Differential-geometric normal forms and structure-preserving control of single-input underactuated Euler–Lagrange systems under disturbances

This paper develops a differential-geometric and Lyapunov-based framework for the stabilization of single-input underactuated Euler–Lagrange systems in the presence of both matched and mismatched disturbances. Starting from the control-affine representation, the normal form is constructed via the Pfaffian annihilator of the distribution $$D=\text{span}\{g,[f,g]\}$$ , which is shown to be involutive for all mechanical systems with a single control input. This property ensures the existence of globally consistent internal coordinates in which the zero dynamics retain a Hamiltonian and symplectic structure. A structure-preserving feedback law is then derived that introduces damping through a projected energy term $$N\nabla H(\eta ),$$ thereby maintaining the mechanical character of the system while guaranteeing Lyapunov stability. Matched disturbances are shown to yield input-to-state stability, while mismatched perturbations lead to uniform ultimate boundedness. The proposed framework unifies geometric control, Hamiltonian preservation, and robustness analysis under a single Lyapunov energy inequality. Its effectiveness is demonstrated on the disturbed cart–pendulum benchmark, confirming both asymptotic and bounded-stability regimes. The approach provides a bridge between nonlinear geometric control and practical energy-based design for underactuated systems.