Mathematical analysis of steady walking states in underactuated limit cycle walking

This paper presents mathematical analyses of steady walking states in underactuated limit cycle walking. We introduce an underactuated rimless wheel with a torso to analyze target steady walking states. Then the formula of boundary conditions are derived and the properties of the steady gaits are analyzed. The transition function of state error is derived from the properties of the steady gaits, and an optimal steady walking state is generated. In addition, we generate the target steady walking speed on the underactuated rimless wheel model and verify our results using numerical simulations. As a conclusion, optimal steady states can be discovered and generated based on the mathematical analysis of dynamic walking states.