Flip bifurcation and gait energetics of a bipedal walker with asymmetric leg movements powered by nonlinear pulse thrust

This article presents a walking bipedal model with asymmetric leg movements through a four-phase gait planning, represented as a nonlinear impulsive hybrid system. To achieve forward movement of the bipedal walker on the horizontal surface, we introduce a nonlinear pulse thrust in relation to the walking state at the heel strike to push the supporting leg off. By linearizing the continuous dynamics, a Poincaré map with explicit form is obtained analytically. The conditions for the stability of the period-1 gait are determined, the bifurcation of the periodic orbits is investigated, and then the gait energetics are analyzed. The theoretical analysis and numerical results show that with the change of the parameters of the pulse thrust, the biped walking exhibits the dynamic behaviors of a flip bifurcation path to chaos and generates a period-2 gait. Within the parameter range of the period-1 gait, a slight increase in the pulse thrust can significantly reduce the energy input of other parts of the bipedal system. It indicates that a well-designed pulse thrust can improve the energy efficiency of the desired periodic gait.