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 |  موضوع: كتاب Multi-Arm Cooperating Robots - Dynamics and Control الأحد 23 مايو 2021, 2:00 am | |
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أحضرت لكم كتاب
Multi-Arm Cooperating Robots - Dynamics and Control
edited by
M.D. ZIVANOVIC
and
M.K. VUKOBRATOVIC
Robotics Center,
Mihajlo Pupin Institute, Belgrade,
Serbia and Montenegro
Robotics Center,
Mihajlo Pupin Institute, Belgrade,
Serbia and Montenegro
و المحتوى كما يلي :
TABLE OF CONTENTS
LIST OF FIGURES ix
PREFACE xi
1. INTRODUCTION TO COOPERATIVE MANIPULATION 1
1.1 Cooperative Systems – Manipulation Systems 1
1.2 Contact in the Cooperative Manipulation 4
1.3 The Nature of Contact 4
1.4 Introducing Coordinate Frames 7
1.5 General Convention on Symbols and Quantity Designations 16
1.6 Relation to Contact Tasks Involving One Manipulator 18
2. PROBLEMS IN COOPERATIVE WORK 19
2.1 Kinematic Uncertainty 19
2.1.1 Kinematic uncertainty due to manipulator redundancy 19
2.1.2 Kinematic uncertainty due to contact characteristics 21
2.2 Force Uncertainty 22
2.3 Summary of Uncertainty Problems in Cooperative Work 24
2.4 The Problem of Control 25
3. INTRODUCTION TO MATHEMATICAL MODELING OF
COOPERATIVE SYSTEMS 27
3.1 Some Known Solutions to Cooperative Manipulation Models 28
3.2 A Method to Model Cooperative Manipulation 30
3.3 Illustration of the Correct Modeling Procedure 37
vvi Table of Contents
3.4 Simulation of the Motion of a Linear Cooperative System 51
3.5 Summary of the Problem of Mathematical Modeling 54
4. MATHEMATICAL MODELS OF COOPERATIVE SYSTEMS 57
4.1 Introductory Remarks 57
4.2 Setting Up the Problem of Mathematical Modeling of a Complex
Cooperative System 65
4.3 Theoretical Bases of the Modeling of an Elastic System 66
4.4 Elastic System Deformations as a Function of Absolute Coordinates 74
4.5 Model of Elastic System Dynamics for the Immobile Unloaded State 82
4.6 Model of Elastic System Dynamics for a Mobile Unloaded State 86
4.7 Properties of the Potential Energy and Elasticity Force of the
Elastic System 89
4.7.1 Properties of potential energy and elasticity force of the elastic
system in the loaded state translation 91
4.7.2 Properties of potential energy and elasticity force of the elastic
system during its rotation in the loaded state 94
4.8 Model of Manipulator Dynamics 100
4.9 Kinematic Relations 101
4.10 Model of Cooperative System Dynamics for the Immobile
Unloaded State 102
4.11 Model of Cooperative System Dynamics for the Mobile
Unloaded State 104
4.12 Forms of the Motion Equations of Cooperative System 106
4.13 Stationary and Equilibrium States of the Cooperative System 118
4.14 Example 123
5. SYNTHESIS OF NOMINALS 137
5.1 Introduction – Problem Definition 138
5.2 Elastic System Nominals 142
5.2.1 Nominal gripping of the elastic system 142
5.2.2 Nominal motion of the elastic system 153
5.3 Nominal Driving Torques 165
5.4 Algorithms to Calculate the Nominal Motion in Cooperative
Manipulation 166
5.4.1 Algorithm to calculate the nominal motion in gripping for
the conditions given for the manipulated object MC 167Table of Contents vii
5.4.2 Algorithm to calculate the nominal motion in gripping for
the conditions of a selected contact point 168
5.4.3 Algorithm to calculate the nominal general motion for
the conditions given for the manipulated object MC 171
5.4.4 Algorithm to calculate the nominal general motion for
the conditions given for one contact point 173
5.4.5 Example of the algorithm for determining the nominal motion 176
