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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مُساهمةموضوع: كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control    كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control  Emptyالخميس 17 أغسطس 2023, 2:30 am

أخواني في الله
أحضرت لكم كتاب
Theory of Applied Robotics - Kinematics, Dynamics, and Control
Reza N. lazar
DepartmentofMechanical Engineering
Manhattan College
Riverdale, NY

كتاب Theory of Applied Robotics - Kinematics, Dynamics, and Control  T_o_a_11
و المحتوى كما يلي :


Contents
1 Introduction 1
1.1 Historical Development 1
1.2 Components and Mechanisms of a Robotic System 2
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 . 7
1.3.1 Geometry . 8
1.3.2 Worksp ace 11
1.3.3 Actuation 12
1.3.4 Control 13
1.3.5 Application 13
1.4 Introduction to Robot's Kinematics, Dynamics, and Control 14
1.4.1 Triad 15
1.4.2 Unit Vectors 16
1.4.3 Reference Frame and Coordinate System 16
1.4.4 Vector Function . . . 19
1.5 Problems of Robot Dynamics . . . . . 19
1.6 Preview of Covered Topics . . . . . . . 21
1.7 Robots as Multi-disciplinary Machines 22
1.8 Summary 22
Exercises 25
I Kinematics
2 Rotation Kinematics
2.1 Rotation About Global Cartesian Axes .
2.2 Successive Rotation About Global Cartesian Axes
2.3 Global Roll-Pitch-Yaw Angles .
2.4 Rotation About Local Cartesian Axes .
2.5 Successive Rotation About Local Cartesian Axes
47xviii Contents
2.6 Euler Angles 48
2.7 Local Roll-Pitch-Yaw Angles 59
2.8 Local Axes Rotation Versus Global Axes Rotation 61
2.9 General Transformation . . . . . . 63
2.10 Active and Passive Transformation 71
2.11 Summary 73
Exercises 75
3 Orientation K inematics 81
3.1 Axis-angle Rotation 81
3.2 * Euler Parameters . . . . . . . . . . 88
3.3 * Determination of Euler Parameters 96
3.4 * Quaternions . . . . . . . . . . . . . 99
3.5 * Spinors and Rot ators . . . . . . . . 102
3.6 * Problems in Representing Rotations . 105
3.6.1 Rotation matrix 105
3.6.2 Angle-axis . . 106
3.6.3 Euler angles . . . 107
3.6.4 Quaternion . . . 109
3.6.5 Euler parameters 111
3.7 * Composition and Decomposition of Rotations 113
3.8 Summary 118
Exercises 119
4 Motion Kinematics 127
4.1 Rigid Body Motion . . . . . . . . . . . 127
4.2 Homogeneous Transformation . . . . . 131
4.3 Inverse Homogeneous Transformation 139
4.4 Compound Homogeneous Transformation 145
4.5 * Screw Coordinates 154
4.6 * Inverse Screw . . . . . . . . . . . 169
4.7 * Compound Screw Transformation 170
4.8 * The Plucker Line Coordinate . . . 173
4.9 * The Geometry of Plane and Line 180
4.9.1 * Moment . . . . . . 180
4.9.2 * Angle and Distance . . 181
4.9.3 * Plane and Line . . . . 181
4.10 * Screw and Plucker Coordinate 186
4.11 Summary 188
Exercises 191
5 Forward K inematics 199
5.1 Denavit-Hartenberg Notation . . . . . . . . . . . . . . . .. 199
5.2 Transformation Between Two Adjacent Coordinate Frames 208
5.3 Forward Position Kinematics of Robots 226Contents xix
5.4 * Coordinate Transformation Using Screws 242
5.5 * Sheth Method 247
5.6 Summary 253
Exercises 255
6 Inverse Kinematics 263
