Dynamic interpolation for obstacle avoidance on Riemannian manifolds

TL;DR

This paper develops a framework for dynamic interpolation that ensures obstacle avoidance on Riemannian manifolds, deriving optimality conditions and applying them to various mechanical systems.

Contribution

It introduces a novel approach to obstacle avoidance using energy functionals on Riemannian manifolds, including Lie groups and sub-Riemannian settings, with explicit differential equations for optimal curves.

Findings

Derived first-order necessary conditions for optimality.

Applied the framework to rigid body and vehicle examples.

Demonstrated effectiveness through illustrative examples.

Abstract

This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing 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. We illustrate the results with examples of rigid bodies, both planar and spatial, and underactuated vehicles including a unicycle and an underactuated unmanned vehicle.