اسم المؤلف
Reza N. Jazar
التاريخ
15 أكتوبر 2021
المشاهدات
التقييم
(لا توجد تقييمات)

Advanced Dynamics Rigid Body, Multibody, and Aerospace Applications
Reza N. Jazar
School of Aerospace, Mechanical, and Manufacturing Engineering
RMIT University
Melbourne, Australia
Contents
Preface xiii
Part I Fundamentals 1
1 Fundamentals of Kinematics 3
1.1 Coordinate Frame and Position Vector 3
1.1.2 Coordinate Frame and Position Vector 4
1.1.3  Vector Definition 10
1.2 Vector Algebra 12
1.2.2 Vector Multiplication 17
1.2.3  Index Notation 26
1.3 Orthogonal Coordinate Frames 31
1.3.1 Orthogonality Condition 31
1.3.2 Unit Vector 34
1.3.3 Direction of Unit Vectors 36
1.4 Differential Geometry 37
1.4.1 Space Curve 38
1.4.2 Surface and Plane 43
1.5 Motion Path Kinematics 46
1.5.1 Vector Function and Derivative 46
1.5.2 Velocity and Acceleration 51
1.5.3  Natural Coordinate Frame 54
1.6 Fields 77
1.6.1 Surface and Orthogonal Mesh 78
1.6.2 Scalar Field and Derivative 85
1.6.3 Vector Field and Derivative 92
Key Symbols 100
Exercises 103
2 Fundamentals of Dynamics 114
2.1 Laws of Motion 114
2.2 Equation of Motion 119
2.2.1 Force and Moment 120
2.2.2 Motion Equation 125
2.3 Special Solutions 131
2.3.1 Force Is a Function of Time, F = F(t) 132
2.3.2 Force Is a Function of Position, F = F(x) 141
2.3.3  Elliptic Functions 148
2.3.4 Force Is a Function of Velocity, F = F(v) 156
2.4 Spatial and Temporal Integrals 165
2.4.1 Spatial Integral: Work and Energy 165
2.4.2 Temporal Integral: Impulse and Momentum 176
2.5  Application of Dynamics 188
2.5.1  Modeling 189
2.5.2  Equations of Motion 197
2.5.3  Dynamic Behavior and Methods of Solution 200
Key Symbols 223
Exercises 226
Part II Geometric Kinematics 241
3 Coordinate Systems 243
3.1 Cartesian Coordinate System 243
3.2 Cylindrical Coordinate System 250
3.3 Spherical Coordinate System 263
3.4  Nonorthogonal Coordinate Frames 269
3.4.1  Reciprocal Base Vectors 269
3.4.2  Reciprocal Coordinate Frame 278
3.4.3  Inner and Outer Vector Product 285
3.4.4  Kinematics in Oblique Coordinate Frames 298
3.5  Curvilinear Coordinate System 300
3.5.1  Principal and Reciprocal Base Vectors 301
3.5.2  Principal–Reciprocal Transformation 311
3.5.3  Curvilinear Geometry 320
3.5.4  Curvilinear Kinematics 325
3.5.5  Kinematics in Curvilinear Coordinates 335
Key Symbols 346
Exercises 347
4 Rotation Kinematics 357
4.1 Rotation About Global Cartesian Axes 357
4.2 Successive Rotations About Global Axes 363
4.3 Global Roll–Pitch–Yaw Angles 370
4.4 Rotation About Local Cartesian Axes 373
4.5 Successive Rotations About Local Axes 376
4.6 Euler Angles 379Contents ix
4.7 Local Roll–Pitch–Yaw Angles 391
4.8 Local versus Global Rotation 395
4.9 General Rotation 397
4.10 Active and Passive Rotations 409
4.11  Rotation of Rotated Body 411
Key Symbols 415
Exercises 416
5 Orientation Kinematics 422
5.1 Axis–Angle Rotation 422
5.2 Euler Parameters 438
5.3  Quaternion 449
5.4  Spinors and Rotators 457
5.5  Problems in Representing Rotations 459
5.5.1  Rotation Matrix 460
5.5.2  Axis–Angle 461
5.5.3  Euler Angles 462
5.5.4  Quaternion and Euler Parameters 463
5.6 Composition and Decomposition of Rotations 465
5.6.1 Composition of Rotations 466
5.6.2  Decomposition of Rotations 468
Key Symbols 470
Exercises 471
6 Motion Kinematics 477
6.1 Rigid-Body Motion 477
6.2 Homogeneous Transformation 481
6.3 Inverse and Reverse Homogeneous Transformation 494
6.4 Compound Homogeneous Transformation 500
6.5  Screw Motion 517
6.6  Inverse Screw 529
6.7  Compound Screw Transformation 531
6.8  Plucker Line Coordinate 534 ¨
6.9  Geometry of Plane and Line 540
6.9.1  Moment 540
6.9.2  Angle and Distance 541
6.9.3  Plane and Line 541
6.10  Screw and Plucker Coordinate 545 ¨
Key Symbols 547
Exercises 548x Contents
7 Multibody Kinematics 555
7.1 Multibody Connection 555
7.2 Denavit–Hartenberg Rule 563
7.3 Forward Kinematics 584
7.4 Assembling Kinematics 615
7.5  Order-Free Rotation 628
7.6  Order-Free Transformation 635
7.7  Forward Kinematics by Screw 643
7.8  Caster Theory in Vehicles 649
7.9 Inverse Kinematics 662
Key Symbols 684
Exercises 686
Part III Derivative Kinematics 693
8 Velocity Kinematics 695
8.1 Angular Velocity 695
8.2 Time Derivative and Coordinate Frames 718
8.3 Multibody Velocity 727
8.4 Velocity Transformation Matrix 739
8.5  Derivative of a Homogeneous Transformation Matrix 748
8.6  Multibody Velocity 754
8.7  Forward-Velocity Kinematics 757
8.8  Jacobian-Generating Vector 765
8.9  Inverse-Velocity Kinematics 778
Key Symbols 782
Exercises 783
9 Acceleration Kinematics 788
9.1 Angular Acceleration 788
9.2 Second Derivative and Coordinate Frames 810
9.3 Multibody Acceleration 823
9.4 Particle Acceleration 830
9.5  Mixed Double Derivative 858
9.6  Acceleration Transformation Matrix 864
9.7  Forward-Acceleration Kinematics 872
9.8  Inverse-Acceleration Kinematics 874
Key Symbols 877
Exercises 878Contents xi
10 Constraints 887
10.1 Homogeneity and Isotropy 887
10.2 Describing Space 890
10.2.1 Configuration Space 890
10.2.2 Event Space 896
10.2.3 State Space 900
10.2.4 State–Time Space 908
10.2.5  Kinematic Spaces 910
10.3 Holonomic Constraint 913
10.4 Generalized Coordinate 923
10.5 Constraint Force 932
