Robot Modeling and Control

"The coverage is unparalleled in both depth and breadth. No other text that I have seen offers a better complete overview of modern robotic manipulation and robot control." -- Bradley Bishop, United States Naval Academy Based on the highly successful classic, Robot Dynamics and Control, by Spong and Vidyasagar (Wiley, 1989), Robot Modeling and Control offers a thoroughly up-to-date, self-contained introduction to the field. The text presents basic and advanced material in a style that is at once readable and mathematically rigorous. Key Features * A step-by-step computational approach helps you derive and compute the forward kinematics, inverse kinematics, and Jacobians for the most common robot designs. * Detailed coverage of vision and visual servo control enables you to program robots to manipulate objects sensed by cameras. * An entire chapter on dynamics prepares you to compute the dynamics of the most common manipulator designs. * The most common motion planning and trajectory generation algorithms are presented in an elementary style. * The comprehensive treatment of motion and force control includes both basic and advanced methods. * The text's treatment of geometric nonlinear control is more readable than in more advanced texts. * Many worked examples and an extensive list of problems illustrate all aspects of the theory. About the authors Mark W. Spong is Donald Biggar Willett Professor of Engineering at the University of Illinois at Urbana-Champaign. Dr. Spong is the 2005 President of the IEEE Control Systems Society and past Editor-in-Chief of the IEEE Transactions on Control Systems Technology. Seth Hutchinson is currently a Professor at the University of Illinois in Urbana-Champaign, and a senior editor of the IEEE Transactions on Robotics and Automation. He has published extensively on the topics of robotics and computer vision. Mathukumalli Vidyasagar is currently Executive Vice President in charge of Advanced Technology at Tata Consultancy Services (TCS), India's largest IT firm. Dr. Vidyasagar was formerly the director of the Centre for Artificial Intelligence and Robotics (CAIR), under Government of India's Ministry of Defense. Preface 3 TABLE OF CONTENTS 3 INTRODUCTION 13 Mathematical Modeling of Robots 15 Symbolic Representation of Robots 15 The Configuration Space 16 The State Space 17 The Workspace 17 Robots as Mechanical Devices 17 Classification of Robotic Manipulators 17 Robotic Systems 19 Accuracy and Repeatability 19 Wrists and End-Effectors 20 Common Kinematic Arrangements of Manipulators 21 Articulated manipulator (RRR) 22 Spherical Manipulator (RRP) 23 SCARA Manipulator (RRP) 24 Cylindrical Manipulator (RPP) 25 Cartesian manipulator (PPP) 26 Parallel Manipulator 27 Outline of the Text 28 Chapter Summary 36 Problems 36 RIGID MOTIONS AND HOMOGENEOUS TRANSFORMATIONS 41 Representing Positions 42 Representing Rotations 44 Rotation in the plane 44 Rotations in three dimensions 47 Rotational Transformations 49 Similarity Transformations 53 Composition of Rotations 54 Rotation with respect to the current frame 54 Rotation with respect to the fixed frame 56 Parameterizations of Rotations 58 Euler Angles 59 Roll, Pitch, Yaw Angles 61 Axis/Angle Representation 62 Rigid Motions 65 Homogeneous Transformations 66 Chapter Summary 69 FORWARD AND INVERSE KINEMATICS 77 Kinematic Chains 77 Forward Kinematics: The Denavit-Hartenberg Convention 80 Existence and uniqueness issues 81 Assigning the coordinate frames 84 Examples 87 Inverse Kinematics 97 The General Inverse Kinematics Problem 97 Kinematic Decoupling 99 Inverse Position: A Geometric Approach 101 Inverse Orientation 109 Examples 110 Chapter Summary 112 Notes and References 114 Problems 114 VELOCITY KINEMATICS -- THE MANIPULATOR JACOBIAN 125 Angular Velocity: The Fixed Axis Case 126 Skew Symmetric Matrices 127 Properties of Skew Symmetric Matrices 128 The Derivative of a Rotation Matrix 129 Angular Velocity: The General Case 130 Addition of Angular Velocities 131 Linear Velocity of a Point Attached to a Moving Frame 133 Derivation of the Jacobian 134 Angular Velocity 135 Linear Velocity 136 Combining the Angular and Linear Jacobians 138 Examples 139 The Analytical Jacobian 143 Singularities 144 Decoupling of Singularities 145 Wrist Singularities 146 Arm Singularities 146 Inverse Velocity and Acceleration 151 Manipulability 153 Chapter Summary 156 Problems 157 PATH AND TRAJECTORY PLANNING 161 The Configuration Space 162 Path Planning Using Configuration Space Potential Fields 166 The Attractive Field 166 The Repulsive field 168 Gradient Descent Planning 169 Planning Using Workspace Potential Fields 170 Defining Workspace Potential Fields 171 Mapping workspace forces to joint forces and torques 173 Motion Planning Algorithm 177 Using Random Motions to Escape Local Minima 178 Probabilistic Roadmap Methods 179 Sampling the configuration space 181 Connecting Pairs of Configurations 181 Enhancement 182 Path Smoothing 182 trajectory planning 183 Trajectories for Point to Point