as F = [0, τ2 , . . . , τn ]T . For i = 1, 2, . . . , n, one can assume that xi = qi , xn+i = q˙i , and the dynamics of the n-link VUM in the state-space are ⎧ ⎪ ⎨ x˙i = xn+i , ⎪ ⎩ x˙n+i = f i1 (x) +

n 

f iμ (x)τμ ,

(3.138)

μ=2

where x = [x1 , x2 , . . . , x2n ]T , and f i1 (x) is shown as follows: ⎡ ⎢ ⎢ ⎢ ⎣

⎤ f 12 (x) f 13 (x) · · · f 1n (x)   f 22 (x) f 23 (x) · · · f 2n (x) ⎥ 01×(n−1) ⎥ −1 = M , (q) ⎥ .. .. .. In−1 . . . ⎦ f n2 (x) f n3 (x) · · · f nn (x)

in which M(q) ∈ Rn×n is the positive-definite inertia matrix.

(3.139)

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3 Control of Vertical Underactuated Manipulator

To design a swing-up controller that drives the n-link VUM into the attractive area, a candidate Lyapunov is chosen as k1 2 1  2 1  2 x + ki xn+i , E + 2 x 2 i=2 i 2 i=2 n

V (x) =

n

(3.140)

where E x = E(x) − E 0 , E(x) is the total energy of the n-link VUM, E 0 = E(0), and ki > 0 (i = 1, 2, . . . , n) are constants. The Lyapunov function V (x) has the same expression as V (x) in (3.102). Combining (3.138), (3.140), and E˙ x = xn+2 τ2 + · · · x2n τn yields dV (x) = dt

n  i=2

⎧ ⎨ xn+i

⎩

[k1 E x + ki f ii (x)] τi + ki f i1 (x) + xi + ki

n 

f iρ (x)τρ

ρ=2,ρ=i

⎫ ⎬ ⎭

.

(3.141) Now, virtual coupling between τ a (= [τ3 , τ4 , . . . , τn ]T ) and τ2 is created: τa = Λ−1 (x) [Φ1 (x) + Φ2 (x)τ2 ] ,

(3.142)

where ⎤ k1 E x + k3 f 33 (x) k3 f 34 (x) ··· k3 f 3n (x) ⎥ ⎢ k1 E x + k4 f 44 (x) · · · k4 f 4n (x) k4 f 43 (x) ⎥ ⎢ Λ(x) = ⎢ ⎥ ∈ R(n−2)×(n−2) , .. .. .. ⎦ ⎣ . . . kn f n4 (x) · · · k1 E x + kn f nn (x) kn f n3 (x) ⎡

(3.143) ⎡

⎤

λ3 xn+3 + k3 f 31 (x) + x3 ⎢ λ4 xn+4 + k4 f 41 (x) + x4 ⎥ ⎥ ⎢ Φ1 (x) = − ⎢ ⎥ ∈ R(n−2)×1 , .. ⎦ ⎣ . λn x2n + kn f n1 (x) + xn T  Φ2 (x) = − k3 f 32 (x), k4 f 42 (x), . . . , kn f n2 (x) ∈ R(n−2)×1 ,

(3.144)

(3.145)

and λ j > 0 ( j = 3, 4, . . . , n) are constants. For a positive constant λ2 , τ2 is chosen as λ2 xn+2 + k2 f 21 (x) + x2 + Φ3 (x)Λ−1 (x)Φ1 (x) , (3.146) τ2 = − Υ (x) where

Υ (x) = k1 E x + k2 f 22 (x) + Φ3 (x)Λ−1 (x)Φ2 (x), Φ3 (x) = [k2 f 23 (x), k2 f 24 (x), . . . , k2 f 2n (x)] ∈ R1×(n−2) .

3.3 Stable Control of Three-Link Underactuated Manipulators

93

Substituting (3.142) and (3.146) into (3.101) yields dV (x) 2 2 2 = −λ2 xn+2 − λ3 xn+3 . . . − λn x2n  0. dt

(3.147)

From the analysis procedure in Sect. 3.3.2, (3.147) guarantees E(x) → E 0 , xi → 0, xn+i → 0, i = 2, 3, . . . , n. So, the n-link VUM approaches the attractive area with a naturally stretched-out posture under the controller (3.142) and (3.146). In Sect. 3.3.3, selecting the parameters ki (i = 1, 2, . . . , n) under the constraints ensures the absence of singularities in controllers (3.142) and (3.146).

3.3.4 Balance Controller Design The swing-up controllers (3.104) and (3.109) ensure that the UTGR enters the attractive area Σb with a naturally extended posture. Because Σb is a small area around x = 0, a linear time-invariant approximate model at the straight-up position is good enough to describe its behavior. Meanwhile, a linear-feedback balance controller is used to stabilize the UTGR at x = 0 in Σb [48]. The linear approximate model of the UTGR around x = 0 is calculated: x˙ = Ax + Bu, where

 u = [τ2 , τ3 ]T , A =

  T I3 −1 0 1 0 " , B = M , " −1 G " 0 001 M 0

⎤ ⎡ β1 + β2 + β3 β2 + β3 β3  " = M(q) " = ⎣ β2 + β3 β2 + β3 β3 ⎦ . M ,G q2 =q3 =0 β3 β3 β3

(3.148)

(3.149)

(3.150)

The optimal performance index is defined as # J=

∞



 x T Qx + u T Ru dt,

(3.151)

0

where Q and R are positive-definite weight matrices. The linear quadratic regulator (LQR) optimal balance controller is designed as u = −K x, K = R −1 B T P.

(3.152)

94

3 Control of Vertical Underactuated Manipulator

where P = P T > 0 is a positive solution of the Riccati equation AT P + P A − P B R −1 B T P + Q = 0. The LQR optimal balance controller (3.152) realize the asymptotic stability of the system (3.148) at x = 0. As a result, the control objective of stabilizing the UTGR at the straight-up position is achieved by sequentially applying first the swing-up controller (3.104) and (3.109) and then the balance controller (3.152).

3.3.5 Simulation Results Simulink is used to establish the model of UTGR to verify the effectiveness of the control strategy. Physical parameters are shown in Table 3.3. A simple calculation gives E 0 = 47.9783 J, c3m = 5.2625, b2m = 2.9311, γ1 = 0.7919, γ2 = 0.1949, γ3 = 0.0028. The parameters are chosen to be 

ε1 = ε2 = ε3 = π/6, ε4 = 0.1, λ1 = 0.5, λ2 = 1.5, Q = I6 , R = I2 , k1 = 0.01, k2 = k3 = 1.

(3.153)

It is easy to prove that ki (i = 1, 2, 3) in (3.153) satisfy the constraints (3.127), (3.134), and (3.136). Using LQR function in MATLAB to calculate the feedback gain K in (3.152):  K =

 −436.6618 −169.9226 −51.5224 −111.4562 −50.6082 −17.0817 . −341.8794 −149.3111 −38.0969 87.6154 −41.2934 −12.8450

The initial conditions for the simulations is set as x0 = [π, 0, π/6, 0, 0, 0]T . The simulation results (Figs. 3.14 and 3.15) show that the UTGR is quickly swung up through the swing-up controller. At t = 6 s, the UTGR enters Σb , and the controller switches to (3.152), which stabilizes the UTGR at x = 0. The control time is less than 10 s, and the maximum values of τ2 and τ3 are less than 45 N · m and 15 N · m, respectively. The effectiveness of the presented method is demonstrated. In addition,

Table 3.3 Physical parameters of UTGR Link i m i /kg L i /m i =1 i =2 i =3

1.258 5.686 2.162

0.34 0.29 0.52

L ci /m

Ji /(kg · m2 )

0.17 0.145 0.26

0.0121 0.0398 0.0487

3.3 Stable Control of Three-Link Underactuated Manipulators

95

Fig. 3.14 Simulation results for q and q˙

x = [q T , q˙ T ]T and τ j ( j = 2, 3) change suddenly and slightly at t = 6 s, which is the result of the controller being switched from a nonlinear one to a linear one at this time. In order to prove the practicability of the proposed strategy, the cases in which the UTGR has uncertainties (±5% of their nominal values in Table 3.3) and white noise in the measured q and q˙ (peak values: ±0.1) are simulated. A verification example (Fig. 3.16) shows the results when m 1 , m 2 , J1 , and J2 are 5% smaller than their nominal values, m 3 and J3 are 5% larger than their nominal values, and there is white noise with a peak value of ±0.1. The results show that the proposed strategy is effective even in this case.

96

3 Control of Vertical Underactuated Manipulator

Fig. 3.15 Simulation results for τ j ( j = 2, 3) and E(x)

3.4 Stable Control of n-Link Underactuated Manipulators This section discusses the motion control of the n-link VUM with any one joint beings passive. A reduced order method is proposed to realize its swing-up and balance control. The proposed method consists of two stages. In Stage 1, n−2 controllers are designed to force the angles and angular velocities of n−2 active link to converge to zero, which ensures that the n-link VUM is simplified to a 2-link VUM. Then, the controller of the rest active link is temporarily set as a constant to simplify the controller design. In Stage 2, the state of the simplified system is guaranteed by maintaining the n−2 controller designed in Stage 1. At the same time, the swing-up controller and balance controller are designed for the rest active link. Combined with the n−2 nonlinear controller designed in the Stage 1, the n-link VUM is swung up rapidly and is stabilized effectively at the straight-up position.

3.4.1 Dynamic Model and Analysis Figure 3.17 shows the model of a general n-link VUM with the i 0 th (1  i 0  n) joint being passive. The dynamics of such system is given by M(q)q¨ + C(q, q) ˙ + G(q) = τ,

(3.154)

3.4 Stable Control of n-Link Underactuated Manipulators

Fig. 3.16 Simulation results with uncertainties and measured white noise Fig. 3.17 Model of an n-link VUM with i 0 th joint being passive

97

98

3 Control of Vertical Underactuated Manipulator

where M(q) ∈ Rn×n , C(q, q) ˙ ∈ Rn and G(q) ∈ Rn are the inertia matrix, combination matrix of Coriolis force and centrifugal force and gravity matrix respectively. T  Assuming X = X 1T , X 2T , where X 1 = [x11 , . . . , x1n ]T = [q1 , . . . , qn ]T , X 2 = [x21 , . . . , x2n ]T = [q˙1 , . . . , q˙n ]T , and (3.154) can be written as the state space form:  X˙ 1 = X 2 , (3.155) X˙ 2 = F(X ) + B(X )U, where U = [τ1 , . . . , τi0 −1 , τi0 +1 , . . . , τn ]T ∈ Rn−1 , F(X ) ∈ Rn , and B(X ) ∈ Rn×(n−1) , which are defined as F(X ) = [ f 1 , . . . , f n ]T = M −1 (X ) [−C(X ) − G(X )] , ⎡

b11 · · · b1(i0 −1) b1(i0 +1) · · · ⎢ b21 · · · b2(i0 −1) b1(i0 +1) · · · ⎢ B(X ) = ⎢ . .. .. ⎣ .. . . bn1 · · · bn(i0 −1) bn(i0 +1) · · ·

(3.156)

⎤ b1n ⎤ ⎡ 0(i0 −1)×(n−i0 ) Ii0 −1 b2n ⎥ ⎥ −1 01×(n−i0 ) ⎦ . .. ⎥ = M (X ) ⎣ 01×(i0 −1) . ⎦ 0(n−i0 )×(i0 −1) In−i0 bnn

(3.157) Remark 3.5 For a fully actuated n-link manipulator, the state space equations are 

X˙ 1 = X 2 , X˙ 2 = F(X ) + B f (X )τ,

(3.158)

where τ = [τ1 , . . . , τn ]T ∈ Rn , and ⎡

b11 · · · ⎢ .. B f (X ) = ⎣ .

