# Geometry of Radial Basis Neural Networks for Safety Biased Approximation of Unsafe Regions

arXiv: https://arxiv.org/abs/2210.05596 · HTML: https://mlsanity.com/paper/2210.05596/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 11 Oct 2022.

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

- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [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)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach](https://mlsanity.com/paper/2003.04950/) (2020)
- [Learning Barrier Functions with Memory for Robust Safe Navigation](https://mlsanity.com/paper/2011.01899/) (2020)
- [Control Barriers in Bayesian Learning of System Dynamics](https://mlsanity.com/paper/2012.14964/) (2020)
- [Backup Control Barrier Functions: Formulation and Comparative Study](https://mlsanity.com/paper/2104.11332/) (2021)
- [Control of Mobile Robots Using Barrier Functions Under Temporal Logic Specifications](https://mlsanity.com/paper/1908.04903/) (2019)
- [Reactive and Safe Road User Simulations using Neural Barrier Certificates](https://mlsanity.com/paper/2109.06689/) (2021)
- [Safe Control Synthesis with Uncertain Dynamics and Constraints](https://mlsanity.com/paper/2202.09557/) (2022)
- [Safe Certificate-Based Maneuvers for Teams of Quadrotors Using Differential Flatness](https://mlsanity.com/paper/1702.01075/) (2017)
- [Polynomial Invariant Generation for Non-deterministic Recursive Programs](https://mlsanity.com/paper/1902.04373/) (2019)
- [Decentralized Control Barrier Functions for Coupled Multi-Agent Systems under Signal Temporal Logic Tasks](https://mlsanity.com/paper/2011.12775/) (2020)
- [A Sequential Composition Framework for Coordinating Multi-Robot Behaviors](https://mlsanity.com/paper/1907.07718/) (2019)

## Abstract (arXiv)

Barrier function-based inequality constraints are a means to enforce safety specifications for control systems. When used in conjunction with a convex optimization program, they provide a computationally efficient method to enforce safety for the general class of control-affine systems. One of the main assumptions when taking this approach is the a priori knowledge of the barrier function itself, i.e., knowledge of the safe set. In the context of navigation through unknown environments where the locally safe set evolves with time, such knowledge does not exist. This manuscript focuses on the synthesis of a zeroing barrier function characterizing the safe set based on safe and unsafe sample measurements, e.g., from perception data in navigation applications. Prior work formulated a supervised machine learning algorithm whose solution guaranteed the construction of a zeroing barrier function with specific level-set properties. However, it did not explore the geometry of the neural network design used for the synthesis process. This manuscript describes the specific geometry of the neural network used for zeroing barrier function synthesis, and shows how the network provides the necessary representation for splitting the state space into safe and unsafe regions.