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

arXiv: https://arxiv.org/abs/2108.03191 · HTML: https://mlsanity.com/paper/2108.03191/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2021.

## Built on by (most cited first, 1 of 1)

- [The Emergence of Deep Reinforcement Learning for Path Planning](https://mlsanity.com/paper/2507.15469/) (2025)

## Abstract (arXiv)

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≤ p≤ ∞, 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.