0
TECHNICAL PAPERS

Advances in Polynomial Continuation for Solving Problems in Kinematics

[+] Author and Article Information
Andrew J. Sommese

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618e-mail: sommese@nd.edu, URL: http://www.nd.edu/∼sommese

Jan Verschelde

Department of Mathematics, Statistics, and Computer Science, 851 S. Morgan St. (MC 249) University of Illinois at Chicago, Chicago, IL 60607-7045e-mail: jan@math.uic.edu, URL: http://www.math.uic.edu/∼jan

Charles W. Wampler

General Motors R&D Center, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055e-mail: charles.w.wampler@gm.com

J. Mech. Des 126(2), 262-268 (May 05, 2004) (7 pages) doi:10.1115/1.1649965 History: Received July 01, 2002; Revised February 01, 2003; Online May 05, 2004
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.

References

Roth,  B., and Freudenstein,  F., 1963, “Synthesis of Path-generating Mechanisms by Numerical Methods,” ASME J. Eng. Ind., 85B-3, pp. 298–306.
Roth,  B., and Freudenstein,  F., 1963, “Numerical Solution of Systems of Nonlinear Equations,” J. Assoc. Comput. Mach., 10, pp. 550–556.
Tsai,  L.-W., and Morgan,  A. P., 1985, “Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods,” ASME J. Mech. Des., 107, pp. 189–200.
Morgan, A. P., 1987, Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems, Prentice-Hall, Englewood Cliffs, NJ.
Wampler,  C. W., Morgan,  A. P., and Sommese,  A. J., 1990, “Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics,” ASME J. Mech. Des., 112, pp. 59–68.
Li,  T. Y., 1997, “Numerical Solution of Multivariate Polynomial Systems by Homotopy Continuation Methods,” Acta Numerica, 6, pp. 399–436.
Raghavan,  M., 1993, “The Stewart Platform of General Geometry has 40 Configurations,” ASME J. Mech. Des., 115, pp. 277–282, June.
Raghavan,  M., and Roth,  B., 1995, “Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators,” ASME J. Mech. Des., 117(B), pp. 71–79.
Waldron,  K. J., and Sreenivasen,  S. V., 1996, “A Study of the Solvability of the Position Problem for Multi-Circuit Mechanisms by Way of Example of the Double Butterfly Linkage,” ASME J. Mech. Des., 118(3), pp. 390–395.
Wampler,  C. W., Morgan,  A. P., and Sommese,  A. J., 1992, “Complete Solution of the Nine-point Path Synthesis Problem for Four-bar Linkages,” ASME J. Mech. Des., 114, pp. 153–159.
Morgan,  A. P., Sommese,  A. J., and Watson,  L. T., 1989, “Finding All Isolated Solutions to Polynomial Systems Using HOMPACK,” ACM Trans. Math. Softw., 15, pp. 93–122.
Verschelde,  J., 1999, “Algorithm 795: PHCpack: A General-purpose Solver for Polynomial Systems by Homotopy Continuation,” ACM Trans. Math. Softw., 25(2), pp. 251–276. Software available at http://www.math.uic.edu/∼jan.
Sommese,  A. J., Verschelde,  J., and Wampler,  C. W., 2001, “Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 38(6), pp. 2022–2046.
Sommese,  A. J., and Verschelde,  J., 2000, “Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets,” Journal of Complexity, 16(3), pp. 572–602.
Sommese, A. J., Verschelde, J., and Wampler, C. W., 2001, “Numerical Irreducible Decomposition Using Projections from Points on the Components,” in E. L. Green, S. Hosten, R. C. Laubenbacher, and V. Powers, ed., Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, vol. 286 of Contemporary Mathematics, pp. 37–51. Amer. Math. Soc.
Sommese, A. J., Verschelde, J., and Wampler, C. W., 2001, “Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components,” C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher, eds, Application of Algebraic Geometry to Coding Theory, Physics and Computation, pp. 297–315. Kluwer Academic Publishers.
Sommese,  A. J., Verschelde,  J., and Wampler,  C. W., 2002, “Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 40(6), pp. 2026–2046.
Sommese, A. J., and Wampler, C. W., 1996, “Numerical Algebraic Geometry,” J. Renegar, M. Shub, and S. Smale, eds, The Mathematics of Numerical Analysis, Vol. 32 of Lectures in Applied Mathematics, pp. 749–763. Amer. Math. Soc.
Sommese, A. J., Verschelde, J., and Wampler, C. W., 2003, “Numerical Irreducible Decomposition Using PHCpack,” M. Joswig, and N. Takayama, eds, Algebra, Geometry and Software Systems, pp. 109–130, Springer-Verlag.
Wampler,  C. W., 1999, “Solving the Kinematics of Planar Mechanisms,” ASME J. Mech. Des., 121, pp. 387–391.
Innocenti,  C., 1995, “Polynomial Solution to the Position Analysis of the 7-link Assur Kinematic Chain with One Quaternary Link,” Mech. Mach. Theory, 30(8), pp. 1295–1303.
Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland, Amsterdam.
Lazard, D., 1992, “Stewart Platform and Gröbner Basis,” Proc. ARK, pp. 136–142, Ferrare, September.
Mourrain, B., 1993, “The 40 Generic Positions of a Parallel Robot,” Proc. ISSAC’93, pp. 173–182, Kiev (Ukraine), July, ACM press.
Ronga, F., and Vust, T., 1992 “Stewart Platforms without Computer?” Proc. Conf. Real Analytic and Algebraic Geometry, Trento, pp. 197–212.
Husty,  M. L., 1996, “An Algorithm for Solving the Direct Kinematics of General Stewart-Gough Platforms,” Mech. Mach. Theory, 31(4), pp. 365–380.
Wampler,  C. W., 1996, “Forward Displacement Analysis of General Six-in-Parallel SPS (Stewart) Platform Manipulators Using Soma Coordinates,” Mech. Mach. Theory, 31(3), pp. 331–337.
Griffis, M., and Duffy, J., 1993, “Method and Apparatus for Controlling Geometrically Simple Parallel Mechanisms with Distictive Connections,” US Patent 5,179,525.
Husty, M. L., and Karger, A., 2000, “Self-motions of Griffis-Duffy Type Parallel Manipulators,” Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, CA, April.
Innocenti,  C., 1995, “Polynomial Solution of the Spatial Burmester Problem,” ASME J. Mech. Des., 117(1), pp. 64–68, March.
Morgan,  A. P., and Sommese,  A. J., 1987, “A Homotopy for Solving General Polynomial Systems that Respects m-homogeneous Structures,” Appl. Math. Comput., 24, pp. 101–113.

Figures

Grahic Jump Location
Top: Find all possible assemblies of these pieces into a 7-bar mechanism. Bottom: One such assembly.
Grahic Jump Location
Griffis-Duffy platform. Both base and endplate are equilateral triangles.
Grahic Jump Location
Assemblies of a seven-bar linkage: (a) a solution curve of degree six and (b) one of six isolated solutions.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In