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Loading...Theory of Applied Robotics – Kinematics, Dynamics, and Control
Second Edition
Reza N. Jazar
Contents
1 Introduction 1
1.1 Historical Development 2
1.2 Robot Components . 3
1.2.1 Link . 3
1.2.2 Joint 3
1.2.3 Manipulator . 5
1.2.4 Wrist 5
1.2.5 End-effector . 6
1.2.6 Actuators 7
1.2.7 Sensors . 7
1.2.8 Controller 7
1.3 Robot Classifications 8
1.3.1 Geometry 8
1.3.2 Workspace . 13
1.3.3 Actuation . 13
1.3.4 Control 13
1.3.5 Application 14
1.4 Introduction to Robot’s Kinematics, Dynamics, and Control 15
1.4.1 F Triad . 16
1.4.2 Unit Vectors 16
1.4.3 Reference Frame and Coordinate System 17
1.4.4 Vector Function 20
1.5 Problems of Robot Dynamics . 20
1.6 Preview of Covered Topics . 22
1.7 Robots as Multi-disciplinary Machines 23
1.8 Summary 24
Exercises . 25
I Kinematics 29
2 Rotation Kinematics 33
2.1 Rotation About Global Cartesian Axes 33
2.2 Successive Rotation About Global Cartesian Axes . 40
2.3 Global Roll-Pitch-Yaw Angles . 44
2.4 Rotation About Local Cartesian Axes 46
2.5 Successive Rotation About Local Cartesian Axes 502.6 Euler Angles 52
2.7 Local Roll-Pitch-Yaw Angles . 62
2.8 Local Axes Versus Global Axes Rotation . 63
2.9 General Transformation 65
2.10 Active and Passive Transformation 73
2.11 Summary 77
2.12 Key Symbols 79
Exercises . 81
3 Orientation Kinematics 91
3.1 Axis-angle Rotation 91
3.2 F Euler Parameters 102
3.3 F Determination of Euler Parameters 110
3.4 F Quaternions . 112
3.5 F Spinors and Rotators 116
3.6 F Problems in Representing Rotations 118
3.6.1 F Rotation matrix . 119
3.6.2 F Angle-axis 120
3.6.3 F Euler angles . 121
3.6.4 F Quaternion . 122
3.6.5 F Euler parameters 124
3.7 F Composition and Decomposition of Rotations 126
3.8 Summary 133
3.9 Key Symbols 135
Exercises . 137
4 Motion Kinematics 149
4.1 Rigid Body Motion . 149
4.2 Homogeneous Transformation . 154
4.3 Inverse Homogeneous Transformation 162
4.4 Compound Homogeneous Transformation 168
4.5 F Screw Coordinates . 178
4.6 F Inverse Screw 195
4.7 F Compound Screw Transformation . 198
4.8 F The Plücker Line Coordinate 201
4.9 F The Geometry of Plane and Line . 208
4.9.1 F Moment . 208
4.9.2 F Angle and Distance . 209
4.9.3 F Plane and Line . 209
4.10 F Screw and Plücker Coordinate . 214
4.11 Summary 217
4.12 Key Symbols 219
Exercises . 221
xx Contents5.1 Denavit-Hartenberg Notation . 233
5.2 Transformation Between Two Adjacent Coordinate Frames 242
5.3 Forward Position Kinematics of Robots . 259
5.4 Spherical Wrist . 270
5.5 Assembling Kinematics . 280
5.6 F Coordinate Transformation Using Screws . 292
5.7 F Non Denavit-Hartenberg Methods . 297
5.8 Summary 305
5.9 Key Symbols 307
Exercises . 309
6 Inverse Kinematics 325
6.1 Decoupling Technique . 325
6.2 Inverse Transformation Technique 341
6.3 F Iterative Technique . 357
6.4 F Comparison of the Inverse Kinematics Techniques 361
6.4.1 F Existence and Uniqueness of Solution . 361
6.4.2 F Inverse Kinematics Techniques . 362
6.5 F Singular Configuration . 363
6.6 Summary 367
6.7 Key Symbols 369
Exercises . 371
7 Angular Velocity 381
7.1 Angular Velocity Vector and Matrix . 381
7.2 F Time Derivative and Coordinate Frames . 393
7.3 Rigid Body Velocity 403
7.4 F Velocity Transformation Matrix 409
7.5 Derivative of a Homogeneous Transformation Matrix 417
7.6 Summary 425
7.7 Key Symbols 427
Exercises . 429
8 Velocity Kinematics 437
8.1 F Rigid Link Velocity . 437
8.2 Forward Velocity Kinematics . 442
8.3 Jacobian Generating Vectors . 452
8.4 Inverse Velocity Kinematics 465
8.5 Summary 473
8.6 Key Symbols 475
Exercises . 477
9 Numerical Methods in Kinematics 485
9.1 Linear Algebraic Equations 485
9.2 Matrix Inversion 497
5 Forward Kinematics 233
Contents xxi9.3 Nonlinear Algebraic Equations 503
9.4 F Jacobian Matrix From Link Transformation Matrices 510
9.5 Summary 518
9.6 Key Symbols 519
Exercises . 521
II Dynamics 525
10 Acceleration Kinematics 529
10.1 Angular Acceleration Vector and Matrix . 529
10.2 Rigid Body Acceleration 538
10.3 F Acceleration Transformation Matrix 541
10.4 Forward Acceleration Kinematics . 549
10.5 Inverse Acceleration Kinematics 552
10.6 F Rigid Link Recursive Acceleration . 556
10.7 Summary 567
10.8 Key Symbols 569
Exercises . 571
11 Motion Dynamics 581
11.1 Force and Moment . 581
11.2 Rigid Body Translational Kinetics 586
11.3 Rigid Body Rotational Kinetics 588
