On the Boundaries of the Wrench-Closure Workspace of Planar Parallel Cable-Driven Mechanisms

The wrench-closure workspace of parallel cable-driven mechanisms is the set of poses for which any wrench can be produced at the end-effector by a set of a non-negative cable tensions. It is already known that the boundary of the constant-orientation wrench-closure workspace of a planar parallel cable-driven mechanism is composed of segments of conic sections. However, the relationship between the geometry of the mechanism and the types of these conic sections is unknown. This paper proposes a graphical method for determining the types of these conic sections. It is also shown that the proposed method can be applied to find the constant-orientation singularities of a 3-RPR planar parallel robot, since these contours correspond to the boundary segment of the analogous three-cable driven planar parallel mechanism.