Constant-Sum High-Order Barrier Functions for Safety Between Parallel Boundaries

TL;DR

This paper introduces a novel approach for constructing separate control barrier functions for parallel safety boundaries, extending to higher relative degrees, and demonstrates its application on a unicycle system.

Contribution

It proposes a new method to construct individual CBFs for parallel boundaries and extends the approach to higher relative degrees using backstepping.

Findings

Successfully applied to a unicycle system

Derived verifiable safety conditions

Extended to higher relative degrees

Abstract

This paper takes a step towards addressing the difficulty of constructing Control Barrier Functions (CBFs) for parallel safety boundaries. A single CBF for both boundaries has been reported to be difficult to validate for safety, and we identify why this challenge is inherent. To overcome this, the proposed method constructs separate CBFs for each boundary. We begin by presenting results for the relative degree one case and then extend these to higher relative degrees using the CBF backstepping technique, establishing conditions that guarantee safety. Finally, we showcase our method by applying it to a unicycle system, deriving a simple, verifiable condition to validate the target CBFs for direct implementation of our results.