Reconfigurable origami-inspired multistable metamorphous structures
可重构折纸启发的多稳态变质结构

The origami unit is a combination of two similar origami cells. The crease pattern of the first one, referred to as cell I, is given in Fig. 1A. Black solid lines are mountain creases, whereas dashed lines are valley creases. All the inclined creases are parallel, and the rest are horizontal. It is a rigid origami pattern with a single degree of freedom (DOF). θ, the dihedral angle between two triangular facets on either side of the diagonal of the central rhombuses, could be treated as an input that uniquely determines the folded shapes of the cells, as given below the pattern. The key geometrical parameters are dimensions a and b. Here, we first take a = b, and α = π/4. Moreover, we let β = α. The crease pattern for cell II is shown in Fig. 1B. It has the identical geometry parameters but opposite fold assignments to those in cell I except those around two shaded central rhombuses. The edges of the rhombuses are dormant creases. To meet the rigid foldable conditions, the creases along their diagonals become valley creases. Its partially folded shape is displayed beneath the pattern. The behavior of both origami cells is demonstrated in movie S1.
折纸单元是两个相似的折纸单元的组合。第一个折痕模式，称为单元格 I，如图 1A 所示。黑色实线是山地折痕，而虚线是山谷折痕。所有倾斜的折痕都是平行的，其余的都是水平的。它是一种具有单一自由度 （DOF） 的刚性折纸图案。θ 是中央菱形对角线两侧两个三角形刻面之间的二面角，可以被视为唯一确定细胞折叠形状的输入，如图案下方给出的那样。关键的几何参数是尺寸 a 和 b。在这里，我们首先取 a = b，α = π/4。此外，我们让 β = α。细胞 II 的折痕模式如图 1B 所示。它具有相同的几何参数，但与单元格 I 中的折叠分配相反，除了两个阴影中心菱形周围的折叠分配。菱形的边缘是休眠的折痕。为了满足刚性可折叠条件，沿其对角线的折痕成为谷状折痕。其部分折叠的形状显示在图案下方。电影 S1 中展示了两种折纸细胞的行为。

The origami unit, shown in Fig. 1C, is created by stacking cell II on top of cell I and then bonding the shaded central rhombuses of two cells and their corresponding left and right edges together, respectively. This is possible when θ angles in both cells are made identical. The unit remains to be a flat-foldable rigid origami with a single DOF with θ angles, now merged together, being taken as the input, and it deploys and collapses easily.
折纸单元，如图 1C 所示，是通过将单元格 II 堆叠在单元格 I 的顶部，然后分别将两个单元格的阴影中心菱形及其相应的左右边缘粘合在一起而创建的。当两个单元格中的 θ 角相同时，这是可能的。该单元仍然是一个平面可折叠的刚性折纸，具有具有 θ 角的单个景深，现在合并在一起，作为输入，它很容易展开和折叠。

The relationships between θ and δ_I as well as δ_II, which are the dihedral angles between a triangular facet and its adjacent parallelogram facet along one of the edges of central rhombuses for cells I and II, respectively, can be obtained (text S1). We plot them in Fig. 1E. It can be seen that when θ = π, both cells are completely flat. Decreasing θ results in cell I being folded up with δ_I reducing steadily along the blue path while δ_II remaining at π. If, instead, θ is increased, i.e., the valley creases along the diagonal of rhombuses in cell I are inverted to mountain creases, the dormant creases along the edges of the central rhombuses are activated. δ_II decreases along the black path, while δ_I stays unchanged as a constant π, indicating that the creases along the edges of rhombuses of cell I become dormant. Effectively, cells I and II are switched over, as shown in Fig. 1 (C and D). This transition only occurs at θ = π. In other words, θ = π is a kinematic bifurcation point.
可以得到 θ 和 δ_I 以及 δ_II 之间的关系，即三角形面与其相邻的平行四边形面之间的二面角，分别是单元格 I 和 II 的中心菱形边之一（文本 S1）。我们在图1E中绘制了它们。可以看出，当 θ = π时，两个电池都是完全平坦的。减小 θ 导致细胞 I 折叠，δ_I 沿蓝色路径稳步减小，而_II δ保持在π。相反，如果 θ 增加，即单元 I 中沿菱形对角线的山谷折痕与山形折痕倒置，则沿中央菱形边缘的休眠折痕被激活。δ_II 沿黑色路径减少，而 δ_I 作为恒定π保持不变，表明细胞 I 菱形边缘的折痕处于休眠状态。实际上，单元 I 和 II 被切换，如图 1（C 和 D）所示。这种转变仅在 θ = π 时发生。换句话说，θ = π 是一个运动学分岔点。

