Dynamic Minkowski sum of convex shapes

Computing the Minkowski sums of rotating ob jects has always been done naively by re-computing every Minkowski sum from scratch. The correspondences between the Minkowski sums are typically completely ignored. We propose a method, called DYMSUM, that can efficiently update the Minkowski sums of rotating convex polyhedra. We show that DYMSUM is significantly more efficient than the traditional approach, in particular when the size of the input polyhedra are large and when the rotation is small between frames. From our experimental results, we show that the computation time of the proposed method grows slowly with respect to the size of the input comparing to the naive approach.

I. Introduction

The Minkowski sum is an important operation in robotics due to its fundamental role in providing the geometric-reasoning ability to the robots, such as configuration space mapping, collision detection, and penetration depth estimation. The Minkowski sum of two shapes and is:

$$P\oplus Q=\{p+q\mid p\in P, q\in Q\}. \eqno{\hbox{(1)}}$$