Theory of Applied Robotics Reza N Jazar Theory of Applied Robotics Kinematics, Dynamics, and Control Second Edition 123 Prof Reza N Jazar School of Aerospace, Mechanical, and Manufacturing Engineering RMIT University Melbourne, Victoria Australia reza.reza.jazar@rmit.edu.au ISBN 978-1-4419-1749-2 e-ISBN 978-1-4419-1750-8 DOI 10.1007/978-1-4419-1750-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 201092 033 c Springer Science+Business Media, LLC 2006, 2010 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Cover illustration c Konstantin Inozemtsev Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to my wife, Mojgan and our children, Vazan and Kavosh I am Cyrus, king of the world, great king, mighty king, king of Babylon, king of Sumer and Akkad, king of the four quarters I ordered to write books, many books, books to teach my people, I ordered to make schools, many schools, to educate my people Marduk, the lord of the gods, said burning books is the greatest sin I, Cyrus, and my people, and my army will protect books and schools They will fight whoever burns books and burns schools, the great sin Cyrus the great Preface to the Second Edition The second edition of this book would not have been possible without the comments and suggestions from my students, especially those at Columbia University Many of the new topics introduced here are a direct result of student feedback that helped me refine and clarify the material My intention when writing this book was to develop material that I would have liked to had available as a student Hopefully, I have succeeded in developing a reference that covers all aspects of robotics with sufficient detail and explanation The first edition of this book was published in 2007 and soon after its publication it became a very popular reference in the field of robotics I wish to thank the many students and instructors who have used the book or referenced it Your questions, comments and suggestions have helped me create the second edition Appendix D Trigonometric Formula Definitions in terms of exponentials cos z = eiz + e−iz (D.1) sin z = eiz − e−iz 2i (D.2) eiz − e−iz i (eiz + e−iz ) (D.3) tan z = eiz = cos z + i sin z (D.4) e−iz = cos z − i sin z (D.5) sin(α ± β) = sin α cos β ± cos α sin β (D.6) cos(α ± β) = cos α cos β ∓ sin α sin β (D.7) Angle sum and difference tan(α ± β) = cot(α ± β) = Symmetry tan α ± tan β ∓ tan α tan β cot α cot β ∓ cot β ± cot α (D.8) (D.9) sin(−α) = − sin α (D.10) cos(−α) = cos α (D.11) tan(−α) = − tan α (D.12) Multiple angle sin(2α) = sin α cos α = tan α + tan2 α cos(2α) = cos2 α − = − sin2 α = cos2 α − sin2 α tan(2α) = cot(2α) = tan α − tan2 α cot2 α − cot α (D.13) (D.14) (D.15) (D.16) 870 Appendix D Trigonometric Formula sin(3α) = −4 sin3 α + sin α (D.17) cos(3α) = cos3 α − cos α (D.18) − tan α + tan α −3 tan2 α + (D.19) tan(3α) = sin(4α) = −8 sin3 α cos α + sin α cos α (D.20) cos(4α) = cos α − cos α + (D.21) tan(4α) = (D.22) −4 tan3 α + tan α tan4 α − tan2 α + sin(5α) = 16 sin5 α − 20 sin3 α + sin α cos(5α) = 16 cos5 α − 20 cos3 α + cos α (D.24) sin(nα) = sin((n − 1)α) cos α − sin((n − 2)α) (D.25) cos(nα) = cos((n − 1)α) cos α − cos((n − 2)α) (D.26) tan(nα) = Half angle cos sin tan (D.23) ³α´ = tan((n − 1)α) + tan α − tan((n − 1)α) tan α ³α´ =± ³α´ =± r r + cos α − cos α r − cos α sin α = =± sin α + cos α (D.27) (D.28) (D.29) − cos α + cos α (D.30) sin α = tan α2 + tan2 α2 (D.31) cos α = − tan2 + tan2 α α (D.32) Powers of functions (1 − cos(2α)) sin α cos α = sin(2α) cos2 α = (1 + cos(2α)) sin3 α = (3 sin(α) − sin(3α)) sin2 α = (D.33) (D.34) (D.35) (D.36) Appendix D Trigonometric Formula (cos α − cos(3α)) sin α cos2 α = (sin α + sin(3α)) cos3 α = (cos(3α) + cos α)) sin4 α = (3 − cos(2α) + cos(4α)) sin3 α cos