Symbolic computation for the Determination of the Minimal direct kinematics Polynomial and the Singular configurations of parallel manipulators

  • J.-P. Merlet
Conference paper


In this paper we will address the problems of parallel manipulator’s direct kinematics (i.e. find the position and orientation of the mobile plate as a function of the articular coordinates) and the determination of their singular configurations. We will show how symbolic computation has been used to determine the minimal degree of a polynomial formulation of the direct kinematic problem together with this polynomial and to calculate all the necessary and sufficient conditions which are to be satisfied by the position and orientation of the mobile plate to get a singular configuration.

We consider a 6 d.o.f. manipulator in the case where the mobile plate is a triangle and the links lengths are time-varying. Using geometrical considerations and symbolic computation we are able to show that an upper-bound of the maximum number of assembly-modes and thus the maximum number of solutions of the direct kinematics problem, is 16. We describe then how symbolic computation has been used to reduce the direct kinematics problem as the solution of a sixteenth order polynomial in one variable. Using a numerical procedure we show that this polynomial may have 16 real roots and we exhibit one example for which the maximum number of assembly mode is reached.

Then we consider shortly the problem of singular configuration which can be reduced to determine some special geometric conditions to be fulfilled by the link’s lines. We show how a geometric package can be used to determine necessary and sufficient conditions for the position and orientation of the mobile plate so that the manipulator is in a singular configuration.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Merlet J-P. 1987 (March).Parallel Manipulator, Part 1: Theory, Design, Kinematics and Control. INRIA Research Report n°646.Google Scholar
  2. [2]
    Gosselin. C. 1988. Kinematic analysis, optimization and programming of parallel robotic manipulators. Ph.D. thesis, McGill University, Montréal, Québec, Canada.Google Scholar
  3. [3]
    Merlet J-P. 1989 (February). Parallel Manipulator, Part 2: Singular configurations and Grassmann geometry. INRIA Research Report, n°791.Google Scholar
  4. [4]
    Nanua P., Waldron K.J. 1989 (May 14–19). Direct kinematic Solution of a Stewart Platform. IEEE Int. Conf. on Robotics and Automation, Scotsdale, Arizona, pp. 431–437.Google Scholar
  5. [5]
    Fichter E.F. A Stewart platform based manipulator: general theory and practical construction.The Int. J. of Robotics Research5(2):157-181.Google Scholar
  6. [6]
    Koliskor, 1986 (April 22–25). The I-coordinate approach to the industrial robot design. V IFAC/IFIP/IMACS/IFORS Symposium., Suzdal, USSR, pp. 108–115.Google Scholar
  7. [7]
    Charentus S., Renaud M. 1989 (July). Calcul du modèle géométrique direct de la plateforme de Stewart. Report n° 89260 LAAS, Toulouse, France.Google Scholar
  8. [8]
    Charentus S., Renaud M. 1989 (June 19–21). Modelling and control of a modular, redundant robot manipulator. 1st Int. Symp. on Experimental Robotics, Montréal, Canada.Google Scholar
  9. [9]
    Hunt K.H. Structural kinematics of in Parallel Actuated Robot Arms. Trans. of the ASME, J. of Mechanisms,Transmissions, and Automation in design (105): 705–712.Google Scholar
  10. [10]
    Reboulet C., Robert A. Hybrid control of a manipulator with an active compliant wrist. Proc 3th ISRR, Gouvieux, France, 7–11 Oct.1985, pp. 76–80.Google Scholar
  11. [11]
    Zamanov V.B, Sotirov Z.M. Structures and kinematics of parallel topology manipulating systems. Proc.Int. Symp. on Design and Synthesis, Tokyo, July 11–13 1984, pp. 453–458.Google Scholar
  12. [12]
    Mohamed M.G., Duffy J. A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators. Trans. of the ASME, J. of Mechanisms, Transmissions, and Automation in design, Vol 107: 226–229.Google Scholar
  13. [13]
    Hunt K.H. 1978. Kinematic geometry of mechanisms. Oxford: Clarendon Press.zbMATHGoogle Scholar
  14. [14]
    Merlet J-P. 1989 (December). Manipulateurs parallèles, 4eme partie: Mode d’assemblage et cinématique directe sous forme polynomiale. INRIA Research Report, n°1135.Google Scholar
  15. [15]
    Merlet J-P. 1989 (October). Parallel Manipulator, Part 2: Singular configurations of parallel manipulators and Grassmann geometry. The Int. J. of Robotics Research, Vol 8, n°5,pp. 45–56Google Scholar
  16. [16]
    Merlet J-P. 1989 (March). Manipulateurs parallèles. 3eme partie: Applications. INRIA Research Report n°1003.Google Scholar

Copyright information

© Springer-Verlag Wien 1991

Authors and Affiliations

  • J.-P. Merlet
    • 1
  1. 1.INRIA Centre de Sophia-AntipolisValbonne CedexFrance

Personalised recommendations