Robot Modeling and Control
Description
In the 2nd Edition of Robot Modeling and Control, students will cover the theoretical fundamentals and the latest technological advances in robot kinematics. With so much advancement in technology, from robotics to motion planning, society can implement more powerful and dynamic algorithms than ever before. This in-depth reference guide educates readers in four distinct parts; the first two serve as a guide to the fundamentals of robotics and motion control, while the last two dive more in-depth into control theory and nonlinear system analysis.
With the new edition, readers gain access to new case studies and thoroughly researched information covering topics such as:
* Motion-planning, collision avoidance, trajectory optimization, and control of robots
* Popular topics within the robotics industry and how they apply to various technologies
* An expanded set of examples, simulations, problems, and case studies
* Open-ended suggestions for students to apply the knowledge to real-life situations
A four-part reference essential for both undergraduate and graduate students, Robot Modeling and Control serves as a foundation for a solid education in robotics and motion planning.
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Persons
MARK W. SPONG has been researching and teaching robotics for over 35 years. He currently serves as a Professor, Excellence in Education Chair, in the Department of Systems Engineering at the University of Texas at Dallas. He has been recognized for outstanding achievements including the John R. Ragazzini Award for Control Education and the IEEE RAS Pioneer in Robotics Award. He is currently a Fellow of both IEEE and IFAC.
SETH HUTCHINSON received his Ph.D. from Purdue University in 1988, and is currently Professor and KUKA Chair for Robotics in the School of Interactive Computing at the Georgia Institute of Technology, where he also serves as Executive Director of the Institute for Robotics and Intelligent Machines. He was the Founding Editor-in-Chief of the IEEE Robotics and Automation Society's Conference Editorial Board, Editor-in-Chief of the IEEE Transactions on Robotics, and is a Fellow of the IEEE. His research in robotics spans the areas of planning, sensing, and control.
MATHUKUMALLI VIDYASAGAR received his Ph.D. in electrical engineering in 1969 from the University of Wisconsin in Madison. During his fifty-year career, he has worked in control theory, machine learning, robotics and cancer biology. Among the many honors he has received are Fellowship in The Royal Society and the IEEE Control Systems Award. At present he is a Distinguished Professor at the Indian Institute of Technology Hyderabad.
Content
Preface v
1 Introduction 1
1.1 Mathematical Modeling of Robots 5
1.1.1 Symbolic Representation of Robot Manipulators 5
1.1.2 The Configuration Space 5
1.1.3 The State Space 6
1.1.4 The Workspace 7
1.2 Robots as Mechanical Devices 7
1.2.1 Classification of Robotic Manipulators 8
1.2.2 Robotic Systems 10
1.2.3 Accuracy and Repeatability 10
1.2.4 Wrists and End Effectors 12
1.3 Common Kinematic Arrangements 13
1.3.1 Articulated Manipulator (RRR) 13
1.3.2 Spherical Manipulator (RRP) 14
1.3.3 SCARA Manipulator (RRP) 14
1.3.4 Cylindrical Manipulator (RPP) 15
1.3.5 Cartesian Manipulator (PPP) 15
1.3.6 Parallel Manipulator 18
1.4 Outline of the Text 18
1.4.1 Manipulator Arms 18
1.4.2 Underactuated and Mobile Robots 27
Problems 27
Notes and References 29
I The Geometry of Robots 33
2 Rigid Motions 35
2.1 Representing Positions 36
2.2 Representing Rotations 38
2.2.1 Rotation in the Plane 38
2.2.2 Rotations in Three Dimensions 41
2.3 Rotational Transformations 44
2.4 Composition of Rotations 48
2.4.1 Rotation with Respect to the Current Frame 48
2.4.2 Rotation with Respect to the Fixed Frame 50
2.4.3 Rules for Composition of Rotations 51
2.5 Parameterizations of Rotations 52
2.5.1 Euler Angles 53
2.5.2 Roll, Pitch, Yaw Angles 55
2.5.3 Axis-Angle Representation 57