α = (2 sin(2α) − sin(4α)) sin2 α cos2 α = (1 − cos(4α)) sin α cos3 α = (2 sin(2α) + sin(4α)) cos4 α = (3 + cos(2α) + cos(4α)) sin5 α = (10 sin α − sin(3α) + sin(5α)) 16 sin4 α cos α = (2 cos α − cos(3α) + cos(5α)) 16 sin3 α cos2 α = (2 sin α + sin(3α) − sin(5α)) 16 sin2 α cos3 α = (2 cos α − cos(3α) − cos(5α)) 16 sin α cos4 α = (2 sin α + sin(3α) + sin(5α)) 16 cos5 α = (10 cos α + cos(3α) + cos(5α)) 16 − cos(2α) tan2 α = + cos(2α) sin2 α cos α = 871 (D.37) (D.38) (D.39) (D.40) (D.41) (D.42) (D.43) (D.44) (D.45) (D.46) (D.47) (D.48) (D.49) (D.50) (D.51) Products of sin and cos 1 cos(α − β) + cos(α + β) 2 1 sin α sin β = cos(α − β) − cos(α + β) 2 1 sin α cos β = sin(α − β) + sin(α + β) 2 1 cos α sin β = sin(α + β) − sin(α − β) 2 cos α cos β = sin(α + β) sin(α − β) = cos2 β − cos2 α = sin2 α − sin2 β (D.52) (D.53) (D.54) (D.55) (D.56) 872 Appendix D Trigonometric Formula cos(α + β) cos(α − β) = cos2 β + sin2 α (D.57) Sum of functions α±β α±β cos 2 α+β α−β cos α + cos β = cos cos 2 α+β α−β cos α − cos β = −2 sin sin 2 sin(α ± β) tan α ± tan β = cos α cos β sin α ± sin β = sin cot α ± cot β = sin(β ± α) sin α sin β (D.58) (D.59) (D.60) (D.61) (D.62) tan α+β sin α + sin β = α−+β sin α − sin β tan (D.63) sin α + sin β −α + β = cot cos α − cos β (D.64) sin α + sin β α+β = tan cos α + cos β (D.65) sin α − sin β α−β = tan cos α + cos β (D.66) sin2 α − sin2 β = sin(α + β) sin(α − β) (D.67) cos α − cos β = − sin(α + β) sin(α − β) (D.68) Trigonometric relations 2 Index 2R planar manipulator acceleration analysis, 543 assembling, 281 control, 837 DH transformation matrix, 246 dynamics, 622, 695 elbow down, 331 elbow up, 331 equations of motion, 625 forward acceleration, 550 ideal, 622 inverse acceleration, 554 inverse kinematics, 331, 359, 505 inverse velocity, 466, 468 Jacobian matrix, 448, 450 joint acceleration, 540 joint forces, 660 joint path, 750 kinematic motion, 332 kinetic energy, 623 Lagrange dynamics, 675, 696 Lagrangean, 624 line path, 752 Newton-Euler dynamics, 651, 653, 655, 680 potential energy, 623 recursive dynamics, 664 time-optimal control, 812 velocity analysis, 413 with massive joints, 653, 655, 680 with massive links, 696 3R planar manipulator DH transformation matrix, 238 forward kinematics, 260 4R planar manipulator statics, 703 Acceleration angular, 529, 534, 536, 538, 539 bias vector, 553 body point, 399, 539, 541, 584 centripetal, 536, 539 constant parabola, 755 constant path, 738 Coriolis, 585 discontinuous path, 745 discrete equation, 803, 813 end-effector, 535 forward kinematics, 549, 550 gravitational, 671, 692, 703 inverse kinematics, 552 jump, 731 matrix, 530, 541, 548, 566 recursive, 557, 560, 641 rotational transformation, 530, 535 sensors, 844 tangential, 536, 539 transformation matrix, 541, 542 Active transformation, 73 Actuator, 7, 13 force and torque, 643, 668, 707 optimal torque, 814, 815 torque equation, 652, 812 Algorithm floating-time, 801, 811 inverse kinematics, 358 LU factorization, 488 LU solution, 488 Newton-Raphson, 504 Angular acceleration, 529, 538, 539 combination, 534 874 Index end-effector, 535 Euler parameters, 536, 538 matrix, 530 quaternions, 538 recursive, 565 vector, 530 Angular momentum link manipulator, 594 Angular velocity, 56, 59, 60, 98, 381 alternative definition, 400 combination, 387 coordinate transformation, 389 decomposition, 387 elements of matrix, 393 Euler frequencies, 388 Euler parameters, 391 instantaneous, 383 instantaneous axis, 382, 384 matrix, 382 principal matrix, 385 quaternions, 390 rate, 382 recursive, 440, 559 rotation matrix, 388 vector, 382 Articulated arm, 9, 262, 265 manipulator, 9, 262, 265, 333, 456 Articulated manipulator equations of motion, 686 inverse kinematics, 328, 330, 343 inverse velocity, 470 Jacobian matrix, 450, 514 left shoulder configuration, 349 right shoulder configuration, 349 Atan2 function, 339 Automorphism, 115 Axis-angle rotation, 91, 94—96, 103— 105, 107, 120 bac-cab rule, 143 Block diagram, 828 Brachistochrone, 798, 809 Bryant angles, 61 Cardan angles, 61 frequencies, 61 Cartesian angular velocity, 59 end-effector position, 464 end-effector velocity, 466 manipulator, 9, 10 path, 754 Central difference, 805 Centroid, 407 Chasles theorem, 178, 192 Christoffel operator, 619, 677 Christoffel symbol, 677 Co-state variable, 792 Control adaptive, 833 admissible, 800 bang-bang, 791, 792 characteristic equation, 830 closed-loop, 827 command, 827 computed force, 835 computed torque, 833 derivative, 839 desired path, 827 error, 828 feedback, 828 feedback command, 835 feedback linearization, 833, 835 feedforward command, 835 gain, 828 gain-scheduling, 833 input, 834 integral, 839 linear, 833, 838 minimum time, 791 modified PD, 841 open-loop, 827, 834 path points, 757 PD, 841 Index proportional, 839 robots, 13 sensing, 842 stability of linear, 829 time-optimal, 801, 804, 811, 812, 815 time-optimal description, 800 time-optimal path, 809 Controller, Coordinate cylindrical, 176 frame, 18 non-Cartesian, 618 non-orthogonal, 130 parabolic, 618 spherical, 177, 413 system, 18 Coriolis acceleration, 534, 541 effect, 585 force, 585 Cycloid, 799 Denavit-Hartenberg, 31 method, 233, 236, 297 nonstandard method, 257, 355 notation, 233 parameters, 233, 419, 422, 438, 560, 702 transformation, 242, 246—252, 254, 256, 292 Derivative coordinate frames, 393 transformation formula, 399 Differential transformation matrix, 420 Differential manifold, 72 Differentiating B-derivative, 393, 395, 397 coordinate frame, 393 G-derivative, 393, 399 second, 402 transformation formula, 399 Direction cosines, 48 Distal end, 233, 702 875 Dynamics, 527, 556, 641 2R planar manipulator, 651, 653, 655, 664, 680 bar linkage, 646 actuator’s force and torque, 668 backward Newton-Euler, 661 forward Newton-Euler, 663 global Newton-Euler, 642 Lagrange, 669 motion, 581 Newton-Euler, 641 one-link manipulator, 644 recursive Newton-Euler, 642, 661 robots, 641 Earth effect of rotation, 585 kinetic energy, 617 revolution, 617 rotation, 617 rotation effect, 534 Eigenvalue rotation matrix, 98 Eigenvector rotation matrix, 98 Ellipsoid energy, 596 momentum, 596 End-effector, acceleration, 549 angular acceleration, 535 angular velocity, 463 articulated robot, 333 configuration vector, 512, 549 configuration velocity, 549 force, 663 frame, 240 inverse kinematics, 325 kinematics, 291 link, 233 orientation, 338, 464 path, 749, 763 position kinematics, 259 876 Index position vector, 458 rotation, 759 SCARA position, 172 SCARA robot, 268 space station manipulator, 270 speed vector, 442, 443 spherical robot, 296 time optimal control, 791 velocity, 454, 465 Energy Earth kinetic, 617 kinetic rigid body, 593 kinetic rotational, 589 link’s kinetic, 669, 692 link’s potential, 671 mechanical, 617 point kinetic, 583 potential, 620 robot kinetic, 670, 692 robot potential, 671, 692 Euler -Lexell-Rodriguez formula, 93 angles, 19, 54, 56, 120 integrability, 60 coordinate frame, 59 equation of motion, 588, 592, 597, 599, 643, 662 frequencies, 56, 59, 388 inverse matrix, 71 parameters, 102, 103, 105, 110, 111, 113, 124, 391 rotation matrix, 54, 71 theorem, 51, 102 Euler angles, 52 Euler equation body frame, 592, 599 Euler-Lagrange equation of motion, 796, 798 Eulerian viewpoint, 407 Final rotation formula, 101 Floating time, 802 DOF algorithm, 801 analytic calculation, 809 backward path, 804 convergence, 807 forward path, 803 method, 801 multi DOF algorithm, 811 multiple switching, 815 path planning, 809 robot control, 811 Force, 581 action, 642 