One way to create a quadruped galloping robot from scratch is to design a brick-shaped body and utilize relatively simple open-chain leg mechanisms controlled with relatively complex control algorithms. Alternatively, we can look at how nature solved the same task designing fast mammals such as cheetah, and by means of morphological computation, we can design a complex mechanical system that has much of the desired behavior within inherent dynamics and only a little control effort is needed to stabilize or augment the motion.In this paper, we have analyzed a real cheetah motion using video tracking and looked for a way to match the dynamic model parameters of the real cheetah with a galloping robot with an elastic spine. We believe the elastic spine is the essential feature for a fast-running energy-efficient galloping robot. Within this paper, we are focused on the flying stage when the elastic spine affects the motion of the robot's front and rear bodies. We have found how to optimize mass distribution and elasticity in the spine in order to get the cheetah-like galloping motion of a quadruped robot.