Etude et representation des lieux de singularite des manipulateurs parallelles spheriques a trois degres de liberte avec actionneurs prismatiques

Abstract

In this paper, an algorithm for the determination of the singularity loci of spherical three-degree-of-freedom parallel manipulators is presented. These singularity loci, which are obtained as curves or surfaces in the Cartesian space, are of great interest in the context of kinematic design. Indeed, it has been shown elsewhere that parallel manipulators lead to a special type of singularity which is located inside the Cartesian workspace and for which the end-effector becomes uncontrollable. This type of singularity is associated with a degeneracy of the direct kinematic problem. The manipulatord treated here are actuated with prismatic joints. Hence, the other type of singularity, for which the inverse kinematic problem degenerates, becomes trivial and will not be dealt with here. Finally, the third type of singularity that can be identified in the context of parallel manipulators, also termed architecture singularity, is assumed to be avoided by the introduction of a few simple constraints on the kinematic parameters which are readily available. The algorithm presented is based on analytical expressions of the determinant of a Jacobian matrix, a quantity that is known to vanish in the singular configurations. A general spherical three-degree-of-freedom parallel manipulator with prismatic actuators is first studied. Then, several particular designs are investigated. For each case, an analytical expression of the singularity locus is derived. A graphical representation in the Cartesian space is then obtained.