Motion 185 Trajectories for Paths Specified by Via Points 194 Historical Perspective 196 Problems 196 DYNAMICS 199 The Euler-Lagrange Equations 200 One Dimensional System 200 The General Case 202 General Expressions for Kinetic and Potential Energy 208 The Inertia Tensor 209 Kinetic Energy for an n-Link Robot 211 Potential Energy for an n-Link Robot 212 Equations of Motion 212 Some Common Configurations 214 Properties of Robot Dynamic Equations 223 The Skew Symmetry and Passivity Properties 224 Bounds on the Inertia Matrix 225 Linearity in the Parameters 226 Newton-Euler Formulation 227 Planar Elbow Manipulator Revisited 234 Problems 236 INDEPENDENT JOINT CONTROL 241 Introduction 241 Actuator Dynamics 243 Set-Point Tracking 249 PD Compensator 250 Performance of PD Compensators 251 PID Compensator 252 Saturation 254 Feedforward Control and Computed Torque 256 Drive Train Dynamics 260 State Space Design 263 State Feedback Compensator 266 Observers 268 Problems 270 MULTIVARIABLE CONTROL 275 Introduction 275 PD Control Revisited 276 Inverse Dynamics 278 Task Space Inverse Dynamics 281 Robust and Adaptive Motion Control 283 Robust Feedback Linearization 283 Passivity Based Robust Control 287 Passivity Based Adaptive Control 289 Problems 290 FORCE CONTROL 293 Introduction 293 Coordinate Frames and Constraints 294 Natural and Artificial Constraints 296 Network Models and Impedance 297 Impedance Operators 300 Classification of Impedance Operators 300 Thévenin and Norton Equivalents 301 Task Space Dynamics and Control 302 Static Force/Torque Relationships 302 Task Space Dynamics 303 Impedance Control 304 Hybrid Impedance Control 305 Problems 308 GEOMETRIC NONLINEAR CONTROL 311 Introduction 311 Background 312 The Frobenius Theorem 316 Feedback Linearization 318 Single-Input Systems 320 Feedback Linearization for n-Link Robots 327 Nonholonomic Systems 330 Involutivity and Holonomy 331 Driftless Control Systems 332 Examples of Nonholonomic Systems 332 Chow's Theorem and Controllability of Driftless Systems 336 Problems 339 COMPUTER VISION 343 The Geometry of Image Formation 344 The Camera Coordinate Frame 344 Perspective Projection 345 The Image Plane and the Sensor Array 346 Camera Calibration 346 Extrinsic Camera Parameters 347 Intrinsic Camera Parameters 347 Determining the Camera Parameters 348 Segmentation by Thresholding 350 A Brief Statistics Review 351 Automatic Threshold Selection 353 Connected Components 358 Position and Orientation 360 Moments 361 The Centroid of an Object 361 The Orientation of an Object 362 Problems 364 VISION-BASED CONTROL 367 Approaches to vision based-control 368 Where to put the camera 368 How to use the image data 369 Camera Motion and Interaction Matrix 369 Interaction matrix vs. Image Jacobian 370 The interaction matrix for points 371 Velocity of a fixed point relative to a moving camera 372 Constructing the Interaction Matrix 373 Properties of the Interaction Matrix for Points 375 The Interaction Matrix for Multiple Points 375 Image-Based Control Laws 376 Computing Camera Motion 377 Proportional Control Schemes 378 The relationship between end effector and camera motions 379 Partitioned Approaches 381 Motion Perceptibility 384 Chapter Summary 386 Problems 386 Appendix A Geometry and Trigonometry 389 Trigonometry 389 Atan2 389 Reduction formulas 390 Double angle identitites 390 Law of cosines 390 Appendix B Linear Algebra 391 Differentiation of Vectors 393 Linear Independence 394 Change of Coordinates 395 Eigenvalues and Eigenvectors 395 Singular Value Decomposition (SVD) 395 Appendix C Lyapunov Stability 399 Quadratic Forms and Lyapunov Functions 401 Lyapunov Stability 402 Lyapunov Stability for Linear Systems 403 LaSalle's Theorem 404 Appendix D State Space Theory of Dynamical Systems 405 State Space Representation of Linear Systems 407 409 415 Robot Modeling and Control introduces the fundamentals of robot modeling and control and provides background material on terminology, linear algebra, dynamical systems and stability theory, followed by detailed coverage of forward and in-verse kinematics, Jacobians, Lagrangian dynamics, motion planning, robust and adaptive motion and force control, and com-puter vision. Both basic and advanced material is presented in a style that is readable and mathematically rigorous. The book provides relevant applications from industrial robotics and mobile robotics. Suitable for a one or two term course, this text is appropriate for undergraduate and graduate students from electrical engineering, mechanical engineering, computer science, and mathematics and can be used as a research reference. Many detailed worked examples and extensive problems illustrate theory and point the reader to more advanced topics. The field of robotics is rapidly expanding with renewed interest in industrial and commercial applications such as robot vacuums and space explorers. This book provides readers with in-depth information on such areas as velocity, kinematics, path and trajectory planning, independent joint control and more