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Auxiliary-Variable Adaptive Control Barrier Functions

Shuo Liu, Wei Xiao, Calin A. Belta

TL;DR

This work introduces Auxiliary-Variable Adaptive Control Barrier Functions (AVCBFs) to address feasibility gaps in safety-critical control caused by mixed or high relative-degree constraints and time-varying control bounds. By embedding auxiliary variables and auxiliary dynamics, AVCBFs ensure all control inputs appear in the safety constraints and enable automatic hyperparameter tuning through a rollback-based parametrization method. The approach preserves safety via forward invariance guarantees while markedly improving feasibility over conventional CBF methods, including HOCBFs and PACBFs, across adaptive cruise control and obstacle-avoidance scenarios. The results demonstrate reduced infeasibility, enhanced adaptive safety, and diminished conservatism, with extensions to reduced relative-degree cases. The work lays a foundation for integrating AVCBFs with differentiable optimization frameworks and learning-based improvements for scalable safety-critical control.

Abstract

This paper addresses the challenge of ensuring safety and feasibility in control systems using Control Barrier Functions (CBFs). Existing CBF-based Quadratic Programs (CBF-QPs) often encounter feasibility issues due to mixed relative degree constraints, input nullification problems, and the presence of tight or time-varying control bounds, which can lead to infeasible solutions and compromised safety. To address these challenges, we propose Auxiliary-Variable Adaptive Control Barrier Functions (AVCBFs), a novel framework that introduces auxiliary variables in auxiliary functions to dynamically adjust CBF constraints without the need of excessive additional constraints. The AVCBF method ensures that all components of the control input explicitly appear in the desired-order safety constraint, thereby improving feasibility while maintaining safety guarantees. Additionally, we introduce an automatic tuning method that iteratively adjusts AVCBF hyperparameters to ensure feasibility and safety with less conservatism. We demonstrate the effectiveness of the proposed approach in adaptive cruise control and obstacle avoidance scenarios, showing that AVCBFs outperform existing CBF methods by reducing infeasibility and enhancing adaptive safety control under tight or time-varying control bounds.

Auxiliary-Variable Adaptive Control Barrier Functions

TL;DR

This work introduces Auxiliary-Variable Adaptive Control Barrier Functions (AVCBFs) to address feasibility gaps in safety-critical control caused by mixed or high relative-degree constraints and time-varying control bounds. By embedding auxiliary variables and auxiliary dynamics, AVCBFs ensure all control inputs appear in the safety constraints and enable automatic hyperparameter tuning through a rollback-based parametrization method. The approach preserves safety via forward invariance guarantees while markedly improving feasibility over conventional CBF methods, including HOCBFs and PACBFs, across adaptive cruise control and obstacle-avoidance scenarios. The results demonstrate reduced infeasibility, enhanced adaptive safety, and diminished conservatism, with extensions to reduced relative-degree cases. The work lays a foundation for integrating AVCBFs with differentiable optimization frameworks and learning-based improvements for scalable safety-critical control.

Abstract

This paper addresses the challenge of ensuring safety and feasibility in control systems using Control Barrier Functions (CBFs). Existing CBF-based Quadratic Programs (CBF-QPs) often encounter feasibility issues due to mixed relative degree constraints, input nullification problems, and the presence of tight or time-varying control bounds, which can lead to infeasible solutions and compromised safety. To address these challenges, we propose Auxiliary-Variable Adaptive Control Barrier Functions (AVCBFs), a novel framework that introduces auxiliary variables in auxiliary functions to dynamically adjust CBF constraints without the need of excessive additional constraints. The AVCBF method ensures that all components of the control input explicitly appear in the desired-order safety constraint, thereby improving feasibility while maintaining safety guarantees. Additionally, we introduce an automatic tuning method that iteratively adjusts AVCBF hyperparameters to ensure feasibility and safety with less conservatism. We demonstrate the effectiveness of the proposed approach in adaptive cruise control and obstacle avoidance scenarios, showing that AVCBFs outperform existing CBF methods by reducing infeasibility and enhancing adaptive safety control under tight or time-varying control bounds.

