小车链摆的动力学与控制
Dynamics and Control of a Chain Pendulum on a Cart
摘要：
本文给出了链摆的欧拉-拉格朗日方程的几何形式。链摆是由球铰连接的$n$刚性连杆串联而成的，它连接在刚性小车上。小车可以在水平控制力作用下在水平面内平移，而链摆可以由于重力而在3D中进行复杂运动。系统配置在$(\Sph^2)^n \times \Re^2$中。我们研究了不受控系统动力学的丰富结构：系统的平衡对应于$2^n$不同链摆配置和任何推车位置中的任何一个。对每个平衡点进行线性化，并给出相应的能控性判据。我们还表明，任何平衡点都可以渐近稳定的比例微分型控制器，我们提供了一些数值例子。
英文摘要：
A geometric form of Euler-Lagrange equations is developed for a chain pendulum, a serial connection of $n$ rigid links connected by spherical joints, that is attached to a rigid cart. The cart can translate in a horizontal plane acted on by a horizontal control force while the chain pendulum can undergo complex motion in 3D due to gravity. The configuration of the system is in $(\Sph^2)^n \times \Re^2$. We examine the rich structure of the uncontrolled system dynamics: the equilibria of the system correspond to any one of $2^n$ different chain pendulum configurations and any cart location. A linearization about each equilibrium, and the corresponding controllability criterion is provided. We also show that any equilibrium can be asymptotically stabilized by using a proportional-derivative type controller, and we provide a few numerical examples.
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