Path optimization using sub-Riemannian manifolds with applications to astrodynamics

Abstract

Differential geometry provides mechanisms for finding shortest paths in metric spaces. This work describes a procedure for creating a metric space from a path optimization problem description so that the formalism of differential geometry can be applied to find the optimal paths. Most path optimization problems will generate a sub-Riemannian manifold. This work describes an algorithm which approximates a sub-Riemannian manifold as a Riemannian manifold using a penalty metric so that Riemannian geodesic solvers can be used to find the solutions to the path optimization problem. This new method for solving path optimization problems shows promise to be faster than other methods, in part because it can easily run on parallel processing units. It also provides some geometrical insights into path optimization problems which could provide a new way to categorize path optimization problems. Some simple path optimization problems are described to provide an understandable example of how the method works and an application to astrodynamics is also given.