Parallel Robots Mechanics and Control H AMID D TAG HI RAD CRC Press Taylor & Francis Group Boca Raton London NewYoric CRC Press Is an Imprint of the Taylor & Francis Croup, an informs business
Contents Preface xi 1 Introduction 1 11 What Is a Robot? 1 12 Robot Components 3 13 Robot Degrees-of-Freedom 6 14 Robot Classification 10 141 Serial Robots 10 142 Parallel Robots 13 1421 The Stewart-Gough Platform 14 1422 The Delta Robot 17 143 Cable-Driven Parallel Robots 18 15 The Aims and Scope of This Book 21 2 Motion Representation 23 21 Spatial Motion Representation 23 211 Position of a Point 24 212 Orientation of a Rigid Body 24 2121 Rotation Matrix 25 2122 Rotation Matrix Properties 27 2123 Screw Axis Representation 31 2124 Euler Angles 35 22 Motion of a Rigid Body 42 23 Homogeneous 231 Homogeneous 232 Homogeneous Transformations 44 Coordinates 44 Transformation Matrix 45 233 Screw Displacement 47 234 Transformation Arithmetics 49 2341 Consecutive Transformations 49 2342 Inverse Transformation 52 Problems 53 3 Kinematics 59 31 Introduction 59 32 Loop Closure Method 61 33 Kinematic Analysis of a Planar Manipulator 62 331 Mechanism Description 62 332 Geometry of the Manipulator 63 333 Inverse Kinematics 63 334 Forward Kinematics 65 335 Simulations,66
yi Contents 34 Kinematic Analysis of Shoulder Manipulator 69 341 Mechanism Description 69 342 Geometry of the Manipulator 71 343 Inverse Kinematics 73 344 Forward Kinematics 74 345 Simulations 75 35 Kinematic Analysis of Stewart-Gough Platform 77 351 Mechanism Description 77 352 Geometry of the Manipulator 78 353 Inverse Kinematics 79 354 Forward Kinematics 80 3541 Background Literature 80 3542 Analytical Solution 82 3543 Numerical Solution 87 355 Simulations 88 3551 Analytical Solution 89 3552 Numerical Solution 91 Problems 95 : 127 4 Jacobians: Velocities and Static Forces Ill 41 Introduction Ill 42 Angular and Linear Velocities 112 421 Angular Velocity of a Rigid Body 112 4211 Angular Velocity and Rotation Matrix Rate 113 4212 Angular Velocity and Euler Angles Rate 114 422 Linear Velocity of a Point 114 423 Screw Coordinates 116 43 Jacobian Matrices of a Parallel Manipulator 118 44 Velocity Loop Closure 119 45 Singularity Analysis of Parallel Manipulators 120 451 Inverse Kinematic Singularity 121 452 Forward Kinematic Singularity 121 453 Combined Singularity 122 46 Jacobian Analysis of a Planar Manipulator 122 461 Velocity Loop Closure 122 462 Singularity Analysis 125 463 Sensitivity Analysis 126 47 Jacobian Analysis of Shoulder Manipulator 471 Velocity Loop Closure 128 4711 Jacobian of the Actuated Limbs 128 4712 Jacobian of the Passive Limb 129 472 Singularity Analysis 130 48 Jacobian Analysis of the Stewart-Gough Platform 132 481 Velocity Loop Closure 132 482 Singularity Analysis 134 4821 Background Literature 134 4822 A 3-6 Stewart-Gough Platform 136 49 Static Forces in Parallel Manipulators 139 491 Free-Body Diagram Approach 139
Contents vii 492 Virtual Work Approach 140 493 Static Forces of a Planar Manipulator 142 494 Static Forces of Shoulder Manipulator 144 495 Static Forces of the Stewart-Gough Platform 146 410 Stiffness Analysis of Parallel Manipulators 147 4101 Stiffness and Compliance Matrices 148 4102 Transformation Ellipsoid 149 4103 Stiffness Analysis of a Planar Manipulator 151 4104 Stiffness Analysis of Shoulder Manipulator 153 4105 Stiffness Analysis of the Stewart-Gough Platform 155 Problems 159 5 Dynamics 167 51 Introduction 167 52 Dynamics of Rigid Bodies: A Review 169 521 Acceleration of Rigid Bodies 169 5211 Angular Acceleration of a Rigid Body 170 5212 Linear Acceleration of a Point 170 522 Mass Properties 171 5221 Center of Mass 172 5222 Moments of Inertia 172 5223 Principal Axes 173 5224 Inertia Matrix Transformations 173 523 Momentum and Kinetic Energy 174 5231 Linear Momentum 174 5232 Angular Momentum 175 5233 Kinetic Energy 176 524 Newton-Euler Laws 177 525 Variable-Mass Systems 178 53 Newton-Euler Formulation 180 531 Dynamic Formulation of a Planar Manipulator: Constant Mass Treatment 181 5311 Acceleration Analysis 181 5312 Dynamic Formulation of the Limbs 182 5313 Dynamic Formulation of the Moving Platform 184 5314 Forward Dynamics Simulations 185 5315 Inverse Dynamics Simulation 190 532 Dynamic Formulation of a Planar Manipulator: Variable-Mass Treatment 193 5321 Acceleration Analysis 193 5322 Dynamic Analysis of the Limbs 196 533 Dynamic Formulation of the Stewart-Gough Platform 199 5331 Acceleration Analysis 199 5332 Dynamic Formulation of the Limbs 202 5333 Dynamic Formulation of the Moving Platform 206 534 Closed-Form Dynamics 207 5341 Closed-Form Dynamics of the Limbs 207 5342 Closed-Form Dynamics of the Moving Platform 209