Paper Structure

This paper contains 24 sections, 4 theorems, 63 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation and Approach
  4. Auxiliary-Variable Adaptive Control Barrier Functions
  5. Motivation Example: Simplified Adaptive Cruise Control
  6. Adaptive HOCBFs for Safety: AVCBFs
  7. Optimal Control with AVCBFs: A Parametrization Method for Tuning Hyperparameters
  8. Implementation and Results
  9. Adaptive Cruise Control for High-Order Relative Degree Systems
  10. Vehicle Dynamics
  11. Vehicle Limitations
  12. Implementation with AVCBFs
  13. Implementation with PACBFs
  14. Comparison between AVCBFs and PACBFs
  15. Implementation with AVCBFs with Reduced Relative Degree
  16. ...and 9 more sections

Key Result

Lemma 1

Let $b : [t_0, t_1] \to \mathbb{R}$ be a continuously differentiable function. If $\dot{b}(t) \geq -\alpha(b(t))$ for all $t \in [t_0, t_1]$, where $\alpha$ is a class $\mathcal{K}$ function of its argument, and $b(t_0) \geq 0$, then $b(t) \geq 0$ for all $t \in [t_0, t_1]$.

Figures (11)

  • Figure 1: Control input $u(t)$ varies as $b(\boldsymbol{x}(t))$ goes to 0 under different lower control bounds. The arrows denote the changing trend for $b(\boldsymbol{x}(t))$ and $c_{d}(t)$ over 50 seconds. $b(\boldsymbol{x}(0))=90$ and $b(\boldsymbol{x}(t))\ge 0$ implies safety. Different sets of hyperparameters are tested.
  • Figure 2: Control input $u(t)$, velocity $v(t)$, time-varying $p_{1}(t), \frac{\dot{a}_{1}(t)}{a_{1}(t)}$ and distance between two vehicles $b(\boldsymbol{x}(t))$ over 30 seconds for AVCBFs and PACBFs. $b(\boldsymbol{x}(t))\ge 0$ implies safety. Solid curves denote AVCBFs and dashed curve denotes PACBFs.
  • Figure 3: Control input $u(t)$ varies with $b(\boldsymbol{x}(0))$ under different lower control bounds. The arrows denote the changing trend for $b(\boldsymbol{x}(t))$ and $c_{d}(t)$ over 30 seconds. $b(\boldsymbol{x}(0))=90$ and $b(\boldsymbol{x}(t))\ge 0$ implies safety. Different sets of hyperparameters are tested.
  • Figure 4: Closed-loop trajectories over time with different controllers: several safe closed-loop trajectories starting at the initial locations (depicted by asterisk, magenta: $(x(0), y(0))=(-3,0)$; orange: $(x(0), y(0))=(-2.5,0)$; blue: $(x(0), y(0))=(-2,0)$) terminates within the target areas (depicted by green circle, solid: $(x_{d}, y_{d})=(1.5,0)$; dashed: $(x_{d}, y_{d})=(3,0)$). The cross symbol indicates that the QP is infeasible at this time step. Different sets of hyperparameters are tested.
  • Figure 5: Analysis of safety and feasibility for different controllers (Fig. \ref{['fig:cbf-candidate']} and Fig. \ref{['fig:first-order-candidate']}), and hyperparameter adaptation in AVCBF-P (Fig. \ref{['fig:avcbf-parameter']} and Fig. \ref{['fig:avcbf-iteration']}), where $x(0)=-3m,x_{d}=1.5m,k_{1}=k_{2}=10, a_{1,w}(0)=0$.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Definition 1: Class $\kappa$ function Khalil:1173048
  • Definition 2
  • Definition 3
  • Lemma 1: glotfelter2017nonsmooth
  • Definition 4: HOCBF xiao2021high
  • Theorem 1: Safety Guarantee xiao2021high
  • Definition 5: CLF ames2012control
  • Definition 6: Minimum Relative Degree
  • Remark 1: Auxiliary Functions with Reduced Relative Degree
  • Remark 2
  • ...and 8 more