具有三阶幻方拓扑构型的厚板折纸机构运动相容性分析

doi:10.3901/JME.2023.11.201

机械工程学报 ›› 2023, Vol. 59 ›› Issue (11): 201-212.doi: 10.3901/JME.2023.11.201

扫码分享

具有三阶幻方拓扑构型的厚板折纸机构运动相容性分析

畅博彦^1,2, 张浩楠¹, 梁栋^1,2, 金国光^1,2

1. 天津工业大学机械工程学院天津 300387;
2. 天津市现代机电装备技术重点实验室天津 300387

收稿日期:2022-06-05 修回日期:2022-12-13 出版日期:2023-06-05 发布日期:2023-07-19
通讯作者: 金国光(通信作者),男,1963年出生,博士,教授,博士研究生导师。主要研究方向为机械设计及理论、机械系统动力学与控制。E-mail:jinguoguang@tiangong.edu.cn
作者简介:畅博彦,男,1985年出生,博士,副教授。主要研究方向为机构学和机械系统动力学。E-mail:mmts_tjpu@126.com;张浩楠,男,1997年出生,硕士研究生。主要研究方向为折纸机构结构学和运动学。E-mail:haonan_zh@126.com;梁栋,男,1985年出生,博士,副教授。主要研究方向为机器人机构及优化理论、多柔体系统动力学与控制,E-mail:dongliang@tiangong.edu.cn
基金资助:
国家自然科学基金(52005368，52175243)和天津市高等学校创新团队培养计划(TD13-5037)资助项目

Foldability Analysis of Thick-panel Origami Mechanism with Third-order Magic Square Topology

CHANG Boyan^1,2, ZHANG Haonan¹, LIANG Dong^1,2, JIN Guoguang^1,2

1. School of Mechanical Engineering, Tiangong University, Tianjin 300387;
2. Tianjin Key Laboratory of Advanced Mechatronics Equipment Technology, Tianjin 300387

Received:2022-06-05 Revised:2022-12-13 Online:2023-06-05 Published:2023-07-19

摘要/Abstract

摘要： 折纸机构是折展机构的一种，折痕分布对折纸机构的刚性折展运动有直接影响。首先，提出一类具有三阶幻方拓扑构型的折纸机构，运用拓扑学理论对折纸机构的拓扑组成元素进行分析，通过构造网孔环路矢量和折痕分布矩阵，形成一种可用于描述折纸机构折痕分布特征的矩阵表示方法。其次，建立拓扑图旋转、翻转和镜像操作过程的矩阵运算数学模型，对256种三阶幻方折纸机构拓扑构型进行同构判定后，将得到的23种拓扑构型分为方扭型、轴对称型和不规则型。最后，在考虑板件厚度的同时，为了判断机构是否可刚性折展，将三阶幻方中每一个网孔环路处的单顶点四折痕机构等效为Bennett机构，通过引入分位角参数矩阵、转角特征矩阵和分位角特征矩阵，建立运动相容性判定方程的通用表达式，提出运动相容性系数的概念，并将其应用于厚板机构的刚性折展性分析，发现23种拓扑构型中有19种可用于厚板折展单元的设计，以折展单元与其镜像体进行模块化扩展为例，说明了此类折展单元在构造大尺度折展机构时的应用前景。

关键词: 可展机构, 折纸机构, 拓扑学, 刚性可折性, 运动相容性

Abstract: Rigid foldability of the origami mechanism is directly affected by the crease pattern. Firstly, components of the origami crease pattern with third-order magic square topology are examined with topology theory. To represent the crease assignment features of the origami topology, the mesh circuit vector and the crease assignment matrix are constructed. Secondly, the origami topology is identical before and after the operations of rotating, flipping, and mirroring for two hundred and fifty-six topology configurations. Following isomorphism determination, twenty-three topological configurations are obtained and classified as three types named square-twist type, axisymmetric type, and irregular type. Finally, to determine whether the thick-panel origami mechanism can achieve rigid folding, the single-vertex four-crease mechanism is equivalent with the Bennett linkages kinematically. A general formula of kinematic compatibility is established by introducing the sector angle parameter matrix, the dihedral angle eigen matrix, and the sector angle eigen matrix. The coefficient of kinematic compatibility is proposed and it can be used to determine whether a thick-panel origami mechanism is rigidly foldable or not. Nineteen of the twenty-three topological configurations are available for rigidly foldable thick-panel modules. These modules can be used to construct large scale deployable mechanisms with their mirrored counterpart which have a great prospect.

