Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T11:21:19.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularity-theoretic methods in robot kinematics

Published online by Cambridge University Press:  01 November 2007

P. S. Donelan
P. S. Donelan*
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, PO Box 600, Wellington, New Zealand.
*
*Corresponding author. E-mail: peter.donelan@vuw.ac.nz
Get access Rights & Permissions [Opens in a new window]

Summary

The significance of singularities in the design and control of robot manipulators is well known, and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators—indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

Keywords

Singularity theoryRobot manipulatorTransversalityLie groupGenericityScrew systems
Type
Article
Information
Robotica , Volume 25 , Issue 6 , November 2007 , pp. 641 - 659
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.Whitney, D. E., “Resolved motion rate control of manipulators and human prostheses,” IEEE Trans. Man–Mach. Syst. 10, 47–53, (1969).CrossRefGoogle Scholar
2.Featherstone, R., “Position and velocity transformations between robot end-effector coordinates and joint angles,” Int. J. Robot. Res. 2, 35–45, (1983).CrossRefGoogle Scholar
3.Hunt, K. H., Kinematic Geometry of Mechanisms (Clarendon Press, Oxford, UK, 1978).Google Scholar
4.Sugimoto, K., Duffy, J. and Hunt, K. H., “Special configurations of spatial mechanisms and robot arms,” Mech. Mach. Theory 17, 119–132, (1982).CrossRefGoogle Scholar
5.Hunt, K. H., “Special configurations of robot arms via screw theory–-Part I: The Jacobian and its matrix cofactors,” Robotica 4, 171–179, (1986).CrossRefGoogle Scholar
6.Hunt, K. H., “Special configurations of robot arms via screw theory–-Part II: Available end-effector displacements,” Robotica 5, 17–22, (1987).CrossRefGoogle Scholar
7.Litvin, F. L. and Parenti-Castelli, V., “Configurations of robot manipulators and their identification and the execution of prescribed trajectories,” Trans. ASME J. Mech., Transm. Autom. Design 107, 170–188, (1985).CrossRefGoogle Scholar
8.Litvin, F. L., Yi, Z., Parenti-Castelli, V. and Innocenti, C., “Singularities, configurations and displacement functions for manipulators,” Int. J. Robot. Res. 5, 66–74, (1986).CrossRefGoogle Scholar
9.Wang, S. L. and Waldron, K. J., “A study of the singular configurations of serial manipulators,” Trans. ASME J. Mech. Transm. Autom. Design 109, 14–20, (1987).CrossRefGoogle Scholar
10.Gosselin, C. and Angeles, J., “Singularity analysis of closed-loop kinematic chains,” IEEE Trans. Robot. Autom. 6, 281–290, (1990).CrossRefGoogle Scholar
11.Merlet, J. P., “Singular configurations of parallel manipulators and Grassmann geometry,” Int. J. Robot. Res. 8, 45–56, (1992).CrossRefGoogle Scholar
12.Di Gregorio, R. and Parenti-Castelli, V., “Mobility Analysis of the 3-UPU Parallel Mechanism Assembled for a Pure Translational Motion,” Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, Georgia (1999) pp. 520–525.Google Scholar
13.Zlatanov, D., Bonev, I. A. and Gosselin, C. M., “Constraint Singularities of Parallel Mechanisms,” Proceedings of the IEEE International Conference on Robotics and Automation (2002) pp. 496–502.Google Scholar
14.Stanišić, M. M. and Engelberth, J. W., “A Geometric Description of Manipulator Singularities in Terms of Singular Surfaces,” Proceedings of the 1st International Workshop on Advances in Robot Kinematics, Ljubljana, Slovenia (1988) pp. 132–141.Google Scholar
15.Hayes, M. J. D., Husty, M. L. and Zsombor-Murray, P. J., “Singular configurations of wrist-partitioned 6R serial robots: A geometric perspective for users,” Trans. Canadian Soc. Mech. Eng. 26, 41–55, (2002).CrossRefGoogle Scholar
16.Burdick, J. W., “A classification of 3R regional manipulator singularities and geometries,” Mech. Mach. Theory 30, 71–89, (1995).CrossRefGoogle Scholar
17.Wenger, P. and El Omri, J., “Changing Posture for Cuspidal Robot Manipulators,” Proceedings of the IEEE International Conference on Robotics and Automation (1996) pp. 3173–3178.Google Scholar
18.Pieper, D. and Roth, B., “The Kinematics of Manipulators Under Computer Control,” Proceedings of the 2nd World Congress on the Theory of Machines and Mechanisms, Zakopane, Poland 2, (1969) pp. 159–169.Google Scholar
19.Stanišić, M. M. and Goehler, C. M., “Singular planes of serial wrist-partitioned manipulators and their singularity mertrics,” Mech. Mach. Theory, 42, 889–902, (2007).CrossRefGoogle Scholar
20.Klein, C. A. and Huang, C. H., “Review of pseudoinverse control for use with kinematically redundant manipulators,” IEEE Trans. Syst., Man Cybern. 3, 245–250, (1983).CrossRefGoogle Scholar
21.Shamir, T. and Yomdin, Y., “Repeatability of redundant manipulators: Mathematical solution of the problem,” IEEE Trans. Autom. Control 33, 1004–1009, (1988).CrossRefGoogle Scholar
22.Kieffer, J. and Lenarčič, J., “On the Exploitation of Mechanical Advantage Near Robot Singularities,” Proceedings of the 3rd International Workshop on Advances in Robot Kinematics, Ferrara, Italy (1992) pp. 65–72.Google Scholar
23.Paul, R. P. and Stevenson, C. N., “Kinematics of robot wrists,” Int. J. Robot. Res. 2, 31–38, (1983).CrossRefGoogle Scholar
24.Gottlieb, D. H., “Robots and fibre bundles,” Bull. Soc. Math. Belg. 38, 219–223, (1986).Google Scholar
25.Baker, D. R. and Wampler, C. W., “On the inverse kinematics of redundant manipulators,” Int. J. Robot. Res. 7, 3–21, (1988).CrossRefGoogle Scholar
26.Tchoń, K., “Towards a Differential Topological Classification of Robot Manipulators,” In Progress in Systems and Control Theory: Robust Control of Linear Systems and Nonlinear Control (Kaashoek, M. et al. eds.) (Birkhäuser, Boston, 1990). pp. 565–574.CrossRefGoogle Scholar
27.Tchoń, K., “Differential topology of the inverse kinematic problem for redundant robot manipulators,” Int. J. Robot. Res. 10, 492–504, (1991).CrossRefGoogle Scholar
28.Tchoń, K., “Calibration of manipulator's kinematics: A singularity theory approach,” IEEE Trans. Robot. Autom. 8, 671–678, (1992).CrossRefGoogle Scholar
29.Tchoń, K. and Urban, P., “Classification of kinematic singularities in planar robot manipulators,” Syst. Control Lett. 19, 293–302, (1992).CrossRefGoogle Scholar
30.Tchoń, K., “Normal forms of singular kinematics of 3R robot manipulators,” Appl. Math. Comput. Sci. 5, 391–407, (1995).Google Scholar
31.Tchoń, K., “A normal form of singular kinematics of robot manipulators with smallest degeneracy,” IEEE Trans. Robot. Autom. 11, 401–404, (1995).CrossRefGoogle Scholar
32.Tchoń, K. and Matuszok, A., “On avoiding singularities in redundant robot kinematics,” Robotica 13, 599–606, (1995).CrossRefGoogle Scholar
33.Tchoń, K. and Muszynski, R., “Singularities of nonredundant robot kinematics,” Int. J. Robot. Res. 16, 71–89, (1997).CrossRefGoogle Scholar
34.Tchoń, K., “Singularities of the Euler wrist,” Mech. Mach. Theory 35, 505–515, (2000).CrossRefGoogle Scholar
35.Pai, D. K. and Leu, M. C., “Genericity and singularities of robot manipulators,” IEEE Trans. Robot. Autom. 8, 545–559, (1992).CrossRefGoogle Scholar
36.Lerbet, J. and Hao, K., “Kinematics of mechanisms to the second order–-Application to the closed mechanisms,” Acta Appl. Math. 59, 1–19, (1999).CrossRefGoogle Scholar
37.Lerbet, J., “Singular set of four screws: An intrinsic solution,” Mech. Mach. Theory 38, 179–194, (2003).CrossRefGoogle Scholar
38.Wenger, P., “Classification of 3R positioning manipulators,” ASME J. Mech. Design 120, 327–332, (1998).CrossRefGoogle Scholar
39.Baili, M., Wenger, P. and Chablat, D., “Classification of One Family of 3R Positioning Manipulators,” Proceedings of the 11th International Conference on Advanced Robotics, Coimbra, Portugal (2003).Google Scholar
