On the complexity of the free space of a translating box in R^3

TL;DR

The paper analyzes the worst-case combinatorial complexity of the free space for a translating box-shaped robot (and more generally fully-parallel convex polygons) among fixed polyhedral obstacles in R^3. By bounding the number of triple vertex contacts through a refined envelope-based framework and a novel separation of segment families, it eliminates the inverse Ackermann factor that appears in prior bounds, proving an optimal $O(n^2)$ bound in these cases. This settles the asymptotic worst-case complexity for boxes and fully-parallel polygons and demonstrates the effectiveness of a parametric-plane, envelope-based approach in 3D translational motion planning. The techniques bridge Minkowski-sum configuration-space reductions with careful counting of contact configurations to yield tight, components-dependent bounds with potential relevance to algorithmic planning in robotics.

Abstract

Consider a convex polyhedral robot $B$ that can translate (without rotating) amidst a finite set of non-moving polyhedral obstacles in $\mathbb R^3$. The "free space" $\mathcal F$ of $B$ is the set of all positions in which $B$ is disjoint from the interior of every obstacle. Aronov and Sharir (1997) derived an upper bound of $O(n^2\log n)$ for the combinatorial complexity of $\mathcal F$, where $n$ is the total number of vertices of the obstacles, and the complexity of $B$ is assumed constant. Halperin and Yap (1993) showed that, if $B$ is either a box or a "flat" convex polygon, then a tighter bound of $O(n^2α(n))$ holds. Here $α(n)$ is the inverse Ackermann function. In this paper we prove that if $B$ is a box, then the complexity of $\mathcal F$ is $O(n^2)$. Furthermore, if $B$ is a convex polygon whose edges come in parallel pairs, then the complexity of $\mathcal F$ is $O(n^2)$ as well. These results settle the question of the asymptotical worst-case complexity of $\mathcal F$ for a box, as well as for all convex polygons.

Figures (13)

• Figure 1: Configuration that achieves $\Omega(n^2\alpha(n))$ triple contacts with a triangular robot. The robot is parallel to the $xz$-plane. There is a configuration of $n/2$ obstacle edges parallel to the $xy$-plane that form $\Omega(n\alpha(n))$ lower-envelope intersections, in which the edges are given slightly different $z$-coordinates ($a$). There are another $n/2$ obstacle edges parallel to the $y$-axis with different $z$-coordinates ($b$).
• Figure 2: A fully-parallel polygon.
• Figure 3: A placement of $B$ (here a cube) making three contacts with obstacles, in this case a vertex contact, an edge contact, and a face contact. This placement of $B$ corresponds to a vertex of the free space.
• Figure 4: The lower envelope of a set of segments, decomposed into concave chains.
• Figure 5: A "flat" triangular robot $B$ in $\mathbb R^3$ making three vertex contacts.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• Remark
• Corollary 5
• proof
• Lemma 6
• proof