Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T08:49:11.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Cuspidal and noncuspidal robot manipulators

Published online by Cambridge University Press:  01 November 2007

Philippe Wenger
Philippe Wenger*
Affiliation:
Institut de Recherche en Communications et Cybernétique de Nantes 1, rue de la Noë, 44321 Nantes, France
*
E-mail: philippe.wenger@irccyn.ec-nantes.fr
Get access Rights & Permissions [Opens in a new window]

Summary

This article synthezises the most important results on the kinematics of cuspidal manipulators i.e. nonredundant manipulators that can change posture without meeting a singularity. The characteristic surfaces, the uniqueness domains and the regions of feasible paths in the workspace are defined. Then, several sufficient geometric conditions for a manipulator to be noncuspidal are enumerated and a general necessary and sufficient condition for a manipulator to be cuspidal is provided. An explicit DH-parameter-based condition for an orthogonal manipulator to be cuspidal is derived. The full classification of 3R orthogonal manipulators is provided and all types of cuspidal and noncuspidal orthogonal manipulators are enumerated. Finally, some facts about cuspidal and noncuspidal 6R manipulators are reported.

Keywords

Cuspidal manipulatorSingularityPostureWorkspaceCusp pointClassification
Type
Article
Information
Robotica , Volume 25 , Issue 6 , November 2007 , pp. 677 - 689
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Borrel, P. and Liegeois, A., “A Study of Manipulator Inverse Kinematic Solutions With Application to Trajectory Planning and Workspace Determination,” Proceedings of the IEEE International Conference on Robotics and Automation (1986) pp. 1180–1185.Google Scholar
2.Parenti, C.V. and Innocenti, C., “Position Analysis of Robot Manipulators : Regions and Subregions,” Proceedings of the International Conference on Advances in Robot Kinematics (1988) pp. 150–158.Google Scholar
3.Burdick, J. W., “Kinematic Analysis and Design of Redundant Manipulators” Ph.D. Dissertation (Stanford, CA: Stanford University, 1988).Google Scholar
4.Wenger, P., “A New General Formalism for the Kinematic Analysis of All Non-Redundant Manipulators,” Proceedings of the IEEE International Conference on Robotics and Automation (1992) pp. 442–447.Google Scholar
5.Tsai, K. Y. and Kholi, D., “Trajectory Planning in Task Space for General Manipulators,” ASME J. Mech. Des., 115, 915–921, (1993).CrossRefGoogle Scholar
6.Smith, D. R. and Lipkin, H., “Higher Order Singularities of Regional Manipulators,” Proceedings of the IEEE International Conference on Robotics and Automation, Atlanta, GA (1993) pp. 194–199.Google Scholar
7.Burdick, J. W., “A classification of 3R regional manipulator singularities and geometries,” Mech. Mach. Theory, 30 (1), 71–89 (1995).CrossRefGoogle Scholar
8.Wenger, P., “Design of Cuspidal and Noncuspidal Manipulators,” Proceedings of the IEEE International Conference on Robotics and Automation (1997) pp. 2172–2177.Google Scholar
9.El Omri, J. and Wenger, P., “How to Recognize Simply a Nonsingular Posture Changing 3-DOF Manipulator,” Proceedings of the 7th International Conference on Advanced Robotics (1995) pp. 215–222.Google Scholar
10.Arnold, V. I., Singularity Theory (Cambridge University Press, Cambridge, UK, 1981).CrossRefGoogle Scholar
11.Wenger, P. and El Omri, J., “Changing Posture for Cuspidal Robots,” Proceedings of the IEEE International Conference on Robotics and Automation (1996) pp. 3173–3178.Google Scholar
12.Wenger, P., “Classification of 3R positioning manipulators,” ASME J. Mech. Des., 120 (2), 327–332 (1998).CrossRefGoogle Scholar
13.Wenger, P., “Some guidelines for the kinematic design of new Manipulators,” Mech. Mach. Theory, 35 (3), 437–449 (2000).CrossRefGoogle Scholar
