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Probabilistic Control Barrier Functions: Safety in Probability for Discrete-Time Stochastic Systems

Pol Mestres, Blake Werner, Ryan K. Cosner, Aaron D. Ames

TL;DR

The paper addresses safety for discrete-time systems subject to stochastic disturbances by introducing δ-probabilistic control barrier functions (CBFs) that guarantee safety over a finite horizon with prescribed probability. It develops both moment-based (Markov-based and Cantelli-based) and data-driven (Hoeffding, scenario, conformal prediction) conditions to certify probabilistic CBFs, with convexity considerations to support optimization-based control. The authors validate the approach through extensive simulation and hardware experiments on a quadruped robot, showing reduced safety violations and practicality under uncertainty. This framework enables finite-horizon probabilistic safety guarantees in settings with unbounded or uncertain disturbances, offering tractable tools for safe controller design in robotics and related domains.

Abstract

Control systems operating in the real world face countless sources of unpredictable uncertainties. These random disturbances can render deterministic guarantees inapplicable and cause catastrophic safety failures. To overcome this, this paper proposes a method for designing safe controllers for discrete-time stochastic systems that retain probabilistic guarantees of safety. To do this we modify the traditional notion of a control barrier function (CBF) to explicitly account for these stochastic uncertainties and call these new modified functions probabilistic CBFs. We show that probabilistic CBFs can be used to design controllers that guarantee safety over a finite number of time steps with a prescribed probability. Next, by leveraging various uncertainty quantification methods, such as concentration inequalities, the scenario approach, and conformal prediction, we provide a variety of sufficient conditions that result in computationally tractable controllers with tunable probabilistic guarantees across a plethora of practical scenarios. Finally, we showcase the applicability of our results in simulation and hardware for the control of a quadruped robot.

Probabilistic Control Barrier Functions: Safety in Probability for Discrete-Time Stochastic Systems

TL;DR

The paper addresses safety for discrete-time systems subject to stochastic disturbances by introducing δ-probabilistic control barrier functions (CBFs) that guarantee safety over a finite horizon with prescribed probability. It develops both moment-based (Markov-based and Cantelli-based) and data-driven (Hoeffding, scenario, conformal prediction) conditions to certify probabilistic CBFs, with convexity considerations to support optimization-based control. The authors validate the approach through extensive simulation and hardware experiments on a quadruped robot, showing reduced safety violations and practicality under uncertainty. This framework enables finite-horizon probabilistic safety guarantees in settings with unbounded or uncertain disturbances, offering tractable tools for safe controller design in robotics and related domains.

Abstract

Control systems operating in the real world face countless sources of unpredictable uncertainties. These random disturbances can render deterministic guarantees inapplicable and cause catastrophic safety failures. To overcome this, this paper proposes a method for designing safe controllers for discrete-time stochastic systems that retain probabilistic guarantees of safety. To do this we modify the traditional notion of a control barrier function (CBF) to explicitly account for these stochastic uncertainties and call these new modified functions probabilistic CBFs. We show that probabilistic CBFs can be used to design controllers that guarantee safety over a finite number of time steps with a prescribed probability. Next, by leveraging various uncertainty quantification methods, such as concentration inequalities, the scenario approach, and conformal prediction, we provide a variety of sufficient conditions that result in computationally tractable controllers with tunable probabilistic guarantees across a plethora of practical scenarios. Finally, we showcase the applicability of our results in simulation and hardware for the control of a quadruped robot.

Paper Structure

This paper contains 11 sections, 9 theorems, 36 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Statement of Contributions
  4. Notation
  5. Motivation
  6. Verification of Probabilistic CBFs using known Moments
  7. Verification of Probabilistic CBFs Using Data
  8. Experimental Validation
  9. Simulation
  10. Hardware
  11. Conclusion

Key Result

Proposition 1

(Probabilistic guarantees over a finite time horizon): Let ${\mathbf{x}}_0\in{\mathcal{C}}$, $H\in\mathbb{N}$, and ${\mathbf{k}}:\mathbb{R}^n\to\mathbb{R}^m$ be such that ${\mathbf{u}} = {\mathbf{k}}({\mathbf{x}})$ satisfies eq:probability-h-geq0 for each ${\mathbf{x}}\in{\mathcal{C}}$. Then, for th In particular, for any $\epsilon > 0$, if then ${\mathbf{k}}$ is $\epsilon$-safe over a time horiz

Figures (4)

  • Figure 1: (left) Number of unsafe trajectories for different values of the disturbance variance $\sigma$. (right) Number of unsafe trajectories for different values of the trajectory length. No filter refers to the implementation of the nominal controller, whereas Markov, Cantelli, Hoeffding, Scenario, and Conformal refer to the approaches in Propositions \ref{['prop:markov-based-condition']}, \ref{['prop:probabilistic-cbfs-give-confidence-hoeffding']}, \ref{['prop:probabilistic-CBFs-given-confidence-scenario-approach']}, \ref{['prop:probabilistic-cbfs-with-given-confidence-using-conformal-prediction']}, respectively.
  • Figure 2: Evolution of 400 different trajectories with initial condition at ${\mathbf{x}}_0 = [0, 0, 0]$ and $\sigma = 0.06$ and trajectory length $20$ for different methods introduced in the paper, along with the nominal controller (left).
  • Figure 3: Experimental Setup
  • Figure 4: Evolution of the value of $h$ for two different trajectories for each of the different approaches considered in the hardware experiment.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • proof
  • Remark 1
  • Proposition 4
  • ...and 14 more