Understanding the common principle underlying passive dynamic walking and running

In this study, we discuss the common stabilization mechanism underlying passive dynamic walking (PDW) and passive dynamic running (PDR), focusing on the feedback structures in analytical Poincare¿ maps. To this end, we have derived linearized analytical Poincare¿ maps for PDW and PDR, and analyzed these stabilities on two models, namely models with elastic elements and with stiff legs. Through our theoretical analysis, we have found an ¿implicit two-delay feedback structure¿, which can be seen as a certain type of two-delay input digital feedback control developed as an artificial control structure in the field of control theory, is an inherent stabilization mechanism in PDR appearing from the model with elastic elements, and two-period and four-period PDW appearing from with stiff legs. This mechanism is the key to adaptive function underlying phase transition phenomenon between PDW and PDR and period-doubling bifurcation phenomenon in PDW. To the best of our knowledge, this has not yet to be addressed and studied so far. Our results shed new light on the common underlying principle of passive dynamic locomotion, including biped PDW and PDR.