We compute the lowest stable-equilibrium pose of a rigid body suspended in space by an arbitrary number of cables, being given the cable lengths and the attachment-point positions on the fixed frame and on the rigid body. This fundamental problem of mechanics if of interest in the fields of underconstrained cable-driven parallel robots and cooperative towing. The approach of the present work is very similar to one that is reported in a previous paper by the authors. Indeed, the problem is formulated as a potential energy minimization, and is solved using a branch-and-bound algorithm. Hence, we report mainly on improvements in the branching and bounding parts of the algorithm. In short, the idea is to search for the optimum rigid-body pose by partitioning only the rotation subgroup of rigid-body displacements. This is done here by dividing the four-dimensional space of Euler-Rodrigues parameters with polyhedral cones instead of boxes, the latter being normally used for this type of problem. The advantage is that cones conformbetter to the four-dimensional unit sphere of Euler-Rodrigues parameters. The convex relaxations of the original optimization problem are then adapted to the newly defined conical subsets. Besides resulting in a more elegant algorithm, this new conical branch-and-bound method leads to a higher efficiency in the case of reported examples.