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摘要: 随着21世纪折纸工程学的发展, 折纸不再仅仅是一项民间艺术, 一方面数学家前期的大量工作随之浮出水面, 另一方面工程应用对折纸结构折叠过程的描述与分析都提出了新的挑战. 同时, 折纸的对象也不再局限于简单的纸张, 工程中存在大量的厚度不可忽略的平板结构, 他们的折叠展开问题一直困扰着相关的工程应用. 近几年超材料的发展给模块化折纸带来了一次从玩具到高科技的飞跃, 然而如何协调地安排这些折纸模块使得整体结构展现出超常且可变化调控的性能是折纸领域的新热点. 由此可见, 折纸运动学在诸多应用与探索方面都起到了决定性的基础作用. 本文重点介绍了已有的机构学理论与方法及其在各种折纸结构分析设计中的应用, 旨在为折纸工程学的发展提供坚实的理论技术基础.Abstract: Origami is no longer a folk art with the development of origami engineering in this century, in which a lot of previous work of mathematicians has surfaced and the engineering applications present new challenges to describe the folding process of rigid origami. Meanwhile, origami is not limited to the thin sheet, the folding and unfolding of thick panels have always hindered their related engineering applications. Recently, the development of metamaterials has brought a leap from toys to high technology for modular origami. How to coordinately set up the origami modules to enable the entire structure with excellent and tunable performance is a new hotspot in modular origami. Hence, origami kinematics has played a decisive role in many applications and explorations. This review focuses on the developed mechanism theories and their applications in the analysis and design of origami structures, which hopefully could be able to provide the theoretical foundation for origami engineering.
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Key words:
- origami /
- origami kinematics /
- rigid origami /
- thick-panel origami /
- modular origami
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图 1 平面机构及其矢量法表达(You & Chen 2011)
图 2 (a)四折痕顶点折纸图案的平面构型, (b)三维构型中的特征向量, (c)旋转序列(Wu & You 2010)
图 3 D-H标注法. (a)空间机构与, (b)球面机构(Denavit & Hartenberg 1955)
图 5 连杆桁架理论. (a) 常见情况, (b) 两个相交的转动轴,(c)两个具有瞬时可动性的平行转动轴,(d)两个平行的转动轴(Yang et al. 2016a)
图 6 迭代步骤中的预测-校正算法(Kumar & Pellegrino 2000)
图 7 (a)折纸顶点, (b)四边形网格折纸(Tachi 2009)
图 8 “MERLIN 2”典型的工作流程(Liu & Paulino 2018)
图 9 四折痕顶点及其等效的球面四杆机构(Chen et al. 2019b)
图 10 三浦折纸(Miura 1985)
图 11 四边形网格刚性折纸的设计与分析方法. (a) Kokotsakis折纸整合法(He & Guest 2020), (b) 变形梯度法(Feng et al. 2020a), (c) 离散微分几何法(Stachel 2010), (d) 球面机构网格的运动协调法(Chen et al. 2019b)
图 12 山谷折痕(M-V)分配. (a)双波纹折纸单元折痕分布(Peng et al. 2018), (b) Eggbox和Miura-ori模式的切换(Pratapa et al. 2019)
图 13 扭转图案. (a)广义三角形扭转图案(Feng et al. 2018b), (b)Plane linkage折纸单元(Chen et al. 2019a). 方形扭转图案. (c)刚性图案, (d)非刚性图案, (e)修正的刚性图案(Feng et al. 2020c, Ma et al. 2021)
图 14 缠绕折纸图案. (a) 机构运动阶段的方形缠绕折纸(Xu et al. 2018), (b)引入切割线的单自由度缠绕折纸(Lang et al. 2016)
图 15 Resch折纸图案. (a) 施加节点轨迹后的运动路径(Magliozzi et al. 2017), (b) 在目标函数下运动方向(Li 2020), (c) 等效杆件模型(Mazzucchi 2018), (d) 两种典型顶点的运动学解析(Yang et al. 2022a)
图 16 Waterbomb折纸图案. (a)对称折叠模式(Chen et al. 2016), (b)曲面构型(Deng et al. 2022),(c)刚性运动阶段的管状结构(Ma et al. 2020), (d)复杂轴对称构型(Zhao et al. 2021)
图 17 Yoshimura折纸图案. (a)原始图案, (b)节点拆分后的四、六折痕混合图案(Zhang & Chen 2019), (c)其管状结构对称与非对称的拼接方式(Lang 2017)
图 18 Kresling折纸图案. (a)倾斜对称形式(Zhai et al. 2018); (b)水平对称形式(Georgakopoulos et al. 2021)
图 19 可刚性平折的管状结构及其组合结构. (a)平行四边形截面的管状结构(Tachi 2010d), (b) 筝形截面的管状结构(Liu et al. 2016), (c)不规则截面的管状结构(Chen et al. 2017), (d)刚性折纸管耦合组装以增加结构刚度(Filipov et al. 2015), (e)模块化管状折纸结构(Mousanezhad et al. 2017)
图 20 立体折纸结构. (a) 刚性平折的高购物袋(Wu & You 2011), (b)折纸盒子机构(Wei 2014), (c) 单自由度刚性可平折立方体(Gu & Chen 2020), (d)可折展长方体剪纸结构(Zhang et al. 2022)
图 21 基于球面机构的板厚容纳方法. (a)楔形面板技术(Tachi 2011), (b)偏置面板技术(Edmondson et al. 2014), (c)节点拆分技术(Tolman et al. 2017)
图 22 基于空间机构的板厚容纳方法. (a)偏置铰链技术(Chen et al. 2015), (b)辅助面板技术(Gu et al. 2021), (c)双铰链技术(Ku & Demaine 2016), (d)剪纸技术(Yang et al. 2022b)
图 23 基于铰链变化的板厚容纳方法. (a)滚动接触技术(Lang et al. 2017), (b)柔性铰链技术(Pehrson et al. 2016); (c)薄膜技术(Zirbel et al. 2013)
图 24 2D 模块化折纸. (a)正方形镶嵌图案(Resch 1965), (b)非规则菱形平铺(Warisaya et al. 2021), (c)广义剪纸图案(Choi et al. 2019), (d)具有多步自引导折叠路径的可变形材料(Coulais et al. 2018)
图 25 3D 模块化折纸. (a) 三维运动学超材料(Yang et al. 2021), (b) 三维模块化超结构(Ma et al. 2022b), (c)三维可编程超材料(Liu et al. 2022), (d)可转换架构(Li & Yin 2021), (e) 可重构棱柱材料(Overvelde et al. 2017)
表 1 折纸运动学理论方法在各类折纸结构中的应用
折纸结构 矢量法 四元数法 矩阵法 旋量法 桁架法 SVD法 程序法 刚性折纸 折纸顶点 
四边形网格折纸 扭转图案 缠绕图案 Resch图案 Waterbomb图案 Yoshimura图案 Kresling图案 管状折纸结构 立体折纸结构 厚板折纸 
模块化折纸 2D模块化折纸 3D模块化折纸 
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