Approximating Robot Configuration Spaces with few Convex Sets using Clique Covers of Visibility Graphs

TL;DR

This work tackles the problem of efficiently covering the robot's collision-free configuration space $\\mathcal{C}^{\\mathrm{free}}$ with a small number of convex regions to speed up planning. It introduces the Visibility Clique Cover (VCC) method, which samples configurations, builds a visibility graph, and uses a truncated clique cover to summarize geometry with ellipsoids that seed a region-inflation process akin to IRIS. By inflating polytopes around the ellipsoids and iterating until a coverage threshold $\\alpha$ is met, VCC achieves large coverage with fewer regions and faster computation than prior seeding techniques, including in high-DoF robots. The approach is probabilistically complete and demonstrates substantial practical gains across multiple robotic platforms, highlighting the usefulness of clique-based structure in visibility graphs for convex decomposition of complex configuration spaces.

Abstract

Many computations in robotics can be dramatically accelerated if the robot configuration space is described as a collection of simple sets. For example, recently developed motion planners rely on a convex decomposition of the free space to design collision-free trajectories using fast convex optimization. In this work, we present an efficient method for approximately covering complex configuration spaces with a small number of polytopes. The approach constructs a visibility graph using sampling and generates a clique cover of this graph to find clusters of samples that have mutual line of sight. These clusters are then inflated into large, full-dimensional, polytopes. We evaluate our method on a variety of robotic systems and show that it consistently covers larger portions of free configuration space, with fewer polytopes, and in a fraction of the time compared to previous methods.

Figures (6)

• Figure 1: The collision-free configuration space of a simple robot is decomposed into 7 polytopes, achieving around 92% coverage. Left: Robot with with 3 revolute joints $q_1$ to $q_3$. Center: Visualization of the full collision-free configuration space $\mathcal{C}^\mathrm{free}$, given by the interior of the green mesh. Right: Approximate convex cover of $\mathcal{C}^\mathrm{free}$ generated with the proposed method. See also: https://sites.google.com/view/cspacevcc/home, https://www.youtube.com/watch?v=x37fPVST6Zk.
• Figure 2: Sketch of the proposed algorithm on a simple example. First four figures: Samples are drawn uniformly from $\mathcal{C}^\mathrm{free}$ to build a visibility graph. The visibility graph is decomposed into five cliques. The principal directions and locations of the cliques are used to direct a region-inflation algorithm. Remaining two figures: This process is repeated until sufficient coverage is obtained by drawing new samples from the remaining free space, and repeating the previous steps.
• Figure 3: The growth direction of an IRIS region can be guided by the initial distance metric. IRIS is initialized with three ellipsoids with same center but different principal axes, resulting in polytopes that cover different portions of $\mathcal{C}^\mathrm{free}$.
• Figure 4: Maximum cliques of infinitely dense visibility graphs can enclose holes, and do not necessarily correspond to collision-free convex sets. The largest collision-free convex region (green trapezoid) has a smaller area than the union of red parallelograms when $0<\varepsilon\leq 1 - \sqrt{5/6}$. In this case, the convex hull of the maximum clique must enclose the hole.
• Figure 5: A visibility graph with 100 random vertices in the triangular environment from Figure \ref{['fig:Maxclique']}. Solving the maximum clique problem \ref{['eqn:maxclique']} with the additional constraints \ref{['eqn:maxcliqueconvhull']} yields a clique with 56 vertices (shown in green), that closely approximates the corresponding convex set in Figure \ref{['fig:Maxclique']}. Solving only problem \ref{['eqn:maxclique']}, yields a clique with 63 vertices (shown in red) that, however, encloses the central hole.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5