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

arXiv: https://arxiv.org/abs/2302.07469 · HTML: https://mlsanity.com/paper/2302.07469/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2023.

## Built on by (most cited first, 5 of 5)

- [Contextual Safety Reasoning and Grounding for Open-World Robots](https://mlsanity.com/paper/2602.19983/) (2026)
- [Sequential QCQP for Bilevel Optimization with Line Search](https://mlsanity.com/paper/2505.14647/) (2025)
- [Distributed Risk-Sensitive Safety Filters for Uncertain Discrete-Time Systems](https://mlsanity.com/paper/2506.07347/) (2025)
- [Safe Probabilistic Planning for Human-Robot Interaction using Conformal Risk Control](https://mlsanity.com/paper/2603.10392/) (2026)
- [Runtime Safety Filtering for Learned Small UAS Separation Policies under GNSS Degradation](https://mlsanity.com/paper/2607.10014/) (2026)

## Abstract (arXiv)

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.