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 |  موضوع: كتاب Dynamics and Control of Robotic Systems السبت 14 نوفمبر 2020, 11:43 am | |
| 
أخوانى فى الله
أحضرت لكم كتاب
Dynamics and Control of Robotic Systems
Andrew J. Kurdila and Pinhas Ben-Tzvi
Virginia Tech
Virginia
USA
و المحتوى كما يلي :
Contents
Preface xiii
Acknowledgment xv
About the Companion Website xvii
1 Introduction 1
1.1 Motivation 1
1.2 Origins of Robotic Systems 5
1.3 General Structure of Robotic Systems 7
1.4 Robotic Manipulators 9
1.4.1 Typical Structure of Robotic Manipulators 9
1.4.2 Classification of Robotic Manipulators 11
1.4.2.1 Classification by Motion Characteristics 11
1.4.2.2 Classification by Degrees of Freedom 12
1.4.2.3 Classification by Driver Technology and Drive
Power 12
1.4.2.4 Classification by Kinematic Structure 12
1.4.2.5 Classification by Workspace Geometry 14
1.4.3 Examples of Robotic Manipulators 14
1.4.3.1 Cartesian Robotic Manipulator 15
1.4.3.2 Cylindrical Robotic Manipulator 16
1.4.3.3 SCARA Robotic Manipulator 16
1.4.3.4 Spherical Robotic Manipulator 17
1.4.3.5 PUMA Robotic Manipulator 18
1.4.4 Spherical Wrist 18
1.4.5 Articulated Robot 20
1.5 Mobile Robotics 20
1.5.1 Humanoid Robots 20
1.5.2 Autonomous Ground Vehicles 22
1.5.3 Autonomous Air Vehicles 23
1.5.4 Autonomous Marine Vehicles 25
1.6 An Overview of Robotics Dynamics and Control Problems 26
1.6.1 Forward Kinematics 27
1.6.2 Inverse Kinematics 28
1.6.3 Forward Dynamics 28
1.6.4 Inverse Dynamics and Feedback Control 29viii Contents
1.6.5 Dynamics and Control of Robotic Vehicles 30
1.7 Organization of the Book 31
1.8 Problems for Chapter 1 33
2 Fundamentals of Kinematics 35
2.1 Bases and Coordinate Systems 35
2.1.1 N-Tuples and M × N Arrays 35
2.1.2 Vectors, Bases and Frames 39
2.1.2.1 Vectors 40
2.1.2.2 Bases and Frames 41
2.2 Rotation Matrices 49
2.3 Parameterizations of Rotation Matrices 52
2.3.1 Single Axis Rotations 52
2.3.2 Cascades of Rotation Matrices 56
2.3.2.1 Cascade Rotations about Moving Axes 56
2.3.2.2 Cascade Rotations about Fixed Axes 57
2.3.3 Euler Angles 57
2.3.3.1 The 3-2-1 Yaw-Pitch-Roll Euler Angles 58
2.3.3.2 The 3-1-3 Precession-Nutation-Spin Euler Angles 62
2.3.4 Axis Angle Parameterization 65
2.4 Position, Velocity, and Acceleration 68
2.5 Angular Velocity and Angular Acceleration 77
2.5.1 Angular Velocity 77
2.5.2 Angular Acceleration 83
2.6 Theorems of Kinematics 84
2.6.1 Addition of Angular Velocities 84
2.6.2 Relative Velocity 87
2.6.3 Relative Acceleration 88
2.6.4 Common Coordinate Systems 91
2.6.4.1 Cartesian Coordinates 91
2.6.4.2 Cylindrical Coordinates 92
2.6.4.3 Spherical Coordinates 94
2.7 Problems for Chapter 2, Kinematics 96
2.7.1 Problems on N-tuples and M × N Arrays 96
2.7.2 Problems on Vectors, Bases, and Frames 97
2.7.3 Problems on Rotation Matrices 98
2.7.4 Problems on Position, Velocity, and Acceleration 102
2.7.5 Problems on Angular Velocity 104
2.7.6 Problems on the Theorems of Kinematics 104
2.7.6.1 Problems on the Addition of Angular Velocities 104
2.7.7 Problems on Relative Velocity and Acceleration 105
2.7.8 Problems on Common Coordinate Systems 108
3 Kinematics of Robotic Systems 109
3.1 Homogeneous Transformations and Rigid Motion 109
3.2 Ideal Joints 115
3.2.1 The Prismatic Joint 116Contents ix
