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Disappearance of chaotic attractor of passive dynamic walking by stretch-bending deformation in basin of attraction

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how stable walking is generated from a dynamic viewpoint, which is useful to provide design principles for developing energy-efficient biped robots. However, the basin of attraction is very small and thin, and it has a fractal-like complicated shape. This makes it difficult to produce stable walking. Furthermore, the passive dynamic walking shows chaotic attractor through a period-doubling cascade by increasing the slope angle, and the chaotic attractor suddenly disappears at a critical slope angle. These make it further difficult to produce stable walking. In our previous work, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics by the impact at foot contact. We elucidated that the fractal-like basin of attraction is generated through iterative stretch and bending deformations of the domain of the Poincaré map by sequential inverse images of the Poincaré map. In this study, we investigated the mechanism for the disappearance of the chaotic attractor by improving our previous analysis. In particular, we focused on the range of the Poincaré map to specify the regions to be stretched and bent by the inverse image of the Poincaré map. We clarified the condition for the chaotic attractor to disappear and the mechanism why the chaotic attractor disappears based on the stretch-bending deformation in the basin of attraction.