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This paper proposes a collision avoidance method for ellipsoidal rigidbodies, which utilizes a control barrier function (CBF) designed from asupporting hyperplane. We formulate the problem in the Special Euclidean GroupSE(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 byemploying a signed distance from a supporting hyperplane of a rigid body to theother rigid body. Although the positive value of this signed distance impliesthat two rigid bodies are collision-free, a naively prepared supportinghyperplane yields a smaller value than the actual distance. To avoid such aconservative evaluation, the supporting hyperplane is rotated so that thesigned distance from the supporting hyperplane to the other rigid body ismaximized. We prove that the maximum value of this optimization problem isequal to the actual distance between two ellipsoidal rigid bodies, henceeliminating excessive conservativeness. We leverage this signed distance as aCBF to prevent collision while the supporting hyperplane is rotated via agradient-based input. The designed CBF is integrated into a quadraticprogramming (QP) problem, where each rigid body calculates its collision-freeinput in a distributed manner, given communication among rigid bodies. Theproposed method is demonstrated with simulations. Finally, we exemplify ourmethod can be extended to a vehicle having nonholonomic dynamics.

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