6.1 Decoupling Technique . . . . . . . 263
6.2 Inverse Transformation Technique 270
6.3 Iterat ive Technique . . . . . . . . . 282
6.4 * Comparison of the Inverse Kinematics Techniques 287
6.4.1 * Existence and Uniqueness of Solution . 287
6.4.2 * Inverse Kinemati cs Techniques. 288
6.5 * Singular Configuration 289
6.6 Summary 291
Exercises 293
7 Angular Velocity 297
7.1 Angular Velocity Vector and Matrix 297
7.2 Time Derivative and Coordinate Frames 310
7.3 Rigid Body Velocity . . . . . . . . . . . 320
7.4 Velocity Transformation Matrix. . . . . 325
7.5 Derivative of a Homogeneous Transformation Matrix 330
7.6 Summary 336
Exercises 339
8 Velocity Kinematics 343
8.1 Rigid Link Velocity . . . . . . . . . . . . . . . . . . . . 343
8.2 Forward Velocity Kinematic s and the Jacobian Matrix 346
8.3 Jacobian Generating Vectors 351
8.4 Inverse Velocity Kinematics 363
8.5 Summary 368
Exercises 371
9 Numerical Methods in Kinematics 375
9.1 Linear Algebraic Equations . . 375
9.2 Matrix Inversion . . . . . . . . . . 388
9.3 Nonlinear Algebraic Equations . . 395
9.4 * Jacobian Matrix From Link Transformation Matrices 402
9.5 * Kinematics Recursive Equations . . . . . . . . 410
9.5.1 * Recursive Velocity in Base Frame . . . 410
9.5.2 * Recursive Acceleration in Base Frame . 412
9.6 Summary 412
Exercises 413xx Contents
II Dynamics 417
10 Acceleration Kinematics 421
10.1 Angular Acceleration Vector and Matrix 421
10.2 Rigid Body Acceleration . . . . . . . 429
10.3 Acceleration Transformation Matrix 432
10.4 Forward Acceleration Kinematics . 435
10.5 * Inverse Acceleration Kinematics 437
10.6 Summary 440
Exercises 443
11 Motion Dynamics 447
11.1 Force and Moment . . . . . . . . . 447
11.2 Rigid Body Translational Kinetics 453
11.3 Rigid Body Rotational Kinetics . . 455
11.4 Mass Moment of Inertia Matrix . . 466
11.5 Lagrange's Form of Newton 's Equations of Motion 478
11.6 Lagrangian Mechanics 487
11.7 Summary 492
Exercises 495
12 Robot Dynamics 505
12.1 Rigid Link Recursive Acceleration 505
12.2 Rigid Link Newton-Euler Dynamics 509
12.3 Recursive Newton-Euler Dynamics . 520
12.4 Robot Lagrange Dynamics . . . . . . 528
12.5 * Lagrange Equations and Link Transformation Matrices 534
12.6 Robot Statics 544
12.7 Summary 551
Exercises .. . .. 557
III Control
13 Path Planning
13.1 Joint Cubic Path .
13.2 Higher Polynomial Path . . . .
13.3 Non-Polynomial Path Planning
13.4 Manipulator Motion by Joint Path
13.5 Cartesian Pat h .
13.6 * Rotational Path . . . . . . . . .
13.7 Manipulator Motion by End-Effector Path .
13.8 Summary
Exercises .
60114 * Time Optimal Control
14.1 * Minimum Time and Bang-Bang Contro l
14.2 * Floating Time Method .
14.3 * Time-Opt imal Contro l for Robots
14.4 Summary
Exercises .
15 Control Techniques
15.1 Open and Closed-Loop Control
15.2 Computed Torque Contro l . .
15.3 Linear Control Technique . .
15.3.1 Proportional Contro l .
15.3.2 Integral Control .
15.3.3 Derivative Control .
15.4 Sensing and Control . . ..
15.4.1 Position Sensors..
15.4.2 Speed Sensors. . .
15.4.3 Acceleration Sensors. .
15.5 Summary
Exercises .
