Warming up a Backup Control Barrier Function

Backup control barrier function (Backup CBF) is a tractable formulation that ensure the feasibility of CBF-based quadratic programming (QP) problems through implicitly defined control invariant set. Backup CBF is based on two fundamental ingredients: a known backup set and a fixed backup policy, and the implicitly defined control invariant set is evaluated through forward integration by backup policy in a finite horizon online. However, there is a lack of systematic methods for providing the backup set and backup policy for a system with input constraints, making it difficult to synthesize a backup CBF. In this paper, we propose a method for second-order systems to obtain backup policy and backup set practically, rather than empirical choice, by constructing a more conservative explicit CBF-QP with auxiliary velocity constraints integrated. The proposed method warms up a backup CBF by an explicit conservative CBF-QP under mild assumptions, while without excessive conservatism, allowing the tasks of safety and feasibility to be guaranteed theoretically. We demonstrate the efficacy of the proposed method in simulation.