Key words: deployable mechanism, origami mechanism, topology, rigid foldability, kinematic compatibility

中图分类号:

TH112

畅博彦, 张浩楠, 梁栋, 金国光. 具有三阶幻方拓扑构型的厚板折纸机构运动相容性分析[J]. 机械工程学报, 2023, 59(11): 201-212.

CHANG Boyan, ZHANG Haonan, LIANG Dong, JIN Guoguang. Foldability Analysis of Thick-panel Origami Mechanism with Third-order Magic Square Topology[J]. Journal of Mechanical Engineering, 2023, 59(11): 201-212.

参考文献

[1] 邓宗全. 空间折展机构设计[M]. 哈尔滨:哈尔滨工业大学出版社，2013. DENG Zongquan. Design of space deployable and foldable mechanism[M]. Harbin:Harbin Institute of Technology Press，2013.
[2] 杨慧，冯健，刘永斌，等. 百米级多超弹性铰链抛物柱面天线折展机构设计与运动学分析[J]. 机械工程学报，2022，58(3):75-83. YANG Hui，FENG Jian，LIU Yongbin，et al. Design and kinematic analysis of a large deployable mechanism of the parabolic cylindrical antenna with multi tape-spring hinges[J]. Journal of Mechanical Engineering，2022，58(3):75-83.
[3] 畅博彦，杨帅，金国光，等. 基于直线驱动的空间可展机构运动分析[J]. 机械工程学报，2020，56(5):192-201. CHANG Boyan，YANG Shuai，JIN Guoguang，et al. Motion analysis of spatial deployable mechanism driven in straight line[J]. Journal of Mechanical Engineering，2020，56(5):192-201.
[4] 李明，蒋延达，崔琦峰，等. 折纸衍生空间可展结构研究回顾与展望刍议[J]. 机械工程学报，2021，57(23):53-65. LI Ming，JIANG Yanda，CUI Qifeng，et al. Immature state-of-the-art review on origami-inspired spaceborne deployable structures[J]. Journal of Mechanical Engineering，2021，57(23):53-65.
[5] 张霄，李明，崔琦峰，等. 基于正六边形折纸的单自由度可展结构[J]. 机械工程学报，2021，57(11):153-164. ZHANG Xiao，LI Ming，CUI Qifeng，et al. Regularly hexagonal origami pattern inspired deployable structure with single degree of freedom[J]. Journal of Mechanical Engineering，2021，57(11):153-164.
[6] DUFOUR L，OWEN K，MINTCHEV S，et al. A drone with insect-inspired folding wings[C]// Institute of Electrical and Electronics Engineers. 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS)，October 9-14，2016，Daejeon，Korea. New York:IEEE，2016:1576-1581.
[7] FABER J A，ARRIETA A F，STUDART A R. Bioinspired spring origami[J]. Science，2018，359(6382):1386-1391.
[8] 杨名远，马家耀，李建民，等. 基于厚板折纸理论的微创手术钳[J]. 机械工程学报，2018，54(17):36-45. YANG Mingyuan，MA Jiayao，LI Jianmin，et al. Thick-panel origami inspired forceps for minimally invasive surgery[J]. Journal of Mechanical Engineering，2018，54(17):36-45.
[9] EDMONDSON B J，BOWEN L A，GRAMES C L，et al. Oriceps:Origami-inspired forceps[C]// American Society of Mechanical Engineers. Proceedings of the ASME 2013 Conference on Smart Materials，Adaptive Structures and Intelligent Systems，September 16-18，2013，Snowbird Utah，USA. SMASIS2013-3299.
[10] RUS D，SUNG C. Spotlight on origami robots[J]. Science Robotics，2018，3(15):eaat0938.
[11] 冯慧娟，杨名远，姚国强，等. 折纸机器人[J]. 中国科学:技术科学，2018，48:1259-1274. FENG Huijuan，YANG Mingyuan，YAO Guoqiang，et al. Origami robots[J]. Scientia Sinica Technologica，2018，48:1259-1274.
[12] 陈焱. 基于机构运动的大变形超材料[J]. 机械工程学报，2020，56(19):2-13. CHEN Yan. Review on kinematic metamaterials[J]. Journal of Mechanical Engineering，2020，56(19):2-13.
[13] 冯慧娟，马家耀，陈焱. 广义Waterbomb折纸管的刚性折叠运动特性[J]. 机械工程学报，2020，56(19):143-159. FENG Huijuan，MA Jiayao，CHEN Yan. Rigid folding of generalized waterbomb origami tubes[J]. Journal of Mechanical Engineering，2020，56(19):143-159.
[14] 方虹斌，吴海平，刘作林，等. 折纸结构和折纸超材料动力学研究进展[J]. 力学学报，2022，54(1):1-38. FANG Hongbin，WU Haiping，LIU Zuolin，et al. Advances in the dynamics of origami structures and origami metamaterials[J]. Chinese Journal of Theoretical and Applied Mechanics，2022，54(1):1-38.
[15] CHEN Y，FAN L，FENG J. Kinematic of symmetric deployable scissor-hinge structures with integral mechanism mode[J]. Computers and Structures，2017，191:140-152.
[16] WEI G，CHEN Y，DAI J S. Synthesis，mobility，and multifurcation of deployable polyhedral mechanisms with radially reciprocating motion[J]. Journal of Mechanical Design，2014，136(9):91003-91015.