actuator, 668 conservative, 620 Coriolis, 585 driven, 642 driving, 642 generalized, 614, 671 gravitational vector, 672 potential, 620 potential field, 616 reaction, 642 sensors, 844 shaking, 648 time varying, 586 Forward kinematics, 32 Frame base, 239 central, 587 final, 240 goal, 240 neshin, 280 principal, 589 reference, 17 special, 239 station, 239 takht, 280 tool, 240 world, 239 wrist, 240 Generalized coordinate, 611, 614, 615, 621 force, 613, 614, 616, 618, 620, 622, 625, 669 inverse Jacobian, 509 Grassmanian, 205 Group properties, 72 Index Hamiltonian, 792 Hayati-Roberts method, 303 Helix, 178 Homogeneous compound transformation, 168 coordinate, 155, 161 direction, 161 general transformation, 162, 166 inverse transformation, 162, 164, 165, 169 position vector, 155 scale factor, 155 transformation, 154, 156, 158— 162, 165 Integrability, 60 Inverse Kinematics comparison of techniques, 361 techniques, 362 Inverse kinematics, 32, 325 articulated manipulator, 343 decoupling technique, 325 Euler angles matrix, 352, 353 general formulas, 340 inverse transformation technique, 341 iterative algorithm, 358 iterative technique, 357 nonstandard DH, 355 Pieper technique, 343 spherical robot, 346 Inverted pendulum, 836 Jacobian analytical, 464, 465 angular, 464 displacement matrix, 442 elements, 463 generating vector, 452, 455, 511 geometrical, 464, 465 inverse, 359, 509 matrix, 358, 359, 362, 364, 442, 443, 450, 454, 456, 877 460, 461, 465, 469, 504, 507, 510, 514, 549, 551, 554, 676 of link, 670 polar manipulator, 446, 555 rotational matrix, 443 spherical wrist, 469 Jerk angular, 537 matrix, 548 rotational transformation, 537 transformation, 547, 549 transformation matrix, 547 zero path, 737 Joint, acceleration vector, 549 active, angle, 235 axis, coordinate, cylindrical, 301 distance, 235 free, inactive, orthogonal, parallel, parameters, 235 passive, path, 749 perpendicular, prismatic, revolute, rotary, screw, speed vector, 442, 454 spherical, 270 translatory, variable, Kinematic length, 235 Kinematics, 31 acceleration, 529 assembling, 280 direct, 259 forward, 32, 233, 259 878 Index forward acceleration, 549 forward velocity, 442 inverse, 32, 325, 341 inverse acceleration, 552 inverse velocity, 465 motion, 149 numerical methods, 485 orientation, 91 rigid body, 149 rotation, 33 surgery, 287 velocity, 437 Kinetic energy, 583 Earth, 617 link, 692 parabolic coordinate, 618 rigid body, 593 robot, 670, 692 rotational body, 589 Kronecker delta, 109 Kronecker’s delta, 68, 589, 609 Lagrange dynamics, 669 equation, 690 equation of motion, 611, 620 mechanics, 620 multiplier, 799 Lagrange equation explicit form, 619 Lagrangean, 620, 693 robot, 693 Lagrangean viewpoint, 407 Law motion, 582 motion second, 582, 586 motion third, 582 robotics, Levi-Civita density, 109 Lie group, 72 Link, angular velocity, 439 class and 2, 247 class 11 and 12, 252 class and 4, 248 class and 6, 249 class and 8, 250 class and 10, 251 classification, 253 end-effector, 233 Euler equation, 662 kinetic energy, 669 length, 235 Newton-Euler dynamics, 642 offset, 235 parameters, 235 recursive acceleration, 556, 560 recursive Newton-Euler dynamics, 661 recursive velocity, 559 rotational acceleration, 557 translational acceleration, 557 translational velocity, 440 twist, 235 velocity, 437 Location vector, 180, 182 LU factorization method, 485, 499 Manipulator 2R planar, 622, 675 3R planar, 260 articulated, 9, 238 Cartesian, cylindrical, definition, inertia matrix, 670 one-link, 621 one-link control, 840 one-link dynamics, 644 planar polar, 674 PUMA, 238 SCARA, space station, 268, 270 spherical, transformation matrix, 333 Mass center, 582, 583, 587 Matrix skew symmetric, 70, 71, 92, 103 Method Index Hayati-Roberts, 303 non