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

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
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 258
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

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

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

l
labview 3
Lagrange multipliers 331, 332
Lagrange's equations 303–307, 321, 340, 343
largest weakly invariant subset 365
LaSalle's invariance principle 363
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

p
pantograph 11
parallel axis theorem 214, 221, 223, 224, 228, 229, 280
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 formulations 143, 157, 158
recursive formulations 138
recursive order algorithms 271
recursive order 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, 390
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