Nonholonomic mechanical systems with symmetry
Nonholonomic mechanical systems with symmetry
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DOI：
10.1007/bf02199365
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发表时间：
1996
影响因子：
2.5
通讯作者：
A. Bloch;P. Krishnaprasad;J. Marsden;R. Murray
中科院分区：
文献类型：
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作者：
A. Bloch;P. Krishnaprasad;J. Marsden;R. Murray
这项工作开发的几何和力学系统的非完整约束和对称性的拉格朗日力学的角度来看，并以控制理论的应用。基本方法是几何力学应用到拉格朗日-达朗贝尔制定，推广使用的连接和动量映射与一个给定的对称群，这种情况下。我们开始制定的非完整系统的力学使用Ehresmann连接模型的约束，并显示如何进入拉格朗日方程的曲率连接。与标准位形空间约束的情形不同，非完整情形中对称性的存在可能导致也可能不导致守恒律。然而，由对称群确定的动量映射仍然满足一个有用的微分方程，该方程从群变量中推导出来。这个动量方程，它在控制问题中起着重要的作用，涉及到平行运输运营商和计算明确的坐标。使用“物体参考系”的另一种描述将动量方程的一部分与欧拉-庞加莱方程的分量沿着与约束一致的对称方向联系起来。本文的目的之一是导出动量的演化方程，并从几何和力学上区分动量守恒和不守恒的情况。前者的一个例子是在平面上滚动的球或垂直盘，后者的一个例子是蛇板，一种使用动量耦合产生运动的滑板的修改版本。我们构建了一个综合的机械连接和Ehresmann连接定义的约束，获得一个重要的新对象，我们称之为非完整连接。当非完整连接是一个主连接给定的对称群，我们展示了如何进行拉格朗日约化的存在下，非完整约束，推广以前的结果，只在特殊情况下举行。文中给出了几个具体的例子来说明这一理论。
This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d'Alembert formulation, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration-space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a “body reference frame” relates part of the momentum equation to the components of the Euler-Poincaré equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory.