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Bounding Stochastic Safety: Leveraging Freedman's Inequality with Discrete-Time Control Barrier Functions

Ryan K. Cosner, Preston Culbertson, Aaron D. Ames

TL;DR

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.

Abstract

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.

Bounding Stochastic Safety: Leveraging Freedman's Inequality with Discrete-Time Control Barrier Functions

TL;DR

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.

Abstract

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.
Paper Structure (18 sections, 12 theorems, 65 equations, 3 figures)

This paper contains 18 sections, 12 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Safety and Discrete-Time Control Barrier Functions
  4. Existing Martingale-based Safety Methods
  5. Safety Guarantees using Freedman's Inequality
  6. Main Result: Freedman's Inequality for Safety
  7. Bound Tightness Comparison
  8. Extending Input-to-State Safety
  9. Case Study: Bipedal Obstacle Avoidance
  10. Conclusion
  11. Proof of Ville's Inequality
  12. Proof of Theorem \ref{['thm:rss']}
  13. Ville's-based Safety Theorems from santoyo_barrier_2021 and steinhardt_finite-time_2012
  14. Proof of Freedman's Inequality, (Theorem \ref{['thm:azuma_hoeffding']})
  15. Proof of Proposition \ref{['thm:comparison']}
  16. ...and 3 more sections

Key Result

Lemma 1

If $W_k$is a nonnegative supermartingale, then for all $\lambda > 0$,

Figures (3)

  • Figure 1: Safety results for a bipedal robot navigating around an obstacle using our method. Details are provided in Section \ref{['example:hlip']}. (Top) Visualization of the Hybrid Linear Inverted Pendulum (HLIP) model. Yellow indicates the center-of-mass (COM), blue is the stance foot, and red is the swing foot. The states $\mathbf{ x }_k$ are the global COM position, the relative COM position, and COM velocity, and the input is the relative position of the feet at impact. (Bottom) A table with variable maximum disturbance value $(d_\textup{max})$ and controller parameter ($\alpha$) shows our (dashed lines) theoretical bound on safety failure from Thm. \ref{['thm:main']}, (dotted lines) the shortest first-violation time based on the worst-case disturbance approximation, and (solid lines) approximated probabilities from 5000 trials (lower is safer). On the left, the trajectories of the COM are shown walking from bottom left towards the top right while avoiding the obstacle with each color corresponding to a different $d_\textup{max}$. The robot attempts to avoid the obstacle (black). Code to reproduce this plot can be found at codebase.
  • Figure 2: Comparison for Prop. \ref{['thm:comparison']} with $B=10, K = 100, \delta = 1,$ and varying $\sigma$ and $\lambda$. The Freedman-based bounds are shown in green when the conditions of Prop.\ref{['thm:comparison']} hold and blue when they do not. The Ville's-based bound is shown in red. Code to reproduce this plot can be found at codebase
  • Figure 3: Probability that the system is unsafe: our bound from Cor. \ref{['cor:issf']} (blue), ISSf bound (red). The $x$-axis is the level set expansion $-\epsilon$ and the $y$-axis is the failure probability (lower is better). The plots from left to right indicate safety for $K = 1, 100, 200, 300,$ and $400$ steps. Simulations where $\mathbb{E}[h(\mathbf{ x }_k) | \mathscr{F}_{k-1}] = \alpha h(\mathbf{ x }_k)$ and approximate probabilities from 1000 samples are shown for simulations where $h(\mathbf{ x }_k)$ is sampled from 3 different conditional distributions: uniform (pink), truncated Gaussian (green), and a categorical (yellow) all which satisfy Cor. \ref{['cor:issf']}. Code for these plots is can be found at codebase.

Theorems & Definitions (26)

  • Definition 1: Forward Invariance and Safety
  • Definition 2: Discrete-Time Control Barrier Function (DT-CBF) agrawal_discrete_2017
  • Definition 3: $K$-step Exit Probability
  • Definition 4: Martingale grimmett_probability_2020, steinhardt_finite-time_2012
  • Lemma 1: Ville's Inequality ville1939etude
  • Theorem 1: Safety using Ville's Inequality, cosner_robust_2023steinhardt_finite-time_2012kushner_stochastic_1967santoyo_barrier_2021
  • Definition 5: Predictable Quadratic Variation (PQV) grimmett_probability_2020
  • Theorem 2: Freedman's Inequality freedman1975tail
  • Theorem 3
  • proof
  • ...and 16 more