3.2.2 The Revolute Joint 117
3.2.3 Other Ideal Joints 119
3.3 The Denavit–Hartenberg Convention 121
3.3.1 Kinematic Chains and Numbering in the DH Convention 121
3.3.2 Definition of Frames in the DH Convention 123
3.3.3 Homogeneous Transforms in the DH Convention 124
3.3.4 The DH Procedure 127
3.3.5 Angular Velocity and Velocity in the DH Convention 133
3.4 Recursive O(N) Formulation of Forward Kinematics 138
3.4.1 Recursive Calculation of Velocity and Angular Velocity 140
3.4.2 Efficiency and Computational Cost 143
3.4.3 Recursive Calculation of Acceleration and Angular
Acceleration 147
3.5 Inverse Kinematics 160
3.5.1 Solvability 160
3.5.2 Analytical Methods 163
3.5.2.1 Algebraic Methods 163
3.5.2.2 Geometric Methods 174
3.5.3 Optimization Methods 176
3.5.4 Inverse Velocity Kinematics 184
3.5.4.1 Singularity 185
3.6 Problems for Chapter 3, Kinematics of Robotic Systems 186
3.6.1 Problems on Homogeneous Transformations 186
3.6.2 Problems on Ideal Joints and Constraints 188
3.6.3 Problems on the DH Convention 188
3.6.4 Problems on Angular Velocity and Velocity for Kinematic
Chains 190
3.6.5 Problems on Inverse Kinematics 195
4 Newton–Euler Formulations 197
4.1 Linear Momentum of Rigid Bodies 197
4.2 Angular Momentum of Rigid Bodies 203
4.2.1 First Principles 203
4.2.2 Angular Momentum and Inertia 208
4.2.3 Calculation of the Inertia Matrix 214
4.2.3.1 The Inertia Rotation Transformation Law 214
4.2.3.2 Principal Axes of Inertia 218
4.2.3.3 The Parallel Axis Theorem 221
4.2.3.4 Symmetry and Inertia 224
4.3 The Newton–Euler Equations 229
4.4 Euler’s Equation for a Rigid Body 233
4.5 Equations of Motion for Mechanical Systems 235
4.5.1 The General Strategy 235
4.5.2 Free Body Diagrams 236
4.6 Structure of Governing Equations: Newton–Euler Formulations 258
4.6.1 Differential Algebraic Equations (DAEs) 258
4.6.2 Ordinary Differential Equations (ODEs) 260x Contents
4.7 Recursive Newton–Euler Formulations 262
4.8 Recursive Derivation of the Equations of Motion 271
4.9 Problems for Chapter 4, Newton–Euler Equations 274
4.9.1 Problems on Linear Momentum 274
4.9.2 Problems on the Center of Mass 277
4.9.3 Problems on the Inertia Matrix 279
4.9.4 Problems on Angular Momentum 281
4.9.5 Problems on the Newton–Euler Equations 282
5 Analytical Mechanics 285
5.1 Hamilton’s Principle 285
5.1.1 Generalized Coordinates 285
5.1.2 Functionals and the Calculus of Variations 288
5.1.3 Hamilton’s Principle for Conservative Systems 292
5.1.4 Kinetic Energy for Rigid Bodies 299
5.2 Lagrange’s Equations for Conservative Systems 303
5.3 Hamilton’s Extended Principle 307
5.3.1 Virtual Work Formulations 307
5.4 Lagrange’s Equations for Robotic Systems 322
5.4.1 Natural Systems 322
5.4.2 Lagrange’s Equations and the Denavit–Hartenberg
Convention 326
5.5 Constrained Systems 329
5.6 Problems for Chapter 5, Analytical Mechanics 334
5.6.1 Problems on Hamilton’s Principle 334
5.6.2 Problems on Lagrange’s Equations 337
5.6.3 Problems on Hamilton’s Extended Principle 339
5.6.4 Problems on Constrained Systems 345
6 Control of Robotic Systems 347
6.1 The Structure of Control Problems 347
6.1.1 Setpoint and Tracking Feedback Control Problems 348
6.1.2 Open Loop and Closed Loop Control 349
6.1.3 Linear and Nonlinear Control 349
6.2 Fundamentals of Stability Theory 350