A card model of the unit was built in which the diagonal creases of the two central rhombuses were cut open to reduce the influence of the material thickness as the cells are bonded together at these rhombuses when forming the unit, making these areas rather thick. The transition of the unit between different states is shown in Fig. 1F and movie S1. Two observations were made. First, by poking the convex central vertex of the unit in a configuration close to the bifurcation state with a probe, the cells snapped from one configuration to another. Second, although it is theoretically possible both cells deploy to an identical shape after passing the bifurcation state, as indicated in Fig. 1D, this never occurred in practice for reasons which we shall discuss next.
构建了该单元的卡片模型，其中两个中心菱形的对角线折痕被切开，以减少材料厚度的影响，因为在形成单元时，单元在这些菱形上粘合在一起，使这些区域相当厚。单元在不同状态之间的转换如图 1F 和视频 S1 所示。会上提出了两点意见。首先，通过用探针在接近分叉状态的配置中戳入单元的凸中心顶点，单元从一个配置捕捉到另一个配置。其次，尽管理论上两个电池在通过分叉状态后可能以相同的形状展开，如图 1D 所示，但由于我们接下来将讨论的原因，这种情况在实践中从未发生过。

In most origami models, the creases are not perfect mechanical joints that can rotate freely. Rather, they exhibit certain stiffness. If the creases are assumed to behave like linear elastic torsional springs with a stiffness k per unit length whereas the facets are rigid (5, 32, 38), the potential energies for each cell are ${\mathrm{\Pi }}_{\mathrm{C}-\mathrm{I}}={\displaystyle \sum _{i=1}^m}\frac{1}{2}{\mathit{kL}}_i {({\mathrm{\varphi }}_i-{\mathrm{\varphi }}_{i0})}^2$ and ${\mathrm{\Pi }}_{\mathrm{C}-\text{II}}={\displaystyle \sum _{j=1}^n}\frac{1}{2}{\mathit{kL}}_j {({\mathrm{\varphi }}_j-{\mathrm{\varphi }}_{j0})}^2$ , where Π_C-I and Π_C-II represent the potential energy of cell I and cell II, respectively, m and n are total number of creases in each cell, L is the length of a crease, and ϕ and ϕ₀ are final and initial rest dihedral angles of a crease, which include θ and δ used earlier to describe motions of cells. Note that the diagonal creases of the two central rhombuses are not included in the calculation since they were slit open when forming the unit. Hence, the total potential energy of a unit Π = Π_C-I + Π_C-II + Π_E in which Π_E is the potential energy of the edge creases formed when the edges of two cells are bonded together, and this energy can be calculated using a formula similar to Π_C-I or Π_C-II. Obviously, the rest dihedral angles ϕ₀ are set by the initial state of a cell (text S2).
在大多数折纸模型中，折痕并不是可以自由旋转的完美机械关节。相反，它们表现出一定的刚度。如果假设折痕的行为类似于线弹性扭转弹簧，每单位长度的刚度为 k，而刻面是刚性的 （5， 32， 38），则每个单元的势能为 和 ，其中 Π_C-I 和 Π_C-II 分别表示单元 I 和单元 II 的势能，m 和 n 是每个单元中的折痕总数， L 是折痕的长度，φ 和 φ₀ 是折痕的最终和初始静止二面角，其中包括 θ 和 δ 之前用于描述细胞运动的 θ 和 。请注意，两个中央菱形的对角线折痕不包括在计算中，因为它们在形成单元时被切开。因此，单位 Π 的总势能 = Π_C-I + Π_C-II + Π_E，其中 Π_E 是两个电池的边缘粘合在一起时形成的边缘折痕的势能，该能量可以使用类似于 Π_CI 或 Π_C-II 的公式计算。显然，φ₀ 的其余二面角是由单元格的初始状态（文本 S2）设置的。

