A Collaborative Solution Methodology for Inverse Position Problem

  • Chandra Sekhar Pedamallu
  • Linet Ozdamar
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 85)


In this study, a difficult advanced kinematics problem is presented and solved with a collaborative methodology that integrates an effective symbolic inference scheme and a local search method into a global interval partitioning algorithm. The resulting methodology proves to be very effective in discarding infeasible sub-spaces, because with some exceptions, the symbolic inference scheme guarantees to reduce infeasibility in each re-partitioning iteration. Thus, the local search method is called much less frequently as compared to a subdivision scheme without symbolic inference. Empirical results are obtained on two applications, the 6R inverse position and modified kinematics problems. The first test problem is quite difficult to solve as compared to the second one, however, our available commercial solvers were not able to solve any of them. The proposed collaborative methodology is generic and can handle any Constraint Satisfaction Problem where the goal might be to cover all solutions or identify a first feasible solution.

Key words

Inverse Kinematics Interval Analysis Symbolic-Interval cooperation Sequential Quadratic Programming 


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  1. Albala, H., and Angeles, J., 1979, Numerical Solution to the Input-Output displacement equation of the general 7R Spatial Mechanism, Proceedings of 5 th world congress on Theory of Machines and Mechanisms, pp. 1008–1011.Google Scholar
  2. Allgower, E. L., and Georg, K., 1980, Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Review. 22:28–85.MATHMathSciNetCrossRefGoogle Scholar
  3. Benhamou, F., and McAllester, D., and Van Hentenryck, P., 1994, CLP(Intervals) Revisited, Proceedings of ILPS’94, International Logic Programming Symposium, pp. 124–138.Google Scholar
  4. Benhamou, F., and Older, W. J., 1997, Applying Interval Arithmetic to Real, Integer and Boolean Constraints, Journal of Logic Programming. 32:1–24.MATHMathSciNetCrossRefGoogle Scholar
  5. Byrne, R. P., and Bogle, I. D. L., 1996, Solving Nonconvex Process optimization problems using Interval Subdivision Algorithms, Global Optimization in Engineering Design, I. E. Grossmann, ed., Kluwer Academic publisher, USA, pp. 155–173.Google Scholar
  6. Casado, L.G., Garcia, I., and Csendes, T., 2000, A new multi-section technique in interval methods for global optimization, Computing, 65:263–269.MATHMathSciNetCrossRefGoogle Scholar
  7. Ceberio, M., and Granvilliers, L., 2000, Solving Nonlinear Systems by Constraint Inversion and Interval Arithmetic, Lecture Notes in Artificial Intelligence. 1930:127–141.MathSciNetGoogle Scholar
  8. Chase, M. A., 1963, Vector Analysis, ASME Journal of Engineering for Industry. 85:289–297.Google Scholar
  9. Chow, S. N., Mallet-paret, J., and Yorke, J. A., 1979, A Homotopy Method for locating all zeros of a system of polynomials: In Functional Differential Equations and Approximation of Fixed Points, H.O. Peitgen, and H.O. Walther ed., Springer-Verlg Lecture Notes in Mathematics, Springer-Verlag, New York, pp. 228–237.Google Scholar
  10. Cleary, J. G., 1987, Logical Arithmetic, Future Computing Systems. 2:125–149.Google Scholar
  11. Coprin., 2004, http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/.Google Scholar
  12. Drexler, F. J., 1978, A homotopy method for the calculation of zeros of zero-dimensional polynomial ideals. In Continuation Methods, H.G. Wacker, ed., Academic Press, New York, pp. 69–93.Google Scholar
  13. Drud, S., 1995, CONOPT: A System for Large Scale Nonlinear Optimization, Tutorial for CON OPT Subroutine Library, ARKI Consulting and Development A/S, Bagsvaerd, Denmark.Google Scholar
  14. Duffy, J., and Crane, C, 1980, A displacement analysis of the general spatial 7R Mechanism, Mechanism and Machine Theory. 15:153–169.CrossRefGoogle Scholar
  15. Duffy, J., 1980, Analysis of Mechanisms and Robot Manipulators, John Wiley Publishers, New York.Google Scholar
  16. Faltings, B., 1994, Arc consistency for continuous variables, Artificial Intelligence. 65:363–376.MATHMathSciNetCrossRefGoogle Scholar
