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

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.

How to Use

• For the spring-mass system, run run_spring_mass.py, or run plot_spring_mass.py directly using the saved data.
• For corridor navigation, run run_corridor_1.py to compare aTTCBF with TTCBF, or run plot_corridor_1.py directly using the saved data. Run run_corridor_2.py to compare aTTCBF with PACBF and RACBF, or run plot_corridor_2.py directly using the saved data.

Simulation Videos

• Spring-Mass System: Relative Degree Six

Our aTTCBF

Nominal Controller

• Corridor Navigation: Comparing aTTCBF with TTCBF

TTCBF and aTTCBF (Linear Class K)

TTCBF and aTTCBF (Exponential Class K)

TTCBF and aTTCBF (Rational Class K)

• Corridor Navigation: Comparing Our aTTCBF with PACBF and RACBF:

Adaptive CBFs