10.6 Virtual and Actual Works 935
10.7  Nonholonomic Constraint 952
10.7.1  Nonintegrable Constraint 952
10.7.2  Inequality Constraint 962
10.8  Differential Constraint 966
10.9 Generalized Mechanics 970
10.10  Integral of Motion 976
10.11  Methods of Dynamics 996
10.11.1  Lagrange Method 996
10.11.2  Gauss Method 999
10.11.3  Hamilton Method 1002
10.11.4  Gibbs–Appell Method 1009
10.11.5  Kane Method 1013
10.11.6  Nielsen Method 1017
Key Symbols 1021
Exercises 1024
Part IV Dynamics 1031
11 Rigid Body and Mass Moment 1033
11.1 Rigid Body 1033
11.2 Elements of the Mass Moment Matrix 1035
11.3 Transformation of Mass Moment Matrix 1044
11.4 Principal Mass Moments 1058
Key Symbols 1065
Exercises 1066
12 Rigid-Body Dynamics 1072
12.1 Rigid-Body Rotational Cartesian Dynamics 1072
12.2  Rigid-Body Rotational Eulerian Dynamics 1096
12.3 Rigid-Body Translational Dynamics 1101xii Contents
12.4 Classical Problems of Rigid Bodies 1112
12.4.1 Torque-Free Motion 1112
12.4.2 Spherical Torque-Free Rigid Body 1115
12.4.3 Axisymmetric Torque-Free Rigid Body 1116
12.4.4  Asymmetric Torque-Free Rigid Body 1128
12.4.5 General Motion 1141
12.5 Multibody Dynamics 1157
12.6  Recursive Multibody Dynamics 1170
Key Symbols 1177
Exercises 1179
13 Lagrange Dynamics 1189
13.1 Lagrange Form of Newton Equations 1189
13.2 Lagrange Equation and Potential Force 1203
13.3  Variational Dynamics 1215
13.4  Hamilton Principle 1228
13.5  Lagrange Equation and Constraints 1232
13.6 Conservation Laws 1240
13.6.1 Conservation of Energy 1241
13.6.2 Conservation of Momentum 1243
13.7  Generalized Coordinate System 1244
13.8  Multibody Lagrangian Dynamics 1251
Key Symbols 1262
Exercises 1264
References 1280
A Global Frame Triple Rotation 1287
B Local Frame Triple Rotation 1289
C Principal Central Screw Triple Combination 1291
D Industrial Link DH Matrices 1293
E Trigonometric Formula 1300
Index 1305
Index
2R planar manipulator
dynamics, 1207, 1258
equations of motion, 1209
forward acceleration, 873
general dynamics, 1258
ideal, 1207
inverse acceleration, 876
inverse kinematics, 664
inverse velocity, 778, 780
Jacobian matrix, 761, 763
kinetic energy, 1208
Lagrangian, 1208
mass moments, 1051
Newton–Euler dynamics, 1168
potential energy, 1208
recursive dynamics, 1172
dynamics, 1163
spatial, 560, 602
A
Abel, Niels Henrik, 149
Acatastatic constraint, 966
Acceleration
angular, 789, 796, 809, 824
applied, 851
bias vector, 875
body point, 725, 824
centripetal, 726, 824, 832
Coriolis, 812, 832, 859
definition, 51
double mixed, 861
energy, 1009, 1011
forward kinematics, 872, 873
gravitational, 1254
inverse kinematics, 874
kinematics, 788
local, 832
matrix, 788, 864, 865
mixed, 816, 817
mixed Coriolis, 859
mixed double, 823, 859
multibody, 823
particle, 830
Razi, 860, 861, 886
tangential, 726, 812, 824, 832, 859
tidal, 190
transformation, 792, 818
Active transformation, 409
Actuator
force and torque, 1159, 1177
torque equation, 1169
Air resistance, 163
Angle
attitude, 829
cruise, 829
Euler, 379
nutation, 379
precession, 379
sideslip, 829
spin, 379
zenith, 509
Angular acceleration, 788, 789, 805, 823,
824
decomposition, 795
Euler parameters, 796, 809
matrix, 788, 808
natural frame, 809
quaternion, 809
relative, 792, 799
Rodriguez formula, 799
rotational transformation, 788, 791
vector, 788
Angular jerk, 808
Cartesian, 880
matrix, 808
Angular velocity, 385, 387, 388, 432, 695,
697, 699–702, 706, 789, 794
alternative definition, 715
13051306 Index
Angular velocity (continued)
combination, 699
coordinate transformation, 709
decomposition, 705
elements of matrix, 714
Euler parameters, 713
instantaneous, 697, 698
instantaneous axis, 699
matrix, 695
principal matrix, 702
quaternion, 711
rotation matrix, 707
Appell function, 1012
Appell, Paul Emile, 1011
Applied force, 933
Arc length, 40
Articulated arm, 666, 671, 769
Assembling kinematics, 615
Atan2 function, 665
Atwood, George, 1001
Atwood machine, 1000
Australia, 109, 885
Automorphism, 457
Axis–angle of rotation, 422, 425, 426,
440, 442, 446, 461
B
bac–cab rule, 22, 29
Basic lemma, 1217
Bernoulli equation, 931
Bernoulli, Johann, 942
Bernoulli, Johann and Jacob, 1221
Bipolar coordinate system, 341
Bipolarcylindrical coordinate system, 341
Bispherical coordinate system, 340, 342
Bong, 53
Book-stacking problem, 193
Boom, 1126
Brachistochrone, 1220
Bragg condition, 296
Broucke, Roger, 30
Bryant angles, 390
Bushehr, 858
C
Camber theory, 658
Canonical equation, 1003
Cardan
angles, 390
frequencies, 390
Cardioid, 258
Cardioidal coordinate system, 342
Cardioidcylindrical coordinate system, 342
Cartesian
angular velocity, 387
coordinate system, 32, 243
end-effector position, 777
end-effector velocity, 778
orthogonality condition, 32
unit vectors, 34
Casscylindrical coordinate system, 342
Caster theory, 649
Catastatic constraint, 966
Center point, 903
Central force, 993
Central frame, 1038
Central principle, 1230, 1231, 1239
Centrifugal moment, 1033
Chasles, Michel, 121
Chasles theorem, 517, 526
Christoffel, Elwin Bruno, 328
Christoffel operator, 334
Christoffel symbol, 325, 327, 333, 334,
1203, 1211
first kind, 327
second kind, 327
third kind, 335
Circular integrals, 153
Collision, 177, 181
elastic, 177, 178
inelastic, 181, 182