  17. Granvilliers, L., and Benhamou, F., 2001, Progress in the Solving of a Circuit Design Problem, J. Global Optim. 20:155–168.MathSciNetCrossRefGoogle Scholar
  18. Granvilliers, L., Monfroy, E., and Benhamou, F., 2001, Symbolic-interval cooperation in constraint programming, Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, London, Ontario, Canada.Google Scholar
  19. Granvilliers, L., 2004, An Interval Component for Continuous Constraints, J. Comput. Appl. Math. 162:79–92.MATHMathSciNetCrossRefGoogle Scholar
  20. Garcia, C. B., and Zangwill, W. I., 1979, Finding all solutions to polynomial systems and other systems of equations. Math. Program. 16:159–176.MATHMathSciNetCrossRefGoogle Scholar
  21. Hartenberg, R. S., and Denavit, J., 1964, Kinematic synthesis of linkages, McGraw-Hill, New York.Google Scholar
  22. Hansen, E., 1992, Global Optimization Using Interval Analysis, Marcel Dekker Inc, New York.MATHGoogle Scholar
  23. Hesse, R., 1973, A Heuristic Search Procedure for Estimating a Global Solution of Nonconvex Programming Problems, Op. Res. 21:1267.MATHCrossRefGoogle Scholar
  24. Kearfott, R. B., 1996, Test results for an interval branch and bound algorithm for equality constrained optimization, State of the art in global optimization, Kluwer, Dordrecht, pp. 181–199.Google Scholar
  25. Kohi, D., and Somi, A. H., 1975, Kinematic analysis of spatial mechanisms via successive screw displacements, ASME Journal of Engineering for Industry. 97:739–747.Google Scholar
  26. Knuppel, O., 1994, PROFIL/BIAS — A Fast Interval Library, Computing, 53:277–287.MathSciNetCrossRefGoogle Scholar
  27. Lawrence, C. T., Zhou, J. L., and Tits, A, L., 1997, User’s Guide for CFSQP version 2.5: A Code for Solving (Large Scale) Constrained Nonlinear (minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints, Institute for Systems Research, University of Maryland, College Park, MD.Google Scholar
  28. Lee, H.-Y., and Liang, C-G., 1988, Displacement analysis of the general spatial 7-link 7R mechanism, Mechanism and Machine Theory. 23:219–226.CrossRefGoogle Scholar
  29. Leykin, A., and Verschelde, J., 2004, PHCmaple; A Maple Interface to the Numerical Homotopy Algorithms in PHCpack, In Proceedings of the Tenth International Conference on Applications of Computer Algebra (ACA’2004), pp. 139–147.Google Scholar
  30. Lhomme, O., Gotlieb, A., and Rueher, M., 1998, Dynamic Optimization of Interval Narrowing Algorithms, Journal of Logic Programming. 37:165–183.MATHMathSciNetCrossRefGoogle Scholar
  31. Manocha, D., and Canny, J. F., 1994, Efficient inverse kinematics for general 6R manipulators. IEEE Journal on Robotics and Automation. 10:648–657.CrossRefGoogle Scholar
  32. Morgan, A., 1983a, A method for computing all solutions to Systems of polynomial equations, ACM Transactions on Mathematical Software. 2(1): 1–17.CrossRefGoogle Scholar
  33. Morgan, A., 1983b, Solving Systems of Polynomial Equations Using Homogenous Coordinates, GMResearch Publication GMR-4220.Google Scholar
  34. Markot, M. Cs., Fernandez, J., Casado, L. G., and T. Csendes, T., New interval methods for constrained global optimization, Conditionally accepted for publication in Mathematical Programming.Google Scholar
  35. Morgan, A., 1987a, Solving Polynomial Systems Using Continuation for Scientific and Engineering Problems. Prentice-Hall, Englewood Cliffs, New Jersey.MATHGoogle Scholar
  36. Morgan, A. P. 1987b, Computing All Solutions To Polynomial Systems Using Homotopy Continuation. Appl. Math. Comput. 24:115–138.MATHMathSciNetCrossRefGoogle Scholar
  37. Morgan, A. P., and Sommese, A. J., 1989, Coefficient-parameter polynomial continuation, Appl. Math. Comput. 29:23–160.MathSciNetCrossRefGoogle Scholar
  38. Morgan, A. P., Sommese, A. J., and Watson, L. T., 1989a, Finding all isolated solutions to polynomial systems using HOMPACK, ACM Trans. Math. Softw. 15:93–122.MATHMathSciNetCrossRefGoogle Scholar
  39. Neumaier, A., 1990, Interval Methods for Systems of Equations Encyclopedia of Mathematics and its Applications 37, Cambridge University Press, Cambridge.Google Scholar
