L-CSS 2024: Bounding Stochastic Safety:: Leveraging Freedman's INequality with Discrete-Time Control Barrier Functions

Ryan K. Cosner^*, Preston Culbertson, and Aaron D. Ames.

Accepted as a L-CSS paper with a presentation at CDC 2024

Links:

• Official IEEE Xplore Version
• Free Extended arXiv Version

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Abstract

When deployed in the real world, safe con- trol methods must be robust to unstructured uncertainties such as modeling error and external disturbances. Typical robust safety methods achieve their guarantees by always assuming that the worst-case disturbance will occur.

In contrast, this paper utilizes Freedman’s inequality in the context of discrete-time control barrier functions (DTCBFs) and c-martingales 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 useful for intuitive safety 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 stronger. Finally, we present simulation examples, including a bipedal walking robot, that demonstrate the utility and tightness of our safety guarantee.

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