Robot Modeling and Control

A New Edition Featuring Case Studies and Examples of the Fundamentals of Robot Kinematics, Dynamics, and Control

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.

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.

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

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