References
A Global Frame Triple Rotation
B Local Frame Triple Rotation
C Principal Central Screws Triple Combination
D Trigonometric Formula
Index
Contents xxi
Index
2R planar manipulat or
cont rol, 652
DR transformation matrix, 212
dynamics, 491, 540
equations of motion, 494
forward acceleration, 439
ideal, 491
inverse acceleration, 441
inverse kinematics, 281, 286,
399
inverse velocity, 366, 368
Jacobian matrix, 350, 352
joint 2 acceleration, 433
joint path , 591
kinetic energy, 492
Lagrange dynamics , 533, 542
Lagrangean, 493
Newton-Euler dynamics, 516
potential energy, 493
recursive dynamics, 524
time-optimal control, 630
with massive links, 542
3R planar manipulato r
DR transformation matrix, 204
forward kinematics, 227
4R planar manipulat or
statics, 548
Acceleration
angular, 423, 428, 429, 431,
432
bias vector, 440
body point , 317, 432, 434, 452
centripetal, 432
constant parabola, 593
constant path , 580
Coriolis, 454
discontinuous path, 588
discrete equation, 620, 631
end-effector, 429
forward kinematics, 437, 439
gravitational, 532, 538, 549
inverse kinematics, 439
jump, 573
matrix, 414, 423, 434-436
recursive, 507, 510
sensors, 659
tangential, 432
Active transformation, 72
Actuator, 7, 12
force and torque, 513, 529,
553
optimal torque, 632, 633
torque equation, 518, 630
Algorithm
floating-time, 619, 629
inverse kinemati cs, 286
LV factorization, 380
LV solut ion, 380
Newton-Raphson, 398
Angle-axis rotation, 106
Angular accelerat ion, 423, 431, 432
combination, 428
end-effector, 429
in terms of Euler parameters,
429, 431
in terms of quaternion, 431
recursive, 414
Angular momentum
2 link manipulator, 462
Angular velocity, 53, 56, 57, 86,
299,306
alternative definition, 318
combination, 305
coordinate transformat ion, 308
decomposition, 305686 Index
elements of matrix, 311
in terms of Euler parameters,
310
in terms of quaternion, 309
in terms of rotation matrix,
307
instantaneous, 301
instantaneous axis, 302
matrix, 300
principal matrix, 304
recursive, 412, 509
Articulated arm, 8, 231, 267, 357,
408
Atan2 function, 272
Automorphism, 102
Axis-angle rotation, 81, 84, 85, 90,
91, 94
Block diagram , 644
Brachisto chrone, 616, 627
Bryant angles, 58
Cardan
angles, 58
frequencies, 58
Cartesian
angular velocity, 56
end-effector position, 365
end-effector velocity, 366
manipulat or, 8, 11
path, 592
Central difference, 625
Chasles theorem, 154, 166
Christoffel operator, 488, 535
Co-state variable, 610
Control
adapt ive, 649
admissible, 618
bang-bang , 609, 610
characteristic equation, 646
closed-loop, 643
command, 643
computed force, 651
computed torque, 648, 649
derivative, 655
desired path , 643
error, 643
feedback, 644
feedback command, 651
feedback linearization, 648, 651
feedforward command, 651
gain-scheduling, 649
input, 650
integral, 655
linear, 649, 654
minimum time, 609
modified PD , 657
open-loop, 643, 650
path points, 595
PD ,657
proportional, 654
robots, 13
sensing, 657
stability of linear, 646
time-opt imal, 618, 622, 629,
630, 633
time-opt imal description, 618
time-opt imal path , 627
Contro ller, 7
Coordinate
cylindrical, 152
frame, 17
non-Cartesian , 487
non-orthogonal, 117
parabolic, 487
spherical, 153, 332
system, 17
Coriolis
acceleration, 428, 434
effect, 454
force, 453
Cycloid, 617
Denavit-Hartenberg, 31
meth od, 199, 202, 248
nonstand ard method, 223, 283
notation, 199
parameters, 199, 334, 345, 510,
548transformation, 208, 212-21 8,
220, 222, 243
Differential manifold, 71
Differentiating, 312
B-derivative, 312, 314
G-derivative, 312, 317
second, 320
transformation formula, 317
Distal end, 199, 548
Dynamics , 421, 507
2R planar manipulator, 516,
524
4 bar linkage, 514
actuator's force and torque,
529
