To address the requirements of high speed and mobility for quadruped robots under unstructured environments, a structure model of hydraulically actuated single leg is proposed. For vertical hopping, a two-phase kinematic model and a three-phase dynamic model of a single leg are established under the structure model. After that, the state analysis on the vertical hopping is conducted, and the inverse kinematics solution and the simulation of the hopping are also implemented for a single leg mass-center's instant vertical hopping with 1.5m/s. Then, the ground impact on the body structure, the operation characteristics and the output force of the hip joint and knee joint's hydraulic cylinder, as well as the hydraulic system design during vertical hopping are discussed. Meanwhile, taking the periodicity of vertical hopping of a single leg into consideration, a PD (proportional-derivative) iterative learning control algorithm is applied to joint trajectory tracking, based on single leg's dynamic model with hydraulic offset. The simulation results indicate that rapid and robust convergence is achieved in trajectory tracking using the presented model, which offers the design and control references for the succeeding prototype development.