oblique, 187
plastic, 182
restitution coefficient, 182
Concave surface, 79
Condition
Bragg, 296
orthogonality, 31, 32
reciprocality, 271
Configuration
coordinate, 924
degree of freedom, 923, 930
path, 47Index 1307
space, 890
trajectory, 890
Confocalellip coordinate system, 342
Confocalparab coordinate system, 343
Conical coordinate system, 343
Conservation
energy, 1241
Jacobi Integral, 1242
laws, 1240
momentum, 1243
Conservative force, 174
Constant of motion, 977, 1241
Constraint
acatastatic, 966
acceleration, 922
catastatic, 966
constraint-free space, 929
force, 932, 933, 937, 943
frozen, 915
generalized, 975
holonomic, 913
inequality, 962
jerk, 923
just-, 928, 929
least, 999
limit, 962
motion, 999
multiple, 919
nonholonomic, 952
nonintegrable, 952
over-, 928, 929
Pfaffian forms, 966, 967, 1238
plane, 916
rheonomic, 913
rolling, 956–958
scleronomic, 913
slip, 962
total, 952
under-, 928, 929
unicycle, 955
Contravariant
metric, 279
vector components, 278
Control
bang-bang, 1225
directional control system, 594
minimum time, 1225
Convex surface, 79
Coordinate
cyclic, 1243
cylindrical, 506
ignorable, 1243
non-Cartesian, 1201
nonorthogonal, 32
parabolic, 1201
spherical, 744
Coordinate frame, 3, 5
Cartesian, 282
curvilinear, 300, 309
reciprocal base vector, 302
reciprocal unit vector, 302
space scale factor, 302
natural, 54, 55
neshin, 616
nonorthogonal, 24, 274, 282, 284, 285,
300
vector product, 285, 289, 291
oblique, 294, 298, 300
acceleration, 299
velocity, 299
origin, 5
orthogonal, 5, 31
orthogonality condition, 31
principal, 284, 1058
transformation, 284
reciprocal, 278, 284
transformation, 284
rim, 586
takht, 616
tire, 608
vehicle, 608
wheel, 608
wheel–body, 608
Coordinate system, 4, 243
bipolar, 341
bipolarcylindrical, 341
bispherical, 340, 342
cardioidal, 342
cardioidcylindrical, 342
Cartesian, 243
casscylindrical, 342
confocalellip, 342
confocalparab, 343
conical, 3431308 Index
Coordinate system (continued)
cylindrical, 250, 253, 257, 258
orthogonality, 253
ellcylindrical, 343
ellipsoidal, 343
elliptic-hyperbolic cylindrical, 267
hypercylindrical, 343
invcasscylindrical, 344
invellcylindrical, 344
invoblspheroidal, 344
logcoshcylindrical, 344
logcylindrical, 344
maxwellcylindrical, 344
natural, 334
nonorthogonal, 269
oblatespheroidal, 345
oblique, 277
orthogonal, 243, 341
parabolic cylindrical, 261
paraboloidal1, 345
paraboloidal2, 345
paracylindrical, 345
principal, 278
rosecylindrical, 345
sixsphere, 345
spherical, 263, 265, 266, 339
orthogonality, 265
tangentcylindrical, 345
tangentsphere, 345
toroidal, 346
Coriolis
acceleration, 802, 832, 833
force, 833, 836
Coriolis, Gaspard Gustave de, 835
Costate variable, 1225
Coulomb, Charles Augustin de, 938
Covariant
metric, 279
vector components, 278
Crackle, 53
Curl, 93
potential force, 173
Curvature, 70
center, 77
plane curve, 61
principal, 84
surface, 84
vector, 63, 64
vectorial expression, 62
Curvilinear
acceleration, 331
base vector, 301
coordinate frame, 309
coordinate system, 293, 295, 300–302,
309, 320, 1245
reason, 334
vector product, 319
work, 331
differential, 339
geometry, 320
kinematics, 325, 335
natural coordinate, 330, 331
parallelepiped, 323, 326, 329
velocity, 331
Cyclic coordinate, 1243
Cycloid, 247, 586, 1220
curtate, 247, 586
ordinary, 247
prolate, 247, 586
D
D’Alembert, Jean Le Rond, 942
D’Alembert principle, 943
Lagrange’s form, 942
Da Vinci, Leonardo, 938
Darboux, Jean Gaston, 70
Darboux vector, 70
Degree of freedom, 923
configuration, 891, 919, 923, 928, 952,
963
event, 896
generalized, 1250
joint, 555
rigid body, 1034
state, 901, 923, 980
state–time, 909
Del, 86
Denavit–Hartenberg
method, 563, 571
notation, 563
parameters, 563, 750, 754
rule, 563Index 1309
transformation, 566–568, 574, 575,
643, 1293–1296, 1298, 1299
Derivative
mixed double, 822, 858–860
mixed second, 819
transformation
mixed, 724, 725
simple, 724
transformation formula, 724, 819
Deviation moment, 1033
Dido problem, 1221
Differential
manifold, 408
Differential geometry, 37
immersed surface, 80
parametric line, 41
space curve, 38, 40, 41
surface and plane, 43, 44
surface expression, 81
tangent line, 41
Differentiating, 718
B-derivative, 718, 722, 814
G-derivative, 718, 725
second, 726
second derivative, 810
transformation formula, 724
Directional
angle, 572
control system, 594, 595, 629
cosine, 6–8, 18, 38, 55, 57, 58, 81, 92,
398–400, 422, 457, 460, 482, 581
curvature, 85
derivative, 89, 91, 92, 257
line, 5, 120, 126
Displacement
actual, 916, 935, 936
impossible, 916, 935, 936
possible, 916, 935, 936
virtual, 935–937
Distance
topocentric, 508
Divergence, 93
Double factorial, 153
Double pendulum, 232, 926, 1016, 1018
Dubai, 109
Duffing equation, 1136, 1137
Dynamic
coupling, 1211
decoupling, 1211
Dynamics, 114, 1157
2R planar manipulator, 1168, 1172
actuator’s force and torque, 1177
application, 188
backward Newton–Euler, 1170
definition, 130
forward Newton–Euler, 1172
fundamentals, 114
global Newton–Euler, 1157
isolated system, 116
modeling, 189
multibody, 1251
Newton–Euler, 1157
Newtonian, 118
recursive Newton–Euler, 1157, 1170
special solutions, 131
E
Earth
free fall, 840, 846, 848, 849
kinetic energy, 1200
moving vehicle, 801
revolution energy, 1200
