On the Stability of the Quadruple Solutions of the Forward Kinematic Problem in Analytic Parallel Robots

  • Adrián Peidró
  • Arturo Gil
  • José María Marín
  • Luis Payá
  • Óscar Reinoso
Article

Abstract

Many parallel robots can change between different assembly modes (solutions of the forward kinematic problem) without crossing singularities, either by enclosing cusps or alpha-curves of the planar sections of their singularity loci. Both the cusps and the alpha-curves are stable singularities, which do not disappear under small perturbations of the geometry of the robot. Recently, it has been shown that some analytic parallel robots can also perform these nonsingular changes of assembly mode by encircling isolated points of their singularity loci at which the forward kinematic problem admits solutions with multiplicity four. In this paper, we study the stability of these quadruple solutions when the design of the robot deviates from the analytic geometry, and we show that such quadruple solutions are not stable since the isolated singular points at which they occur degenerate into closed deltoid curves. However, we also demonstrate that, although the quadruple solutions are unstable, the behavior of the robot when moving around them is practically unaffected by the perturbations from the analytic geometry. This means that the robot preserves its ability to perform nonsingular transitions by enclosing the quadruple solutions, even when its geometry is not exactly analytic due to small manufacturing tolerances.

Keywords

Parallel robots Singularities Forward kinematics Quadruple solutions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bamberger, H., Wolf, A., Shoham, M.: Assembly mode changing in parallel mechanisms. IEEE Trans. Robot. 24(4), 765–772 (2008)CrossRefGoogle Scholar
  2. 2.
    Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W: Bertini: Software for Numerical Algebraic Geometry. Available at bertini.nd.edu with permanent doi:10.7274/R0H41PB5
  3. 3.
    Caro, S., Wenger, P., Chablat, D.: Non-Singular Assembly Mode Changing Trajectories of a 6-DOF Parallel Robot. In: The ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, pp 1–10 (2012)Google Scholar
  4. 4.
    Coste, M.: Asymptotic singularities of planar parallel 3-RPR manipulators. In: Lenarcic, J., Husty, M. (eds.) Latest Advances in Robot Kinematics, pp 35–42. Springer, Netherlands (2012)Google Scholar
  5. 5.
    Coste, M.: A simple proof that generic 3-R,PR manipulators have two aspects. J. Mech. Robot. 4 (1), 011,008 (2012)CrossRefGoogle Scholar
  6. 6.
    Coste, M., Chablat, D., Wenger, P.: New Trends in Mechanism and Machine Science: Theory and Applications in Engineering, Chap. Perturbation of Symmetric 3-RPR Manipulators and Asymptotic Singularities, pp 23–31. Springer, Netherlands (2013)CrossRefGoogle Scholar
  7. 7.
    Coste, M., Chablat, D., Wenger, P.: Nonsingular Change of Assembly Mode without Any Cusp. In: Lenarčič, J., Khatib, O. (eds.) Advances in robot kinematics, pp 105–112. Springer International Publishing, Switzerland (2014)Google Scholar
  8. 8.
    DallaLibera, F., Ishiguro, H.: Non-singular transitions between assembly modes of 2-DOF planar parallel manipulators with a passive leg. Mech. Mach. Theory 77, 182–197 (2014)CrossRefGoogle Scholar
  9. 9.
    Gosselin, C. M., Merlet, J. P.: The direct kinematics of planar parallel manipulators: special architectures and number of solutions. Mech. Mach. Theory 29(8), 1083–1097 (1994)CrossRefGoogle Scholar
  10. 10.
    Hernández, A., Altuzarra, O., Petuya, V., Macho, E.: Defining conditions for nonsingular transitions between assembly modes. IEEE Trans. Robot. 25(6), 1438–1447 (2009)CrossRefGoogle Scholar