14.Corvez, S. and Rouillier, F., “Using Computer Algebra Tools to Classify Serial Manipulators,” In: Automated Deduction in Geometry, Lecture Notes in Computer Science (Springer, Berlin, Germany, 2004) Vol. 2930, pp. 31–43.Google Scholar
15.Baili, M., Wenger, P. and Chablat, D., “Classification of one Family of 3R Positioning Manipulators,” Proceedings of the 11th International Conference on Advanced Robotics (2003) pp. 1849–1854.Google Scholar
16.Pai, D. K. and Leu, M. C., “Genericity and singularities of robot manipulators,” IEEE Trans. Robot. Autom., 8 (5), 545–559 (1991).CrossRefGoogle Scholar
17.Baili, M., Wenger, P. and Chablat, D., “A Classification of 3R Orthogonal Manipulators by the Topology of Their Workspace,” Proceedings of the IEEE International Conference on Robotics and Automation, (2004) pp. 1933–1938.Google Scholar
18.Wenger, P., Baili, M. and Chablat, D., “Workspace Classification of 3R Orthogonal Manipulators,” In: Advances in Robot Kinematics (Kluwer Academic, Norwell, MA, 2004), pp. 219–228.Google Scholar
19.Wenger, P., “Uniqueness domains and regions of feasible continuous paths for cuspidal manipulators,” IEEE Trans. Robot., 20 (4) (Aug. 2004) pp. 745–750.CrossRefGoogle Scholar
20.Wenger, P., Chablat, P. D. and Baili, M., “A DH-parameter based condition for 3R orthogonal manipulators to have 4 distinct inverse kinematic solutions,” J. Mech. Des. 127 (1), 150–155, (2005).Google Scholar
21.Ranjbaran, F. and Angeles, J., “On Positioning Singularities of 3-Revolute Robotic Manipulators,” Proceedings of the 12th Symposium on Engineering Applications in Mechanics, Montreal, CA, (1994) pp. 273–282.Google Scholar
22.Urbanic, A. and Lenarcic, J., “Kinematic Considerations on General 3R Manipulators,” Proceedings of the 3rd International Workshop on Robotic in the Alpe-Adria, Slovenia (1994) pp. 293–298.Google Scholar
23.Burdick, J. W., “A recursive method for finding revolute-jointed manipulator singularities,” ASME J. Mech. Des., 117 (1), 55–63 (1995).Google Scholar
24.Khalil, W. and Creusot, D., “SYMORO+, a system for the modeling of robots,” Robotica, 15 (2) (Mar. 1997) pp. 153–161.CrossRefGoogle Scholar
25.Roth, B., “Performance Evaluation of Manipulators From a Kinematic Viewpoint,” In: Performance Evaluation of Programmable Robots and Manipulators, NBS Special Publication (NBS, Gaithersburg, MD, 1975).Google Scholar
26.Kumar, A. and Waldron, K. J., “The workspace of a mechanical manipulator,” ASME J. Mech. Des., 103, 665–672, (1981).Google Scholar
27.Gupta, K. C. and Roth, B., “Design considerations for manipulator workspaces,” ASME J. Mech. Des., 104, 704–711, (1982).Google Scholar
28.Yang, D. C. H and Lee, T. W., “On the workspace of mechanical manipulators,” ASME J. Mech. Des., 105, 62–69, (1983).Google Scholar
29.Freudenstein, F. and Primrose, E. J. F., “On the analysis and synthesis of the workspace of a three-link, turning-pair connected robot arm,” ASME J. Mech. Transm. Autom. Des. 106, 365–370, (1984).CrossRefGoogle Scholar
30.Kholi, D. and Spanos, J., “Workspace analysis of mechanical manipulators using polynomial discriminant,” ASME J. Mech. Transm. Autom. Des., 107, 209–215, (1985).CrossRefGoogle Scholar
31.Rastegar, J. and Deravi, P., “Methods to determine workspace with different numbers of configurations and all the possible configurations of a manipulator,” J. Mech. Mach. Theory, 22 (4)343–350 (1987).CrossRefGoogle Scholar
32.Kohli, D. and Hsu, M. S., “The Jacobian analysis of workspaces of mechanical manipulators,” Mech. Mach. Theory, 22 (3), 265–275 (1987).CrossRefGoogle Scholar
33.Ceccarelli, M., “A formulation for the workspace boundary of general n-revolute manipulators,” Mech. Mach. Theory, 31, 637–646 (1996).CrossRefGoogle Scholar
34.Ottaviano, E., Husty, M. and Ceccarelli, M., “A Cartesian Representation for the Boundary Workspace of 3R Manipulators,” In: Advances in Robot Kinematics (Levante, S., ed.) (Kluwer Academic, Norwell, MA, 2004) pp. 247–254.Google Scholar
