On the forward kinematics of cable-driven parallel robots

Most cable-driven parallel robots are kinematically over-constrained mechanisms. This results in a non-trivial computation of the forward kinematic transformation. It is well known that the forward kinematics of parallel robots may have multiple solutions and in general the convergence of numerical methods is unknown. In recent works, it was proposed to formulate the forward kinematics as optimization problem that models the cables as linear springs in order to compute the platform pose which has minimal potential energy in the cables. In this paper, we analyzed this objective function. Using the Hessian matrix, we show that under certain conditions the problem at hand is convex and we can expect a unique and stable minimum. The computations are exemplified for point-shaped platforms as well as for the planar case. For the spatial case, we present an encouraging numerical study. An ordinary least squares method is then applied to find a position approximation and an improvement to previous methods is demonstrated.