⎤ b1n .. ⎥ = M −1 (X ). . ⎦

(3.159)

bn1 · · · bnn

Notice that B(X ) in (3.157) is obtained from B f (X ) by removing the i 0 th column. In addition, since M(X ) is symmetric positive definite, B f (X ) in (3.159) is also symmetric positive definite. According to the property of positive definite matrix, all determinants of the principal submatrices of B f (X ) are positive. These are recalled as they will be used in the following subsection. Compared with the two or three-link VUM, the n-link VUM is a kind of higherDOF nonlinear system, which makes its swing-up and balance control more challenging. In order to simplify the control problem, this subsection proposes a reduced order approach to achieve the control objective. The idea of controlling the n-link VUM is to reduce it to a 2-link one first, that is: (1) When i 0 = 1, the manipulator is reduced to be an Acrobot-like system,

3.4 Stable Control of n-Link Underactuated Manipulators

99

(2) When i 0 > 1, the manipulator is reduced to be a Pendubot-like system. Then an effective control strategy is designed for the simplified system to achieve the system control objectives. For easier description, a break-joint v is defined, where v ∈ {2, 3, . . . , n} for i 0 = 1 and v = i 0 for i 0 > 1. Besides Joint 1 and Joint v (one of them is actuated and the other is passive), the rest n − 2 actuated joints are defined as k, where k = 2, . . . , v − 1, v + 1, . . . , n.

(3.160)

First, the corresponding n − 2 active links are driven to stretch out toward their former links in a natural way. That is, the break-joint v separates the n-link VUM into two parts. The angles and angular velocities of the first part links from Link 1 to Link v − 1 are controlled to converge to zero, thus those links are reduced to Composite Link 1. In the same way, the second part links from Link v to Link n are reduced to Composite Link 2. Therefore, the n-link VUM will be reduced to be a 2-link one. Next, some characteristics of B(X ) in (3.155) are introduced. The matrix B(X ) in (3.155) is rewritten as  ˜ ), i 0 = 1, B(X (3.161) B(X ) = ¯ B(X ), i 0 > 1, ˜ ), B(X ¯ ) can be written as the following block matrixes: where B(X ⎡

b12 b22 .. .

· · · b1(v−1) · · · b2(v−1) .. .

b1v b2v .. .

b1(v+1) b2(v+1) .. .

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ b(v−1)2 · · · b(v−1)(v−1) b(v−1)v b(v−1)(v+1) ˜ B(X ) = ⎢ ⎢ bv2 · · · bv(v−1) bvv bv(v+1) ⎢ ⎢ b(v+1)2 · · · b(v+1)(v−1) b(v+1)v b(v+1)(v+1) ⎢ ⎢ . .. .. .. ⎣ .. . . . b1v bnv b1v bn2 · · · ⎡

b11 b21 .. .

b12 b22 .. .

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ b(i0 −1)1 b(i0 −1)2 ¯ B(X ) = ⎢ ⎢ bi 1 bi0 2 0 ⎢ ⎢ b(i +1)1 b(i +1)2 0 ⎢ 0 ⎢ . .. ⎣ .. . bn2 bn1

··· ···

b1(i0 −1) b2(i0 −1) .. .

b1(i0 +1) b2(i0 +1) .. .

· · · b1n · · · b2n .. . ··· ··· ··· .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ b(v−1)n ⎥ ⎥ bvn ⎥ ⎥ b(v+1)n ⎥ ⎥ ⎥ ⎦

· · · bnn

··· ···

b1n b2n .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ · · · b(i0 −1)n ⎥ ⎥ · · · bi0 n ⎥ ⎥ · · · b(i0 +1)n ⎥ ⎥ .. ⎥ . ⎦

· · · b(i0 −1)(i0 −1) b(i0 −1)(i0 +1) · · · bi0 (i0 −1) bi0 (i0 +1) · · · b(i0 +1)(i0 −1) b(i0 +1)(i0 +1) .. .. . . · · · bn(i0 −1) bn(i0 +1) · · ·

bnn

100

3 Control of Vertical Underactuated Manipulator

˜ ), B(X ¯ ) can be further expressed as Then, B(X ⎡ ˜ B11 (X ) ⎢ B˜ 21 (X ) ˜ )=⎢ B(X ⎣ B˜ 31 (X ) B˜ 41 (X )

B˜ 12 (X ) B˜ 22 (X ) B˜ 32 (X ) B˜ 42 (X )

⎤ ⎡ B˜ 13 (X ) B¯ 11 (X ) ⎥ ⎢ ¯ ˜ B23 (X ) ⎥ ¯ ) = ⎢ B21 (X ) , B(X ⎦ ⎣ ˜ B¯ 31 (X ) B33 (X ) ˜ B¯ 41 (X ) B43 (X )

⎤ B¯ 12 (X ) B¯ 22 (X ) ⎥ ⎥. B¯ 32 (X ) ⎦ B¯ 42 (X )

(3.162)

Thus, a submatrix B1 (X ) ∈ R(n−2)×(n−2) is constructed as ⎧  B˜ 21 (X ) B˜ 23 (X ) ⎪ ⎪ , i 0 = 1, ⎨ ˜ B (X ) B˜ 43 (X ) B1 (X ) =  ¯ 41 ⎪ B (X ) ⎪ ⎩ ¯ 22 , i 0 > 1. B42 (X )

(3.163)

Note that, in the case of i 0 = 1, B1 (X ) is obtained by removing the 1st and vth rows, and the 1st and vth columns of B f (X ) in (3.159). It is an (n − 2) × (n − 2) principle submatrix of B f (X ). In other cases, i.e., i 0 > 1, B1 (X ) is obtained by removing the 1st and i 0 th rows, and the same 1st and i 0 th columns of B f (X ). It is also an (n − 2) × (n − 2) principle submatrix of B f (X ). According to the property of B f (X ) mentioned in Remark 3.5, it is known that |B1 (X )| = 0.

(3.164)

which guarantees that no singularity arises when the controllers are designed for the n − 2 actuated links. Finally, U ∈ Rn−1 is divided into U1 ∈ Rn−2 and U2 ∈ R1 , where U1 = ((τk )1×(n−2) )T and  U2 =

τv , i 0 = 1, τ1 , i 0 > 1.

(3.165)

Thus, the whole control procedure is divided into 2 stages, where U1 and U2 are designed to perform the assignments given as follows. (1) For the n − 2 active links given in (3.160), design controllers for U1 and U2 to stretch the links (3.160) out toward their former links in a natural way. By this means, both the angles and angular velocities of these n − 2 links converge to zero. Consequently, the n-link VUM is reduced to an Acrobot-like (i 0 = 1) or Pendubotlike system (i 0 > 1). Specifically, U1 is designed based on n − 2 Lyapunov functions constructed by the angles and angular velocities of n − 2 actuated links, and U2 is set to constant for this stage to simplify the controller design. (2) For the reduced 2-link system, U1 is still employed to hold the states of the n − 2 actuated links. Design a swing-up controller and a balance controller for U2 based on the reduced order system respectively to achieve the control objective. During the above control procedures, Definition 3.4 is given to define the motion space.

3.4 Stable Control of n-Link Underactuated Manipulators

101

Definition 3.4 Let the whole motion space be Σ, and define Stage 2 to be ⎧ ⎨

 x

1k min mod , Σ2 : 2π ⎩ |x2k |  ρ2k ,

  $   mod x1k   ρ1k ,  . −2π 

(3.166)

Then, Stage 1 is Σ1 : Σ − Σ2 .

(3.167)

And Stage 2 is divided into the swing-up area Σ21 and the attractive area Σ22 , which satisfy

Σ22

⎧ ⎪ ⎪ X ∈ Σ2 ,   $ ⎪ ⎪ x11  x11  ⎪ ⎪ , mod ⎨ min mod   ρ11 ,  −2π $  2π :   x11 + x1v ⎪ ⎪ mod x11 + x1v   ρ1v , , min mod ⎪ ⎪   ⎪ 2π −2π ⎪ ⎩ |E(X ) − E 0 |  ρe , Σ21 : Σ2 − Σ22 .

(3.168)

(3.169)

In (3.166)–(3.169), ρ1k , ρ2k , ρ11 , ρ1v , ρe > 0, mod(x/y) is the remainder of x divided by y, and its sign coincides with y, E 0 is the potential energy at the straight-up position.

3.4.2 Controller Design in Stage 1 As is mentioned above, the purpose of Stage 1 is to make the n − 2 actuated links given in (3.160) stretch out toward their former links in a natural way. Based on the n − 2 active links in (3.160), the following n − 2 Lyapunov functions are constructed Vk (X ) =

1 2 1 x + x2 . 2 1k 2 2k

(3.170)

Differentiating (3.170) with respect to time yields V˙k (X ) = x2k [x1k + f k + R (B(X ), k) U ] , where R(B(X ), k) ∈ R1×(n−1) is the kth row of B(X ). If design U as x1k + f k + R (B(X ), k) U = −γk x2k ,

(3.171)

(3.172)

102

3 Control of Vertical Underactuated Manipulator

where γk > 0, then it follows that 2  0. V˙k (X ) = −γk x2k

(3.173)

Defining Xˆ 1 = ((x1k )1×(n−2) )T and Xˆ 2 = ((x2k )1×(n−2) )T , (3.172) is written as: ˆ ) + B1 (X )U1 + B2 (X )U2 = −Γ Xˆ 2 , Xˆ 1 + F(X

(3.174)

ˆ ) = (( f k )1×(n−2) )T , and where F(X  B2 (X ) =

(bkv )(n−2)×1 , i 0 = 1, (bk1 )(n−2)×1 , i 0 > 1.

Γ = diag{γ2 , . . . , γv−1 , γv+1 , . . . , γn }.

(3.175) (3.176)

Since B1 (X ) is a positive definite matrix given in (3.164), so the solution of (3.174) is ˆ ) + B2 (X )U2 ]. (3.177) U1 = −B1−1 (X )[ Xˆ 1 + Γ Xˆ 2 + F(X Note that (3.177) shows a certain relationship between U1 and U2 , which indicates that U1 and U2 are coupling. The control U1 can be designed only when U2 is known in advance. In this case, one feasible treatment for this stage is to assume U2 be constant first. For simplification, set (3.178) U2 = 0. Thus, the controller used to control the n-link VUM in Stage 1 is 

ˆ ) + B2 (X )U2 ], U1 = −B1−1 (X )[ Xˆ 1 + Γ Xˆ 2 + F(X U2 = 0.

(3.179)

Note from (3.173) that, although Vk (X ) monotonically decreases, it cannot guarantee that the states x1k and x2k converge to zero. For example, Vk (X ) may remain unchanged when x2k = 0. Thus, the stability of the system needs to be analyzed in Stage 1 by using LaSalle’s invariance theorem [47]. It is known that Vk (X ) are continuously differentiable, and under the controllers (3.177) and (3.178), they are WCLFs [49]. Substituting (3.179) into (3.155) yields: X˙ = Fa (X ).

(3.180)

Since V˙k (X )  0 in (3.173), Vk (X ) are bounded. Define Φ1 = {X ∈ R2n | Vk (X )  ck },

(3.181)

3.4 Stable Control of n-Link Underactuated Manipulators

103

where ck > 0 for the kth Lyapunov function. Then any solution X of (3.180) starting in Φ1 remains in Φ1 for all t  0. Let Ψ1 be an invariant set of (3.180), which is Ψ1 = {X ∈ Φ1 | V˙k (X ) = 0}.

(3.182)

When V˙k (X ) = 0, x2k = 0. Hence, Xˆ 2 = 0. Thus, according to (3.155), the following equation can be obtained ˆ ) + B(X ˆ )U = 0. F(X (3.183) ˆ ) ∈ R(n−2)×(n−1) is similar to B(X ) in (3.161), in which the 1st and vth where B(X rows are removed. By dividing U into U1 and U2 , (3.183) is rewritten as ˆ ) + B1 (X )U1 + B2 (X )U2 = 0. F(X

(3.184)

Substituting (3.177) into (3.184) yields ˆ ) + B2 (X )U2 ] + B2 (X )U2 = 0. ˆ ) − [ Xˆ 1 + Γ Xˆ 2 + F(X F(X Simplifying (3.185) yields

Xˆ 1 + Γ Xˆ 2 = 0.