  11. 11.
    Husty, M.: Non-Singular Assembly Mode Change in 3-RPR-Parallel Manipulators. In: Kecskeméthy, A., Müller, A. (eds.) Computational Kinematics, pp 51–60. Springer, Berlin (2009)Google Scholar
  12. 12.
    Husty, M., Schadlbauer, J., Caro, S., Wenger, P.: The 3-RPS Manipulator Can have Non-Singular Assembly-Mode Changes. In: Thomas, F., Pérez Gracia, A. (eds.) Computational Kinematics, Mechanisms and Machine Science, vol. 15, pp 339–348. Springer, Netherlands (2014)Google Scholar
  13. 13.
    Innocenti, C., Parenti-Castelli, V.: Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators. J. Mech. Design 120(1), 73–79 (1998)CrossRefGoogle Scholar
  14. 14.
    Kong, X., Gosselin, C. M.: Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators. Mech. Mach. Theory 36(9), 1009–1018 (2001)MATHCrossRefGoogle Scholar
  15. 15.
    Macho, E., Altuzarra, O., Pinto, C., Hernández, A.: Transitions between Multiple Solutions of the Direct Kinematic Problem. In: Lenarčič, J., Wenger, P. (eds.) Advances in robot kinematics: analysis and design, pp 301–310. Springer, Netherlands (2008)Google Scholar
  16. 16.
    McAree, P., Daniel, R.: An explanation of never-special assembly changing motions for 3-3 parallel manipulators. Int. J. Robot. Res. 18(6), 556–574 (1999)CrossRefGoogle Scholar
  17. 17.
    Moroz, G., Chablat, D., Wenger, P., Rouiller, F.: Cusp points in the parameter space of RPR-2PRR parallel manipulators. In: Pisla, D., Ceccarelli, M., Husty, M., Corves, B. (eds.) New Trends in Mechanism Science, Mechanisms and Machine Science, vol. 5, pp 29–37. Springer, Netherlands (2010)Google Scholar
  18. 18.
    Moroz, G., Rouiller, F., Chablat, D., Wenger, P.: On the determination of cusp points of 3-RPR parallel manipulators. Mech. Mach. Theory 45(11), 1555–1567 (2010)MATHCrossRefGoogle Scholar
  19. 19.
    Peidró, A., Marín, J., Gil, A., Reinoso, O.: Performing nonsingular transitions between assembly modes in analytic parallel manipulators by enclosing quadruple solutions. ASME Journal of Mechanical Design 137(12), 122,302 (2015)CrossRefGoogle Scholar
  20. 20.
    Peidró, A., Reinoso, O., Gil, A., Marín, J., Payá, L.: A virtual laboratory to simulate the control of parallel robots. IFAC-PapersOnLine 48(29), 19–24 (2015)CrossRefGoogle Scholar
  21. 21.
    Thomas, F., Wenger, P.: On the Topological Characterization of Robot Singularity Loci. A Catastrophe-Theoretic Approach. In: Proceedings of the 2011 IEEE International Conference on Robotics and Automation, pp 3940–3945 (2011)Google Scholar
  22. 22.
    Urízar, M., Petuya, V., Altuzarra, O., Hernández, A.: Researching into Non-Singular Transitions in the Joint Space. In: Lenarcic, J., Stanisic, M.M. (eds.) Advances in Robot Kinematics: Motion in Man and Machine, pp 45–52. Springer, Netherlands (2010)Google Scholar
  23. 23.
    Urízar, M., Petuya, V., Altuzarra, O., Hernández, A.: Assembly mode changing in the cuspidal analytic 3-RPR. IEEE Trans. Robot. 28(2), 506–513 (2012)CrossRefGoogle Scholar
  24. 24.
    Zein, M., Wenger, P., Chablat, D.: Singular curves in the joint space and cusp points of 3-RPR parallel manipulators. Robotica 25(6), 717–724 (2007)CrossRefGoogle Scholar
  25. 25.
    Zein, M., Wenger, P., Chablat, D.: Non-singular assembly-mode changing motions for 3-RPR parallel manipulators. Mech. Mach. Theory 43(4), 480–490 (2008)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Adrián Peidró
    • 1
  • Arturo Gil
    • 1
  • José María Marín
    • 1
  • Luis Payá
    • 1
  • Óscar Reinoso
    • 1
  1. 1.Systems Engineering and Automation DepartmentMiguel Hernández UniversityElcheSpain

Personalised recommendations