2.5.4 Exponential Coordinates 59
2.6 Rigid Motions 61
2.6.1 Homogeneous Transformations 62
2.6.2 Exponential Coordinates for General Rigid Motions 65
2.7 Chapter Summary 65
Problems 67
Notes and References 73
3 Forward Kinematics 75
3.1 Kinematic Chains 75
3.2 The Denavit-Hartenberg Convention 78
3.2.1 Existence and Uniqueness 80
3.2.2 Assigning the Coordinate Frames 83
3.3 Examples 87
3.3.1 Planar Elbow Manipulator 87
3.3.2 Three-Link Cylindrical Robot 89
3.3.3 The Spherical Wrist 90
3.3.4 Cylindrical Manipulator with Spherical Wrist 91
3.3.5 Stanford Manipulator 93
3.3.6 SCARA Manipulator 95
3.4 Chapter Summary 96
Problems 96
Notes and References 99
4 Velocity Kinematics 101
4.1 Angular Velocity: The Fixed Axis Case 102
4.2 Skew-Symmetric Matrices 103
4.2.1 Properties of Skew-Symmetric Matrices 104
4.2.2 The Derivative of a Rotation Matrix 105
4.3 Angular Velocity: The General Case 107
4.4 Addition of Angular Velocities 108
4.5 Linear Velocity of a Point Attached to a Moving Frame 110
4.6 Derivation of the Jacobian 111
4.6.1 Angular Velocity 112
4.6.2 Linear Velocity 113
4.6.3 Combining the Linear and Angular Velocity Jacobians 115
4.7 The Tool Velocity 119
4.8 The Analytical Jacobian 121
4.9 Singularities 122
4.9.1 Decoupling of Singularities 123
4.9.2 Wrist Singularities 125
4.9.3 Arm Singularities 125
4.10 Static Force/Torque Relationships 129
4.11 Inverse Velocity and Acceleration 131
4.12 Manipulability 133
4.13 Chapter Summary 136
Problems 138
Notes and References 140
5 Inverse Kinematics 141
5.1 The General Inverse Kinematics Problem 141
5.2 Kinematic Decoupling 143
5.3 Inverse Position: A Geometric Approach 145
5.3.1 Spherical Configuration 146
5.3.2 Articulated Configuration 148
5.4 Inverse Orientation 151
5.5 Numerical Inverse Kinematics 156
5.6 Chapter Summary 158
Problems 160
Notes and References 162
II Dynamics and Motion Planning 163
6 Dynamics 165
6.1 The Euler-Lagrange Equations 166
6.1.1 Motivation 166
6.1.2 Holonomic Constraints and Virtual Work 170
6.1.3 D'Alembert's Principle 174
6.2 Kinetic and Potential Energy 177
6.2.1 The Inertia Tensor 178
6.2.2 Kinetic Energy for an n-Link Robot 180
6.2.3 Potential Energy for an n-Link Robot 181
6.3 Equations of Motion 181
6.4 Some Common Configurations 184
6.5 Properties of Robot Dynamic Equations 194
6.5.1 Skew Symmetry and Passivity 194
6.5.2 Bounds on the Inertia Matrix 196
6.5.3 Linearity in the Parameters 196
6.6 Newton-Euler Formulation 198
6.6.1 Planar Elbow Manipulator Revisited 206
6.7 Chapter Summary 209
Problems 211
Notes and References 214
7 Path and Trajectory Planning 215
7.1 The Configuration Space 216
7.1.1 Representing the Configuration Space 217
7.1.2 Configuration Space Obstacles 218
7.1.3 Paths in the Configuration Space 221
7.2 Path Planning for Q = R2 221
7.2.1 The Visibility Graph 222
7.2.2 The Generalized Voronoi Diagram 224
7.2.3 Trapezoidal Decompositions 226
7.3 Artificial Potential Fields 229
7.3.1 Artificial Potential Fields for Q = Rn 230
7.3.2 Potential Fields for Q ¿ Rn 235
7.4 Sampling-Based Methods 245
7.4.1 Probabilistic Roadmaps (PRM) 246
7.4.2 Rapidly-Exploring Random Trees (RRTs) 250
7.5 Trajectory Planning 252
7.5.1 Trajectories for Point-to-Point Motion 253
7.5.2 Trajectories for Paths Specified by Via Points 261
7.6 Chapter Summary 263
Problems 265
Notes and References 267
III Control of Manipulators 269
8 Independent Joint Control 271
8.1 Introduction 271
8.2 Actuator Dynamics 273
8.3 Load Dynamics 276
8.4 Independent Joint Model 278
8.5 PID Control 281
8.6 Feedforward Control 288
8.6.1 Trajectory Tracking 289
8.6.2 The Method of Computed Torque 291
8.7 Drive-Train Dynamics 292
8.8 State Space Design 297
8.8.1 State Feedback Control 299
8.8.2 Observers 301
8.9 Chapter Summary 304
Problems 307
Notes and References 309
9 Nonlinear and Multivariable Control 311
9.1 Introduction 311
9.2 PD Control Revisited 313
9.3 Inverse Dynamics 317
9.3.1 Joint Space Inverse Dynamics 317
9.3.2 Task Space Inverse Dynamics 320
9.3.3 Robust Inverse Dynamics 322
