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أخوانى فى الله أحضرت لكم كتاب Dynamics and Control of Robotic Systems Andrew J. Kurdila and Pinhas Ben-Tzvi Virginia Tech Virginia USA
![كتاب Dynamics and Control of Robotic Systems D_a_c_11](https://i.servimg.com/u/f61/15/36/74/14/d_a_c_11.jpg) و المحتوى كما يلي :
Contents Preface xiii Acknowledgment xv About the Companion Website xvii 1 Introduction 1 1.1 Motivation 1 1.2 Origins of Robotic Systems 5 1.3 General Structure of Robotic Systems 7 1.4 Robotic Manipulators 9 1.4.1 Typical Structure of Robotic Manipulators 9 1.4.2 Classification of Robotic Manipulators 11 1.4.2.1 Classification by Motion Characteristics 11 1.4.2.2 Classification by Degrees of Freedom 12 1.4.2.3 Classification by Driver Technology and Drive Power 12 1.4.2.4 Classification by Kinematic Structure 12 1.4.2.5 Classification by Workspace Geometry 14 1.4.3 Examples of Robotic Manipulators 14 1.4.3.1 Cartesian Robotic Manipulator 15 1.4.3.2 Cylindrical Robotic Manipulator 16 1.4.3.3 SCARA Robotic Manipulator 16 1.4.3.4 Spherical Robotic Manipulator 17 1.4.3.5 PUMA Robotic Manipulator 18 1.4.4 Spherical Wrist 18 1.4.5 Articulated Robot 20 1.5 Mobile Robotics 20 1.5.1 Humanoid Robots 20 1.5.2 Autonomous Ground Vehicles 22 1.5.3 Autonomous Air Vehicles 23 1.5.4 Autonomous Marine Vehicles 25 1.6 An Overview of Robotics Dynamics and Control Problems 26 1.6.1 Forward Kinematics 27 1.6.2 Inverse Kinematics 28 1.6.3 Forward Dynamics 28 1.6.4 Inverse Dynamics and Feedback Control 29viii Contents 1.6.5 Dynamics and Control of Robotic Vehicles 30 1.7 Organization of the Book 31 1.8 Problems for Chapter 1 33 2 Fundamentals of Kinematics 35 2.1 Bases and Coordinate Systems 35 2.1.1 N-Tuples and M × N Arrays 35 2.1.2 Vectors, Bases and Frames 39 2.1.2.1 Vectors 40 2.1.2.2 Bases and Frames 41 2.2 Rotation Matrices 49 2.3 Parameterizations of Rotation Matrices 52 2.3.1 Single Axis Rotations 52 2.3.2 Cascades of Rotation Matrices 56 2.3.2.1 Cascade Rotations about Moving Axes 56 2.3.2.2 Cascade Rotations about Fixed Axes 57 2.3.3 Euler Angles 57 2.3.3.1 The 3-2-1 Yaw-Pitch-Roll Euler Angles 58 2.3.3.2 The 3-1-3 Precession-Nutation-Spin Euler Angles 62 2.3.4 Axis Angle Parameterization 65 2.4 Position, Velocity, and Acceleration 68 2.5 Angular Velocity and Angular Acceleration 77 2.5.1 Angular Velocity 77 2.5.2 Angular Acceleration 83 2.6 Theorems of Kinematics 84 2.6.1 Addition of Angular Velocities 84 2.6.2 Relative Velocity 87 2.6.3 Relative Acceleration 88 2.6.4 Common Coordinate Systems 91 2.6.4.1 Cartesian Coordinates 91 2.6.4.2 Cylindrical Coordinates 92 2.6.4.3 Spherical Coordinates 94 2.7 Problems for Chapter 2, Kinematics 96 2.7.1 Problems on N-tuples and M × N Arrays 96 2.7.2 Problems on Vectors, Bases, and Frames 97 2.7.3 Problems on Rotation Matrices 98 2.7.4 Problems on Position, Velocity, and Acceleration 102 2.7.5 Problems on Angular Velocity 104 2.7.6 Problems on the Theorems of Kinematics 104 2.7.6.1 Problems on the Addition of Angular Velocities 104 2.7.7 Problems on Relative Velocity and Acceleration 105 2.7.8 Problems on Common Coordinate Systems 108 3 Kinematics of Robotic Systems 109 3.1 Homogeneous Transformations and Rigid Motion 109 3.2 Ideal Joints 115 3.2.1 The Prismatic Joint 116Contents ix 3.2.2 The Revolute Joint 117 3.2.3 Other Ideal Joints 119 3.3 The Denavit–Hartenberg Convention 121 3.3.1 Kinematic Chains and Numbering in the DH Convention 121 3.3.2 Definition of Frames in the DH Convention 123 3.3.3 Homogeneous Transforms in the DH Convention 124 3.3.4 The DH Procedure 127 3.3.5 Angular Velocity and Velocity in the DH Convention 133 3.4 Recursive O(N) Formulation of Forward Kinematics 138 3.4.1 Recursive Calculation of Velocity and Angular Velocity 140 3.4.2 Efficiency and Computational