# Backup Control Barrier Functions: Formulation and Comparative Study

arXiv: https://arxiv.org/abs/2104.11332 · HTML: https://mlsanity.com/paper/2104.11332/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2021.

## Built on by (most cited first, 12 of 12)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Safe Control Under Input Limits with Neural Control Barrier Functions](https://mlsanity.com/paper/2211.11056/) (2022)
- [Real-Time Out-of-Distribution Failure Prevention via Multi-Modal Reasoning](https://mlsanity.com/paper/2505.10547/) (2025)
- [Discrete GCBF Proximal Policy Optimization for Multi-agent Safe Optimal Control](https://mlsanity.com/paper/2502.03640/) (2025)
- [Back to Base: Towards Hands-Off Learning via Safe Resets with Reach-Avoid Safety Filters](https://mlsanity.com/paper/2501.02620/) (2025)
- [Safe Beyond the Horizon: Efficient Sampling-based MPC with Neural Control Barrier Functions](https://mlsanity.com/paper/2502.15006/) (2025)
- [Provably Safe, Yet Scalable Reinforcement Learning](https://mlsanity.com/paper/2606.14536/) (2026)
- [Geometry of Radial Basis Neural Networks for Safety Biased Approximation of Unsafe Regions](https://mlsanity.com/paper/2210.05596/) (2022)
- [Space Processor Computation Time Analysis for Reinforcement Learning and Run Time Assurance Control Policies](https://mlsanity.com/paper/2405.06771/) (2024)
- [Uncertainty Aware-Predictive Control Barrier Functions: Safer Human Robot Interaction through Probabilistic Motion Forecasting](https://mlsanity.com/paper/2508.20812/) (2025)
- [LEAP-CBF: A Safety Filter for Uncertain Systems with Least-Effort Adversarial Potentials](https://mlsanity.com/paper/2609.28364/) (2026)
- [Predictive Semantic Safety: From Visual Physical Reasoning to Safety-Critical Control](https://mlsanity.com/paper/2609.34356/) (2026)

## Abstract (arXiv)

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.