Nonsmooth Control Barrier Functions for Obstacle Avoidance between Convex Regions

In this paper, we focus on non-conservative obstacle avoidance between robots with control affine dynamics with strictly convex and polytopic shapes. The core challenge for this obstacle avoidance problem is that the minimum distance between strictly convex regions or polytopes is generally implicit and non-smooth, such that distance constraints cannot be enforced directly in the optimization problem. To handle this challenge, we employ non-smooth control barrier functions to reformulate the avoidance problem in the dual space, with the positivity of the minimum distance between robots equivalently expressed using a quadratic program. Our approach is proven to guarantee system safety. We theoretically analyze the smoothness properties of the minimum distance quadratic program and its KKT conditions. We validate our approach by demonstrating computationally-efficient obstacle avoidance for multi-agent robotic systems with strictly convex and polytopic shapes. To our best knowledge, this is the first time a real-time QP problem can be formulated for general non-conservative avoidance between strictly convex shapes and polytopes.