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عدد المساهمات : 18959 التقييم : 35383 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
| موضوع: كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control الخميس 24 أغسطس 2023, 3:43 am | |
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أخواني في الله أحضرت لكم كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control Third Edition
و المحتوى كما يلي :
Contents 1 Introduction 1 1.1 Historical Development . 2 1.2 Robot Components . 2 1.2.1 Link 2 1.2.2 Joint 3 1.2.3 Manipulator . 5 1.2.4 Wrist . 6 1.2.5 End-Effector . 7 1.2.6 Actuators 7 1.2.7 Sensors . 8 1.2.8 Controller . 8 1.3 Robot Classifications . 8 1.3.1 Geometry . 9 1.3.2 Workspace 11 1.3.3 Actuation . 12 1.3.4 Control 12 1.3.5 Application . 13 1.4 Robot’s Kinematics, Dynamics, and Control 13 1.5 Principle of Kinematics . 14 1.5.1 Triad . 14 1.5.2 Unit Vectors . 15 1.5.3 Orthogonality Condition . 20 1.5.4 Coordinate Frame and Transformation . 20 1.5.5 Vector Definition . 22 1.5.6 Vector Function 25 1.6 Summary . 28 1.7 Key Symbols 29 Exercises . 31 Part I Kinematics 2 Rotation Kinematics 37 2.1 Rotation About Global Cartesian Axes 37 2.2 Successive Rotation About Global Cartesian Axes . 43 2.3 Rotation About Local Cartesian Axes . 52 2.4 Successive Rotation About Local Cartesian Axes 55 2.5 Euler Angles 59 2.6 Local Axes Versus Global Axes Rotation 69 2.7 General Transformation . 71 2.8 Active and Passive Transformation . 79 xxixxii Contents 2.9 Summary . 80 2.10 Key Symbols 82 Exercises . 83 3 Orientation Kinematics 91 3.1 Axis–Angle Rotation . 91 3.2 Order-Free Rotation 102 3.3 Euler Parameters 109 3.4 Quaternions . 117 3.5 Spinors and Rotators . 124 3.6 Problems in Representing Rotations 126 3.6.1 Rotation Matrix 127 3.6.2 Axis–Angle . 127 3.6.3 Euler Angles 128 3.6.4 Quaternion 129 3.6.5 Euler Parameters . 130 3.7 Composition and Decomposition of Rotations 132 3.7.1 Composition of Rotations 132 3.7.2 Decomposition of Rotations 134 3.8 Summary . 137 3.9 Key Symbols 139 Exercises . 140 4 Motion Kinematics 149 4.1 Rigid Body Motion . 149 4.2 Homogenous Transformation . 153 4.3 Inverse and Reverse Homogenous Transformation . 162 4.4 Combined Homogenous Transformation . 167 4.5 Order-Free Transformation 176 4.6 Screw Coordinates . 182 4.7 Inverse Screw . 197 4.8 Combined Screw Transformation . 200 4.9 The Plücker Line Coordinate 202 4.10 The Geometry of Plane and Line . 207 4.10.1 Moment 208 4.10.2 Angle and Distance . 208 4.10.3 Plane and Line . 209 4.11 Screw and Plücker Coordinate . 212 4.12 Summary . 213 4.13 Key Symbols 215 Exercises . 216 5 Forward Kinematics . 225 5.1 Denavit–Hartenberg Notation 225 5.2 Transformation Between Adjacent Coordinate Frames 232 5.3 Forward Position Kinematics of Robots 247 5.4 Spherical Wrist 269 5.5 Assembling Kinematics . 276 5.6 Coordinate Transformation Using Screws 289 5.7 Non-Denavit–Hartenberg Methods . 293 5.8 Summary . 299 5.9 Key Symbols 300 Exercises . 302Contents xxiii 6 Inverse Kinematics 313 6.1 Decoupling Technique 313 6.2 Inverse Transformation Technique 329 6.3 Iterative Technique 343 6.4 Comparison of the Inverse Kinematics Techniques 347 6.4.1 Existence and Uniqueness of Solution . 347 6.4.2 Inverse Kinematics Techniques . 347 6.5 Singular Configuration . 347 6.6 Summary . 349 6.7 Key Symbols 350 Exercises . 352 Part II Derivative Kinematics 7 Angular Velocity 361 7.1 Angular Velocity Vector and Matrix . 361 7.2 Time Derivative and Coordinate Frames . 376 7.3 Rigid Body Velocity 388 7.4 Velocity Transformation Matrix 392 7.5 Derivative of a Homogenous Transformation Matrix . 400 7.6 Summary . 405 7.7 Key Symbols 407 Exercises . 409 8 Velocity Kinematics . 415 8.1 Rigid Link Velocity 415 8.2 Forward Velocity Kinematics 419 8.3 Jacobian Generating Vectors . 429 8.4 Inverse Velocity Kinematics . 442 8.5 Linear Algebraic Equations . 448 8.6 Matrix Inversion . 458 8.7 Nonlinear Algebraic Equations . 463 8.8 Jacobian Matrix From Link Transformation Matrices 469 8.9 Summary . 476 8.10 Key Symbols 477 Exercises . 479 9 Acceleration Kinematics . 489 9.1 Angular Acceleration Vector and Matrix . 489 9.2 Rigid Body Acceleration 508 9.3 Acceleration Transformation Matrix 510 9.4 Forward Acceleration Kinematics . 518 9.5 Inverse Acceleration Kinematics 520 9.6 Rigid Link Recursive Acceleration . 526 9.7 Second Derivative and Coordinate Frames . 534 9.8 Summary . 543 9.9 Key Symbols 545 Exercises . 547xxiv Contents Part III Dynamics 10 Applied Dynamics . 557 10.1 Force and Moment . 