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 robust safety methods achieve their guarantees by always assuming that the worst-case disturbance will occur. In contrast, this paper utilizes...
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Zusammenfassung: | When deployed in the real world, safe control 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 directly 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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DOI: | 10.48550/arxiv.2403.05745 |