Robot Modeling and Control

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 v1 Introduction 11.1 Mathematical Modeling of Robots 51.1.1 Symbolic Representation of Robot Manipulators 51.1.2 The Configuration Space 51.1.3 The State Space 61.1.4 The Workspace 71.2 Robots as Mechanical Devices 71.2.1 Classification of Robotic Manipulators 81.2.2 Robotic Systems 101.2.3 Accuracy and Repeatability 101.2.4 Wrists and End Effectors 121.3 Common Kinematic Arrangements 131.3.1 Articulated Manipulator (RRR) 131.3.2 Spherical Manipulator (RRP) 141.3.3 SCARA Manipulator (RRP) 141.3.4 Cylindrical Manipulator (RPP) 151.3.5 Cartesian Manipulator (PPP) 151.3.6 Parallel Manipulator 181.4 Outline of the Text 181.4.1 Manipulator Arms 181.4.2 Underactuated and Mobile Robots 27Problems 27Notes and References 29I The Geometry of Robots 332 Rigid Motions 352.1 Representing Positions 362.2 Representing Rotations 382.2.1 Rotation in the Plane 382.2.2 Rotations in Three Dimensions 412.3 Rotational Transformations 442.4 Composition of Rotations 482.4.1 Rotation with Respect to the Current Frame 482.4.2 Rotation with Respect to the Fixed Frame 502.4.3 Rules for Composition of Rotations 512.5 Parameterizations of Rotations 522.5.1 Euler Angles 532.5.2 Roll, Pitch, Yaw Angles 552.5.3 Axis-Angle Representation 572.5.4 Exponential Coordinates 592.6 Rigid Motions 612.6.1 Homogeneous Transformations 622.6.2 Exponential Coordinates for General Rigid Motions 652.7 Chapter Summary 65Problems 67Notes and References 733 Forward Kinematics 753.1 Kinematic Chains 753.2 The Denavit-Hartenberg Convention 783.2.1 Existence and Uniqueness 803.2.2 Assigning the Coordinate Frames 833.3 Examples 873.3.1 Planar Elbow Manipulator 873.3.2 Three-Link Cylindrical Robot 893.3.3 The Spherical Wrist 903.3.4 Cylindrical Manipulator with Spherical Wrist 913.3.5 Stanford Manipulator 933.3.6 SCARA Manipulator 953.4 Chapter Summary 96Problems 96Notes and References 994 Velocity Kinematics 1014.1 Angular Velocity: The Fixed Axis Case 1024.2 Skew-Symmetric Matrices 1034.2.1 Properties of Skew-Symmetric Matrices 1044.2.2 The Derivative of a Rotation Matrix 1054.3 Angular Velocity: The General Case 1074.4 Addition of Angular Velocities 1084.5 Linear Velocity of a Point Attached to a Moving Frame 1104.6 Derivation of the Jacobian 1114.6.1 Angular Velocity 1124.6.2 Linear Velocity 1134.6.3 Combining the Linear and Angular Velocity Jacobians 1154.7 The Tool Velocity 1194.8 The Analytical Jacobian 1214.9 Singularities 1224.9.1 Decoupling of Singularities 1234.9.2 Wrist Singularities 1254.9.3 Arm Singularities 1254.10 Static Force/Torque Relationships 1294.11 Inverse Velocity and Acceleration 1314.12 Manipulability 1334.13 Chapter Summary 136Problems 138Notes and References 1405 Inverse Kinematics 1415.1 The General Inverse Kinematics Problem 1415.2 Kinematic Decoupling 1435.3 Inverse Position: A Geometric Approach 1455.3.1 Spherical Configuration 1465.3.2 Articulated Configuration 1485.4 Inverse Orientation 1515.5 Numerical Inverse Kinematics 1565.6 Chapter Summary 158Problems 160Notes and References 162II Dynamics and Motion Planning 1636 Dynamics 1656.1 The Euler-Lagrange Equations 1666.1.1 Motivation 1666.1.2 Holonomic Constraints and Virtual Work 1706.1.3 D'Alembert's Principle 1746.2 Kinetic and Potential Energy 1776.2.1 The Inertia Tensor 1786.2.2 Kinetic Energy for an n-Link Robot 1806.2.3 Potential Energy for an n-Link Robot 1816.3 Equations of Motion 1816.4 Some Common Configurations 1846.5 Properties of Robot Dynamic Equations 1946.5.1 Skew Symmetry and Passivity 1946.5.2 Bounds on the Inertia Matrix 1966.5.3 Linearity in the Parameters 1966.6 Newton-Euler Formulation 1986.6.1 Planar Elbow Manipulator Revisited 2066.7 Chapter Summary 209Problems 211Notes and References 2147 Path and Trajectory Planning 2157.1 The Configuration Space 2167.1.1 Representing the Configuration Space 2177.1.2 Configuration Space Obstacles 2187.1.3 Paths in the Configuration Space 2217.2 Path Planning for Q = ℝ2 2217.2.1 The Visibility Graph 2227.2.2 The Generalized Voronoi Diagram 2247.2.3 Trapezoidal Decompositions 2267.3 Artificial Potential Fields 2297.3.1 Artificial Potential Fields for Q = ℝn 2307.3.2 Potential Fields for Q ≠ ℝn 2357.4 Sampling-Based Methods 2457.4.1 Probabilistic Roadmaps (PRM) 2467.4.2 Rapidly-Exploring Random Trees (RRTs) 2507.5 Trajectory Planning 2527.5.1 Trajectories for Point-to-Point Motion 2537.5.2 Trajectories for Paths Specified by Via Points 2617.6 Chapter Summary 263Problems 265Notes and References 267III Control of Manipulators 2698 Independent Joint Control 2718.1 Introduction 2718.2 Actuator Dynamics 2738.3 Load Dynamics 