Direct Geometrico-static Problem of Underconstrained Cable-Driven Parallel Robots With <inline-formula> <tex-math notation="LaTeX">$n$</tex-math></inline-formula> Cables

Abstract

This paper studies the direct geometrico-static problem of underconstrained cable-driven parallel robots supported by $n$ cables, with $n\le 6$ . The task consists in identifying the equilibrium poses of the end-effector when cable lengths are specified. The problem is challenging, because the end-effector preserves some freedoms after cable lengths are assigned by motors. Hence, kinematics and statics are coupled, and they must be tackled simultaneously. A general method is presented to model the problem by a set of algebraic equations, and a least-degree univariate polynomial in the corresponding ideal is numerically found for any value of $n$ . For the efficient computation of the solution set, a software package is developed, which implements an algorithm based on homotopy continuation. Distinctive features of the code are that it finds all problem solutions, including those with slack cables, and stability analysis is integrated in order to identify feasible configurations.