Computing the lowest equilibrium pose of a cable-suspended rigid body

Abstract

We solve the problem of finding the lowest stable-equilibrium pose of a rigid body subjected to gravity and suspended in space by an arbitrary number of cables. Besides representing a contribution to fundamental rigid-body mechanics, this solution finds application in two areas of robotics research: underconstrained cable-driven parallel robots and cooperative towing. The proposed approach consists in globally minimizing the rigid-body potential energy. This is done by applying a branch-and-bound algorithm over the group of rotations, which is partitioned into boxes in the space of Euler-Rodrigues parameters. The lower bound on the objective is obtained through a semidefinite relaxation of the optimization problem, whereas the upper bound is obtained by solving the same problem for a fixed orientation. The resulting algorithm is applied to several examples drawn from the literature. The reported Matlab implementation converges to the lowest stable equilibrium pose generally in a few seconds for cable-robot applications. Interestingly, the proposed method is only mildly sensitive to the number of suspending cables, which is shown by solving an example with 1000 cables in two hours.