6. COOPERATIVE SYSTEM CONTROL 189
6.1 Introduction to the Problem of Cooperative System Control 189
6.2 Classification of Control Tasks 191
6.2.1 Basic assumptions 191
6.2.2 Classification of the tasks 202
6.3 Choice of Control Tasks in Cooperative Manipulation 207
6.4 Control Laws 212
6.4.1 Mathematical model 212
6.4.2 Illustration of the application of the input calculation method 213
6.4.3 Control laws for tracking the nominal trajectory of the
manipulated object MC and nominal trajectories of contact
points of the followers 216
6.4.4 Behavior of the non-controlled quantities in tracking the
manipulated object MC and nominal trajectories of contact
points of the followers 223
6.4.5 Control laws to track the nominal trajectory of the manipulated
object MC and nominal contact forces of the followers 229
6.4.6 Behavior of the non-controlled quantities in tracking the
trajectory of the manipulated object MC and nominal contact
forces of the followers 234
6.5 Examples of Selected Control Laws 236
7. CONCLUSION: LOOKING BACK ON THE PRESENTED
RESULTS 251
7.1 An Overview of the Introductory Considerations 251
7.2 On Mathematical Modeling 252
7.3 Cooperative System Nominals 254
7.4 Cooperative System Control Laws 256viii Table of Contents
7.5 General Conclusions about the Study of Cooperative Manipulation 257
7.6 Possible Directions of Further Research 258
APPENDIX A: ELASTIC SYSTEM MODEL FOR THE IMMOBILE
UNLOADED STATE 261
APPENDIX B: ELASTIC SYSTEM MODEL FOR THE MOBILE
UNLOADED STATE 269
REFERENCES 277
INDEX 283LIST OF FIGURES
1 Cooperative manipulation system 3
2 Contact 6
3 Cooperative work of the fingers on an immobile object 8
4 Kinematic uncertainty due to contact 22
5 Cooperative work of two manipulators on the object 23
6 Reducing the cooperative system to a grid 31
7 Approximation of the cooperative system by a grid 32
8 Linear elastic system 37
9 Approximating a linear elastic system 44
10 Block diagram of the model of a cooperative system without
force uncertainty 51
11 Results of simulation of a ‘linear’ elastic system 54
12 Elastic system 63
13 Displacements of the elastic system nodes – the notation system 66
14 Angular displacements of the elastic system 76
15 Displacements of the elastic system 78
16 Planar deformation of the elastic system 83
17 Rotation of the loaded elastic system 95
18 Block diagram of the cooperative system model 106
19 Elastic system of two springs 113
20 Initial position of the cooperative system 123
21a Simulation results for τ j
i = 0, i, j = 1, 2, 3 127
21b Simulation results for τ j
i = 0, i, j = 1, 2, 3 128
22a Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 129
22b Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 130
22c Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 131
22d Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 132
22e Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 133
ixx List of Figures
22f Simulation results for τ 1
1 = 50 [Nm] and τ21 = −50 [Nm] 134
22g Simulation results for τ11 = 50 [Nm] and τ21 = −50 [Nm] 135
23 Nominal trajectory of the object MC 143
24 Elastic deviations from the nominal trajectory 146
25 Nominal trajectory of a contact point 163
26 ‘Linear’ cooperative system 177
27 Nominals for gripping a manipulated object 181
28 Nominal input to a closed-loop cooperative system for gripping 182
29 Simulation results for gripping (open-loop cooperative system) 183
30 Nominals for manipulated object general motion 184
