Collision Avoidance for Ellipsoidal Rigid Bodies with Control Barrier Functions Designed from Rotating Supporting Hyperplanes

TL;DR

This paper introduces a novel collision avoidance approach for ellipsoidal rigid bodies using a control barrier function based on rotating supporting hyperplanes, formulated within rigid body motion dynamics and verified through simulations.

Contribution

It develops a non-conservative signed distance measure via hyperplane rotation, enabling distributed collision avoidance for rigid bodies with potential extension to nonholonomic vehicles.

Findings

Maximized signed distance equals actual distance, reducing conservativeness.

Distributed quadratic programming enables collision-free control inputs.

Method successfully demonstrated in simulation scenarios.

Abstract

This paper proposes a collision avoidance method for ellipsoidal rigid bodies, which utilizes a control barrier function (CBF) designed from a supporting hyperplane. We formulate the problem in the Special Euclidean Group SE(2) and SE(3), where the dynamics are described as rigid body motion (RBM). Then, we consider the condition for separating two ellipsoidal rigid bodies by employing a signed distance from a supporting hyperplane of a rigid body to the other rigid body. Although the positive value of this signed distance implies that two rigid bodies are collision-free, a naively prepared supporting hyperplane yields a smaller value than the actual distance. To avoid such a conservative evaluation, the supporting hyperplane is rotated so that the signed distance from the supporting hyperplane to the other rigid body is maximized. We prove that the maximum value of this optimization problem is equal to the actual distance between two ellipsoidal rigid bodies, hence eliminating excessive conservativeness. We leverage this signed distance as a CBF to prevent collision while the supporting hyperplane is rotated via a gradient-based input. The designed CBF is integrated into a quadratic programming (QP) problem, where each rigid body calculates its collision-free input in a distributed manner, given communication among rigid bodies. The proposed method is demonstrated with simulations. Finally, we exemplify our method can be extended to a vehicle having nonholonomic dynamics.