Geometric Method for Type Synthesis of Parallel Manipulators

242 Pages·2020·11.943 MB·English

by  Qinchuan Li, Jacques M. Hervé, Wei Ye

Springer Tracts in Mechanical Engineering
Qinchuan Li
Jacques M. Hervé
Wei Ye
Geometric
Method for
Type Synthesis
of Parallel
Manipulators
Springer Tracts in Mechanical Engineering
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é
Qinchuan Li Jacques M. Herv Wei Ye
(cid:129) (cid:129)
Geometric Method for Type
Synthesis of Parallel
Manipulators
123
Qinchuan Li JacquesM. Hervé
Faculty of MechanicalEngineering EcoleCentrale Paris
andAutomation Chatenay-Malabry, France
ZhejiangSci-Tech University
Hangzhou, China
Wei Ye
Faculty of MechanicalEngineering
andAutomation
ZhejiangSci-Tech University
Hangzhou, China
ISSN 2195-9862 ISSN 2195-9870 (electronic)
SpringerTracts inMechanical Engineering
ISBN978-981-13-8754-8 ISBN978-981-13-8755-5 (eBook)
https://doi.org/10.1007/978-981-13-8755-5
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Brief Introduction
This book intends to introduce the type synthesis of lower mobility parallel
manipulators using the group theory-based method. The book collects synthesis
method, expressions, and operations about displacement sets of rigid bodies.
Furthermore, the research results previously scattered in many journals and con-
ferences worldwide are methodically edited and presented in a unified form. The
book is likely to be of interest to university researchers, graduate students in the
area of parallel manipulators, parallel kinematic machine, and creative mechanism
design who wish to learn the synthesis methodology of parallel manipulators and
general mechanisms. This work was supported by the National Natural Science
Foundation of China (NSFC) under Grant Nos. 51525504, 51475431 and
51075369.
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 History and Application of Parallel Mechanisms. . . . . . . . . . . . 1
1.2 Type Synthesis of Parallel Mechanisms . . . . . . . . . . . . . . . . . . 6
1.2.1 The Motion-Based Methods . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Constraint-Based Methods . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Other Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Objective and Organization of This Book. . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Fundamental of Group Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Group and Subgroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Geometry in Nonrelativistic Mechanics . . . . . . . . . . . . . . . . . . 28
2.4.1 The Projective Space and Group . . . . . . . . . . . . . . . . . 28
2.4.2 Affine Space and Group . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Euclidean Affine Space and Group . . . . . . . . . . . . . . . 33
3 Rotation and Displacements of Rigid Body. . . . . . . . . . . . . . . . . . . 37
3.1 Vector Products and Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Rotation of Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Operator of Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Axis of a Finite Screw Motion . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Lie Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 The Displacement Lie Subgroups. . . . . . . . . . . . . . . . . . . . . . . 52
4 Lie Group Based Method for Type Synthesis of Parallel
Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Kinematic Pairs and Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Composition of Kinematic Bonds . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Displacement Subgroup of Primitive Mechanical
Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
vii
viii Contents
4.4 Intersection of Kinematic Bonds . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Procedures of Type Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Type Synthesis of 5-DOF 3R2T Parallel Mechanism . . . . . . . . . . . 73
5.1 Kinematic Bond Between the Base and the Moving
Platform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Limb Kinematic Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Mechanical Generators of Limb Kinematic Bonds . . . . . . . . . . 76
5.3.1 Mechanical Generators of fTðPvwÞgfSðNÞg. . . . . . . . . 76
5.3.2 Mechanical Generators of fGðuÞgfSðNÞg . . . . . . . . . . 76
5.3.3 Mechanical Generators of fG2ðuÞgfSðNÞg and
fGðuÞgfS2ðNÞg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.4 Generation of 2-DOF Joints . . . . . . . . . . . . . . . . . . . . 78
5.4 Generation of Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Input Selection Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Type Synthesis of 4-DOF 2R2T Parallel Mechanisms. . . . . . . . . . . 87
6.1 Kinematic Bond Between the Base and the Moving
Platform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Limb Kinematic Bond and a Configurable Platform . . . . . . . . . 88
6.3 Mechanical Generators of Limb Kinematic Bonds . . . . . . . . . . 92
6.4 Generation of Parallel Mechanisms . . . . . . . . . . . . . . . . . . . . . 94
6.4.1 Conventional Parallel Mechanisms . . . . . . . . . . . . . . . 94
6.4.2 Parallel Mechanisms with a Configurable Platform. . . . 95
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Type Synthesis of 4-DOF Parallel Mechanisms with Bifurcation
of Schoenflies Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1 Preliminaries and Notations of Displacement Group . . . . . . . . . 101
7.1.1 Displacement Subgroup . . . . . . . . . . . . . . . . . . . . . . . 101
7.1.2 {G(y)} and {G − 1(y)} . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Bifurcation of Schoenflies Motion in PMs . . . . . . . . . . . . . . . . 104
7.2.1 Displacement Set of PMs with Bifurcation
of Schoenflies Motion. . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2.2 Bifurcation of 1-DOF Rotation Motion . . . . . . . . . . . . 105
