Exploring Topological and Geometric Combination Indexfor Overhand Knot Formation
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摘要: 打结过程绳带拓扑与几何形态控制是新型打结机构研制的关键问题。基于纽结理论Reidemeister基本变换提出交腕结成形绳带的等效操作和成形原理，构建以非完整空心齿轮盘驱动绳带旋转缠绕、弧形立面凸轮支撑绳带、可动支臂夹爪拾取固定绳带的新型打结机构。提出以绳带交叉点与绳带悬链线极值点构造的拓扑与几何形态组合指标，将绳带成结过程划分为4个关键阶段，利用悬链线理论求解各阶段边界瞬态绳带几何形态。实验结果表明，对边界瞬态绳带拓扑与几何形态控制，实现了绳带在新型打结机构成结过程中的形态保持。
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关键词:
- 纽结理论 /
- 交腕结 /
- 打结机构 /
- 悬链线理论 /
- 拓扑与几何形态组合指标
Abstract: The topological and geometric control of the morphology of a rope belt in the knotting process iskeyto the development of a new knotting mechanism. Based on the Reidemeister fundamental transformation of the knot theory, the equivalent operation and formation principles of the rope beltduring the overhand knot forming process are proposed. A new knotting mechanism is built with the non-integral hollow gear disk that drives rope belt winding and the arc facade cam thatsupports the rope belt. The movable arm gripper picks up and fixes the rope belt. The topological and geometric combination index is constructed by using the cross-points of the rope belt and the extreme points of the catenary, and the knotting process is divided into four key stages. The geometric form of the boundary transient rope belt is solved by using the catenary theory. The experimental results show that the topological and geometriccontrol of the morphology of the boundary transient rope belt is realized and that the shape of the rope belt is maintained in the knotting process of the new knotting mechanism. -
表 1 绳带成结运动阶段划分表
绳带打结阶段 组合划分指标 绳带拓扑结构表示 核心机构运动示意图 第一阶段：
起始状态非完整空心齿轮盘缺口与固定板缺口重合；交叉点数为0、极值点数为1，组合指标为$In{d_C} = \left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]$ 
第二阶段：
无缠绕状态缠绕绳带转至即将与中心绳带接触；交叉点数为1、极值点数为1，组合指标为$In{d_C} = \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]$ 
第三阶段：
半缠绕状态缠绕绳带与中心绳带接触；交叉点数为2、极值点数为2，组合指标为$In{d_C} = \left[ {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right]$ 
第四阶段：
全缠绕状态齿轮旋转一周，缠绕绳带绕中心绳带旋转并穿越绳圈；交叉点数为3、极值点数为2，组合指标为$In{d_C} = \left[ {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right]$ 
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表 2 绳带打结阶段特征瞬态已知条件
运动阶段 组合指标 支撑点坐标及各段悬链线长度$S$/mm
（几何形态）悬链平面旋转角度/（°）
（拓扑形态）第一阶段 $\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]$ $A\left( {0,18,0} \right)$，$B\left( {7.43,9,36.99} \right)$；${S_{AB}} = 42$ ${\varphi _{AB}}{\rm{ = }}78.5^\circ $ 第二阶段 $\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]$ $A\left( {0,18,0} \right)$，$B\left( {28.31,9,42.74} \right)$，$C\left( {19.25,11.24,29.14} \right)$；${S_{AB}} = 56$ ${\varphi _{AB}}{\rm{ = 56}}{\rm{.5}}^\circ $ 第三阶段 $\left[ {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right]$ $A\left( {0,18,0} \right)$，$B\left( {36.99,9,29.14} \right)$，$C\left( {19.25,11.24,29.14} \right)$，${C_0}\left( {19.25,0,29.14} \right)$；${S_{AC}} = 38$，${S_{CB}} = 19$ ${\varphi _{AC}} = 56.5^\circ $，${\varphi _{CB}} = 0^\circ $ 第四阶段 $\left[ {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right]$ $A\left( {0,18,0} \right)$，$B\left( {7.43,9,36.99} \right)$，$C\left( {19.25,11.24,29.14} \right)$，$D\left( {22.45,10.7,29.14} \right)$，$E\left( {20.92,10.2,26.74} \right)$，${E_0}\left( {20.92,0,26.74} \right)$；${S_{AC}} = 38$，${S_{EB}} = 18$ ${\varphi _{AC}} = 56.5^\circ $，${\varphi _{EB}} = 142.8^\circ $
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