557 10.1.1 Force and Moment 557 10.1.2 Momentum 558 10.1.3 Equation of Motion . 558 10.1.4 Work and Energy . 559 10.2 Rigid Body Translational Kinetics 565 10.3 Rigid Body Rotational Kinetics . 567 10.4 Mass Moment Matrix . 576 10.5 Lagrange’s Form of Newton’s Equations . 584 10.6 Lagrangian Mechanics 591 10.7 Summary . 597 10.8 Key Symbols 600 Exercises . 602 11 Robot Dynamics 609 11.1 Rigid Link Newton–Euler Dynamics 609 11.2 Recursive Newton–Euler Dynamics . 626 11.3 Robot Lagrange Dynamics . 632 11.4 Lagrange Equations and Link Transformation Matrices 659 11.5 Robot Statics 667 11.6 Summary . 673 11.7 Key Symbols 676 Exercises . 678 Part IV Control 12 Path Planning 687 12.1 Cubic Path 687 12.2 Polynomial Path . 692 12.3 Non-polynomial Path Planning . 702 12.4 Spatial Path Design 705 12.5 Forward Path Robot Motion 708 12.6 Inverse Path Robot Motion . 712 12.7 Rotational Path 720 12.8 Summary . 724 12.9 Key Symbols 725 Exercises . 726 13 Time Optimal Control 731 13.1 Minimum Time and Bang-Bang Control . 731 13.2 Floating Time Method 738 13.3 Time Optimal Control for Robots . 746 13.4 Summary . 752 13.5 Key Symbols 753 Exercises . 754Contents xxv 14 Control Techniques . 759 14.1 Open- and Closed-Loop Control 759 14.2 Computed Torque Control . 764 14.3 Linear Control Technique . 767 14.3.1 Proportional Control 768 14.3.2 Integral Control 768 14.3.3 Derivative Control 768 14.4 Sensing and Control 770 14.4.1 Position Sensors 771 14.4.2 Speed Sensors 771 14.4.3 Acceleration Sensors 771 14.5 Summary . 772 14.6 Key Symbols 773 Exercises . 774 A Global Frame Triple Rotation 777 B Local Frame Triple Rotation . 779 C Principal Central Screws Triple Combination 781 D Industrial Link DH Matrices . 783 E Matrix Calculus 791 F Trigonometric Formula 797 G Algebraic Formula 805 H Unit Conversions . 807 Bibliography . 809 Index 815 Index A Acceleration angular, 489, 491, 501, 507, 508 bias vector, 521 body point, 384, 508, 510 centripetal, 500, 504, 508, 510 constant parabola, 705 constant path, 694 Coriolis, 510, 536, 563 definition, 27 discontinuous path, 700 discrete equation, 740, 748 end-effector, 499 forward kinematics, 518, 519 gravitational, 634, 660, 669 inverse kinematics, 520 jump, 689 matrix, 490, 510, 516, 533 mixed, 538, 540 mixed Coriolis, 543 mixed double, 543 Razi, 543, 552 recursive, 526, 528, 609 rotational transformation, 490, 499 sensors, 771 simple, 534 tangential, 500, 504, 508, 510, 536, 543 transformation, 494, 541 transformation matrix, 510 Accessory, 260 Active transformation, 79 Actuator, 7, 12 force and torque, 610, 632, 673 optimal torque, 750, 751 torque equation, 617, 642, 747 Admissible path, 735 Algorithm floating-time, 739, 746 inverse kinematics, 343 LU factorization, 451 LU solution, 451 Newton-Raphson, 464 Angle Euler, 59 nutation, 59 precession, 59 spin, 59 Angular acceleration, 489, 491, 508 B-expression, 502 end-effector, 499 Euler parameters, 501, 507 matrix, 490 principal, 502 quaternions, 507 recursive, 533 relative, 494, 500 rotational transformation, 490, 492 technical point, 504 vector, 490 Angular jerk, 506 Angular momentum, 569–571 2 link manipulator, 574 Angular velocity, 64, 66, 67, 97, 361, 364, 372, 491, 495, 502 alternative definition, 384, 386 combination, 364 coordinate transformation, 372 decomposition, 370 elements of matrix, 376 Euler frequencies, 370 Euler parameters, 375 instantaneous, 363 instantaneous axis, 361, 364, 372, 495 matrix, 362 principal matrix, 369 quaternions, 374 rate, 361 recursive, 418, 527 rom B-frame, 363 rom G-frame, 363 rotation matrix, 371 skew symmetric, 363 vector, 361 Anthropomorphic hand, 7 Arm, 2 Articulated arm, 9, 252, 254 manipulator, 9, 252, 254, 325, 433 robot, 252 Articulated manipulator equations of motion, 647 inverse kinematics, 319, 322, 323, 331 inverse velocity, 447 Jacobian matrix, 428, 472 left shoulder configuration, 336 right shoulder configuration, 336 Asimov, Isaac, 1 Associativity property, 76 Atan2 function, 317 Attitude angle, 58 Automorphism, 123 property, 122 Auxiliary frame, 285 Avicenna, 278 Axis-angle Index A Acceleration angular, 489, 491, 501, 507, 508 bias vector, 521 body point, 384, 508, 510 centripetal, 500, 504, 508, 510 constant parabola, 705 constant path, 694 Coriolis, 510, 536, 563 definition, 27 discontinuous path, 700 discrete equation, 740, 748 end-effector, 499 forward kinematics, 518, 519 gravitational, 634, 660, 669 inverse kinematics, 520 jump, 689 matrix, 490, 510, 516, 533 mixed, 538, 540 mixed Coriolis, 543 mixed double, 543 Razi, 543, 552 recursive, 526, 528, 609 rotational transformation, 