Gaussian Control Barrier Functions: Non-Parametric Paradigm to Safety

Abstract

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). A dynamical system is defined to be safe if a subset of its states remains within the prescribed set, also called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safe set</i> . CBFs achieve safety by designing a candidate function a priori. However, designing such a function can be challenging. Consider designing a CBF in a disaster recovery scenario where safe and navigable regions need to be determined. The decision boundary for safety here is unknown and cannot be designed a priori. Moreover, CBFs employ a parametric design approach and cannot handle arbitrary changes to the safe set in practice. In our approach, we work with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safety samples</i> to construct the CBF online by assuming a flexible GP prior on these samples, and term our formulation as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gaussian</i> CBF. GPs have favorable properties such as analytical tractability and robust uncertainty estimation. This allows realizing the posterior with high safety guarantees while also computing associated partial derivatives analytically for safe control. Moreover, Gaussian CBFs can change the safe set arbitrarily based on sampled data, thus allowing non-convex safe sets. We validated experimentally on a quadrotor by demonstrating safe control for 1) arbitrary safe sets, 2) collision avoidance with online safe set synthesis, 3) and juxtaposed Gaussian CBFs with CBFs in the presence of noisy states. The experiment video link is: https://youtu.be/HX6uokvCiGk.