كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis
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 كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis

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مُساهمةموضوع: كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis    كتاب Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis  Emptyالسبت 14 يناير 2023, 10:07 pm

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Kinematics of Mechanical Systems - Fundamentals, Analysis and Synthesis
Jorge Angeles, Shaoping Bai

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Contents
1 Introduction to Kinematic Synthesis . 1
1.1 The Role of Kinematic Synthesis in Mechanical Design 1
1.2 Glossary 5
1.3 Kinematic Analysis Versus Kinematic Synthesis 8
1.3.1 A Summary of Systems of Algebraic Equations 10
1.4 Algebraic and Computational Tools . 10
1.4.1 The Two-Dimensional Representation of the Cross
Product . 11
1.4.2 Algebra of 2 × 2 Matrices . 13
1.4.3 Algebra of 3 × 3 Matrices . 14
1.4.4 Linear-Equation Solving: Determined Systems . 14
1.4.5 Linear-Equation Solving: Overdetermined Systems . 18
1.5 Nonlinear-Equation Solving: The Determined Case 29
1.5.1 The Newton–Raphson Method . 31
1.6 Overdetermined Nonlinear Systems of Equations . 32
1.6.1 The Newton–Gauss Method . 33
1.7 Packages Relevant to Linkage Synthesis . 36
References . 38
2 The Qualitative Synthesis of Kinematic Chains 41
2.1 Notation 41
2.2 Background . 42
2.3 Kinematic Pairs 47
2.3.1 The Kinematic Pair 49
2.4 Graph Representation of Kinematic Chains . 50
2.5 Groups of Displacements 53
2.5.1 Displacement Subgroups 56
2.6 Kinematic Bonds . 60
viiviii Contents
2.7 The Chebyshev–Grübler–Kutzbach–Hervé Formula . 63
2.7.1 Trivial Chains . 63
2.7.2 Exceptional Chains . 66
2.7.3 Paradoxical Chains . 69
2.8 Applications to Robotics 69
2.8.1 The Synthesis of Robotic Architectures and Their
Drives 69
References . 73
3 Linkage Synthesis for Function Generation . 75
3.1 Introduction . 75
3.2 Input-Output (IO) Functions . 76
3.2.1 Planar Four-Bar Linkages . 76
3.2.2 The Denavit-Hartenberg Notation 79
3.2.3 Spherical Four-Bar-Linkages 80
3.2.4 Spatial Four-Bar-Linkages . 86
3.3 Exact Synthesis 91
3.3.1 Planar Linkages 91
3.3.2 Spherical Linkages . 96
3.3.3 Spatial Linkages . 98
3.4 Analysis of the Synthesized Linkage 99
3.4.1 Planar Linkages 99
3.4.2 Spherical Four-Bar Linkages 110
3.4.3 Spatial Four-Bar Linkages . 113
3.5 Approximate Synthesis 120
3.5.1 The Approximate Synthesis of Planar Four-Bar
Linkages 123
3.5.2 The Approximate Synthesis of Spherical Linkages 125
3.5.3 The Approximate Synthesis of Spatial Linkages 127
3.6 Linkage Performance Evaluation . 131
3.6.1 Planar Linkages: Transmission Angle
and Transmission Quality . 131
3.6.2 Spherical Linkages: Transmission Angle
and Transmission Quality . 136
3.6.3 Spatial Linkages: Transmission Angle
and Transmission Quality . 137
3.7 Design Error Versus Structural Error 139
3.7.1 Minimizing the Structural Error 142
3.7.2 Branch-Switching Detection . 144
3.7.3 Introducing a Massive Number of Data Points 145
3.8 Synthesis Under Mobility Constraints . 145
References . 148Contents ix
4 Motion Generation . 151
4.1 Introduction . 151
4.2 Planar Four-Bar Linkages 151
4.2.1 Dyad Synthesis for Three Poses 153
4.2.2 Dyad Synthesis for Four Poses . 154
4.2.3 Dyad Synthesis for Five Poses . 156
4.2.4 Case Study: Synthesis of a Landing Gear Mechanism . 158
4.2.5 The Presence of a P Joint in dyad Synthesis 165
4.2.6 Approximate Synthesis . 170
4.3 Spherical Four-Bar Linkages . 174
4.3.1 Dyad Synthesis for Three Attitudes . 177
4.3.2 Dyad Synthesis for Four Attitudes 178
