Quadruped bounding control with variable duty cycle via vertical impulse scaling

This paper introduces a bounding gait control algorithm that allows a successful implementation of duty cycle modulation in the MIT Cheetah 2. Instead of controlling leg stiffness to emulate a `springy leg' inspired from the Spring-Loaded-Inverted-Pendulum (SLIP) model, the algorithm prescribes vertical impulse by generating scaled ground reaction forces at each step to achieve the desired stance and total stride duration. Therefore, we can control the duty cycle: the percentage of the stance phase over the entire cycle. By prescribing the required vertical impulse of the ground reaction force at each step, the algorithm can adapt to variable duty cycles attributed to variations in running speed. Following linear momentum conservation law, in order to achieve a limit-cycle gait, the sum of all vertical ground reaction forces must match vertical momentum created by gravity during a cycle. In addition, we added a virtual compliance control in the vertical direction to enhance stability. The stiffness of the virtual compliance is selected based on the eigenvalue analysis of the linearized Poincaré map and the chosen stiffness is 700 N/m, which corresponds to around 12% of the stiffness used in the previous trotting experiments of the MIT Cheetah, where the ground reaction forces are purely caused by the impedance controller with equilibrium point trajectories. This indicates that the virtual compliance control does not significantly contributes to generating ground reaction forces, but to stability. The experimental results show that the algorithm successfully prescribes the duty cycle for stable bounding gaits. This new approach can shed a light on variable speed running control algorithm.