Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

doi:10.1016/j.cad.2011.06.021

Computer-Aided Design

Volume 43, Issue 10, October 2011, Pages 1258-1269

https://doi.org/10.1016/j.cad.2011.06.021 Get rights and content

Abstract

We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in $R^{3}$ , where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph structure of the Minkowski sum of the two polytopes, while one of them rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis.

Our work targets applications that require exact collision detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in $R^{3}$ , and it well balances between preprocessing time and space on the one hand, and query time on the other.

We use Cgal arrangements and in particular the support for spherical Gaussian maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments (i) to verify the correctness of the algorithm and its implementation, and (ii) to compare its efficiency with an alternative (static) exact method. The results are reported.

Highlights

► Compute fast the appropriate Minkowski sum for a given rotation angle. ► Examine structural changes in the Minkowski sum of two rotating polytopes. ► Handle degenerate input, Exact results, sqrt-extension efficient number type. ► The results are 4–5 times better than the corresponding static method. ► Dynamic approach and arbitrary axis enhancements improve the results.

Section snippets

Introduction and related work

Let $P$ and $Q$ be two polyhedra in $R^{d}$ . The Minkowski sum of $P$ and $Q$ is defined as $P \oplus Q = {p + q ∣ p \in P, q \in Q}$ . Assume that a bounded polyhedron, referred to as polytope, $R$ is moving in three-dimensional space. It is well known that $R$ collides with another static polytope $Q$ , if and only if the origin is in $- R \oplus Q$ , where $- R$ is a copy of $R$ reflected about the origin. This observation is the basis of the intensive use of Minkowski sums in motion planning and many other related problems; see, e.g., the

Preliminaries

In this section we introduce the extended Gaussian map—a unique dual representation of a polytope. An extended Gaussian map is in turn represented by an arrangement of geodesic arcs embedded on a sphere. This data structure is supported by the 2D Arrangements package of Cgal [20]. The package also supports the overlay operation of such arrangements used to compute the Minkowski sum of polytopes represented by the arrangements; see [21], [22], [23] and Section 2.2 for more details.

Cgal in

Rotation about one axis

Let $P$ and $Q$ be two polytopes with $m$ and $n$ vertices, respectively, and let $M$ be their Minkowski sum, which is known to have at most $O (m n)$ vertices. Without loss of generality, we assume that $P$ rotates counterclockwise about the $z$ -axis, looking toward the origin from $z = \infty$ , and $Q$ is stationary. When $P$ rotates but rotation movement is sufficiently “small” (an exact definition for “small” is provided below) the combinatorial structure of the Minkowski sum with $Q$ does not change. Our work is based on

Enhancements

The algorithm described in Section 3 performs well on examples with small numbers of criticalities. When the number of criticalities increase the query time remains almost unchanged, but the preprocessing time and space dramatically increases; see Table 3. In addition, (general) rotation about three axes is computationally limited, namely the tight bound on the number of distinct Minkowski sum is high; see Section 3.3. The following sections describe two enhancements applied to the original

Experiments

We are not aware of any previous work that updates the exact Minkowski sum of rotating polytopes. Recently, Lien [11] suggested an algorithm to handle Minkowski sums of rotating polytopes, calculating their Minkowski sum by updating the orientation of each feature. As Lien’s method does not use exact geometric computation, it presumably cannot identify all possible distinct structures that arise during the rotation; therefore, we believe comparing our method with Lien’s is unavailing. For the

Future work

This work was implemented as part of a complete integrated framework for proximity queries using the spherical Gaussian map. It handles the full construction of the structure of Minkowski sums under rotation using an efficient data structure. We plan to design another approach for handling collision detection of translating and rotating polytopes, by implementing an algorithm in the spirit of Lin and Canny [13], in order to complete this framework. For every pair of polytopes we can find the

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^☆: This work has been supported in part by the Israel Science Foundation 236/06, by the German-Israeli Foundation 969/07, by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University, and by the EU Project under Contract No. 255827 (CGL—Computational Geometry Learning).

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Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space☆

Abstract

Highlights

Section snippets

Introduction and related work

Preliminaries

Rotation about one axis

Enhancements

Experiments

Future work

Computer-Aided Design

Computers & Graphics

Journal of Symbolic Computation

Computational Geometry: Theory and Applications

Computational geometry: algorithms and applications

Robot motion planning

Spatial planning: a configuration space approach

IEEE Transactions on Computers

Accurate Minkowski sum approximation of polyhedral models

Minkowski addition of polytopes: computational complexity and applications to Gröbner bases

SIAM Journal on Discrete Mathematics

Exact Minkowksi sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces

Algorithmica

Movie: Minkowski sums of rotating convex polyhedra

Collision and proximity queries

A fast algorithm for incremental distance calculation

Proceedings of the IEEE International Conference on Robotics and Automation