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Learning-based Parameterized Barrier Function for Safety-Critical Control of Unknown Systems

Sihua Zhang, Di-Hua Zhai, Xiaobing Dai, Tzu-yuan Huang, Yuanqing Xia, Sandra Hirche

TL;DR

The paper tackles safety-critical control for unknown disturbances by integrating Gaussian process regression with parameterized high-order control barrier functions. It derives a deterministic GP prediction error bound and uses it to construct a shrunk, parameterized safe set via GP-P-HOCBFs, ensuring forward invariance of the original safety constraints while improving feasibility in the associated quadratic program. The approach provides a tunable balance between safety conservatism and control feasibility and is validated on a 7-DOF Franka Emika manipulator, demonstrating safer obstacle avoidance and trajectory tracking under disturbance uncertainty. This work advances safe control under model mismatch by offering a flexible barrier-function design that directly incorporates learning-induced uncertainty bounds into real-time optimization.

Abstract

With the increasing complexity of real-world systems and varying environmental uncertainties, it is difficult to build an accurate dynamic model, which poses challenges especially for safety-critical control. In this paper, a learning-based control policy is proposed to ensure the safety of systems with unknown disturbances through control barrier functions (CBFs). First, the disturbance is predicted by Gaussian process (GP) regression, whose prediction performance is guaranteed by a deterministic error bound. Then, a novel control strategy using GP-based parameterized high-order control barrier functions (GP-P-HOCBFs) is proposed via a shrunk original safe set based on the prediction error bound. In comparison to existing methods that involve adding strict robust safety terms to the HOCBF condition, the proposed method offers more flexibility to deal with the conservatism and the feasibility of solving quadratic problems within the CBF framework. Finally, the effectiveness of the proposed method is demonstrated by simulations on Franka Emika manipulator.

Learning-based Parameterized Barrier Function for Safety-Critical Control of Unknown Systems

TL;DR

The paper tackles safety-critical control for unknown disturbances by integrating Gaussian process regression with parameterized high-order control barrier functions. It derives a deterministic GP prediction error bound and uses it to construct a shrunk, parameterized safe set via GP-P-HOCBFs, ensuring forward invariance of the original safety constraints while improving feasibility in the associated quadratic program. The approach provides a tunable balance between safety conservatism and control feasibility and is validated on a 7-DOF Franka Emika manipulator, demonstrating safer obstacle avoidance and trajectory tracking under disturbance uncertainty. This work advances safe control under model mismatch by offering a flexible barrier-function design that directly incorporates learning-induced uncertainty bounds into real-time optimization.

Abstract

With the increasing complexity of real-world systems and varying environmental uncertainties, it is difficult to build an accurate dynamic model, which poses challenges especially for safety-critical control. In this paper, a learning-based control policy is proposed to ensure the safety of systems with unknown disturbances through control barrier functions (CBFs). First, the disturbance is predicted by Gaussian process (GP) regression, whose prediction performance is guaranteed by a deterministic error bound. Then, a novel control strategy using GP-based parameterized high-order control barrier functions (GP-P-HOCBFs) is proposed via a shrunk original safe set based on the prediction error bound. In comparison to existing methods that involve adding strict robust safety terms to the HOCBF condition, the proposed method offers more flexibility to deal with the conservatism and the feasibility of solving quadratic problems within the CBF framework. Finally, the effectiveness of the proposed method is demonstrated by simulations on Franka Emika manipulator.
Paper Structure (10 sections, 4 theorems, 28 equations, 3 figures)

This paper contains 10 sections, 4 theorems, 28 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Control barrier function
  4. High order control barrier function
  5. PROBLEM STATEMENT
  6. MAIN RESULT
  7. Gaussian Process
  8. GP-based parameterized HOCBF
  9. SIMULATION
  10. CONCLUSIONS

Key Result

Lemma 1

Given the set $C$ defined by C for a continuous differentiable function $h(\bm{x})$, if $h(\bm{x})$ is a CBF, then Lipschitz continuous control input $\bm{u}(t) \in K_{cbf}(\bm{x}) = \{ \bm{u} \in \mathbb{U}: \frac{\partial h(\bm{x})}{\partial \bm{x}} (\bm{f}(\bm{x})+\bm{g}(\bm{x})\bm{u} + \bm{d}(

Figures (3)

  • Figure 1: The trajectory of end-effector and safety constraint function $h(\bm{q})$. (a)The trajectory of end-effector under different controllers. (b)The curves of $h(\bm{q})$ under different controllers.
  • Figure 2: The curves of $h(\bm{q})$ and $h^*(\bm{q})$ with different $\lambda$.
  • Figure 3: The control input of the second joint $\bm{u}_2$ solved by different control methods.

Theorems & Definitions (13)

  • Definition 1: Class $\mathcal{K}$ Function hassan2002nonlinear
  • Definition 2: Relative Degree hassan2002nonlinear
  • Definition 3: Control Barrier Function
  • Lemma 1: jankovic2018robust
  • Definition 4: HOCBF xiao2021high
  • Lemma 2: xiao2021high
  • Lemma 3: hashimoto2022learning
  • Definition 5: GP-P-HOCBF
  • Theorem 1
  • proof
  • ...and 3 more