Etude et representation des lieux de singularite de la plate-forme de Gough-Stewart

Abstract

Dans ce document, les lieux de singularite des manipulateurs paralleles spatiaux a six degres de liberte avec actionneurs prismatiques--communement appeles plate-formes de Gough-Stewart--sont etudies. Une representation graphique de ces lieux dans l'espace de travail du manipulateur etudie est egalement obtenue. Tel que mentionne dans la litterature, trois types de singularite peuvent survenir et on s'interesse ici aux singularites de type II, c'est-a-dire celles pour lesquelles le nombre de solutions au probleme gemetrique direct degenere. La methode utilisee ici est basee sur l'expression analytique du determinant de la matrice jacobienne du manipulateur en question. Il est montre que pour une orientation donnee de la plate-forme, l'expression des lieux de singularite dans l'espace cartesien est representee par un polynome de degre quatre ou de degre trois. Un algorithme de travail est alors introduit afin de rendre systematiquement utilisable ce resultat en temps reel. Enfin, une comparaison avec une approche geometrique--la geometrie de Grassmann--est investiguee et des exemples illustrant l'interet de cette etude pour l'optimisation et la conception de manipulateurs paralleles sont presentes.

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In this paper, the singularity loci of six-degree-of-freedom spatial parallel manipulators with prismatic actuators--commonly called Gough-Stewart platforms--are studied. A graphical representation of these places in the workspace of the manipulator studied is also obtained. As mentioned in the literature, three types of singularities can arise and we are interested here in type II singularities, that is to say those for which the number of solutions to the direct geometric problem degenerates. The method used here is based on the analytical expression of the determinant of the Jacobian matrix of the manipulator in question. It is shown that for a given orientation of the platform, the expression of the singularity places in Cartesian space is represented by a polynomial of degree four or degree three. A working algorithm is then introduced in order to make this result systematically usable in real time. Finally, a comparison with a geometric approach--Grassmann geometry--is investigated and examples illustrating the interest of this study for the optimization and design of parallel manipulators are presented.