角刚度及其在稳定平面地层中的应用
Angle rigidity and its usage to stabilize planar formations
摘要：
移动机器人团队的编队稳定性问题具有挑战性，当没有距离或相对位移的测量是可用的，每个机器人只能测量的光线对它的邻居形成的角度中的一些，我们开发的“角度刚性”的概念的多点框架，命名为“angularity”，由一组节点嵌入在一个欧几里得空间和一组角度约束。与相对于全局轴定义的方位角或角度不同，我们使用的角度不依赖于全局坐标系的知识，并且根据逆时针方向进行标记。这里，\emph{angle rigidity}是指指定在适当的角度约束下，当一个或多个节点局部扰动时，倾斜度只能作为整体平移、旋转或缩放的属性。我们首先证明，与文献中的轴承刚度或其他报告的与框架角度相关的刚度相比，这种角度刚度属性是\emph{not}的全局属性，因为角度刚性倾斜度可能会导致弯曲模糊。然后，我们构造了必要和充分条件，\emph{infinitesimal}角刚度检查一个angularity的刚度矩阵的秩。给出了无穷小极小角刚性的一个组合必要条件。利用所发展的理论，编队稳定算法设计的机器人团队，以实现全球角度刚性编队，其中只有角度测量是必需的。
英文摘要：
Motivated by the challenging formation stabilization problem for mobile robotic teams when no distance or relative displacement measurements are available and each robot can only measure some of those angles formed by rays towards its neighbors, we develop the notion of ``angle rigidity" for a multi-point framework, named ``angularity", consisting of a set of nodes embedded in a Euclidean space and a set of angle constraints among them. Different from bearings or angles defined with respect to a global axis, the angles we use do not rely on the knowledge of a global coordinate system and are signed according to the counter-clockwise direction. Here \emph{angle rigidity} refers to the property specifying that under proper angle constraints, the angularity can only translate, rotate or scale as a whole when one or more of its nodes are perturbed locally. We first demonstrate that this angle rigidity property, in sharp comparison to bearing rigidity or other reported rigidity related to angles of frameworks in the literature, is \emph{not} a global property since an angle rigid angularity may allow flex ambiguity. We then construct necessary and sufficient conditions for \emph{infinitesimal} angle rigidity by checking the rank of an angularity's rigidity matrix. We develop a combinatorial necessary condition for infinitesimal minimal angle rigidity. Using the developed theories, a formation stabilization algorithm is designed for a robotic team to achieve a globally angle rigid formation, in which only angle measurements are needed.
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