# ProBF: Learning Probabilistic Safety Certificates with Barrier Functions

arXiv: https://arxiv.org/abs/2112.12210 · HTML: https://mlsanity.com/paper/2112.12210/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 22 Dec 2021.

## Builds on (most cited first, 16 of 16)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Reinforcement Learning for Safety-Critical Control under Model Uncertainty, using Control Lyapunov Functions and Control Barrier Functions](https://mlsanity.com/paper/2004.07584/) (2020)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [Episodic Learning with Control Lyapunov Functions for Uncertain Robotic Systems](https://mlsanity.com/paper/1903.01577/) (2019)
- [Multi-Robot Collision Avoidance under Uncertainty with Probabilistic Safety Barrier Certificates](https://mlsanity.com/paper/1912.09957/) (2019)
- [Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty](https://mlsanity.com/paper/2106.07108/) (2021)
- [Gaussian Process-based Min-norm Stabilizing Controller for Control-Affine Systems with Uncertain Input Effects and Dynamics](https://mlsanity.com/paper/2011.07183/) (2020)
- [Probabilistic Safety Constraints for Learned High Relative Degree System Dynamics](https://mlsanity.com/paper/1912.10116/) (2019)
- [Learning Control Barrier Functions with High Relative Degree for Safety-Critical Control](https://mlsanity.com/paper/2011.10721/) (2020)

## Abstract (arXiv)

Safety-critical applications require controllers/policies that can guarantee safety with high confidence. The control barrier function is a useful tool to guarantee safety if we have access to the ground-truth system dynamics. In practice, we have inaccurate knowledge of the system dynamics, which can lead to unsafe behaviors due to unmodeled residual dynamics. Learning the residual dynamics with deterministic machine learning models can prevent the unsafe behavior but can fail when the predictions are imperfect. In this situation, a probabilistic learning method that reasons about the uncertainty of its predictions can help provide robust safety margins. In this work, we use a Gaussian process to model the projection of the residual dynamics onto a control barrier function. We propose a novel optimization procedure to generate safe controls that can guarantee safety with high probability. The safety filter is provided with the ability to reason about the uncertainty of the predictions from the GP. We show the efficacy of this method through experiments on Segway and Quadrotor simulations. Our proposed probabilistic approach is able to reduce the number of safety violations significantly as compared to the deterministic approach with a neural network.