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.

翻译：本文提出一种适用于椭球刚体的碰撞避免方法，该方法采用基于支撑超平面设计的控制障碍函数（CBF）。我们在特殊欧几里得群SE(2)和SE(3)中建立问题模型，将刚体运动（RBM）作为动力学描述。随后，通过计算某一刚体支撑超平面到另一刚体的有符号距离，考虑分离两椭球刚体的条件。尽管该有符号距离的正值表明两刚体无碰撞，但直觉选取的支撑超平面会导致所获距离小于实际距离。为避免这种保守性评估，我们旋转支撑超平面，使得该超平面到另一刚体的有符号距离最大化。理论证明该优化问题的最大值等于两椭球刚体间的实际距离，从而消除过度保守性。我们利用该有符号距离作为CBF，在通过梯度驱动输入旋转支撑超平面的同时实现碰撞预防。所设计的CBF被集成至二次规划（QP）问题中，每刚体通过刚体间通信以分布式方式计算其无碰撞输入。通过仿真验证了所提方法的有效性，最后示例说明该方法可扩展至具有非完整动力学的车辆。