Control Barrier Functions for Collision Avoidance between Strongly Convex Regions

In this paper, we focus on nonconservative, continuous-time collision avoidance between robots and obstacles with continuous control-affine dynamics and convex state-dependent safe regions using control barrier functions (CBFs). System safety is defined using the minimum distance between the safe regions associated with robots and obstacles. However, continuous-time collision avoidance using the minimum distance as a CBF can pose challenges since the distance function is given by a parametric optimization problem and nonsmooth in general. We identify a class of state-dependent convex sets, defined as strongly convex maps, for which the minimum distance is continuously differentiable. We show that the directional derivatives of the minimum distance KKT solution for strongly convex maps can be efficiently computed using a linear complementarity problem and used for fast KKT solution computation in real-time implementations. Lastly, we propose a CBF-based optimal feedback control law and prove strong safety for the closed-loop system, while allowing for discontinuous (in state) and nonconvex (in control) cost functions. We validate our approach in simulation on a quadrotor system navigating through an obstacle-filled corridor and demonstrate that CBF constraints can be enforced in real-time for state-dependent convex sets without overapproximations.