[17] LEE T U，GATTAS J M. Geometric design and construction of structurally stabilized accordion shelters[J]. Journal of Mechanisms and Robotics，2016，8(3):031009.
[18] 畅博彦，金国光，戴建生. 基于变约束旋量原理的变胞机构构型综合[J]. 机械工程学报，2014，50(5):17-25. CHANG Boyan，JIN Guoguang，DAI Jiansheng. Type synthesis of metamorphic mechanism based on variable constraint screw theory[J]. Journal of Mechanical Engineering，2014，50(5):17-25.
[19] KAWASAKI T，YOSHIDA M. Crystallographic flat origamis[J]. Mathematics，1988，42(2):153-157.
[20] HULL T. Project origami:Activities for exploring mathematics[M]. Boca Raton:CRC Press，2012.
[21] HULL T. Counting mountain-valley assignments for flat folds[J]. Ars Combinatoria，2003，67:175-188.
[22] GREENBERG H C，GONG M L，MAGLEBY S P，et al. Identifying links between origami and compliant mechanisms[J]. Mechanical Sciences，2011，2:217-225.
[23] PENG R，MA J，CHEN Y. The effect of mountain-valley folds on the rigid foldability of double corrugated pattern[J]. Machine and Mechanism Theory，2018，128:461-474.
[24] FENG H，PENG R，MA J，et al. Rigid foldability of generalized triangle twist origami pattern and its derived 6R linkages[J]. Machine and Mechanism Theory，2020，10(5):051003.
[25] FENG F，DANG X，JAMES R D，et al. The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami[J]. Journal of the Mechanics and Physics of Solids，2020，142:104018.
[26] EVANS T A，LANG R J，MAGLEBY S P，et al. Rigidly foldable origami twists[M]. Providence:American Mathematical Society，2015.
[27] MA J，CHAI S，CHEN Y. Geometric design，deformation mode，and energy absorption of patterned thin-walled structures[J]. Mechanics of Materials，2022，168:104269.
[28] LIU X，GATTAS J M，CHEN Y. One-DOF superimposed rigid origami with multiple states[J]. Scientific Reports，2016，6(1):36883.
[29] CHEN Y，FENG H，MA J，et al. Symmetric waterbomb origami[J]. Proceedings of the Royal Society A，2016，472(2190):20150846.
[30] EDMONDSON B J，LANG R J，MAGLEBY S P，et al. An offset panel technique for thick rigidly foldable origami[C]// American Society of Mechanical Engineers. Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference，August 17-20，2014，Buffalo，New York，USA. 2014:DETC2014-35606.
[31] TACHI T. Rigid-foldable thick origami[J]. Origami，2011，5:253-264.
[32] HOBERMAN C. Folding structures made of thick hinged sheets:US，2006026806[P]. 2007-01-18.
[33] CHEN Y，PENG R，YOU Z. Origami of thick panels[J]. Science，2015，349(6246):396-400.
[34] BENNETT G T. A new mechanism[J]. Engineering，1903，76(12):777-778.
[35] BENNETT G T. The skew isogram mechanism[J]. Proceedings of the London Mathematical Society，1914，2(1):151-173.
[36] MYARD F E. Contribution to the geometry of articulated systems[J]. Bulletin de la Société Mathématique de France，1931，59:183-210.
[37] BAKER J E. An analysis of the Bricard linkages[J]. Mechanism and Machine Theory，1980，15(4):267-286.
[38] JOSEPH O. How to fold it:The mathematics of linkages，origami，and polyhedra[M]. London:the British Library，2011.
[39] ZHANG X，NIE R，CHEN Y，et al. Deployable structures:Structural design and static/dynamic analysis[J]. Journal of Elasticity，2021，146(2):199-235.
[40] WANG C，LI J，ZHANG D. Optimization design method for kirigami-inspired space deployable structures with cylindrical surfaces[J]. Applied Mathematical Modelling，2021，89:1575-1598.
[41] CHEN X，FENG H，MA J，et al. A plane linkage and its tessellation for deployable structure[J]. Mechanism and Mahcine Theory，2019，142:103605.
[42] 畅博彦，徐鑫，梁栋，等. 厚板三浦折展机构的几何设计与运动分析[J]. 中国空间科学技术，2022，42(4):146-157. CHANG Boyan，XU Xin，LIANG Dong，et al. Geometric design and motion analysis of Miura-Ori mechanismwith thick panels[J]. Chinese Space Science and Technology，2022，42(4):146-157.

