mls94-complete[1]

versus Jacobian of a mapping, 115, 120

and manipulability measures, 128 for mapping forces, 121–123, 130 for parallel manipulators, 133

for redundant manipulators, 130 relationship between body and spa-

tial, 117

for SCARA manipulator, 118, 122 singularities, see kinematic singular-

ities spatial, 116

for Stanford manipulator, 119 manipulator workspace, see workspace mass matrix, see manipulator inertia ma-

trix Mathematica, 435

matrix exponential, 19, 27, 40 properties of, 74

maximally independent contact regions, 233

medical robotics, 398 metric tensor, 244 microrobotics, 399

minimally invasive surgery, 398 Motoman, 282

multifingered grasp, 237 zz, see also grasping

multifingered hand, 8

limitations and advantages, 212

Newton’s law, 157, 159, 166, 167 Newton-Euler equations, 165–167, 314 nilpotent Lie algebra, 344, 376 nonholonomic constraints, 268, 274–276,

318 classification, 340

versus holonomic constraints, 274 integrating, 319

zz, see also Pfa an constraints nonholonomic motion planning, 319, 331 nonmanipulable grasps, 239, 290

normal vector, 244 normalized Gauss frame, 245

numbering conventions for a robot, 83

ω limit set, 188 one-forms, 326, 408 open loop control, 190

open-chain manipulators, 82 optimal manipulator design, 432 optimal steering, 371

orthogonal coordinate chart, 244 orthogonal matrices, see rotation matri-

ces

Paden-Kahan subproblems, 99–103, 147– 148

solving inverse kinematics using, 104 palm frame, 215

parallel manipulators, 132–142 inverse kinematics, 133, 140 kinematic singularities, 134

zz, see also four bar linkage, Stewart platform

passivity, 172, 187, 209 PD control, 193–195

perspective transformations, 37 Pfa an constraints, 266–268

converting to control system, 320, 327

integrability conditions, 328 Philip Hall basis, 344

pitch of a screw, 45 pitch of a twist, 47, 427 pitch of a wrench, 65

planar grasping, 222, 231, 232, 262 planar joint, 82

planar rigid body transformations, 76 planar rotational motion, 75

planar Stewart platform, 141 plane contact, 260

Poinsot’s theorem, 19, 64, 65 point contact with friction, 217 points

rigid transformation of, 35, 36, 417 rotational transformation of, 25 versus vectors, 21, 36, 322

position control, 189–198 positive definite functions, 182 positive span, 225, 230 positively dependent, 225

potential energy for an open-chain manipulator, 169

prehensile grasp, 260 prismatic joint, 40, 81, 84

twist associated with, 48, 87 product of exponentials formula, 82, 85–

91

basic formula, 87 choice of base frame, 91

versus Denavit-Hartenberg parameters, 93

dynamics using, 175, 207 independence of order of joint mo-

tions, 146

independence on order of joint motions, 86

manipulator Jacobian using, 116 projection maps, 75

prosthetic hands, 10

pseudo-inverse for resolving redundancy, 130

pull back map, 407, 408 PUMA manipulator, 4, 6

zz, see also elbow manipulator pure quaternion, 74

pure rolling, 249, 252, 338 push forward map, 407

quaternions, 33–34, 74

rank of structure equations, 134 rate of convergence, 181, 184 reachable set, 318, 320 reachable workspace, 95 reciprocal product, 66 reciprocal screws, 66–69

definition, 66 systems of, 69, 78

use in analyzing mechanisms, 67, 69, 126

redundant manipulators, 122 dynamics, 286–290

in grasping, 238

kinematic versus actuator redundancy, 286

kinematics, 129–132 reference configuration, 87

choice of, 91 regular distribution, 325 regular filtration, 340

relative curvature form, 250 relative growth vector, 341

relative motion, representation using the exponential map, 42

revolute joint, 81, 84

twist associated with, 48, 87 right-handed coordinate frame, 22 rigid bodies, 20

dynamics, 165–167

inertial properties, 160–163 kinetic energy, 161

rigid body motion, 34–50 definition of, 20

representation using SE(3), 35, 416 representation using body-fixed frame,

22

rigid body transformations, 20–22 actions on points and vectors, 21,

35–37, 417 composition rule, 37 formal definition, 21 group properties, 37

homogeneous representation, 36 planar, 76

rigid body velocity, 53–61, 418–420

rigid displacement, 20

rigid transformations, see rigid body transformations

robot, origin of word, 1 robustness of control laws, 190 Rodrigues’ formula, 28, 76 roll, pitch, yaw angles, 32

