Lyapunov-Based Control of Robotic Systems (Automation and Control Engineering)

384 Pages·2009·5.5 MB·English

by  Aman Behal

Lyapunov-Based
Control of
Robotic Systems
© 2010 by Taylor and Francis Group, LLC
AUTOMATION AND CONTROLENGINEERING
ASeries of Reference Books and Textbooks
Series Editors
FRANK L. LEWIS, PH.D., SHUZHI SAMGE, PH.D.,
FELLOWIEEE, FELLOWIFAC FELLOWIEEE
Professor Professor
Automation and Robotics Research Institute Interactive Digital Media Institute
The University ofTexas at Arlington The National University ofSingapore
Lyapunov-Based Control of Robotic Systems, Aman Behal, Warren Dixon,
Darren M. Dawson, and Bin Xian
System Modeling and Control with Resource-Oriented Petri Nets,
Naiqi Wu and MengChu Zhou
Sliding Mode Control in Electro-Mechanical Systems, Second Edition,
Vadim Utkin, Jürgen Guldner, and Jingxin Shi
Optimal Control: Weakly Coupled Systems and Applications,
Zoran Gajic´, Myo-Taeg Lim, Dobrila Skataric´, Wu-Chung Su,
and Vojislav Kecman
Intelligent Systems: Modeling, Optimization, and Control,Yung C. Shin
and Chengying Xu
Optimal and Robust Estimation: With an Introduction to Stochastic Control
Theory, Second Edition,Frank L. Lewis; Lihua Xie and Dan Popa
Feedback Control of Dynamic Bipedal Robot Locomotion,
Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi,
and Benjamin Morris
Intelligent Freight Transportation,edited by Petros A. Ioannou
Modeling and Control of Complex Systems,edited by Petros A. Ioannou
and Andreas Pitsillides
Wireless Ad Hoc and Sensor Networks: Protocols, Performance,
and Control,Jagannathan Sarangapani
Stochastic Hybrid Systems,edited by Christos G. Cassandras
and John Lygeros
Hard Disk Drive: Mechatronics and Control,Abdullah Al Mamun,
Guo Xiao Guo, and Chao Bi
Autonomous Mobile Robots: Sensing, Control, Decision Making
and Applications, edited by Shuzhi Sam Ge and Frank L. Lewis
Neural Network Control of Nonlinear Discrete-Time Systems,
Jagannathan Sarangapani
Quantitative Feedback Theory: Fundamentals and Applications,
Second Edition, Constantine H. Houpis, Steven J. Rasmussen,
and Mario Garcia-Sanz
Fuzzy Controller Design: Theory and Applications, Zdenko Kovacic
and Stjepan Bogdan
© 2010 by Taylor and Francis Group, LLC
Lyapunov-Based
Control of
Robotic Systems
Aman Behal
University of Central Florida
Orlando, Florida, U.S.A.
Warren Dixon
University of Florida
Gainesville, Florida, U.S.A.
Darren M. Dawson
Clemson University
Clemson, South Carolina, U.S.A.
Bin Xian
Tianjin University
Tianjin, China
Boca Raton London New York
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Library of Congress Cataloging‑in‑Publication Data
Lyapunov-based control of robotic systems / Aman Behal … [et al.].
p. cm. -- (Automation and control engineering)
Includes bibliographical references and index.
ISBN 978-0-8493-7025-0 (hardcover : alk. paper)
1. Robots--Control systems. 2. Nonlinear control theory. 3. Lyapunov functions. I.
Behal, Aman.
TJ211.35.L83 1009
629.8’92--dc22 2009040275
Visit the Taylor & Francis Web site at
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© 2010 by Taylor and Francis Group, LLC
To my loving wife,
Hina Behal
A.B.
To my parents,
Dwight and Belinda Dixon
W.E.D
To my children,
Jacklyn and David
D.M.D.
To my parents,
Kaiyong Xian and Yanfang Liu
B.X.