(3.185)

(3.186)

Since Xˆ 2 = 0, Xˆ 1 = 0, the largest invariant set for the n-link VUM in Stage 1 can be expressed as (3.187) M1 = {X ∈ Ψ | Xˆ 1 = 0, Xˆ 2 = 0}. According to LaSalle’s invariance theorem [50], every solution X of (3.180) starting in Φ1 approaches to M1 as t → +∞. The above obtained results in this section are summarized by the following theorem. Theorem 3.4 Consider the n-link VUM (3.155). Let M1 be the largest invariant set of the system (3.180). Vk (X ) are WCLFs, Φ1 is a compact closed and bounded set that contains all the initial states of the system (3.180), and Ψ1 is a set with states in Φ1 where V˙k (X ) = 0. If the controller (3.179) is employed, then every solution X of the closed-loop system (3.180) converges to the invariant set M1 defined in (3.187) as t → +∞. Remark 3.6 Note that the invariant set M1 in (3.187) shows the convergence of the states of the n − 2 actuated links. In order to guarantee the system (3.180) is stable in the sense of Lyapunov, an insight into the energy variation of Stage 1 is given now. From (2.99), it is known that ˙ ) = q˙ T τ = Xˆ 2T U1 + q˙a U2 , E(X

(3.188)

104

3 Control of Vertical Underactuated Manipulator



where a=

v, i 0 = 1, 1, i 0 > 1.

(3.189)

Since U2 is assumed to be zero in this stage, and the states of the n − 2 links in ˙ is finally equal to zero, which means (3.160) converge to M1 , E(x) E(x) = λ,

(3.190)

where λ is a constant. Note that λ will be limited, but the exact value of λ cannot be ˙ determined because E(x) in (3.188) may be positive or negative. General speaking, λ is different under different initial state. In most cases, the system is expected to start with small angular velocities or even a static state, which means Xˆ 2 is small or zero. Meanwhile, U2 = 0. Thus, the energy of the first stage will not change unlimitedly or largely, but converge to a constant λ.

3.4.3 Controller Design in Stage 2 From Theorem 3.4, all states of the n − 2 active links in (3.160) converge to zero by using the controller (3.179) as t → +∞. However, to realize the overall control objective, the control strategy is switched to Stage 2 when (3.166) is satisfied. Since ρ1k and ρ2k in (3.166) are defined as very small values, when (3.166) is satisfied, x1k and x2k are close to zero. At this point, the n-link VUM is reduced to be an Acrobot-like or a Pendubot-like system, which are shown in Fig. 3.18. It is noted that the reduced 2-link manipulator will be maintained if the control U1 in (3.177) holds. So in this subsection, it is assumed that (3.177) is still employed

Fig. 3.18 Model of an n-link underactuated manipulator being reduced to a an Acrobot (i 0 = 1), b a Pendubot (i 0 > 1, v = i 0 )

3.4 Stable Control of n-Link Underactuated Manipulators

105

for U1 , then a swing-up controller and a balance controller are designed for U2 respectively based on the reduced system. For the reduced 2-link manipulator (Fig. 3.18), the new links are defined as follows: Composite Link 1 is a single link which starts from Link 1 to Link v − 1, whereas Composite Link 2 is a single link which starts from Link v to Link n. For i = 1, 2, m˜ i , L˜ i , L˜ ci and I˜i are defined as the mass, the length, the center of mass and the moment of inertia of the composite link i, respectively. For v−1 n   an uniform rigid body system, they are given by m˜ 1 = m j , m˜ 2 = m j , L˜ 1 = j=1 v−1  j=1

L j , L˜ 2 =

n  j=v z 4 ]T

j=v

L j , L˜ ci = 0.5 L˜ i , I˜i = 13 m˜ i L˜ 2ci , (i = 1, 2). Thus, by assuming z =

= [x11 , x1v , x21 , x2v ]T = [q1 , qv , q˙1 , q˙v ]T , one can obtain the [z 1 , z 2 , z 3 , following reduced model: ⎧ z˙ 1 ⎪ ⎪ ⎨ z˙ 2 z ˙ ⎪ 3 ⎪ ⎩ z˙ 4

= z3, = z4, = f˜1 (z) + b˜1 (z)U2 , = f˜2 (z) + b˜2 (z)U2 ,

(3.191)

where f˜1 (z), f˜2 (z), b˜1 (z) and b˜2 (z) can be expressed with the new parameters m˜ j , L˜ j , L˜ cj , I˜j ( j = 1, 2) and coordinate z. More details of f˜1 (z), f˜2 (z), b˜1 (z) and b˜2 (z) are shown in Sect. 3.1. Then, the candidate Lyapunov function is constructed Va (z) =

μ1 2 μ2 2 1 2 E + z + β(z)z a+2 , 2 z 2 a 2

(3.192)

where μ1 , μ2 > 0, and a is given by (3.189), β(z) > 0 is a time-varying design parameter, and E z = E(z) − E 0 = E(X ) − E 0 .

(3.193)

Since the states of the n − 2 actuated links are maintained as those in Stage 1 through (3.177), i.e., (3.194) Xˆ 2 = 0. Hence, from (3.188), (3.193) and (3.194), the following equation can be obtained ˙ ) = z a+2 U2 . E˙ z = E(X

(3.195)

˙ ) is dependent on the n − 1 control inputs and the angular Note from (3.188), E(X velocities of the n − 1 active links, which makes the energy-based controller design very complicated. However, from (3.195), through the reduced control in Stage 1, ˙ ) only contains z a+2 and U2 , which greatly simplifies the energy control. E(X

106

3 Control of Vertical Underactuated Manipulator

Hence, from (3.191), (3.192) and (3.195), the time derivative of Vc (z) is & % % & ˙ z a+2 + μ1 E z + β(z)b˜a (z) z a+2 U2 . V˙a (z) = μ2 z a + β(z) f˜a (z) + 0.5z a+2 β(z) (3.196) When (3.197) μ1 E z + β(z)b˜2 (z) = 0, the controller U2 is U2 =

˙ − γc z a+2 −μ2 z a − β(z) f˜a (z) − 0.5z a+2 β(z) , ˜ μ1 E z + β(z)ba (z)

(3.198)

where γc > 0. Clearly, (3.198) ensures that 2  0. V˙a (z) = −γc z a+2

(3.199)

In order to avoid the singularities, β(z) is chosen as β(z) =

η b˜a (z)

,

(3.200)

where η > 2μ1 E 0 > 0. Consequently, μ1 E z + β(z)b˜a (z) > 0.

(3.201)

To guarantee the stability of (3.191) in the swing-up area, one can employ the similar analytical method presented in Sect. 3.1 to verify that the states of (3.191) converge to the largest invariant set M2 = {z ∈ R4 | E z = 0, z a = 0, z a+2 = 0} through the controller (3.198). In this situation, the passive link travels in a periodic circle orbit, which can be derived from (3.190) to be (1) Acrobot (θ1 + θ2 + 2θ3 )z 32 = 2(θ4 + θ5 )g(1 − cos z 1 ),

(3.202)

θ2 z 42 = 2θ5 g(1 − cos z 2 ),

(3.203)

(2) Pendubot

where 

θ1 = m˜ 1 L˜ 2c1 + m˜ 2 L˜ 21 + I˜1 , θ2 = m˜ 2 L˜ 2c2 + I˜2 , θ3 = m˜ 2 L˜ 1 L˜ c2 , θ4 = m˜ 1 L˜ c1 + m˜ 2 L˜ 1 , θ5 = m˜ 2 L˜ c2 ,

(3.204)

3.4 Stable Control of n-Link Underactuated Manipulators

107

Therefore, the controller is designed as ⎧ −1 ˆ ) + B2 (X )U2 ], ⎪ ⎨ U1 = −B1 (X )[ Xˆ 1 + Γ Xˆ 2 + F(X ˜ ˙ − γc z a+2 −μ2 z a − β(z) f a (z) − 0.5z a+2 β(z) ⎪ , ⎩ U2 = μ1 E z + β(z)b˜a (z)

(3.205)

to control the n-link VUM in the swing-up area. Substituting (3.205) into (3.155) obtains the following closed-loop system X˙ = Fb (X ).

(3.206)

Φ = {X ∈ R2n | Vk (X )  ck , Va (z)  ca },

(3.207)

Define where k is given as (3.160), ca > 0 is a constant. Combined with Theorem 3.4 and the stability of (3.191) in the swing-up area, the following stability theorem is presented for the n-link VUM in the swing-up area. Theorem 3.5 Consider the n-link VUM (3.154) in the swing-up area. Let M = {X ∈ R2n | E(X ) = 0, Xˆ 1 = 0, Xˆ 2 = 0, x1a = 0, x2a = 0},

(3.208)

be the largest invariant set of the system (3.206). Φ contains all the initial states of the system (3.206), and Ψ is a set with states in Φ where V˙k (X ) = 0 and V˙a (x) = 0. Vk (X ) and Va (x) are WCLFs. If the controller (3.205) is employed, then every solution X of the closed-loop system (3.206) converges to the invariant set M. That is, the states of all the actuated links converge to zero whereas the passive link travels in a periodic circle orbit given as (3.202) or (3.203). The proof of Theorem 3.5 is similar to that of Theorem 3.4 presented in Sect. 3.1. Thus, it is omitted. A common balance controller is usually designed based on the approximate model by linearizing (3.155) at X = 0. During the linearization, some approximations are taken as follows:  $  ⎧ x   11 ⎪ mod x11   ρ11 , ⎪ , min mod ⎪   ⎪ 2π −2π ⎪ ⎪ ⎪ ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎧ ⎛ i ⎞  ⎞⎫ ⎛ i ⎪ ⎪    ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ x x ⎪ ⎪ ⎪ ⎜ j=1 1 j ⎟  ⎨ ⎜ j=1 1 j ⎟⎪ ⎬ ⎪ ⎪ ⎜ ⎟ ⎟  ⎜  ⎪ ⎪ min mod ⎜ , mod ⎜  ρ1i , ⎟ ⎟  ⎪ ⎪ ⎪ ⎝ 2π ⎠  ⎝ −2π ⎠⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ ⎩ ⎭  

⎪ .. ⎪ ⎪ ⎪ .⎧ ⎪ ⎛ n ⎞  ⎞⎫ ⎛ n ⎪ ⎪     ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x1 j ⎟  x1 j ⎟⎪ ⎪ ⎪ ⎪ ⎨ ⎜ ⎜ ⎬ ⎪ ⎪ j=1 j=1 ⎜ ⎟  ⎟ ⎜ ⎪ ⎪ min mod , mod  ρ1n , ⎜ ⎟ ⎟  ⎜  ⎪ ⎪ ⎪ ⎝ 2π ⎠  ⎝ −2π ⎠⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎩ ⎭ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ |x |  ρ , i = 1, 2, . . . , n. 2i 2i ⎪ ⎩ |E(X ) − E 0 |  ρe .

(3.209)

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3 Control of Vertical Underactuated Manipulator

In (3.209), ρ1i , ρ2i > 0 are small real numbers (i = 1, 2, . . . , n). However, because of the motion complexity of the n-link VUM, it is hard to design the controller to meet the condition (3.209). In this part, an integrated balance method with linear and nonlinear control is proposed. Assuming the states of the reduced 2-link manipulator are guaranteed through the controllers (3.177), (3.191) linearized at z = 0, which is equivalent to X = 0 for(3.155), the variation of z can be calculated as follows z˙ = Az + BU2 .

(3.210)

Based on (3.210), an optimization objective is defined as # J= 0

∞

(z T Qz + RU22 )dt,

(3.211)

where Q > 0 and R > 0. Then, the optimal controller for U2 is U2 = −K z,

(3.212)

where K = R −1 B T P, and P = P T > 0 is the solution to the Riccati equation AT P + P A − P B R −1 B T P + Q = 0.

(3.213)

Combined with (3.177), the balance controller of the n-link VUM is 

ˆ ) + B2 (X )U2 ], U1 = −B1−1 (X )[ Xˆ 1 + Γ Xˆ 2 + F(X U2 = −K z,

(3.214)

Note that (3.177) is a group of nonlinear controllers, while (3.212) is a linear controller. Equation (3.177) guarantee that the n − 2 active links stretch out all the time even when the control U2 switches from (3.198) to (3.212). Thus, the n-link VUM can have a suitable posture and move into the attractive area smoothly. Remark 3.7 According to the division of the motion space given in (3.166)–(3.169), the definition of the attractive area is the combination condition of (3.166) and (3.168), which should be satisfied at the same time. In addition, ρ1k and ρ2k in (3.161) are much smaller than those in (3.209), which means the definition of the attractive area given in Definition 3.4 is more strict than that of (3.209). However, due to the reduced control of Stage 1, the manipulator can still enter the attractive area easily. In other words, the balance controller designed based on the reduced control of Stage 1 is more efficient than simple linear approximate control.

