Sampling-Aware Control Barrier Functions for Safety-Critical and Finite-Time Constrained Control

TL;DR

This work addresses safety and feasibility for nonlinear systems under sampled-data control by introducing Sampling-Aware Control Barrier Functions (SACBFs) that explicitly account for inter-sampling effects and high relative-degree constraints. SACBFs use Taylor-based upper bounds on barrier evolution between sampling instants to guarantee continuous-time forward invariance under zero-order-hold control, and a relaxation variable (r-SACBF) is added to improve feasibility when multiple constraints are enforced. The approach is validated on a unicycle robot, showing that SACBFs achieve safety and finite-time reach-and-remain where traditional HOCBF methods can fail due to inter-sampling effects, while r-SACBFs maintain feasibility in multi-constraint scenarios. The results have practical significance for safety-critical and real-time robotic systems, and future work will address stochastic disturbances and learning-based estimation of the bounds to reduce conservatism and enhance scalability.

Abstract

In safety-critical control systems, ensuring both safety and feasibility under sampled-data implementations is crucial for practical deployment. Existing Control Barrier Function (CBF) frameworks, such as High-Order CBFs (HOCBFs), effectively guarantee safety in continuous time but may become unsafe when executed under zero-order-hold (ZOH) controllers due to inter-sampling effects. Moreover, they do not explicitly handle finite-time reach-and-remain requirements or multiple simultaneous constraints, which often lead to conflicts between safety and reach-and-remain objectives, resulting in feasibility issues during control synthesis. This paper introduces Sampling-Aware Control Barrier Functions (SACBFs), a unified framework that accounts for sampling effects and high relative-degree constraints by estimating and incorporating Taylor-based upper bounds on barrier evolution between sampling instants. The proposed method guarantees continuous-time forward invariance of safety and finite-time reach-and-remain sets under ZOH control. To further improve feasibility, a relaxed variant (r-SACBF) introduces slack variables for handling multiple constraints realized through time-varying CBFs. Simulation studies on a unicycle robot demonstrate that SACBFs achieve safe and feasible performance in scenarios where traditional HOCBF methods fail.

Figures (4)

• Figure 1: Time evolution of decision variables under r-SACBF, SACBF, and HOCBF methods when initial heading angle is $\frac{\pi}{12}$. The cross symbol "$\times$" indicates that the QP is infeasible at that time step.
• Figure 2: Closed-loop trajectories with controllers derived using r-SACBF (blue), SACBF (magenta) and HOCBF (cyan) with different initial heading angle. The cross symbol "$\times$" indicates that the QP is infeasible at that location.
• Figure 3: Closed-loop trajectory generated by the r-SACBF method for the multi-constraint task. The robot avoids three obstacles and reaches the target regions at 5s, 18s, and 22s as indicated.
• Figure 4: Time evolution of the SACBF functions. The top plot shows the zero-order SACBFs, and the bottom plot shows the first-order SACBFs. The four curves correspond to three safety constraints (blue, magenta, red) and one reach-and-remain constraint (black). All functions stay positive over the 22s horizon.

Theorems & Definitions (10)

• Definition 1: Lipschitz continuity rockafellar1998variational
• Definition 2: Class $\kappa$ function Khalil:1173048
• Definition 3
• Definition 4
• Definition 5: Time-varying High Order Control Barrier Function (HOCBF) xiao2021high
• Theorem 1: Safety Guarantee xiao2021high
• Definition 6: Sampling-Aware Control Barrier Function (SACBF)
• Theorem 2
• proof
• Remark 1