Probabilistic motion planning for non-Euclidean and multi-vehicle problems

TL;DR

This paper extends the theoretical guarantees of sample-based motion planning algorithms like PRM* and RRT* to non-Euclidean and multi-vehicle systems, ensuring asymptotic optimality in complex state spaces.

Contribution

It introduces a flexible framework that proves asymptotic optimality of these algorithms in non-Euclidean and collaborative multi-vehicle settings, including a new joint state space construction.

Findings

Algorithms are asymptotically optimal in non-Euclidean spaces.

Framework applies to collaborative multi-vehicle systems.

Validated on coupled systems, fleets of Reeds-Shepp vehicles, and fractal spaces.

Abstract

Trajectory planning tasks for non-holonomic or collaborative systems are naturally modeled by state spaces with non-Euclidean metrics. However, existing proofs of convergence for sample-based motion planners only consider the setting of Euclidean state spaces. We resolve this issue by formulating a flexible framework and set of assumptions for which the widely-used PRM*, RRT, and RRT* algorithms remain asymptotically optimal in the non-Euclidean setting. The framework is compatible with collaborative trajectory planning: given a fleet of robotic systems that individually satisfy our assumptions, we show that the corresponding collaborative system again satisfies the assumptions and therefore has guaranteed convergence for the trajectory-finding methods. Our joint state space construction builds in a coupling parameter $1\leq p\leq \infty$, which interpolates between a preference for minimizing total energy at one extreme and a preference for minimizing the travel time at the opposite extreme. We illustrate our theory with trajectory planning for simple coupled systems, fleets of Reeds-Shepp vehicles, and a highly non-Euclidean fractal space.