rolling contact kinematics, 242–255 rolling penny, see disk rolling on a plane rotation about a line, 38, 87, 99

as a screw motion, 49 twist coordinates, 43

rotation about two axes, 100 rotation group, 24

rotation matrices, 23

actions on points and vectors, 25 eigenvalues of, 30

properties of, 23–26, 73 rotation to a given distance, 102 rotational motion, 22–34

composition rule, 25

equivalent axis representation, 31 Euler angle representation, 31 exponential coordinates, 27–31 about a fixed axis, 27, 29 parameterization singularities, 31, 32 planar, 75

quaternion representation, 33 representation using rotation matri-

ces, 23 rotational velocity, 51–53

body versus spatial, 52

Salisbury Hand, 11, 12 SCARA manipulator, 6, 83

dynamics, 177

forward kinematics, 87, 92 grasp using, 240, 291 inverse kinematics, 106

manipulator Jacobian, 118, 120, 122 screw motions, 19, 45, 46

instantaneous velocity of, 57 screw system, 68

screw theory advantages of, 20 origins of, 19

screws, 45–50

associated with wrenches, 64 Chasles’ theorem, 49 geometric attributes of, 45–46 infinite pitch, 48

rigid body transformations associated with, 46

twists associated with, 48 SE(3), 35, 409

bi-invariant quadratic forms, 425

bi-invariant volume forms, 431 hyperbolic metric, 426 invariant metrics, 423

lack of bi-invariant metric, 427 metric properties, 422

se(3), 40, 411

second fundamental form, 245 second-order controllable systems, 361 self-motion manifold, 130

separating hyperplane, 226 setpoint stabilization, 193 singular configurations, 123, 151

for parallel manipulators, 134 singular values of a matrix, 128, 148 singularities, see kinematic singularities skew-symmetric matrices, 27

properties of, 26, 28, 73

for representing cross product, 26 slider-crank mechanism, 151, 203, 314 sliding, 249, 268

small-time locally controllable, see locally controllable

SO(3), 24, 409

zz, see also rotation matrices so(3), 28, 411

zz, see also skew-symmetric matrices

soft-finger contact, 217

space robots, 334, 342, 351, 396 spatial angular velocity, 52 spatial frame, 51

spatial manipulator Jacobian, see manipulator Jacobian

spatial operator algebra, 207 spatial velocity, 54, 419

addition of, 58, 422 geometric interpretation, 54

relationship with body velocity, 55, 56, 61, 420

transformation of, 58, 421 spatial wrench, 63

special Euclidean group, see SE(3) special orthogonal group, see SO(3) sphere rolling on a plane, 252, 338, 343 sphere rolling on a sphere, 349 spherical joint, 81, 138

spherical wrist, 125

e ect on workspace, 96 versus spherical joint, 139

spring mass system, 185, 187, 189 stability by linearization, 184 stability definitions, 179–181 stable, 179

Stanford manipulator, 2, 4, 147 manipulator Jacobian, 119

Stanford/JPL hand, see Salisbury Hand

Steinitz’s theorem, 230, 299 Stewart platform, 138–142, 153 strictly internal forces, 223

structurally dependent forces, 122, 239 structure equations, 132–134

for four-bar mechanism, 136 for Stewart platform, 140

supporting hyperplane, 226 surface models, 243

tangent space, 243, 404 teleoperation, 395

tendon kinematics, 293–300 tool frame, 84

torsion form, 246

trajectory generation, using manipulator Jacobian, 117

trajectory tracking, see position control translational motion, 34, 48

transpose of Jacobian, see Jacobian transpose

twists, 19, 417 definition of, 41

geometric attributes, 45–50, 427 Lie bracket between, 175

parameterizing manipulators via, 91– 95

reciprocal to a wrench, 66

for revolute and prismatic joints, 87 screw coordinates, 47

screw motions corresponding to, 48 transformation of, 59, 77

twist coordinates, 41 two-link planar manipulator

constrained, 315 dynamics, 164 inverse kinematics, 97 moving in a slot, 203

U-joint, 153

uncertainty configuration, 137 underwater robots, 397 uniform stability, 179, 185 unit quaternions, 34, 74

unit twist, 49

Utah/MIT hand, 10, 12, 212

variable geometry truss, 152 vector field, 322, 406 vectors, 21

versus points, 21, 36

rigid transformation of, 21, 37, 417 rotational transformation of, 25

velocity

end-e ector, 115

rigid body, see rigid body velocity