Asymptotically Optimal Path Planning With an Approximation of the Omniscient Set

TL;DR

The paper tackles slow convergence in asymptotically optimal RRT*-based planning by approximating the omniscient set with two complementary sampling spaces. It introduces Locally Informed Sampling Space (S_l), composed of multiple small hyperellipsoids derived from subsections of the current best path, and Convex Sampling Space (S_c), defined as the convex hull of the current path rotated about the start-goal axis; a combined space S_{cl} = S_l ∩ S_c is also discussed. The authors prove that sampling from S_l preserves asymptotic optimality and provide efficient sampling procedures, while S_c and S_{cl} offer faster convergence at the cost of requiring occasional sampling from the full space to maintain guarantees. Extensive experiments in 2D, 3D, and 6D demonstrate faster convergence and shorter paths compared to state-of-the-art planners such as Informed-RRT*, RRT*, RRTX, and RRT#, across multiple environments, with some limitations in highly distinct homotopy scenarios. These methods offer practical acceleration for high-dimensional motion planning while remaining compatible with existing RRT*-based frameworks.

Abstract

The asymptotically optimal version of Rapidly-exploring Random Tree (RRT*) is often used to find optimal paths in a high-dimensional configuration space. The well-known issue of RRT* is its slow convergence towards the optimal solution. A possible solution is to draw random samples only from a subset of the configuration space that is known to contain configurations that can improve the cost of the path (omniscient set). A fast convergence rate may be achieved by approximating the omniscient with a low-volume set. In this letter, we propose new methods to approximate the omniscient set and methods for their effective sampling. First, we propose to approximate the omniscient set using several (small) hyperellipsoids defined by sections of the current best solution. The second approach approximates the omniscient set by a convex hull computed from the current solution. Both approaches ensure asymptotical optimality and work in a general n-dimensional configuration space. The experiments have shown superior performance of our approaches in multiple scenarios in 3D and 6D configuration spaces.

Figures (16)

• Figure 1: Examples of trees with $7,000$ nodes (red) with the current best solution (blue path). RRT* samples the whole space (a), and Informed-RRT* samples from one (blue) hyperellipsoid (b). Proposed PI-RRT* samples from multiple small hyperellipsoids (c) and proposed C-RRT* samples from a convex set around the current best solution (d).
• Figure 2: Comparison of $\mathcal{C}_{free}$ (white) and the sampling spaces (blue).
• Figure 3: Visualization of one $s_{j,k}$ in blue on the path $\mathcal{P}$.
• Figure 4: An example of the convex sampling space $\mathcal{S}_c$ for a path $P = (p_1, \ldots, p_5) = ((-3,0,0), (0,-2,-2), (2,2,0), (3,2,2), (5,0,0))$ (a). Sampling space $\mathcal{S}_c$ (visualized using the blue mesh) is defined by $\mathcal{P}$ rotated around the SG-axis (blue) (b). The path can be transformed using Eq. \ref{['eq:transf']} to a plane. The 2D convex hull of the transformed points defines the slice (blue polygon). The convex hull of the slice is described by $V = (v_1,\ldots, v_4)$ (c).
• Figure 5: Local Informed Sampling