This paper proposes a novel legged locomotion principle for tracking the quadruped-robot base spatial motion which can generate automatically the trotting and walking gaits without flight phases. The principle is based on a new legged locomotion generator using the robot centroidal properties without a massless-leg assumption. Our quadruped robot is newly designed with symmetric legged dynamics properties using parallelogram mechanisms, and the corresponding generalized dynamics model is built. The centroidal momentum and dynamics models are established by combining the centroidal properties of the base and each leg, in which the virtual joint between the base and each leg CoM is used instead of the real joints. The stance legs follow the base motion with respect to contact constraints. The swing legs compensate for the delay influences (to centroidal motion) by the stance legs. By combining the whole-body kinematics model and the kinematics constraints relating to the contact constraints and the parallelogram mechanism, the adaptive swing-leg motion is achieved in the running process without pre-defined trajectories. When two swing legs land at different timings, the walking gait is applied. We also propose a new state machine for switching the locomotion gaits between walking and trotting, which also serves the torque controller which determines the dimension of the legged control states and the contact Jacobian. We apply our hierarchical torque controller to enable the robot with compliant dynamics properties. Therefore, our new locomotion principle integrates the system design, motion generation and whole-body torque control together, so that this one common framework is able to handle several locomotion gaits. To verify the usefulness and validity of our new locomotion principle based legged motion generator, we run several simulations: