论文标题
受控的Lagrangians和Euler的稳定 - 与对称性破裂I：动力学形状的机械系统
Controlled Lagrangians and Stabilization of Euler--Poincaré Mechanical Systems with Broken Symmetry I: Kinetic Shaping
论文作者
论文摘要
我们将具有动力学成型的受控拉格朗日人的方法扩展到具有损坏对称性的半领产品的那些机械系统，更具体地说，是Euler-Poincaré方程，带有反位的参数。我们为此类系统的受控Lagrangian找到了一个匹配条件，该系统的配置歧管是一般的半领产品lie组$ \ mathsf {g} \ ltimes v $。我们激励人心的例子是底层水下的底层水下车辆和可移动基地上的顶部旋转。他们的配置空间是特殊的Euclidean组$ \ MATHSF {se}（3）= \ Mathsf {so}（3）（3）\ ltimes \ Mathbb {r}^{3} $，其中$ \ Mathsf {sesf {se}（se}（3）$ - symmetry被Gravity打破了。匹配条件产生的控件稳定了这些示例的不稳定平衡。此外，匹配有助于我们找到渐近稳定那些不稳定平衡的其他耗散控制。
We extend the method of controlled Lagrangians with kinetic shaping to those mechanical systems on semidirect product Lie groups with broken symmetry, more specifically to the Euler--Poincaré equations with advected parameters. We find a matching condition for the controlled Lagrangian for such systems whose configuration manifold is a general semidirect product Lie group $\mathsf{G} \ltimes V$. Our motivating examples are a bottom-heavy underwater vehicle and a top spinning on a movable base. Their configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$, where the $\mathsf{SE}(3)$-symmetry is broken by the gravity. The controls resulting from the matching condition stabilize unstable equilibria of these examples. Furthermore, the matching helps us find additional dissipative controls that asymptotically stabilize those unstable equilibria.
