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 [23], 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 [32].
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 the vector fields predicted by this conjecture are given for SE(2) equipped with the left-invariant Riemannian (resp. minimum swept-volume) distance introduced in [35] (resp. [23]) 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.
Origine | Fichiers produits par l'(les) auteur(s) |
---|