490, 499 sensors, 771 simple, 534 tangential, 500, 504, 508, 510, 536, 543 transformation, 494, 541 transformation matrix, 510 Accessory, 260 Active transformation, 79 Actuator, 7, 12 force and torque, 610, 632, 673 optimal torque, 750, 751 torque equation, 617, 642, 747 Admissible path, 735 Algorithm floating-time, 739, 746 inverse kinematics, 343 LU factorization, 451 LU solution, 451 Newton-Raphson, 464 Angle Euler, 59 nutation, 59 precession, 59 spin, 59 Angular acceleration, 489, 491, 508 B-expression, 502 end-effector, 499 Euler parameters, 501, 507 matrix, 490 principal, 502 quaternions, 507 recursive, 533 relative, 494, 500 rotational transformation, 490, 492 technical point, 504 vector, 490 Angular jerk, 506 Angular momentum, 569–571 2 link manipulator, 574 Angular velocity, 64, 66, 67, 97, 361, 364, 372, 491, 495, 502 alternative definition, 384, 386 combination, 364 coordinate transformation, 372 decomposition, 370 elements of matrix, 376 Euler frequencies, 370 Euler parameters, 375 instantaneous, 363 instantaneous axis, 361, 364, 372, 495 matrix, 362 principal matrix, 369 quaternions, 374 rate, 361 recursive, 418, 527 rom B-frame, 363 rom G-frame, 363 rotation matrix, 371 skew symmetric, 363 vector, 361 Anthropomorphic hand, 7 Arm, 2 Articulated arm, 9, 252, 254 manipulator, 9, 252, 254, 325, 433 robot, 252 Articulated manipulator equations of motion, 647 inverse kinematics, 319, 322, 323, 331 inverse velocity, 447 Jacobian matrix, 428, 472 left shoulder configuration, 336 right shoulder configuration, 336 Asimov, Isaac, 1 Associativity property, 76 Atan2 function, 317 Attitude angle, 58 Automorphism, 123 property, 122 Auxiliary frame, 285 Avicenna, 278 Axis-angle matrix, 91 rotation, 91 transformation, 91 Axis-angle rotation, 91, 94–96, 110, 112, 114, 127 B bac-cab rule, 19, 147 Bang-bang control, 731 Bank angle, 58 Bar, 2 Bernoulli, Jacob, 738 Bernoulli, Johann, 738 Binary link, 228 Block diagram, 760 Body frame, 14 Bong, 28 Brachistochrone, 737, 745 Bryant angles, 68 C Camera inspection, 260, 263 SSRMS, 260 vision, 263 Capek, Karel, 1 Cardan angles, 68 frequencies, 68 Cardano, Geronimo, 68 Cartesian angular velocity, 66 coordinate space, 247 coordinate system, 20 end-effector position, 441 end-effector velocity, 442 manipulator, 9, 11 orthogonality condition, 20 path, 705 Cartesian manipulator equations of motion, 650 Catapults, 597 Central difference, 743 Centrifugal moments, 576 Centroid, 391 Chasles, Michel, 186 Chasles theorem, 187 China, 597 Christoffel Elwin Bruno, 658 Christoffel operator, 658 Christoffel symbol, 658 Classification industrial links, 236 Closure property, 75 Co-state variable, 732 Common normal, 226 Compound link, 2 Condition orthogonality, 20 Configuration path, 27 Control adaptive, 763 admissible, 733 bang-bang, 731, 733 characteristic equation, 762 closed-loop, 759 command, 759 computed force, 765 computed torque, 763, 764 derivative, 768 desired path, 759 directional control system, 263 error, 759 feedback, 760 feedback command, 765 feedback linearization, 763, 765 feedforward command, 765 gain, 760 gain-scheduling, 763 input, 764 integral, 768 linear, 763, 767 minimum time, 731 modified PD, 770 open-loop, 759, 764 path points, 707 PD, 770 proportional, 768 robots, 12 sensing, 770 stability of linear, 761 time-optimal, 738, 741, 746, 747, 751 time-optimal description, 747 time-optimal path, 745 Controller, 8 Control unit, 8 Coordinate cylindrical, 174 non-Cartesian, 589 non-orthogonal, 136 parabolic, 589 spherical, 175, 396 system, 21 Coordinate frame, 14, 16, 21 dummy, 278 extra, 278 global, 21 intermediate, 278 jump, 278 local, 21 main, 240, 278 neshin, 277 origin, 16 orthogonal, 16, 20 orthogonality condition, 20 Sina, 240, 278 spare, 278 takht, 277 temporary, 278 Coordinate space Cartesian, 247 joint variable, 247 Coordinate system, 14, 21 Coriolis acceleration, 498, 510 effect, 563 force, 563 Coriolis, Gaspard, 563 Couple, 558 Crackle, 28Index 817 Critically-damped, 762 Cyclic interchanging, 14, 19 Cycloid, 738 D Damping ratio, 762 Dead frame, 269 Decoupling technique, 313 inverse kinematics, 313 inverse orientation, 314 inverse position, 314 Denavit-Hartenberg, 35 construction steps, 231 industrial links, 236 linkage, 245 mechanism, 245 method, 225, 228, 293 non-classical, 242 nonstandard method, 245, 341 notation, 225 parameters, 225, 401, 405, 416, 528, 668 proof of equation, 233, 234 shortcomings, 232 spherical robot, 244 Stanford arm, 230 3R PUMA robot, 229 3R planar manipulator, 229 transformation, 232, 235–239, 242, 