Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics

TL;DR

This paper develops a rigorous framework for assessing the optimality of sampling-based motion planning algorithms for systems with momentum, introducing an asymptotically optimal algorithm with convergence bounds for linear affine dynamics.

Contribution

It introduces the Differential Fast Marching Tree algorithm, proving its asymptotic optimality and providing convergence rate bounds for drift control systems with linear affine dynamics.

Findings

The algorithm is asymptotically optimal for linear affine systems.

Concrete bounds on convergence rate are established.

The framework applies to systems like double-integrators and aids future nonlinear system planning.

Abstract

In this paper we provide a thorough, rigorous theoretical framework to assess optimality guarantees of sampling-based algorithms for drift control systems: systems that, loosely speaking, can not stop instantaneously due to momentum. We exploit this framework to design and analyze a sampling-based algorithm (the Differential Fast Marching Tree algorithm) that is asymptotically optimal, that is, it is guaranteed to converge, as the number of samples increases, to an optimal solution. In addition, our approach allows us to provide concrete bounds on the rate of this convergence. The focus of this paper is on mixed time/control energy cost functions and on linear affine dynamical systems, which encompass a range of models of interest to applications (e.g., double-integrators) and represent a necessary step to design, via successive linearization, sampling-based and provably-correct algorithms for non-linear drift control systems. Our analysis relies on an original perturbation analysis for two-point boundary value problems, which could be of independent interest.