7.2.3 A 2-PPPRR PM with Bifurcation of Schoenflies
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3 Type Synthesis of PMs with Bifurcation of Schoenflies
Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.1 Geometric Conditions for PMs with Bifurcation
of Schoenflies Motion. . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.2 {X(y)}{X(x)}: General Representation of Limb
Bonds for PMs with Bifurcation of Schoenflies
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents ix
7.3.3 {X − i(y)} and {X − j(x)} . . . . . . . . . . . . . . . . . . . . . 110
7.3.4 Category 1: For i = 0, fXðyÞgfXðxÞg¼
fXðyÞgfX(cid:2)3ðxÞg . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3.5 Category II: For i = 1, fXðyÞgfXðxÞg¼
fX(cid:2)1ðyÞgfX(cid:2)2ðxÞg. . . . . . . . . . . . . . . . . . . . . . . . 113
7.3.6 Category III: For i = 2, fXðyÞgfXðxÞg¼
fX(cid:2)2ðyÞgfX(cid:2)1ðxÞg. . . . . . . . . . . . . . . . . . . . . . . . 120
7.3.7 Category IV: For i = 3, fXðyÞgfXðxÞg¼
fX(cid:2)3ðyÞgfXðxÞg . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3.8 Implementation of 2-DOF Joints: C and U Joint . . . . . 121
7.4 Partitioned Mobility and Input Selection. . . . . . . . . . . . . . . . . . 123
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8 Type Synthesis of 3-DOF RPR-Equivalent Parallel
Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.1 RPR Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Limb Bond of RPR-Equivalent PMs . . . . . . . . . . . . . . . . . . . . 129
8.2.1 Displacement Set of the RPR-Equivalent PM. . . . . . . . 129
8.2.2 Limb Bond of an RPR-Equivalent PM . . . . . . . . . . . . 129
8.2.3 Parallel Arrangements of Three Limbs. . . . . . . . . . . . . 132
8.3 Overconstrained RPR-Equivalent PMs . . . . . . . . . . . . . . . . . . . 134
8.3.1 Subcategory 4-4-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.2 Subcategory 4-4-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.3 Subcategory 5-5-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.4 Non-overconstrained RPR-Equivalent PMs. . . . . . . . . . . . . . . . 141
8.4.1 Subcategory 1 of Non-overconstrained RPR-
Equivalent PM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4.2 Subcategory 2 of Non-overconstrained RPR-
Equivalent PM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9 Type Synthesis of 3-DOF PU-Equivalent Parallel Mechanisms. . . . 151
9.1 General and Special aTbR Motion. . . . . . . . . . . . . . . . . . . . . . 151
9.1.1 General aTbR Motion and Parasitic Motion. . . . . . . . . 151
9.1.2 Special aTbR Motion and Parasitic Motion . . . . . . . . . 153
9.1.3 Special Case: A 1T2R PM with Rotation
Bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.2 Non-overconstrained 1T2R PM Without Parasitic Motion . . . . . 157
9.2.1 Definition of a 1T2R PM Without Parasitic Motion . . . 157
9.2.2 Limb Bond of a 1T2R PM Without Parasitic
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9.2.3 Geometrical Condition of a 1T2R PM Without
Parasitic Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
x Contents
9.2.4 Enumeration of Non-overconstrained 1T2R PM
Without Parasitic Motion . . . . . . . . . . . . . . . . . . . . . . 160
9.3 Overconstrained 1T2R PM Without Parasitic Motion . . . . . . . . 162
9.4 Parasitic Motion Comparison of 3-PRS PMs with Different
Limb Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.4.1 Parasitic Motion of 3-PRS PMs in Category 1. . . . . . . 166
9.4.2 Parasitic Motion of 3-PRS PMs in Category 2. . . . . . . 169
9.4.3 Parasitic Motion of 3-PRS PMs in Category 3. . . . . . . 173
9.4.4 Parasitic Motion of 3-PRS PMs in Category 4. . . . . . . 175
9.4.5 Parasitic Motion of 3-PRS PMs in All Categories . . . . 177
9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10 Type Synthesis of a Special Family of Remote Center-of-Motion
ParallelManipulatorswithFixedLinearActuatorsforMinimally
Invasive Surgery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.1 Kinematic Bonds and Mechanical Generations . . . . . . . . . . . . . 181
10.1.1 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.1.2 {G(u)} and Its Mechanical Generators. . . . . . . . . . . . . 183
10.1.3 {C(N, v)} and Its Mechanical Generators . . . . . . . . . . 184
10.2 Serial Generators of SP Equivalent . . . . . . . . . . . . . . . . . . . . . 186
10.3 Parallel Generators of SP Equivalent . . . . . . . . . . . . . . . . . . . . 186
10.3.1 General Considerations. . . . . . . . . . . . . . . . . . . . . . . . 186
10.3.2 A Family of 5-DOF Limbs . . . . . . . . . . . . . . . . . . . . . 186
10.3.3 A New Family of 5-DOF Limbs . . . . . . . . . . . . . . . . . 187
10.3.4 Elimination of the Independent Local Rotations . . . . . . 188
10.3.5 Subfamily 1: {R(O, u)}{R(A, v)}{R(B, v)}
i i i
{C(O, w)} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.3.6 Subfamily 2: {C(O, u)}{R(A, v)}{R(B, v)}
i i i
{R(O, w)} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.3.7 A Special Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.3.8 Subfamily 3: {C(O, u)}{R(A, v)}{C(O, w)} . . . . . . . 195
i i
10.4 Parallel Generators of SP-Equivalent Motion . . . . . . . . . . . . . . 197
10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11 Type Synthesis of Non-overconstrained 3-DOF Translational
Parallel Mechanisms with Less Structural Shakiness . . . . . . . . . . . 201
11.1 Number of Infinities of Rotation Axes and Motion Type. . . . . . 201
11.1.1 Definition of Number of Infinities of Rotation
Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.1.2 Number of Rotation Axes of 2T1R Motion . . . . . . . . . 202
11.1.3 Number of Rotation Axes of 3T1R Motion . . . . . . . . . 204
11.1.4 Number of Rotation Axes of 3T2R Motion . . . . . . . . . 205