Generalizations of Backup Control Barrier Functions: Expansion and Adaptation for Input-Bounded Safety-Critical Control

TL;DR

This paper generalizes backup control barrier functions by separating expansion and backup controllers, allowing broader safe set expansion and online adaptation while maintaining formal safety guarantees for input-bounded nonlinear systems.

Contribution

It introduces a generalized bCBF framework that separates expansion and backup controllers, enabling broader safe set expansion and online adaptation with safety guarantees.

Findings

The generalized framework recovers existing bCBF methods as special cases.

Sufficient conditions for forward invariance of the safe set are established.

Extension to parameterized controllers allows online adaptation under input bounds.

Abstract

Guaranteeing the safety of nonlinear systems with bounded inputs remains a key challenge in safe autonomy. Backup control barrier functions (bCBFs) provide a powerful mechanism for constructing controlled invariant sets by propagating trajectories under a pre-verified backup controller to a forward invariant backup set. While effective, the standard bCBF method utilizes the same backup controller for both set expansion and safety certification, which can restrict the expanded safe set and lead to conservative dynamic behavior. In this study, we generalize the bCBF framework by separating the set-expanding controller from the verified backup controller, thereby enabling a broader class of expansion strategies while preserving formal safety guarantees. We establish sufficient conditions for forward invariance of the resulting implicit safe set and show how the generalized construction recovers existing bCBF methods as special cases. Moreover, we extend the proposed framework to parameterized controller families, enabling online adaptation of the expansion controller while maintaining safety guarantees in the presence of input bounds.