Reciprocal screw-based force-closure of an n-DOF open chain: Minimum number of cables required to fully constrain it

Due to the unilateral driving property of cables, it has been mathematically proven that a 6-DOF single rigid-bodied cable-driven platform requires a minimum of 7 cables with positive tension to fully constrain it. However, force-closure analysis of open chains driven by cables is still an open question. For the case of an n-DOF open chain driven by cables, two important questions arise: (i) Are n+1 cables sufficient to fully constrain the entire chain? (ii) How can force-closure analysis be carried out for a given cable routing configuration while retaining the geometric insights of the problem? This paper will address these issues and propose a systematic and novel approach based on the reciprocal screw theory. The analysis shows that the entire n-DOF open chain requires a minimum of n+1 cables to fully constrain it and the proposed methodology can analyze any cable routing configuration.