289, 783–788 2R planar manipulator, 235 Derivative coordinate frames, 376 mixed, 377, 534 mixed double, 543 mixed second, 542 simple, 376, 534 transformation mixed, 387, 388 transformation formula, 383, 541 Deviation moments, 576 Devol, George, 2 Differential transformation matrix, 402 Differential equation angular velocity, 383 first-order vectorial kinematic, 383 rotation matrix, 383 vectorial kinematic, 383 Differential manifold, 76 Differentiating B-derivative, 376, 379, 381, 538 coordinate frame, 376 G-derivative, 376, 384 second, 386 second derivative, 534 transformation formula, 383 Direct dynamics, 555 Direct kinematics, 247 Directed line, 202 Direction cosines, 53 Directional control system, 104, 263 cosine, 18, 25, 72, 73, 127, 243 Directional cosine, 17 Directional line, 16 Directions principal, 578 Displacement, 23 Distal end, 225, 669 Dynamics, 13, 526, 555, 609 actuator’s force and torque, 632 backward Newton-Euler, 626 direct, 13, 555 forward Newton-Euler, 627 4 bar linkage, 624 global Newton-Euler, 609 inverse, 13, 555 Lagrange, 632 motion, 557 Newton-Euler, 609 Newtonian, 563 one-link manipulator, 611 recursive Newton-Euler, 609, 626 robots, 609 2R planar manipulator, 616–618, 628, 637 E Earth effect of rotation, 563 kinetic energy, 589 moving vehicle, 498 revolution, 589 rotating, 498 rotation, 589 rotation effect, 498 Eigenvalue rotation matrix, 98, 142 Eigenvector rotation matrix, 98, 142 End-effector, 7 acceleration, 518 angular acceleration, 499 angular velocity, 440 articulated robot, 325 configuration vector, 470, 518 configuration velocity, 518 force, 628 frame, 231 inverse kinematics, 313 kinematics, 288 link, 225 orientation, 328, 441 path, 708, 712 position kinematics, 247 position vector, 435 rotation, 720 SCARA position, 172 SCARA robot, 256 space station manipulator, 257 speed vector, 419, 423 spherical robot, 292 time optimal control, 731 velocity, 431, 442 Energy Earth kinetic, 589 kinetic, 559, 572 kinetic rotational, 568 link’s kinetic, 633, 660 link’s potential, 634 mechanical, 589 potential, 591 robot kinetic, 633, 660818 Index robot potential, 634, 660 Engelberger, Joseph, 2 Equation the most important, 19 Euclidean space, 126 Euler angles, 22, 62–64, 128 integrability, 67 coordinate frame, 66 equation of motion, 567, 571, 611, 627 frequencies, 64, 66, 370 inverse matrix, 62 -Lexell-Rodriguez formula, 93 parameters, 109, 110, 112, 113, 116, 117, 119, 130, 374, 375 rotation matrix, 62 rotation theorem, 102 theorem, 56, 102, 109 Euler angles, 59, 338 Euler-Chasles theorem, 187 Euler equation body frame, 571 Eulerian viewpoint, 391 Eulerian wrist, 105 Euler-Lagrange equation of motion, 735, 736 Euler, Leonhard, 56, 93, 130, 383 Euler theorem, 153 Exponent -1, 47 Exteroceptors sensor, 770 F Feedback control, 760 Final rotation formula, 100, 101 First variation, 736 Fixed frame, 14, 21 Fleming, Sir John Ambrose, 15 Floating bar, 266 Floating time, 739 analytic calculation, 744 backward path, 741 convergence, 743 forward path, 741 method, 738 multi DOF algorithm, 746 multiple switching, 751 1 DOF algorithm, 739 path planning, 745 robot control, 746 Force, 557, 558 action, 610 actuator, 632 body, 557 conservative, 591 contact, 557 Coriolis, 563 driven, 610 driving, 610 external, 557 function, 563 generalized, 586, 591, 635 gravitational vector, 635 internal, 557 moment of, 558 potential, 591 potential field, 588 reaction, 610 resultant, 557 sensors, 771 shaking, 626 total, 557 Force system, 557, 558 equivalent, 558 Formula derivative transformation, 383, 387, 388, 541–543 derivative transport, 502 geometric transformation, 359 mixed-derivative, 388 relative acceleration, 493, 494, 500 relative angular velocity, 365 Rodriguez, 132, 364 Forward kinematics, 36, 247 4 bar linkage dynamics, 624 4R planar manipulator statics, 669 Frame base, 231 central, 565 final, 231 fixed, 21 global, 21 goal, 231 moving, 21 neshin, 277 principal, 568, 571, 577, 578 reference, 20 special, 231 station, 231 takht, 277 tool, 231 transformation, 21 world, 231 wrist, 231 Function Heaviside, 322 G Galilei, 738 Gauss, Johann Carl Friedrich, 130 Generalized coordinate, 584, 586, 588, 592 force, 586, 588, 590, 591, 593, 595, 632 inverse Jacobian, 467 Globalframe, 14 Grassmann, Hermann, 205 Grassmannian, 205 Gripper, 7 Gripper axis, 269, 270 Gripper frame, 269 Gripper wall, 270 Group properties, 75 H Hamilton, Sir William Rowan, 120 Hamilton, William, 130 Hamiltonian, 731, 732 Hand, 7 Hayati-Roberts method, 298 Heading angle, 58 Heaviside, Oliver, 322 Heaviside function, 