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```Robot Modeling and
Control
First Edition
Mark W. Spong, Seth Hutchinson, and M. Vidyasagar
JOHN WILEY & SONS, INC.
New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
Preface
TO APPEAR
i
Contents
Preface i
1 INTRODUCTION 1
1.1 Mathematical Modeling of Robots 3
1.1.1 Symbolic Representation of Robots 3
1.1.2 The Configuration Space 4
1.1.3 The State Space 5
1.1.4 The Workspace 5
1.2 Robots as Mechanical Devices 5
1.2.1 Classification of Robotic Manipulators 5
1.2.2 Robotic Systems 7
1.2.3 Accuracy and Repeatability 7
1.2.4 Wrists and End-Effectors 8
1.3 Common Kinematic Arrangements of Manipulators
9
1.3.1 Articulated manipulator (RRR) 10
1.3.2 Spherical Manipulator (RRP) 11
1.3.3 SCARA Manipulator (RRP) 12
iii
iv CONTENTS
1.3.4 Cylindrical Manipulator (RPP) 13
1.3.5 Cartesian manipulator (PPP) 14
1.3.6 Parallel Manipulator 15
1.4 Outline of the Text 16
1.5 Chapter Summary 24
Problems 26
2 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS
29
2.1 Representing Positions 30
2.2 Representing Rotations 32
2.2.1 Rotation in the plane 32
2.2.2 Rotations in three dimensions 35
2.3 Rotational Transformations 37
2.3.1 Similarity Transformations 41
2.4 Composition of Rotations 42
2.4.1 Rotation with respect to the current frame 42
2.4.2 Rotation with respect to the fixed frame 44
2.5 Parameterizations of Rotations 46
2.5.1 Euler Angles 47
2.5.2 Roll, Pitch, Yaw Angles 49
2.5.3 Axis/Angle Representation 50
2.6 Rigid Motions 53
2.7 Homogeneous Transformations 54
2.8 Chapter Summary 57
3 FORWARD AND INVERSE KINEMATICS 65
3.1 Kinematic Chains 65
3.2 Forward Kinematics: The Denavit-Hartenberg
Convention 68
3.2.1 Existence and uniqueness issues 69
3.2.2 Assigning the coordinate frames 72
3.2.3 Examples 75
3.3 Inverse Kinematics 85
3.3.1 The General Inverse Kinematics Problem 85
3.3.2 Kinematic Decoupling 87
3.3.3 Inverse Position: A Geometric Approach 89
3.3.4 Inverse Orientation 97
3.3.5 Examples 98
CONTENTS v
3.4 Chapter Summary 100
3.5 Notes and References 102
Problems 103
4 VELOCITY KINEMATICS – THE MANIPULATOR JACOBIAN113
4.1 Angular Velocity: The Fixed Axis Case 114
4.2 Skew Symmetric Matrices 115
4.2.1 Properties of Skew Symmetric Matrices 116
4.2.2 The Derivative of a Rotation Matrix 117
4.3 Angular Velocity: The General Case 118
4.4 Addition of Angular Velocities 119
4.5 Linear Velocity of a Point Attached to a Moving
Frame 121
4.6 Derivation of the Jacobian 122
4.6.1 Angular Velocity 123
4.6.2 Linear Velocity 124
4.6.3 Combining the Angular and Linear
Jacobians 126
4.7 Examples 127
4.8 The Analytical Jacobian 131
4.9 Singularities 132
4.9.1 Decoupling of Singularities 133
4.9.2 Wrist Singularities 134
4.9.3 Arm Singularities 134
4.10 Inverse Velocity and Acceleration 139
4.11 Manipulability 141
4.12 Chapter Summary 144
Problems 146
5 PATH AND TRAJECTORY PLANNING 149
5.1 The Configuration Space 150
5.2 Path Planning Using Configuration Space
Potential Fields 154
5.2.1 The Attractive Field 154
5.2.2 The Repulsive field 156
5.3 Planning Using Workspace Potential Fields 158
5.3.1 Defining Workspace Potential Fields 159
vi CONTENTS
5.3.2 Mapping workspace forces to joint forces
and torques 161
5.3.3 Motion Planning Algorithm 165
5.4 Using Random Motions to Escape Local Minima 166
5.5.1 Sampling the configuration space 169
5.5.2 Connecting Pairs of Configurations 169
5.5.3 Enhancement 170
5.5.4 Path Smoothing 170
5.6 trajectory planning 171
5.6.1 Trajectories for Point to Point Motion 173
5.6.2 Trajectories for Paths Specified by Via
Points 182
5.7 Historical Perspective 184
Problems 186
6 DYNAMICS 187
6.1 The Euler-Lagrange Equations 188
6.1.1 One Dimensional System 188
6.1.2 The General Case 190
6.2 General Expressions for Kinetic and Potential
