Abstract: Robots deployed in unstructured, real-world environments operate under considerable uncertainty due to imperfect state estimates, model error, and disturbances. The goal of this paper is to develop controllers that are provably safe under uncertainties. To this end, we leverage Control Barrier Functions (CBFs) which guarantee that a robot remains in a ``safe set’’ during its operation—yet CBFs (and their guarantees) are traditionally studied in the context of continuous-time, deterministic systems with bounded uncertainties. In this work, we study the safety properties of discrete-time CBFs (DTCBFs) for systems with discrete-time dynamics and unbounded stochastic disturbances. Using tools from martingale theory, we develop bounds for the finite-time safety of systems whose dynamics satisfy the discrete-time barrier function condition in expectation, and analyze the effect of Jensen’s inequality on DTCBF-based controllers. Finally we present several examples of our method synthesizing safe control inputs for systems subject to significant process noise, including an inverted pendulum, a double integrator, and a quadruped locomoting on a narrow path.
