A simple method for computing Minkowski sum boundary in 3D using collision detection

Abstract

Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3-d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the difficulty of implementing the existing methods. There are two main approaches for computing Minkowski sums: divide-and-conquer and convolution. The first approach decomposes the input polyhedra into convex pieces, computes the Minkowski sums between a pair of convex pieces, and unites all the pairwise Minkowski sums. Although conceptually simple, the major problems of this approach include: (1) The size of the decomposition and the pairwise Minkowski sums can be extremely large and (2) robustly computing the union of a large number of components can be very tricky. On the other hand, convolving two polyhedra can be done more efficiently. The resulting convolution is a superset of the Minkowski sum boundary. For non-convex inputs, filtering or trimming is needed. This usually involves computing (1) the arrangement of the convolution and its substructures and (2) the winding numbers for the arrangement subdivisions. Both computations are difficult to implement robustly in 3-d. In this paper we present a new approach that is simple to implement and can efficiently generate accurate Minkowski sum boundary. Our method is convolution based but it avoids computing the 3-d arrangement and the winding numbers. The premise of our method is to reduce the trimming problem to the problems of computing 2-d arrangements and collision detection, which are much better understood in the literature. To maintain the simplicity, we intentionally sacrifice the exactness. While our method generates exact solutions in most cases, it does not produce low dimensional boundaries, e.g., boundaries enclosing zero volume. We classify our method as 'nearly exact' to distinguish it from the exact and approximate methods. © 2009 Springer-Verlag.