Collision Avoidance for Ellipsoidal Rigid Bodies With Control Barrier Functions Designed From Rotating Supporting Hyperplanes

This article proposes a collision avoidance method for ellipsoidal rigid bodies that utilizes a control barrier function (CBF) designed from a supporting hyperplane. We formulate the problem in the special Euclidean groups <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${SE}(2)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${SE}(3)$ </tex-math></inline-formula>, where the kinematics are described as rigid body motion (RBM). 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 a positive value of this signed distance implies that the two rigid bodies are collision-free, a naively designed 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. We exemplify that our method can be extended to other kinematic models. The proposed method is demonstrated through simulation results.