A bounded input control method based on biologically inspired models is proposed for underactuated horizontal translational oscillators with rotating actuator (TORA) to realize the stabilization control of the system with actuator saturation constraint. Firstly, the passivity property is analyzed based on the dynamics of the TORA system, and the control objective is given. Next, a novel passivity-based Lyapunov function is constructed, and a simple nonlinear state feedback controller is designed straightforwardly. Then, considering the actuator saturation constraint, a biologically inspired neural dynamic model is introduced, and an improved state feedback controller is designed by using the bounded smooth output characteristics of the neural dynamics. Finally, the stability of the control system is strictly analyzed and proven using the LaSalle invariance principle. Compared with other methods, the proposed method not only takes the actuator saturation constraint into account, but also provides a simple but efficient control algorithm, which is easy to implement in engineering. Simulation and comparison results show that the proposed method has better control performance.