Visual synthesis of RRR- and RPR-legged planar parallel manipulators using constraint manifold geometry

In this paper, we present our ongoing work on development of a visual design tool for designing planar parallel manipulators that satisfy a given rational motion. Although in this paper, we have restricted ourselves to RRR- and RPR-type legs, the approach presented here is general enough to accommodate other leg topologies. The basic idea is to represent the kinematic constraint of such parallel manipulators as an algebraic manifold and the given motion as a one-parameter curve in the image space of planar displacements. The algebraic manifold is projected in the three-dimensional space and a simple set of relationships are obtained that couple the geometry of the projected manifold to the design parameters of the parallel manipulators. Simple geometric transformations in the projected space allow a user to visually contain the image curve inside the manifold, thus satisfying the kinematic constraints. This interactive process, at the end, gives the dimensions of the links of the legs and the location of the fixed and moving frames. This is an extension of our previous work on the dimensional synthesis of planar 6R closed chains.