To address the requirements of high speed and mobility for quadruped robots galloping, a composite rigid-flexible structure model is proposed for a single leg of quadruped robots. The composite rigid-flexible structure is designed for a single leg of quadruped robots, and operating characteristics with or without impact loading, stiffness characteristics, D-H kinematics and toe work space are analyzed. After that, taking noise factor such as ground impact under dynamic motion into consideration, linear quadratic Gaussian control is applied to driver control of a single leg of quadruped robots. Then, with vertical hopping as an important basic motion prototype for galloping, analysis and simulation of equal-height jumping control under disabled environment are conducted based on finite state machine and peak height feedback. Finally, continuous vertical hopping experiment of a single leg with composite rigid-flexible structure of quadruped robots is conducted. Analysis and experiment results indicate the rationality of the structure design and the effectiveness of the control scheme.