三阶魔方机构的拓扑与运动简图

doi:10.3901/JME.2020.23.089

机械工程学报 ›› 2020, Vol. 56 ›› Issue (23): 89-97.doi: 10.3901/JME.2020.23.089

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三阶魔方机构的拓扑与运动简图

卢文娟^1,2, 吴华芳^1,2, 刘甜璐^1,2, 曾达幸^1,2

1. 燕山大学先进锻压成形技术与科学教育部重点实验室秦皇岛 066004;
2. 燕山大学河北省重型机械流体动力传输与控制重点实验室秦皇岛 066004

收稿日期:2019-06-27 修回日期:2019-12-11 出版日期:2020-12-05 发布日期:2021-01-11
作者简介:卢文娟,女,1983年出生,博士,副教授,硕士研究生导师。主要研究方向为机构学理论及应用、机器人技术。E-mail:luwenjuan@ysu.edu.cn;曾达幸(通信作者),男,1978年出生,博士,教授,博士研究生导师。主要研究方向为并联机器人构型分析、型综合理论及应用。E-mail:roboms@ysu.edu.cn
基金资助:
国家自然科学基金青年科学基金（51905464）、国家重点研发计划（2017YFB1300203）、国家自然科学基金（51775473）和河北省自然科学基金（E2018203140）资助项目。

Topology and Motion Diagram of the Third-order Cube Mechanism

LU Wenjuan^1,2, WU Huafang^1,2, LIU Tianlu^1,2, ZENG Daxing^1,2

1. Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education, Yanshan University, Qinhuangdao 066004;
2. Hebei Key Laboratory of Heavy Mechinery Fluid Power Transimission and Control, Yanshan University, Qinhuangdao 066004

Received:2019-06-27 Revised:2019-12-11 Online:2020-12-05 Published:2021-01-11

摘要/Abstract

摘要： 表面上简单、紧凑的魔方机构，实际上却可以实现着运动的千变万化，这种奇妙的模型远超出了传统机构学的概念和设计方法。以经典三阶立方体魔方为对象，从建立魔方机构的运动简图为突破口探索魔方世界中的机构学问题。分析魔方机构中的构件和运动副，提出了两种新的运动副以描述魔方在运动过程中构件间的连接在可动与相对固定之间切换的问题；提出了正位状态和非正位状态的概念，表达魔方机构在转动过程中构态的变化，用机构拓扑图分别描述该两种构态下构件间的连接关系，采用邻接矩阵反映构件间的位置关系和拓扑特征，从而将魔方机构的图表达转换为数学表达的方法。为将复杂的魔方机构模块化，采取对魔方整体的约束关系进行内外划分并提取魔方运动单元机构的方法，结合抽离出的单元机构子邻接矩阵，以第一卦限单元机构为例，分别建立了正位和非正位状态时外部环路魔方机构的运动简图。研究工作为非传统、变拓扑、多构态、多环路机构的结构分析提供参考，为魔方机构后续的自由度分析与运动特性分析提供模型。

关键词: 魔方机构, 多构态, 拓扑图, 邻接矩阵, 运动简图

Abstract: The seemingly simple and compact Rubik's cube mechanism can actually realize the ever-changing movement, this wonderful model is beyond the traditional concept of mechanism and mechanism design method. The classical third-order cubic cube is taken as the research object, from the establishment of the motion diagram of the Rubik's cube mechanism as a breakthrough to explore the problem of mechanism in the world of Rubik's cube. The components and kinematic pairs in the Rubik's cube mechanism are analyzed, and two new kinematic pairs are defined to describe the problem of switching between movable and relatively fixed connections between components in the process of Rubik's cube movement. The aligned state and non-aligned state are defined to express the change of the configurations of the Rubik's cube mechanism in the process of rotation, the connection relationship between components in these two configurations is described by the mechanism topology diagram, and the adjacency matrix is used to reflect the location relationship and topological characteristics of components, so that the graphic expression of Rubik's cube mechanism can be transformed into mathematical expression. Then, in order to modularize the complex Rubik's cube mechanism, the method of dividing the internal and external constraints of the whole Rubik's cube and extracting the kinematic unit mechanism of the Rubik's cube is adopted, combining with the sub-adjacency matrix of the unit mechanism extracted from the corresponding configuration adjacency matrix of the Rubik's cube, taking the first octant unit mechanism as an example, the motion diagram of the external loop Rubik's cube mechanism in the aligned and non-aligned states are established respectively. The research work in this paper provides a reference for the structural analysis of non-traditional, variable topology, multi-configuration and multi-loop mechanisms, and provides a model for the follow-up analysis of freedom and motion characteristics of Rubik's cube mechanisms.

Key words: Rubik's cube mechanism, multi-configuration, topology diagram, adjacency matrix, motion diagram

中图分类号:

TH112

卢文娟, 吴华芳, 刘甜璐, 曾达幸. 三阶魔方机构的拓扑与运动简图[J]. 机械工程学报, 2020, 56(23): 89-97.

LU Wenjuan, WU Huafang, LIU Tianlu, ZENG Daxing. Topology and Motion Diagram of the Third-order Cube Mechanism[J]. Journal of Mechanical Engineering, 2020, 56(23): 89-97.

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三阶魔方机构的拓扑与运动简图

Topology and Motion Diagram of the Third-order Cube Mechanism

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