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

Jianye Xu, Bassam Alrifaee

TL;DR

The paper addresses the complexity of High-Order Control Barrier Functions (HOCBFs) when enforcing safety for high-relativedegree constraints by introducing a Truncated Taylor-based CBF (TTCBF) that uses a Taylor expansion of the forward-step change in the barrier function. It provides an explicit form and lower bound for the standard HOCBF condition, clarifying how the $\mathcal{K}$-function parameters and initial derivatives shape safety, and then shows that a truncated Taylor expansion yields a sufficient discrete-time CBF condition that requires only a single $\mathcal{K}$-function. The TTCBF reduces tuning effort and design complexity while preserving safety, as demonstrated in numerical collision-avoidance experiments where the analytical lower bounds align with observed behavior. Overall, the approach offers a scalable alternative for high relative-degree safety constraints in real-time control tasks with limited parameter tuning. The practical impact lies in easier deployment of safe controllers for robots and other dynamical systems encountering complex constraints.

Abstract

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.

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

TL;DR

The paper addresses the complexity of High-Order Control Barrier Functions (HOCBFs) when enforcing safety for high-relativedegree constraints by introducing a Truncated Taylor-based CBF (TTCBF) that uses a Taylor expansion of the forward-step change in the barrier function. It provides an explicit form and lower bound for the standard HOCBF condition, clarifying how the -function parameters and initial derivatives shape safety, and then shows that a truncated Taylor expansion yields a sufficient discrete-time CBF condition that requires only a single -function. The TTCBF reduces tuning effort and design complexity while preserving safety, as demonstrated in numerical collision-avoidance experiments where the analytical lower bounds align with observed behavior. Overall, the approach offers a scalable alternative for high relative-degree safety constraints in real-time control tasks with limited parameter tuning. The practical impact lies in easier deployment of safe controllers for robots and other dynamical systems encountering complex constraints.

Abstract

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.

Paper Structure

This paper contains 11 sections, 4 theorems, 26 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Insights Into hocbf
  5. Truncated Taylor cbf
  6. Numerical Experiments
  7. qp Formulation
  8. First Experiment: hocbf Approach
  9. Second Experiment: Our ttcbf Approach
  10. Discussions
  11. Conclusions

Key Result

Lemma 1

Consider an hocbf $h$ in def:continuous-t-hocbf with relative degree $r$. For each $i \in \{1, \ldots, r\}$, the corresponding auxiliary function defined in eq:chain-hocbf with a linear class $\mathcal{K}$ function can be flattened and equivalently expressed as Here, $e_{i-j} \bigl( \lambda_{1}, \ldots, \lambda_{i} \bigr) \coloneqq \sum_{\substack{I \subseteq \{1, \ldots, i \} \\ | I |=i-j}} \pro

Figures (3)

  • Figure 1: An obstacle-avoidance example with three different class $\mathcal{K}$ functions: conservative, moderate, and aggressive. Footprints in circles; trajectories in solid lines.
  • Figure 2: Experimental results from the standard hocbf approach with different parameters for the class $\mathcal{K}$ functions.
  • Figure 3: Experimental results from our Taylor-based approach with different parameters for the class $\mathcal{K}$ function.

Theorems & Definitions (16)

  • Definition 1: Class $\mathcal{K}$ functions
  • Definition 2: Forward invariant set
  • Definition 3: cbf ames2019control
  • Definition 4: Discrete-Time cbf agrawal2017discrete
  • Definition 5: Relative degree
  • Definition 6: hocbf xiao2019control
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 6 more