Preface ............................9
Notation ............................11
1 Introduction............................13
1.1 Characteristics of classical robots ............................13
1.2 Other types of architecture............................16
1.3 Needs for robotics............................20
1.4 Parallel robots: definition............................20
1.4.1 Generalized parallel manipulators: definition............................21
1.4.2 Parallel manipulators............................21
1.4.3 Fully parallel manipulators............................21
1.4.4 Fully parallel manipulators: analysis............................22
1.4.4.1 Planar robots............................22
1.4.4.2 General case............................23
1.5 Contents............................24
1.6 Exercises............................25
2 Architectures............................27
2.1 Introduction............................27
2.2 Planar robots............................27
2.2.1 2 dof manipulators............................27
2.2.2 3 dof manipulators............................29
2.3 Spatial motion robots............................30
2.3.1 Joints............................31
2.3.2 Actuators............................31
2.3.3 Classification of parallel robots.............................32
2.3.4 Synthesis of architectures ............................33
2.3.4.1 Generator combinations and intersections ............................33
2.3.4.2 Motion group generator ............................34
2.3.5 3 dof manipulators............................34
2.3.5.1 Translation manipulators............................35
2.3.5.2 Orientation manipulators............................39
2.3.5.3 Complex degrees of freedom manipulators............................42
2.3.6 4 dof manipulators............................46
2.3.7 5 dof manipulators............................46
2.3.8 6 dof manipulators............................48
2.3.8.1 RRP S chain robot............................48
2.3.8.2 P RRS chain robots............................52
2.3.8.3 R RRS chain robots............................54
2.3.8.4 Exotic chain robots............................55
2.3.8.5 Decoupled robots............................60
2.4 Articulated truss............................62
2.5 Examples of applications............................64
2.5.1 Spatial applications............................64
2.5.2 Medical applications............................66
2.5.3 Industrial applications............................66
2.5.4 Joysticks............................68
2.5.5 Simulators............................69
2.5.6 Special applications............................71
2.6 Notion of standard manipulators............................73
2.7 Exercises............................74
3 Jacobian and inverse kinematics............................77
3.1 Inverse kinematics............................77
3.1.1 General method............................77
3.1.1.1 Planar manipulators............................78
3.1.1.2 RRP S manipulators ............................79
3.1.1.3 P RRS manipulators............................80
3.1.1.4 R RRS manipulators............................81
3.1.1.5 Spherical manipulators............................82
3.1.2 Extrema of the articular coordinates ............................83
3.1.2.1 Extrema for a cartesian box ............................83
3.1.2.2 Extrema for a sphere............................84
3.1.2.3 Extrema for any space............................84
3.1.2.4 Exact and approximate computation ............................85
3.2 Inverse jacobian matrix ............................86
3.2.1 Euler angles inverse jacobian............................87
3.2.1.1 Example: RRP S manipulators............................87
3.2.2 Inverse kinematic jacobian............................88
3.2.2.1 Example: RRP S manipulators............................89
3.2.2.2 Example: P RRS manipulators............................90
3.2.2.3 Example: R RRS manipulators............................91
3.2.2.4 Other example............................91
3.2.3 Isotropy............................92
3.3 Jacobian matrix............................94
3.3.1 Direct calculation of the Jacobian............................94
3.4 Jacobian matrix and internal sensors............................95
3.4.1 Practical examples............................96
3.5 Calibration............................97
3.6 Exercises............................100
4 Direct kinematics............................103
4.1 Planar mechanism ............................103
4.1.1 The 4-bar mechanism............................103
4.1.2 Coupler curve and circularity............................104
4.1.3 Direct kinematics of the 3-RP R robot............................105
4.1.3.1 Assembly modes............................106
4.1.3.2 Polynomial direct kinematics............................106
4.1.3.3 Particular cases............................108
4.1.4 Other planar robots............................109
4.2 Mechanisms for translations in space ............................110
4.3 Spherical mechanisms............................110
4.4 6 degrees of freedom mechanisms............................112
4.4.1 Example of analysis: the TSSM ............................112
4.4.1.1 Upper bound on the number of assembly modes............................112
4.4.1.2 Polynomial formulation............................114
4.4.1.3 Example of TSSM with 16 assembly modes............................116
4.4.2 Analysis of other space mechanisms............................118
4.4.2.1 MSSM ............................119
4.4.2.2 Active wrist............................120
4.4.2.3 Stewart platform ............................121
4.4.2.4 3 degrees of freedom wrist............................122
4.4.3 Main results............................125
4.4.3.1 6-5 manipulators ............................125