35.Miko, P., The Closed Form Solution of the Inverse Kinematic Problem of 3R Positioning Manipulators Ph.D. Thesie (Lausanne, Switzerland: Ecole Polytechnique Fédérale de Lausanne 2005).Google Scholar
36.Bergamaschi, P. R., Nogueira, A. C. and Saramago, S. D. P., “Design and optimization of 3R manipulators using the workspace features,” Appl. Math. Comput. 172 (1), 439–463 (2006).Google Scholar
37.Ottaviano, E., Husty, M. and Ceccarelli, M., “Level-Set Method for Workspace Analysis of Serial Manipulators,” In: Advances in Robot Kinematics, Mechanisms and Motion (Springer, Berlin, Germany, 2006) 307–315.Google Scholar
38.El Omri, J., Kinematic Analysis of Robot Manipulator, Ph.D. Thesis (in French) (Nantes, France: Ecole Centrale de Nantes, 1996).Google Scholar
39.Wenger, P. and El Omri, J., “On the Kinematics of Singular and Nonsingular Posture Changing Manipulators,” Proceedings of the Advances in Robot Kinematics, ARK'94, Ljubljana (1994) pp. 29–38.Google Scholar
40.Pieper, D. L., The Kinematics of Manipulators Under Computer Control, Ph.D. Thesis (Stanford, CA: Stanford University, 1968).Google Scholar
41.Lazard, D. and Rouillier, F., “Solving parametric polynomial systems,” Technical Report, RR-5322, INRIA (Oct. 2004).Google Scholar
42.Collins, G. E., “Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition,” In: Springer Lecture Notes in Computer Science (Springer, Berlin, Germany, 1975) No. 3, pp. 515–532.Google Scholar
43.Baili, M., “Analysis and Classification of 3R Orthogonal Manipulators, Ph.D. Thesis (in French) (Nantes, France: Ecole Centrale de Nantes, 2004).Google Scholar
44.Gibson, C. G. “Kinematics From the Singular Viewpoint,” In: Geometrical Foundation of Robotics (Selig, J. M. ed.) (World Scientific, Singapore, 2000) pp. 61–79.CrossRefGoogle Scholar
45.Zein, M., Wenger, M. P. and Chablat, D., “An exhaustive study of the workspace topologies of all 3R orthogonal manipulators with geometric simplifications,” Mech. Mach. Theory, 41 (8), 971–986 (2006).CrossRefGoogle Scholar
46.Hemmingson, E., Ellqvist, S. and Pauxels, J., “New robot improves cost-efficiency of spot welding,” ABB Rev. [Spot Welding Robots], vol. 3, 4–6 (1996).Google Scholar
47.Zoppi, M., “Effective Backward Kinematics for an Industrial 6R Robot,” Proceedings of the ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Montreal, Canada, (2002).CrossRefGoogle Scholar
48.Mavroidis, C. and Roth, B., “Structural parameters which reduce the number of manipulator configurations,” ASME J. Mech. Des., 116, 3–10 (1994).Google Scholar
49.Donelan, P. S. and Gibson, C. G., “Singular Phenomena in Kinematics,” In: Singularity Theory (Bruce, B. and Mond, D., eds.) (Cambridge University Press, Cambridge, UK, 1999) pp. 379–402.Google Scholar
50.Innocenti, C. and Parenti-Castelli, V., “Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators,” J. Mech. Des., 120, 73–99 (1998).Google Scholar
51.Wenger, P. and Chablat, D., “Workspace and Assembly-Modes in Fully Parallel Manipulators: A Descriptive Study,” In: Advances on Robot Kinematics (Kluwer Academic Norwell, MA, 1998) pp. 117–126.CrossRefGoogle Scholar
52.Mcaree, P. R. and Daniel, R. W., “An explanation of never-special assembly changing motions for 3-3 parallel manipulators,” Int. J. Robot. Res., 18 (6), 556–574 (1999).CrossRefGoogle Scholar
53.Zein, M., Wenger, P. and Chablat, D., “Non-Singular Assembly-Mode Changing Motions for 3-RPR Parallel Manipulators,” In: IFToMM Mechanism and Machine Theory. (in press).Google Scholar
54.Kong, X. and Gosselin, C. M., “Determination of the Uniqueness Domains of 3-RPR Planar Parallel Manipulators With Similar Platforms,” Proceedings of the 2000 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Baltimore, MD (2000).CrossRefGoogle Scholar
55.Wenger, P., and Chablat, D., “The Kinematic Analysis of a Symmetrical Three-Degree-of-Freedom Planar Parallel Manipulator,” Proceedings of the CISM-IFToMM Symposium on Robot Design, Dynamics and Control, Montreal, (2004). pp. 1–7.Google Scholar