# Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety

arXiv: https://arxiv.org/abs/2203.15474 · HTML: https://mlsanity.com/paper/2203.15474/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 45 references that are in the map.
- First posted on arXiv 29 Mar 2022.

## Builds on (most cited first, 19 of 19)

- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Learning Control Barrier Functions from Expert Demonstrations](https://mlsanity.com/paper/2004.03315/) (2020)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach](https://mlsanity.com/paper/2003.04950/) (2020)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Scaling Gaussian Process Regression with Derivatives](https://mlsanity.com/paper/1810.12283/) (2018)
- [Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach](https://mlsanity.com/paper/2006.04361/) (2020)
- [DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization](https://mlsanity.com/paper/1706.02586/) (2017)
- [Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximation](https://mlsanity.com/paper/2109.13359/) (2021)
- [Control Barrier Functions for Stochastic Systems](https://mlsanity.com/paper/2003.03498/) (2020)
- [Learning Robust Hybrid Control Barrier Functions for Uncertain Systems](https://mlsanity.com/paper/2101.06492/) (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)
- [Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making](https://mlsanity.com/paper/1504.06002/) (2015)
- [Safety-Critical Control and Planning for Obstacle Avoidance between Polytopes with Control Barrier Functions](https://mlsanity.com/paper/2109.12313/) (2021)
- [Safe Certificate-Based Maneuvers for Teams of Quadrotors Using Differential Flatness](https://mlsanity.com/paper/1702.01075/) (2017)
- [Autonomous Racing with Multiple Vehicles using a Parallelized Optimization with Safety Guarantee using Control Barrier Functions](https://mlsanity.com/paper/2112.06435/) (2021)
- [Distributed Collision-Free Motion Coordination on a Sphere: A Conic Control Barrier Function Approach](https://mlsanity.com/paper/2003.13363/) (2020)

## Built on by (most cited first, 2 of 2)

- [Learning Control Barrier Functions and their application in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2404.16879/) (2024)
- [Uncertainty Aware-Predictive Control Barrier Functions: Safer Human Robot Interaction through Probabilistic Motion Forecasting](https://mlsanity.com/paper/2508.20812/) (2025)

## Abstract (arXiv)

Inspired by the success of control barrier functions (CBFs) in addressing safety, and the rise of data-driven techniques for modeling functions, we propose a non-parametric approach for online synthesis of CBFs using Gaussian Processes (GPs). Mathematical constructs such as CBFs have achieved safety by designing a candidate function a priori. However, designing such a candidate function can be challenging. A practical example of such a setting would be to design a CBF in a disaster recovery scenario where safe and navigable regions need to be determined. The decision boundary for safety in such an example is unknown and cannot be designed a priori. In our approach, we work with safety samples or observations to construct the CBF online by assuming a flexible GP prior on these samples, and term our formulation as a Gaussian CBF. GPs have favorable properties, in addition to being non-parametric, such as analytical tractability and robust uncertainty estimation. This allows realizing the posterior components with high safety guarantees by incorporating variance estimation, while also computing associated partial derivatives in closed-form to achieve safe control. Moreover, the synthesized safety function from our approach allows changing the corresponding safe set arbitrarily based on the data, thus allowing non-convex safe sets. We validate our approach experimentally on a quadrotor by demonstrating safe control for fixed but arbitrary safe sets and collision avoidance where the safe set is constructed online. Finally, we juxtapose Gaussian CBFs with regular CBFs in the presence of noisy states to highlight its flexibility and robustness to noise. The experiment video can be seen at: https://youtu.be/HX6uokvCiGk.