Locomotion stability is essential for controlling quadruped robots and adapting them to unstructured terrain. We propose a control strategy with center-of-mass (CoM) dynamic planning for the stable locomotion of these robots. The motion trajectories of the swing legs are synchronized with the CoM of the robot. To implement the synchronous control scheme, we adjusted the swing legs to form a support triangle. The strategy is applicable to both static walk gait and dynamic trot gait. In the motion control processes of the robot legs, the distribution of the ground reaction forces is optimized to minimize joint torque and locomotion energy consumption. We also used an improved joint-torque controller with varied controller coefficients in the stance and swing phases. The simulation and experimental results demonstrate that the robot can complete omnidirectional locomotion in both walk and trot gaits. At a given locomotion speed, the stability margins for the robot during walking and trotting were 27.25% and 37.25% higher, respectively, than in the scheme without CoM planning. The control strategy with energy consumption optimization (ECO) reduced the energy consumption of the robot in walk and trot gaits by 11.25% and 13.83%, respectively, from those of the control scheme without ECO.