Cost 143 3.4.3 Recursive Calculation of Acceleration and Angular Acceleration 147 3.5 Inverse Kinematics 160 3.5.1 Solvability 160 3.5.2 Analytical Methods 163 3.5.2.1 Algebraic Methods 163 3.5.2.2 Geometric Methods 174 3.5.3 Optimization Methods 176 3.5.4 Inverse Velocity Kinematics 184 3.5.4.1 Singularity 185 3.6 Problems for Chapter 3, Kinematics of Robotic Systems 186 3.6.1 Problems on Homogeneous Transformations 186 3.6.2 Problems on Ideal Joints and Constraints 188 3.6.3 Problems on the DH Convention 188 3.6.4 Problems on Angular Velocity and Velocity for Kinematic Chains 190 3.6.5 Problems on Inverse Kinematics 195 4 Newton–Euler Formulations 197 4.1 Linear Momentum of Rigid Bodies 197 4.2 Angular Momentum of Rigid Bodies 203 4.2.1 First Principles 203 4.2.2 Angular Momentum and Inertia 208 4.2.3 Calculation of the Inertia Matrix 214 4.2.3.1 The Inertia Rotation Transformation Law 214 4.2.3.2 Principal Axes of Inertia 218 4.2.3.3 The Parallel Axis Theorem 221 4.2.3.4 Symmetry and Inertia 224 4.3 The Newton–Euler Equations 229 4.4 Euler’s Equation for a Rigid Body 233 4.5 Equations of Motion for Mechanical Systems 235 4.5.1 The General Strategy 235 4.5.2 Free Body Diagrams 236 4.6 Structure of Governing Equations: Newton–Euler Formulations 258 4.6.1 Differential Algebraic Equations (DAEs) 258 4.6.2 Ordinary Differential Equations (ODEs) 260x Contents 4.7 Recursive Newton–Euler Formulations 262 4.8 Recursive Derivation of the Equations of Motion 271 4.9 Problems for Chapter 4, Newton–Euler Equations 274 4.9.1 Problems on Linear Momentum 274 4.9.2 Problems on the Center of Mass 277 4.9.3 Problems on the Inertia Matrix 279 4.9.4 Problems on Angular Momentum 281 4.9.5 Problems on the Newton–Euler Equations 282 5 Analytical Mechanics 285 5.1 Hamilton’s Principle 285 5.1.1 Generalized Coordinates 285 5.1.2 Functionals and the Calculus of Variations 288 5.1.3 Hamilton’s Principle for Conservative Systems 292 5.1.4 Kinetic Energy for Rigid Bodies 299 5.2 Lagrange’s Equations for Conservative Systems 303 5.3 Hamilton’s Extended Principle 307 5.3.1 Virtual Work Formulations 307 5.4 Lagrange’s Equations for Robotic Systems 322 5.4.1 Natural Systems 322 5.4.2 Lagrange’s Equations and the Denavit–Hartenberg Convention 326 5.5 Constrained Systems 329 5.6 Problems for Chapter 5, Analytical Mechanics 334 5.6.1 Problems on Hamilton’s Principle 334 5.6.2 Problems on Lagrange’s Equations 337 5.6.3 Problems on Hamilton’s Extended Principle 339 5.6.4 Problems on Constrained Systems 345 6 Control of Robotic Systems 347 6.1 The Structure of Control Problems 347 6.1.1 Setpoint and Tracking Feedback Control Problems 348 6.1.2 Open Loop and Closed Loop Control 349 6.1.3 Linear and Nonlinear Control 349 6.2 Fundamentals of Stability Theory 350 6.3 Advanced Techniques of Stability Theory 357 6.4 Lyapunov’s Direct Method 358 6.5 The Invariance Principle 361 6.6 Dynamic Inversion or Computed Torque Control 366 6.7 Approximate Dynamic Inversion and Uncertainty 376 6.8 Controllers Based on Passivity 389 6.9 Actuator Models 393 6.9.1 Electric Motors 393 6.9.2 Linear Actuators 400 6.10 Backstepping Control and Actuator Dynamics 404 6.11 Problems for Chapter 6, control of Robotic Systems 407Contents xi 6.11.1 Problems on Gravity Compensation and PD Setpoint Control 407 6.11.2 Problems on Computed Torque Tracking Control 412 6.11.3 Problems on Dissipativity Based Tracking Control 413 7 Image Based Control of Robotic Systems 415 7.1 The Geometry of Camera Measurements 415 7.1.1 Perspective Projection and Pinhole Camera Models 415 7.1.2 Pixel Coordinates and CCD Cameras 418 7.1.3 The Interaction Matrix 419 7.2 Image Based Visual Servo Control 423 7.2.1 Control Synthesis and Closed Loop Equations 424 7.2.2 Calculation of Initial Conditions 427 7.3 Task