4.3.3 Dyad Synthesis for Five Attitudes 179
4.3.4 Spherical dyads with a P Joint . 181
4.3.5 Approximate dyad Synthesis . 181
4.3.6 Examples . 189
4.4 Spatial Four-Bar Linkages . 195
4.4.1 Geometric Constraints of CC and RC dyads 196
4.4.2 The Synthesis of the CC dyad 198
4.4.3 The Synthesis of the RC dyad 200
4.4.4 Synthesis of Four-Bar Linkages 202
4.4.5 A Semigraphical Solution of the Direction Equations 202
4.4.6 Solving the Linear Equations of the Moment Variables 204
4.4.7 Congruences of the Fixed and the Moving Axes 204
4.4.8 Examples . 205
4.4.9 Summary . 208
References . 210
5 Trajectory Generation 213
5.1 Planar Linkages 214
5.1.1 Planar Path Generation with Prescribed Timing . 214
5.1.2 Coupler-Curve Synthesis of Planar Four-Bar Linkages 221
5.2 Trajectory Generation for Spherical Four-Bar Linkages 234
5.2.1 Path Synthesis with Discrete Positions 235
5.2.2 Path Synthesis with Prescribed Timing 236
5.2.3 Examples . 237
5.2.4 Summary . 240
5.3 Path Generation for RCCC Linkages 240
5.3.1 A Generic RCCC Linkage . 240
5.3.2 An Alternative Coordinate Frame 242
5.3.3 Constraint Equations of the CC and RC dyads 243
5.3.4 N-point Path Synthesis 245
5.3.5 Example 247
5.3.6 Summary . 249
References . 249x Contents
6 Synthesis of Planar Complex Linkages . 251
6.1 Six-Bar Linkages . 251
6.2 Synthesis of the Stephenson-III Linkage . 253
6.2.1 Elimination of Motion Variables . 254
6.3 Extension to Stephenson-I and II Linkages . 255
6.3.1 Stephenson-II Linkage 256
6.3.2 Parameterized Coordinates of Point D 256
6.3.3 Synthesis Equations for the Stephenson-II Linkage . 258
6.3.4 Stephenson-I Linkage 259
6.4 Synthesis in the Presence of Incomplete Data Sets . 260
6.4.1 A Design Example . 262
6.5 A 10-Bar Linkage 262
6.6 Synthesis for m Prescribed Poses . 265
6.7 Solution of the 10-Posture Synthesis Equations . 266
6.8 A Synthesis Example . 268
References . 272
Appendix A: A Summary of Dual Algebra . 275
Appendix B: Coupler Curves . 313
Index . 32
Index
A
Algebraic system
degree, 10
eliminant, 10
resolvent, 10
Alternative coordinate frame, 242
Alternative coordinates, 242
Analysis, 8
Angular-velocity matrix, 292
Approximate synthesis
function generation, 120
planar four-bar linkage, 123
spatial linkage, 127
spherical linkage, 125
Assemblability, 70
Augmented synthesis equations, 125
Axial vector, 285
B
Bezout number, 10, 171, 173, 185, 206, 270
Bicycle, 75
Bimodal linkage, 101
Bivariate equations, 9
Branching defect, 191
Branch switching, 144
Burmester problem
planar, 151
spatial, 195
spherical, 176
Burmester-Roth Theorem, 180
Burmester theory, 314
C
Canonical equation, 116
Cartesian decomposition, 44
Cayley–Hamilton Theorem, 285
CC dyad synthesis
five poses, 198, 205
four poses, 199, 207
Centerpoint, 154, 175
Centerpoint curve, 156, 161, 179
Chain
exceptional, 63, 66
paradoxical, 63, 69
trivial, 63
Chain rule, 21
Chebyshev-Grübler-Kutzbach formula, 64
Chebyshev-Grübler-Kutzbach-Hervé Formula, 63
Chebyshev norm, 15
Chirality, 42
Cholesky decomposition, 18
Circle of sliders, 167
Circlepoint, 154, 175
Circlepoint curve, 155, 161, 173, 179, 204
Circular guideway, 181, 193
Cognate linkages, 229, 233
Conceptual design, 3
Condition number, 15
Cone axis, 174
Cone element, 174
Congruence, 204, 207
Conical cubic surface, 180
Conical surface, 174
Conjugate branches, 101
Conjugate postures, 102
Constrained minimization, 182
Constrained optimization problem, 182
Constraint Jacobian, 183
Contour-intersection, 85
Convex combination, 127
algebraic equation, 221
bivariate polynomial, 223
exact synthesis, 221, 228
four-bar linkage, 221
RCCC linkage, 248
