Closed-form Minkowski sums of convex bodies with smooth positively curved boundaries. (February 2022) - Ruan, Sipu; Chirikjian, Gregory S.
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 Minkowski sum boundaries 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. Highlights: Minkowski sums of convex bodies with smooth boundary are expressed in closed form. The bodies are parameterized by either outward normal or un-normalized gradients. The closed-form expressions only depend on the parameterization of one body. General linear transformations are allowed to extend the closed-form expressions.