Safety for Time-Varying Parameterized Sets Using Control Barrier Function Methods

A fundamental and classical problem in mobile autonomous systems is maintaining the safety of autonomous agents during deployment. Prior literature has presented techniques using control barrier functions (CBFs) to achieve this goal. These prior techniques utilize CBFs to keep an isolated point in state space away from the unsafe set. However, various situations require a non-singleton set of states to be kept away from an unsafe set. Prior literature has addressed this problem using nonsmooth CBF methods, but no prior work has solved this problem using only "smooth" CBF methods. This paper addresses this gap by presenting a novel method of applying CBF methods to non-singleton parameterized convex sets. The method ensures differentiability of the squared distance function between ego and obstacle sets by leveraging a form of the log-sum-exp function to form strictly convex, arbitrarily tight overapproximations of these sets. Safety-preserving control inputs can be computed via convex optimization formulations. The efficacy of our results is demonstrated through multi-agent simulations.