40.Zein, M., Wenger, P. and Chablat, D., “An exhaustive study of the workspace topologies of all 3R orthogonal manipulators with geometric simplifications,” Mech. Mach. Theory 41, 971–986, (2006).CrossRefGoogle Scholar
41.Karger, A., “Classification of robot-manipulators with only singular configurations,” Mech. Mach. Theory 30, 727–736, (1995).CrossRefGoogle Scholar
42.Karger, A., “Classification of serial robot-manipulators with non-removable singularities,” Trans. ASME J. Mech. Design 118, 202–208, (1996).CrossRefGoogle Scholar
43.Karger, A., “Singularity analysis of serial robot-manipulators,” Trans. ASME J. Mech. Design 118, 520–525, (1996).CrossRefGoogle Scholar
44.Ghosal, A. and Ravani, B., “A differential-geometric analysis of singularities of point trajectories of serial and parallel manipulators,” Trans. ASME J. Mech. Design 123, 80–89, (2001).CrossRefGoogle Scholar
45.Donelan, P. S., “Generic properties of Euclidean kinematics,” Acta Appl. Math. 12, 265–286, (1988).CrossRefGoogle Scholar
46.Gibson, C. G. and Newstead, P. E., “On the geometry of the planar 4-bar mechanism,” Acta Appl. Math. 7, 113–135, (1986).CrossRefGoogle Scholar
47.Gibson, C. G. and Selig, J. M., “Movable hinged spherical quadrilaterals I and II,” Mech. Mach. Theory 23, 13–24, (1988).CrossRefGoogle Scholar
48.Gibson, C. G. and Marsh, D., “Concerning cranks and rockers,” Mech. Mach. Theory 23, 355–360, (1988).CrossRefGoogle Scholar
49.Gibson, C. G. and Selig, J. M., “On the linkage varieties of Watt 6-bar mechanisms I and II,” Mech. Mach. Theory 24, 106–113, (1989).Google Scholar
50.Gibson, C. G. and Hobbs, C. A., “Simple singularities of space curves,” Math. Proc. Camb. Phil. Soc. 113, 297–310, (1992).CrossRefGoogle Scholar
51.Gibson, C. G., “Kinematic Singularities–-A New Mathematical Tool,” Proceedings of the 3rd International Workshop on Advances in Robot Kinematics, Ferrara, Italy (1992) pp. 209–215.Google Scholar
52.Donelan, P. S., Gibson, C. G. and Hawes, W., “Trajectory singularities of general planar motions, Proc. Royal Soc. Edinburgh 129A, 37–55 (1999).CrossRefGoogle Scholar
53.Gibson, C. G., Hawes, W. and Hobbs, C. A., “Local Pictures for General Two–Parameter Motions of the Plane,” In: Advances in Robot Kinematics and Computational Geometry (Kluwer Academic Publishers, Norwel, MA, 1994) pp. 49–58.CrossRefGoogle Scholar
54.Gibson, C. G. and Hobbs, C. A., “Local models for general one-parameter motions of the plane and space,” Proc. Royal Soc. Edinburgh, 125A, 639–656, (1995).CrossRefGoogle Scholar
55.Gibson, C. G. and Hobbs, C. A., “Singularity and bifurcation for general two-dimensional planar motions,” New Zealand J. Math., 25, 141–163, (1996).Google Scholar
56.Gibson, C. G., Hobbs, C. A. and Marar, W. L., “On versal unfoldings of singularities for general two-dimensional spatial motions,” Acta Appl. Math. 47, 221–242, (1996).CrossRefGoogle Scholar
57.Gibson, C. G., Marsh, D. and Xiang, Y., “Singular aspects of generic planar motions with two degrees of freedom,” Int. J. Robot. Res. 17, 1068–1080, (1998).CrossRefGoogle Scholar
58.Hines, R., Marsh, D. and Duffy, J., “Catastrophe analysis of the planar two-spring mechanism,” Int. J. Robot. Res. 17, 89–101, (1998).CrossRefGoogle Scholar
59.Yin, J. P., Marsh, D. and Duffy, J., “Catastrophe Analysis of Planar Three-Spring Systems,” Proceedings of the ASME Design Engineering Conference, Atlanta (1998).CrossRefGoogle Scholar
60.Carricato, M., Duffy, J. and Parenti-Castelli, V., “Catastrophe analysis of a planar system with flexural pivots,” Mech. Mach. Theory 37, 693–716, (2002).CrossRefGoogle Scholar
61.Murray, R. M., Li, Z. and Shastry, S. S., A Mathematical Introduction to Robotic Manipulation (CRC Press, Boca Raton, 1994.Google Scholar
