The constant-orientation dimensional synthesis of planar cable-driven parallel mechanisms through convex relaxations

Abstract

The wrench-closure workspace (WCW) of cable-driven parallel 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 of finding a geometry for a planar cable-driven parallel mechanism (PCDPM) whose constant orientation wrench closure workspace (COWCW) contains a prescribed workspace. To this end, we first introduce a linear program to verify whether a given pose is inside or outside the WCW of a given PCDPM. 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 COWCW. We extend this feasibility problem to find a PCDPM geometry whose COWCWs include a given set of boxes. These multiple boxes may represent an estimate of the prescribed workspace, which may be obtained through interval analysis. Finally, we introduce a nonlinear program through which the PCDPM 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 COWCW of the resulting PCDPM contains the set of boxes. Otherwise, the COWCW generally offers a good coverage of the set of boxes.