Composition of Control Barrier Functions With Differing Relative Degrees for Safety Under Input Constraints

TL;DR

This paper introduces a novel method to combine multiple control barrier functions with different relative degrees, ensuring safety under input constraints through a composite soft-minimum approach and real-time optimization, demonstrated on a ground robot.

Contribution

It develops a new construction for combining multiple CBFs with differing relative degrees and extends this to systems with input constraints, enabling guaranteed safety.

Findings

The composite soft-minimum CBF ensures safety within the union of individual CBFs.

The approach guarantees state safety through real-time optimization control.

Demonstrated effectiveness on a nonholonomic ground robot example.

Abstract

This paper presents a new approach for guaranteed safety subject to input constraints (e.g., actuator limits) using a composition of multiple control barrier functions (CBFs). First, we present a method for constructing a single CBF from multiple CBFs, which can have different relative degrees. This construction relies on a soft minimum function and yields a CBF whose $0$-superlevel set is a subset of the union of the $0$-superlevel sets of all the CBFs used in the construction. Next, we extend the approach to systems with input constraints. Specifically, we introduce control dynamics that allow us to express the input constraints as CBFs in the closed-loop state (i.e., the state of the system and the controller). The CBFs constructed from input constraints do not have the same relative degree as the safety constraints. Thus, the composite soft-minimum CBF construction is used to combine the input-constraint CBFs with the safety-constraint CBFs. Finally, we present a feasible real-time-optimization control that guarantees that the state remains in the $0$-superlevel set of the composite soft-minimum CBF. We demonstrate these approaches on a nonholonomic ground robot example.