6.3 Advanced Techniques of Stability Theory 357
6.4 Lyapunov’s Direct Method 358
6.5 The Invariance Principle 361
6.6 Dynamic Inversion or Computed Torque Control 366
6.7 Approximate Dynamic Inversion and Uncertainty 376
6.8 Controllers Based on Passivity 389
6.9 Actuator Models 393
6.9.1 Electric Motors 393
6.9.2 Linear Actuators 400
6.10 Backstepping Control and Actuator Dynamics 404
6.11 Problems for Chapter 6, control of Robotic Systems 407Contents xi
6.11.1 Problems on Gravity Compensation and PD Setpoint
Control 407
6.11.2 Problems on Computed Torque Tracking Control 412
6.11.3 Problems on Dissipativity Based Tracking Control 413
7 Image Based Control of Robotic Systems 415
7.1 The Geometry of Camera Measurements 415
7.1.1 Perspective Projection and Pinhole Camera Models 415
7.1.2 Pixel Coordinates and CCD Cameras 418
7.1.3 The Interaction Matrix 419
7.2 Image Based Visual Servo Control 423
7.2.1 Control Synthesis and Closed Loop Equations 424
7.2.2 Calculation of Initial Conditions 427
7.3 Task Space Control 441
7.4 Task Space and Visual Control 447
7.5 Problems for Chapter 7 459
A Appendix 465
A.1 Fundamentals of Linear Algebra 465
A.1.1 Solution of Matrix Equations 467
A.1.2 Linear Independence and Rank 468
A.1.3 Invertibility and Rank 470
A.1.4 Least Squares Approximation 470
A.1.5 Rank Conditions and the Interaction Matrix 475
A.2 The Algebraic Eigenvalue Problem 475
A.2.1 Self-adjoint Matrices 476
A.2.2 Jordan Canonical Form 478
A.3 Gauss Transformations and LU Factorizations 479
References 485
Index 489
Index
a
a priori 420
accelerometers 415
action functional 294, 298, 303, 304, 321,
331
actual motion 292, 307
actual trajectory 292
actuation moment 312
actuation moments 309
actuation torques 311
actuators 7, 311
addition theorem for angular velocities 84
admissible direction 307
admissible directions 304, 305
admissible variations 305
algebraic eigenvalue problem 219
algebraic unknowns 247, 253, 258
analog-to-digital 3
analytical mechanics 204, 288, 289, 293
angle encoders 415
angular acceleration 77, 83
angular momenta 203
angular momentum 203, 204, 208–210,
215, 230–232, 234–236, 282
angular velocities 236
angular velocity 71, 77, 79, 81, 95, 210,
215, 300, 315, 326, 420, 421, 424
anthropomorphic 20
anthropomorphic arm 20
anthropomorphic robot arm 20
anthropomorphic robot arms 20
anthropomorphic robots 20
approximate dynamic inversion 377–379,
389
arm sweep 20
armature 394
articulated robot arm 20
asymptotic stability 349, 354, 361, 362,
366
asymptotically stable 354, 355, 359, 362,
363, 365, 390
autonomous 364, 365
autonomous ground vehicles 22
axis angle parameterization 65
b
back electromotive force (EMF) 395
back EMF 395
back EMF constant 395
backstepping control 405
base body 138
base frame 122
bases 35
basis 40–42
basis fixed derivative 71
brushes 394
c
calculus of variations 288, 305, 307, 321
calibrated coordinates 416
calibration constants 419
calibration matrix 419
calibration parameter 418
camera coordinate trajectories 433
camera coordinates 416–419, 422, 428,
431, 433
camera extrinsic parameters 424
camera frame 415, 416, 420, 421, 424, 427,
429
camera frame coordinates 417, 421
Dynamics and Control of Robotic Systems, First Edition. Andrew J. Kurdila and Pinhas Ben-Tzvi.
