Complex walking behaviours, chaos and bifurcations of a simple passive compass-gait biped model suffering from leg length asymmetry

This paper is concerned with the analysis of the displayed nonlinear phenomena, chaos and bifurcations, in the planar passive dynamic walking of the planar compass-gait biped model under a leg length asymmetry as it goes down an inclined surface. The passive dynamic walking of the compass-gait model is modelled with an impulsive hybrid nonlinear dynamics. In this work, we present a normalised dynamics expressed in terms of dimensionless ratios. Our analysis and simulation of the passive bipedal gaits is realised mainly through bifurcation diagrams where a normalised leg length discrepancy is adopted as the bifurcation parameter. We report the exhibition of complex behaviours, namely the period-doubling bifurcation (PDB), the cyclic-fold bifurcation (CFB), the period-doubling route to chaos, the period-remerging scheme, the boundary crisis (BC), etc. We demonstrate also the exhibition of the Neimark-Sacker-2 bifurcation by investigating the tendency of the characteristic multipliers of the Jacobian matrix of the Poincare map.