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TTCBF: A Truncated Taylor Control Barrier Function for High-Order Safety Constraints

Jianye Xu, Bassam Alrifaee

TL;DR

Safety-critical control hinges on keeping the state inside a forward-invariant safe set defined by a barrier function $h$. This paper introduces TTCBF, a Truncated Taylor CBF, which handles high-relative-degree constraints in discrete time using a single class $\mathcal{K}$ function and a truncated Taylor expansion with $\\Delta T = r\\Delta t$, while bounding the remainder to guarantee forward invariance. An adaptive variant, aTTCBF, adds online gain $\\eta(t_k)$ with a coefficient-free $\\hat{\\alpha}$ to further improve adaptability, reducing design complexity relative to existing adaptive high-order CBFs. The authors formulate a CLF-CBF-QP that minimizes deviation from a nominal controller while enforcing safety and stability, and validate on a relative-degree-$6$ spring-mass system and in cluttered corridor navigation, showing improved speed tracking, reduced control effort, and fewer design parameters. Overall, TTCBF and aTTCBF offer practical, scalable safety guarantees for high-order constraints in robotics.

Abstract

Control Barrier Functions (CBFs) enforce safety by rendering a prescribed safe set forward invariant. However, standard CBFs are limited to safety constraints with relative degree one, while High-Order CBF (HOCBF) methods address higher relative degree at the cost of introducing a chain of auxiliary functions and multiple class K functions whose tuning scales with the relative degree. In this paper, we introduce a Truncated Taylor Control Barrier Function (TTCBF), which generalizes standard discrete-time CBFs to consider high-order safety constraints and requires only one class K function, independent of the relative degree. We also propose an adaptive variant, adaptive TTCBF (aTTCBF), that optimizes an online gain on the class K function to improve adaptability, while requiring fewer control design parameters than existing adaptive HOCBF variants. Numerical experiments in a relative-degree-six spring-mass system and a cluttered corridor navigation validate the above theoretical findings.

TTCBF: A Truncated Taylor Control Barrier Function for High-Order Safety Constraints

TL;DR

Safety-critical control hinges on keeping the state inside a forward-invariant safe set defined by a barrier function . This paper introduces TTCBF, a Truncated Taylor CBF, which handles high-relative-degree constraints in discrete time using a single class function and a truncated Taylor expansion with , while bounding the remainder to guarantee forward invariance. An adaptive variant, aTTCBF, adds online gain with a coefficient-free to further improve adaptability, reducing design complexity relative to existing adaptive high-order CBFs. The authors formulate a CLF-CBF-QP that minimizes deviation from a nominal controller while enforcing safety and stability, and validate on a relative-degree- spring-mass system and in cluttered corridor navigation, showing improved speed tracking, reduced control effort, and fewer design parameters. Overall, TTCBF and aTTCBF offer practical, scalable safety guarantees for high-order constraints in robotics.

Abstract

Control Barrier Functions (CBFs) enforce safety by rendering a prescribed safe set forward invariant. However, standard CBFs are limited to safety constraints with relative degree one, while High-Order CBF (HOCBF) methods address higher relative degree at the cost of introducing a chain of auxiliary functions and multiple class K functions whose tuning scales with the relative degree. In this paper, we introduce a Truncated Taylor Control Barrier Function (TTCBF), which generalizes standard discrete-time CBFs to consider high-order safety constraints and requires only one class K function, independent of the relative degree. We also propose an adaptive variant, adaptive TTCBF (aTTCBF), that optimizes an online gain on the class K function to improve adaptability, while requiring fewer control design parameters than existing adaptive HOCBF variants. Numerical experiments in a relative-degree-six spring-mass system and a cluttered corridor navigation validate the above theoretical findings.
Paper Structure (14 sections, 36 equations, 5 figures, 1 table)

This paper contains 14 sections, 36 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Truncated Taylor-Based Formulation
  4. Truncated Taylor CBF (TTCBF)
  5. Adaptive TTCBF (aTTCBF)
  6. Optimal Control Problem Formulation
  7. Numerical Experiments
  8. Spring-Mass System: Relative Degree Six
  9. Corridor Navigation
  10. Comparing aTTCBF with TTCBF
  11. Comparing aTTCBF with PACBF and RACBF
  12. Discussions
  13. Conclusions
  14. Computing Control-Design Parameters

Figures (5)

  • Figure 1: An obstacle-avoidance example with three different parameters of a class $\mathcal{K}$ function: conservative, moderate, and aggressive. Footprints: circles; trajectories: solid lines.
  • Figure 2: Spring-mass system controlled by the nominal and our controller. Forbidden areas are shown in red.
  • Figure 3: Corridor navigation with our TTCBF and aTTCBF with a linear (1st column), an exponential (2nd column), and a rational (3rd column) class $\mathcal{K}$ functions. The 1st row: robot trajectories; 2nd row: steering rate $u_1$. 3rd row: acceleration $u_2$. 4th row: speed $v$. 5th row: normalized control effort $\overline{|\hat{u}|}$ and average speed $\overline{v}$.
  • Figure 4: Comparing aTTCBF (our), PACBF, and RACBF: (a) trajectories, (b-c) control inputs, (d-e) control performance.
  • Figure 5: Comparing aTTCBF (our), PACBF, and RACBF: (a-d) adaptive parameters of five selected safety constraints (two corridor boundaries and three obstacles).