Backup Control Barrier Functions: Formulation and Comparative Study

TL;DR

This paper provides a detailed tutorial and comparison of backup control barrier functions (CBFs), demonstrating their effectiveness in guaranteeing control invariance and comparing them with other methods like Hamilton Jacobi PDE and Sum-of-Squares.

Contribution

It offers a comprehensive tutorial on backup CBFs, proves their relative degree, and compares their performance with benchmarks for computing control invariant sets.

Findings

Backup CBFs always have relative degree 1 under mild assumptions.

Backup CBFs can approximate maximum control invariant sets effectively.

Comparison shows backup CBFs are computationally efficient and practical for control invariant set computation.

Abstract

The backup control barrier function (CBF) was recently proposed as a tractable formulation that guarantees the feasibility of the CBF quadratic programming (QP) via an implicitly defined control invariant set. The control invariant set is based on a fixed backup policy and evaluated online by forward integrating the dynamics under the backup policy. This paper is intended as a tutorial of the backup CBF approach and a comparative study to some benchmarks. First, the backup CBF approach is presented step by step with the underlying math explained in detail. Second, we prove that the backup CBF always has a relative degree 1 under mild assumptions. Third, the backup CBF approach is compared with benchmarks such as Hamilton Jacobi PDE and Sum-of-Squares on the computation of control invariant sets, which shows that one can obtain a control invariant set close to the maximum control invariant set under a good backup policy for many practical problems.