Skip to main content
Log in

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

  • Published:
Journal of Intelligent & Robotic Systems Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Bamberger, H., Wolf, A., Shoham, M.: Assembly mode changing in parallel mechanisms. IEEE Trans. Robot. 24(4), 765–772 (2008)

    Article  Google Scholar 

  2. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W: Bertini: Software for Numerical Algebraic Geometry. Available at with permanent doi:10.7274/R0H41PB5

  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)

  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)

  5. Coste, M.: A simple proof that generic 3-R,PR manipulators have two aspects. J. Mech. Robot. 4 (1), 011,008 (2012)

    Article  Google Scholar 

  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)

    Book  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Adrián Peidró.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peidró, A., Gil, A., Marín, J.M. et al. On the Stability of the Quadruple Solutions of the Forward Kinematic Problem in Analytic Parallel Robots. J Intell Robot Syst 86, 381–396 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: