对称性破缺的Euler-Poincaré力学系统的受控Lagrange量与镇定II：势成形
Controlled Lagrangians and Stabilization of Euler--Poincaré Mechanical Systems with Broken Symmetry II: Potential Shaping
摘要：
本文将势成形控制拉格朗日方法应用于对称性破缺的Euler-Poincaré力学系统。我们假设位形空间是一个一般的半直积李群$\mathsf{G} \ltimes V$，特别关注那些位形空间是特殊欧几里德群$\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$的系统。工作背后的关键思想是使用$\mathsf{G} \ltimes V$及其相关平流参数的表示。具体而言，我们推导出匹配条件的修改后的潜在的利用的表示和平流参数。我们的激励例子是一个沉重的顶部旋转的可移动的基础和水下航行器与非重合的重心和浮力。我们考虑这些系统的几个不同的控制问题，并表明，我们的结果给出了一个一般的框架，再现我们以前的工作，前一个例子，也是伦纳德对后者。此外，在后者的情况下，我们展示了我们的代表为基础的方法的优势，给出了一个更简单，更简洁的制定的问题。
英文摘要：
We apply the method of controlled Lagrangians by potential shaping to Euler--Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group $\mathsf{G} \ltimes V$ with a particular interest in those systems whose configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$. The key idea behind the work is the use of representations of $\mathsf{G} \ltimes V$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting the representations and advected parameters. Our motivating examples are a heavy top spinning on a movable base and an underwater vehicle with non-coincident centers of gravity and buoyancy. We consider a few different control problems for these systems, and show that our results give a general framework that reproduces our previous work on the former example and also those of Leonard on the latter. Also, in one of the latter cases, we demonstrate the advantage of our representation-based approach by giving a simpler and more succinct formulation of the problem.
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