Variational Obstacle Avoidance with Applications to Interpolation Problems in Hybrid Systems

TL;DR

This paper develops a variational framework for obstacle avoidance on Riemannian manifolds and applies it to interpolate points in hybrid systems, ensuring safety and optimality.

Contribution

It introduces a novel variational approach for obstacle avoidance and interpolation in hybrid systems, with rigorous existence proofs and safety conditions.

Findings

Derived dynamical equations for obstacle avoidance extrema.

Proved existence of minimizers using weak convergence arguments.

Provided conditions for safe obstacle avoidance within tolerance.

Abstract

We study variational obstacle avoidance problems on complete Riemannian manifolds and apply the results to the construction of piecewise smooth curves interpolating a set of knot points in systems with impulse effects. We derive the dynamical equations for extrema in the variational problem, and show the existence of minimizers by using lower-continuity arguments for weak convergence on an infinite-dimensional Hilbert manifold. We then provide conditions under which it is possible to ensure that the extrema will safely avoid a given obstacle within some desired tolerance.