Bifurcations and chaos in passive dynamic walking: A review

A different switching surface stabilizing an existing unstable periodic gait: an analysis based on perturbation theory

“…A difference has appeared all over the interval since both ψ and ε are relatively out of the accepted region for perturbation analysis albeit cyclic-fold bifurcation has been reported too [10]. It has been shown that the bifurcation point is affected by the parameters of the system, such as the slope angle, mass ratio, and the length of the upper body [3,8,9,18]. This part aims to briefly show that the switching surface is another important element influencing the bifurcation diagram.…”

“…Alongside the construction of many passivity-based bipedal robots around the world, the theory of PDW has been developed as well. Now, it is well accepted that a passive walker would present a chaotic motion in addition to limit cycle walking (for more details refer to [18] and the references therein). The foot contact model, foot shape model, body optimization, and the development of stability analysis on uneven terrains are some of the other important improvements in the subject [4,5,11,14,19,22,23,26].…”

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

“…Iqbal et al provided in [76] an overview of the previous literature on the chaotic behavior exhibited in the passive dynamic walking of biped robots. The walking gait model of passive biped robots is described by an impulsive hybrid nonlinear dynamics and hence a discontinuous (non-smooth) dynamics.…”

“…The simplest model of passive biped robot is known as the compass-gait model, where its impulsive hybrid dynamics is composed of a nonlinear differential equation modeling the swing motion and a nonlinear algebraic equation for the impact phase. The passive compass-gait model has been wildly studied in terms of chaos/bifurcations analysis and less studied from chaos/bifurcations control point-of-view [76]. It has been known that the passive dynamic walking of biped robots down inclined surfaces displays a cascade of local perioddoubling bifurcations as route to chaos [76].…”

“…The passive compass-gait model has been wildly studied in terms of chaos/bifurcations analysis and less studied from chaos/bifurcations control point-of-view [76]. It has been known that the passive dynamic walking of biped robots down inclined surfaces displays a cascade of local perioddoubling bifurcations as route to chaos [76]. We showed recently that the passive dynamic walking of a compass-gait model and the semi-passive dynamic walking of a torsodriven biped model exhibit also the cyclic-fold bifurcations and hence a period-three route to chaos [68,70,71,74], the intermittency route [69] and global bifurcations, namely the interior and boundary crisis [69,70].…”