Denavit-Hartenberg, 297 parametrically continuous convention, 303 Moment, 581 action, 642 driven, 642 driving, 642 reaction, 642 Moment of inertia about a line, 610 about a plane, 610 about a point, 610 characteristic equation, 608 diagonal elements, 607 Huygens-Steiner theorem, 602 matrix, 599 parallel-axes theorem, 600 polar, 599 principal, 600 principal axes, 589 principal invariants, 608 product, 599 pseudo matrix, 600 rigid body, 588 rotated-axes theorem, 600 Moment of momentum, 582 Momentum, 582 angular, 582 ellipsoid, 596 translational, 582 Motion, 15 Newton equation of motion, 611 Newton equation body frame, 588 global frame, 587 Lagrange form, 613 rotating frame, 585 Newton-Euler backward equations, 661 equation of motion, 662 equations of motion, 642 forward equations, 662, 663 879 global equations, 641 recursive equations, 661 Non Denavit-Hartenberg methods, 297 Non-standard Denavit-Hartenberg method, 257 Numerical methods, 485 analytic inversion, 500 Cayley-Hamilton inversion, 502 condition number, 495 ill-conditioned, 494 Jacobian matrix, 510 LU factorization, 485 LU factorization with pivoting, 491 matrix inversion, 497 Newton-Raphson, 504, 506 nonlinear equations, 503 norm of a matrix, 496 partitioning inversion, 500 pivot element, 491 uniqueness of solution, 494 well-conditioned, 494 Nutation, 52 Object manipulation, 174 Optimal control, 791 a linear system, 792 description, 800 first variation, 797 Hamiltonian, 792, 796 Lagrange equation, 796 objective function, 791, 795 performance index, 795 second variation, 797 switching point, 793 Orthogonality condition, 67 Passive transformation, 73 Path Brachistochrone, 809 Cartesian, 754 constant acceleration, 738 880 Index constant angular acceleration, 761 control points, 757 cubic, 729 cycloid, 749 harmonic, 748 higher polynomial, 735 jerk zero, 737 joint space, 749 non-polynomial, 747 planning, 729, 754 point sequence, 739 quadratic, 734 quintic, 736 rest-to-rest, 731, 732 rotational, 759 splitting, 741 to-rest, 732 Pendulum control, 836 inverted, 836, 842 linear control, 840 oscillating, 615 simple, 532, 614 spherical, 621 Permutation symbol, 109 Phase plane, 793 Pieper technique, 343 Plücker angle, 209 axis coordinate, 205 classification coordinate, 206 distance, 209 line coordinate, 201—205, 209, 213—215, 296, 297 moment, 208 ray coordinate, 203, 205 reciprocal product, 209 screw, 214 virtual product, 209 Poinsot’s construction, 596 Point at infinity, 161 Polar manipulator inverse acceleration, 555 Pole, 189 Position sensors, 843 Positioning, 15 Potential force, 620 Potential energy robot, 671, 692 Precession, 52 Proximal end, 233, 702 Quaternions, 112, 122 addition, 112 composition rotation, 115 flag form, 112 inverse rotation, 114 matrix, 123 multiplication, 112 rotation, 113 unit, 124 Rigid body acceleration, 538, 558 angular momentum, 590 angular velocity, 98 Euler equation of motion, 592, 597 kinematics, 149 kinetic energy, 593 moment of inertia, 588 motion, 149 motion classification, 193 motion composition, 153 principal rotation matrix, 606 rotational kinetics, 588 steady rotation, 593 translational kinetics, 586 velocity, 403, 404 Robot application, 14 articulated, 9, 262, 265, 281, 333, 456, 461 Cartesian, 10 classification, control, 13, 15 control algorithms, 833 cylindrical, 10, 318 Index dynamics, 15, 20, 556, 641, 672, 675 end-effector path, 763 equation of motion, 694 forward kinematics, 259, 295 gravitational vector, 672 inertia matrix, 670 kinematics, 15 kinetic energy, 670, 692 Lagrange dynamics, 669, 690 Lagrange equation, 672 Lagrangean, 671, 678 link classification, 294 modified PD control, 841 Newton-Euler dynamics, 641 PD control, 841 potential energy, 671, 692 recursive Newton-Euler dynamics, 661 rest