This paper presents a bio-inspired quadruped controller that allows variable-speed galloping. The controller design is inspired by observations from biological runners. Quadrupedal animals increase the vertical impulse that is generated by ground reaction forces at each stride as running speed increases and the duration of each stance phase reduces, whereas the swing phase stays relatively constant. Inspired by this observation, the presented controller estimates the required vertical impulse at each stride by applying the linear momentum conservation principle in the vertical direction and prescribes the ground reaction forces at each stride. The design process begins with deriving a planar model from the MIT Cheetah 2 robot. A baseline periodic limit cycle is obtained by optimizing ground reaction force profiles and the temporal gait pattern (timing and duration of gait phases). To stabilize the optimized limit cycle, the obtained limit cycle is converted to a state feedback controller by representing the obtained ground reaction force profiles as functions of the state variable, which is monotonically increasing throughout the gait, adding impedance control around the height and pitch trajectories of the obtained limit cycle and introducing a finite state machine and a pattern stabilizer to enforce the optimized gait pattern. The controller that achieves a stable 3 m s(-1) gallop successfully adapts the speed change by scaling the vertical ground reaction force to match the momentum lost by gravity and adding a simple speed controller that controls horizontal speed. Without requiring additional gait optimization processes, the controller achieves galloping at speeds ranging from 3 m s(-1) to 14.9 m s(-1) while respecting the torque limit of the motor used in the MIT Cheetah 2 robot. The robustness of the controller is verified by demonstrating stable running during various disturbances, including 1.49 m step down and 0.18 m step up, as well as random ground height and model parameter variations.