Robust Safety under Stochastic Uncertainty with Discrete-Time Control Barrier Functions

TL;DR

This paper develops probabilistic safety guarantees for discrete-time control barrier functions in stochastic systems, enabling safe robot operation under uncertainty with theoretical bounds and practical examples.

Contribution

It extends control barrier functions to discrete-time stochastic systems, providing probabilistic safety bounds and analyzing the impact of Jensen's inequality.

Findings

Probabilistic safety bounds for stochastic systems using DTCBFs

Analysis of Jensen's inequality effect on control performance

Successful safe control synthesis for robots with significant noise

Abstract

Robots deployed in unstructured, real-world environments operate under considerable uncertainty due to imperfect state estimates, model error, and disturbances. Given this real-world context, 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 associated 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 probabilistic bounds for the safety (over a finite time horizon) 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.