Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Rise of the Neimark-Sacker bifurcation

This paper continues our research work on analysis of passive dynamic walking of the two-degree-of-freedom planar compass-gait biped robot under the OGY-based state-feedback control. We showed, in our previous works, that the walking dynamics of the compass-gait model under control exhibits chaos and periodic-doubling and cyclic-fold bifurcations. In a first part, our analysis of the walking behavior was achieved using the impulsive hybrid continuous-time nonlinear dynamics of the compass-gait model under the OGY-based control. In a second part, our study of the controlled gait and the displayed local bifurcations was realized via the controlled hybrid Poincaré map. In the present work, we demonstrate, for the first time, the appearance of the Neimark-Sacker bifurcation in the controlled dynamic walking of the compass-gait model. Our investigation is achieved via the controlled hybrid Poincaré map instead of the impulsive continuous dynamics model of the bipedal walking. For such study, we mainly use bifurcation diagrams and 2D phase portraits of the discrete Poincaré map. We show that such Neimark-Sacker bifurcation is generated from a period-1 gait and is localized in a small range of the bifurcation parameter, the slope angle. We introduce a two-parameter bifurcation diagram to study occurrence of the Neimark-Sacker bifurcation.