Dept. of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, Japan.
Bioinspir Biomim. 2020 Jul 15;15(5):055002. doi: 10.1088/1748-3190/ab9283.
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 humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, 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 caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal.
被动式动态步行是一种无需任何控制或输入即可在缓坡上行走的模型。该模型已被广泛用于研究人类如何以低能耗行走，并为节能双足机器人提供了设计原则。然而，吸引盆非常小而薄，形状具有分形般的复杂，这使得产生稳定的行走变得困难。在我们之前的研究中，我们使用了最简单的行走模型，并基于动力系统理论研究了分形吸引盆，重点研究了由具有鞍点双曲性的连续动力学和由于脚部接触而引起的不连续动力学组成的模型的混合动力学。我们阐明了分形吸引盆是通过顺序逆像对 Poincaré 映射的域进行迭代拉伸和弯曲变形产生的。然而，分形吸引盆是否实际上是分形的，即吸引盆中是否嵌入了无限多的自相似模式，取决于坡度角，其机制尚不清楚。在本研究中，我们改进了之前的分析，以阐明这种机制。特别是，我们新关注 Poincaré 映射的范围，并指定由 Poincaré 映射的顺序逆像拉伸和弯曲的区域。通过对指定区域的分析，我们阐明了吸引盆具有分形所需的条件和机制。