62.Selig, J., Geometrical Fundamentals of Robotics (Springer Verlag, New York, 2005.Google Scholar
63.Munkres, J., Topology, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 2000).Google Scholar
64.Guillemin, V. and Pollack, A., Differential Topology (Prentice Hall, Englewood Cliffs, NJ, 1974).Google Scholar
65.Lee, J. M., Introduction to Smooth Manifolds (Springer, New York, 2002).Google Scholar
66.Gibson, C. G., Singular Points of Smooth Mappings, Research Notes in Mathematics 25 (Pitman, London, 1979).Google Scholar
67.Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Singularities of Differentiable Maps, Vol. 1 (Birkhauser, Cambridge MA, 1985.CrossRefGoogle Scholar
68.Martinet, J., Singularities of Smooth Functions and Maps, LMS Lecture Note Series 58 (Cambridge University Press, Cambridge, UK, 1982).Google Scholar
69.Golubitsky, M. and Guillemin, V., Stable Mappings and Their Singularities (Springer Verlag, New York, 1973.CrossRefGoogle Scholar
70.Burdick, J. W., “On the Inverse Kinematics of Redundant Manipulators: Characterization of the Self-Motion Manifolds,” Proceedings of the IEEE International Conference on Robotics and Automation (1989) pp. 264–270.Google Scholar
71.Brockett, R., “Robotic Manipulators and the Product of Exponentials Formula,” Proceedings of the Mathematical Theory of Networks and Systems, Beer-Sheva, Israel (1984) pp. 120–129.Google Scholar
72.Denavit, J. and Hartenberg, R. S., “A kinematic notation for lower pair mechanisms based on matrices,” ASME J. Appl. Mech. 22, 215–221, (1955).CrossRefGoogle Scholar
73.Hervé, J. M., “Analyse structurelle de méchanismes par group des déplacement,” Mech. Mach. Theory 13, 437–450, (1978).CrossRefGoogle Scholar
74.Gibson, C. G. and Hunt, K. H., “Geometry of screw systems I & II,” Mech. Mach. Theory 25, 1–27, (1990).CrossRefGoogle Scholar
75.Donelan, P. S. and Gibson, C. G., “First-order invariants of Euclidean motions,” Acta Appl. Math. 24, 233–251, (1991).CrossRefGoogle Scholar
76.Donelan, P. S. and Gibson, C. G., “On the hierarchy of screw systems,” Acta Appl. Math. 32, 267–296, (1993).CrossRefGoogle Scholar
77.Gottlieb, D. H., “Topology and the robot arm,” Acta Appl. Math. 11, 117–121, (1988).CrossRefGoogle Scholar
78.Bruce, J. W., Kirk, N. P. and Plessis, A. A. du, “Complete transversals and the classification of singularities,” Nonlinearity 10, 253–275, (1997).CrossRefGoogle Scholar
79.Wall, C. T. C., “Regular Stratifications,” In: Dynamical Systems, Warwick, 1974 (A. Manning ed.) Lecture Notes in Mathematics 468, (Springer, New York, 1975) pp. 332–344.CrossRefGoogle Scholar
80.Lipkin, H. and Pohl, E., “Enumeration of singular configurations for robotic manipulators,” Trans. ASME J. Mech. Design 113, 272–279, (1991).CrossRefGoogle Scholar
81.Gibson, C. G., Wirthmüller, K., Plessis, A. A. du and Looijenga, E. J. N., Topological Stability of Smooth Mappings, Lecture Notes in Mathematics 552 (Springer, Berlin, Germany, 1976).CrossRefGoogle Scholar
82.Cocke, M. W., Donelan, P. S. and Gibson, C. G., “Trajectory Singularities for a Class of Parallel Mechanisms,” Proceedings of the 8th International Workshop on Real and Complex Singularities, Luminy, France, (Birkhäuser, Basel, 2006) pp. 53–70.CrossRefGoogle Scholar
83.Nevins, J. L. and Whitney, D. E., “Assembly research,” Automation 16, 595–613, (1980).CrossRefGoogle Scholar
84.Whitney, D. E. and Nevins, J. L., “What is the RCC and what can it do?” In: Robot Sensors, Tactile and Non-Vision (A. Pugh ed.) (IFS Publications, Bedford, UK, 1986) pp. 3–15.Google Scholar
85.Cocke, M. W., Donelan, P. S. and Gibson, C. G., “Instantaneous Singular Sets Associated to Spatial Motions,” In: Real and Complex Singularities, São Carlos, 1998, (Tari, F. and Bruce, J. W., eds.) Research Notes in Mathematics 412 (Chapman and Hall/CRC Press, Boca Raton, 2000) pp. 147–163.Google Scholar