9.3.4 Adaptive Inverse Dynamics 327
9.4 Passivity-Based Control 329
9.4.1 Passivity-Based Robust Control 331
9.4.2 Passivity-Based Adaptive Control 332
9.5 Torque Optimization 333
9.6 Chapter Summary 337
Problems 341
Notes and References 343
10 Force Control 345
10.1 Coordinate Frames and Constraints 347
10.1.1 Reciprocal Bases 347
10.1.2 Natural and Artificial Constraints 349
10.2 Network Models and Impedance 351
10.2.1 Impedance Operators 353
10.2.2 Classification of Impedance Operators 354
10.2.3 Thévenin and Norton Equivalents 355
10.3 Task Space Dynamics and Control 355
10.3.1 Impedance Control 356
10.3.2 Hybrid Impedance Control 358
10.4 Chapter Summary 361
Problems 362
Notes and References 364
11 Vision-Based Control 365
11.1 Design Considerations 366
11.1.1 Camera Configuration 366
11.1.2 Image-Based vs. Position-Based Approaches 367
11.2 Computer Vision for Vision-Based Control 368
11.2.1 The Geometry of Image Formation 369
11.2.2 Image Features 373
11.3 Camera Motion and the Interaction Matrix 378
11.4 The Interaction Matrix for Point Features 379
11.4.1 Velocity Relative to a Moving Frame 380
11.4.2 Constructing the Interaction Matrix 381
11.4.3 Properties of the Interaction Matrix for Points 384
11.4.4 The Interaction Matrix for Multiple Points 385
11.5 Image-Based Control Laws 386
11.5.1 Computing Camera Motion 387
11.5.2 Proportional Control Schemes 389
11.5.3 Performance of Image-Based Control Systems 390
11.6 End Effector and Camera Motions 393
11.7 Partitioned Approaches 394
11.8 Motion Perceptibility 397
11.9 Summary 399
Problems 401
Notes and References 405
12 Feedback Linearization 409
12.1 Background 410
12.1.1 Manifolds, Vector Fields, and Distributions 410
12.1.2 The Frobenius Theorem 414
12.2 Feedback Linearization 417
12.3 Single-Input Systems 419
12.4 Multi-Input Systems 429
12.5 Chapter Summary 433
Problems 433
Notes and References 435
IV Control of Underactuated Systems 437
13 Underactuated Robots 439
13.1 Introduction 439
13.2 Modeling 440
13.3 Examples of Underactuated Robots 443
13.3.1 The Cart-Pole System 443
13.3.2 The Acrobot 445
13.3.3 The Pendubot 446
13.3.4 The Reaction-Wheel Pendulum 447
13.4 Equilibria and Linear Controllability 448
13.4.1 Linear Controllability 450
13.5 Partial Feedback Linearization 456
13.5.1 Collocated Partial Feedback Linearization 457
13.5.2 Noncollocated Partial Feedback Linearization 459
13.6 Output Feedback Linearization 461
13.6.1 Computation of the Zero Dynamics 463
13.6.2 Virtual Holonomic Constraints 466
13.7 Passivity-Based Control 466
13.7.1 The Simple Pendulum 467
13.7.2 The Reaction-Wheel Pendulum 471
13.7.3 Swingup and Balance of The Acrobot 473
13.8 Chapter Summary 474
Problems 476
Notes and References 477
14 Mobile Robots 479
14.1 Nonholonomic Constraints 480
14.2 Involutivity and Holonomy 484
14.3 Examples of Nonholonomic Systems 487
14.4 Dynamic Extension 493
14.5 Controllability of Driftless Systems 495
14.6 Motion Planning 499
14.6.1 Conversion to Chained Forms 499
14.6.2 Differential Flatness 506
14.7 Feedback Control of Driftless Systems 509
14.7.1 Stabilizability 509
14.7.2 Nonsmooth Control 511
14.7.3 Trajectory Tracking 513
14.7.4 Feedback Linearization 515
14.8 Chapter Summary 519
Problems 520
Notes and References 521
A Trigonometry 523
A.1 The Two-Argument Arctangent Function 523
A.2 Useful Trigonometric Formulas 523
B Linear Algebra 525
B.1 Vectors 525
B.2 Inner Product Spaces 526
B.3 Matrices 528
B.4 Eigenvalues and Eigenvectors 530
B.5 Differentiation of Vectors 533
B.6 The Matrix Exponential 534
B.7 Lie Groups and Lie Algebras 534
B.8 Matrix Pseudoinverse 536
B.9 Schur Complement 536
B.10 Singular Value Decomposition (SVD) 537
C Lyapunov Stability 539
C.1 Continuity and Differentiability 539
C.2 Vector Fields and Equilibria 541
C.3 Lyapunov Functions 545
C.4 Stability Criteria 545
C.5 Global and Exponential Stability 546
C.6 Stability of Linear Systems 547
C.7 LaSalle's Theorem 548
C.8 Barbalat's Lemma 549
D Optimization 551
