Advertisement

A Fast Branch-and-Prune Algorithm for the Position Analysis of Spherical Mechanisms

  • Arya ShabaniEmail author
  • Soheil SarabandiEmail author
  • Josep M. PortaEmail author
  • Federico ThomasEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)

Abstract

Different branch-and-prune schemes can be found in the literature for numerically solving the position analysis of spherical mechanisms. For the prune operation, they all rely on the propagation of motion intervals. They differ in the way the problem is algebraically formulated. This paper exploits the fact that spherical kinematic loop equations can be formulated as sets of 3 multi-affine polynomials. Multi-affinity has an important impact on how the propagation of motion intervals can be performed because a multi-affine polynomial is uniquely determined by its values at the vertices of a closed hyperbox defined in its domain.

Keywords

multi-affine polynomials parallel spherical robots forward kinematics position analysis branch-and-prune algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Raghavan and B. Roth, “Inverse kinematics of the general 6R manipulator and related linkages,” ASME Journal of Mechanical Design, Vol. 115, pp. 502-508, 1993.CrossRefGoogle Scholar
  2. 2.
    M.L.Husty, “An algorithm for solving the direct kinematics of general Stewart-Gough platforms,”’ Mechanism and Machine Theory, Vol. 31, No. 4, pp. 365-379, 1996.CrossRefGoogle Scholar
  3. 3.
    J.M. Porta and F. Thomas, “Yet another approach to the Gough-Stewart platform forward kinematics,” Proc. of the IEEE International Conference in Robotics and Automation, Brisbane, Australia, 2018.Google Scholar
  4. 4.
    C. Wampler, A. Morgan, and A. Sommese, “Numerical continuation methods for solving polynomial systems arising in kinematics,” ASME Journal Mechanical De- sign, Vol. 112, pp. 59-68, 1990.CrossRefGoogle Scholar
  5. 5.
    A. Castellet and F. Thomas, “An algorithm for the solution of inverse kinematics problems based on an interval method,” Advances in Robot Kinematics, M. Husty and J. Lenarcic (Eds.), pp. 393-403, Kluwer, 1998.Google Scholar
  6. 6.
    C.W. Wampler, “Displacement analysis of spherical mechanisms having three or fewer loops,” ASME Journal of Mechanical Design, Vol. 126, No. 1, pp. 93-100, 2004.CrossRefGoogle Scholar
  7. 7.
    S. Bai, M. R.Hansen, and J. Angeles, “A robust forward-displacement analysis of spherical parallel robots,” Mechanism and Machine Theory, Vol. 44, No. 12, pp. 2204-2216, 2009.CrossRefGoogle Scholar
  8. 8.
    C. Bomb´ın, L. Ros, and F. Thomas, “A concise Bezier clipping technique for solving inverse kinematics problems,” Advances in Robot Kinematics, J. Lenarcic and M. Stanisic (Eds.), pp.53-61, Kluwer, 2000.Google Scholar
  9. 9.
    C. Bomb´ın, L. Ros, and F. Thomas, “On the computation of the direct kinematics of parallel spherical mechanisms using Bernstein polynomials,” Proc. IEEE Int. Conf. on Robotics and Automation, Seoul, Korea, 2001.Google Scholar
  10. 10.
    E. Celaya, “Interval propagation for solving parallel spherical mechanisms,” Advances in Robot Kinematics, J. Lenarcic and F. Thomas (Eds.), pp. 415-422, Springer, 2002.Google Scholar
  11. 11.
    J. Gallier, Curves and Surfaces in Geometric Modeling: Theory And Algorithms, Morgan Kaufmann, 1999.Google Scholar
  12. 12.
    J.M. Porta, L. Ros, F. Thomas, and C. Torras, “A branch-and-prune solver for distance constraints,” IEEE Transactions on Robotics, Vol. 21, No. 2, pp. 176-187, 2005.CrossRefGoogle Scholar
  13. 13.
    A. Rikun, A convex envelope formula for multilinear functions, Journal of Global Optimization, Vol. 10, No. 4, pp.425-437, 1970.Google Scholar
  14. 14.
    R. Wagner, “Multi-linear interpolation,” Web document available at: http://rjwagner49.com/Mathematics/Interpolation.pdf, 2004.
  15. 15.
    X. Kong and C.M. Gosselin, “Type synthesis of 3-DOF spherical parallel manipulators based on Screw Theory,” ASME Journal Mechanical Design, Vol. 126, No. 1, pp. 101-108, 2004.CrossRefGoogle Scholar
  16. 16.
    J. Borr´as, R. Di Gregorio, “Direct position analysis of a large family of spherical and planar parallel manipulators with four loops,” Second International Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, pp. 19-29, 2008.Google Scholar
  17. 17.
    A. Morgan and V. Shapiro, “Box-bisection for solving second-degree systems and the problem of clustering,” ACM Transactions on Mathematical Software, Vol. 13, No. 2, pp. 152-167, 1987.MathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Sherbrooke and N. Patrikalakis, “Computation of the solutions of nonlinear polynomial systems,” Computer Aided Geometric Design, Vol. 10, No. 5, pp. 379-405, 1993.MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. Thomas, “Approaching dual quaternions from matrix algebra,” IEEE Transactions on Robotics, Vol. 30, No. 5, pp. 1037-1048, 2014.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Robotica i Informatica IndustrialCSIC-UPCBarcelonaSpain

Personalised recommendations