Makarenkov Oleg
Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080, USA.
Proc Math Phys Eng Sci. 2020 Jan;476(2233):20190450. doi: 10.1098/rspa.2019.0450. Epub 2020 Jan 8.
We consider the simplest model of a passive biped walking down a slope given by the equations of switched coupled pendula (McGeer T. 1990 Passive dynamic walking. , 62-82. (doi:10.1177/027836499000900206)). Following the fundamental work by Garcia (Garcia . 1998 . , 281. (doi:10.1115/1.2798313)), we view the slope of the ground as a small parameter ≥ 0. When = 0, the system can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed in Garcia (Garcia . 1998 . , 281. (doi:10.1115/1.2798313)), the family of cycles disappears when increases and only isolated asymptotically stable cycles (walking cycles) persist. However, no mathematically complete proofs of the existence and stability of walking cycles have been reported in the literature to date. The present paper proves the existence and stability of a walking cycle (long-period gait cycle, as termed by McGeer) by using the methods of perturbation theory for maps. In particular, we derive a perturbation theorem for the occurrence of stable fixed points from 1-parameter families in two-dimensional maps that can be of independent interest in applied sciences.
我们考虑由切换耦合摆方程给出的、沿斜坡下行的被动双足机器人的最简单模型（McGeer T. 1990被动动态行走。 ，62 - 82。(doi:10.1177/027836499000900206)）。继Garcia的基础工作（Garcia. 1998. ，281。(doi:10.1115/1.2798313)）之后，我们将地面的斜率视为一个小参数 ≥ 0。当 = 0时，系统可以用封闭形式求解，并且可以用封闭形式计算一族周期（即潜在行走周期）的存在性。正如在Garcia中所观察到的（Garcia. 1998. ，281。(doi:10.1115/1.2798313)），当 增加时，这族周期消失，只有孤立的渐近稳定周期（行走周期）持续存在。然而，迄今为止，文献中尚未报道关于行走周期的存在性和稳定性的数学上完整的证明。本文通过使用映射的微扰理论方法，证明了一个行走周期（McGeer所称的长周期步态周期）的存在性和稳定性。特别地，我们从二维映射中的单参数族导出了一个关于稳定不动点出现的微扰定理，该定理在应用科学中可能具有独立的研究价值。
