Aiming at the problem that the stability of the quadruped robot is decreased as its leg momentum is too high, a stable balance adjustment structure of the quadruped robot based on the bionic lateral swing posture is proposed. First, the leg structure of the quadruped robot is improved and designed by using the mechanism of the lateral swing posture of the leg of the hoof animal. Then, the D-H method is used to construct the corresponding leg kinematics model and determine the generalized coordinates of the leg joints in the lateral swing posture. The torque expression of the quadruped robot when it is tilted is established. Based on the differential equation of momentum of the hip joint and its static stability analysis, the static stability conditions in the upright posture and the bionic lateral swing posture are given. Finally, the experimental simulation and comparative analysis of the upright posture and the lateral swing posture of the quadruped robot are proposed by using the Adams virtual prototype technology. The simulation results show that as the angle of lateral swing increases, the peak value of the positive flip torque of the quadruped robot body increases accordingly, while the degree of tilt decreases accordingly, which shows that the bionic lateral swing posture of the quadruped robot has higher static stability than the traditional upright posture. This research provides a technical reference for the design and optimization of the offline continuous gait of the robot and the improvement of stability.