Now consider cell I with a rest angle θ_I0 = π/2. Its normalized potential energy (NPE) versus θ curve is plotted in blue in Fig. 2A. It is noticeable that the energy curve has two wells at θ = θ_I0 = π/2 and θ = 3π/2. This indicates that cell I is a bistable structure with two stable states. If cell II is constructed exactly the same as cell I, i.e., θ_II0 = 3π/2 because of the way that we define θ in cell II, it will also exhibit two stable states, one at θ = θ_II0 = 3π/2 and the other at θ = π/2, drawn as the green curve in Fig. 2A. The second stable state of cell II matches the first stable position of cell I. Hence, we can fold cell II to reach the configuration with θ = π/2 and then bond both cells together to create a unit. The rest angles at the edge creases of the unit are set as the angles when bonding of the edges takes place, and the associated energy is plotted in red in Fig. 2A. The overall potential energy of the unit, Π, is given in Fig. 2B. It is symmetrical about the bifurcation state (θ = π), and Π still has two wells at θ₁ = π/2 and θ₂ = 3π/2, respectively, indicating that there are two stable states for the unit as expected. If we select θ_I0 = π/3 and θ_II0 = 3π/2, the overall energy curve is no longer symmetrical, but it remains to have two wells at θ₁ = 0.419π and θ₂ = 1.557π, different from those angles of each cell (see Fig. 2C). When θ_I0 = 2π/3 and θ_II0 = 3π/2, the energy curve shown in Fig. 2D is similar to those of other units where two energy wells appear at θ₁ = 0.586π and θ₂ = 1.442π. It can be concluded that if the creases behave like a linear elastic rotational hinge, the origami unit becomes a bistable structure whose stable states are related to but slightly different from the stable states of each cell. Figure 2E gives contours of the angle θ at two stable states when different initial rest states of the cells are selected.
现在考虑静止角为 θ_I0 = π/2 的单元格 I。其归一化势能 （NPE） 与 θ 的关系曲线在图 2A 中以蓝色绘制。值得注意的是，能量曲线在 θ = θ_I0 = π/2 和 θ = 3π/2 处有两个井。这表明细胞 I 是具有两种稳定状态的双稳态结构。如果单元 II 的构造与单元 I 完全相同，即 θ_II0 = 3π/2，因为我们在单元 II 中定义 θ 的方式，它也将表现出两种稳定状态，一种在 θ = θ_II0 = 3π/2，另一种在 θ = π/2 处，如图 2A 中的绿色曲线所示。细胞 II 的第二个稳定状态与细胞 I 的第一个稳定位置匹配。 因此，我们可以折叠细胞 II 以达到 θ = π/2 的构型，然后将两个细胞粘合在一起以创建一个单元。单元边缘折痕处的其余角度设置为边缘发生粘合时的角度，相关能量在图 2A 中以红色绘制。该单位的总势能 Π 如图 2B 所示。分岔态（θ = π）对称，Π分别在θ₁ = π/2和θ₂ = 3π/2处有两个井，表明该单元如预期的那样存在两种稳定状态。如果我们选择 θ_I0 = π/3 和 θ_II0 = 3π/2，则总能量曲线不再对称，但在 θ₁ = 0.419π 和 θ₂ = 1.557π 处仍然有两个阱，与每个单元的角度不同（见图 2C）。当 θ_I0 = 2π/3 和 θ_II0 = 3π/2 时，图 2D 中显示的能量曲线与其他单元的能量曲线相似，其中两个能量阱出现在 θ₁ = 0.586π 和 θ₂ = 1.442π。 可以得出结论，如果折痕表现得像线性弹性旋转铰链，折纸单元就会变成一个双稳态结构，其稳定状态与每个细胞的稳定状态相关，但略有不同。图 2E 给出了在选择不同的细胞初始静止状态时，两种稳定状态下的角度 θ 的轮廓。

For all units, there always exists a localized NPE peak when θ = π, e.g., the one shown in Fig. 2B. However, it is unlikely that the switch between two stable states will follow the NPE curve over this peak. Any small perturbation will result in the structure rapidly snap from one stable state to another, similar to the snap-through behavior in buckling of curved elastic beam systems (43). This is confirmed by our observation of the card model shown in Fig. 1F and movie S1. Moreover, the two cells will never take the same shape as doing so will require higher NPE (text S2).
对于所有单元，当θ = π时，始终存在局部NPE峰，例如，图2B中所示的那个。然而，两个稳定状态之间的切换不太可能在这个峰值上遵循NPE曲线。任何微小的扰动都会导致结构迅速从一种稳定状态卡入到另一种稳定状态，类似于弯曲弹性梁系统屈曲的卡扣行为（43）。我们对图 1F 和电影 S1 中所示的卡片模型的观察证实了这一点。此外，这两个单元格永远不会采用相同的形状，因为这样做将需要更高的 NPE（文本 S2）。

Three sample units, SU1, SU2 and SU3, were constructed. First, three sets of cells were manufactured in their unstrained forms using a 0.6-mm-thick elastomeric material, in which θ_I0 = π/2, π/3, and 2π/3, respectively, while θ_II0 was kept unchanged at 3π/2, and 0.4-mm-thick carbon fiber laminate sheet was cut and bonded to each facet to maintain rigid origami and confine all deformation to the creases (see Fig. 3A). Each cell II was then deformed to a shape with its θ close to θ_I0 using a rig and then bonded with cell I to form a unit. The edges of both cells were made slightly wider so that the corresponding edges could be bonded together to form edge creases of the unit (text S3).
构建了SU1、SU2和SU3三个样本单元。首先，使用 0.6 mm 厚的弹性材料以未应变形式制造三组电池，其中 θ_I0 分别 = π/2、π/3 和 2π/3，而 θ_II0 保持在 3π/2 不变，并将 0.4 mm 厚的碳纤维层压板切割并粘合到每个刻面上，以保持刚性折纸并将所有变形限制在折痕上（见图 3A）。然后使用钻机将每个单元 II 变形为其 θ 接近 θ_I0 的形状，然后与单元 I 结合形成一个单元。两个单元格的边缘都稍微宽一些，以便相应的边缘可以粘合在一起以形成单元的边缘折痕（文本 S3）。

