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According to biological evidences, central pattern generators (CPGs) are neural networks responsible for the generation of rhythmic movements for animals, such as breathing, heartbeat, and locomotion, even when isolated from the brain and sensory inputs. Inspired by this mechanism, researchers have proposed the CPG driven control method as a new way to generate rhythmic control policies for the locomotion of legged robots. In this work, we design a CPG control construction for controlling the locomotion of a quadruped robot, which is capable of realizing the different gaits and gait transitions. Firstly, a body CPG network is constructed by mutually coupled phase oscillators, which can produce multiple phase-locked oscillation patterns that correspond to the four basic quadruped gaits. The gait transitions can be realized by altering the internal oscillator parameters. Then, we design a robotic platform based on Webots and AIBO, and realize dynamic locomotion with the designed CPG network for AIBO. The simulation and experimental results demonstrate the proposed CPG network is effective to generate gait patterns for quadruped robots.