Energy 196
6.2.1 The Inertia Tensor 197
6.2.2 Kinetic Energy for an n-Link Robot 199
6.2.3 Potential Energy for an n-Link Robot 200
6.3 Equations of Motion 200
6.4 Some Common Configurations 202
6.5 Properties of Robot Dynamic Equations 211
6.5.1 The Skew Symmetry and Passivity
Properties 212
6.5.2 Bounds on the Inertia Matrix 213
6.5.3 Linearity in the Parameters 214
6.6 Newton-Euler Formulation 215
6.7 Planar Elbow Manipulator Revisited 222
Problems 225
7 INDEPENDENT JOINT CONTROL 229
7.1 Introduction 229
7.2 Actuator Dynamics 231
CONTENTS vii
7.3 Set-Point Tracking 237
7.3.1 PD Compensator 238
7.3.2 Performance of PD Compensators 239
7.3.3 PID Compensator 240
7.3.4 Saturation 242
7.4 Feedforward Control and Computed Torque 244
7.5 Drive Train Dynamics 248
7.6 State Space Design 251
7.6.1 State Feedback Compensator 254
7.6.2 Observers 256
Problems 259
8 MULTIVARIABLE CONTROL 263
8.1 Introduction 263
8.2 PD Control Revisited 264
8.3 Inverse Dynamics 266
8.3.1 Task Space Inverse Dynamics 269
8.4 Robust and Adaptive Motion Control 271
8.4.1 Robust Feedback Linearization 271
8.4.2 Passivity Based Robust Control 275
8.4.3 Passivity Based Adaptive Control 277
Problems 279
9 FORCE CONTROL 281
9.1 Introduction 281
9.2 Coordinate Frames and Constraints 282
9.2.1 Natural and Artificial Constraints 284
9.3 Network Models and Impedance 285
9.3.1 Impedance Operators 288
9.3.2 Classification of Impedance Operators 288
9.3.3 The´venin and Norton Equivalents 289
9.4 Task Space Dynamics and Control 290
9.4.1 Static Force/Torque Relationships 290
9.4.3 Impedance Control 292
9.4.4 Hybrid Impedance Control 293
Problems 297
10 GEOMETRIC NONLINEAR CONTROL 299
viii CONTENTS
10.1 Introduction 299
10.2 Background 300
10.2.1 The Frobenius Theorem 304
10.3 Feedback Linearization 306
10.4 Single-Input Systems 308
10.5 Feedback Linearization for n-Link Robots 315
10.6 Nonholonomic Systems 318
10.6.1 Involutivity and Holonomy 319
10.6.2 Driftless Control Systems 320
10.6.3 Examples of Nonholonomic Systems 320
10.7 Chow’s Theorem and Controllability of Driftless
Systems 324
Problems 328
11 COMPUTER VISION 331
11.1 The Geometry of Image Formation 332
11.1.1 The Camera Coordinate Frame 332
11.1.2 Perspective Projection 333
11.1.3 The Image Plane and the Sensor Array 334
11.2 Camera Calibration 334
11.2.1 Extrinsic Camera Parameters 335
11.2.2 Intrinsic Camera Parameters 335
11.2.3 Determining the Camera Parameters 336
11.3 Segmentation by Thresholding 338
11.3.1 A Brief Statistics Review 339
11.3.2 Automatic Threshold Selection 341
11.4 Connected Components 346
11.5 Position and Orientation 348
11.5.1 Moments 349
11.5.2 The Centroid of an Object 349
11.5.3 The Orientation of an Object 350
Problems 353
12 VISION-BASED CONTROL 355
12.1 Approaches to vision based-control 356
12.1.1 Where to put the camera 356
12.1.2 How to use the image data 357
12.2 Camera Motion and Interaction Matrix 357
12.2.1 Interaction matrix vs. Image Jacobian 358
CONTENTS ix
12.3 The interaction matrix for points 359
12.3.1 Velocity of a fixed point relative to a moving
camera 360
12.3.2 Constructing the Interaction Matrix 361
12.3.3 Properties of the Interaction Matrix for
Points 363
12.3.4 The Interaction Matrix for Multiple Points 363
12.4 Image-Based Control Laws 364
12.4.1 Computing Camera Motion 365
12.4.2 Proportional Control Schemes 366
12.5 The relationship between end effector and camera
motions 367
12.6 Partitioned Approaches 369
12.7 Motion Perceptibility 372
12.8 Chapter Summary 374
Problems 375
Appendix A Geometry and Trigonometry 377
A.1 Trigonometry 377
A.1.1 Atan2 377
A.1.2 Reduction formulas 378
A.1.3 Double angle identitites 378
A.1.4 Law of cosines 378
Appendix B Linear Algebra 379
B.1 Differentiation of Vectors```