rotation effect, 802
rotation energy, 1200
shrinking, 185
spherical, 185
Eddington, Arthur Stanley, 115
e-delta identity, 26, 29
Eigenvalue, 433
Eigenvector, 433
Einstein summation convention, 28,
301
Ellcylindrical coordinate system, 343
Ellipsoid
energy, 1092
momentum, 1092
Ellipsoidal coordinate system, 343
Elliptic function, 148, 149, 1129, 1130,
1132
limiting, 153
period, 1531310 Index
Elliptic integral, 148, 1132
arc length, 155
complete first kind, 148
complete second kind, 149
complete third kind, 149
first kind, 148
second kind, 148
third kind, 149
End-effector
acceleration, 872
angular velocity, 776
configuration vector, 872
configuration velocity, 872
force, 1172
orientation, 777
position kinematics, 584
position vector, 771
SCARA position, 505
space station manipulator, 580
time-optimal control, 1225
velocity, 757, 766, 778
velocity vector, 757
Energy
classical form equation, 950
Earth kinetic, 1200
first-form equation, 950
kinetic, 165
kinetic rigid body, 1036, 1037, 1077
kinetic rotational, 1073
mechanical, 166, 1200
multibody, 1252
potential, 166, 1203
primitive-form equation, 950
second-form equation, 950
Envelope, 136
Epitrochoid, 247
Equation
Bernoulli, 931
inhomogeneous, 931
linear ODE, 931, 962
Equation of motion, 119, 197
integral form, 119
Equivalent Lagrangian, 1213
Euclidean space, 887, 888
Euler
angles, 379, 382, 384, 462
integrability, 388
body frame equation, 1076, 1095
equation, 126
coordinate frame, 388
equation, 1076, 1093, 1095, 1159,
1171
equation of motion, 1072
frequencies, 385, 387, 706, 805
general equation, 1098
global rotation matrix, 403
inverse matrix, 403
–Lexell-Rodriguez formula, 424
local rotation matrix, 403
parameters, 439, 442, 443, 448, 449,
451, 463, 711, 713
rotation matrix, 382, 403
theorem, 378, 437–439
Euler-Chasles theorem, 438
Eulerian viewpoint, 737
Euler-Lagrange
equation of motion, 1215, 1217
Euler, Leonhard, 126, 463
Event space, 896
Event trajectory, 896
F
Fargo, ND, 885
Ferris wheel, 249
Field, 77
curl, 93
cylindrical, 258
derivative, 89, 94, 95, 98
directional derivative, 91
divergence, 93
index notation, 99
isosurface, 86, 92
isovalue, 86
Laplacian, 94
scalar, 77, 85, 99
second derivative, 97
spherical, 267
stationary, 77
tensor, 99
timeinvariant, 77
total derivative, 89
vector, 77, 88, 92, 97–99Index 1311
Final rotation formula, 435, 436
First integral, 977
First variation, 1216, 1224
Fixed frame, 4
Flash space, 910
Flash–time space, 910
Flicker space, 911
Flicker–time space, 911
Focus point, 903
Forbidden umbrella, 136, 137
Force
action, 1158
actuator, 1177
body, 120
central, 993
conservative, 166, 1203
constraint, 932, 937, 943
contact, 120
Coriolis, 833, 836
derivative, 131
driven, 1158
driving, 1158
effective, 839
external, 120
friction, 937
function, 117
general, 941
generalized, 1192, 1254
gravitational, 994
gravitational vector, 1255
internal, 120
Magnus, 1110
potential, 166, 1203
potential field, 1200
reaction, 1158
resultant, 120
system, 120
total, 120
Formula
axis–angle, 520
derivative transformation, 724, 725,
819, 820, 822, 859, 864
derivative transport, 797
final rotation, 435, 436
Frenet–Serret, 64, 65
geometric transformation, 693
Gibbs, 447
Grubler, 559, 560 ¨
inversion, 530
Kutzbach, 560
parallel axes, 1047
relative acceleration, 792, 795, 799
relative angular velocity, 700, 705
Rodriguez, 447, 456, 463, 466,
475, 477, 520, 699, 754,
785, 799
trigonometric, 1300
Forward kinematics, 584
Foucault pendulum, 853, 855
Frame
central, 1102
principal, 1073
Free-body diagram, 156
Freezing process, 915
Frenet
frame, 58
trihedron, 58
Frenet, Jean Fred ´ eric, 65 ´
Frenet–Serret formula, 64
Friction
force, 938
law, 938
Fundamental equation, 941, 942, 950,
996, 997, 1014, 1233, 1240, 1242
first form, 942, 1014
second form, 950
third form, 950
G
Galilean transformation, 890
Galilei, Galileo, 1221
Gauss
equation, 999
function, 999
Gauss dynamics, 999
Gauss, Johann Carl Friedrich, 464, 999
Generalized
applied force, 1013–1015
constraint, 973, 975
coordinate, 923, 926–929, 975, 1189,
1192, 1194, 1205
force, 986, 1002, 1191, 1192, 1194,
1202, 1203, 1206, 1209, 1246,
12521312 Index
Generalized (continued)
inertia force, 1013–1015
kinetic energy, 971
mechanics, 970
momenta, 1248
momentum, 986, 1002, 1004, 1192,
1248, 1268
potential force, 972
space, 928, 929, 974
velocity, 1249
work, 1247
Geodesics problem, 1218
Gibbs
matrix, 447
vector, 447
Gibbs–Appell
dynamics, 1009
equation, 1009, 1011, 1012
function, 1009, 1011
Gibbs, Josiah Willard, 1011
Gorz, 131
definition, 89
operator, 86
Gram determinant, 295
Grassmanian, 537
Gravitation
Newton law, 121
universal constant, 121
Gravity center, 123
Group properties, 408
Grubler formula, 560 ¨
Guldin, Paul, 226, 227
Guldin theorem, 226, 227
H
Hamilton
dynamics, 1002
equation, 1002, 1003
function, 1002
Hamiltonian, 1225
dynamics, 1002
equation, 1002
function, 1002
method, 1002
system, 1006
Hamilton principle, 1228, 1230
general form, 1228
Hamilton, William, 464
Harmonic oscillator, 142
Helix, 51, 517
Hertz, Heinrich Rudolf, 954
Hodograph, 47, 49
Homogeneity, 887, 888
position, 889
time, 888
Homogeneous
compound transformation, 500
coordinate, 482, 489
direction, 489
general transformation, 490, 498