3.4 Stable Control of n-Link Underactuated Manipulators

109

3.4.4 Simulation Results In this subsection, two simulation results of simplifying the 4-link VUM into an Acrobot-like and Pendubot-like system are given to prove the effectiveness of the proposed method. The physical parameters are shown in Table 3.4, and the parameters in (3.166) and (3.169) are ρ1k = 10−5 rad, ρ2k = 10−5 rad/s, ρ11 = ρ1v = π/6 rad, ρe = 0.5 J. (1) Case of Being Reduced to Acrobot-like System The case i 0 = 1 and the break-joint v = 2 are considered. According to (3.160), k = 3, 4, which means that in Stage 1, the 3rd and 4th links are first driven to stretch out toward their former links in a natural way. That is, the links from Link 2 to Link 4 are reduced to Composite Link 2 while Link 1 is Composite Link 1. In such case, the 4-link VUM is reduced to be an Acrobot-like system as shown in Fig. 3.18a. The parameters of (3.172), (3.198), and (3.200) are γ3 = 2.25, γ4 = 1.58, μ1 = 0.28, μ2 = 20, γc = 118, η = 32642, and E 0 = 63.63 J. B1 (x) is invertible, thus the designed controllers for U1 = [τ3 , τ4 ]T are  −1          $   b b x13 γ 0 x23 f b τ3 = − 33 34 + 3 + 3 + 32 τ2 . τ4 b43 b44 x14 f4 b42 0 γ4 x24

(3.215)

The parameters of the reduced Acrobot are m˜ 1 = 1.250 kg, m˜ 2 = 8.968 kg, L˜ 1 = 0.340 m, L˜ 2 = 1.330 m, L˜ c1 = 0.170 m, L˜ c2 = 0.665 m, I˜1 = 0.0121 kg · m2 , I˜2 = 1.3220 kg · m2 , and Q = I4 , R = 0.5 are selected for designing the statefeedback controller (3.212), which is K = [−631.179, −313.731, −183.149, −104.573].

(3.216)

The initial state is X (0) = [π, 0, π/8, π/8, 0, 0, 0, 0]T . The simulation results (Fig. 3.19) show that there exist two switches (t = 14.42 s and t = 19.85 s) in the control process. Before the first switching time, the angles and angular velocities of the 3rd and 4th link are forced toward zero by using the controllers (3.177) and (3.178). At t = 14.42 s, the manipulator is reduced to be an Acrobot-like system, and the first switching condition (3.166) is satisfied. The controller of U2 is switched from (3.178) to (3.198) to increase the energy and stretch out the 2nd link. Meanwhile,

Table 3.4 Physical parameters of a 4-link underactuated manipulator Segment m i /kg L i /m L ci /m Link 1 Link 2 Link 3 Link 4

1.250 5.686 2.162 1.120

0.340 0.290 0.520 0.540

0.170 0.145 0.260 0.270

Ii /(kg · m2 ) 0.0121 0.0398 0.0487 0.0272

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3 Control of Vertical Underactuated Manipulator

Fig. 3.19 Simulation results of a 4-link manipulator with i 0 = 1 being reduced to an Acrobot

the controllers (3.177) are remained to keep the states of the 3rd and 4th link to be zero. At t = 19.85 s, (3.166) and (3.168) for the attractive area are satisfied, and the balance controllers (3.177) and (3.212) take over. Finally, the manipulator is balanced at the straight-up position at about 22.0 s.

3.5 Conclusions

111

(2) Case of Being Reduced to Pendubot-like System Next, the case i 0 = 3 is considered. Then the break-joint v = i 0 = 3, and according to (3.160), k = 2, 4. Thus, in Stage 1, the 2rd and 4th links are first forced to stretch out toward their former links in a natural way, and the links from Link 1 to Link 2 are reduced to Composite Link 1 while the links from Link 3 to Link 4 are reduced to Composite Link 2, as shown in Fig. 3.18b. At this point, the 4-link manipulator is reduced to be a Pendubot-like manipulator. The parameters of (3.172), (3.198), and (3.200) are γ2 = 1.7, γ4 = 0.41, μ1 = 0.05, μ2 = 195, γc = 83, η = 8271, and E 0 = 63.63 J. B1 (x) is invertible, then the designed controllers for U1 = [τ2 , τ4 ]T are  −1          $   b22 b24 x12 γ2 0 x22 f2 b21 τ2 =− + + + τ . (3.217) τ4 b42 b44 x14 x24 f4 b41 1 0 γ4 The parameters of the reduced Pendubot are: m˜ 1 = 6.936 kg, m˜ 2 = 3.282 kg, L˜ 1 = 0.63 m, L˜ 2 = 1.06 m, L˜ c1 = 0.315 m, L˜ c2 = 0.530 m, I˜1 = 0.2294 kg · m2 , I˜2 = 0.3073 kg · m2 . Q = I4 , R = 0.5 are selected for designing the state-feedback controller (3.212), and the state-feedback gain K = [−337.011, −337.034, −128.512, −82.206].

(3.218)

The initial condition is X (0) = [π, π/8, 0, π/8, 0, 0, 0, 0]T . Simulation results (Fig. 3.20) show that there also exist two switches (t = 13.32 s and t = 14.91 s)in the whole control process. Before t = 13.32 s, the manipulator is controlled through (3.177) and (3.178), and the angles and angular velocities of the 2nd and 4th links are forced toward zero. Up to t = 13.32 s, the manipulator is reduced to a Pendubot-like system, and the controller of U2 is switched from (3.178) to (3.198). At t = 14.91 s, the balance controllers (3.177) and (3.212) take over, and the 4-link manipulator is finally balanced at the straight-up position at about 16.0 s. In addition, from the energy curves of the two cases, the energy before the first switching time (t = 14.42 for i 0 = 1, t = 13.32 for i 0 > 1) has no large sudden change, but converges to a constant. That verifies the energy analysis in Remark 3.6.

3.5 Conclusions This chapter discusses the motion control of the VUMs, and proposes different control strategies to achieve their control objective. First, a unified partition control method is presented for two-link vertical underactuated manipulator to realize the stable control of the system by switching the controller. Then, in order to avoid designing multiple controllers, the trajectory planning method is introduced into the control strategy design, and the system position control target is realized by the trajectory tracking method. Next, the partition control method is applied to the UTGR, and the control

112

3 Control of Vertical Underactuated Manipulator

Fig. 3.20 Simulation results of a 4-link manipulator with i 0 = 3 being reduced to a Pendubot (a: The energy, b: The angle of Joint 1, c: The torque of Joint 2, d: The angle of Joint 2, e: The torque of Joint 3, f: The angle of Joint 3, g: The torque of Joint 4, h: The angle of Joint 4)

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objective is realized. Singularities in the swing-up area is avoided by adjusting the controller parameters. For an n-link underactuated manipulator with any one of its joints being passive, a reduced order approach is proposed to achieve the swing-up control and balance control. Simulation results demonstrate the effectiveness of the proposed control methods.

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Chapter 4

Control of Planar Underactuated Manipulator with a Passive First Joint

Different from the VUMs discussed in the last section, a planar underactuated manipulator (PUM) moves in a horizontal plane without the gravity. Both space manipulators [1] and underwater vehicles [2] belong to this kind of system. The study on this kind of system is meaningful for the aeronautics and astronautics engineering and deep-sea detecting engineering. For such a system, since the linear approximation model of its equilibrium point is uncontrollable [3–5], control methods for the VUM are not suitable for the PUM. This chapter mainly discusses the planar n-link (n  2) manipulators with a passive first joint (i.e., planar PAn−1 (n  2) manipulator ) [6]. Such planar manipulators can be divided into two kinds. One is the planar two-link underactuated manipulator (i.e., planar Acrobot ), which is a holonomic system [7]. The other is the planar manipulators with three or more links, which can be classified as the first-order nonholonomic system [8]. The motion-state constraints and stable control methods of these two kinds of manipulators are different. This chapter will also provide the motion-state constraint analysis and the respective control methods in details.

4.1 Motion-State Constraint Analysis Figure 4.1 shows the physical structure model of the planar PAn−1 manipulator. For the ith link (i = 1, 2, . . . , n), m i is the mass, L i is the length, li is the distance from its joint to its center of mass, qi is the angle of the ith joint, τi is the driving torque applied to the ith link, and Ji is the moment of inertia of the ith link about its centroid. (x, y) is the position of the endpoint. The dynamic equation of the system is M (q) q¨ + H (q, q) ˙ = τ,

© Science Press 2023 J. Wu et al., Control of Underactuated Manipulators, https://doi.org/10.1007/978-981-99-0890-5_4

(4.1)

117

118

4 Control of Planar Underactuated Manipulator with a Passive First Joint

Fig. 4.1 Physical structure of planar n-link manipulator with a passive first joint

where q = [q1 , q2 , . . . , qn ]T ∈ Rn×1 is the angle vector, τ = [0, τ2 , . . . , τn ]T ∈ Rn×1 is the input torque vector, M(q) ∈ Rn×n is  a symmetric positive definite matrix and H (q, q) ˙ = M˙ (q) q˙ − 0.5∂ q˙ T M (q) q˙ ∂q ∈ Rn×1 is the combination of Coriolis and the centrifugal forces. The dynamic equations (4.1) can be divided into two parts: the passive part and the active part, ˙ = 0, M p (q) q¨ + H p (q, q) Ma (q) q¨ + Ha (q, q) ˙ = τa ,

(4.2a) (4.2b)

where τa = [τ2 , τ3 , . . . , τn ]T ∈ R(n−1)×1 , M p (q) = [m 11 , m 12 , . . . , m 1n ] ∈ R1×n , ⎤ ⎡ m 21 · · · m 2n   1 ∂ ⎢ .. ⎥ ∈ R(n−1)×n and q˙ T M (q) q˙ , Ma (q) = ⎣ ... H p (q, q) ˙ = M˙ p (q) q˙ − . ⎦ 2 ∂q1 m n1 · · · m nn  1 ∂  T (n−1)×1 ˙ q˙ M (q) q˙ ∈ R , qa = [q2 , q3 , . . . , qn ]T . ˙ = Ma (q) q˙ − Ha (q, q) 2 ∂qa

The geometric constraint between all links’ angles and the end-point position is obtained according to the Fig. 4.2. n  ⎧ n   ⎪ ⎪ qi , X¯ i = −L 1 sin q1 − · · · − L n sin ⎨x = i=1 i=1   n n   ⎪ ⎪ qi , Y¯i = L 1 cos q1 − · · · − L n cos ⎩y = i=1

(4.3)

i=1

where, X¯ 1 , X¯ 2 , . . . , X¯ n are the projections of the 1th, 2nd, . . ., nth links on the x-axis, and Y¯1 , Y¯2 , . . . , Y¯n are their projections on the y-axis.

4.1 Motion-State Constraint Analysis

119

Fig. 4.2 Structural diagram of planar n-link system

4.1.1 Integrability The integrable conditions [7] for the underactuated manipulator, including partially integrable conditions and complete integrable conditions, are given as follows. The partially integrable conditions are listed as follows. (1) The gravitational torque is constant. (2) A variable related to the underactuated joint does not appear in M. If an underactuated manipulator meets the partially integrable conditions, then it has the first-order nonholonomic constraint, that is the motion-state constraint on the angular velocities. The complete integrable conditions are listed as follows. (1) The system is partially integrable. (2) The null space distribution of the underactuated inertia matrix is involutive. If an underactuated manipulator meets the complete integrable conditions, then it has the holonomic constraint, that is the motion-state constraint on the angles.

4.1.2 Motion-State Constraint on Angular Velocities According to the dynamic equation (4.2a), the constraint on the passive joint is m 11 q¨1 + m 12 q¨2 + · · · + m 1n q¨n + H1 = 0.

(4.4)

For PUM, the term for gravity in the constraint equation (4.4) is zero, and the variable for the passive joint, q1 , does not appear in the inertia matrix, M, so H1 = m˙ 11 q˙1 +m˙ 12 q˙2 + · · · + m˙ 1n q˙n ,

(4.5)

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

where m˙ 11 , m˙ 12 , and m˙ 1n are time derivatives of m 11 , m 12 , and m 1n respectively. According to the integrable conditions, (4.4) is partially integrable. Therefore, the following state constraint on the angular velocities can be derived from (4.4),

=



t



0 t

(m 11 q¨1 + m 12 q¨2 + · · · + m 1n q¨n + H1 )dt  t  t (m 11 q¨1 + m˙ 11 q˙1 )dt + (m 12 q¨2 + m˙ 12 q˙2 )dt + · · · + (m 1n q¨n + m˙ 1n q˙n )dt.