11.4 Mass Moment of Inertia Matrix 599
11.5 Lagrange’s Form of Newton’s Equations . 611
11.6 Lagrangian Mechanics . 620
11.7 Summary 627
11.8 Key Symbols 629
Exercises . 631
12 Robot Dynamics 641
12.1 Rigid Link Newton-Euler Dynamics . 641
12.2 F Recursive Newton-Euler Dynamics 661
12.3 Robot Lagrange Dynamics . 669
12.4 F Lagrange Equations and Link Transformation Matrices . 690
12.5 Robot Statics 700
12.6 Summary 709
12.7 Key Symbols 713
Exercises . 715
III Control 725
13 Path Planning 729
ii Contents
13.1 Cubic Path . 729
13.2 Polynomial Path 735
xx13.3 F Non-Polynomial Path Planning 747
13.4 Manipulator Motion by Joint Path 749
13.5 Cartesian Path . 754
13.6 F Rotational Path . 759
13.7 Manipulator Motion by End-Effector Path 763
13.8 Summary 777
13.9 Key Symbols 779
Exercises . 781
14 F Time Optimal Control 791
14.1 F Minimum Time and Bang-Bang Control . 791
14.2 F Floating Time Method . 801
14.3 F Time-Optimal Control for Robots . 811
14.4 Summary 817
14.5 Key Symbols 819
Exercises . 821
15 Control Techniques 827
15.1 Open and Closed-Loop Control 827
15.2 Computed Torque Control . 833
15.3 Linear Control Technique . 838
15.3.1 Proportional Control 839
15.3.2 Integral Control 839
15.3.3 Derivative Control . 839
15.4 Sensing and Control 842
15.4.1 Position Sensors 843
15.4.2 Speed Sensors 843
15.4.3 Acceleration Sensors 844
15.5 Summary 845
15.6 Key Symbols 847
Exercises . 849
References 853
A Global Frame Triple Rotation 863
B Local Frame Triple Rotation 865
C Principal Central Screws Triple Combination 867
D Trigonometric Formula 869
Index 873
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 2 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, 534874
end-effector, 535
Euler parameters, 536, 538
matrix, 530
quaternions, 538
recursive, 565
vector, 530
Angular momentum
2 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
Index875
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, 7
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
Dynamics, 527, 556, 641
2R planar manipulator, 651,
653, 655, 664, 680
4 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, 6
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
Index876
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
1 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
Index877
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,
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, 3
acceleration vector, 549
active, 4
angle, 235
axis, 4
coordinate, 4
cylindrical, 301
distance, 235
free, 4
inactive, 4
orthogonal, 8
parallel, 8
parameters, 235
passive, 4
path, 749
perpendicular, 8
prismatic, 3
revolute, 3
rotary, 3
screw, 4
speed vector, 442, 454
spherical, 270
translatory, 3
variable, 4
Kinematic length, 235
Kinematics, 31
acceleration, 529
assembling, 280
direct, 259
forward, 32, 233, 259
Index878
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, 1
Levi-Civita density, 109
Lie group, 72
Link, 3
angular velocity, 439
class 1 and 2, 247
class 11 and 12, 252
class 3 and 4, 248
class 5 and 6, 249
class 7 and 8, 250
class 9 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, 9
cylindrical, 9
definition, 5
inertia matrix, 670
one-link, 621
one-link control, 840
one-link dynamics, 644
planar polar, 674
PUMA, 238
SCARA, 9
space station, 268, 270
spherical, 9
transformation matrix, 333
Mass center, 582, 583, 587
Matrix
skew symmetric, 70, 71, 92,
103
Method
Index879
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
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
Index880
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, 8
control, 13, 15
control algorithms, 833
cylindrical, 10, 318
Index881
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, 8
history, 2
laws, 1
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
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, 9
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
Index882
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,
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
Translation, 32
Triad, 16
Trigonometric equation, 338
Turn vector, 275
Twist vector, 275
Unit system,
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
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
xix
xix
Index883
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
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