In this paper, a leg structure of quadruped robot was designed, based on mechanism synthesis, and aims to allow a system for the complex control of a quadruped robot. This structure is composed of a crank linkage mechanism and quadrilateral mechanism. The innovativeness of this robot is that it can achieve a special gait driven only by single motor. In this research, we set up the kinematic model of leg mechanism through an analytic method. Then we draw the foot track of multiple groups of parameters and select the best parameter group by comparing multiple groups of parameters. After that, the robot model was built and analyzed for its walking ability and climbing ability with virtual prototype technology. Finally, we made an experimental prototype to trial the motion ability of the robot. Simulation and experimental analysis prove that the mechanism is reasonable, reliable and can meet the requirements of linear walking and climbing. The conclusions of this research will be useful for application in the field of single actuated quadruped robots.