viii Contents 5343 Closed-Form Dynamics of the Stewart-Gough Manipulator 210 5344 Forward Dynamics Simulations 211 5345 Inverse Dynamics Simulation 215 54 Virtual Work Formulation 541 D'Alembert's Principle 221 542 Principle of Virtual Work 543 Dynamic Formulation of a Planar Manipulator: Constant Mass Treatment 544 Formulation Verification 545 Dynamic Formulation of a Planar Manipulator: Variable Mass Treatment 546 Dynamic Formulation of the Stewart-Gough Platform 229 55 Lagrange Formulation 221 222 225 227 228 551 Generalized Coordinates 232 552 Lagrange Equations of the Second Kind 234 553 Lagrange Equations of the First Kind 236 554 Dynamic Formulation Properties 237 5541 Mass Matrix Properties 238 5542 Linearity in Parameters 5543 Coriolis and Centrifugal Vector Properties 239 555 Dynamic Formulation of a Planar Manipulator 243 5551 Dynamic Formulation of the Limbs 245 5552 Dynamic Formulation of the Moving Platform 249 5553 Dynamic Formulation of the Whole Manipulator 250 556 Dynamic Analysis of the Stewart-Gough Platform 252 5561 Dynamic Formulation of the Limbs 253 5562 Dynamic Formulation of the Moving Platform 258 232 239 Problems 5563 Dynamic Formulation of the Whole Manipulator 260 260 6 Motion Control 61 Introduction 62 Controller Topology 270 63 Motion Control in Task Space 274 631 Decentralized PD Control 632 Feed Forward Control 633 Inverse Dynamics Control 634 Partial Linearization IDC 64 Robust and Adaptive Control 641 Robust Inverse Dynamics Control 642 Adaptive Inverse Dynamics Control ' 269 269 274 275 277 279 65 Motion Control in Joint Space 292 651 Dynamic Formulation in the Joint Space 293 652 Decentralized PD Control 294 653 Feed Forward Control 295 654 Inverse Dynamics Control 297 66 Summary of Motion Control Techniques 299 661 Dynamic Formulations 662 Decentralized PD Control 282 283 289 300 300
Contents ix 300 310 316 323 663 Feed Forward Control 664 Inverse Dynamics Control 301 665 Partial Linearization IDC 301 666 Robust Inverse Dynamics Control 301 667 Adaptive Inverse Dynamics Control 302 303 67 Redundancy Resolution 671 Introduction 303 672 Problem Formulation 304 673 Lagrange and Karush-Kuhn-Tucker Multipliers 307 674 Iterative Solutions 310 6741 Numerical Methods 6742 An Iterative-Analytical Method 312 68 Motion Control of a Planar Manipulator 681 Decentralized PD Control 316 682 Feed Forward Control 683 Inverse Dynamics Control ; 327 684 Partial Linearization IDC 333 685 Robust Inverse Dynamics Control 335 686 Adaptive Inverse Dynamics Control 338 343 687 Motion Control in Joint Space 69 Motion Control of the Stewart-Gough Platform 343 691 Decentralized PD Control 344 692 Feed Forward Control 351 693 Inverse Dynamics Control 354 694 Partial Linearization IDC 358 695 Robust Inverse Dynamics Control 360 696 Motion Control in Joint Space 365 6961 Decentralized PD Control 365 6962 Feed Forward Control 373 6963 Inverse Dynamics Control 376 Problems 382 7 Force Control 391 71 Introduction 391 72 Controller Topology 392 721 Cascade Control 394 722 Force Feedback in Outer Loop 395 723 Force Feedback in Inner Loop 397 73 Stiffness Control 400 731 Single-Degree-of-Freedom Stiffness Control 401 732 General Stiffness Control 405 733 Stiffness Control of a Planar Manipulator 408 734 Stiffness Control of the Stewart-Gough Platform 416 74 Direct Force Control 423 741 Force Control of a Planar Manipulator 428 742 Force Control of the Stewart-Gough Platform 433 75 Impedance Control 441 751 Impedance 752 Impedance Control Concept 443 445
x Contents 753 Impedance Control Structure 448 754 Impedance Control of a Planar Manipulator 451 755 Impedance Control of the Stewart-Gough Platform 456 Problems 464 Appendix A: Linear Algebra 477 Al Vectors and Matrices 477 A2 Vector and Matrix Operations 478 A3 Eigenvalues and Singular Values 480 A4 Pseudo-Inverse 482 A41 Pseudo-Inverse Properties 483 A42 Linear Inverse Problems 484 A 5 Kronecker Product 485 Appendix B: Trajectory Planning 487 B l Point-to-Point Motion 487 Bll Cubic Polynomials 487 Bl2 Quintic Polynomials 488 B13 Linear Segments with Parabolic Blends 489 B14 Minimum Time Trajectory 490 B2 Specified Path with Via Points 491 Appendix C: Nonlinear Control Review 493 C l Dynamical Systems 493 C2 Stability Definitions 494 C3 Lyapunov Stability 495 C4 Krasovskii-Lasalle Theorem 498 References 501 Index 511