D.1 Unconstrained Optimization 551
D.2 Constrained Optimization 552
E Camera Calibration 555
E.1 The Image Plane and the Sensor Array 555
E.2 Extrinsic Camera Parameters 556
E.3 Intrinsic Camera Parameters 557
E.4 Determining the Camera Parameters 557
Bibliography 561
Index 576
Preface
This text is a second edition of our book, Robot Modeling and Control, John Wiley & Sons, Inc., 2006, which grew out of the earlier text, M.W. Spong and M. Vidyasagar, Robot Dynamics and Control, John Wiley & Sons, Inc., 1989. The second edition reflects some of the changes that have occurred in robotics and robotics education in the past decade. In particular, many courses are now treating mobile robots on an equal footing with robot manipulators. As a result, we have expanded the discussion on mobile robots into a full chapter. In addition, we have added a new chapter on underactuated robots. We have also revised the material on vision, vision-based control, and motion planning to reflect changes in those topics.
Organization of the Text
After the introductory first chapter, which introduces the terminology and history of robotics and discusses the most common robot design and applications, the text is organized into four parts. Part I consists of four chapters dealing with the geometry of rigid motions and the kinematics of manipulators.
Chapter 2 presents the mathematics of rigid motions; rotations, translations, and homogeneous transformations.
Chapter 3 presents solutions to the forward kinematics problem using the Denavit-Hartenberg representation, which gives a very straightforward and systematic way to describe the forward kinematics of manipulators.
Chapter 4 discuses velocity kinematics and the manipulator Jacobian. The geometric Jacobian is derived in the cross product form. We also introduce the so-called analytical Jacobian for later use in task space control. We have reversed the order of our treatment of velocity kinematics and inverse kinematics from the presentation in the first edition in order to include a new section in Chapter 5 on numerical inverse kinematics algorithms, which rely on the Jacobian for their implementation.
Chapter 5 deals with the inverse kinematics problem using the geometric approach, which is especially suited for manipulators with spherical wrists. We show how to solve the inverse kinematics in closed form for the most common manipulator designs. We also discuss numerical search algorithms for solving inverse kinematics. Numerical algorithms are increasingly popular because of both the increasing power of computers and the availability of open-source software for numerical algorithms.
Part II deals with dynamics and motion planning and consists of two chapters.
Chapter 6 is a detailed account of robot dynamics. The Euler-Lagrange equations are derived from first principles and their structural properties are discussed in detail. The recursive Newton-Euler formulation of robot dynamics is also presented.
Chapter 7 is an introduction to the problems of path and trajectory planning. Several of the most popular methods for motion planning and obstacle avoidance are presented, including the method of artificial potential fields, randomized algorithms, and probabilistic roadmap methods. The problem of trajectory generation is presented as essentially a problem of polynomial spline interpolation. Trajectory generation based on cubic and quintic polynomials as well as trapezoidal velocity trajectories are derived for interpolation in joint space.
Part III deals with the control of manipulators.
Chapter 8 is an introduction to independent joint control. Linear models and linear control methods based on PD, PID, and state space methods are presented for set-point regulation, trajectory tracking, and disturbance rejection. The concept of feedforward control, including the method of computed torque control, is introduced as a method for nonlinear disturbance rejection and for tracking of time-varying reference trajectories.
