On the Ability of a Cable-Driven Robot to Generate a Prescribed Set of Wrenches

This paper presents a new geometry-based method to determine if a cable-driven robot operating in a $d$-degree-of-freedom workspace $(2≤d≤6)$ with $n≥d$ cables can generate a given set of wrenches in a given pose, considering acceptable minimum and maximum tensions in the cables. To this end, the fundamental nature of the available wrench set is studied. The latter concept, defined here, is closely related to similar sets introduced by Ebert-Uphoff and co-workers (2004, “Force-Feasible Workspace Analysis for Underconstrained, Point-Mass Cable Robots,” IEEE Trans. Rob. Autom., 5, pp. 4956–4962; 2007, “Workspace Optimization of a Very Large Cable-Driven Parallel Mechanism for a Radiotelescope Application,” Proceedings of the ASME IDETC/CIE Mechanics and Robotics Conference, Las Vegas, NV). It is shown that the available wrench set can be represented mathematically by a zonotope, a special class of convex polytopes. Using the properties of zonotopes, two methods to construct the available wrench set are discussed. From the representation of the available wrench set, computationally efficient and noniterative tests are presented to verify if this set includes the task wrench set, the set of wrenches needed for a given task.