Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions

Learning-based control has demonstrated great promise for handling complex tasks in various applications. However, ensuring system safety under uncertain dynamics remains a significant challenge. Control Barrier Functions (CBFs) offer mathematical tools for enforcing safety constraints given known system dynamics, yet their guarantees can be lost in the presence of model errors. In this article, we present a framework that combines model-based safety methods with data-driven techniques to guarantee safety recursively for systems with uncertain dynamics. We build upon our previous work, where Gaussian Process (GP) regression was utilized to quantify uncertainty in model-based CBF constraints, resulting in a second-order cone program (SOCP) controller. When the SOCP is feasible at a state, it provides a pointwise probabilistic safety guarantee. A critical innovation we develop further in this work is an eventtriggered online data collection algorithm that actively and safely gathers data to provide the recursive feasibility of the SOCP-based controller. By continuously assessing the sufficiency of data based on the feasibility measure of the SOCP, our method triggers safe exploratory actions when necessary to reduce the uncertainty in critical control directions. This approach ensures that a feasible, safety-preserving control input is always available, thereby establishing forward invariance of the safe set with high probability, even in previously unexplored regions. We validate the proposed framework through two numerical simulation experiments.