inverse transformation, 494, 496, 498,
501
position vector, 483
scale factor, 482
transformation, 482, 487–489, 494, 497
Huygens–Steiner theorem, 1047
Hyperbolic integrals, 153
Hypercylindrical coordinate system, 343
Hypergeometric function, 152, 153
Hypertrochoid, 247
Hypotrochoid, 247
I
Ignorable coordinate, 1243
Impenetrability, 889
Impossible displacement, 936
Impulse, 119, 176, 177
Impulsive point, 893
Ince, Edward Lindsay, 970
Index notation, 26
India, 858
Inequality constraint, 962
Inertial frame, 889
Inhomogeneous equation, 931
Integrability, 388
Integral of motion, 976, 977, 1241
Integrating factor, 960
Interpolant, 14
Interpolation, 14
linear, 14
problem, 14
vector, 14Index 1313
Invariant vector, 244
Invcasscylindrical coordinate system, 344
Invellcylindrical coordinate system, 344
Inverse kinematics
Pieper technique, 676
transformation technique, 675
Inverse Lagrangian, 1214
Invoblspheroidal coordinate system, 344
Iran, 109, 858, 885
Isotropy, 887, 888
position, 889
time, 888
J
Jacobi, Carl Gustav Jacob, 149
Jacobi function, 148, 149
Jacobian, 315
analytical, 778
displacement matrix, 757
elements, 776
generating vector, 765, 767
geometrical, 778
matrix, 757, 758, 763, 767, 769, 773,
774, 778, 781, 872, 873, 877
polar manipulator, 764, 874
rotational matrix, 757
Jeeq, 53
Jerk, 53, 807, 808
angular, 808
body point, 808
definition, 51
global, 807
matrix, 865
rotational transformation, 807
transformation, 866, 867
Jerkmeter, 53
Joint, 555
acceleration vector, 872
active, 555
angle, 565
distance, 565
equivalent spherical, 562
inactive, 555
multiple, 561
parameters, 565
passive, 555
prismatic, 555
revolute, 555
rotary, 555
translatory, 555
universal, 560
variable vector, 757
velocity vector, 757, 767
Jolt, 53
Jounce, 53
Just constraint, 928, 929
Just rigid, 1033
K
Kane
dynamics, 1013
equation, 1013, 1014
Kane equation, 1016
Kennedy theorem, 734
Kinematic
length, 565
space, 910
Kinematics
acceleration, 789
assembling, 615
foot–leg, 513
forward, 584
forward acceleration, 872
forward velocity, 757
inverse, 675
inverse acceleration, 874
inverse velocity, 778
velocity, 754
Kinetic energy, 165
Earth, 1200
parabolic coordinate, 1202
rigid body, 1036, 1037, 1077
rotational body, 1073
Kinetic potential, 984
Kronecker delta, 26, 681, 1042
L
Lagrange
dynamics, 996, 1251
equation, 1217
equation of motion, 1189, 1203, 1215,
1234, 1238, 1239
explicit equation, 1202, 12101314 Index
Lagrange (continued)
mechanics, 1203
multiplier, 933, 1221
Lagrange, Joseph-Louis, 213
Lagrangian, 984, 1203, 1241
dynamics, 1189
equivalent, 1213
form invariant, 1214
fundamental theorem, 1234
generality, 1238
inverse, 1214
viewpoint, 737
Laplacian, 94, 98
Larz, 53
Law
conservation, 176, 1240
cosine, 105
force, 114
gravitational, 121, 122
inertia, 890
inverse square, 994
Kepler’s second, 994
Kepler’s third, 419
motion, 114, 126
motion second, 1101
Newton’s first, 114
Newton’s second, 114
Newton’s third, 114
rotational motion, 1077
sine, 508
Laws of motion, 114
Least constraint, 999
Levi-Civita symbol, 26, 27, 448
Levi-Civita, Tullio, 900
Libration, 142
Lie group, 408
Limit constraint, 962
Linear space, 15
angular velocity, 756
class 1 and 2, 1293
class 3 and 4, 1294
class 5 and 6, 1295
class 7 and 8, 1296
class 9 and 10, 1298
class 11 and 12, 1299
classification, 573
compound, 561
end effector, 563
Euler equation, 1171
kinetic energy, 1252
length, 565
Newton–Euler dynamics, 1157
offset, 565
parameters, 565
recursive dynamics, 1170
translational velocity, 756
twist, 565
velocity, 754
Lissajous curves, 144
Lituus, 260
Lobachevsky space, 207
Location vector, 519, 521, 650
Logcoshcylindrical coordinate
system, 344
Logcylindrical coordinate system, 344
M
Manipulator
2R planar, 1207
planar polar, 1257
SCARA, 505
Mass, 115
Mass center, 122, 126, 1039, 1040, 1102
Mass moment, 1033
characteristic equation, 1062
diagonal elements, 1041
matrix, 1035
parallel-axes theorem, 1044
polar, 1035
principal, 1058
principal directions, 1059
principal frame, 1058
principal invariants, 1062
principal planes, 1065
product, 1035
pseudomatrix, 1037
rigid body, 1072Index 1315
rotated-axes theorem, 1044
transformation, 1044
Matrix
orthogonality condition, 399
skew symmetric, 423, 425, 439
Maxwellcylindrical coordinate system, 344
Measure number, 5
Mechanical energy, 166
Mechanics
Lagrangian, 1189
Newtonian, 118
Mechanism
3D slider–crank, 598
4-bar, 560, 1163
inverted slider–crank, 731
slider–crank, 597
Melbourne, 109, 885
Methods of dynamics, 996
Gauss method, 999
Gibbs–Appell method, 1009
Hamilton method, 1002
Kane method, 1013
Lagrange method, 996, 998
Mangerone–Deleanu method, 1018
Nielsen method, 1017, 1018, 1020
Tzenoff method, 1020, 1021 ´
Methods of solution, 200
perturbation methods, 200
series methods, 200
Metric tensor, 279
Moment
action, 1158
applied, 126
driven, 1158
driving, 1158
of a force, 120
reaction, 1158
resultant, 120
total, 120
Moment of inertia, 1033
characteristic equation, 1062
diagonal elements, 1041
Huygens–Steiner theorem, 1047
parallel-axes theorem, 1044
polar, 1035
principal, 1058
principal axes, 1073
principal invariants, 1062
product, 1035
pseudomatrix, 1037
rigid body, 1072
rotated-axes theorem, 1044
Moment of momentum, 126
