Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions

Abstract

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 such, this algorithm combines features of both single-query algorithms (chiefly RRT) and multiple-query algorithms (chiefly PRM), and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. 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+ρ), where n is the number of sampled points, d is the dimension of the configuration space, and ρ 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 the-oretical 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.

Effective Metrics for Multi-Robot Motion-Planning

“…In this work we assessed metrics using RRT-style planners, such as dRRT (see Section III). Although we do not believe that our reported results are biased towards these specific types of planners, it would be interesting to see whether the conclusions can be reproduced for other planners, that operate differently than RRT, e.g., PRM*, RRT* [27] and FMT* [25]. This also leads to the question of the effect metrics have on the quality of the solution in MRMP.…”

The VFO-Driven Motion Planning and Feedback Control in Polygonal Worlds for a Unicycle with Bounded Curvature of Motion

“…Recently, we observe fast development of sampling-based motion planners building upon the concept of RRT (Rapidly Random exploring Trees) [11,15,17,20,23,30] and other closely related probabilistic approaches such as [13]. Their controller-driven variants are usually obtained by integration of a specific extend procedure into the planner.…”

“…Proof One can obtain relation (17), corresponding to the curvature of integral curve f i+1 i y i+1 , p i by considering (3) under nominal conditions as follows:…”

“…given by (7), substituting θ i+1 = θ i+1 ai (upon the assumption of nominal conditions) and x i+1 = f i+1 i y i+1 , p i (by virtue of Property 1) into the resulting expression, one applies some algebra to obtain (17).…”

Optimal randomized path planning for redundant manipulators based on Memory-Goal-Biasing

“…In response to this challenge, researchers have presented sampling-based algorithms that are capable of realizing high-dimensional path planning, [3][4][5][6][7][8] such as probabilistic roadmaps, Rapidly-exploring Random Tree (RRT), and their variants. Despite their feasibility for probabilistic completeness, it is difficult to achieve an expected balance between optimization performance and planning efficiency.…”