backward Newton-Euler, 522
forward Newton-Euler, 529
global Newton-Eul er, 511
Lagrange , 530
Newton-Euler, 511
one-link manipulator, 513
recursive Newton-Euler, 511,
522
Earth
effect of rotation, 453
kinetic energy, 486
revolution, 486
rot ation , 486
rotation effect, 428
Eigenvalue, 87
Eigenvector , 87
Ellipsoid
energy, 465
momentum, 464
End-effector , 6
acceleration, 437
angular acceleration , 429
angular velocity, 363
articulated robot, 267
configuration vector , 348, 405,
437
configuration velocity, 437
force, 530
frame, 207, 231
Index 687
inverse kinematics, 265
kinematics, 237
link, 199
orientation, 271, 364
path , 591, 600
position, 231
position kinematics, 226
position vector, 358
rotation, 597
SCARA position, 149
SCARA robot, 240
space station manipulator, 243
spherical robot, 247
time optimal control, 609
velocity, 348, 354, 365
velocity vector, 348
Energy
Ear th kinetic, 486
kinetic rigid body, 461
kinetic rotational, 458
link's kinetic, 531, 537
link's potential, 532
mechanical, 486
point kinetic, 451
potential, 489
robot kinetic, 531, 538
robot potential, 532, 538
Euler
-Lexell-Rodriguez formula, 83
angles, 18, 48, 51, 53, 107
integrability, 57
coordinate frame, 56
equation of motion , 457, 460,
461, 466, 467, 513, 523
frequencies, 53, 56, 306
inverse matrix, 69
parameters, 88- 92, 96-98 , 100,
111,309,310
rotation matrix, 51, 69
theorem, 48, 88
Euler equation
body frame, 460, 467
Euler-Lagrange
equat ion of motion , 614, 615
Eulerian688 Index
viewpoint , 326
Floating time , 620
1 DOF algorithm, 619
analytic calculation, 627
backward path, 622
convergence, 625
forward path, 621
method,618
multi DOF algorithm, 629
multiple switching, 633
path planning, 627
robot control, 629
Force, 449
action, 512
actuator, 529
conservative, 489
Coriolis, 454
driven , 512
driving, 512
generalized , 483, 532
gravitational vector, 533
potential, 489
potential field, 485
reaction, 512
sensors, 660
shaking, 516
time varying, 454
Forward kinematics, 32
Frame
central, 455
final, 207
goal, 207
principal, 457
reference, 16
special, 206
station, 206
tool, 207
world, 206
wrist, 207
Generalized
coordinate, 480, 483,484, 490
force, 482,483,485,487,489,
491, 494, 530
inverse Jacobian, 403
Grassmanian, 177
Group properties, 70
Hamiltonian, 610
Hand , 231
Hayati-Roberts notation, 224
Helix, 154
Homogeneous
compound transformation, 145
coordinate, 133, 138
direction, 138
general transformation, 139,
143
inverse transformation, 139,
141, 142, 146
position vector , 133
scale factor , 133
transformation, 131, 134-137,
139, 141
Integrability, 57
Inverse kinematics, 32, 265
decoupling technique, 265
inverse transformation technique,272
iterative technique, 284
Pieper technique, 274
Inverted pendulum, 652
Jacobian
analytical, 365
elements, 363
generating vector, 353, 355,
404
geometrical, 365
inverse, 287, 403
matrix, 285, 287, 290, 292,
348, 352, 355, 357, 361,
365, 368, 397, 401, 404,
407, 408, 437, 439, 442,
534
oflink,531
polar manipulator, 349
Jerkangul ar , 430
matrix, 436
transformation, 435, 437
zero path, 579
Joint, 3
acceleration vector, 437
active, 4
coordinate, 4
cylindrical, 252
inactive, 4
orthogonal, 8
parallel, 8
passive , 4
path, 591
perp endicular , 8
screw, 4
variable vector, 348
velocity vector, 348, 355
Joint angle, 200
Joint distance, 200
Joint parameters, 202
Kinematic length, 200
Kinematics, 31
acceleration, 423
forward , 32, 226
forward acceleration, 437
forward velocity, 348
inverse, 32, 265, 272
inverse acceleration, 439
inverse velocity, 365
numerical methods, 377
velocity, 345
Kinetic energy, 451
Earth, 486
link, 537
parabolic coordinate, 487
rigid body, 461
robot, 531, 538
rotational body, 458
Kronecker's delta, 65, 457, 479