Space Control 441 7.4 Task Space and Visual Control 447 7.5 Problems for Chapter 7 459 A Appendix 465 A.1 Fundamentals of Linear Algebra 465 A.1.1 Solution of Matrix Equations 467 A.1.2 Linear Independence and Rank 468 A.1.3 Invertibility and Rank 470 A.1.4 Least Squares Approximation 470 A.1.5 Rank Conditions and the Interaction Matrix 475 A.2 The Algebraic Eigenvalue Problem 475 A.2.1 Self-adjoint Matrices 476 A.2.2 Jordan Canonical Form 478 A.3 Gauss Transformations and LU Factorizations 479 References 485 Index 489 Index a a priori 420 accelerometers 415 action functional 294, 298, 303, 304, 321, 331 actual motion 292, 307 actual trajectory 292 actuation moment 312 actuation moments 309 actuation torques 311 actuators 7, 311 addition theorem for angular velocities 84 admissible direction 307 admissible directions 304, 305 admissible variations 305 algebraic eigenvalue problem 219 algebraic unknowns 247, 253, 258 analog-to-digital 3 analytical mechanics 204, 288, 289, 293 angle encoders 415 angular acceleration 77, 83 angular momenta 203 angular momentum 203, 204, 208–210, 215, 230–232, 234–236, 282 angular velocities 236 angular velocity 71, 77, 79, 81, 95, 210, 215, 300, 315, 326, 420, 421, 424 anthropomorphic 20 anthropomorphic arm 20 anthropomorphic robot arm 20 anthropomorphic robot arms 20 anthropomorphic robots 20 approximate dynamic inversion 377–379, 389 arm sweep 20 armature 394 articulated robot arm 20 asymptotic stability 349, 354, 361, 362, 366 asymptotically stable 354, 355, 359, 362, 363, 365, 390 autonomous 364, 365 autonomous ground vehicles 22 axis angle parameterization 65 b back electromotive force (EMF) 395 back EMF 395 back EMF constant 395 backstepping control 405 base body 138 base frame 122 bases 35 basis 40–42 basis fixed derivative 71 brushes 394 c calculus of variations 288, 305, 307, 321 calibrated coordinates 416 calibration constants 419 calibration matrix 419 calibration parameter 418 camera coordinate trajectories 433 camera coordinates 416–419, 422, 428, 431, 433 camera extrinsic parameters 424 camera frame 415, 416, 420, 421, 424, 427, 429 camera frame coordinates 417, 421 Dynamics and Control of Robotic Systems, First Edition. Andrew J. Kurdila and Pinhas Ben-Tzvi.
2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/kurdila/robotic-systems490 Index camera intrinsic parameter matrix 419 camera model 415 canonical image plane coordinates 416 Capek, Karel 5 Cartesian manipulator 18 Cartesian robot 15, 16 Cartesian robotic manipulator 16 CCD 418 CCD arrays 418 center of mass 198, 199, 202–204, 209, 210, 214, 217, 218, 221–223, 228, 230–232, 235, 280, 300, 314–316, 326–328, 337 change of basis 40 change of basis formula 49 charge coupled device 418 chattering solutions 379 closed loop 424 closed loop connectivity 13 closed loop control 349 closed loop equations 431 closed loop system 424–426 closed loop topology 13 closed set of ordinary differential equations 426 closed system of ordinary differential equations 426 communicator 7 commutator 394 components 40, 45 computed control torques 376 computed torque control 350, 356, 369, 376, 377, 391 computed torque control law 366 computed torque controllers 348 concatenates 57 configuration 28 configuration space 292, 293 conformal partitions 39 connectivity topology 12 conservation of energy 325 conservative 304 conservative mechanical system 293, 304, 305, 331 conservative mechanical systems 285, 293, 303, 307 constrained optimization 177 constraint force 309 contemporaneous 307 control input 424, 425 control law 424 control synthesis 1 control unit 7 controllability 350 conventional manipulator 12 coordinate plane of symmetry 225 coordinate systems 35, 41 coordinates 40, 45 coordinates of the offset 428 core frame 122 Coriolis centripetal