A new stabilization control algorithm based on controlled Lagrangian method for a class of mechanical systems with underactuation degree one is presented in this paper. Firstly, a desired controlled system with the Lagrangian structure and desired properties is constructed. By equating the underactuated system with the desired system, the matching condition and controller structure are determined. A sufficient condition for the matching condition to be held is derived, and from the sufficient condition desired kinetic energy, potential energy, gyroscopic forces and dissipative forces of the desired system can be solved explicitly. Compared with the existing matching conditions, to solve proposed sufficient condition at most two partial differential equations needs to be solved, and the rests are all algebraic equations, which is easier to solve.
An algorithm to solve this sufficient condition is given in detail, and a nonlinear smooth feedback control law can be obtained to stabilize the underactuated systems. Finally, the novel control algorithm is applied to achieve almost global stability for a vertical takeoff and landing aircraft and to locally stabilize a Pendubot with two degrees of freedom at the highest point. Simulation results demonstrate effectiveness of the proposed method.