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

TL;DR

This paper investigates the geometric properties of Radial Basis Function neural networks for synthesizing zeroing barrier functions, enabling safety verification in control systems with unknown or evolving safe sets.

Contribution

It introduces a novel neural network geometry tailored for zeroing barrier function synthesis, facilitating safety region classification from perception data.

Findings

Neural network geometry effectively splits safe and unsafe regions.

The approach enables safety verification without prior knowledge of the safe set.

The method adapts to evolving safe regions in navigation tasks.

Abstract

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.