Output tracking and safety-critical control for affine nonlinear systems with high-relative-degree outputs and safety constraints

This paper introduces a dual superlevel sets framework for affine nonlinear systems with high-relative-degree outputs and safety-critical constraints, where output tracking (treated as soft constraints) and safety enforcement (treated as hard constraints) are unified via hierarchical control barrier functions (CBFs). By constructing cascaded high-order derivative sets initialized from these super-level sets, convergence and safety conditions are rigorously established. To circumvent the computational complexity of traditional high-order CBF methods, an online Lyapunov-like condition solver is first proposed by exploiting the Faà-di-Bruno's formula, which explicitly encodes high-order derivative constraints (up to the system relative degree) into a real-time quadratic programming (QP) problem. The resulting QP problem incorporates n-th-order Lie derivatives of safety and output constraints as algebraic conditions, while controller feasibility is ensured through the introduction of a slack variable. Finally, the effectiveness of the proposed method is demonstrated through single-link manipulator.