Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization

Abstract

This paper studies the narrow phase collision detection problem for two general unions of convex bodies encapsulated by smooth surfaces. The approach, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CFC</i> (Closed-Form Contact space), is based on parameterizing their contact space in closed-form. The first body is dilated to form the contact space while the second is shrunk to a point. Then, the collision detection is formulated as finding the closest point on the parametric contact space with the center of the second body. Numerical solutions are proposed based on the point-to-surface distance as well as the common-normal concept. Furthermore, when the two bodies are moving or under linear deformations, their first time of contact is solved continuously along the time-parameterized trajectories. Benchmark studies are conducted for the proposed algorithms in terms of solution stability and computational cost. Applications of the sampling-based motion planning for robot manipulators are demonstrated.