Fixed time convergence guarantees for Higher Order Control Barrier Functions

TL;DR

This paper introduces a new method for designing higher-order Control Barrier Functions that guarantee convergence to a safe set within a fixed, user-defined time, enhancing safety in time-critical robotic applications.

Contribution

It proposes a structured differential constraint using repeated roots in the characteristic polynomial to achieve fixed-time convergence for higher-order CBFs, with explicit formulations and conditions.

Findings

Ensures fixed-time convergence for second-order systems.

Demonstrates reliability across robotic models.

Outperforms traditional HOCBFs in fixed-time guarantees.

Abstract

We present a novel method for designing higher-order Control Barrier Functions (CBFs) that guarantee convergence to a safe set within a user-specified finite. Traditional Higher Order CBFs (HOCBFs) ensure asymptotic safety but lack mechanisms for fixed-time convergence, which is critical in time-sensitive and safety-critical applications such as autonomous navigation. In contrast, our approach imposes a structured differential constraint using repeated roots in the characteristic polynomial, enabling closed-form polynomial solutions with exact convergence at a prescribed time. We derive conditions on the barrier function and its derivatives that ensure forward invariance and fixed-time reachability, and we provide an explicit formulation for second-order systems. Our method is evaluated on three robotic systems - a point-mass model, a unicycle, and a bicycle model and benchmarked against existing HOCBF approaches. Results demonstrate that our formulation reliably enforces convergence within the desired time, even when traditional methods fail. This work provides a tractable and robust framework for real-time control with provable finite-time safety guarantees.