 2020 John Wiley & Sons Ltd. Published 2020 by John Wiley & Sons Ltd.
Companion website: www.wiley.com/go/kurdila/robotic-systems490 Index
camera intrinsic parameter matrix 419
camera model 415
canonical image plane coordinates 416
Capek, Karel 5
Cartesian manipulator 18
Cartesian robot 15, 16
Cartesian robotic manipulator 16
CCD 418
CCD arrays 418
center of mass 198, 199, 202–204, 209,
210, 214, 217, 218, 221–223, 228,
230–232, 235, 280, 300, 314–316,
326–328, 337
change of basis 40
change of basis formula 49
charge coupled device 418
chattering solutions 379
closed loop 424
closed loop connectivity 13
closed loop control 349
closed loop equations 431
closed loop system 424–426
closed loop topology 13
closed set of ordinary differential equations
426
closed system of ordinary differential
equations 426
communicator 7
commutator 394
components 40, 45
computed control torques 376
computed torque control 350, 356, 369,
376, 377, 391
computed torque control law 366
computed torque controllers 348
concatenates 57
configuration 28
configuration space 292, 293
conformal partitions 39
connectivity topology 12
conservation of energy 325
conservative 304
conservative mechanical system 293, 304,
305, 331
conservative mechanical systems 285,
293, 303, 307
constrained optimization 177
constraint force 309
contemporaneous 307
control input 424, 425
control law 424
control synthesis 1
control unit 7
controllability 350
conventional manipulator 12
coordinate plane of symmetry 225
coordinate systems 35, 41
coordinates 40, 45
coordinates of the offset 428
core frame 122
Coriolis centripetal matrix 326
couple 311, 312
coupled, nonlinear ordinary differential
equations 426
cross product 41
cross products of inertia 204, 206, 208,
221, 222, 224, 225, 227
cylindrical manipulator 12
cylindrical robot 16
d
DAEs 235, 258
DC motor 393
decrescent 359
deficient 12
degrees of freedom 10, 11, 28, 282
Denavit–Hartenberg (DH) convention 28,
121, 326, 327
dependent variations 332
derivative feedback 367
derivative of unit vectors 71
derivative Theorem 420
derivative with respect to an observer 71
desired image point locations 424
Devol, George 5
dextral 41, 42
dextrous workspace 14
DH convention 121–125, 127, 130, 133,
190, 192
DH Convention 191
DH procedure 127, 128
diagonal matrix 38
diagonalizable 97
differential algebraic equations 258Index 491
differential unknowns 247, 253, 258
differential-algebraic equations 235
differentiation of rotation matrices 77
differentiation of unit vectors 77
digital-to-analog 2
direct drive manipulator 12
direction 293
direction cosine matrices 47
directional derivative 290, 291
dot product 37
driven joint 14
dynamic inversion 350, 356, 368, 369,
376
e
(3-2-1) Euler angles 58
eigenvectors 219
electric robot 12
electromagnetic induction 393
electromechanical linear motors 400
end effector 7
equilibrium 351, 359, 362–364
Euclidean norm 37
Euler angles 57, 58, 62, 65
Euler’s equations 233
Euler’s first law 230–232, 235
Euler’s laws 211, 229,
235
Euler’s second law 209, 230–232, 235
exactly determined 425
exponential stability 354
extrema 288
extremization 288
extremization problems 288
f
feature points 416, 420, 422, 423, 424, 429,
431–433, 438
feedback control 26, 27, 349
feedback control law 426
feedback controllers 349
feedback function 424
feedback linearization 350
feedforward control 367
final camera coordinates 431
final camera frame 427, 428, 430, 432, 433,
438
floating point operation 143
flops 143
focal coordinates 418
focal length 415, 419–421, 429, 463
focal plane 415–418, 431, 433
focal plane coordinates 417, 418, 420, 429,
431
focal plane trajectories 433, 438
forward dynamics 1, 12, 26, 27, 29, 348,
368, 369
forward kinematics 1, 26–28, 160
frame 40–42
frames of reference 35
free body diagrams 235, 236–238
full matrix 38
functionals 288
fundamental theorem of variational
calculus 305
g
G-derivative 288, 289
G-differentiable 289
gain matrices 367
Gateaux derivative 288, 289, 304
general purpose robot 12
generalized coordinates 285, 286, 288,
289, 292–294, 297, 303, 304,
306–310, 314, 316, 317, 319, 321,
322, 326, 329, 330, 339, 340
generalized displacements 308
generalized forces 308
generalized inertia 326
generalized inertia or mass matrix 322
generalized mass or inertia matrix 306
global asymptotic stability 359
global minimizer 177
