The Critical Radius in Sampling-based Motion Planning

TL;DR

This paper introduces a new analysis of sampling-based motion planning, identifying a critical connection radius that guarantees success and near-optimality with fewer connections, improving upon previous results.

Contribution

It establishes a precise critical radius for sampling-based planners and demonstrates near-optimality with only a constant number of neighbors, extending analysis to deterministic and sparsified samples.

Findings

Critical radius proportional to n^{-1/d} guarantees planner success.

Near-optimality achieved with Θ(1) neighbors per sample.

Analysis applies to various PRM-based planners, including RRG, FMT*, and BTT.

Abstract

We develop a new analysis of sampling-based motion planning in Euclidean space with uniform random sampling, which significantly improves upon the celebrated result of Karaman and Frazzoli (2011) and subsequent work. Particularly, we prove the existence of a critical connection radius proportional to ${\Theta(n^{-1/d})}$ for $n$ samples and ${d}$ dimensions: Below this value the planner is guaranteed to fail (similarly shown by the aforementioned work, ibid.). More importantly, for larger radius values the planner is asymptotically (near-)optimal. Furthermore, our analysis yields an explicit lower bound of ${1-O( n^{-1})}$ on the probability of success. A practical implication of our work is that asymptotic (near-)optimality is achieved when each sample is connected to only ${\Theta(1)}$ neighbors. This is in stark contrast to previous work which requires ${\Theta(\log n)}$ connections, that are induced by a radius of order ${\left(\frac{\log n}{n}\right)^{1/d}}$. Our analysis is not restricted to PRM and applies to a variety of PRM-based planners, including RRG, FMT* and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for all the aforementioned planners when constructed with deterministic samples, which are then sparsified in a randomized fashion. We believe that this new model, and its analysis, is interesting in its own right.