At present, most quadruped robots can move quickly and steadily on both flat and undulating ground; however, natural environments are complex and changeable, so it is important for a quadruped robot to be able to jump over obstacles immediately. Inspired by the jumping movement of quadruped animals, we present aerial body posture adjustment laws and generate animal-like jumping trajectories for a quadruped robot. Then, the bionic reference trajectories are optimized to build a trajectory library of a variety of jumping motions based on the kinematic and dynamic constraints of the quadruped robot. The model predictive control (MPC) method is employed by the quadruped robot to track the optimized trajectory to achieve jumping behavior. The simulations show that the quadruped robot can jump over an obstacle of 40 cm in height. The effectiveness of the animal-like jump control method is verified.