4.4.3.2 6-4 manipulators ............................126
4.4.3.3 6-3 manipulators ............................127
4.4.3.4 5-5 manipulators ............................127
4.4.3.5 5-4 manipulators ............................127
4.4.3.6 4-4 manipulators ............................129
4.4.3.7 Manipulators with 5 aligned points ............................130
4.4.3.8 Manipulators PPP-3S,PRR-3S,PPR-3S ............................130
4.5 Nair systematic method ............................131
4.5.1 Principle of the method............................131
4.5.1.1 The linear system............................132
4.5.1.2 Closure equations............................133
4.5.1.3 Resolution............................133
4.5.2 Example: wrist with 3 degrees of freedom............................134
4.5.3 Results of Nair's method............................136
4.5.3.1 9 link manipulators ............................136
4.5.3.2 7 and 8 link manipulators ............................137
4.5.3.3 R-R manipulators ............................137
4.5.4 Conclusion............................138
4.6 Case of the general robot............................138
4.6.1 The SSM ............................138
4.6.1.1 Maximum number of assembly modes............................138
4.6.1.2 Determination of the solutions............................139
4.6.2 General robots ............................140
4.6.2.1 Maximum number of assembly modes............................140
4.6.2.2 Determination of the solutions............................140
4.7 Summary of results and conclusion............................142
4.8 Fast numerical methods............................145
4.8.1 Iterative methods ............................145
4.8.2 Methods efficiency and computation time............................149
4.8.3 Convergence of the iterative methods............................149
4.8.4 Drawbacks of iterative methods and real-time issues............................152
4.9 Direct kinematics with extra sensors............................152
4.9.1 Type and location of the extra sensors............................153
4.9.2 Maximal number of sensors............................153
4.9.2.1 Addition of angular sensors............................154
4.9.2.2 Addition of linear sensors............................154
4.9.3 Relationship between sensors accuracy and pose accuracy............................154
4.10 Conclusions............................156
4.11 Exercises............................157
5 Singular configurations............................161
5.1 Introduction ............................161
5.1.1 Singularities and velocity............................161
5.1.2 Singularities and statics............................162
5.1.3 Singularities and kinematics............................162
5.2 State of the art............................163
5.3 Grassmann geometry............................164
5.3.1 Variety and geometry............................165
5.3.2 Principle governing the search of singularities............................167
5.3.3 Analytical examples............................167
5.3.3.1 Planar manipulator............................168
5.3.3.2 MSSM............................169
5.3.4 Analysis of other spatial manipulators............................179
5.3.4.1 TSSM............................180
5.3.4.2 3 dof wrist............................181
5.3.4.3 INRIA active wrist............................182
5.4 Degrees of freedom associated with singularities............................182
5.4.1 Example: the MSSM............................183
5.4.1.1 Type 3d configuration ............................183
5.4.1.2 Type 5a configuration ............................184
5.4.1.3 Type 5b configuration............................186
5.5 Manipulability and condition number............................187
5.6 Practical search for singularities............................189
5.6.1 Semi-jacobian method............................190
5.6.1.1 Singularity in a cartesian box............................191
5.6.1.2 Singularities in an articular workspace ............................192
5.7 Mechanisms in permanent singularity............................192
5.8 Singularity-free path-planning............................193
5.9 Exercises............................193
6 Workspace............................195
6.1 Workspace limits and representation............................195
6.1.1 The different types of workspaces............................195
6.1.2 State of the art............................196
6.2 Calculation of the constant orientation workspace ............................199
6.3 Planar manipulator............................201
6.3.1 Constant orientation workspace............................201
6.3.1.1 Articular coordinates limits............................202
6.3.1.2 Mechanical limits on the passive joints............................202
6.3.2 Orientation workspace............................203
6.3.3 Dextrous workspace............................203
6.3.4 Maximal workspace............................204
6.3.5 Inclusive orientation workspace............................207
6.3.6 Total orientation workspace............................209
6.4 6 dof manipulators............................211
6.4.1 Cross-sections of the constant orientation workspace............................211
6.4.2 3D constant orientation workspace............................213
6.4.2.1 Workspace area and volume............................215
6.4.2.2 Mechanical limits on the joints............................216
6.4.2.2.1 Model for the mechanical limits............................217
6.4.2.2.2 Possible locations of the centers of the joints............................217
6.4.2.2.3 Joints on the moving platform............................218
6.4.2.2.4 Examples............................219
6.4.2.3 Interference between links............................221
