Bipedal walking is a complex phenomenon that is not fully understood. Simplified models make it easier to highlight the important features. Here, the variable length inverted pendulum (VLIP) model is used, which has the particularity of taking into account the vertical oscillations of the center of mass (CoM). When the desired walking gait is defined as virtual constraints, i.e., as functions of a phasing variable and not on time, for the evolution of the swing foot and the vertical oscillation of the CoM, the walk will asymptotically converge to the periodic motion under disturbance with proper choice of the virtual constraints, thus a self-stabilization is obtained. It is shown that the vertical CoM oscillation, positions of the swing foot and the choice of the switching condition play crucial roles in stability. Moreover, a PI controller of the CoM velocity along the sagittal axis is also proposed such that the walking speed of the robot can converge to another periodic motion with a different walking speed. In this way, a natural walking gait is illustrated as well as the possibility of velocity adaptation as observed in human walking.

中文翻译：

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虚拟约束控制的变长倒立摆的行走稳定性

双足行走是一种尚未完全了解的复杂现象。简化的模型更容易突出重要特征。这里使用了可变长度倒立摆（VLIP）模型，该模型具有考虑质心（CoM）的垂直振荡的特殊性。当所需的步行步态被定义为虚拟约束时，即作为相位变量的函数而不是时间的函数，对于摆动脚的演变和 CoM 的垂直振荡，步行将在扰动下渐近收敛到周期性运动通过正确选择虚拟约束，因此获得了自稳定化。结果表明，垂直 CoM 振荡、摆脚位置和切换条件的选择对稳定性起着至关重要的作用。而且，还提出了沿矢状轴的 CoM 速度的 PI 控制器，使得机器人的步行速度可以收敛到具有不同步行速度的另一个周期运动。通过这种方式，说明了自然步行步态以及在人类步行中观察到的速度适应的可能性。