Chapter 9 discusses nonlinear and multivariable control. This chapter summarizes much of the research in robot control that took place in the late 1980s and early 1990s. Simple derivations of the most common robust and adaptive control algorithms are presented that prepare the reader for the extensive literature in robot control.
Chapter 10 treats the force control problem. Both impedance control and hybrid control are discussed. We also present the lesser known hybrid impedance control method, which allows one to control impedance and regulate motion and force at the same time. To our knowledge this is the first textbook that discusses the hybrid impedance control approach to robot force control.
Chapter 11 is an introduction to visual servo control, which is the problem of controlling robots using feedback from cameras mounted either on the robot or in the workspace. We present those aspects of vision that are most useful for vision-based control applications, such as imaging geometry and feature extraction. We then develop the differential kinematics that relate camera motion to changes in extracted features and we discuss the main concepts in visual servo control.
Chapter 12 is a tutorial overview of geometric nonlinear control and the method of feedback linearization of nonlinear systems. Feedback linearization generalizes the methods of computed torque and inverse dynamics control that are covered in Chapters 8 and 9. We derive and prove the necessary and sufficient conditions for local feedback linearization of single-input/single-output nonlinear systems, which we then apply to the flexible joint control problem. We also introduce the notion of nonlinear observers with output injection.
Part IV is a completely new addition to the second edition and treats the control problems for underactuated robots and nonholonomic systems.
Chapter 13 deals with underactuated serial-link robots. Underactuation arises in applications such as bipedal locomotion and gymnastic robots. In fact, the flexible-joint robot models presented in Chapters 8 and 12 are also examples of underactuated robots. We present the ideas of partial feedback linearization and transformation to normal forms, which are useful for controller design. We also discuss energy and passivity methods to control this class of systems.
Chapter 14 deals primarily with wheeled mobile robots, which are examples of systems subject to nonholonomic constraints. Many of the control design methods presented in the chapters leading up to Chapter 14 do not apply to nonholonomic systems. Thus, we cover some new techniques applicable to these systems. We present two fundamental results, namely Chow's theorem and Brockett's theorem, that provide conditions for controllability and stabilizability, respectively, of mobile robots.
Finally, the appendices have been expanded to give much of the necessary background mathematics to be able to follow the development of the concepts in the text.
A Note to the Instructor
This text is suitable for several quarter-long or semester-long courses in robotics, either as a two- or three- course sequence or as stand-alone courses. The first five chapters can be used for a junior/senior-level introduction to robotics for students with at least a minimal background in linear algebra. Chapter 8 may also be included in an introductory course for students with some exposure to linear control systems. The independent joint control problem largely involves the control of actuator and drive-train dynamics; hence most of the subject can be taught without prior knowledge of Euler-Lagrange dynamics.
A graduate-level course on robot dynamics and control can be taught using all or parts of Chapters 6 through 12.
Finally, one or more special topics courses can be taught using Chapters 9 through 14. Below we outline several possible courses that can be taught from this book:
Course 1: Introduction to Robotics
Level: Junior/Senior undergraduate
For a one quarter course (10 weeks):
- Chapter 1: Introduction
- Chapter 2: Rigid Motions and Homogeneous Transformations
- Chapter 3: Forward Kinematics
- Chapter 4: Velocity Kinematics and Jacobians
- Chapter 5: Inverse Kinematics
For a one semester course (16 weeks) add:
- Chapter 7: Motion Planning and Trajectory Generation
- Chapter 8: Independent Joint Control
Course 2: Robot Dynamics and Control
Level: Senior undergraduate/graduate
For a one quarter course (10 weeks):
- Chapters 1-5: Rapid Review of Kinematics (selected sections)
- Chapter 6: Dynamics
- Chapter 7: Path and Trajectory Planning
- Chapter 9: Nonlinear and Multivariable Control
- Chapter 10: Force Control
For a one semester course (16 weeks) add:
- Chapter 11: Vision-Based Control
- Chapter 12: Feedback Linearization
Course 3: Advanced Topics in Robot Control
Level: Graduate
For a one semester course (16 weeks):
- Chapter 6: Dynamics
- Chapter 7: Motion Planning and Trajectory Generation
- Chapter 9: Nonlinear and Multivariable Control
- Chapter 11: Vision-Based Control
- Chapter 12: Feedback Linearization
- Chapter 13: Underactuated Robots
- Chapter 14: Mobile Robots
The instructor may wish to supplement the...
