Bounding Stochastic Safety: Leveraging Freedman's Inequality with Discrete-Time Control Barrier Functions

When deployed in the real world, safe control methods must be robust to unstructured uncertainties such as modeling error and external disturbances. Typical approaches either leverage worst-case uncertainty bounds to provide safety guarantees for an expanded (overly conservative) safe set or synthesize controllers that always assume the worst-case disturbance will occur. In contrast, this paper utilizes Freedman's inequality in the context of discrete-time control barrier functions (DTCBFs) to provide stronger (less conservative) safety guarantees for stochastic systems. Our approach accounts for the underlying disturbance distribution instead of relying exclusively on its worst-case bound and does not require the barrier function to be upper-bounded, which makes the resulting safety probability bounds more directly useful for intuitive constraints such as signed distance. We compare our results with existing safety guarantees, such as Input-to-State-Safety (ISSf) and martingale results that rely on Ville's inequality. When the assumptions for all methods hold, we provide a range of parameters for which our guarantee is less conservative. Finally, we present simulation examples, including a bipedal walking robot, that demonstrate the utility and tightness of our safety guarantee.