0

0

0

(4.6)

Integrating (4.6) from 0 to t yields (m 11 q˙1 + m 12 q˙2 + · · · + m 1n q˙n )|t0 = 0.

(4.7)

m 11 q˙1 + m 12 q˙2 + · · · + m 1n q˙n − n 1 = 0,

(4.8)

n 1 = f (q(0))q(0). ˙

(4.9)

Rewrite (4.7) as

where

Equation (4.8) is the state constraint on the angular velocities of the planar PAn−1 system. The system usually starts from a stationary position, so it can be assumed that the initial angular velocities are all zero, that is q(0) ˙ = 0.

(4.10)

Combining (4.9) and (4.10), n 1 = 0. Thus, the state constraint on the angular velocities becomes (4.11) m 11 q˙1 + m 12 q˙2 + · · · + m 1n q˙n = 0, that is M p q˙ = 0.

(4.12)

Note that, according to (4.11), when the active links of the planar PAn−1 (n  2) manipulator are static, the passive link must also be static.

4.1.3 Motion-State Constraint on Angles In the above subsection, the motion-state constraint on angular velocities of the planar PAn−1 manipulator has been analyzed and obtained. Next, the motion-state constraint on angles is analyzed taking the planar Acrobot and the planar three-link manipulator (i.e., planar PAA manipulator) as examples.

4.1 Motion-State Constraint Analysis

121

(1) Planar Acrobot For the planar Acrobot, the motion-state constraint on angular velocities is m 11 (q)q˙1 + m 12 (q)q˙2 = 0.

(4.13)

m 11 (q) = a1 + a2 + 2a3 cos q2 , m 12 (q) = a2 + a3 cos q2 ,

(4.14)

where in which ai (i = 1, 2, 3) are the structural parameters and their specific expressions are given in (2.17). It is easy to check that the null space distribution of the two dimensional matrix M p = [m 11 , m 12 ] is involutive. According to the integrable conditions showed in Sect. 4.1.1, (4.13) is integrable. That means the motion-state constraint of the planar Acrobot on angles can be derived by integration. From (4.13) and (4.14), one gets q˙1 = −

(a2 + a3 cos q2 ) q˙2 . a1 + a2 + 2a3 cos q2

(4.15)

Since the inertia matrix is positive, it is not difficult to show that a1 + a2 + 2a3 cos q2 > 0. Integrating (4.15) from 0 to t yields  q1 (t) − q1 (0) = −

q2 (t)

q2 (0)

a2 + a3 cos q2 dq2 . a1 + a2 + 2a3 cos q2

(4.16)

Note that there exists a constant term on the right side of the above equation if |q2 (t) − q2 (0)| > 2π. To calculate it precisely, assume that q2 (0) ∈ [−π, π] , q2 (t) ∈ [−π + 2kπ, π + 2kπ] ,

(4.17)

where k ∈ Z is an integer. When k  0, rewrite (4.16) as  q1 (t) − q1 (0) = −

π

q2 (0) 2π



a2 + a3 cos q2 dq2 a1 + a2 + 2a3 cos q2

a2 + a3 cos q2 dq2 a + a2 + 2a3 cos q2 1 π  q2 (t) a2 + a3 cos q2 − ··· − dq2 . −π+2kπ a1 + a2 + 2a3 cos q2 −

(4.18)

122

4 Control of Planar Underactuated Manipulator with a Passive First Joint

Defining μ = tan

q2 , one has 2 cos q2 =

1 − μ2 2dμ , dq2 = . 1 + μ2 1 + μ2

(4.19)

Substituting (4.19) into (4.18) yields 

+∞

q1 (t) − q1 (0) = −  −

tan

[ f 1 (μ) + f 2 (μ)]dμ

q2 (0) 2

+∞ −∞

− ··· −

[ f 1 (μ) + f 2 (μ)]dμ 

tan

−∞

q2 (t) 2

[ f 1 (μ) + f 2 (μ)]dμ,

(4.20)

where f 1 (μ) =

a2 − a1 1 , f 2 (μ) = . a1 + a2 + 2a3 + (a1 + a2 + 2a3 )μ2 1 + μ2

(4.21)

Let F1 (μ) and F2 (μ) be the primitive functions of f 1 (μ) and f 2 (μ), respectively. Integrating (4.21) yields F1 (μ) = γ g(μ) + C1 ,

(4.22)

F2 (μ) = arctan(μ) + C2 ,

(4.23)

where C1 and C2 are constants, and γ =



a2 − a1 (a1 + a2 )2 − 4a32

, g(μ) = arctan(ρμ), ρ =

a1 + a2 − 2a3 . a1 + a2 + 2a3

(4.24)

Note that a1 + a2 + 2a3 cos q2 > 0, (a1 + a2 )2 − 4a32 > 0, and ρ > 0. Calculating (4.20) and incorporating (4.21)–(4.23)and (4.24) into it yield   q (t) tan 22 q1 (t) − q1 (0) = − F1 (μ)|+∞q2 (0) + F1 (μ)|+∞ + · · · + F (μ)| 1 −∞ −∞ tan 2   q (t) tan 22 − F2 (μ)|+∞q2 (0) + F2 (μ)|+∞ −∞ + · · · + F2 (μ)|−∞ tan 2       q2 (0) q2 (0) q2 (t) q2 (t) − g tan + kπ + =− − γ g tan . (4.25) 2 2 2 2

On the other hand, when k < 0, let β(0) = −q2 (0), β(t) = −q2 (t).

(4.26)

4.1 Motion-State Constraint Analysis

123

Then, β(0) ∈ [−π, π], β(t) ∈ [−π − 2kπ, π − 2kπ]. (4.16) can be rewritten as  q1 (t) − q1 (0) =

β(t)

β(0)

a2 + a3 cos β dβ. a1 + a2 + 2a3 cos β

(4.27)

Following the same line as in (4.18)–(4.25), calculate (4.27) and obtain q1 (t) − q1 (0) =

      β(0) β(0) β(t) β(t) − g tan − kπ − + γ g tan . 2 2 2 2

(4.28)

That is, q1 (t) − q1 (0) = −

      q2 (t) q2 (t) q2 (0) q2 (0) − γ g tan − g tan + kπ + . (4.29) 2 2 2 2

Equation (4.25) and (4.29) show that, for k ∈ Z, the constraint on the angles is q1 (t) = −

      q2 (t) q2 (t) q2 (0) − γ g tan − g tan + kπ + η2 , 2 2 2

(4.30)

where γ and η2 (= q1 (0) + q2 (0)/2) are constants related to the system parameters and the initial angles. (2) Planar PAA Manipulator For the planar PAA manipulator, the motion-state constraint on angular velocities is m 11 q˙1 + m 12 q˙2 + m 13 q˙3 = 0,

(4.31)

M p q˙ = 0.

(4.32)

that is where M p = [m 11 , m 12 , m 13 ] and their specific expressions are given in (2.31). Next, the integrability of (4.32) will be analyzed according to the integrable conditions showed in Sect. 4.1.1. The null space distribution of M p is chosen to be Δ = span{θ1 , θ2 },

(4.33)

where θ1 = [−m 12 , m 11 , 0]T , θ2 = [−m 13 , 0, m 11 ]T . If matrix A = [θ1 , θ2 , [θ1 , θ2 ]] does not have full rank, then the Lie bracket [θ1 , θ2 ] ∈ Δ, and the null space distribution Δ is involutive. Calculating the Lie bracket [θ1 , θ2 ] gives ⎡ ⎤ −2a6 sin q3 ∂θ1 ∂θ2 θ1 − T θ2 = ⎣ 2a5 sin(q2 + q3 ) + 2a6 sin q3 ⎦ M11 . [θ1 , θ2 ] = ∂q T ∂q −2a sin q − 2a sin(q + q ) 3

2

5

2

3

(4.34)

124

4 Control of Planar Underactuated Manipulator with a Passive First Joint

The determinant of matrix A is det A = (m 11 )2 z(q2 , q3 ),

(4.35)

where z(q2 , q3 ) = M11 (−2a6 sin q3 ) + M12 [2a5 sin(q2 + q3 ) + 2a6 sin q3 ] −M13 [2a3 sin q2 + 2a5 sin(q2 + q3 )] .

(4.36)

Substituting m 11 , m 12 and m 13 into (4.36) yields z(q2 , q3 ) = (2a5 a6 − 2a3 a4 ) sin q2 + (2a3 a5 − 2a1 a6 ) sin q3 + (2a2 a5 − 2a3 a6 ) sin(q2 + q3 ).

(4.37)

If there is a state vector of the system angles that makes the determinant of matrix A non-zero, then the null space distribution, Δ, is not involutive. Assume that q2 = q3 = π/2, and (4.37) can be rewritten as z(q2 , q3 ) = (2a5 a6 − 2a3 a4 ) + (2a3 a5 − 2a1 a6 ).

(4.38)

From (2.31), one finds that a5 a6 − a3 a4 < 0, a3 a5 − a1 a6 < 0.

(4.39)

z(q2 , q3 ) < 0.

(4.40)

Thus, According to (4.40) and m 211 > 0, the determinant of matrix A is less than zero, and / Δ, and the null space distribution, Δ, matrix A has full rank. Therefore, [θ1 , θ2 ] ∈ is not involutive. Thus, the constraint equation (4.31) is not completely integrable according to the integrable conditions. Similarly, it can be derived that the planar PAn−1 (n > 2) manipulators with a passive first joint have only motion-state constraint on the angular velocities and have no motion-state constraint on the angles.

4.2 Motion-State Constraint-Based Control of Planar Acrobot This section presents a novel control strategy for stabilizing the planar Acrobot in a horizontal plane at a target position [9]. First, it is shown that the positioning control is implemented by using the angular constraint to control the active joint. Next, the target position is calculated by formulating the problem as a motion optimization

4.2 Motion-State Constraint-Based Control of Planar Acrobot

125

problem, and the algorithm that performs such calculations is presented. Then, a positioning controller based on a Lyapunov function is designed to achieve the control objective.

4.2.1 Motion Characteristic Analysis Equation (4.30) gives the relationship between the angles at a time t1 q1 (t1 ) = −

      q2 (t1 ) q2 (0) q2 (t1 ) − γ g tan − g tan + k(t1 )π + η2 , (4.41) 2 2 2

where k(t1 ) is determined by q2 (t1 ). Let t2 be the time after t1 at which the second link completes one rotation. Then, q2 (t2 ) = q2 (t1 ) + 2π, and the relationship between q1 (t2 ) and q2 (t2 ) is q1 (t2 ) = −

      q2 (0) q2 (t1 ) + 2π q2 (t1 ) + 2π + η2 − γ g tan − g tan + [k(t1 ) + 1]π . 2 2 2

(4.42) From (4.41) and (4.42), the angular displacement of the first link during the time period t2 − t1 is       q2 (t1 ) + 2π q2 (t1 ) ε = q1 (t2 ) − q1 (t1 ) = −π − γ g tan − g tan +π 2 2 (4.43) for a complete rotation of the second link. From the fact that tan(x + π) = tan(x) and from (4.24), one has     q2 (t1 ) + 2π q2 (t1 ) g tan = g tan . 2 2 So, (4.43) implies ε = − (γ + 1) π.

