Geodesic turnpikes for robot motion planning
Propriété de turnpike pour les géodésiques en planification de mouvement
Résumé
Endowing the configuration space of a robot with an appropriate metric structure and characterizing and computing the corresponding geodesics are central issues in motion planning. As recently observed in [1], the geodesics of SE(2) equipped with the so-called minimum swept-volume distance exhibit in practice a behavior akin to the turnpike property in optimal control, with transient phases separated by a longer steady state close to prototypical trajectories, the turnpikes [2]. This presentation gives a theoretical counterpoint to this empirical observation with a formal definition of geodesic turnpikes using vector fields on Finsler manifolds, a simple differential characterization of geodesics in the case where the manifold is a Lie group and the Finsler distance is left-invariant, and, in the case where the corresponding operator is also reversible, a conjecture characterizing the turnpikes by vector fields satisfying simple conditions in the corresponding Lie algebras. As a proof of concept, closed-form (resp. numerical) procedures for computing these vector fields according to this conjecture are given for SE(2) equipped with the left-invariant Riemannian (resp. minimum swept-volume) distance introduced in [3] (resp. [1]) for rectangular shapes. The solutions empirically match, in both cases, the observed turnpike behavior of the corresponding geodesics. In the minimum swept- volume distance case, using the turnpikes for initialization also yields an order-of-magnitude speedup in computing geodesics.
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