2768.4 Independent Joint Model 2788.5 PID Control 2818.6 Feedforward Control 2888.6.1 Trajectory Tracking 2898.6.2 The Method of Computed Torque 2918.7 Drive-Train Dynamics 2928.8 State Space Design 2978.8.1 State Feedback Control 2998.8.2 Observers 3018.9 Chapter Summary 304Problems 307Notes and References 3099 Nonlinear and Multivariable Control 3119.1 Introduction 3119.2 PD Control Revisited 3139.3 Inverse Dynamics 3179.3.1 Joint Space Inverse Dynamics 3179.3.2 Task Space Inverse Dynamics 3209.3.3 Robust Inverse Dynamics 3229.3.4 Adaptive Inverse Dynamics 3279.4 Passivity-Based Control 3299.4.1 Passivity-Based Robust Control 3319.4.2 Passivity-Based Adaptive Control 3329.5 Torque Optimization 3339.6 Chapter Summary 337Problems 341Notes and References 34310 Force Control 34510.1 Coordinate Frames and Constraints 34710.1.1 Reciprocal Bases 34710.1.2 Natural and Artificial Constraints 34910.2 Network Models and Impedance 35110.2.1 Impedance Operators 35310.2.2 Classification of Impedance Operators 35410.2.3 Thévenin and Norton Equivalents 35510.3 Task Space Dynamics and Control 35510.3.1 Impedance Control 35610.3.2 Hybrid Impedance Control 35810.4 Chapter Summary 361Problems 362Notes and References 36411 Vision-Based Control 36511.1 Design Considerations 36611.1.1 Camera Configuration 36611.1.2 Image-Based vs. Position-Based Approaches 36711.2 Computer Vision for Vision-Based Control 36811.2.1 The Geometry of Image Formation 36911.2.2 Image Features 37311.3 Camera Motion and the Interaction Matrix 37811.4 The Interaction Matrix for Point Features 37911.4.1 Velocity Relative to a Moving Frame 38011.4.2 Constructing the Interaction Matrix 38111.4.3 Properties of the Interaction Matrix for Points 38411.4.4 The Interaction Matrix for Multiple Points 38511.5 Image-Based Control Laws 38611.5.1 Computing Camera Motion 38711.5.2 Proportional Control Schemes 38911.5.3 Performance of Image-Based Control Systems 39011.6 End Effector and Camera Motions 39311.7 Partitioned Approaches 39411.8 Motion Perceptibility 39711.9 Summary 399Problems 401Notes and References 40512 Feedback Linearization 40912.1 Background 41012.1.1 Manifolds, Vector Fields, and Distributions 41012.1.2 The Frobenius Theorem 41412.2 Feedback Linearization 41712.3 Single-Input Systems 41912.4 Multi-Input Systems 42912.5 Chapter Summary 433Problems 433Notes and References 435IV Control of Underactuated Systems 43713 Underactuated Robots 43913.1 Introduction 43913.2 Modeling 44013.3 Examples of Underactuated Robots 44313.3.1 The Cart-Pole System 44313.3.2 The Acrobot 44513.3.3 The Pendubot 44613.3.4 The Reaction-Wheel Pendulum 44713.4 Equilibria and Linear Controllability 44813.4.1 Linear Controllability 45013.5 Partial Feedback Linearization 45613.5.1 Collocated Partial Feedback Linearization 45713.5.2 Noncollocated Partial Feedback Linearization 45913.6 Output Feedback Linearization 46113.6.1 Computation of the Zero Dynamics 46313.6.2 Virtual Holonomic Constraints 46613.7 Passivity-Based Control 46613.7.1 The Simple Pendulum 46713.7.2 The Reaction-Wheel Pendulum 47113.7.3 Swingup and Balance of The Acrobot 47313.8 Chapter Summary 474Problems 476Notes and References 47714 Mobile Robots 47914.1 Nonholonomic Constraints 48014.2 Involutivity and Holonomy 48414.3 Examples of Nonholonomic Systems 48714.4 Dynamic Extension 49314.5 Controllability of Driftless Systems 49514.6 Motion Planning 49914.6.1 Conversion to Chained Forms 49914.6.2 Differential Flatness 50614.7 Feedback Control of Driftless Systems 50914.7.1 Stabilizability 50914.7.2 Nonsmooth Control 51114.7.3 Trajectory Tracking 51314.7.4 Feedback Linearization 51514.8 Chapter Summary 519Problems 520Notes and References 521A Trigonometry 523A.1 The Two-Argument Arctangent Function 523A.2 Useful Trigonometric Formulas 523B Linear Algebra 525B.1 Vectors 525B.2 Inner Product Spaces 526B.3 Matrices 528B.4 Eigenvalues and Eigenvectors 530B.5 Differentiation of Vectors 533B.6 The Matrix Exponential 534B.7 Lie Groups and Lie Algebras 534B.8 Matrix Pseudoinverse 536B.9 Schur Complement 536B.10 Singular Value Decomposition (SVD) 537C Lyapunov Stability 539C.1 Continuity and Differentiability 539C.2 Vector Fields and Equilibria 541C.3 Lyapunov Functions 545C.4 Stability Criteria 545C.5 Global and Exponential Stability 546C.6 Stability of Linear Systems 547C.7 LaSalle's Theorem 548C.8 Barbalat's Lemma 549D Optimization 551D.1 Unconstrained Optimization 551D.2 Constrained Optimization 552E Camera Calibration 555E.1 The Image Plane and the Sensor Array 555E.2 Extrinsic Camera Parameters 556E.3 Intrinsic Camera Parameters 557E.4 Determining the Camera Parameters 557Bibliography 561Index 576