Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid Poincaré map

In our previous work, we have analyzed the passive dynamic walking of the compass-gait biped model under the OGY-based state-feedback control using the impulsive hybrid nonlinear dynamics. Such study was carried out through bifurcation diagrams. It was shown that the controlled bipedal gait exhibits attractive nonlinear phenomena such as the cyclic-fold (saddle-node) bifurcation, the period-doubling (flip) bifurcation and chaos. Moreover, we revealed that, using the controlled continuous-time dynamics, we encountered a problem in finding, identifying and hence following branches of (un)stable solutions in order to characterize local bifurcations. The present paper solves such problem and then provides a further investigation of the controlled bipedal walking dynamics using the developed analytical expression of the controlled hybrid Poincaré map. Thus, we show that analysis via such Poincaré map allows to follow branches of both stable and unstable fixed points in bifurcation diagrams and hence to explore the complete dynamics of the controlled compass-gait biped model. We demonstrate the generation, other than the conventional local bifurcations in bipedal walking, i.e. the flip bifurcation and the saddle-node bifurcation, of a saddle-saddle bifurcation, a subcritical flip bifurcation and a new type of a local bifurcation, the saddle-flip bifurcation. In addition, to further understand the occurrence of the local bifurcations, we present an analysis with a two-parameter bifurcation diagram. Some new hidden walking dynamics are identified.