Direct Geometrico-Static Analysis of Under-Constrained Cable-Driven Parallel Robots with 4 Cables

This paper studies the direct geometrico-static problem of under-constrained parallel robots suspended by 4 cables. The task consists in determining the end-effector pose and the cable tensions when the cable lengths are assigned. The problem is challenging, because kinematics and statics are coupled and they must be solved simultaneously. An effective elimination procedure is presented that provides the complete solution set, thus proving that, when all cables are in tension, 216 potential solutions exists in the complex field. A least-degree univariate polynomial free of spurious factors is obtained in the ideal governing the problem and solutions are numerically computed via both an eigenvalue formulation and homotopy continuation. Equilibrium configurations with slack cables are also considered.