The Dimensional Synthesis of Planar Parallel Cable-Driven Mechanisms Through Convex Relaxations

Abstract

The wrench-closure workspace (WCW) 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 positive cable tensions. In this paper, we tackle the dimensional synthesis problem, namely, that of finding a geometry for a planar parallel cable-driven mechanism (PPCDM) whose WCW contains a prescribed workspace. To this end, we first recall a linear program to determine whether a given pose is inside or outside the WCW of a given PPCDM. The relaxation of this linear program over a box leads to a nonlinear feasibility problem that can only be satisfied when this box is completely inside the WCW. We extend this feasibility problem to find a PPCDM geometry whose WCW includes a given set of boxes. These boxes represent the prescribed workspace or an estimate thereof, which may be obtained through interval analysis. Finally, we introduce a nonlinear program through which the PPCDM geometry is changed while maximizing the scaling factor of the prescribed set of boxes. When the optimum scaling factor is greater or equal to one, the WCW of the resulting PPCDM contains the set of boxes.