spherical , 235
synthesis, 222
approximate, 222
exact, 225, 229
univariate polynomial, 225
Coupler-curve equation, 320
planar four-bar linkage, 316
spatial four-bar linkage, 247
spherical four-bar linkage, 322
Crank, 105
Cross product, 11
Cross-product matrix, 43
Cubic of stationary curvature, 214
Cylindrical pair, 49
D
Damping, 34
Damping factor, 34
Data-conditioning, 158
Deadpoint, 102
Delta robot, 69
Denavit-Hartenberg
frames, 80
notation, 79
Denavit-Hartenberg parameters, 5
Descartes’ rules, 231
Design
functions, 3
specifications, 3
variants, 3
Design error, 120, 139, 145
least-square, 121
Design-error vector, 120
Design parameter, 266
Detailed design, 3
Determined system, 29
Dialytic elimination, 186
Dimensionally homogeneous, 170
Dimensional synthesis, 8
Dimensioning, 5
functional, 5
mechanical, 5
Direction equation, 197
Displacement subgroups, 56
Dual angle, 89, 139, 277
Dual angular velocity, 304
matrix, 304
vector, 305
Dual Euler–Rodrigues parameters, 299
Dual exponential, 276
Dual Freudenstein parameters, 89
Dualization, 88
Dualization operation, 88
Dual matrix, 278
determinant, 279
eigenvalue, 279
eigenvector, 279
inverse, 278
pure, 278
square, 278
Dual-matrix exponential, 275
Dual number, 275
Dual quaternion, 276
Dual rotation matrix, 292, 294
Dual trigonometric function, 277
Dual unit, 88
Dual vector, 196
dual part, 282
Dwell, 213
Dyad
CC, 196, 244
definition, 151
PR, 165
RC, 196, 243
Dyad synthesis
approximate, 181
CC dyad, 198
planar
five poses, 156, 163, 169
four poses, 154, 161, 168
three poses, 153, 160, 167
RC dyad, 200
E
Embodiment design, 3
Engineering design, 1
process, 3
Error vector, 19
Euclidean norm, 15, 77
Euler–Rodrigues parameters, 304
Euler’s Theorem, 284
Exact synthesis, 266
function generation, 91
planar linkage, 91
spherical four-bar linkage, 96
F
First-order normality conditions, 170Index 327
Floating-point operation, 22
Flop, see floating-point operation
Four-bar linkage
path generation
with timing, 214
path synthesis, 214
planar, 76
spatial, 86, 113
spherical, 80
spherical linkage, 110
4C linkage, 202
4C linkage synthesis
five poses, 205
Fractal, 32
Franz Reuleaux, 3
Freudenstein equation, 84
planar linkages, 79
Freudenstein parameters, 76, 78
spherical linkage, 83
Frobenius norm, 15
Function generation, 8, 75
Function-generation problem, 76
G
Generalized CGK formula, 64
Global minimum, 174
Gnomonic projection, 323
Gram-Schmidt orthogonalization, 24, 122
Graph, 50
Groups of displacement, 41, 53, 63
H
Householder reflection, 26, 122
Hyperbolic paraboloid, 116
I 
kinematic pair, 49
Incomplete data set, 260
Infinity norm, 15
Inflection circle, 214
Input-output equation
spherical linkage, 84
Input-output functions, 76
Instant-screw parameters, 305
IO equation, 77
Isotropic matrix, 17, 217
J
Jacobian matrix, 31
K
Kinematic bond, 60
Kinematic chain
architecture, 80
multiloop, 41
Kinematic synthesis, 4, 8
Kinetostatics, 11
L
Lagrange multipliers, 182
Lamé curve, 158
Landing gear, 158
Latitude, 186
Least-square approximation, 158, 219
Least-square error, 21, 33
normality condition, 19
Least-square solution, 33
Lemniscate, 232
Linear invariants, 294
Linkage, 7
Linkage discriminant, 106, 136
Linkage feasibility, 105
Local maximum, 173
Local minimum, 173
Longitude, 186
Lower Kinematic Pairs (LKP), 41
LU-decomposition, 17
M
Machine, 1
function, 1
Manifold, 269
Marcus Vitruvius Polio, 3
Mechanical system, 1
Mechanical-system gesture, 8
Mechanical-system posture, 8
Mechanism, 7
MEL microfinger, 71
Minimum-size linkage, 128
Mobility constraints, 145
Moment equation, 197
Motion generation, 8, 151
Motion representation, 41
Motion variable, 266
