Differentially Flat Design of Bipeds Ensuring Limit-Cycles

In bipedal walking, a trajectory is acceptable as long as it is repetitive and allows the foot to clear the ground, while allowing the biped to move forward. Since the actual trajectory followed by a biped is not as important, a biped having more than one passive joints can also meet the motion requirements. Due to physical constraints, a biped is under-actuated at the ground contact with the feet. A biped should exhibit limit cycles when moving continuously in an environment. In general, it is difficult to prove existence of limit cycles for nonlinear systems. In this work, we generate limit cycles for a class of nonlinear underactuated bipeds using differential flatness. A specific inertia distribution renders the biped design differentially flat. Differential Flatness allows generation of a family of limit cycles amenable to numerical optimization. The results are illustrated by two DOF biped.