A Scalable Controlled Set Invariance Framework with Practical Safety Guarantees

Most existing methods for guaranteeing safety within robotics require time-consuming set-based computations, which greatly limit their applicability to real-world systems. In recent work, the authors have proposed a novel controlled set invariance framework to tackle this limitation. The framework uses a classical barrier function formulation, but replaces the difficult task of computing large control invariant sets with the more tractable tasks of (i) finding a controller that stabilizes the system to a backup set, and (ii) verifying that this backup set is invariant under the stabilizing controller. In this paper, we build upon these results to show that the requirement of proving invariance of the backup set can be relaxed at the expense of providing weaker guarantees on the safety of the system. This trade-off is shown to be favorable in practice, as the theoretically weaker safety guarantees are sufficient in many practical applications. The end result is a framework with a computational complexity that scales quadratically. The effectiveness of the approach is demonstrated in simulation on a Segway.