Lagrange
dynamics, 530
equation, 536
Index 689
equation of motion, 480, 489
mechanics , 489
multiplier, 617
Lagrange equation
explicit form, 488
Lagrangean , 489, 538
robot, 538
viewpoint, 326
Law
motion , 450
motion second, 450, 455
motion third, 450
robotics, 1
Levi-Civita density, 96
Lie group, 71
Link, 3
angular velocity, 346
class 1 and 2, 213
class 11 and 12, 218
class 3 and 4, 214
class 5 and 6, 215
class 7 and 8, 216
class 9 and 10, 217
classification, 219
end-effector, 199
Euler equation, 523
kinetic energy, 531
Newton-Euler dynamics, 511
recursive acceleration, 507, 510
recursive Newton-Euler dynamics, 522
recursive velocity, 509, 510
rotational acceleration, 508
translational acceleration, 508
translational velocity, 347
velocity, 345
Link length, 200
Link offset , 200
Link parameters, 202
Link twist, 200
Location vector, 156, 158
LV factorization method, 377, 392
Manipulator
2R planar , 491, 533690 Index
3R planar , 227
articulat ed, 205
definition, 5
inertia matri x, 532
one-link, 490
one-link control, 655
one-link dynamics, 513
PUMA, 204
SCARA, 8
transformat ion matri x, 267
Mass center, 450, 451, 455
Matrix
skew symmet ric, 68, 69, 82,
89
Moment , 449
action, 512
driven, 512
driving, 512
reaction, 512
Moment of inertia
about a line, 479
ab out a plane, 479
about a point , 479
characteristic equation, 477
diagonal elements , 477
Huygens-Steiner theorem, 471
matri x, 468
parallel-axes theorem, 469
polar, 468
principal, 469
principal axes, 458
principal invariants, 477
product , 468
pseudo matrix, 469
rigid body, 457
rotated-axes theorem, 469
Moment of momentum, 450
Moment um, 450
angular, 450
ellipsoid, 464
linear, 450
Motion, 14
Newton
equation of motion, 480
Newton equation
body frame, 456
global frame, 455
Lagrange form, 482
rotatin g frame, 453
Newton-Euler
backward equations, 522
equation of motion, 523
forward equations, 529
global equations, 511
recursive equations, 522
Numerical methods, 377
analytic inversion, 394
Cayley-Hamilton inversion, 395
condition number, 388
ill-conditioned, 388
Jacobian matrix, 404
LU factorization, 377
LU factorization with pivoting, 383
matrix inversion, 390
Newton-Ra phson, 398, 400
nonlinear equations, 397
norm of a matrix, 389
partition ing inversion, 393
uniqueness of solut ion, 387
well-conditioned, 388
Optim al control, 609
a linear syste m, 610
descript ion, 618
first variation, 615
Hamiltonian, 610, 613
Lagrange equation, 614
objective function, 609, 613
performance index, 613
second variation, 615
switching point , 611
Orthogonality condition, 64
Passive transformation, 72
Path
Brachistochrone, 627
Cartesian, 592
constant acceleration, 580constant angular acceleration,
599
cont rol points, 595
cubic , 571
cycloid, 590
har monic, 589
higher polynomial, 578
jerk zero, 579
joint space, 591
non-polynomial, 589
planning, 592
point sequence, 582
quadratic, 577
quintic, 578
rest-to-rest, 573, 574
rotati onal, 597
splitting, 584
to- rest , 573
Pendulum
cont rol, 652
inverted, 652, 657
linear contro l, 655
oscillat ing, 484
simple, 425, 483
spherical, 490
Permutation symbol, 96
Phase plane, 611
Pieper technique, 274
Plucker
angle, 181
classification coord inate, 178
distance, 181
line coordinate, 173, 175- 177,
181, 185-187, 247, 248
moment , 180
ray coordinate, 175, 177
reciprocal produ ct , 181
screw , 186
virtual product , 181
Poinsot 's construction, 464
Point at infinity, 138
Pole, 163
Position sensors, 658
Positioning, 14
Potent ial
Index 691
force, 489
Potential energy
robot, 532, 538
Proximal end , 199, 548
Quaternions, 99
addition, 99
composition rotation, 102
flag form, 99
inverse rotation, 101
multiplication, 99
rotation, 100
Rigid body
acceleration, 431, 508
