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.
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Notes
The cross-product matrix of r∈ℝ3 is defined as cpm(r)≡∂(r×x)/∂ x for any x∈ℝ3.
By convention, the minimum objective value of an infeasible problem is set to infinity.
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Acknowledgements
The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chair program as well as a FSR postdoctoral grant from the Université Catholique de Louvain.
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Appendix
Appendix
The constraints required to tighten the relaxation (SDR-1) are obtained from the “reformulation-linearization technique” (Sherali and Tuncbilek 1992). First consider the outer products
where, in this case, ≥ denotes the componentwise inequality between the left- and right-hand-side matrices. Upon substituting (7) into (16), we obtain the additional (convex) linear inequalities appearing in problem (SDR-2), namely,
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Collard, JF., Cardou, P. Computing the lowest equilibrium pose of a cable-suspended rigid body. Optim Eng 14, 457–476 (2013). https://doi.org/10.1007/s11081-012-9191-5
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DOI: https://doi.org/10.1007/s11081-012-9191-5