كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control

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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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