322Index 819 Helix, 28, 182 Hermann Grassmann, 205 Hodograph, 27 Home configuration, 228 Homogenous combined transformation, 167 coordinate, 154, 159 direction, 159 general transformation, 159, 166 inverse transformation, 162, 164, 165, 168 position vector, 154 reverse transformation, 162 scale factor, 154 transformation, 153–155, 157, 158, 162, 164 I Identity matrix, 198 Identity property, 75 Industrial robot classification, 236 Inner automorphism, 122 Inspection camera, 260, 263 Integrability, 67 Inverse function symbol, 47 trigonometric functions, 45, 46 Inverse dynamics, 555 Inverse kinematics, 36, 313 articulated manipulator, 319, 322, 323, 331 comparison of techniques, 347 decoupling technique, 313 Euler angles, 338 Euler angles matrix, 340 existence, 347 general formulas, 328 inverse transformation technique, 329 iterative algorithm, 343 iterative technique, 343 multiple solutions, 315 Newton-Raphson method, 343, 346 nonstandard DH, 341 numerical solution, 323 Pieper technique, 331 spherical robot, 334 techniques, 347 2R planar manipulator, 315 uniqueness, 347 Inverse orientation, 314 Inverse position, 314 Inverse property, 76 Inverse symbol, 47 Inverse velocity, 442 Inverted pendulum, 765 Iteration technique, 346 Iterative technique, 343 J Jacobi identity, 19 Jacobian, 343 analytical, 441, 442 angular, 441 displacement matrix, 420, 423 elements, 440 generating rate vector, 533 generating vector, 429–431, 469, 533 geometrical, 441 inverse, 344, 467 of link, 633 matrix, 344, 346, 348, 419, 420, 423, 427, 429, 431, 433, 437, 438, 442, 446, 463, 466, 469, 471, 472, 518, 519, 523, 644 polar manipulator, 423, 524 rotational matrix, 420, 423 spherical wrist, 446 2R manipulator, 424 Jacobian matrix, 419, 420, 423 displacement, 420 rotational, 420–422 systematic method, 422, 423 Jeeq, 28 Jerk, 506 angular, 505 body point, 506 definition, 27 global, 506 matrix, 516 rotational transformation, 505, 506 transformation, 516, 517 transformation matrix, 516 zero path, 694 Johns Hopkins beast, 2 Joint, 3 acceleration vector, 518 active, 4 angle, 227 axis, 3 coordinate, 3 cylindrical, 296 degree of freedom, 4 distance, 227 elbow, 229 free, 4 inactive, 4 orthogonal, 9 parallel, 9 parameters, 228, 289 passive, 4 path, 708 perpendicular, 9 prismatic, 3 revolute, 3 rotary, 3 shoulder, 229 speed vector, 419, 431 spherical, 269 translatory, 3 variable, 3 waist, 229 Joint variable angle, 3 coordinate space, 247 distance, 3 Jolt, 28 Jounce, 28 K Kinematic length, 227 Kinematic operation, 285 Kinematic pair, 3 Kinematic surgery, 285820 Index Kinematics, 13, 35 acceleration, 489, 491 assembling, 276, 277 direct, 247 forward, 13, 36, 225, 247 forward acceleration, 518 forward velocity, 419 inverse, 13, 36, 313, 329 inverse acceleration, 520 inverse velocity, 442 motion, 149 numerical methods, 448 operation, 285 orientation, 91 rigid body, 149 rotation, 37 surgery, 285 velocity, 415 Kinetic energy, 559 Earth, 589 link, 660 parabolic coordinate, 590 rigid body, 572 robot, 633, 660 rotational body, 568 Kronecker’s delta, 73, 116, 568 L Lagrange dynamics, 632 equation, 659 equation of motion, 584, 591 mechanics, 591 multiplier, 738 Lagrangean, 591, 661 robot, 661 Lagrangean viewpoint, 391 Lagrange formula, 19 Lagrange multiplier, 738 Larz, 28 Law cosine, 33 of motion, 558 motion second, 565 robotics, 1 second of motion, 558, 559 third of motion, 558, 559 Levi-Civita density, 116 Lexell, Anders Johan, 93 Lie group, 76 Linear space, 24 Line of action, 22 Link, 2 angular velocity, 417 binary, 228 class 11 and 12, 238, 788 class 1 and 2, 236, 783 class 3 and 4, 237, 784 class 5 and 6, 237, 785 class 7 and 8, 238, 786 class 9 and 10, 238, 787 classification, 236 compound, 2 end-effector, 225 Euler equation, 627 kinetic energy, 633 length, 227 Newton-Euler dynamics, 609 offset, 227 parameters, 228, 289 recursive acceleration, 526, 528 recursive Newton-Euler dynamics, 626 recursive velocity, 527, 528 rotational acceleration, 526 translational acceleration, 526 translational velocity, 417 twist, 227 velocity, 415 Living frame, 269 Local frame, 14 Location vector, 184, 185 Louis Poinsot, 564 LU factorization method, 449, 452, 453, 456, 460 matrix inversion, 458 pivoting, 453 uniqueness of solution, 456 M Main coordinate frame, 240, 278 Mainframe, 7 Manganic, 597 Manipulator anthropomorphic, 9 articulated, 9, 229 Cartesian, 9 cylindrical, 9 definition, 5 elbow, 9 inertia matrix, 634 one-link, 592 one-link control, 768 one-link dynamics, 611 planar polar, 636 PUMA, 229 SCARA, 9 space station, 257 spherical, 9 3R