global positioning system (GPS) 415
global stability 349–351
globally asymptotically stable 355, 363
globally exponentially stable 357
globally stable 351, 363
gripping device 7
h
Hamilton’s extended principle 319–321
Hamilton’s principle 285, 293, 295, 296,
299, 303, 305, 307, 308, 319, 321492 Index
holonomic constraints 329–331
homogeneous 189
homogeneous coordinates 110, 417, 419,
428
homogeneous transform 117, 118, 124,
417
homogeneous transformations 109, 110,
112, 117, 123, 191, 417, 428
humanoid arms 20
humanoid robots 21
hybrid manipulator 14
hydraulic robot 12
i
IBVS 424, 426
IBVS controller 429, 432
ideal joints 9, 10, 11, 115, 116–119, 236
identity matrix 38
image based visual servo 423
image based visual servo control law 431
image Jacobian 441
image Jacobian matrix 419
image plane 424
image plane coordinates 420–422, 426
image plane tracking error 424
image point 416–418
inclination 63
independent 286, 329
index 35
inertia matrices 215
inertia matrix 208–210, 214, 217, 218,
223, 227, 228, 232, 280, 328
inertia rotation transformation law 214,
217, 223, 224
inertia tensor 209, 231, 300
inertial coordinates 418, 419
inertial frame 211, 230, 329, 420, 424, 429,
432
inertial frame 0 coordinates 417
inertial matrix 209
inertial reference frame 229
inflection points 288
initial camera configuration 431, 433
initial camera coordinates 431
initial camera frame 427, 428, 430, 432,
433, 438
initial condition 427, 432
initial coordinate frame 428
initial focal plane coordinates 428
initial focal plane tracking error 428
interaction matrix 419, 420, 422, 463
intrinsic 418
intrinsic parameter matrix 419
invariant 363, 365
invariant set 363
inventor 3, 4
inverse dynamics 26
inverse kinematics 1, 20, 26–28, 160, 162,
168, 176, 177
inverse matrix 96
invertible 425
j
Jacobian matrices 133
Jacobian matrix 133, 185, 191, 194, 327,
330
joint angles 27
joint coordinate systems 116–118
joint frames 116
joint space control 441
joint variables 27, 28, 117, 118, 122,
133–135, 327
joints 9
k
Karel Capek 5
kinematic chain 13, 14, 122, 123, 133, 326,
327
kinematic chains 13, 14, 121
kinematic constraints 321
kinematic decoupling 168, 176
Kinematic structure 12
kinematically decoupled 172
kinematics 35
kinetic energy 293, 300, 303, 319, 322, 326
kinetics 35
l
labview 3
Lagrange multipliers 331, 332
Lagrange’s equations 303–307, 321, 340,
343
largest weakly invariant subset 365
LaSalle’s invariance principle 363Index 493
line of nodes 62
line-of-sight 416, 418
linear control theory 349, 367
linear matrix equations 38
linear momenta 197
linear momentum 197, 198, 200, 203, 204,
229, 230, 235, 275–277
linear multistep methods 427
linear ODEs 350
linear systems 349, 350
link displacement 124
link offset 124
link parameters 128
link rotation 124
link twist 124
links 9
local minimizer 176
local stability of an equilibrium 351
locally asymptotically stable 355
locally decrescent 359
locally negative definite 358, 359, 362, 363
locally negative semi-definite 365
locally positive definite 358, 359, 360, 362,
364
lower limit 35
Lyapunov functions 358, 360
m
magnetic flux 395
magnetic flux linkage 395
magnetometers 415
main diagonal 38
manifolds 40
manipulator workspace 14
maple 4
mass center 200, 277, 278
mass matrix 293
mathcad 4
mathematica 4
matlab 4
matrix element 37
matrix inverse 38
matrix multiplication 38
maxima 288
measure valued solutions 379
mechatronics 2
method of dynamic inversion 350
minimal 286, 288, 329
minimum singular values 438, 441
modal matrix 219
moments of inertia 204, 206, 208, 217,
221, 224
motion 292
mscadams 3
multibody dynamics 12, 115
multibody system 13
multifunctional manipulator 6
n
natural systems 322, 324, 325, 366, 389
negative definite 359, 363, 380, 390
negative semi-definite 362, 363, 378
Newton’s first law 229
Newton’s second law 229
non-autonomous system 351
non-conservative 321
non-conservative systems 307
nonconservative 319
nonlinear ODEs 350
nonlinear systems 349, 350
norm 36
number of degrees 311
number of degrees of freedom 236, 286
numerical integration methods 427
o
observability 350
odd function 225, 226
ODEs 258, 260
open loop control 349
open loop manipulator 13
optimization 177
optimization theory 176
ordinary differential equations 235, 258,
426, 427, 430
origin offset 432
orthogonal matrix 97
orthonormal 41, 42
