On the characterization of the regions of feasible trajectories in the workspace of parallel manipulators

It was shown recently that parallel manipulators with several inverse kinematic solutions have the ability to avoid parallel singularities [Chablat 1998a] and self-collisions [Chablat 1998b] by choosing appropriate joint configurations for the legs. In effect, depending on the joint configurations of the legs, a given configuration of the end-effector may or may not be free of singularity and collision. Characterization of the collision/singularity-free workspace is useful but may be insufficient since two configurations can be accessible without collisions nor singularities but it may not exist a feasible trajectory between them. The goal of this paper is to define the maximal regions of the workspace where it is possible to execute trajectories. Two different families of regions are defined : 1. those regions where the end-effector can move between any set of points, and 2. the regions where any continuous path can be tracked. These regions are characterized from the notion of aspects and free-aspects recently defined for parallel manipulators [Chablat 1998b]. The construction of these regions is achieved by enrichment techniques and using an extension of the octree structures to spaces of dimension greater than three. Illustrative examples show the interest of this study to the optimization of trajectories and the design of parallel manipulators.