Fast Marching Tree: a Fast Marching Sampling-Based Method for Optimal Motion Planning in Many Dimensions (1306.3532v4)

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 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.

• The paper introduces FMT as a sampling-based motion planning algorithm that converges asymptotically to optimal paths through lazy dynamic programming.
• It employs a method that reduces collision-check complexity to O(n), significantly lowering computational costs compared to similar algorithms.
• Empirical results demonstrate that FMT outperforms PRM* and RRT* in high-dimensional, obstacle-rich environments, ensuring faster convergence.

Fast Marching Tree: An Algorithmic Analysis for Optimal Motion Planning

This essay examines the paper on Fast Marching Tree (FMT), a probabilistic sampling-based motion planning algorithm designed to solve high-dimensional planning problems. The paper presents a thorough theoretical and empirical analysis of FMT, demonstrating its asymptotic optimality and computational efficiency in scenarios where traditional algorithms like PRM$^\star$ and RRT$\star$ might struggle.

Key Contributions

The paper introduces FMT as a novel approach that combines the strengths of both single-query and multiple-query frameworks. The algorithm performs a lazy dynamic programming recursion using probabilistically drawn samples to construct a tree of paths. This approach is particularly effective in high-dimensional spaces and scenarios where collision detection is costly.

Asymptotic Optimality and Convergence

FMT is rigorously proven to be asymptotically optimal, ensuring that as the number of samples increases, the cost of the solution approaches the optimal cost. The revolutionary aspect of FMT lies in its analysis under the notion of convergence in probability rather than almost sure convergence. This flexibility allows the derivation of convergence rate bounds—an advancement in the understanding of sampling-based motion planning.

The paper provides a specific convergence rate for certain parameters and configurations, denoted as $O(n^{-1/d + \rho})$, with $n$ representing the number of sampled points, $d$ the dimensionality, and $\rho$ an arbitrarily small constant.

Computational Complexity

FMT demonstrates an expected computational complexity of $O(n \log n)$, making it comparable to PRM$^\star$ and RRT$\star$ in terms of operations. However, FMT reduces the number of necessary obstacle collision checks to $O(n)$, a significant improvement over the $O(n \log n)$ checks performed by PRM$^\star$. This reduction becomes particularly valuable in obstacle-rich environments.

Numerical Experiments

Empirical results confirm the theoretical predictions, with FMT producing high-quality solutions more quickly than PRM$^\star$ and RRT$\star$ in complex, high-dimensional planning problems. Notably, FMT excels in environments with intricate obstacle arrangements where collision detection is expensive.

Extensions and Implementations

The paper explores several extensions:

• Non-Uniform Sampling: Adapts FMT for scenarios with non-uniform sampling distributions, maintaining asymptotic optimality with minor adjustments.
• General Cost Functions: Extends FMT to accommodate costs beyond simple Euclidean metrics, including line integral costs, potentially enhancing its applicability.
• $k$-Nearest Implementation: Proposes a variant using $k$-nearest neighbors rather than fixed radii, offering adaptability to varying obstacle densities.

Practical Implications and Future Directions

The implications of FMT are noteworthy for practical robotic planning applications, particularly in dynamic and uncertain environments. The algorithm's ability to adapt through extensions and maintain computational efficiency presents it as a robust choice for real-world applications.

Future research may focus on further optimizing FMT’s parameters, developing adaptive sampling strategies, and extending its capabilities to cope with dynamic environments and differential motion constraints.

In summary, FMT represents a significant advancement in optimal motion planning, offering both theoretical insights and practical efficiency. The balance between lazy collision-checking and directed dynamic programming recursion helps FMT achieve rapid convergence to high-quality solutions, positioning it as a valuable tool in the landscape of robotic motion planning.