(4.44)

This shows that the change in q1 (t) is a fixed value when q2 (t) makes a complete rotation. Suppose that the target angles q1d , q2d ∈ [−π, π]. To achieve the target position, due to periodicity of angular movement, the final angles q1∞ and q2∞ have to satisfy 

where n 1 , n 2 ∈ Z.

q1∞ = q1d + 2n 1 π, q2∞ = q2d + 2n 2 π,

(4.45)

126

4 Control of Planar Underactuated Manipulator with a Passive First Joint

Based on the characteristics discussed above, the positioning control can be performed with two steps to achieve the target position (4.45). In the first step, set q2 (t) to q2d to make q1 (t) equal to q1∞ . In the second step, drive q1 (t) to the target value, q1∞ , by changing q2 (t) by 2π rad for each adjustment so as to keep q2 (t) at q2d at the end of each adjustment. This guarantees precise control of q1 (t) through a succession of fixed angular displacements, (4.44). This strategy stabilizes the planar Acrobot at the target position. Let t1 f be the time at the end of the first step. At this time, q2 (t1 f ) = q2d , and q1 (t1 f ) is obtained from (4.30)    q2d  q2 (0) q2d q2 (0) + q1 (0) + − γ g tan − γ g tan . q1 (t1 f ) = − 2 2 2 2

(4.46)

To make q1 (t) reach the target value q1∞ , the number of the round of rotations of q2 (t) in the second step, n 2 , has to satisfy q1 (t1 f ) + n 2 ε = q1∞ ,

(4.47)

where n 2 ε is the change in q1 (t) in the second step. Equation (4.47) shows that the first link is adjusted to the target angle q1∞ , and the angle of the second link has to be q2∞ = q2d + 2n 2 π at the end of the second step. This indicates that the system finally achieves the target position, (4.45). Incorporating (4.45) and (4.46) into (4.47) yields   q2 (0) − q2d (γ + 1) n 2 + 2n 1 π = q1 (0) − q1d + 2     q2 (0) q2d  − g tan . − γ g tan 2 2

(4.48)

This shows that a target position is achievable as long as the integers n 1 and n 2 in (4.48) exist. Therefore, the problem of reaching the target position is equivalent to the problem of finding an integer solution to (4.48). In fact, these two steps can be combined and the positioning control can be performed in just one step because there exists an n 1 that satisfies (4.48) for a given n 2 , and these n 1 and n 2 correspond to q1∞ and q2∞ (4.45) of the target position.

4.2.2 Motion Optimization This subsection discusses the strategy to find an optimal solution (n 1 , n 2 ) to (4.48) that corresponds to an optimal trajectory. Note that n 1 and n 2 are finite integers. So, an enumeration method can be used in a prescribed range.

4.2 Motion-State Constraint-Based Control of Planar Acrobot

127

Assume that n 1 and n 2 are in the ranges of n 1 min  n 1  n 1 max , n 2 min  n 2  n 2 max ,

(4.49)

where n i min and n i max are the minimum and maximum of n i (i = 1, 2), respectively. The control error is   q2d q2 (0) e(n 1 , n 2 ) = (γ + 1)n 2 + 2n 1 π + q1d + − q1 (0) − 2 2      q2 (0) q2d − g tan . + γ g tan 2 2

(4.50)

It is practical to set a bound on the control precision, em2 . Then n 1 and n 2 are obtained by solving the optimization problem min |n 2 |

s.t. e2 (n 1 , n 2 )  em2 under (4.59).

(4.51)

The optimization algorithm is shown in Algorithm 4.1. Algorithm 4.1 Calculation of n 1 and n 2 1: for n 2 = n 2 min : n 2 max do 2: for n 1 = n 1 min : n 1 max do 3: calculate e2 2 then 4: if e2  em 5: save n 1 and n 2 6: break 7: end if 8: end for 9: end for

4.2.3 Controller Design The single actuator in the second joint can be used to carry out positioning control under the conditions, (4.13) and (4.30) for the two links. This subsection provides a strategy to use the Lyapunov function method to design a controller for this purpose. Definition 4.1 (Largest Invariant Set [10]) Assume that there exists a Lyapunov function, V (x), for the system x˙ = f (x). (4.52) V (x) → ∞ when x → ∞. Let Ψ = {x P |V˙ (x) = 0}. If there is a Ω ⊂ Ψ for which x(t) → Ω when t → ∞, then the set Ω is called the LIS of the system (4.52).

128

4 Control of Planar Underactuated Manipulator with a Passive First Joint

Let x = [x1 , x2 , x3 , x4 ]T = [q1 , q2 , q˙1 , q˙2 ]T . The system (4.2) can be rewritten as ⎧ x˙1 ⎪ ⎪ ⎨ x˙2 x˙3 ⎪ ⎪ ⎩ x˙4

= = = =

x3 , x4 , A1 (x) + B1 (x)τ2 , A2 (x) + B2 (x)τ2 ,

(4.53)

where the nonlinear functions Ai (x) and Bi (x) (i = 1, 2) are ⎧ −1       ⎪ 1 m 12 H2 − m 22 H1 −H1 ⎪ A1 (x) = m 11 (q) m 12 (q) ⎪ = , ⎨ A (x) −H2 m 21 (q) m 22 (q) Δ m 12 H1 − m 11 H2 2 (4.54)  −1       ⎪ B1 (x) 1 −m 12 0 m 11 (q) m 12 (q) ⎪ ⎪ , = = ⎩ B2 (x) m 21 (q) m 22 (q) 1 Δ m 11 where Δ = m 11 m 22 − m 212 . The Lyapunov function candidate is chosen as V (x) =

1 1 (x2 − q2∞ )2 + x4 2 . 2 2

(4.55)

The derivative of V (x) with respect to time is V˙ (x) = x4 [x2 − q2∞ + A2 (x) + B2 (x)τ2 ] .

(4.56)

Choose the control law to be τ2 =

−x2 + q2∞ − A2 (x) − λx4 , B2 (x)

(4.57)

where λ > 0 is a constant. According to (4.54), B2 (x) > 0. There does not exist any singular points for the control law, (4.57). Substituting (4.57) into (4.56) can get V˙ (x) = −λx4 2  0.

(4.58)

Note that V˙ (x) is negative semi-definite and V˙ (x) = 0 only when x4 = 0. The LIS is used to determine the final motion state of the system (4.53) for the control law (4.57). For the system (4.53) under the control law (4.57), the largest invariant set is Ω : (x2 , x4 ) = (q2∞ , 0).

(4.59)

Therefore, only the convergence of the system when V˙ (x) = 0 needs to be discussed. If V˙ (x) = 0, then V (x) is a constant, and x4 = 0 according to (4.58). That means that x2 is a constant. There are two cases in which x2 is a constant. (a) x2 = q2∞ and (b) x2 = q2∞ . If x2 = q2∞ and V˙ (x) = 0, then x4 = x˙2 = 0 from (4.53), which is consistent with Ω.

4.2 Motion-State Constraint-Based Control of Planar Acrobot

129

Now, by reduction to absurdity, x2 = q2∞ is true. Assume that the system is stabilized at x2 = q¯2∞ = q2∞ and x4 = 0. It follows from the constraints on the angular velocities that x3 = 0. Meanwhile, according to (2.17) and (4.54), A2 (x) = 0. Thus, (4.57) can be rewritten as τ2 =

−q¯2∞ + q2∞ . B2 (x)

(4.60)

It can be proved that τ2 = 0 because B2 (x)|x2 =q¯2∞ is a positive constant and q¯2∞ = q2∞ . So, x˙4 = 0 from (4.53). This contradicts the assumption that x4 = 0. As a result, x2 can only remain at x2 = q2∞ . It is clear from the above proof process that the system state continuously converges to the LIS, Ω, under the control law (4.57). This achieves the control objective. (4.13) and (4.30) also show true.

4.2.4 Simulation Results A numerical example is used to demonstrate the validity of the control strategy. The parameters of the planar Acrobot are 

m 1 = m 2 = 1.0 kg, l1 = l2 = 1.0 m, lc1 = lc2 = 0.5 m, I1 = I2 = 0.083 kg · m2 .

(4.61)

The initial and the target states are chosen to be 

[q1 (0), q2 (0), q˙1 (0), q˙2 (0)] = [0, 0, 0, 0] , [q1d , q2d , q˙1d , q˙2d ] = [−4.284 rad, 1.54 rad, 0, 0] .

(4.62)

From (4.50) and (4.62), e(n 1 , n 2 ) is e(n 1 , n 2 ) = 0.7839n 2 + 6.2832n 1 + 4.7153.

(4.63)

Set the ranges of n 1 and n 2 to be n 1 min = n 2 min = −40, n 1 max = n 2 max = 40.

(4.64)

And choose em2 = 0.0082 . The optimal solution was found by Algorithm 4.1 to be n 1opt = −1, n 2opt = 2.

(4.65)

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

Fig. 4.3 Simulation results for (4.67)

So, the target position is 

q1 = q1d + 2n 1opt π = −1.999 rad, q2∞ = q2d + 2n 2opt π = 14.106 rad.

(4.66)

And the control objective is 

q˙2 = 0 rad/s, q2∞ = q2d + 2n 2opt π = 14.106 rad.

(4.67)

In addition, select λ = 1.8 and implement the control law (4.57). The simulation results are shown in Fig. 4.3. It can be seen that both the speeds and the positions converge to their target values very smoothly, and the links were stabilized at (4.66). Also, the control torque was less than 3 N·m.

4.3 Stable Control of Planar Three-Link PAA Manipulator Since the state constraint on the angular velocities of the planar PAA manipulator (4.11) is unintegrable, the constraint on angles cannot be immediately obtained. But if the angle of one of the active links is kept constant (that is, its angular velocity is zero), the manipulator can be regarded as a planar Acrobot whose constraint on angles can be obtained directly by using quadrature. Based on such a consideration, this section presents a stable control strategy to realize the positioning of the planar PAA manipulator [11, 12].

4.3 Stable Control of Planar Three-Link PAA Manipulator

131

Control of the manipulator is divided into two stages. In each stage, by keeping the angle of one of the active links be a constant, the system is reduced to a planar Acrobot. Due to the constraint on angles of the planar Acrobot, the passive link can be stabilized at an arbitrary position through control of the other active link. Then, the PSO is used to optimize the target angles associated with the target position. Next, a controller for each stage is designed to enable the control objective to be reached.

4.3.1 Motion Strategy of Angles to Target Values The motion sequence is defined to be (i, j)(i, j = 2, 3). This means that, in the first stage, the angle of one active link, qi (i = 2 or 3), is adjusted to the target value, qid , while the angle of the other active link, q j ( j = 3 or 2), remains constant at the initial value, q j (0). In the second stage, the angle of the link that was changed in stage one, namely qi (i = 3 or 2), remains constant at the target value, qid , while the angle of the other link, q j ( j = 2 or 3), is adjusted to its target value q jd . In addition, the first constraints are defined to be the motion-state constraints on angles between the passive link and each active link, and the second constraint is defined to be the constraint between all the angles and the control objective. The second constraint has been given in (4.3), that is 

x = −L 1 sin q1 − L 2 sin(q1 + q2 ) − L 3 sin(q1 + q2 + q3 ), y = L 1 cos q1 + L 2 cos(q1 + q2 ) + L 3 cos(q1 + q2 + q3 ).

(4.68)

Next, the first constraints are calculated. If the angle of one of the active links is kept constant, the system can be regarded as a planar Acrobot, and the constraint between the other active link and the passive link can be obtained by using quadrature. Below, this subsection illustrates how to use the integral function int() to obtain the constraint for two cases: q1 ∼q2 and q1 ∼q3 . For the first case (q1 ∼q2 ), assume that q3 has a constant value of q3c (which equals either q3 (0) or q3d ). Thus, q˙3 is zero, and the constraint between the angles q1 and q2 can be obtained by using quadrature. Since q3 = q3c and q˙3 = 0, (4.11) can be rewritten as (4.69) m 11 q˙1 + m 12 q˙2 = 0. Integrating (4.69) from 0 to t yields q1 (t) −

η q1 (0)

 =

q2 (t)

q2 (0)

 (−m 12 m 11 )dq2 ,

(4.70)

where η = 1 or 2, q11 (0) is the initial value of q1 in the first stage (q11 (0) = q1 (0)), and q12 (0) is the initial value of q1 in the second stage. The quadrature of the expression (4.70) can be obtained by employing the function int() in MATLAB.

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

Thus, the constraint between the angles q1 and q2 is q1 −

η q1 (0)

−2a6 cos q3c + a1 − a2 − a4 =− √ D1 q2 − − f 1 (q2 (0)), 2

    q2 + π arctan E 1 + π 2π (4.71)

where D1 = (−8a3 a5 + 4a2 a6 + 4a1 a6 + 4a4 a6 ) cos q3c + a12 + a22 + 4a62 cos2 q3c +a42 − 4a52 − 4a32 + 2a1 a2 + 2a1 a4 + 2a2 a4 , E 1 = − √GD1 , 1 −2a6 cos q3c + a1 − a2 − a4 q2 f 1 (q2 ) = − − arctan E 1 , √ 2 D1 G 1 = −(a1 + a2 + a4 + 2a6 cos q3c − 2a5 cos q3c − 2a3 ) tan q22 + 2a5 sin q3c . (4.72) For the second case (q1 ∼q3 ), assume that q2 has a constant value of q2c (which is either q2 (0) or q2d ). Thus, q˙2 is zero, and the constraint between the angles q1 and q3 can be obtained. Then, (4.11) can be rewritten as M11 q˙1 + M13 q˙3 = 0.

