Characterizing Smooth Safety Filters via the Implicit Function Theorem

TL;DR

This paper introduces a general framework for designing smooth safety filters for autonomous systems using the Implicit Function Theorem, enabling safety guarantees with smooth controllers and quantifying their conservatism.

Contribution

It provides a novel characterization of smooth safety filters via the Implicit Function Theorem, offering universal formulas and insights into their conservatism.

Findings

Families of smooth safety-critical controllers derived

Quantification of conservatism in safety filters

Illustrative examples demonstrating utility

Abstract

Optimization-based safety filters, such as control barrier function (CBF) based quadratic programs (QPs), have demonstrated success in controlling autonomous systems to achieve complex goals. These CBF-QPs can be shown to be continuous, but are generally not smooth, let alone continuously differentiable. In this paper, we present a general characterization of smooth safety filters -- smooth controllers that guarantee safety in a minimally invasive fashion -- based on the Implicit Function Theorem. This characterization leads to families of smooth universal formulas for safety-critical controllers that quantify the conservatism of the resulting safety filter, the utility of which is demonstrated through illustrative examples.