Path Planning in a Riemannian Manifold using Optimal Control

TL;DR

This paper explores motion planning on Riemannian manifolds by applying optimal control theory, specifically Pontryagin's principle, to compute geodesics and optimal controls for steering objects efficiently.

Contribution

It introduces a method to compute optimal controls and geodesic trajectories on Riemannian manifolds using Pontryagin's minimization principle and curvature analysis.

Findings

Optimal controls have unit norm, corresponding to arc length parametrization.

Geodesic curvature and Gaussian curvature are computed for the manifold.

Trajectories are shown to be extremal solutions of the Pontryagin principle.

Abstract

We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time t and shown to have norm 1 which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manifold. So we compute the geodesic curvature and the Gaussian curvature of the Riemannian structure.