Bifurcations and chaos in the semi-passive bipedal dynamic walking model under a modified OGY-based control approach

Abstract

This paper aims at investigating the semi-passive dynamic walking of a torso-driven biped robot under the Ott–Grebogi–Yorke (OGY) control approach as it goes down inclined planes. For the control, we used the desired torso angle as the accessible control parameter. Compared with our work in Gritli et al. (Nonlinear Dyn 79(2):1363–1384, 2015), a modified design of the OGY-based controller is proposed in this paper. Such controller is obtained by linearizing the impulsive hybrid nonlinear dynamics of the biped robot around a desired one-periodic hybrid limit cycle. Both the differential equation and the algebraic equation are linearized. As a result, we develop a simple mathematical expression of a controlled hybrid Poincare map. Determination of its fixed point and its Jacobian matrix requires only the knowledge of the nominal impact instant. We show efficiency of the designed OGY controller for the control of chaos in the impulsive hybrid nonlinear dynamics for some desired nominal values of the slope, and the desired torso angle. Furthermore, we analyzed via bifurcation diagrams the displayed behaviors in the controlled semi-passive biped robot as the slope parameter varies. We show the appearance of a Neimark–Sacker bifurcation and a cyclic-fold bifurcation, and also the exhibition of chaos. Our analysis of the controlled semi-passive gait is achieved also by means of the spectrum of Lyapunov exponents. Such study is realized via the controlled hybrid Poincare map where a reduction of its dimension is achieved.