(4.73)

Integrating (4.73) from 0 to t yields q1 (t) −

η q1 (0)

 =

q3 (t)

q3 (0)

 (−M13 M11 )dq3 .

(4.74)

Employing the function int() in MATLAB obtains the constraint between the angles q1 and q3 , that is η

q1 − q1 (0) = −

−2a3 cos q2c + a1 + a2 − a4 √ D2 q3 − − f 2 (q3 (0)), 2



 arctan E 2 +

  q3 + π π 2π (4.75)

where D2 = (4a2 a3 + 4a1 a3 + 4a3 a4 − 8a5 a6 ) cos q2c + a12 + a22 + 4a32 cos2 q2c + a42 − 4a52 − 4a62 + 2a1 a2 + 2a1 a4 + 2a2 a4 , G2 E2 = − √ , D2 q3 2a3 cos q2c + a1 + a2 − a4 arctan E 2 , f 2 (q3 ) = − − √ 2 D2 G 2 = −(a1 + a2 + a4 + 2a3 cos q2c − 2a5 cos q2c − 2a6 ) tan q23 + 2a5 sin q2c . (4.76)

4.3 Stable Control of Planar Three-Link PAA Manipulator

133

The first constraints between the passive link and each active link are (4.71) and (4.75). So, the control of the active links can make the angle of the passive link converge to the target value based on (4.71) and (4.75) and also make its angular velocity converge to zero based on (4.11).

4.3.2 PSO-Based Target Angle Optimization The PSO is an intelligent algorithm that makes a particle swarm move to the best solution space by an iterative process that employs an iterative rule and an evaluation function [13]. The technique has been successfully used in many areas. The basic idea is that a population of particles navigates an N -dimensional search space. The number of variables for each particle is S. The particles are adjusted according to the following iterative rule. 

vkδ (t + 1) = ωvkδ (t) + p1r1 (gkδ − skδ (t)) + p2 r2 (bδ − skδ (t)), skδ (t + 1) = skδ (t) + vkδ (t + 1),

(4.77)

where k = 1, 2, . . . , N . δ = 1, 2, . . . , S. t is the distance traveled by each particle. vkδ and skδ are the speed and position of the kth particle, respectively. ω is inertia weight. p1 and p2 are weighting factors. r1 and r1 are two random numbers in the range [0, 1]. gkδ is the best position of the kth particle. bδ is the best position of the particle swarm. An evaluation function is used to update gkδ and bδ , which is based on the control objective, given as  f (s) =

(x − xd )2 + (y − yd )2 ,

(4.78)

where (x, y) indicates the coordinates of the end-point calculated from (4.68), and (xd , yd ) indicates the target coordinates of the end-point. The PSO procedure is as follows. Step 1: Initialize each particle in the swarm to a random position skδ and speed vkδ . Step 2: Calculate f (s) for skδ and gkδ respectively. If the value of f (s) for skδ is less than the value of f (s) for gkδ , then set gkδ = skδ . Step 3: Calculate f (s) for bδ . If there is a gkδ for which the value of f (s) is less than the value of f (s) for bδ , then set bδ = gkδ . Step 4: If the value of f (s) for bδ is less than a constant, e, stop. Otherwise, use (4.77) to update all the positions and speeds, and go to Step 2. Let sk = (sk1 , sk2 ) be a 2-dimensional vector, where sk1 stands for q2 and sk2 stands for q3 . The procedure below for obtaining all the angles based on the first constraints (4.71), (4.75) takes the two stages of the control process into account. Step 1: Initialize the angles(q1 (0), q2 (0), q3 (0)) to the initial conditions.

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

Step 2: Use the appropriate constraint (4.71) or (4.75) and the values q1 (0), qi (0), 1 in the qi and q j (0) (i = 2 or 3, j = 3 or 2) to calculate the constant value of q1w first stage. 1 ). Use the appropriate constraint (4.75) or (4.71) and the Step 3: Let (q12 (0) = q1w 1 values q1w , qi , q j (0) and q j (i = 2 or 3, j = 3 or 2) to calculate the constant value of q1 in the second stage. The coordinates (x, y) can be calculated by using the angles q1 , q2 , and q3 produced by the above procedure and also (4.68). Those angles also can be used to calculate the evaluation function (4.78). If the result is less than the constant e, then q1 , q2 , and q3 are the target angles. In conclusion, the basis of the controller design is the use of PSO to adjust the angles of the system to the target values. Note that a solution exists only when the target position is in the reachable area.

4.3.3 Target Angle-Based Controllers Design Let x = [x1 , x2 , x3 , x4 , x5 , x6 ]T = [q, q] ˙ T , and then the dynamic equation of the planar PAA manipulator (2.38) can be rewritten as x˙ = f (x) + g(x)τ,

(4.79)

f (x) = [x4 , x5 , x6 , F1 , F2 , F3 ]T , g(x) = [g1 (x), g2 (x)]T .

(4.80)

where

The column vector, which consists of F1 , F2 , and F3 , satisfies ˙ (q, q). ˙ [F1 , F2 , F3 ]T = −M −1 (q)H

(4.81)

g1 (x) is a 3 × 3 null matrix, and g2 (x) satisfies ⎡

⎡ ⎤ ⎤ 0 b1 (x) c1 (x) 000 g2 (x) = ⎣ 0 b2 (x) c2 (x) ⎦ = M −1 (q) ⎣ 0 1 0 ⎦ , 001 0 b3 (x) c3 (x)

(4.82)

where b1 , b2 , b3 , c1 , c2 , and c3 are nonlinear functions. The control objective of the first stage is to make qi (i = 2 or 3) converge to the target angle qid while q j ( j = 3 or 2) is kept constant at the initial angle q j (0). The Lyapunov function V1 (x) is chosen to be 1 1 1 1 Pi−1 (xi − xid )2 + K i−1 xi+3 2 + P j−1 (x j − x j (0))2 + K j−1 x j+3 2 , 2 2 2 2 (4.83) where Pi−1 , K i−1 , P j−1 , and K j−1 are positive constant, xid = qid and x j (0) = q j (0). V1 (x) =

4.3 Stable Control of Planar Three-Link PAA Manipulator

135

The derivative of V1 (x) with respect to time is V˙1 (x) = xi+3 (Pi−1 (xi − xid ) + K i−1 (Fi + bi τ2 + ci τ3 )) +x j+3 (P j−1 (x j − x j (0)) + K j−1 (F j + b j τ2 + c j τ3 )).

(4.84)

If the control sequence is (i = 2, j = 3), then choose the control laws to be ⎧   P1 λ1 ⎪ −1 ⎪ τ = (−x + x ) − F − x − c τ ⎨ 2 2 2d 2 5 2 3 b2 , K K 1   1 P2 λ2 ⎪ ⎪ (−x3 + x3 (0)) − F3 − x6 − b3 τ2 c3 −1 , ⎩ τ3 = K2 K2

(4.85)

and if it is (i = 3, j = 2), then choose the control laws to be ⎧   P1 λ1 ⎪ ⎪ (−x2 + x2 (0)) − F2 − x5 − c2 τ3 b2 −1 , ⎨ τ2 = K1   K1 P2 λ2 ⎪ ⎪ (−x3 + x3d ) − F3 − x6 − b3 τ2 c3 −1 , ⎩ τ3 = K2 K2

(4.86)

where λ1 and λ2 are positive constants that adjust the rate of convergence. From (4.134), one obtains b2 =

m 11 m 33 − m 13 2 m 11 m 22 − m 12 2 , c3 = . det M det M

(4.87)

Since M(q) is a positive-definite symmetric matrix, M −1 (q) is also a positive-definite symmetric matrix. Thus, b2 and c3 are set to values greater than zero to avoid a singularity problem with the control torques τ2 and τ3 . Substituting (4.85) or (4.86) into (4.84) yields V˙1 (x) = −λ1 x5 2 − λ2 x6 2  0.

(4.88)

Note that, from (4.88), although V1 (x) decreases monotonically, there is no guarantee that xi converges to xid or that x j converges to x j (0). For example, V1 (x) may remain unchanged when xi and x j are constant. Therefore, it is necessary to analyze the stability of the system in the first stage by using Lasalle’s invariance principle. V1 (x) is continuously differentiable and it is a WCLF under control law (4.85) or (4.86). Since (4.88), V1 (x) is bounded. Define Ω1 = {x ∈ R6 |V1 (x)  ε1 },

(4.89)

where ε1 is a positive constant for the Lyapunov function V1 (x). Then, any solution, x, of (4.79) that starts in Ω1 remains in Ω1 for all t  0. Let Φ1 be an invariant set with respect to (4.79).

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

 Φ1 = {x(t) ∈ Ω1 V˙1 (x) = 0 }.

(4.90)

When V˙1 (x) = 0, x5 = 0 and x6 = 0. Hence, from (4.11), x4 = 0, and from (4.133), F1 = F2 = F3 = 0.

(4.91)

From (4.79), the following equation can be obtained [x˙4 , x˙5 , x˙6 ]T = [F1 , F2 , F3 ]T + g2 (x)τ.

(4.92)

Combining (4.134), (4.91) and (4.92) gives g2 (x)τ = M −1 (q)τ = 0.

(4.93)

Since matrix M is positive definite, τ = [0, τ2 , τ3 ]T = 0 is obtained from (4.93). 1 Substituting τ = 0 into (4.85) yields (xi , x j ) = (xid , x j (0)). Moreover, x1 = x1d 1 from (4.71) or (4.75), where x1d is the target value of q1 in the first stage. Therefore, the largest invariant set for the first stage is  1 M1 = {x ∈ R6 x1 = q1d , xi = xid , x j = x j (0), x4 = 0, x5 = 0, x6 = 0}. (4.94) According to LaSalle’s invariance principle, every solution, x, of (4.79) that starts in Ω1 approaches M1 as t → +∞. The above results are summarized in the theorem below. Theorem 4.1 Consider a planar three-link underactuated mechanical system in the first stage. Let M1 be the largest invariant set of the system (4.79). V1 (x) is a WCLF, Ω1 is a compact closed and bounded set that contains all the initial states of the system (4.79), and Φ1 is a set with states in Ω1 , where V˙1 (x) = 0. If controller (4.85) or (4.86) is employed, then, as t → +∞, every solution, x, of the closed-loop system (4.79) converges to the invariant set M1 defined in (4.94). The switching conditions that determine when the control law switches from the first stage to the second stage are 

|xi − xid |  e1 , |xi+3  | e2 , x j − x j (0)  e1 , x j+3   e2 ,

(4.95)

where e1 and e2 are small positive numbers. In the first stage, the angle and angular velocity of the jth link remain at the initial values: x j = x j (0) and x j+3 − x j+3 (0) = 0. So, the control law will switch to the second stage only if switching condition (1) is satisfied. Then, the control of the second stage is to stabilize q j ( j = 3 or 2) at q jd while qi (i = 2 or 3) is kept constant at the target angle qid . Based on the angle constraint (4.71) or (4.75), the control objective for the passive link is also achieved.

4.3 Stable Control of Planar Three-Link PAA Manipulator

137

The form of the control law is similar to (4.85) or (4.86) for the first stage. The Lyapunov function V2 (x) is chosen to be 1 1 1 1 Pi−1 (xi − xid )2 + K i−1 xi+3 2 + P j−1 (x j − x jd )2 + K j−1 x j+3 2 , 2 2 2 2 (4.96) where x jd = q jd is the target angle of the jth link. The derivative of V2 (x) with respect to time is V2 (x) =

V˙2 (x) = xi+3 (Pi−1 (xi − xid ) + K i−1 (Fi + bi τ2 + ci τ3 )) + x j+3 (P j−1 (x j − x jd ) + K j−1 (F j + b j τ2 + c j τ3 )).

