Asymptotically-Optimal Multi-Query Path Planning for a Polygonal Robot

TL;DR

The paper addresses fast, high-quality 2D path planning for a translating and rotating polygonal robot by introducing rotation-stacked reduced visibility graphs (RVG). RVG partitions the $SE(2)$ configuration space into rotation layers, builds per-layer reduced visibility graphs using Minkowski sums, and connects layers to allow simultaneous translation and rotation, achieving resolution-complete and asymptotically optimal performance. It provides algorithmic details for building layers, propagating vertices, and performing path searches, along with formal guarantees and extensive computational evaluation showing favorable comparisons to state-of-the-art sampling-based methods. The results indicate RVG offers strong practical performance for multi-query planning and has potential applications in object manipulation and autonomous navigation, with clear avenues for future enhancements such as non-holonomic extensions and parallelization.

Abstract

Shortest-path roadmaps, also known as reduced visibility graphs, provides a highly efficient multi-query method for computing optimal paths in two-dimensional environments. Combined with Minkowski sum computations, shortest-path roadmaps can compute optimal paths for a translating robot in 2D. In this study, we explore the intuitive idea of stacking up a set of reduced visibility graphs at different orientations for a polygonal holonomic robot to support the fast computation of near-optimal paths, allowing simultaneous 2D translation and rotation. The resulting algorithm, rotation-stacked visibility graph (RVG), is shown to be resolution-complete and asymptotically optimal. Extensive computational experiments show RVG significantly outperforms state-of-the-art single- and multi-query sampling-based methods on both computation time and solution optimality fronts.

Figures (13)

• Figure 1: A motivating example in developing RVG. In object rearrangement, high-quality paths often need to be planned to translate and rotate objects in a 2D workspace, e.g., a tabletop. Images are reproduced from huang2023toward.
• Figure 2: [Left] A 2D polygonal environment and the (blue) visibility graph for it. [Right] The (green) reduced visibility graph for the same environment. The additional blue edges do not appear in shortest paths unless the start/goal happens to fall on them (a zero probability event).
• Figure 3: (a)-(c) The semi-algebraic sets composed of the grown obstacles from all layers with resolutions=18, 90, 180 from left to right resulted from 2 rectangles, a robot and an obstacle, shown in Fig \ref{['fig:resolution']}. (d) The semi-algebraic sets projected to the $\mathbb{R}^2$ space when resolution=180.
• Figure 4: Intersections of Minkowski sum boundaries (solid black and brown lines) of the robot at two poses (dashed lines) and an obstacle (partial border showing in red).
• Figure 5: The solutions from our algorithm with different resolutions. The pink, purple, and deep cyan rectangles represent the start configuration, goal configuration, and obstacles(including boundaries). [Left] The map. [Middle] The solution found at resolution $n=36$. [Right] The solution at resolution $n=72$, taking more time to compute.

Theorems & Definitions (2)

• Theorem III.1
• proof : Proof sketch