31 Nominal input to a closed-loop cooperative system for general motion 185
32 Simulation results for motion (open-loop cooperative system) 186
33 Mapping from the domain of inputs to the domain of states 194
34 Mapping from the domain of states to the domain of inputs 195
35 Mapping from the domain of inputs to the domain of outputs 195
36 Mapping from the domain of outputs to the domain of inputs 196
37 Mapping through the domain of states 196
38 Mapping of the control system domain 197
39 Structure of the control system 200
40 Mapping of the control object domain 201
41 Mapping of the cooperative manipulation domain 205
42 Global structure of the closed loop system 215
43 Motion in the plane of the loaded elastic system 224
44 Block diagram of the closed-loop cooperative system 240
45a Gripping – tracking Y20 and Y30 241
45b Gripping – tracking Y20 and Y30 242
46a Gripping – tracking Y20 and Fc02 243
46b Gripping – tracking Y20 and Fc02 244
47a General motion – tracking Y20 and Y30 245
47b General motion – tracking Y20 and Y30 246
48a General motion – tracking Y20 and Fc02 247
48b General motion – tracking Y20 and Fc02
INDEX
absolute coordinate, 63, 64, 71, 73, 74
actuator, 34, 61, 210, 257, 258
angular displacement, 74
approach planning, 2
approach to the object, 2
calculate nominal motion, 166
general motion
contact point, 173
manipulated object MC, 171
gripping
contact point, 168
manipulated object MC, 167
Castigliano principle, 33, 47
closed loop, 215, 240
configuration of contact points, 78
constraint, 5, 65, 140, 151, 160, 228, 261
contact, 4
constraint, 7, 13
elastic, 5
environment, 7
internal load, 5
nature, 4
participant, 4
rigid, 5, 30
sliding, 5
rotational, 5
translational, 5
space, 13
stiff, 5, 30
rotational, 5
translational, 5
surface, 5
transferred load, 4
understood, 4
contact force, 17, 24, 28, 39, 41, 85
control laws, 25, 118, 189–191, 199, 202,
212, 216, 229, 256
controllability, 192, 256
cooperative manipulation, 1, 4, 7, 12, 19, 25,
28, 35, 50, 58, 72, 99, 106, 140,
154, 189, 190, 202, 207, 236, 251,
252, 254, 256, 257
cooperative system, 1
approximation, 31
at rest, 31
contact space, 15
control, 189
control problem, 26
coordinated motion, 154, 254
elastic part, 58, 252
equilibrium state, 118, 120
forms of the motion equations, 106
nominal, 254
rigid part, 58, 252
state space, 15
state vector, 16
stationary state, 118
trajectory, 137
uncontrolled, 118
unloaded state
immobile, 102
mobile, 104
coordinate frame, 7
absolute, 10
external, 9
internal, 10
natural, 10
criterion, 28
damping force, 65
deformation work, 32, 35, 73, 107, 253, 261,
270
directly tracked, 142, 189, 211, 214, 216,
229, 256
displacement, 41
displacement method, 69
dissipation energy, 49, 84, 88, 265, 272
283input, 256
output, 256
state, 256
elastic displacement, 73
elastic environment, 65
elastic force, 35, 39, 47, 55, 83, 98, 141, 142,
208, 261
elastic interconnections, 7, 37, 41, 59, 62, 65,
74, 139, 252
elastic structure, 7
elastic system, 7
DOFs, 62
deformation energy, 45, 46
deformation work, 71
deformations, 74
dissipation energy, 49, 84, 88, 265, 272
elastic forces
properties, 89
internal forces, 45
kinetic energy, 43, 84, 86, 263, 269
kinetic potential, 48
loaded, 77
rotation, 94
translation, 91
node forces, 71
nominals, 142
potential energy, 45, 46, 83, 87, 90, 92,
261, 270
properties, 89
rate of displacement, 77
relative displacements, 44, 76, 77
space, 11
state 0, 74, 265, 274
state vector, 16
unloaded state
immobile, 75, 82, 265
mobile, 75, 86, 274
elasticity force, 65