Mozzi–Chasles axis, 287, 292, 295, 296, 299
Mozzi–Chasles Theorem, 287, 293
N
Natural invariants, 205, 284, 320
Negative-definite, 173
Newton-Gauss method, 33, 185328 Index
Newton-Raphson method, 31, 35, 185
Normality condition, 34
Number synthesis, 8
O
Optimization
multiobjective, 4
Optimum design, 170
Orthogonal complement, 130, 183
Overdetermined system, 18, 32
P
Parallelism condition, 165
Parameterized coordinates, 216, 256, 318
Parametric model, 4
Path generation, 8, 213
Path synthesis
RCCC linkage, 240
five points, 247
nine points, 249
N points, 245
spherical, 234, 238
discrete points, 234, 235
discrete points with timing, 236
great-arc, 234
Pencil of lines, 59
Performance evaluation, 131
Planar four-bar linkage
feasibility condition, 92
PRRR linkage, 92, 108
PRRP linkage, 108
RRRP linkage, 92
Planar linkage
Bloch synthesis, 94
crank-rocker, 105
double crank, 105, 109
double rocker, 105, 110
drag-link, 106
Grashof, 106
non-Grashof, 106
rocker-crank, 105
transmission angle, quality, 131
Planar pair, 49
Plücker coordinates, 195, 287
normalized, 277
Polynomial
bivariate sextic, 313
cubic, 156
dodecic, 206
multivariate, 10
nonic, 172, 174
octic, 270
quadratic, 180, 218
quartic, 183
root finding, 206
sextic, 180, 221, 229
trivariate quartic, 322
univariate, 10, 225, 232
Pose error, 170
Positive-definite, 173
Principle of Transference, 87, 275
Prismatic pair, 48
Projection theorem, 22
Proper dual number, 276
Pure dual number, 276
Putative least-square approximation, 156
Q
QR decomposition, 122, 203
Qualitative synthesis, 69
R
Rank, 18
Rank-nullity theorem, 43
RCCC linkage, 196
Reciprocal bases, 14
Reliability, 213
Repeatability, 213
Revolute pair, 48
Rigid body, 7, 42
Rigid-body guidance, 151
Rigid-body kinematics, 283
Rigid-body pose, 7
Rigid-body twist, 8
Ring, 154
Roberts-Chebyshev Theorem, 233
Robotic gripper, 93
Robustness, 158
Robust selection, 261
Rocker, 101, 105
Roundoff error, 219
Ruled surface, 213
S
Saddle point, 173
Sarrus mechanism, 66
SCARA systems, 57
Schönflies-motion generator, 68
Schwarz’s Theorem, 173
Scissor, 75
Screw pair, 48
Semigraphical method, 85, 202, 207, 208Index 329
Sign-indefinite, 173
Signum function, 28, 100
Singularity, 103
Stephenson-II, 258
Six-bar linkages, 251
Skew-symmetric matrix, 282, 285
Slack variable, 147
Slider-crank mechanism, 67
Spatial four-bar linkage, 195
Spatial linkages, 98
function generation, 118
mobility, 119
Spatial motion synthesis
RR dyad, 201
Spherical centerpoint curve, 204
Spherical dyad, 181
Spherical dyad synthesis
approximate, 195
five attitudes, 179, 191
four attitudes, 178, 190
three attitudes, 177, 189
with P joint, 181, 192
Spherical pair, 49
Stationary point, 34, 173
Stephenson linkage, 216
Stephenson-I, 259
Stephenson-II, 256
Stephenson-III, 253
Structural error, 139, 142, 145
Sylvester’s Theorem, 18, 22
Synthesis, 8
approximate, 9, 79, 170
equations, 9
exact, 9
planr kinkage
11 poses, 174
qualitative, 4, 41
semigraphical methods, 9
Synthesized linkage
analysis, 99
mobility analysis, 113
Synthesized planar linkage
mobility analysis, 106
T
10-bar linkage, 262
10-pose synthesis, 266
3-RRR planr linkage, 262
Trajectory generation, 234
Transcendental, 10
Transmission angle, 136
Transmission defect, 133
Transmission quality, 132, 137
spatial linkage, 137
Triad, 214
RRR, RPR, 214
Trial and error, 127
Type synthesis, 8
U
Unconstrained minimization, 186
Unconstrained nonlinear least squares, 186
Underdeterminacy, 261
W
Watt curve, 232
Weighted error-norm, 126
Weighting matrix, 33
Working region, 158
Y
Yang-and-Freudenstein linkage, 118


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