In this paper, a method to determine the internal and external boundaries of planar workspaces, represented with an ordered set of points, is presented. The sequence of points are grouped and can be interpreted to form a sequence of curves. Three successive curves are used for determining the instantaneous center of rotation for the second one of them. The two extremal points on the curve with respect to the instantaneous center are recognized as singular points. The chronological ordering of these singular points is used to generate the two envelope curves, which are potentially intersecting. Methods have been presented in the paper for the determination of the workspace boundary from the envelope curves. Strategies to deal with the manipulators with joint limits and various degenerate situations have also been discussed. The computational steps being completely geometric, the method does not require the knowledge about the manipulator’s kinematics. Hence, it can be used for the workspace of arbitrary planar manipulators. A number of illustrative examples demonstrate the efficacy of the proposed method.
Issue Section:
Mechanisms and Robotics
Keywords:
manipulators,
workspace boundary,
singularity,
sweep envelope,
planar closed-loop manipulator
Topics:
Manipulators
1.
Roth
, B.
, 1976, “Performance Evaluation of Manipulators From a Kinematic Viewpoint
,” NBS Special Publication No. 459 (Oct. 1976), Performance Evaluation of Programmable Robots and Manipulators, U.S. Government Printing Office, WA, pp. 39
–61
.2.
Sen
, D.
, and Mruthyunjaya
, T. S.
, 1999, “Synthesis of Workspaces of Planar Manipulators With Arbitrary Topology Using Shape Representation and Simulated Annealing
,” Mech. Mach. Theory
0094-114X, 34
(3
), pp. 391
–420
.3.
Cheng
, H. H.
, and Thompson
, S.
, 1999, “Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers
,” ASME J. Mech. Des.
1050-0472, 121
, pp. 200
–205
.4.
Lai
, Z. C.
, and Yang
, D. C. H.
, 1986, “A New Method for the Singularity Analysis of Simple Link Manipulators
,” Int. J. Robot. Res.
0278-3649, 5
(2
), pp. 66
–74
.5.
Lipkin
, H.
, and Pohl
, E.
, 1991, “Enumeration of Singular Configurations for Robotics Manipulators
,” ASME J. Mech. Des.
1050-0472, 113
, pp. 272
–279
.6.
Long
, G. L.
, 1997, “Use of the Cylindroid for the Singularity Analysis of Rank 3 Robot Manipulators
,” Mech. Mach. Theory
0094-114X, 32
(3
), pp. 391
–404
.7.
Park
, F. C.
, and Kim
, J. W.
, 1999, “Singularity Analysis of Closed Kinematic Chains
,” ASME J. Mech. Des.
1050-0472, 121
, pp. 32
–38
.8.
Wang
, S. L.
, and Waldron
, K. J.
, 1987, “A Study of the Singular Configurations of Serial Manipulators
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 109
, pp. 14
–20
.9.
Zlatanov
, D.
, Fenton
, R. G.
, and Benhabib
, B.
, 1995, “A Unifying Framework for Classification and Interpretation of Mechanism Singularities
,” ASME J. Mech. Des.
1050-0472, 117
, pp. 566
–572
.10.
Kumar
, A.
, and Waldron
, K. J.
, 1981, “The Workspaces of a Mechanical Manipulator
,” ASME J. Mech. Des.
0161-8458, 103
, pp. 665
–672
.11.
Mruthyunjaya
, T. S.
, Gobinath
, T.
, and Balakumar
, P.
, 1986, “Extreme Reach Analysis of Manipulators With Revolute or Prismatic Joints
,” ASME Design Engineering Technical Conference
, Columbus, OH, Sept., Paper No. 86-DET-191.12.
Shimano
, B.
, and Roth
, B.
, 1977, “Ranges of Motion of Manipulators
,” Proceedings of the Second International CISM-IFToMM Symposium
, Warsaw, Sept. 1976, PWN-Polish Science, Warszawa, pp. 18
–26
.13.