equilibrium, 30
exchange of the leader’s, 254
external force, 32, 41, 85
external load, 62
feedback loop, 256
feedback loops, 118, 142, 190
finite-element method, 67
follower, 29, 112, 114
followers’ acceleration, 152, 160
force uncertainty, 22, 24, 30, 31, 251
general motion, 63–65, 89
generalized forces, 47, 85, 88
generalized stiffnesses, 45
geometric configuration, 34, 254
global coordinate, 68
grasping, 2
gravitation force, 65
grid, 34, 61, 251
displacements, 34
position, 34
statically undetermined, 31, 251
gripper, 31
gripping, 2
gripping phase, 142, 254
holonomic constraints, 156
individual stiffness, 68
influence numbers, 33
internal force, 24, 28, 41
internal forces, 73
kinematic chain, 7
kinematic instability, 41
kinematic relations, 101
kinematic uncertainty, 19, 24, 251
due to contact, 21
due to manipulator redundancy, 19
kinematically indeterminate, 69
kinematically mobile, 39
kinematically stable, 41
kinematically unstable, 39
kinetic energy, 32, 43, 84, 86, 263, 269
Lagrange equations, 49, 82, 252
leader, 29, 40, 112, 114
leader’s acceleration, 152, 160, 161
leadership principle, 40
lifting, 3
local coordinate, 68
lowering, 4
manipulated object
immobile in space, 110
immobile on the support, 109
in space, 109
mobile in space, 110
mobile on the support, 110
284 Multi-Arm Cooperating Robots
domainon the support, 109
manipulation systems, 1
manipulator tip, 31, 255
mapping, 256
mapping domain, 194, 197
cooperative manipulation, 205
input, 194
output, 194
state, 194
mapping one-to-one, 198
Maxwell’s coefficient, 33, 107
method of deformation work, 32
method of direct stiffness, 68
modes of rigid body, 39, 253
motion equations, 109, 156
natural output space, 194, 256
node, 7, 34, 62, 73
external, 61, 255
internal, 61
nominal, 137, 254
synthesis, 137
nominal gripping, 142
preset y0s , 144
conditions, 145
preset yv = y1, 146
conditions, 149
nominal input, 137
nominal motion, 137, 153
brief procedure, 154
driving torques, 165
initial state, 153
nominal trajectory
manipulated object MC, 155
one contact point, 161
superscript
‘0s’, 154
‘s’, 154
‘u’, 154
nominal trajectory, 137
non-holonomic constraint, 151, 153
non-holonomic constraints, 156
observability, 192, 256
output
controlled, 189, 211, 256
directly tracked, 189, 211, 216
natural space, 256
nominal, 220, 231, 257
non-controlled, 190, 211, 216, 257
potential energy, 32, 45, 46, 83, 87, 90, 92,
261, 270
principle of minimal, 32
quantity designations, 16
quasi-static, 255
realizable, 255
trajectories, 255
realizable nominals, 137
relative displacements, 73
angular, 76
releasing, 2, 4
resultant force, 24
rigid body, 7, 65, 253
rigid manipulator, 65
rigid structure, 7
servoactuator, 258
singularity, 41
six DOFs, 65
slave manipulator, 28
state 0, 12, 82
immobile, 70
unloaded, 70
statically indeterminate, 69
statically transferred, 255
stiffness matrix, 41
stiffness matrix, 39, 253
assembled, 70, 72
disassembled, 68
generalized, 72
structure
damping matrix, 116
stiffness matrix, 115, 117
vector
displacement, 117
force, 117
structure envelope, 7
structures of the matrices, 114
submatrices, 115
subscript, 8, 39, 114, 270
’0’, 8
’c’, 39
’e’, 39
’i’, 270
’ia’, 270
Index 285‘0’, 114
‘c’, 114
‘d’, 114
‘e’, 114
‘s’, 114
‘v’, 114
symbol convention, 16
task space, 10
transferred positions, 255
transferring, 3
unload, 34
unloaded state, 62, 253
unperturbed motion, 137
unperturbed trajectory, 137
unpowered joints, 7
weighting matrix, 28
withdrawing, 2, 4
]
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