[论文解读] Rolling Manifolds: Intrinsic Formulation and Controllability

Yacine Chitour, Petri Kokkonen|arXiv (Cornell University)|Nov 12, 2010

Advanced Differential Geometry Research参考文献 32被引用 24

一句话总结

本文通过联络和控制理论，对无自转（NS）和无滑动（R）的滚动流形给出了内在的几何表述。研究证明：在零曲率情况下，无滑动（R）系统完全可控当且仅当（M,g）的holonomy群为SO(n)；并引入了一种新颖的‘滚动曲率’张量和‘滚动联络’，以刻画非零曲率下的可控性。此外，本文通过扭曲积和接触结构对三维情形下的不可控性进行了分类。

ABSTRACT

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints "without spinning" and "without slipping" in terms of the Levi-Civita connections $ abla^{g}$ and $ abla^{\hg}$. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and $(\hM,\hg)$ respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the \emph{rolling curvature}. In the case where one of the manifolds is a space form (let say $(\hM,\hg)$), we show that it is enough to roll along loops of $(M,g)$ and the resulting orbits carry a structure of principal bundle which preserves the rolling $(R)$ distribution. In the zero curvature case, we deduce that the rolling $(R)$ is completely controllable if and only if the holonomy group of $(M,g)$ is equal to SO(n). In the nonzero curvature case, we prove that the structure group of the principal bundle can be realized as the holonomy group of a connection on $TM\oplus \R$, that we call the rolling connection. We also show, in the case of positive (constant) curvature, that if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian covering, the unit sphere with the metric induced from the Euclidean metric of $\R^{n+1}$. When the two manifolds are three-dimensional, we provide a complete local characterization of the reachable sets when the two manifolds are three-dimensional and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. Besides the trivial case where the manifolds $(M,g)$ and $(\hM,\hg)$ are (locally) isometric, we show that (local) non controllability occurs if and only if $(M,g)$ and $(\hM,\hg)$ are either warped products or contact manifolds with additional restrictions that we precisely describe. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions.

研究动机与目标

• 通过Levi-Civita联络，提供无自转与无滑动滚动的内在、坐标无关的几何表述。
• 利用几何控制理论分析无自转（NS）和无滑动（R）系统的可控性。
• 表征滚动系统的可达集，尤其在非零曲率情形下。
• 识别三维流形中局部与全局可控性的必要与充分条件。
• 将滚动框架推广至不同维度的流形。

提出的方法

• 将状态空间Q构作为流形M和M̂上的标准正交标架主丛，并赋予自然的丛结构。
• 利用Levi-Civita联络∇g和∇ĝ，定义两个分布：DNS（无自转）与DR（滚动）。
• 在状态空间Q上引入‘滚动曲率’张量，该张量控制DR的李括号结构。
• 在TM ⊕ R上构造‘滚动联络’，其holonomy群决定了非零曲率下DR-轨道的结构。
• 通过李括号计算与曲率恒等式，分析DR的轨道结构与可积性。
• 应用Ambrose定理及空间形式的相关结果，表征当其中一个流形为空间形式时的可控性。

实验结果

研究问题

• RQ1 在何种条件下，无滑动（R）系统是完全可控的？
• RQ2 滚动曲率张量如何影响滚动系统的李括号结构与轨道几何？
• RQ3 （M,g）的holonomy群在决定滚动（R）系统的可控性中起何作用？
• RQ4 在三维情形下，局部不可控性何时发生？何种几何结构（如扭曲积、接触流形）导致此现象？
• RQ5 如何将滚动问题推广至不同维度的流形？

主要发现

• 对于无自转（NS）问题，从任意初始状态出发的可达集恰好由（M,g）与（M̂,ĝ）的holonomy群的乘积描述。
• 在零曲率情况下，无滑动（R）系统完全可控当且仅当（M,g）的holonomy群为SO(n)。
• 在非零曲率情况下，DR-轨道的结构群被实现为新联络——‘滚动联络’——在TM ⊕ R上的holonomy群。
• 若滚动联络可约化且目标流形为空间形式且具有正曲率，则（M,g） admits the n-sphere as a Riemannian covering。
• 在三维情形下，局部不可控性发生当且仅当（M,g）与（M̂,ĝ）均为满足特定曲率约束的扭曲积或接触流形。
• 本文对三维情形下可达集提供了完整的局部表征，识别了除等距情形外非开轨道的必要与充分条件。