Stochastic Control Barrier Functions under State Estimation: From Euclidean Space to Lie Groups

Abstract: Ensuring safety for autonomous systems under uncertainty remains challenging, particularly when safety of the true state is required despite the true state not being fully known. Control barrier functions (CBFs) have become widely adopted as safety filters. However, standard CBF formulations do not explicitly account for state estimation uncertainty and its propagation, especially for stochastic systems evolving on manifolds. In this paper, we propose a safety-critical control framework with a provable bound on the finite-time safety probability for stochastic systems under noisy state information. The proposed framework explicitly incorporates the uncertainty arising from both process and measurement noise, and synthesizes controllers that adapt to the level of uncertainty. The framework admits closed-form solutions in linear settings, and experimental results demonstrate its effectiveness on systems whose state spaces range from Euclidean space to Lie groups.