matrix 326 couple 311, 312 coupled, nonlinear ordinary differential equations 426 cross product 41 cross products of inertia 204, 206, 208, 221, 222, 224, 225, 227 cylindrical manipulator 12 cylindrical robot 16 d DAEs 235, 258 DC motor 393 decrescent 359 deficient 12 degrees of freedom 10, 11, 28, 282 Denavit–Hartenberg (DH) convention 28, 121, 326, 327 dependent variations 332 derivative feedback 367 derivative of unit vectors 71 derivative Theorem 420 derivative with respect to an observer 71 desired image point locations 424 Devol, George 5 dextral 41, 42 dextrous workspace 14 DH convention 121–125, 127, 130, 133, 190, 192 DH Convention 191 DH procedure 127, 128 diagonal matrix 38 diagonalizable 97 differential algebraic equations 258Index 491 differential unknowns 247, 253, 258 differential-algebraic equations 235 differentiation of rotation matrices 77 differentiation of unit vectors 77 digital-to-analog 2 direct drive manipulator 12 direction 293 direction cosine matrices 47 directional derivative 290, 291 dot product 37 driven joint 14 dynamic inversion 350, 356, 368, 369, 376 e (3-2-1) Euler angles 58 eigenvectors 219 electric robot 12 electromagnetic induction 393 electromechanical linear motors 400 end effector 7 equilibrium 351, 359, 362–364 Euclidean norm 37 Euler angles 57, 58, 62, 65 Euler’s equations 233 Euler’s first law 230–232, 235 Euler’s laws 211, 229, 235 Euler’s second law 209, 230–232, 235 exactly determined 425 exponential stability 354 extrema 288 extremization 288 extremization problems 288 f feature points 416, 420, 422, 423, 424, 429, 431–433, 438 feedback control 26, 27, 349 feedback control law 426 feedback controllers 349 feedback function 424 feedback linearization 350 feedforward control 367 final camera coordinates 431 final camera frame 427, 428, 430, 432, 433, 438 floating point operation 143 flops 143 focal coordinates 418 focal length 415, 419–421, 429, 463 focal plane 415–418, 431, 433 focal plane coordinates 417, 418, 420, 429, 431 focal plane trajectories 433, 438 forward dynamics 1, 12, 26, 27, 29, 348, 368, 369 forward kinematics 1, 26–28, 160 frame 40–42 frames of reference 35 free body diagrams 235, 236–238 full matrix 38 functionals 288 fundamental theorem of variational calculus 305 g G-derivative 288, 289 G-differentiable 289 gain matrices 367 Gateaux derivative 288, 289, 304 general purpose robot 12 generalized coordinates 285, 286, 288, 289, 292–294, 297, 303, 304, 306–310, 314, 316, 317, 319, 321, 322, 326, 329, 330, 339, 340 generalized displacements 308 generalized forces 308 generalized inertia 326 generalized inertia or mass matrix 322 generalized mass or inertia matrix 306 global asymptotic stability 359 global minimizer 177 global positioning system (GPS) 415 global stability 349–351 globally asymptotically stable 355, 363 globally exponentially stable 357 globally stable 351, 363 gripping device 7 h Hamilton’s extended principle 319–321 Hamilton’s principle 285, 293, 295, 296, 299, 303, 305, 307, 308, 319, 321492 Index holonomic constraints 329–331 homogeneous 189 homogeneous coordinates 110, 417, 419, 428 homogeneous transform 117, 118, 124, 417 homogeneous transformations 109, 110, 112, 117, 123, 191, 417, 428 humanoid arms 20 humanoid robots 21 hybrid manipulator 14 hydraulic robot 12 i IBVS 424, 426 IBVS controller 429, 432 ideal joints 9, 10, 11, 115, 116–119, 236 identity matrix 38 image based visual servo 423 image based visual servo control law 431 image Jacobian 441 image Jacobian matrix 419 image plane 424 image plane coordinates 420–422, 426 image plane tracking error 424 image point 416–418 inclination 63 independent 286, 329 index 35 inertia matrices 215 inertia matrix 208–210, 214, 217, 218, 223, 227, 228, 232, 280, 328 inertia rotation transformation