Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid
Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid
复制标题
自驱动变形体在流体中的运动和控制
DOI：
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发表时间：
2009
影响因子：
3
通讯作者：
A. Munnier
中科院分区：
文献类型：
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作者：
T. Chambrion;A. Munnier
本文研究了在无限大二维理想流体中游动的变形物体的运动。形状的变化被规定为满足约束的时间的函数，以确保它们只产生于内力的工作：运动被称为自推进的必要条件。身体的净刚性运动是由于这些形状变化与周围流体之间的动量交换而产生的。本文的目的有三个：首先，它描述了一个严格的框架，用于研究动物在流体中的运动。我们的模型与以前的模型不同，主要在于与形状变化相关的自由度是无限的。将最小作用量原理应用于系统体液，得到欧拉-拉格朗日方程。分析力学的形式主义提供了一种简单的方法来处理导致形状变化的物体内部动力学与流体动力学之间的强耦合。欧拉-拉格朗日方程采用常微分方程（ODE）和偏微分方程（PDE）的耦合系统的形式。严格证明了该系统解的存在唯一性。其次，我们有兴趣弄清楚形状变化和内力之间的联系。虽然经典的，它可以是相当令人惊讶的选择形状的变化，以发挥控制的作用，因为内力，他们是由于似乎是一个更自然和现实的选择。我们证明，当有关的形状变化的自由度的数量是有限的，这两种选择实际上是等价的，在这个意义上，有一个一对一的关系之间的形状变化和内部forces.Third，我们展示了如何控制问题，包括在关联与每个形状变化的游泳体的轨迹，可以在几何控制理论的框架内进行分析。这使我们能够利用微分几何的强大工具，如李括号或轨道定理的概念，并获得第一个理论结果（据我们所知）的控制游泳机构在一个理想的流体。我们得到了一些有趣和令人惊讶的跟踪特性：例如，对于任何给定的形状变化产生的净位移的流体（说，向前移动），我们证明了其他形状变化任意接近前一个存在，这导致一个完全不同的运动（例如，向后移动）：这种现象将被称为月球漫步。我们的大部分结果都通过数值例子来说明。
In this paper we study the locomotion of a shape-changing body swimming in a two-dimensional perfect fluid of infinite extent. The shape changes are prescribed as functions of time satisfying constraints ensuring that they result from the work of internal forces only: conditions necessary for the locomotion to be termed self-propelled. The net rigid motion of the body results from the exchange of momentum between these shape changes and the surrounding fluid.The aim of this paper is three-fold.First, it describes a rigorous framework for the study of animal locomotion in fluid. Our model differs from previous ones mostly in that the number of degrees of freedom related to the shape changes is infinite. The Euler–Lagrange equation is obtained by applying the least action principle to the system body fluid. The formalism of Analytic Mechanics provides a simple way to handle the strong coupling between the internal dynamics of the body causing the shape changes and the dynamics of the fluid. The Euler–Lagrange equation takes the form of a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs). The existence and uniqueness of solutions for this system are rigorously proved.Second, we are interested in making clear the connection between shape changes and internal forces. Although classical, it can be quite surprising to select the shape changes to play the role of control because the internal forces they are due to seem to be a more natural and realistic choice. We prove that, when the number of degrees of freedom relating to the shape changes is finite, both choices are actually equivalent in the sense that there is a one-to-one relation between shape changes and internal forces.Third, we show how the control problem, consisting in associating with each shape change the resulting trajectory of the swimming body, can be analysed within the framework of geometric control theory. This allows us to take advantage of the powerful tools of differential geometry, such as the notion of Lie brackets or the Orbit Theorem and to obtain the first theoretical result (to our knowledge) of control for a swimming body in an ideal fluid. We derive some interesting and surprising tracking properties: For instance, for any given shape changes producing a net displacement in the fluid (say, moving forward), we prove that other shape changes arbitrarily close to the previous ones exist, which lead to a completely different motion (for instance, moving backward): This phenomenon will be called Moonwalking. Most of our results are illustrated by numerical examples.