© 2010 by Taylor and Francis Group, LLC
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 1
1.1 History of Robotics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lyapunov-Based Control Philosophy . . . . . . . . . . . . . 3
1.3 The Real-Time Computer Revolution. . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Robot Control 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Modeling and Control Objective . . . . . . . . . . . . . . . 10
2.2.1 Robot Manipulator Model and Properties . . . . . . 10
2.2.2 Control Objective . . . . . . . . . . . . . . . . . . . 12
2.3 Computed Torque Control Approaches . . . . . . . . . . . . 12
2.3.1 PD Control . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Robust Control . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Sliding Mode Control . . . . . . . . . . . . . . . . . 16
2.4 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Direct Adaptive Control . . . . . . . . . . . . . . . . 18
2.4.2 Neural Network-Based Control . . . . . . . . . . . . 24
2.5 Task-Space Control and Redundancy . . . . . . . . . . . . . 28
2.5.1 Kinematic Model . . . . . . . . . . . . . . . . . . . . 29
2.5.2 Control Objective and Error System Formulation . . 30
© 2010 by Taylor and Francis Group, LLC
viii Contents
2.5.3 Computed Torque Control Development and Stabil-
ity Analysis . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.4 Adaptive Control Extension . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Vision-Based Systems 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Monocular Image-Based Geometry . . . . . . . . . . . . . . 41
3.2.1 Fixed-Camera Geometry. . . . . . . . . . . . . . . . 41
3.2.2 Euclidean Reconstruction . . . . . . . . . . . . . . . 44
3.2.3 Camera-in-Hand Geometry . . . . . . . . . . . . . . 46
3.2.4 Homography Calculation . . . . . . . . . . . . . . . 47
3.2.5 Virtual Parallax Method . . . . . . . . . . . . . . . . 50
3.3 Visual Servo Tracking . . . . . . . . . . . . . . . . . . . . . 51
3.3.1 Control Objective . . . . . . . . . . . . . . . . . . . 51
3.3.2 Control Formulation . . . . . . . . . . . . . . . . . . 54
3.3.3 Stability Analysis. . . . . . . . . . . . . . . . . . . . 56
3.3.4 Camera-in-Hand Extension . . . . . . . . . . . . . . 57
3.3.5 Simulation Results . . . . . . . . . . . . . . . . . . . 58
3.4 Continuum Robots . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Continuum Robot Kinematics . . . . . . . . . . . . . 69
3.4.2 Joint Variables Extraction . . . . . . . . . . . . . . . 72
3.4.3 Task-Space Kinematic Controller . . . . . . . . . . . 74
3.4.4 Simulations and Discussion . . . . . . . . . . . . . . 76
3.5 Mobile Robot Regulation and Tracking . . . . . . . . . . . 78
3.5.1 Regulation Control . . . . . . . . . . . . . . . . . . . 79
3.5.2 Tracking Control . . . . . . . . . . . . . . . . . . . . 93
3.6 Structure from Motion . . . . . . . . . . . . . . . . . . . . . 107
3.6.1 Object Kinematics . . . . . . . . . . . . . . . . . . . 107
3.6.2 Identification of Velocity . . . . . . . . . . . . . . . . 108
3.6.3 Camera-in-Hand Extension . . . . . . . . . . . . . . 113
3.6.4 Simulations and Experimental Results . . . . . . . . 119
3.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4 Path Planning and Control 141
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2 Velocity Field and Navigation Function Control for Manip-
ulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2.1 System Model. . . . . . . . . . . . . . . . . . . . . . 145
4.2.2 Adaptive VFC Control Objective . . . . . . . . . . . 146
© 2010 by Taylor and Francis Group, LLC
Contents ix
4.2.3 Navigation Function Control Extension . . . . . . . 150
4.2.4 Experimental Verification . . . . . . . . . . . . . . . 154
4.3 Velocity Field and Navigation Function Control for WMRs 163
4.3.1 Kinematic Model . . . . . . . . . . . . . . . . . . . . 163
4.3.2 WMR Velocity Field Control . . . . . . . . . . . . . 164
4.3.3 WMR Navigation Function Control Objective . . . . 174
4.4 Vision Navigation. . . . . . . . . . . . . . . . . . . . . . . . 181
4.4.1 Geometric Modeling . . . . . . . . . . . . . . . . . . 184
4.4.2 Image-Based Path Planning . . . . . . . . . . . . . . 187