planar, 250 transformation matrix, 325 2R planar, 593, 642 Manjanic, 597 Manjaniq, 597 Mass center, 558, 560, 565 Mass moment matrix, 576 principal, 578 Matrix displacement Jacobian, 420, 423 Jacobian, 420 orthogonality condition, 72 rotational Jacobian, 420, 423 skew symmetric, 92, 110, 137 Measure number, 16 Mechanics Newtonian, 563 Mechanism slider-crank, 245 3D slider-crank, 266 Method Hayati-Roberts, 298Index 821 non Denavit-Hartenberg, 293 parametrically continuous convention, 298 Minimum time control, 731 Mixed derivative, 377, 388 Mobius strip, 15 Mobius surface, 15 Modular manipulator, 302, 304–307 Moment, 557, 558 action, 610 driven, 610 driving, 610 external, 571 reaction, 610 resultant, 557, 571 total, 557 Moment of inertia, 576 about a line, 583 about a plane, 583 about a point, 583 characteristic equation, 582 eigenvalues, 578 elements, 576 Huygens-Steiner theorem, 578 matrix, 576 parallel-axes theorem, 576–578 polar, 576 principal, 577 principal axes, 568 principal invariants, 582 product, 576 rigid body, 568, 570 rotated-axes theorem, 576–578 Moment of momentum, 558 Momentum, 558 angular, 558, 569–571 linear, 558 translational, 558 Motion, 13 Moving frame, 14 Mozzi, Giulio, 186 Multibody articulated manipulator, 180 directional control system, 263 order free rotation, 102 order free transformation, 176 spherical robot, 244 spherical wrist, 280 3D slider-crank mechanism, 266 N Natural frequency, 762 Negative triad, 14 Neshin frame, 277, 279, 282 Newton equation of motion, 558, 559, 566, 584 Newton equation body frame, 566 definition, 537 global frame, 565 Lagrange form, 586 rotating frame, 563 Newton-Euler backward equations, 626 equation of motion, 610, 627 forward equations, 626, 627 global equations, 609 recursive equations, 626 Newton-Raphson method, 343 Non-standard Denavit-Hartenberg method, 245 Non Denavit-Hartenberg methods, 293 Norm-infinity, 458 Norm of a vector, 24 Numerical methods, 448 analytic inversion, 460 Cayley–Hamilton inversion, 462 condition number, 457 consistent solution, 456 ill-conditioned, 456 inconsistent solution, 456 Jacobian matrix, 469 LU factorization, 449 LU factorization with pivoting, 453 matrix inversion, 458 Newton-Raphson, 463, 466 nonlinear equations, 463 norm of a matrix, 458 pivot element, 453 undetermined solution, 456 uniqueness of solution, 456 well-conditioned, 456 Nutation, 59 O Object manipulation, 173 Obstacle free path, 716 Onager, 597 Opposite triad, 14 Optimal control, 731 a linear system, 733 description, 747 first variation, 736 Hamiltonian, 732, 735 Lagrange equation, 735 objective function, 731, 735 performance index, 735 second variation, 736 switching point, 733 Order-free theorem, 176 Origin, 16 Orthogonal exes, 226 Orthogonality condition, 20, 72 Orthogonal matrix, 71 Orthogonal transformation, 71 Orthogonal triad, 14 Over-damped, 762 P Parallel exes, 226 Passive transformation, 79 Path Brachistochrone, 745 Cartesian, 705 configuration, 27 constant acceleration, 694 constant angular acceleration, 722 control points, 707 cubic, 687822 Index cycloid, 704 harmonic, 703 higher polynomial, 692 jerk zero, 694 joint space, 708 non-polynomial, 702 planning, 687, 705 point sequence, 696 quintic, 692 rest-to-rest, 689, 690 rotational, 720 segment method, 697, 699, 700 spline, 701 splitting method, 697, 699–701 to-rest, 689 Pendulum control, 765 inverted, 765, 770 linear control, 768 oscillating, 587 simple, 495, 586 spherical, 572, 574, 592 Permutation symbol, 116 Perpendicular exes, 226 Persia, 278, 543 Persian, 278, 597 Phase plane, 734 Physical quantity vectorial, 22 Physical quantity scalaric, 24 vectorial, 23 Pieper technique, 331 Pitch angle, 58 Pivot element, 453 Pivoting method, 453 Plücker angle, 208 axis coordinate, 205 classification coordinate, 206 coordinate, 202, 204, 206, 207 distance, 208 line, 202 line arrangement, 204 line coordinate, 202, 204, 205, 208, 211–213, 293 method, 202 moment, 208 ray coordinate, 204, 205 reciprocal product, 208 screw, 212 vector, 202 virtual product, 208 Plücker, Julius, 204 Plücker coordinate, 207 general case, 206 line at infinity, 206 line through origin, 206 Poinsot, Louis, 564 Poinsot theorem, 187 Point at infinity, 159 Polar manipulator inverse acceleration, 524 Jacobian, 423 Pole, 192, 193, 391 Pontryagin principle, 732 Pop, 28 Positioning, 13 Position sensors, 771 Position vector, 16, 23 Positive triad, 14 Potential energy, 559, 591 field, 559 force, 591 function, 559 Potential energy robot, 634, 660 Precession, 59 Prince of physicians, 278 Principal angular