overdetermined 425
p
pantograph 11
parallel axis theorem 214, 221, 223, 224,
228, 229, 280494 Index
parallel manipulators 13
passive joint 15
passivity 389
passivity principles 348
passivity properties 389
permanent magnet 393
perspective projection 415
pinhole camera 415
pixel array 418
pixel coordinates 418, 419
pixels 418
planar manipulator 11
planes of symmetry 225, 226
Player Piano 5
pneumatic robot 12
position control 348
position feedback 367
position vectors 73, 310, 312, 416
positive definite 359, 362, 363, 377, 380
positive invariant 363, 366
positive invariant set 363, 378
possible motions 292, 293, 307
potential energy 293, 303, 319, 322
power supply 8
PPP manipulators 15
predictor corrector methods 427
principal axes 214, 218, 219, 225, 280
principal diagonal 38
principal moments of inertia 218, 219,
224, 234
principal point 418
princpal axes 218
prismatic joints 9, 14, 15, 27, 116, 117,
119, 122, 134
products of inertia 214
proengineer 3
projection matrix 419
proportional-derivative (PD) control 377
pseudo inverse 425
r
ranges 426
rate gyros 415
reachable workspace 14
reaction forces 309
real time 160
recursive O(N) formulations 143, 157, 158
recursive formulations 138
recursive order (N) algorithms 271
recursive order (N) formulations 209
redundant 12
redundant generalized coordinates 286,
329–331, 333
relative acceleration 88
relative position 76, 77
relative velocity 87
representations 40
resultant moment 231
retinal coordinates 416
revolute joints 9, 14, 15, 44, 117, 118, 119,
122, 134
right ascension 63
right handed 41
right-handed 42
rigid body 300
rigid body motions 109, 189
robotic arm 6, 9
robotic control systems 415
robotic manipulators 9, 14, 15
robotic system 1, 5, 6, 9, 13, 322, 415
robotics 12
root frame 122
Rossum’s Universal Robots 5
rotation matrices 47, 49, 109
rotation matrix 428
rotation matrix parameterizations 52
Runge–Kutta methods 427
s
scalar triple product 97, 300
scale factors 418
scaled coordinates 418
screw joint 9
sensor suites 8
sensors 8, 415
serial manipulator 13
setpoint control 348, 361, 377
setpoint control law 348
single axis rotations 52
single degree of freedom joints 10
singular value decomposition 425
skew 83
skew operator 41
skew symmetric 325, 326, 378, 389, 390Index 495
skew symmetry 389
solidworks 3
sparse matrix 38
spatial manipulator 12
spherical joint 9, 282
spherical manipulator 12, 16, 17
spherical robot 17, 18
spherical robotic manipulator 17
spherical wrist 18, 20, 192, 194
square matrix 38
stability 348, 349, 425
stabilizability 350
stable 351, 359, 362, 424
stable equilibrium 351
stable system 351
state variables 426, 427
stationarity 290
stationarity conditions 303
stationary 290, 304
stator 394
symbolic computations 322
symmetric positive definite 378, 379
system connectivity 12
system feature points 425
system interaction matrix 422, 423, 425,
438
system vectors 272
systems of ordinary differential equations
427
t
Tait–Bryan angles 58
target camera configuration 431
task space controllers 441, 442
task space coordinates 441, 447
task space Jacobian matrix 441
task space variables 441
tensor analysis 40, 214
tensor basis 209
tensor transformation 214
time varying bases 69
topological tree 121
total time derivative 68, 71
tracking control 30, 348, 361
tracking control law 348
tracking control problem 424
tracking error 424
tracking error in the image plane 424
trajectory 292
trajectory tracking 348
transformation laws 40
transport theorem 421
transpose 37
tree topology connectivity 13
true motion 293, 307
u
unconstrained optimization problem 177
underdetermined 425
uniformly ultimately bounded 381
Unimate 5
unit vector 40
universal joint 9, 119, 283
upper limit 35
v
variation 307, 322
variation ????t 321
variation operator 308, 321
vector spaces 39, 40
vectors 39, 40, 45
virtual displacements 307, 308, 309–313,
318, 319, 321
virtual variation 307, 309, 311, 319, 320
virtual variation operator 307–310, 312,
319, 321, 330
virtual variations 307–309, 321, 330
virtual work 307–313, 315, 316, 319, 320
visual servo control 424
visual servo image based control law 431
Vonnegut, Kurt 5
w
weakly invariant 363
weakly invariant set 363
weakly invariant subset 365
workspace 15–18
workspace geometry 14
wrist center 18
y
yaw-pitch-roll angles 58
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