Momentum, 125, 176, 177
angular, 126
definition, 115
ellipsoid, 1092
linear, 125
translational, 125
Motion
central force, 993
impossible, 913
possible, 913
spiral, 245
Moving frame, 4, 58
Multibody
2R manipulator, 676, 1168, 1169, 1172
3D slider–crank mechanism, 598
articulated manipulator, 640, 666, 671
assembling, 615, 616, 621
best spherical wrist, 623
closed-loop mechanism, 597
connection, 555
Denavit–Hartenberg rule, 563
directional control system, 594
direct kinematics, 584
dynamics, 1157, 1251, 1254
forward kinematics, 584, 643
gravitational vector, 1255
inertia matrix, 1253
inverse kinematics, 662
inverse transformation, 674
kinematics, 555
kinetic energy, 1252, 1253
Lagrange dynamics, 1251
Lagrange equation, 1254
Lagrangian, 1254
order-free rotation, 628
order-free transformation, 6351316 Index
Multibody (continued)
potential energy, 1253
recursive dynamics, 1170–1172
rest position, 565
shuttle manipulator, 577, 588
spherical arm, 619
spherical robot, 576, 677
spherical wrist, 616
takht and neshin, 616
tire–wheel–vehicle, 606, 611–614
trebuchet, 581
universal joint, 602
velocity coupling vector, 1255
N
Nabla, 86, 100
identities, 100
Natural coordinate frame, 54, 66, 71
binormal, 57
bivector, 56
curvature, 57
orthogonality, 62
principal normal line, 56
n-body problem, 194, 992
Neshin, 616
Neutron star, 1127
New Delhi, 858
Newton equation, 126
body frame, 1102
definition, 822
global frame, 1101
Lagrange form, 1191
rotating frame, 839
Newton–Euler
backward equations, 1170
equation of motion, 1171
equations of motion, 1157
forward equations, 1171, 1172
global equations, 1157
recursive equations, 1170
Newtonian
dynamics, 922
Newton, Isaac, 126, 992
New York, 885
Nielsen
dynamics, 1017
equation, 1017, 1020
Node point, 903
Nonholonomic Constraint, 952
Nonpotential force, 170
O
Oblatespheroidal coordinate system, 345
Oklahoma City, 109
Optimal control, 1225
Hamiltonian, 1225
Lagrange equation, 1215
linear system, 1225
objective function, 1225
switching point, 1226
Orientable surface, 79
Orlando, 858
Orthogonality condition, 31, 399
Orthogonal mesh, 79
Osculating
plane, 54, 56, 78, 86
sphere, 66
Overconstraint, 928, 929
Overrigid, 1034
P
Paraboloidal1 coordinate system, 345
Paraboloidal2 coordinate system, 345
Paracylindrical coordinate system, 345
Pars, Leopold Alexander, 942
Partial
angular velocity, 1015
derivative, 78, 79, 86, 1245
velocity, 1013, 1014, 1016, 1017
Particle, 889
definition, 115
impenetrability, 115
Passive transformation, 409
Path
configuration, 47
cycloid, 247
frame, 58
kinematics, 46
minimizing, 1215, 1216
motion, 98
optimal, 1217, 1224
projectile, 135, 144, 221
shortest, 1218Index 1317
spiral, 260
variable, 1215
Pendulum, 145, 149, 921
acceleration, 800
compound, 1271
connected, 1269
constraint, 981
double, 1016, 1018
elastic, 1005, 1012, 1206
first integral, 981
flexible support, 1268
Foucault, 853, 855
heavy, 1272
integral of motion, 988
inverted, 1264
in wind, 1264
moving support, 921, 925, 1266, 1269
oscillating support, 1193
planar, 925, 1156
simple, 145, 152, 800, 1192
sliding, 1193
spherical, 229, 803, 804, 933, 988, 1204
turning, 1181, 1267, 1275
variable mass, 1271
velocity, 800
Permutation symbol, 26, 27, 448
Perpendicular plane, 54
Persia, 584, 860
Persian, 616
Persian Gulf, 109, 858, 885
Pfaffian forms, 966, 967, 1238
Pfaff, Johann Friedrich, 969
Pfaff problem, 970
Phase
plane, 901, 902, 906, 1006, 1227
Hamilton, 1006, 1007
modified, 1006, 1007
portrait, 901, 903, 905, 1121
velocity
modified, 1007
Physical
force, 933
quantity
scalaric, 12
vectorial, 10
Pieper technique, 676
Planar curve, 65
Plucker ¨
angle, 541
classification coordinate, 538
distance, 541
line coordinate, 534–537, 541,
544–546, 648
moment, 540
ray coordinate, 535, 537
reciprocal product, 541
screw, 545
virtual product, 541
Poinsot, Louis, 121
Point
at infinity, 489, 490
singular, 902
Point mass, 889
Pole, 493
Pop, 53
Position vector, 5
Potential
energy, 166
force, 166, 1203
Principal
angular acceleration, 794, 795
angular velocity, 702, 794
axes, 461, 486, 1065, 1073, 1133, 1138
base vector, 327
base vectors, 301, 1245
body frame, 990
central screw, 526, 532, 533
components, 283
coordinate frame, 278, 1060, 1073
coordinates, 310, 311
curvature, 84
derivative, 327, 334
differential rotation, 752
direction, 85
frame, 278, 279, 287, 304, 309
invariants, 1062
mass moment, 1058–1060, 1062, 1063,
1065, 1139
metric, 326
nonorthogonal frame, 284
normal line, 56
plane, 243
planes, 1065
Principal (continued)
rotation matrix, 423, 455, 707, 1061
unit vectors, 243, 271, 278, 282, 298,
299, 301, 304, 1245
Principle
angular impulse, 177
angular impulse and moment of
momentum, 177
central, 1230, 1231, 1239
conservation of energy, 165, 166, 985,
1242
conservation of moment of momentum,
177
conservation of momentum, 176
D’Alembert, 942, 943, 1014
decoupling, 663
determinacy, 130
Galileo relativity, 889
Gauss, 197
Hamilton, 1228, 1229, 1231, 1232
impulse and momentum, 176, 183
least constraint, 999
minimum constraint, 197
Newton–Laplace, 130
Pontryagin, 1225
projection, 289
relative velocity, 742
superposition, 118
variational, 1228, 1230
virtual work, 937, 942, 945
work and energy, 165
Problem
antiprojectile, 220
book-stacking, 193