acceleration, 502, 503 angular velocity, 502 Principal directions, 578 Principle conservation of energy, 559 superposition, 564 Prismatic velocity coefficient matrix, 402 velocity transformation, 402 Problem composition, 132 decomposition, 132 open, 57 Proprioceptors sensor, 770 Proximal end, 225, 669 Q Quaternion automorphism, 122 Quaternions, 117, 129 addition, 118 composition rotation, 122 flag form, 117, 123 inverse rotation, 120 matrix, 130 multiplication, 118 rotation, 119 unit, 130 R Rabota, 1 Razi, Zakariya , 543 Razi acceleration, 543 Redundancy, 251 Reference frame, 14 Resolved rates, 442 Rest position, 228, 247 Revolute velocity coefficient matrix, 402 velocity transformation, 402 Right hand rule, 14 Rigid body acceleration, 508, 526 angular momentum, 569, 570 angular velocity, 97 Euler equation, 571 kinematics, 149 kinetic energy, 572 moment of inertia, 568, 570Index 823 motion, 149–151 motion classification, 195 motion composition, 152 non-central rotation, 169, 170, 195 off-center rotation, 169, 170, 195 principal rotation matrix, 581 rotational kinetics, 567 steady rotation, 574 translational kinetics, 565 velocity, 388, 389 Rigid body theorem, 153 Robot application, 13 articulated, 9, 252, 254, 282, 325, 433, 438 Cartesian, 11 classification, 8 closed-loop, 9 control, 12, 13 control algorithms, 763 cylindrical, 9, 307 dynamics, 13, 526, 609, 635, 642 end-effector path, 712 equation of motion, 662 fixed sequence, 8 forward kinematics, 247, 292 gravitational vector, 635 hybrid, 9 inertia matrix, 634 intelligent, 8 kinematics, 13 kinetic energy, 633, 660 Lagrangean, 634, 658 Lagrange dynamics, 632, 659 Lagrange equation, 635 link classification, 291 manual handling, 8 modified PD control, 770 Newton-Euler dynamics, 609 numerical control, 8 open-loop, 9 parallel, 9 PD control, 770 playback, 8 potential energy, 634, 660 recursive Newton-Euler dynamics, 626 rest position, 226, 228, 252, 254, 283 SCARA, 172, 255 serial, 9 spherical, 9, 230, 286, 292, 334, 432 state equation, 734 statics, 667 time-optimal control, 734, 746 variable sequence, 8 velocity coupling vector, 635 Robotic geometry, 9 history, 2 laws, 1 Robotics Institute of America, 1 Rodriguez rotation formula, 93, 95, 111–115, 121, 127, 150, 185, 191, 196, 201, 364, 372, 720 vector, 115, 132, 133 Rodriguez rotation matrix, 116 Rodriguez, Benjamin, 130 Rodriguez, Olinde, 93 Roll angle, 58 Roll-pitch-yaw body rotation matrix, 58 bovy angles, 58 frequency, 58 global angles, 50, 123 global rotation matrix, 50, 123 Rotation, 35 about global axes, 37, 43, 45, 46 about local axes, 52, 55, 56 acceleration transformation, 490, 492 axis-angle, 91, 94–96, 110, 112, 114, 127, 372 composition, 132 decomposition, 132 eigenvalue, 98, 142 eigenvector, 98, 142 exponential form, 114 final formula, 100 general, 71 infinitesimal, 113 instantaneous center, 391 local versus global, 69 matrix, 22, 127 order free, 102 pole, 391 quaternion, 119 representation, 91 Rodriguez formula, 94, 372 Rodriguez matrix, 116 stanley method, 117 Taylor expansion, 130 triple global axes, 45, 46 x-matrix, 38, 52 y-matrix, 38, 52 z-matrix, 38, 52 Rotational jerk, 506 Rotational path, 720 Rotation kinematics, 37 Rotations problems, 126 Rotation theorem, 102 Rotator, 94, 124, 126 Rule bac-cab, 19 relative angular acceleration, 493 relative angular velocity, 493 right-hand, 14 Runaround, 1 S Scalar, 24 equal, 24 equivalent, 24 Scale, 16 SCARA manipulator, 9 robot, 172, 255 Screw, 182, 186, 195 axis, 182, 392 central, 183, 185, 188, 198, 202, 212, 228, 289, 291, 293 combination, 200, 201 coordinate, 182 decomposition, 201, 202 exponential, 200 forward kinematics, 289824 Index general, 199 instantaneous, 213 intersection, 293 inverse, 197–199, 201 left-handed, 182 link classification, 291 location vector, 182, 184 motion, 182, 188, 228, 289, 392 parameters, 182, 193 pitch, 182 Plücker coordinate, 212 principal, 186, 201, 202 reverse central, 183 right-handed, 14, 182 special case, 192 transformation, 185, 194 twist, 182 twist angle, 182 twist axis, 182 Screw motion, 182 Second derivative, 386 Second variation, 736 Sensor acceleration, 771 position, 771 rotary, 771 velocity, 771 Sensors, 8 Shaking force, 626 Sheth method, 293 Sheth notation, 293 Shuttle arm, 2 Simple derivative, 376 Sina, Abu Ali, 278 Sina coordinate frame, 240, 278 Sina frame, 104, 240, 241, 263, 266, 267, 278, 285 Singular configuration, 347 Singularity, 298 Snap, 28 Sooz, 28 Space station arm, 257 Spherical pendulum, 572, 574 wrist, 280 Spherical coordinate, 175 Spherical joint, 269 