Sugimoto
, K.
, and Duffy
, J.
, 1981, “Determination of Extreme Distances of a Robot Hand. Part 1: A General Theory
,” ASME J. Mech. Des.
0161-8458, 103
, pp. 631
–636
.14.
Kohli
, D.
, and Spanos
, J.
, 1985, “Workspace Analysis of Mechanical Manipulators Using Polynomial Discriminants
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 107
, pp. 209
–215
.15.
Kohli
, D.
, and Hsu
, M. S.
, 1987, “The Jocobian Analysis of Workspaces of Mechanical Manipulators
,” Mech. Mach. Theory
0094-114X, 22
(3
), pp. 265
–275
.16.
Jo
, D. Y.
, and Haug
, E. J.
, 1989, “Workspace Analysis of Multibody Mechanical Systems Using Continuation Methods
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 111
, pp. 581
–589
.17.
Abdel-Malek
, K.
, Adkins
, F.
, Yeh
, H. J.
, and Haug
, E. J.
, 1997, “On the Determination of Boundaries to Manipulator Workspaces
,” Rob. Comput.-Integr. Manufact.
0736-5845, 13
(1
), pp. 63
–72
.18.
Abdel-Malek
, K.
, Yeh
, H. J.
, and Othman
, S.
, 2000, “Interior and Exterior Boundaries to the Workspaces of Mechanical Manipulators
,” Rob. Comput.-Integr. Manufact.
0736-5845, 16
(5
), pp. 365
–776
.19.
Ottaviano
, E.
, Husty
, M.
, and Ceccarelli
, M.
, 2006, “Identification of the Workspace Boundary of a General 3-R Manipulator
,” ASME J. Mech. Des.
1050-0472, 128
, pp. 236
–242
.20.
Agrawal
, S. K.
, and Garimella
, R.
, 1994, “Workspace Boundaries of Free-Floating Open and Closed Chain Planar Manipulators
,” ASME J. Mech. Des.
1050-0472, 116
, pp. 105
–110
.21.
Bajpai
, A.
, and Roth
, B.
, 1986, “Workspace and Mobility of a Closed-Loop Manipulator
,” Int. J. Robot. Res.
0278-3649, 5
(2
), pp. 131
–142
.22.
Cerventes-Sanchez
, J. J.
, Harnandez-Rodriguez
, J. C.
, and Rendon-Sanchez
, J. G.
, 2000, “On the Workspace, Assembly Configurations and Singularity Curves of the RRRRR-Type Planar Manipulator
,” Mech. Mach. Theory
0094-114X, 35
(8
), pp. 1117
–1139
.23.
Fallahi
, B.
, Lai
, H. Y.
, Naghiri
, R.
, and Wang
, Y.
, 1994, “A Study of the Workspace of Five-Bar Closed Loop Manipulator
,” Mech. Mach. Theory
0094-114X, 29
(5
), pp. 759
–765
.24.
Gosselin
, C.
, 1990, “Determination of the Workspace of 6-D.O.F. Parallel Manipulators
,” ASME J. Mech. Des.
1050-0472, 112
(3
), pp. 31
–336
.25.
Luh
, C.
, Adkins
, F. A.
Haug
, E. J.
, and Qiu
, C. C.
, 1996, “Working Capability Analysis of Stewart Platforms
,” ASME J. Mech. Des.
1050-0472, 118
(2
), pp. 220
–227
.26.
Merlet
, J. P.
, Gosselin
, C. M.
, and Mouly
, N.
, 1998, “Workspaces of Planar Parallel Manipulators
,” Mech. Mach. Theory
0094-114X, 33
(1
), pp. 7
–20
.27.
Pennock
, G. R.
, and Kassner
, D. J.
, 1993, “The Workspace of a General Geometry Planar Three-Degree-of-Freedom Platform-Type Manipulator
,” ASME J. Mech. Des.
