A generic numerical continuation scheme for solving the direct kinematics of cable-driven parallel robot with deformable cables

Solving the direct kinematics (DK) of parallel robots, i.e. finding all the possible poses of the platform for given input values, is most of the time a difficult problem. This is evidently true for cable-driven parallel robots (CDPR) that are more complex than classical parallel robot because of the unilateral nature of the cable actions but also if cable deformations are taken into account. Furthermore there are many different deformable cable models and developing a DK algorithm for each of them will be a tremendous task. Consequently using a numerical continuation scheme that starts from the DK solutions for non-deformable cables and then moves toward the solution for deformable cables appears to be an interesting approach. To use this scheme we have to assume that the cable model relies on physical parameters that have limit values for which the deformable cable acts like a non-deformable one. Under that assumption we are then able to derive a possible maximum of solutions for the DK, whatever is the cable model. We then apply this approach for a specific complex cable model, the catenary case, to show that this approach is computer efficient but requires to address difficult theoretical issues in order to obtain a solving algorithm that is guaranteed to determine all DK solutions.