Spherical manipulator, 283–285 Spherical robot, 244, 286 Spherical wrist, 105, 269, 280–282, 286, 446 dead frame, 269 forearm, 269 gripper frame, 269 hand, 269 Jacobian, 446 living frame, 269 tilt, 269 tool frame, 269 turn, 269 twist, 269 Spin, 59 Spin angle, 58 Spinor, 94, 124, 126 Spline, 701 SRMS, 257 SSRMS, 257, 260 Camera, 260 dimensions, 257 kinematic parameters, 257 Stanley method, 117 Stark effect, 590 Switching point, 733, 741 Symbols, xi System of particles motion equation, 560 T Takht frame, 180, 277, 279, 283, 332 Tetrad, 15 Theorem Euler, 102 Euler-Chasles, 187 Euler rigid body, 153 Euler rotation, 102 Huygens-Steiner, 578 order free rotations, 103 order free transformations, 176 parallel-axes, 576, 578 Poinsot, 187 rotated-axes, 576 3R planar manipulator DH transformation matrix, 229 forward kinematics, 250 Tilt vector, 274 Time derivative, 376 Time optimal control, 731 Tool frame, 269 Top, 64 Torque, 558 Transformation, 35 active and passive, 79 general, 71 geometric, 504 homogenous, 153, 154 jerk, 516 kinematic, 504 order free, 176 orthogonal, 71 rotational acceleration, 504 rotational jerk, 505, 506 rotational velocity, 504 Transformation matrix derivative, 400 differential, 402, 403 elements, 73 velocity, 392 Translation, 35 Trebuchet, 240, 595 Triad, 14 coordinate frame, 14 left-handed, 14 negative, 14 positive, 14 right-handed, 14 Trigonometric equation, 318, 319 first type, 318, 319 Trigonometric functions inverse, 45, 46 Turn vector, 274 Twist vector, 274 2R planar manipulatorIndex 825 acceleration analysis, 512 assembling, 278 control, 766 DH transformation matrix, 235 dynamics, 593, 666 elbow down, 316 elbow up, 316 equations of motion, 595 forward acceleration, 519 ideal, 593 inverse acceleration, 522 inverse kinematics, 315, 316, 344, 464 inverse velocity, 443–445 Jacobian, 424 Jacobian matrix, 427 joint 2 acceleration, 509 joint forces, 623 joint path, 709 kinematic motion, 317 kinematics, 512 kinetic energy, 594 Lagrangean, 594 Lagrange dynamics, 642, 663 line path, 710 Newton-Euler dynamics, 616–618, 637 potential energy, 594 recursive dynamics, 628 singular configuration, 445 time-optimal control, 747 velocity analysis, 397 with massive joints, 617, 618, 637 with massive links, 663 U Under-damped, 762 Unimates, 2 Unimation, 2 Unit system, xi Unit vectors, 16 V Variable scalar , 26 Variation first, 736 second, 736 Vecface, 23 Vecfree, 23 Veclane, 23 Vecline, 23 Vecpoface, 23 Vecpoint, 23 Vecpolane, 23 Vecpoline, 23 Vecporee, 23 Vector absolute value, 16 addition, 24 associative property, 24 axis, 22 bounded, 23 characteristics, 22 commutative property, 24 comparable, 23 components, 16 decomposed expression, 16, 26 decomposition, 16, 26, 136 definition, 22 derivative, 26 direction, 22, 23 end-effector speed, 419 end point, 22 free, 23 function, 26 gravitational force, 635, 659 inverse element property, 24 joint speed, 419 length, 16, 22 line, 23 line of action, 22 magnitude, 16 modulus, 16 natural expression, 16, 26 null element property, 24 physical quantity, 22 plane, 23 point, 23 point-free, 23 point-line, 23 point-plane, 23 position, 16 requirements, 22 sliding, 23 space, 24 start point, 22 surface, 23 tilt, 274 turn, 274 twist, 274 types, 22 variable, 25 variable direction, 25 variable length, 25 vecface, 23 vecfree, 23 vecline, 23 vecpoface, 23 vecpoint, 23 vecpolane, 23 vecpoline, 23 vecporee, 23 velocity coupling, 635, 659 Vector space, 25 Vector triple product, 19 Velocity coefficient matrix, 402 definition, 27 discrete equation, 740, 748 inverse transformation, 394 matrix, 516 multiple frames, 390 operator matrix, 400 prismatic transformation, 402 revolute angular matrix, 405 revolute transformation, 402 rigid body, 388 sensors, 771 simple, 534 transformation matrix, 392–395, 400 Velocity kinematics, 415826 Index forward, 415 inverse, 415 Vision camera, 263 W Walter William Grey, 2 Work, 559, 561 virtual, 586 Work-energy principle, 559 Working space, 255 Workspace, 12 dexterous, 12 reachable, 12 World War II, 2 Wrench, 564 Wrist, 12, 13, 271 classification, 270 dead frame, 269 decoupling kinematics, 314 design, 276 Eulerian, 272, 273 forward kinematics, 269, 271 frame, 231 kinematics assembly, 282 live frame, 269 Pitch-Yaw-Roll, 272, 275 point, 6, 269, 328 position vector, 327 possible types, 272 Roll-Pitch-Roll, 270–273 Roll-Pitch-Yaw, 272, 274 spherical, 6, 230, 244, 269, 272, 274, 286, 438 transformation matrix, 271, 282, 325 Wrist point, 263 Y Yaw angle, 58 Z Zero velocity point, 391
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