The determination of nonconvex workspaces of generally constrained planar stewart platforms

An extension of a novel optimization approach to the determination of accessible output sets of planar manipulators is presented. This approach provides a general method for the determination of workspaces and has the advantage that it may be easily automated. The method consists of finding a suitable initial radiating point interior to the accessible output set of the manipulator, and then determining the points of intersection of a representative pencil of rays, emanating from this point, with the boundary of the accessible set. The points of intersection are determined by means of an optimization approach in which the dynamic constrained optimization algorithm of Snyman is used. If any segment of the workspace boundary cannot be determined due to nonconvexity, then the missing segment is mapped using a suitably chosen new radiating point. The method is illustrated by its application to the planar Stewart platform. The effects of leg length limits, joint angle limits, leg interference, and singular configurations are considered in the formulation. The method has been implemented in a practical, interactive computer code that has been used to determine convex and nonconvex workspaces of different Stewart platforms of arbitrary geometry.