# High-Order Control Barrier Functions: Insights and a Truncated Taylor-Based Formulation

arXiv: https://arxiv.org/abs/2503.15014 · HTML: https://mlsanity.com/paper/2503.15014/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2025.

## Built on by (most cited first, 3 of 3)

- [Uncertainty-Aware Safety-Critical Decision and Control for Autonomous Vehicles at Unsignalized Intersections](https://mlsanity.com/paper/2505.19939/) (2025)
- [CoMoCAVs: Cohesive Decision-Guided Motion Planning for Connected and Autonomous Vehicles with Multi-Policy Reinforcement Learning](https://mlsanity.com/paper/2507.14903/) (2025)
- [A Risk-Sensitive and Uncertainty-Aware Decision-Making and Control Framework for Safe and Robust Autonomous Driving](https://mlsanity.com/paper/2609.09650/) (2026)

## Abstract (arXiv)

We examine the complexity of the standard High-Order Control Barrier Function (HOCBF) approach and propose a truncated Taylor-based approach that reduces design parameters. First, we derive the explicit inequality condition for the HOCBF approach and show that the corresponding equality condition sets a lower bound on the barrier function value that regulates its decay rate. Next, we present our Truncated Taylor CBF (TTCBF), which uses a truncated Taylor series to approximate the discrete-time CBF condition. While the standard HOCBF approach requires multiple class K functions, leading to more design parameters as the constraint's relative degree increases, our TTCBF approach requires only one. We support our theoretical findings in numerical collision-avoidance experiments and show that our approach ensures safety while reducing design complexity.