On the inverse kinematics of cable-driven parallel robots with up to 6 sagging cables

We consider in this paper cable driven parallel robot (CDPR) with m catenary cables that present a significant deformation due to their elasticity and own mass and we address the inverse kinematics (IK) of such a robot i.e. determining the lengths at rest of the cables for a given pose of the platform. We show that the IK requires to solve a square system of equations if the robot has up to 6 cables and an underconstrained system if the number of cables is larger than 6. In this paper we address the problem of finding all solutions of the IK when the number of cables is lower or equal to 6. We propose an algorithm that allows one to determine all possible solutions. for any number of cables. We illustrate the use of the IK solver on a CDPR with 6 cables and shows that the IK may have no solution even if the pose lie within the envelope of the base points or may have several solutions, in which case some cables may have a length at rest that is much larger than the distance between the anchor points of the cable.