Dynamic interpolation for obstacle avoidance on Riemannian manifolds

ABSTRACT

This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem.

KEYWORDS:

• Lie groups
• interpolation problems
• trajectory planning
• Riemannian and Sub-Riemannian geometry
• obstacle avoidance

Acknowledgments

L.C. wishes to thank CMUC, Universidade de Coimbra for the hospitality received there where the main part of this work was developed. We appreciate comments provided by the reviewer which helped with the technical clarity of the work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of A. Bloch was supported by the NSF (Division of Mathematical Sciences) grant DMS-1613819 and AFOSR grant FA 9550-18-0028. The research of M. Camarinha was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2019 and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. L. Colombo was supported by the MINECO (Spain) grant MTM2016-76072-P and Juan de La Cierva Incorporación fellowship.