position, 234, 237, 263, 264, 284 SCARA, 172, 266 spherical, 9, 239, 288, 295, 346, 455 state equation, 795 statics, 701 time-optimal control, 795, 811 velocity coupling vector, 672 Robotic geometry, history, laws, Rodriguez rotation formula, 93, 95, 103, 104, 106—108, 114, 120, 150, 181, 187, 193, 199, 384, 421, 759 vector, 109, 127 Rodriguez rotation matrix, 109 Roll-pitch-yaw frequency, 62 global angles, 44, 62 global rotation matrix, 44, 62 Rotation, 32 about global axes, 33, 40, 42 881 about local axes, 46, 50, 51 axis-angle, 91, 94—96, 103—105, 107, 120 composition, 126 decomposition, 126 eigenvalue, 98 eigenvector, 98 exponential form, 106 final formula, 101 general, 65 infinitesimal, 106 instantaneous center, 407 local versus global, 63 matrix, 19, 119 pole, 407 quaternion, 113 Rodriguez formula, 94 Rodriguez matrix, 109 stanley method, 111 Taylor expansion, 124 triple global axes, 42 X-matrix, 34 x-matrix, 47 Y-matrix, 34 y-matrix, 47 Z-matrix, 34 z-matrix, 47 Rotational path, 759 Rotations problems, 118 Rotator, 94, 116 SCARA manipulator, robot, 172, 266 Screw, 178, 181, 193 axis, 178, 408 central, 179, 182, 183, 201, 214, 236, 292, 294, 296 combination, 198, 200 coordinate, 178 decomposition, 200, 201 exponential, 199 forward kinematics, 292 instantaneous, 215 882 Index intersection, 297 inverse, 195, 196, 200 left-handed, 178 link classification, 294 location vector, 180 motion, 185, 235, 408 parameters, 179, 190 pitch, 178 Plücker coordinate, 214 principal, 192, 200, 201 reverse central, 179 right-handed, 16, 178 special case, 188 transformation, 181, 191 twist, 178 Second derivative, 402 Sensor acceleration, 844 position, 843 rotary, 843 velocity, 843 Sheth notation, 297 Singular configuration, 363 Singularity, 303 Spherical coordinate, 177 Spin, 52 Spinor, 94, 116 Spline, 745 Stanley method, 111 Stark effect, 618 Symbols, xix Translation, 32 Triad, 16 Trigonometric equation, 338 Turn vector, 275 Twist vector, 275 Tilt vector, 275 Time derivative, 393 Top, 56 Torque, 582 Transformation, 31 active and passive, 73 general, 65 homogeneous, 154 Transformation matrix derivative, 417 differential, 420, 421 elements, 68 velocity, 409 Work, 583, 586 virtual, 614 Work-energy principle, 583 Working space, 266 Workspace, 13 Wrench, 584 Wrist, 13—15, 273 classification, 271 dead frame, 270 decoupling kinematics, 326 design, 279 Eulerian, 276 forward kinematics, 270 Unit system, xix Unit vectors, 16 Vector decomposition, 130 gravitational force, 672, 691 tilt, 275 turn, 275 twist, 275 velocity coupling, 672, 691 Velocity body point, 584 coefficient matrix, 419 discrete equation, 803, 813 inverse transformation, 411 matrix, 548 multiple frames, 405 operator matrix, 417 prismatic transformation, 419 revolute angular matrix, 423 revolute transformation, 419 rigid body, 403 sensors, 843 transformation matrix, 409— 412, 417 Index frame, 240 kinematics assembly, 281 living frame, 270 Pitch-Yaw-Roll, 278 point, 6, 270, 271, 337 position vector, 336 Roll-Pitch-Roll, 276 Roll-Pitch-Yaw, 277 spherical, 6, 239, 270, 271, 274, 275, 288, 461 transformation matrix, 273, 333 Zero velocity point, 407 883 .. .Theory of Applied Robotics Reza N Jazar Theory of Applied Robotics Kinematics, Dynamics, and Control Second Edition 123 Prof Reza N Jazar School of Aerospace, Mechanical, and Manufacturing... kinematics, dynamics, and control Classical kinematics and dynamics of robots has its roots in the work of great scientists of the past four centuries who established the methodology and understanding of. .. review of the historical development and classification of robots Part I ? ?Kinematics, ” presents the forward and inverse kinematics of robots Kinematics analysis refers to position, velocity, and