law 214, 217, 223, 224 inertia tensor 209, 231, 300 inertial coordinates 418, 419 inertial frame 211, 230, 329, 420, 424, 429, 432 inertial frame 0 coordinates 417 inertial matrix 209 inertial reference frame 229 inflection points 288 initial camera configuration 431, 433 initial camera coordinates 431 initial camera frame 427, 428, 430, 432, 433, 438 initial condition 427, 432 initial coordinate frame 428 initial focal plane coordinates 428 initial focal plane tracking error 428 interaction matrix 419, 420, 422, 463 intrinsic 418 intrinsic parameter matrix 419 invariant 363, 365 invariant set 363 inventor 3, 4 inverse dynamics 26 inverse kinematics 1, 20, 26–28, 160, 162, 168, 176, 177 inverse matrix 96 invertible 425 j Jacobian matrices 133 Jacobian matrix 133, 185, 191, 194, 327, 330 joint angles 27 joint coordinate systems 116–118 joint frames 116 joint space control 441 joint variables 27, 28, 117, 118, 122, 133–135, 327 joints 9 k Karel Capek 5 kinematic chain 13, 14, 122, 123, 133, 326, 327 kinematic chains 13, 14, 121 kinematic constraints 321 kinematic decoupling 168, 176 Kinematic structure 12 kinematically decoupled 172 kinematics 35 kinetic energy 293, 300, 303, 319, 322, 326 kinetics 35 l labview 3 Lagrange multipliers 331, 332 Lagrange’s equations 303–307, 321, 340, 343 largest weakly invariant subset 365 LaSalle’s invariance principle 363Index 493 line of nodes 62 line-of-sight 416, 418 linear control theory 349, 367 linear matrix equations 38 linear momenta 197 linear momentum 197, 198, 200, 203, 204, 229, 230, 235, 275–277 linear multistep methods 427 linear ODEs 350 linear systems 349, 350 link displacement 124 link offset 124 link parameters 128 link rotation 124 link twist 124 links 9 local minimizer 176 local stability of an equilibrium 351 locally asymptotically stable 355 locally decrescent 359 locally negative definite 358, 359, 362, 363 locally negative semi-definite 365 locally positive definite 358, 359, 360, 362, 364 lower limit 35 Lyapunov functions 358, 360 m magnetic flux 395 magnetic flux linkage 395 magnetometers 415 main diagonal 38 manifolds 40 manipulator workspace 14 maple 4 mass center 200, 277, 278 mass matrix 293 mathcad 4 mathematica 4 matlab 4 matrix element 37 matrix inverse 38 matrix multiplication 38 maxima 288 measure valued solutions 379 mechatronics 2 method of dynamic inversion 350 minimal 286, 288, 329 minimum singular values 438, 441 modal matrix 219 moments of inertia 204, 206, 208, 217, 221, 224 motion 292 mscadams 3 multibody dynamics 12, 115 multibody system 13 multifunctional manipulator 6 n natural systems 322, 324, 325, 366, 389 negative definite 359, 363, 380, 390 negative semi-definite 362, 363, 378 Newton’s first law 229 Newton’s second law 229 non-autonomous system 351 non-conservative 321 non-conservative systems 307 nonconservative 319 nonlinear ODEs 350 nonlinear systems 349, 350 norm 36 number of degrees 311 number of degrees of freedom 236, 286 numerical integration methods 427 o observability 350 odd function 225, 226 ODEs 258, 260 open loop control 349 open loop manipulator 13 optimization 177 optimization theory 176 ordinary differential equations 235, 258, 426, 427, 430 origin offset 432 orthogonal matrix 97 orthonormal 41, 42 overdetermined 425 p pantograph 11 parallel axis theorem 214, 221, 223, 224, 228, 229, 280494 Index parallel manipulators 13 passive joint 15 passivity 389 passivity principles 348 passivity properties 389 permanent magnet 393 perspective projection 415 pinhole camera 415 pixel array 418 pixel coordinates 418, 419 pixels 418 planar manipulator 11 planes of symmetry 225, 226 Player Piano 5 pneumatic robot 12 position control 348 position feedback 367 position vectors 73, 310, 312, 416 positive definite 359, 362, 363, 377, 380 positive