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Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations

Lars Lindemann, Alexander Robey, Lejun Jiang, Satyajeet Das, Stephen Tu, Nikolai Matni

TL;DR

This work tackles safety-critical control when only partial state information is available and model/estimation errors exist. It introduces Robust Output Control Barrier Functions (ROCBFs) to guarantee forward invariance of a safe set under uncertain dynamics and perception maps, and formulates a constrained optimization to learn ROCBFs from safe expert demonstrations. The authors establish verifiable conditions linking data density, Lipschitz continuity, and error bounds to ROCBF validity, and present an algorithmic pipeline including boundary-point detection and an unconstrained relaxation for practical learning. Validation in CARLA with both state-based and perception-based (RGB image) inputs demonstrates that safe, data-driven output feedback laws can be learned and deployed in perception-rich environments, enabling safer autonomous driving with uncertain sensing and dynamics.

Abstract

This paper addresses learning safe output feedback control laws from partial observations of expert demonstrations. We assume that a model of the system dynamics and a state estimator are available along with corresponding error bounds, e.g., estimated from data in practice. We first propose robust output control barrier functions (ROCBFs) as a means to guarantee safety, as defined through controlled forward invariance of a safe set. We then formulate an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator or an expert controller. When the parametrization of the ROCBF is linear, then we show that, under mild assumptions, the optimization problem is convex. Along with the optimization problem, we provide verifiable conditions in terms of the density of the data, smoothness of the system model and state estimator, and the size of the error bounds that guarantee validity of the obtained ROCBF. Towards obtaining a practical control algorithm, we propose an algorithmic implementation of our theoretical framework that accounts for assumptions made in our framework in practice. We validate our algorithm in the autonomous driving simulator CARLA and demonstrate how to learn safe control laws from simulated RGB camera images.

Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations

TL;DR

This work tackles safety-critical control when only partial state information is available and model/estimation errors exist. It introduces Robust Output Control Barrier Functions (ROCBFs) to guarantee forward invariance of a safe set under uncertain dynamics and perception maps, and formulates a constrained optimization to learn ROCBFs from safe expert demonstrations. The authors establish verifiable conditions linking data density, Lipschitz continuity, and error bounds to ROCBF validity, and present an algorithmic pipeline including boundary-point detection and an unconstrained relaxation for practical learning. Validation in CARLA with both state-based and perception-based (RGB image) inputs demonstrates that safe, data-driven output feedback laws can be learned and deployed in perception-rich environments, enabling safer autonomous driving with uncertain sensing and dynamics.

Abstract

This paper addresses learning safe output feedback control laws from partial observations of expert demonstrations. We assume that a model of the system dynamics and a state estimator are available along with corresponding error bounds, e.g., estimated from data in practice. We first propose robust output control barrier functions (ROCBFs) as a means to guarantee safety, as defined through controlled forward invariance of a safe set. We then formulate an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator or an expert controller. When the parametrization of the ROCBF is linear, then we show that, under mild assumptions, the optimization problem is convex. Along with the optimization problem, we provide verifiable conditions in terms of the density of the data, smoothness of the system model and state estimator, and the size of the error bounds that guarantee validity of the obtained ROCBF. Towards obtaining a practical control algorithm, we propose an algorithmic implementation of our theoretical framework that accounts for assumptions made in our framework in practice. We validate our algorithm in the autonomous driving simulator CARLA and demonstrate how to learn safe control laws from simulated RGB camera images.
Paper Structure (26 sections, 6 theorems, 43 equations, 7 figures, 2 algorithms)

This paper contains 26 sections, 6 theorems, 43 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Background and Problem Formulation
  5. Robust Output Control Barrier Functions
  6. Learning ROCBFs from Expert Demonstrations
  7. The Datasets
  8. The Constrained Optimization Problem
  9. Conditions guaranteeing learned safe ROCBFs
  10. Guaranteeing $\mathcal{C}\subset\mathcal{D}\subseteq\mathcal{S}$
  11. Increasing the volume of $\mathcal{C}$
  12. Guaranteeing that $h(x)$ is a ROCBF
  13. Algorithmic Implementation
  14. The Algorithm
  15. Construction of the Datasets
  16. ...and 11 more sections

Key Result

Theorem 1

Assume that $h(x)$ is a ROCBF on the set $\mathcal{Y}$ that is such that $\mathcal{Y}\supseteq Y(\mathcal{C})$, and assume that the function $U:\mathcal{Y}\times \mathbb{R}_{\ge 0}\to \mathcal{U}$ is continuous in the first and piecewise continuous in the second argument and such that $U(y,t)\in \ma

Figures (7)

  • Figure 1: Uncertain system under consideration.
  • Figure 2: Proposed framework to learn safe control laws.
  • Figure 3: Problem Setup (left): The set of observed safe expert demonstrations $Z_{\mathrm{dyn}}$ (black lines). Also shown, the set of admissible output measurements $\mathcal{Y}$ (orange ring). Transformation into state domain (centre): The geometric safe set $\mathcal{S}$ (red box) and the set of admissible safe states $\mathcal{D}$ (green region) that is defined as the union of $\epsilon$-balls centered at $\hat{X}(y_i)$. Learned safe set (right): The set of as unsafe labeled states ${\mathcal{N}}$ (golden ring) that defines the $\sigma$-layer surrounding $\mathcal{D}$.
  • Figure 4: Simulation environment in CARLA. The cars are tracking desired reference paths on different courses. Left: The training course from which training data, during a left turn, was generated to train and test the ROCBF. Middle: An unknown test course on which the learned ROCBF is tested. Right: Downsampled RGB dashboard camera image.
  • Figure 5: State-based ROCBF controller.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Lemma 1