The Critical Radius in Sampling-based Motion Planning

We develop a new analysis of sampling-based motion planning 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 (AO). 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 AO 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 RGG, FMT$^*$ and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for 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.