  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.