具有三阶幻方拓扑构型的厚板折纸机构运动相容性分析

Foldability Analysis of Thick-panel Origami Mechanism with Third-order Magic Square Topology

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 12

编辑推荐

Metrics

本文评价

[1]	刘恩博, 赵永生, 李永杰, 郭笑宇, 许允斗. 正馈构架可展天线机构卸载方法与实验研究[J]. 机械工程学报, 2024, 60(15): 38-48.
[2]	申理精, 耿坤, 李盘浩, 张静. 索杆桁架式可展开机构设计与力学分析[J]. 机械工程学报, 2022, 58(17): 135-143.
[3]	畅博彦, 杨帅, 金国光, 张转, 朱永杰. 基于直线驱动的空间可展机构运动分析[J]. 机械工程学报, 2020, 56(5): 192-201.
[4]	刘文兰, 许允斗, 郭金伟, 姚建涛, 赵永生. 被动输入过约束四面体可展天线机构动力学分析[J]. 机械工程学报, 2020, 56(5): 181-191.
[5]	杨栋皓, 曹文熬, 丁华锋. 基于新型两层两环连杆的一族伞状可展机构的构型综合[J]. 机械工程学报, 2020, 56(5): 150-160.
[6]	杨廷力, 沈惠平, 刘安心, 杭鲁滨. 机构拓扑学理论的基本思想与数学方法——从方法论角度回顾几种原创性理论与方法[J]. 机械工程学报, 2020, 56(3): 1-15.
[7]	郝艳玲, 李锐明, 孙学敏, 姚燕安. 可重构立方体机构的设计与运动模式分析[J]. 机械工程学报, 2020, 56(13): 120-127.
[8]	郭金伟, 许允斗, 刘文兰, 姚建涛, 赵永生. 基于四面体单元的新型可展机构自由度分析[J]. 机械工程学报, 2019, 55(12): 9-18.
[9]	高慧芳, 刘婧芳, 余跃庆. 一种单自由度对称可展耦合机构的设计[J]. 机械工程学报, 2018, 54(5): 62-73.
[10]	张武翔, 杨毅, 罗均, 谢少荣, 周鑫, 李恒宇. 多环闭链可展机构尺寸调整方法[J]. 机械工程学报, 2015, 51(19): 11-20.
[11]	李端玲;张忠海;于振. 球面剪叉可展机构的运动特性分析[J]. , 2013, 49(13): 1-7.
[12]	杨毅;丁希仑. 基于空间多面体向心机构的伸展臂设计研究[J]. , 2011, 47(5): 26-34.