Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries

TL;DR

This paper presents new closed-form formulas for Minkowski sums of smooth convex bodies with positive curvature, facilitating efficient computation in motion planning and collision detection.

Contribution

It introduces two theorems providing parametric formulas for Minkowski sums of convex bodies with smooth, positively curved boundaries, including a practical non-normalized gradient form.

Findings

Validated formulas through numerical experiments with superquadrics.

Demonstrated applications in motion planning and collision detection.

Matched previous geometric results for ellipsoids.

Abstract

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions, there is a unique relationship between the position of each boundary point and the surface normal. The main results are presented as two theorems. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each surface point. Although simple to express mathematically, such a parameterization is not always practical to obtain computationally. Therefore, the second theorem derives a more useful parametric closed-form expression using the gradient that is not normalized. In the special case of two ellipsoids, the proposed expressions are identical to those derived previously using geometric interpretations. In order to examine the results, numerical validations and comparisons of the Minkowski sums between two superquadric bodies are conducted. Applications to generate configuration space obstacles in motion planning problems and to improve optimization-based collision detection algorithms are introduced and demonstrated.