Stress-Testing Relative-Degree-Two Higher-Order Control Barrier Functions: Authority Limits, Numerical Robustness and Boundary Behavior

Higher-order control barrier functions (HOCBFs) provide a systematic means of enforcing safety constraints of relative degree greater than one. In practice the closed-form HOCBF filter can be compromised by insufficient control authority, numerical integration error, and initialisation on or near the boundary of the safe set. This paper presents a comprehensive stress-test suite for a relative-degree-two HOCBF applied to a three-state nonlinear system. We quantify the minimum actuation magnitude required to preserve forward invariance, demonstrate that a modest safety margin together with tightened solver tolerances eliminates spurious barrier violations, and show that a 25 % increase in the dominant nonlinearity exponent is safely rejected. Boundary initialisation produces a theoretically expected transient overshoot that cannot be instantaneously corrected by a relative-degree-two controller. Open-loop trajectories diverge catastrophically, confirming the necessity of the safety filter. All revised source code, numerical results and recommendations are supplied as supplementary material.