brachistochrone, 1220
bug, 51
central force, 993
central-force motion, 1250
collision, 898
composition, 465
decomposition, 465
Dido, 1221
dynamic, 116
foot–leg kinematic, 513
forward acceleration, 872
forward kinematics, 584
forward velocity, 758
free fall, 847
geodesic, 1218
interpolation, 14
inverse acceleration, 875
inverse kinematics, 662
inverse Lagrangian, 1214
inverse velocity, 778
minimization, 1217
minimum time, 1225
n-body, 30, 194, 992
open, 367, 379
Pfaff, 970
projectile, 134
regularization, 898
resolved rates, 778
rigid body, 1112
rotation, 460
shortest path, 1218
three-body, 29, 213–216, 219
time-optimal, 1225
two-body, 194, 210
vector interpolation, 14
walking, 513
Projectile, 134–141, 156
antiprojectile gun, 220
height, 157
height time, 157
in air, 156, 160
limit velocity, 159
path, 158
range, 157
range time, 157
umberella, 137
variable gravitational acceleration, 198
Q
Quaternions, 449, 463
composition rotation, 454
flag form, 450
inverse rotation, 453
multiplication, 450
rotation, 451
R
principal, 1044Index 1319
Razi acceleration, 860
Razi, Zakariya, 860
Reciprocal
base vectors, 272, 273, 275, 283, 301,
340
components, 281, 283
coordinate frame, 278, 282
coordinates, 310, 311
derivative, 334
frame, 284
geometric interpretation, 277
metric, 335
principal vector, 276
reciprocality condition, 271
unit vectors, 269, 298, 299, 301
vectors, 269, 271, 272
scalar triple product, 271, 276
Rectifying plane, 54, 57, 75
Rectilinear motion, 906
escape, 908
libration, 907
limitation, 907
lost, 908
rest, 906
Reference frame, 4
Rest
point, 893
position, 902
Right-handed convention, 3
Right-hand Rule, 4
Rigid body
acceleration, 823
angular momentum, 1074, 1078
angular velocity, 432
asymmetric, 1112, 1128
axially symmetric, 1116
axisymmetric, 1112, 1116
body frame, 1033
centrosymmetric, 1115
classical problems, 1112
definition, 1033, 1034
Duffing equation, 1136
equimomental, 1112
Euler equation, 1076, 1093
general motion, 1141
just-rigid, 1033
kinematics, 477
kinetic energy, 1036, 1037, 1077
mass moment, 1072
moment of inertia, 1072
motion, 477
motion classification, 527
motion composition, 481
motion condition, 430
motion theorem, 1035
overrigid, 1034
Poinsot interpretation, 1135
principal rotation matrix, 1061
rolling disc, 1143
rotational kinetics, 1072
rotation condition, 430
rotation theorem, 433
spherical, 1039, 1112, 1115
stability, 1138–1141
torque-free, 1112
translational dynamics, 1101
translational kinetics, 1101
velocity, 727, 730
Robot
articulated, 666, 769, 774
recursive dynamics, 1170
rest position, 572
SCARA, 505
spherical, 677, 767
Rocket motion, 183
final velocity, 184
multistage, 185
Rodriguez
rotation formula, 424, 427, 441,
445–447, 456, 461, 467, 477, 520,
532, 699, 754
vector, 446, 447, 466
Rodriguez, Benjamin, 463
Roll angle, 1107
Roller coaster, 42, 72, 77
Rolling constraint, 956–958
Roll–pitch–yaw
frequency, 394
global angles, 455
global rotation matrix, 455
Rosecylindrical coordinate
system, 3451320 Index
Rotation, 425
acceleration transformation, 788, 791
axis–angle, 422, 425, 426, 440, 442,
446, 461
composition, 465
decomposition, 465
eigenvalue, 433
eigenvector, 433
exponential form, 445
general, 397
global axes, 357
infinitesimal, 444
kinematics, 357
local axes, 373, 376, 378
local versus global, 395
matrix, 460
order free, 628
pole, 734
quaternion, 451
reverse, 425
Stanley method, 449
successive global axes, 360,
363–365
successive local axes, 376, 377
X-matrix, 358
x-matrix, 360
Y -matrix, 358
y-matrix, 373
Z-matrix, 357
z-matrix, 373
Rotational jerk, 807
Rotator, 425, 457
Routhian, 1008
Rule
bac–cab, 22, 29
chain, 746
Denavit–Hartenberg, 563, 566–568
homogeneous transformation, 610,
741
Napier, 369
parallel-axes, 1044
relative angular acceleration, 792
relative angular velocity, 792, 860
right-hand, 4, 565
rotated-axes, 1044
Rush space, 910
Rush–time space, 910
S
Scalar, 12
equal, 12
equivalent, 12
triple product, 23, 33
Scale, 5
SCARA robot, 505
Screw, 51, 121, 517, 522, 527
axis, 517
central, 518, 519, 521, 523, 533, 546,
566, 643, 648
combination, 531, 532
coordinate, 517
decomposition, 533
exponential, 532
forward kinematics, 643
instantaneous, 546
intersection, 648
inverse, 529, 530, 532
left-handed, 517
location vector, 519
motion, 565, 738
parameters, 518, 525
pitch, 517
Plucker coordinate, 545 ¨
principal, 526, 532, 533
reverse central, 519
right-handed, 3, 517
special case, 525
transformation, 520, 523
twist, 517
Second derivative, 726
Second variation, 1216, 1224
Series solution, 29, 210, 213
Serret, Joseph Alfred, 65
Shake, 131
Sharang, 131
Shiraz, 885
Sina, Abu Ali, 584
Sixsphere coordinate system, 345
Slip constraint, 962
Snap, 53
Snatch, 131
Sooz, 53
Space
configuration, 890Index 1321
constraint-free, 929
curve, 38, 40, 41, 48
describing, 890
Euclidean, 888
event, 896
flash, 910
flash–time, 910
flicker, 911
flicker–time, 911
kinematic, 910
Lobachevsky, 207
rush, 910
rush–time, 910
spark, 911
sparkle, 911
sparkle–time, 911
spark–time, 911
state, 900
state–time, 908
Sparkle space, 911
Sparkle–time space, 911
Spark space, 911
Spark–time space, 911
Spatial integral, 165
Special solutions, 131
force of position, 141
force of time, 132
Force of velocity, 156