4.4.3 Tracking Control Development . . . . . . . . . . . . 191
4.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . 194
4.5 Optimal Navigation and Obstacle Avoidance . . . . . . . . 209
4.5.1 Illustrative Example: Planar PBVS . . . . . . . . . . 213
4.5.2 6D Visual Servoing: Camera-in-Hand . . . . . . . . . 218
4.6 Background and Notes . . . . . . . . . . . . . . . . . . . . . 222
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5 Human Machine Interaction 233
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2 Exercise Machine . . . . . . . . . . . . . . . . . . . . . . . . 235
5.2.1 Exercise Machine Dynamics . . . . . . . . . . . . . . 236
5.2.2 Control Design with Measurable User Input . . . . . 237
5.2.3 Desired Trajectory Generator . . . . . . . . . . . . . 239
5.2.4 Control Design without Measurable User Input . . . 241
5.2.5 Desired Trajectory Generator . . . . . . . . . . . . . 246
5.2.6 Experimental Results and Discussion . . . . . . . . . 247
5.3 Steer-by-Wire . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.3.1 Control Problem Statement . . . . . . . . . . . . . . 254
5.3.2 Dynamic Model Development . . . . . . . . . . . . . 255
5.3.3 Control Development. . . . . . . . . . . . . . . . . . 258
5.3.4 Stability Analysis. . . . . . . . . . . . . . . . . . . . 259
5.3.5 Elimination of Torque Measurements: Extension . . 260
5.3.6 Numerical Simulation Results . . . . . . . . . . . . . 265
5.3.7 Experimental Results . . . . . . . . . . . . . . . . . 271
5.4 Robot Teleoperation . . . . . . . . . . . . . . . . . . . . . . 274
5.4.1 System Model. . . . . . . . . . . . . . . . . . . . . . 277
5.4.2 MIF Control Development . . . . . . . . . . . . . . . 278
5.4.3 UMIF Control Development . . . . . . . . . . . . . . 284
5.5 Rehabilitation Robot . . . . . . . . . . . . . . . . . . . . . . 295
5.5.1 Robot Dynamics . . . . . . . . . . . . . . . . . . . . 296
5.5.2 Path Planning and Desired Trajectory Generator . . 297
© 2010 by Taylor and Francis Group, LLC
x Contents
5.5.3 Control Problem Formulation . . . . . . . . . . . . . 302
5.5.4 Simulation Results . . . . . . . . . . . . . . . . . . . 307
5.6 Background and Notes . . . . . . . . . . . . . . . . . . . . . 317
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Appendices 326
A Mathematical Background 327
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
B Supplementary Lemmas and Expressions 335
B.1 Chapter 3 Lemmas . . . . . . . . . . . . . . . . . . . . . . . 335
B.1.1 Open-Loop Rotation Error System . . . . . . . . . . 335
B.1.2 Open-Loop Translation Error System . . . . . . . . 337
B.1.3 Persistence of Excitation Proof . . . . . . . . . . . . 337
B.2 Chapter 4 Lemmas and Auxiliary Expressions . . . . . . . . 339
B.2.1 Experimental Velocity Field Selection . . . . . . . . 339
B.2.2 GUB Lemma . . . . . . . . . . . . . . . . . . . . . . 340
B.2.3 Boundedness of θ˙ (t) . . . . . . . . . . . . . . . . . 342
d
B.2.4 Open-Loop Dynamics for Υ(t) . . . . . . . . . . . . 344
B.2.5 Measurable Expression for L (t) . . . . . . . . . . 344
Υd
B.2.6 Development of an Image Space NF and Its Gradient 345
B.2.7 Global Minimum . . . . . . . . . . . . . . . . . . . . 347
B.3 Chapter 5 Lemmas and Auxiliary Expressions . . . . . . . . 347
B.3.1 Numerical Extremum Generation . . . . . . . . . . . 347
B.3.2 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . 349
B.3.3 Definitions from Section 5.3.2 . . . . . . . . . . . . . 350
B.3.4 Upperbound for V (t). . . . . . . . . . . . . . . . . 350
a1
B.3.5 Upper Bound Development for MIF Analysis . . . . 351
B.3.6 Teleoperator — Proof of MIF Controller Stability . . 354
B.3.7 Teleoperator — Proof of MIF Passivity . . . . . . . . 358
B.3.8 Teleoperator — Proof of UMIF Desired Trajectory
Boundedness . . . . . . . . . . . . . . . . . . . . . . 359
B.3.9 Teleoperator — Proof of UMIF Controller Stability . 363
B.3.10 Teleoperator — Proof of UMIF Passivity . . . . . . . 366
B.3.11 Proof of Bound on N˜ . . . . . . . . . . . . . . . . . 367
B.3.12 Calculation of Region of Attraction. . . . . . . . . . 369
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
© 2010 by Taylor and Francis Group, LLC