(4.97)

Control laws for (i = 2, j = 3) and (i = 3, j = 2) are ⎧  P1 ⎪ ⎪ (−x2 + x2d ) − F2 − ⎨ τ2 =  K1 P2 ⎪ ⎪ (−x3 + x3d ) − F3 − ⎩ τ3 = K2

 λ1 x5 − c2 τ3 b2 −1 , K1  λ2 x6 − b3 τ2 c3 −1 . K2

(4.98)

Substituting (4.98) into (4.97) yields V˙2 (x) = −λ1 x5 2 − λ2 x6 2  0.

(4.99)

Note that V˙2 (x) is negative semi-definite, and that V˙2 (x) = 0 if and only if x5 = 0 and x6 = 0. Define (4.100) Ω2 = {x ∈ R6 |V2 (x)  ε2 }, where ε2 is a positive constant for the Lyapunov function V2 (x). Combining this with Theorem 4.1 yields the following stability theorem for the second-stage control of a three-link planar underactuated mechanical system. Theorem 4.2 Consider the second-stage control of the planar three-link underactuated mechanical system. Let M2 be the largest invariant set of the system (4.79) M2 = {x ∈ R6 |x1 = x1d , xi = xid , x j = x jd , x4 = 0, x5 = 0, x6 = 0}. (4.101) V2 (x) is a WCLF, Ω2 is a compact closed and bounded set that contains all the initial states of the system (4.79), and Φ2 is a set with states in Ω2 , where V˙2 (x) = 0. If controller (4.98) is employed, then, as t → +∞, every solution, x, of the closed-loop system (4.79) converges to the invariant set M2 defined in (4.101). The proof of Theorem 4.2 is similar to that for Theorem 4.1 and is thus omitted. In the second stage, controller (4.98) forces the system to the control objective, which means that the angles of all the active links are stabilized at the target values.

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

4.3.4 Simulation Results A model of a planar three-link PAA system is built with MATLAB tools, and two numerical examples are used to demonstrate the validity of the control strategy explained above. The parameters of the system are shown in Table 4.1. The initial state and the target position are chosen to be 

˙ = [0, 0, 0, 0, 0, 0] , [q(0), q(0)] Pd (X d , Yd ) = (1.7, −1.2) m.

(4.102)

The control objective is to move the end-point from any initial position to any target position. For the initial state [q(0), q(0)] ˙ = [0, 0, 0, 0, 0, 0], the initial position is (0, 3) m. So the control objective is to move the end-point from the initial position, (0, 3) m, to the target position, (xd , yd ) = (1.7, −1.2) m. The step size is chosen to be 0.001 s, and the control sequence is (i = 2, j = 3). In the iterative Eq. (4.77), the parameters for PSO are p1 = 2, p2 = 1.8, and ω = 0.53. The search space N = 15, and the number S of variables for each particle is 2. Meantime, considering the multiple solutions and the periodicity of the target angles with respect to the target position, the initial position of each particle is randomly initialized within range of −10 rad to 10 rad. The target angles with respect to the target position calculated by using PSO are [x1d , x2d , x3d ] = [5.505, −7.804, 5.686] rad.

(4.103)

Select P1 = 1, P2 = 1, K 1 = 1, K 2 = 1, λ1 = 1.9 and λ2 = 1.9, and implement control laws (4.85) and (4.98) in the different stages. Let e = 0.001 m, e1 = 0.001 rad, and e2 = 0.001 rad/s. The simulation results (Fig. 4.4) show that both the angles and the angular velocities converged to their target values very smoothly, and that the end-point stabilized at (1.7, −1.2) m. The control torques were less than 5 Nm. These results show that the control process switched from the first stage to the second stage at t = 9.23 s. Before that, the angles of the active links were forced to q2d and q3 (0). Further, consider that a real control problem usually contains external disturbances. Let the torque τ3 exist white noise from 2 s to 8 s. When the control sequence

Table 4.1 Parameters of model of planar three-link PAA system ith link m i /kg L i /m L ci /m i=1 i=2 i=3

0.8 1.2 1.0

0.8 1.2 1.0

0.4 0.6 0.5

Ji /(kg · m2 ) 0.04267 0.14400 0.08333

4.3 Stable Control of Planar Three-Link PAA Manipulator

139

Fig. 4.4 Simulation results for initial state and target position (4.102) with P1 = 1, λ1 = 1.9, K 1 = 1, P2 = 1, λ2 = 1.9, K 2 = 1, for control sequence (i = 2, j = 3)

Fig. 4.5 Simulation results for the initial state and target position (4.102) with white noise and P1 = 1, λ1 = 1.9, K 1 = 1, P2 = 1, λ2 = 1.9, K 2 = 1, for control sequence (i = 2, j = 3)

140

4 Control of Planar Underactuated Manipulator with a Passive First Joint

is (i = 2, j = 3) and the initial state and the target position are (4.102), the simulation results are shown in Fig. 4.5, and the coordinates of the end-point stabilized at (1.7, −1.2) m. The results show that the angle of the third link deviates from the initial value for a while in the first stage, and then the control strategy still realizes the system control objective. Hence, the control strategy has robustness against external disturbances. But the control time is longer than that of Fig. 4.4.

4.4 Two-Stage Control of Planar n-Link (n > 3) PAn−1 Manipulator The control method presented in the last section can be extended directly for the control of the planar PAn−1 (n > 3) underactuated manipulator [14]. But the extension brings a new problem that the control is divided into n − 1 stages, which is harmful to system stability. Thus, [15] presents an improved three-stage control method, in which the planar PAn−1 (n > 3) underactuated manipulator is reduced to the planar virtual PAA one in the first stage and then the two-stage control method presented in the last section is used directly. Further, next a two-stage control method will be developed for quickly positioning the planar PAn−1 manipulator [16]. In stage 1, the system is directly reduced to a planar virtual Acrobot by controlling n−2 active links to their target angles. A hybrid intelligent optimization algorithm, which combines the GA and the PSO algorithm, is used to optimize all links’ target angles. Coordinating GA and PSO, it is ensured that all links’ target angles, the passive link’s angle at the end of stage 1 and the initial angle of the active link of the planar virtual Acrobot meet the angle constraint of the planar virtual Acrobot. So, the position control of the planar PAn−1 manipulator is achieved by controlling the active link of the planar virtual Acrobot to its target angle in stage 2. The active link of the planar virtual Acrobot is defined to be u, u ∈ {2, 3, . . . , n} . The remaining n − 2 active links are defined as b1 , b2 , . . . , bn−2 in sequence. b j ∈ {2, 3, . . . , n} except u, j = 1, 2, . . . , n − 2,

(4.104)

where b1 < b2 < · · · < bn−2 . Here, the initial angles and the target angles of the n − 2 active links are defined as qb1 0 , qb2 0 , . . . , qbn−2 0 and qb1 d , qb2 d , . . . , qbn−2 d , respectively. The initial angle and the target angle of the uth link are defined as qu0 and qud , and the initial angle and the target angle of the passive link are defined as q10 and q1d . The initial position and the target position of the end-point are defined as (x0 , y0 ) and (xd , yd ). In addition, the angle of the passive link at the end of the stage 1 is defined as q1mid . Based on the above definitions and the holonomic constraint of the planar virtual Acrobot, the state constraint between q1 and qu is

4.4 Two-Stage Control of Planar n-Link (n > 3) PAn−1 Manipulator

 q1 (t) =

qu (t)

   −m 1u m 11 dqu − q1mid , t > t1 ,

141

(4.105)

qu0

where t1 is the initial time of stage 2.

4.4.1 Controllers Design of Stage 1 for Model Reduction The state space equation of the planar PAn−1 manipulator is ⎧ x˙11 = x21 , ⎪ ⎪ ⎪ ⎪ x ˙1u = x2u , ⎪ ⎪ ⎪ ˙ ⎨ Xˆ 1 = Xˆ 2 , x˙21 = f 1 + g1u τu + G 1 (X ) U , ⎪ ⎪ ⎪ ⎪ x˙2u = f u + guu τu + G 2 (X ) U , ⎪ ⎪ ⎪ ⎩ ˙ˆ X 2 = Fˆ (X ) + G 3 (X ) τu + G 4 (X ) U ,

(4.106)

where T T   Xˆ 1 = x1b1 , x1b2 , . . . , x1bn−2 , Xˆ 2 = x2b1 , x2b2 , . . . , x2bn−2 , T T   Fˆ (X ) = f b1 , f b2 , . . . , f bn−2 , U = τb1 , τb2 , . . . , τbn−2 ,     G 1 (X ) = g1b1 , g1b2 , . . . , g1bn−2 , G 2 (X ) = gub1 , gub2 , . . . , gubn−2 , T  G 3 (X ) = gb1 u , gb2 u , . . . , gbn−2 u , ⎡ ⎤ gb1 b1 gb1 b2 · · · gb1 bn−2 ⎢ gb2 b1 gb2 b2 · · · gb2 bn−2 ⎥ ⎢ ⎥ G 4 (X ) = ⎢ . ⎥. .. .. ⎣ .. ⎦ . . gbn−2 b1 gbn−2 b2 · · · gbn−2 bn−2 First, control n−2 active links to their target angles qb1 d , qb2 d , . . . , qbn−2 d , and maintain qu in its initial value qu0 . Based on the above control targets, choose V3 (X ) =

bn−2 ! pi μu 2 μi " pu x2u + (x1u − x1u0 )2 + (x1i − x1id )2 + x2i2 , (4.107) 2 2 2 2 i=b 1

where x1u0 = qu0 , x1id = qid (i = b1 , b2 , . . . , bn−2 ), pu , pi , μu and μi are positive constants. The derivative of V4 (X ) with respect to time is V˙3 (X ) = x2u [ p!u (x1u − x1u0 ) +  μu ( f u + guu τu + G 2 (X ) U )] " T + Xˆ 2 P Xˆ 1 − Xˆ 1d +  Fˆ (X ) + G 3 (X ) τu + G 4 (X ) U , (4.108)

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4 Control of Planar Underactuated Manipulator with a Passive First Joint

where Γ = diag(μb1 , μb2 , . . . , μbn−2 ), Xˆ 1d = [x1b1 d , x1b2 d , . . . , x1bn−2 d ]T , and P = diag( pb1 , pb2 , . . . , pbn−2 ). Choose the controllers of stage 1 to be ⎧   pu du ⎪ −1 ⎪ x2u − f u − G 2 (X ) U , (x1u − x1u0 ) − ⎪ τu = guu − ⎨ μu μu ⎪  "  ! ⎪ ⎪ ⎩ U = G −1 (X ) −Γ −1 P Xˆ 1 − Xˆ 1d −  −1 D Xˆ 2 − Fˆ (X ) − G 3 (X ) τu , 4 (4.109)   where D = diag db1 , db2 , . . . , dbn−2 , du and di are positive constants. Clearly, (4.109) guarantees bn−2

2 − V˙3 (X ) = −du x2u

di x2i2  0.

(4.110)

i=b1

Substituting (4.109) into (4.106) yields the closed-loop system X˙ = Fb (X ) .

(4.111)

Let Ψ2 be an invariant set of the closed-loop system (4.111), which is  # $ Ψ2 = X ∈ R2n  V˙4 (X ) = 0 .

(4.112)

When V˙3 (X ) = 0, x22 = x23 = · · · = x2n = 0. According to the motion-state constraint on angular velocities (4.11), x21 = 0. Hence, X 2 = 0. From (4.106), one has F (X ) + G (X ) τ = 0. (4.113) ˙ = 0, Since H (q, q) ˙ = 0 and F (X ) = −M −1 (q) H (q, q) τ = 0.

(4.114)

x1u = x1u0 , Xˆ 1 = Xˆ 1d .

(4.115)

From (4.109) and (4.114), one obtains

Therefore, the largest invariant set of the closed-loop system (4.111) is M4 =

%

& X ∈ Ψ2 | x11 = q1mid , x1u = x1u0 , Xˆ 1 = Xˆ 1d , X 2 = 0 .

(4.116)

According to LaSalle’s invariance theorem, it can be derived that the states of the system in stage 1 approach to the largest invariant set M4 as t → +∞.

4.4 Two-Stage Control of Planar n-Link (n > 3) PAn−1 Manipulator

143

When all states satisfy the switch condition (4.117), the control of the system switches from stage 1 to stage 2. ⎧ |x ⎪  1u − x1u0 | <  e1 , |x2u |