Spherical
arm, 619
coordinate system, 263
Earth, 185
field, 267
kinematics, 265
pendulum, 803, 804
rigid body, 1039
wrist, 616
Spinor, 425, 457
Spiral
Archimedes, 260
Fermat, 260
hyperbolic, 260
logarithmic, 260
motion, 245, 260
Stable
focus point, 903
node point, 904
Stanley method, 449
Stark effect, 1201
State
space, 900
trajectory, 900
State–time
space, 908
trajectory, 908, 909
Steering axis
caster angle, 650
caster plane, 650
forward location, 651
lateral location, 651
lean angle, 650
lean plane, 650
Sundman, Karl Frithiof, 900
Suspension
caster angle, 650
caster plane, 650
forward location, 651
lateral location, 651
lean angle, 650
lean plane, 650
location vector, 651
steering axis, 650
Symbols, xv
System of particles
angular momentum, 128
kinetic energy, 174
motion equation, 127
rotational motion, 129
work, 175
Szebehely, Victor, 419, 900
T
Takht, 616
Tangentcylindrical coordinate system, 345
Tangent plane, 78, 79, 86
Tangentsphere coordinate system, 345
Tavaan, 131
Tehran, 109
Temporal integral, 165, 176
Theorem
Chasles, 121, 517, 526
Euler, 378, 435, 437, 438, 461, 1142
Euler–Chasles, 4381322 Index
Theorem (continued)
fundamental Lagrangian, 1234
Guldin, 226, 227
Huygens–Steiner, 1047
Kennedy, 734
order-free rotations, 628
order-free transformations, 635
parallel-axes, 1044, 1047
Poinsot, 121
rigid-body motion, 1035
rigid-body rotation, 433
rotated-axes, 1044
Three-body problem, 29, 213
series solution, 29–31
Tides, 189
neap, 193
ocean, 189
spring, 193
Time derivative, 718
Time history, 896
Tire
coordinate frame, 608, 609
Tireprint, 607
Top, 384, 990, 1151
dynamics, 1150
Topocentric distance, 508
Toroidal coordinate system, 346
Toronto, 109, 885
Torque, 120
Torsion, 65, 70
Trajectory
actual, 916
configuration, 890
event, 896
possible, 916
state, 900
state–time, 908
Transformation
active and passive, 409
general, 397
homogeneous, 482
order free, 635
tire to vehicle frame, 613
tire to wheel–body frame, 611, 612
tire to wheel frame, 610, 611
wheel to tire frame, 609, 611
wheel to wheel–body frame, 612
wheel–body to vehicle frame, 614
Transformation matrix
derivative, 749
differential, 752, 753
elements, 400
velocity, 739
Trebuchet, 581
coordinate frame, 3
natural, 55
negative, 3
nonorthogonal, 269, 270
opposite, 3
orthogonal, 3
positive, 3
right-handed, 3
standard, 3
Trigonometric equation, 668
Trochoid, 247, 586
Tug, 131
Two-body problem, 210
Tzenoff ´
dynamics, 1021
equation, 1020, 1021
U
Underconstraint, 928, 929
Unicycle constraint, 955
Unit system, xv
Unit vector, 5, 34–36
Cartesian frames, 34
definition, 36
spherical frames, 34
Universal joint, 560, 602, 604, 605
history, 604
speed ratio, 605
Unstable
focus point, 903
node point, 904
USA, 109
V
Variable
scalar, 46
Variation, 940
Vecface, 10
Vecfree, 10Index 1323
Veclane, 10
Vecline, 10
Vecpoface, 10
Vecpoint, 10
Vecpolane, 10
Vecpoline, 10
Vecporee, 11
Vector
absolute value, 5
algebra, 12
angular acceleration, 788
anticommutative property, 17
associative property, 12, 15
axis, 10
bounded, 10
Cartesian, 245
characteristics, 10
commutative property, 12, 15, 17
components, 5
contravariant components, 278
covariant components, 278
decomposed expression, 6, 9
decomposition, 6, 9
definition, 10
derivative, 46, 48
direction, 10
end point, 10
equation, 20
free, 10
function, 46, 51
Gibbs, 447
gravitational force, 1255
inner product, 17, 285, 289, 291
interpolation, 14
invariant, 244
inverse-element property, 16
length, 5, 10
length-invariant property, 402
line, 10
line of action, 10
modulus, 5
multiplication, 17
natural expression, 6, 9
normal, 79
normal base, 312
null-element property, 16
outer product, 17, 285, 289, 291
plane, 10
point, 10
point-free, 11
point-line, 10
point-plane, 10
position, 5
principal base, 301
principal unit, 278, 301
quaternion product, 17
reciprocal, 269–271, 273, 275–277
reciprocal base, 278
reciprocality condition, 271
requirements, 10
reverse, 271
Rodriguez, 447
rotation, 432
scalar triple product, 23, 294
sliding, 10
space, 24
start point, 10
surface, 10
tangent base, 312
triple product, 24
types, 10
unit, 34
variable, 8
variable direction, 8
variable length, 8
vecface, 10
vecfree, 10
vecline, 10
vecpoface, 10
vecpoint, 10
vecpolane, 10
vecpoline, 10
vecporee, 11
velocity coupling, 1255
Vehicle dynamics
Lagrange method, 1091
Newton–Euler equations, 1109
pitch angle, 1107
pitch rate, 1107
roll angle, 1107
roll dynamics, 1107
roll equation, 1109
roll rate, 11071324 Index
Vehicle dynamics (continued)
yaw angle, 1107
yaw rate, 1107
Velocity
coefficient matrix, 751
definition, 51
end effector, 757
hodograph, 49
inverse transformation, 739
limit, 159
matrix, 865
operator matrix, 749
prismatic transformation matrix,
751
revolute transformation matrix,
751
transformation matrix, 739, 743,
749
Virtual
displacement, 936, 937
product, 541
velocity, 938
work, 935, 937, 941
W
Wheel
camber angle, 608
coordinate frame, 608, 609
degrees of freedom, 608
spin, 608
Wheel–body
coordinate frame, 608, 609
kinematics, 606
Work, 165
actual, 935
virtual, 935, 1192
Wrench, 121
Wrist
spherical, 576, 773
transformation matrix, 618
Y
Yank, 131
Z
Zero-velocity point, 733
Zoor, 131

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