invariant 363, 366 positive invariant set 363, 378 possible motions 292, 293, 307 potential energy 293, 303, 319, 322 power supply 8 PPP manipulators 15 predictor corrector methods 427 principal axes 214, 218, 219, 225, 280 principal diagonal 38 principal moments of inertia 218, 219, 224, 234 principal point 418 princpal axes 218 prismatic joints 9, 14, 15, 27, 116, 117, 119, 122, 134 products of inertia 214 proengineer 3 projection matrix 419 proportional-derivative (PD) control 377 pseudo inverse 425 r ranges 426 rate gyros 415 reachable workspace 14 reaction forces 309 real time 160 recursive O(N) formulations 143, 157, 158 recursive formulations 138 recursive order (N) algorithms 271 recursive order (N) formulations 209 redundant 12 redundant generalized coordinates 286, 329–331, 333 relative acceleration 88 relative position 76, 77 relative velocity 87 representations 40 resultant moment 231 retinal coordinates 416 revolute joints 9, 14, 15, 44, 117, 118, 119, 122, 134 right ascension 63 right handed 41 right-handed 42 rigid body 300 rigid body motions 109, 189 robotic arm 6, 9 robotic control systems 415 robotic manipulators 9, 14, 15 robotic system 1, 5, 6, 9, 13, 322, 415 robotics 12 root frame 122 Rossum’s Universal Robots 5 rotation matrices 47, 49, 109 rotation matrix 428 rotation matrix parameterizations 52 Runge–Kutta methods 427 s scalar triple product 97, 300 scale factors 418 scaled coordinates 418 screw joint 9 sensor suites 8 sensors 8, 415 serial manipulator 13 setpoint control 348, 361, 377 setpoint control law 348 single axis rotations 52 single degree of freedom joints 10 singular value decomposition 425 skew 83 skew operator 41 skew symmetric 325, 326, 378, 389, 390Index 495 skew symmetry 389 solidworks 3 sparse matrix 38 spatial manipulator 12 spherical joint 9, 282 spherical manipulator 12, 16, 17 spherical robot 17, 18 spherical robotic manipulator 17 spherical wrist 18, 20, 192, 194 square matrix 38 stability 348, 349, 425 stabilizability 350 stable 351, 359, 362, 424 stable equilibrium 351 stable system 351 state variables 426, 427 stationarity 290 stationarity conditions 303 stationary 290, 304 stator 394 symbolic computations 322 symmetric positive definite 378, 379 system connectivity 12 system feature points 425 system interaction matrix 422, 423, 425, 438 system vectors 272 systems of ordinary differential equations 427 t Tait–Bryan angles 58 target camera configuration 431 task space controllers 441, 442 task space coordinates 441, 447 task space Jacobian matrix 441 task space variables 441 tensor analysis 40, 214 tensor basis 209 tensor transformation 214 time varying bases 69 topological tree 121 total time derivative 68, 71 tracking control 30, 348, 361 tracking control law 348 tracking control problem 424 tracking error 424 tracking error in the image plane 424 trajectory 292 trajectory tracking 348 transformation laws 40 transport theorem 421 transpose 37 tree topology connectivity 13 true motion 293, 307 u unconstrained optimization problem 177 underdetermined 425 uniformly ultimately bounded 381 Unimate 5 unit vector 40 universal joint 9, 119, 283 upper limit 35 v variation 307, 322 variation ????t 321 variation operator 308, 321 vector spaces 39, 40 vectors 39, 40, 45 virtual displacements 307, 308, 309–313, 318, 319, 321 virtual variation 307, 309, 311, 319, 320 virtual variation operator 307–310, 312, 319, 321, 330 virtual variations 307–309, 321, 330 virtual work 307–313, 315, 316, 319, 320 visual servo control 424 visual servo image based control law 431 Vonnegut, Kurt 5 w weakly invariant 363 weakly invariant set 363 weakly invariant subset 365 workspace 15–18 workspace geometry 14 wrist center 18 y yaw-pitch-roll angles 58
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