1050-0472, 115
, pp. 269
–276
.28.
Rastegar
, J.
, and Deravi
, P.
, 1987, “Methods to Determine Workspace, Its Subspaces With Different Numbers of Configurations and All the Possible Configurations of a Manipulator
,” Mech. Mach. Theory
0094-114X, 22
(4
), pp. 343
–350
.29.
Ricard
, R.
, and Gosselin
, C. M.
, 1994, “On the Determination of the Workspace of Complex Planar Robotic Manipulators
,” Proceedings of the 1994 ASME Biennial Technical Conference on Robotics, Kinematics, Dynamics and Controls
, Vol. 72
, pp. 133
–140
.30.
Ricard
, R.
, and Gosselin
, C. M.
, 1998, “On the Determination of the Workspace of Complex Planar Robotics Manipulators
,” ASME J. Mech. Des.
1050-0472, 120
, pp. 269
–278
.31.
Ting
, K.-L.
, 1992, “Gross Motion and Classification of Manipulators With Closed-Loop, Four-Bar Chains
,” Int. J. Robot. Res.
0278-3649, 11
(3
), pp. 238
–247
.32.
Lee
, T. W.
, and Yang
, D. C. H.
, 1983, “On the Evaluation of Manipulator Workspaces
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 105
, pp. 70
–77
.33.
Cwiakala
, M.
, and Lee
, T. W.
, 1985, “Generation and Evaluaton of a Manipulator Workspace Based on Optimum Path Search
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 107
, pp. 245
–255
.34.
Alciatore
, D. G.
, and Ng
, C. D.
, 1974, “Determining Manipulator Workspace Boundaries Using the Monte Carlo Method and Least Squares Segmentation
,” Proceedings of the 1994 ASME Biennial Technical Conference on Robotics, Kinematics, Dynamics and Controls
, Vol. 72
, pp. 141
–146
.35.
Rastegar
, J.
, and Perel
, D.
, 1990, “Generation of Manipulator Workspace Boundary Geometry Using Monte Carlo Method and Interactive Computer Graphics
,” ASME J. Mech. Des.
1050-0472, 112
(3
), pp. 452
–454
.36.
Sen
, D.
, and Mruthyunjaya
, T. S.
, 1998, “A Centro-Based Characterization of Singularities in the Workspace of Planar Closed-Loop Manipulators
,” Mech. Mach. Theory
0094-114X, 33
(8
), pp. 1091
–1104
.37.
Haug
, E. J.
, Luh
, C.-M.
, Adkins
, F. A.
, and Wang
, J.-Y.
, 1996, “Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces
,” ASME J. Mech. Des.
1050-0472, 118
, pp. 228
–234
.38.
Sen
, D.
, and Mruthyunjaya
, T. S.
, 1999, “A Computational Geometry Approach for Determination of Boundary of Planar Manipulators With Arbitrary Topology
,” Mech. Mach. Theory
0094-114X, 34
(1
), pp. 149
–169
.39.
Haug
, E. J.
, Adkins
, F. A.
, and Luh
, C. M.
, 1998, “Operational Envelopes for Working Bodies of Mechanisms and Manipulators
,” ASME J. Mech. Des.
1050-0472, 120
, pp. 84
–91
.40.
Ling
, Z. K.
, and Chase
, T. R.
, 1996, “Generating the Swept Area of a Body Undergoing Planar Motion
,” ASME J. Mech. Des.
1050-0472, 118
, pp. 186
–192
.41.
Sen
, D.
, Chowdhury
, S.
, and Pandey
, S.
, 2004, “Geometric Design of Interference-Free Planar Linkages
,” Mech. Mach. Theory
0094-114X, 39
(7
), pp. 737
–759
.42.
Mortenson
, M. E.
, 1985, Geometric Modeling
, Wiley
, pp. 403
–413
.Copyright © 2008
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