Nonconservativity and noncommutativity in locomotion
Nonconservativity and noncommutativity in locomotion
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DOI：
10.1140/epjst/e2015-50085-y
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发表时间：
2015-12-01
影响因子：
2.8
通讯作者：
Choset, H.
中科院分区：
文献类型：
--
作者：
Hatton, R. L.;Choset, H.
基于李氏支架的几何力学技术提供了运动系统运动能力的高级表征。特别是，它们将它们在循环步态上经历的净位移与约束的面积积分联系起来；因此，绘制这些约束提供了一个视觉“景观”，直观地捕获了系统动态方程的所有可用解。最近，我们发现系统坐标的选择严重影响了这些方法的有效性。这个性质乍一看似乎与微分几何结构应该是坐标不变的原则背道而驰。在本文中，我们提供了李括号技术的教程概述，然后研究了这些系统的坐标无关的非完整性如何具有坐标相关的非保守和非交换分量的分离，它们分别捕获了系统约束如何在构型空间的形状和位置分量上变化。非保守约束变化可以通过Stokes定理进行几何积分，但非交换效应只能用类似的方法近似；因此，选择非完整性主要是非保守性的坐标可以提高几何技术的精度。
Geometric mechanics techniques based on Lie brackets provide high-level characterizations of the motion capabilities of locomoting systems. In particular, they relate the net displacement they experience over cyclic gaits to area integrals of their constraints; plotting these constraints thus provides a visual "landscape" that intuitively captures all available solutions of the system's dynamic equations. Recently, we have found that choices of system coordinates heavily influence the effectiveness of these approaches. This property appears at first to run counter to the principle that differential geometric structures should be coordinate-invariant. In this paper, we provide a tutorial overview of the Lie bracket techniques, then examine how the coordinate-independent nonholonomy of these systems has a coordinate-dependent separation into nonconservative and noncommutative components that respectively capture how the system constraints vary over the shape and position components of the configuration space. Nonconservative constraint variations can be integrated geometrically via Stokes' theorem, but noncommutative effects can only be approximated by similar means; therefore choices of coordinates in which the nonholonomy is primarily nonconservative improve the accuracy of the geometric techniques.