On the Force-Closure Analysis of n-DOF Cable-Driven Open Chains Based on Reciprocal Screw Theory

It has been mathematically proven that a completely restrained n- degree-of-freedom (n-DOF) single rigid-bodied cable-driven platform requires a minimum of n + 1 cables with positive tension to fully constrain it. However, the force-closure analysis of open chains that are driven by cables is still an open question. For the case of an n -DOF cable-driven open chain, the following two important questions arise. 1) How can the force-closure analysis be carried out for a given cable routing configuration, while retaining the geometric insights of the problem? 2) Are n + 1 cables sufficient to fully constrain the entire chain? This paper addresses these issues by proposing a systematic and novel approach based on the reciprocal screw theory. The key idea is to express wrenches acting on the open chain as linear combinations of the reciprocal screws and determine the total required torques at each joint. This is followed by equating the joint torques that are provided by the cable forces with the joint torques, which are required by the external wrenches, and checking for force closure. The proposed methodology can analyze open chains with arbitrary cable routing configuration. The analysis shows that the entire n-DOF open chain requires a minimum of n + 1 cables to fully constrain it.