Stability analysis method independent of numerical integration for limit cycle walking with constraint on impact posture

This paper proposes the method independent of numerical integration for analyzing the stability of a limit cycle walker that falls down as a 1-DOF rigid body in the same posture. We introduce the model of an underactuated rimless wheel with a torso for analysis and show that the transition function of the state error can be analytically derived from the recurrence formula of kinetic energy immediately before impact. We then investigate the accuracy of the derived solution through comparison with the numerical solutions and discuss how the convergence property changes with respect to the control parameter. Furthermore, we extend the method to an underactuated biped and show that the stability analysis can be conducted in a similar manner without performing numerical simulations.