Before revision:  The LQR optimal gain matrix with non-zero off-diagonal elements leverages the coupling of dynamics to compensate for system underactuation , such as a quadruped robot with passive ankles. The specially designed foot with adaptive sole aims at improving the traversability of rough terrains with rocks, loose gravel and rubble by enlarging the contact surfaces with ground . Experiments on the quadruped ANYmal demonstrate the effectiveness of the proposed method for robust dynamic locomotion given external disturbances and environmental uncertainties.
After revision:  The LQR optimal gain matrix with non-zero off-diagonal elements leverages the coupling of dynamics to compensate for system underactuation . Meanwhile, the projected inverse dynamic control complements the LQR to satisfy inequality constraints. In addition to these contributions, we show robustness of our control framework to unmodeled adaptive feet . Experiments on the quadruped ANYmal demonstrate the effectiveness of the proposed method for robust dynamic locomotion given external disturbances and environmental uncertainties.