Fast Marching Tree: a Fast Marching Sampling-Based Method for Optimal Motion Planning in Many Dimensions

In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM and RRT. The FMT algorithm performs a "lazy" dynamic programming recursion on a predetermined number of probabilistically-drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds--the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order $O(n^{{-1/d+\rho})$,} where $n$ is the number of sampled points, $d$ is the dimension of the configuration space, and $\rho$ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.