  40. Panier, E. R., and Tits, A.L., 1993, On Combining Feasibility, Descent and Superlinear Convergence in Inequality Constrained Optimization, Math. Programming. 59:261–276.MATHMathSciNetCrossRefGoogle Scholar
  41. Pieper, D. L., 1968, The kinematics of Manipulators Under Computer Control. Ph.D Thesis, Standford University, USA.Google Scholar
  42. Pieper, D. L., and Roth, B., 1969, The kinematics of manipulators under computer control, Proceedings II-International congress on the Theory of Machines and Mechanisms. 2:159–168.Google Scholar
  43. Ratschek, H., and Rokne, J., 1988, New computer Methods for Global Optimization, John Wiley, New York.MATHGoogle Scholar
  44. Ratschek, H., and Rokne, J., 1995, Interval Methods, Handbook of Global Optimization, R. Horst and P.M. Pardalos, ed., Kluwer Academic publisher, Netherlands, pp. 751–828.Google Scholar
  45. Recio, T., and Gonzalex-Lopez, M. J., 1994, On the symbolic insimplification of the general 6R-manipulator kinematic equations, In Proceedings of the international symposium on Symbolic and algebraic computation, pp. 354–358.Google Scholar
  46. Roth, B., Rasteger, J., and Scheinman, V., 1973, On the design of computer controlled manipultors, First CISM-IFToMM Symposium, pp. 93–113.Google Scholar
  47. Sam-Haroud, D., and Fairings, B., 1996, Consistency techniques for continuous constraints, Constraints, 1:85–118.MathSciNetCrossRefGoogle Scholar
  48. Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.-H., and Nguyen, T.-V., 2002, Benchmarking Global Optimization and Constraint Satisfaction Codes, Global Optimization and Constraint Satisfaction: First International Workshop on Global Constraint Optimization and Constraint Satisfaction, COCOS 2002, Valbonne-Sophia Antipolis, France, 2002.Google Scholar
  49. Sommese, A. J., Verschelde, 1, and Wampler, C. W., 2002, Advances in Polynomial Continuation for solving problems in Kinematics, In Proc. ASME Design Engineering Technical Conf (CDROM), Montreal, Quebec.Google Scholar
  50. 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, Journal of Mechanisms, Transmissions, and Automation in Design. 107:189–200.CrossRefGoogle Scholar
  51. Uicker Jr., Denavir J. J., and Hartenberg R.S., 1964, An iterative method for the displacement analysis of spatial mechanisms, ASME Journal of Applied Mechanics. 31: 309–314.MATHGoogle Scholar
  52. Van-Hentenryck, P., Michel, L., and Deville, Y., 1997a, Numerica: a Modeling Language for Global Optimization, MIT press, London, England.Google Scholar
  53. Van-Hentenryck, P., Mc Allester, D., and Kapur, D., 1997b, Solving polynomial systems using branch and prune approach, SIAM Journal on Numerical Analysis. 34:797–827.MATHMathSciNetCrossRefGoogle Scholar
  54. Verschelde, J., Verlinden, P., and Cools., R., 1994, Homotopies Exploiting Newton Polytopes For Solving Sparse Polynomial Systems, SIAM Journal on Numerical Analysis. 31: 915–930.MATHMathSciNetCrossRefGoogle Scholar
  55. Wampler, C. and Morgan, A., 1991, Solving the 6R Inverse Position Problem using a Generic-Case Solution Methodology, Mech. Mach. Theory. 26:91–106.CrossRefGoogle Scholar
  56. Watson, L. T., 1979, A Global Convergent Algorithm for Computing Fixed points of C2 Maps, Applied Math. Comput. 5:297–311.MATHCrossRefGoogle Scholar
  57. Yang, A. T., and Freudenstein, F., 1964, Application of Dual number quaterian algebra to the analysis of spatial mechanisms, ASME journal of Applied mechanics. 86:300–308.MathSciNetGoogle Scholar
  58. Yuan, M.S.C., and Freudenstei, F., 1971, Kinematic Analysis of Spatial Mechanisms by means of Screw Coordinates (two parts), ASME Journal of Engineering for Industry. 93:61–73.Google Scholar
  59. Zhou, J. L., and Tits, A. L., 1996, An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems with Many Objective Functions, SIAM J. on Optimization. 6:461–487.MATHMathSciNetCrossRefGoogle Scholar
  60. Zhou, J. L., and Tits, A. L., 1993, Nonmonotone Line Search for Minimax Problems, J. Optim. Theory Appl. 76:455–476.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Chandra Sekhar Pedamallu
    • 1
  • Linet Ozdamar
    • 1
  1. 1.School of Mechanical and Production Engineering, Systems and Engineering Management DivisionNanyang Technological UniversitySingapore

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