Direct Geometrico-Static Problem of Underconstrained Cable-Driven Parallel Robots With Three Cables

This paper studies the direct geometrico-static problem of underconstrained cable-driven parallel robots with three cables. The task consists in determining the end-effector pose and the cable tensile forces when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled, and they must be tackled simultaneously. An effective elimination procedure is proposed and a least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem. This is proven to admit 156 solutions in the complex field. Several approaches for the efficient computation of the complete solution set are presented, including an eigenproblem formulation and homotopy continuation.