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Learning high-order CBFs using Gaussian processes for systems in Brunovsky canonical form
A. Begzadić, A. Lederer, J. Cortés, S. Herbert
Proceedings of the IEEE Conference on Decision and Control, Rio de Janeiro, Brazil, 2025, submitted

Abstract

Learning control barrier functions (CBFs) offers a promising approach to enforcing safety but doing so with formal guarantees remains challenging. This is compounded by the complexity of the dynamics considered in this paper, given by systems in Brunovsky canonical form, which naturally require high-order CBFs (HOCBFs). The signed distance function (SDF) naturally encodes HOCBF properties but is often non-smooth, limiting its applicability in learning-based methods and safety-critical control. To address this, we propose a smoothing technique that ensures continuous differentiability with respect to the relative degree of the system dynamics while preserving key safety properties of the SDF. We then leverage Gaussian Process regression to learn an HOCBF candidate from noisy measurements, providing probabilistic safety guarantees through an inner approximation of the safe set. Additionally, we establish formal feasibility guarantees for the HOCBF-based controller and ensure the safety of the resulting closed-loop dynamics with high probability. Our approach enables online adaptation by efficiently updating the learned HOCBF with new data. We demonstrate our approach in simulation on a planar system and robotic manipulator with 2DOF.

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