In order to improve the walking stability and obstacle jumping ability of the robot on the slope, the goat was taken as the bionic prototype, a kind of bionic goat quadruped robot has been designed. By the bionic principle, the mechanical structure of the quadruped robot was designed, and the Denavit-Hartenberg (D-H) method was used to build the kinematic model, through which the forward and inverse kinematics of the robot was calculated, thus the equations of velocity and acceleration of the joint angle change were obtained during the quadruped robot motion, providing control theory foundation for robots. During walk gait planning, a low contact compact foot trajectory planning method using a high order polynomial curve was used to carry out the foot trajectory planning of the robot for the swing phase and stance phase. Through MATLAB software simulation, the simulation results of MATLAB software showed that the trajectory of the foot of the quadruped robot was semi-elliptic in a gait cycle, which were basically consistent with the foot trajectory obtained from the previous goat slope walking test. Therefore, the simulation results by MATLAB software showed that the foot trajectory was reasonable. Based on this, a gait plan was carried out, and the sequence of the lift leg steps of the robot was obtained. The test platform for the whole machine was built. The results showed that the height of the leg of the quadruped robot was 7.37 cm, and the length of the gait was 28.40 cm. Compared with the planned average gait length S=30 cm and average step height H=7 cm, the error of the leg height and the gait length was 5.28% and 5.33%, respectively. The designed gait planning achieved the expected results. The knee joint angle varied from 126.3° to 156.1°, and the hip joint angle varied from 103.9° to 138.4°. The changes in the joint angle of the quadruped robot could prove the correctness of the foot motion trajectory planning, which will provide a theoretical basis for the structural design and gait planning of the quadruped robot.