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

Authors: Lucas Janson, Edward Schmerling, Ashley Clark, Marco Pavone

Published: 2013 (Journal Paper)

Source: The International Journal of Robotics Research (IJRR)

Algorithm: FMT*

arXiv: 1306.3532

DOI: 10.1177/0278364915577958

Summary

FMT* operates on a fundamentally different mechanism than RRT*, yet still achieves asymptotic optimality. FMT* can be faster than RRT* in certain planning regimes, and it is more amenable to parallelization (c.f. Group Marching Tree).

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+r), where n is the number of sampled points, d is the dimension of the configuration space, and r 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.

Primary

Paper arxiv.org

Standard

arXiv Abstract 1306.3532 arXiv PDF 1306.3532 arXiv HTML 1306.3532 DOI 10.1177/0278364915577958

Alternate

web.stanford.edu PDF web.stanford.edu

Tags

  • Motion planning

  • Optimal motion planning

  • Sampling-based planning

  • Asymptotic optimality

  • Fast marching tree

  • FMT

  • FMT*

  • RRT*

  • RRT

  • Graph search

  • Dynamic programming

  • Fast marching method