angular momentum, 458
angular velocity, 86
Euler equation of motion, 461,
466
kinematics, 127
kinetic energy, 461
moment of inertia, 457
motion, 127
mot ion classification, 167
motion composition , 131
principal rotation matrix, 476
rotational kinetics, 457
steady rot ation, 462
translat ional kinetics, 455
velocity, 321, 323
Robot
applicat ion, 13
articulated, 8, 231, 238, 267,
357, 361
Cartesian , 11
classification, 7
control, 13, 14
control algorithms, 648
cylindrical, 11, 259
dynamics, 14, 19, 507, 533
end-effector path, 600
equation of mot ion, 540
forward kinematics, 226, 246
gravitational vector , 533
inertia matrix, 532692 Index
kinemati cs, 14
kinetic energy, 531, 538
Lagrange dynamics, 530, 536
Lagrange equation, 533
Lagrangean, 532, 536
link classification, 245
modified PD control, 657
Newton-Euler dynamics, 511
PD control, 657
potential energy, 532, 538
recursive Newton-Euler dynamics, 522
rest position , 200, 203, 231,
235, 239
SCARA , 149, 239
spherical, 10, 205, 235, 246,
274,355
state equat ion, 613
statics, 546
time-opt imal control, 613, 629
velocity coupling vector, 533
Robotic
geometry, 8
history, 1
law, 1
Rodriguez
rot ation formula, 83, 84, 89,
92- 95, 101, 106, 114, 128,
158, 161, 167, 172, 302,
337, 597
vector, 95, 113
Roll-pitch-yaw
frequency, 60
global angles, 41, 59
global rotation matrix, 41, 59
Rotati on, 32, 83
about global axis, 33, 38, 40
about local axis, 43, 47, 48
angle-axis, 106
axis-angle, 81, 83-85, 90, 91,
94, 106
composition, 113
decomposition, 113
eigenvalue, 87
eigenvector, 87
exponential form, 93
general, 63
infinitesimal, 92
local versus global, 61
matr ix, 18, 105
pole, 326
quaternion, 100
stanley met hod, 98
X-matrix, 33
x-matrix, 43
Y-matrix, 33
y-mat rix, 43
Z-matrix, 33
z-matrix, 43
Rotational path , 597
Rotator, 83, 102
SCARA
manipulator, 8
robot , 149, 239
Screw, 154, 157, 166
axis, 154
central, 155, 156, 159, 160,
173, 187, 202, 243, 245,
247
combination, 170, 172
coordinate, 154
decomposition, 172, 173
exponential, 171
forward kinematics, 243
instantaneous, 187
intersection, 248
inverse, 169, 170, 172
left-handed, 155
link classificati on, 245
location vector, 156
motion, 202, 327
parameters, 155, 164
pitch, 154
Plucker coordinate, 186
principal, 166, 172, 173
reverse central, 156
right-h anded, 15, 155
special case, 162
transformation, 158, 165twist, 154
Second derivative, 320
Sensor
acceleration, 659
position, 658
rotary, 658
velocity, 659
Sheth not ation , 248
Singular configuration, 291
Spherical coordinate, 153
Spinor, 83, 102
Spline, 588
Stanley method, 98
Stark effect, 487
Symbols, xi
Ti lt vector, 231
Time derivative, 312
Top, 53
Transformation , 31
active and passive, 71
general, 63
homogeneous, 131
Transformation matrix
derivative, 332
differential, 336, 337
elements, 66
velocity, 327
Translati on, 32
Triad , 15
Trigonometric equation, 271
Turn vector , 231
Twist vector , 231
Unit system, xi
Unit vectors, 16
Index 693
Vector
gravitational force, 533
velocity coupling, 533
vector
gravitational force, 537
velocity coupling, 536
Vector decomposition, 117
Velocity
body point , 452
discrete equation, 620, 631
end-effector, 348
inverse tr ansformation, 330
matrix, 436
operator matrix, 333
prismatic transformat ion, 335
revolut e transformation, 335
sensors, 659
transformation matrix, 327,
329, 331, 333
Work, 451, 454
virtual, 483
Work-energy principle, 451
Workspace, 11
Wrench, 452
Wrist , 12-14, 231
decoupling kinematics, 266
forward kinematics, 229
frame, 207
kinematics assembly, 238
point , 6, 229, 271
position vector, 270
spherical, 6, 205, 231, 235, 361
transformation matrix 230
, ,
267
Zero velocity point , 326


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