Filtering for Liveness: Recurrence via Backup Reachability Control Barrier Functions

Liveness requires a dynamical system to repeatedly accomplish a desired objective over time. In this work, we consider a particular form of liveness known as recurrence, which requires the closed-loop trajectory to regularly revisit a prescribed set, with no more than a specified amount of time between visits. Inspired by the control barrier function (CBF) safety-filter paradigm, we develop a controller-filtering framework that makes adjustments to a nominal controller only when necessary to guarantee recurrence. Unlike conventional CBF methods, which enforce set forward invariance, our framework allows the trajectory to leave the set while guaranteeing its timely return. To achieve this, our framework combines two components: a backup reachability CBF that preserves the ability to return to the prescribed set within the allowable time, and a timing-based mechanism that invokes a reaching controller only when the nominal behavior would otherwise fail to meet the return-time requirement. The resulting framework guarantees recurrence while preserving the nominal behavior when possible, enabling control objectives that require temporary excursions from the set, such as traversal between disjoint regions. Our method is successfully utilized in a drone racing example with a recurrence constraint on beacon visibility.