6.4.2.3.1 Notion of distance between links............................221
6.4.2.3.2 Interference loci............................221
6.4.3 Orientation workspace............................222
6.4.4 Dextrous workspace............................224
6.4.5 Maximal workspace............................226
6.4.6 Comparison between architectures ............................227
6.5 Trajectory checking............................229
6.5.1 Constant orientation trajectory............................230
6.5.1.1 Limits on the link lengths............................231
6.5.1.2 Mechanical limits on the joints............................232
6.5.1.3 Computation time............................232
6.5.2 Non constant orientation trajectory............................233
6.5.2.1 Articular coordinates limits............................234
6.5.2.2 Computation time and example............................235
6.6 Path-planning............................235
6.6.1 Introduction............................236
6.6.2 Path-planning in a plane............................236
6.6.2.1 Path-Planning with tiling............................236
6.6.2.2 Path-planning with visibility graph............................237
6.6.3 Path-planning in space with tiling............................238
6.6.4 Path-planning for planar robots............................238
6.7 Exercises............................241
7 Velocity and Acceleration............................245
7.1 Relations between the articular velocities and the generalized velocities ............................245
7.1.1 Determination of the articular velocities............................245
7.1.2 Determination of the twist............................245
7.2 Extrema of the generalized velocities............................246
7.2.1 Extrema of the generalized velocities in a pose............................246
7.2.2 Minimum of the cartesian velocity in a translation workspace............................247
7.3 Extrema of the articular velocities in a translation workspace............................251
7.3.1 Particular case............................253
7.3.2 Determination of the motion of the passive joints............................254
7.4 Accelerations............................254
7.4.1 General robot............................255
7.4.2 Active wrist............................256
7.5 Conclusion............................256
7.6 Exercises............................257
8 Static analysis............................259
8.1 Relations between generalized and articular forces............................259
8.1.1 Fundamental relations............................259
8.1.2 Determination of the generalized forces............................259
8.1.3 Determination of the articular forces............................259
8.2 Articular forces and maximal generalized forces............................260
8.2.1 Maximal articular forces in a pose............................260
8.2.2 Maximal articular forces in a translation workspace............................261
8.2.2.1 Extrema on a line segment............................262
8.2.2.2 Extrema for an horizontal rectangle............................263
8.2.2.3 Extrema for a cartesian box............................264
8.2.3 Maximal generalized forces in a pose............................265
8.2.4 Maximal generalized forces in the workspace............................266
8.3 Parallel robots as force sensors............................266
8.4 Stiffness and compliance............................267
8.4.1 Stiffness matrix of a parallel robot............................268
8.4.1.1 Elastic model............................268
8.4.1.2 Beam model............................270
8.4.2 Passive compliance............................270
8.4.3 Stiffness maps............................271
8.4.3.1 Iso-stiffness maps............................272
8.4.3.2 Iso-stiffness of the general robot............................272
8.4.3.3 Iso-stiffness of the active wrist............................275
8.5 Extrema of the stiffnesses in a workspace............................276
8.5.1 Extrema on a line segment............................276
8.5.2 Extrema on a rectangle............................276
8.5.3 Extrema in a cartesian box............................277
8.6 Balancing............................277
8.7 Exercises............................278
9 Dynamics............................281
9.1 Introduction............................281
9.2 MSSM inverse dynamics............................283
9.3 SSM dynamics............................285
9.3.1 Hypothesis and notation............................285
9.3.2 Algorithm principle............................286
9.4 Active wrist dynamics............................289
9.5 Computation time............................290
9.6 Examples............................290
9.6.1 Inverse dynamics............................291
9.6.2 Direct dynamics............................291
9.7 Exercises............................292
10 Design............................295
10.1 Introduction............................295
10.2 Design method............................299
10.2.1 Reduction of the parameters............................300
10.2.2 Restriction of the search space............................301
10.2.2.1 Restriction via the workspace ............................301
10.2.2.2 Restriction via the articular velocities ............................303
10.2.3 Search for the optimal robot............................305
10.2.4 Design examples............................308
10.2.4.1 HFM2 robot............................308
10.2.4.2 HDM1 robot............................310
10.3 Exercises............................310
11 Conclusion............................313
WEB adresses............................317
Index............................321
References............................339
This document was translated from LATEX by HEVEA.