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多边形连杆的库仑控制

Coulomb control of polygonal linkages

研究多边形连杆在顶点电荷库仑势下的平衡，证明四边形连杆凸构型在适当电荷下为全局最小，并推广至等边五边形，给出控制论应用。

On the area of constrained polygonal linkages

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).

多边形连杆机构模空间上的交换作用

Swap action on moduli spaces of polygonal linkages

本文的基本对象是闭多边形连杆机构在$\mathbb{R}^2$或$\mathbb{R}^3$中的模空间$M_{2,3}(L)$。正如G. Khimshiashvili，空间$M_{2}(L)$配备有定向面积函数$A$，而（如本文所建议的）$M_{3}(L)$配备有矢量面积函数$S$。后者是一般的Morse函数，其临界点有一个很好的描述。在预印本中，我们在保持函数$A$和$S$以及Morse点的空间$M_{2,3}(L)$上定义了一个\textit{swap action}（即由边换位生成的某个群的作用）。我们证明了群的交换子的作用是平凡的，给出了一些计算机实验，并提出了一个猜想。

Motion planning and control of a planar polygonal linkage

For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate the configuration space by the vertex-edge graph of its cell decomposition discovered by the first author. The algorithm has three aspects: (1) the number of navigation steps does not exceed 15 (independent of the linkage), (2) each step is a disguised flex of a quadrilateral from one triangular configuration to another, which is a well understood type of flex, and (3) each step can be performed explicitly by adding some extra bars and obtaining a mechanism with one degr…

循环多边形的Morse指标

Morse index of a cyclic polygon

已知平面多边形连杆机构的循环位形是有符号面积函数的临界点。本文给出了一个计算循环构形符号面积的Morse指数的显式公式。它不仅依赖于一个循环构型的组合数学，而且还包括一些度量特征。