Exact Minkowksi Sums of Polyhedra and Exact and Efficient Decomposition of Polyhedra into Convex Pieces

Abstract

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by CGAL. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning.The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small number of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations.The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2 ) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron.

Scalable 3D Tracking of Multiple Interacting Objects

“…(2)) to penalize object collisions, as space cannot be shared among objects. We employ a penalty term which amounts to the total penetration depth for all pairwise collisions in a hypothesized 3D configuration of objects [5]. This should be contrasted to [11] which penalizes penetration between adjacent fingers alone.…”

Robust polyhedral Minkowski sums with GPU implementation

“…This approach is inefficient because a polyhedron with r reflex edges can have Ω(r 2 ) convex pieces, so an input with n edges can entail a union of Ω(n 4 ) component Minkowski sums. Hachenberger [9] provides an exact implementation of this algorithm. The program is very slow [4,2,18].…”

“…We compare our CPU program to the published running times for prior programs adjusted for processor speed, except that we timed Hachenberger's [9] exact convex decomposition [18]. Hachenberger's program took 2,000-4,000 times longer than our program on small tests, 10,000-100,000 times longer on medium tests, and aborted or did not terminate after several hours on the other tests.…”

Dynamic Minkowski sums under scaling

“…Although the main idea of this approach (first proposed by Lozano-Pérez [24]) is simple, the decomposition and the union steps can be very tricky and have become the main focus of several recent works [1,13,18,29,21]. …”