Quadruped bionic robot has a strong adaptability to the environment, compared with wheeled and tracked robots, it has superior motion performance, and has a wide range of application prospects in rescue and disaster relief, ground mine clearance, mountain transportation, so it has become a research hotspot all over the world. Leg structure is an important embodiment of the superior performance of quadruped robot, and it is also the key and difficult point of design. This article proposes a novel quadruped robot with waist structure, which can complete a variety of gait forms. Based on the theory of linkage mechanism, a novel leg structure is designed with anti-parallelogram mechanism, which improves the strength and stiffness of the robot. Using D-H description method, the kinematics analysis of this quadruped robot single leg is carried out. On this basis, in order to ensure the foot contact with the ground and achieve zero impact, polynomial programming is used to plan the foot trajectory of swing phase and support phase. Based on the static stability margin, the optimal static gait of the quadruped robot is planned. A co-simulation study has been carried out to investigate further the validity and effectiveness of the quadruped robot on gait. The simulation results clearly show the robot can walk steadily and its input and output meet the expected requirements. The solid prototype platform is built, and the trajectory planning experiment of single leg is carried out, and the foot trajectory of single leg is obtained by using laser tracker. The gait planning algorithm is applied to the whole robot, and the results show that the robot can walk according to the scheduled gait, which proves the effectiveness of the proposed algorithm.