We then measured the angles corresponding to two stable configurations of the units, whose values are given in Fig. 3A. The actual angles are close to the theoretical ones with a maximum error of 16.7% (θ₂ of SU2).
然后，我们测量了对应于单元的两个稳定配置的角度，其值如图 3A 所示。实际角度接近理论角度，最大误差为16.7%（SU2_的θ2）。

A total of four experiments were conducted for each unit. Starting from each of its two stable states, a unit is compressed to move away from this state either by either a lateral force (increasing θ), or a set of in-plane point forces (decreasing θ) as indicated in Fig. 3B. The energy curves were then obtained through integration of the measured forces over the corresponding displacements. The NPE versus θ plot of SU1 is given in Fig. 3C. The trend of experimental curves matches well with that of analysis, although the computed NPE is overestimated. This is because creases of the prototypes had finite width, whereas the theoretical width of the creases is zero, and thus, the actual model is less stiff than the analytical model.
每个单元总共进行了四次实验。从其两种稳定状态开始，一个单元被压缩以通过侧向力（增加 θ）或一组面内点力（减小 θ）离开该状态，如图 3B 所示。然后，通过对相应位移上的测量力进行积分来获得能量曲线。SU1 的 NPE 与 θ 的关系图如图 3C 所示。实验曲线的趋势与分析曲线的趋势吻合良好，尽管计算出的NPE被高估了。这是因为原型的折痕具有有限的宽度，而折痕的理论宽度为零，因此，实际模型的刚度不如分析模型。

Figure 3D shows the transition process of the first prototype by an external stimulus (movie S2). It was noticed that the unit never folded flat because the flat configuration does not correspond to any stable state, and a snap-through process took place in less than 1 s as shown in the first and second photos of the second row. Two stable states of the unit were symmetric to each other.
图 3D 显示了第一个原型在外部刺激下的过渡过程（视频 S2）。我们注意到，该装置从未折叠平整，因为平坦的配置不对应于任何稳定状态，并且在不到 1 秒的时间内发生了卡通过程，如第二排的第一张和第二张照片所示。该单元的两个稳定状态彼此对称。

The origami unit is scalable and could be used as building blocks for robots. Here, we use a robotic limb to demonstrate its vast possibilities. Figure 4A shows the origami pattern and its partially folded state for a limb consisting of a chain of three units where two units are bridged together by an inverted unit. Note that, although we state that there are three units, some facets of the neighboring units are actually overlapping. In its front view, we use ⊕ or ⊖ to mark the profile when a unit is convex up or down. This multistable robotic limb consists of three main components: the origami skeleton, a set of arc-shaped strips made from shape memory alloy (SMA) that are mounted on the central vertex of each unit for reconfiguration, and a flexible air-tight skin over the entire structure and a pump used to deploy the skeleton in each configuration (text S4). Figure 4B shows the arrangements of SMA strips in a unit in which a pair of them is installed on either side of the central vertex, and thus, there were a total of six SMA strips. Heating up one could force the vertex to invert, resulting in the unit to switch from ⊕ to ⊖ configurations, or vice versa (movie S3). Unlike conventional actuators, an SMA strip does not provide continuous control. Rather, it only provides an impulsive moment to let the unit snap from ⊕ to ⊖ and thus, much simpler in design. Once the skeleton enters into a particular configuration, e.g., ⊖-⊕-⊖, the force generated by the SMA actuators could be removed without causing the configuration change because each configuration is mechanically stable. However, its shape can further change by altering the pressure in the bag. Since the limb has three units, it has a total of eight configurations (2³). Figure 4C shows those configurations.
折纸单元是可扩展的，可以用作机器人的构建块。在这里，我们使用机器人肢体来展示其巨大的可能性。图4A显示了折纸模式及其部分折叠状态，该肢体由三个单元的链组成，其中两个单元通过一个倒置的单元桥接在一起。请注意，尽管我们声明有三个单元，但相邻单元的某些方面实际上是重叠的。在其前视图中，我们使用 ⊕ 或 ⊖ 来标记单元向上或向下凸起时的轮廓。这种多稳态机器人肢体由三个主要组件组成：折纸骨架，一组由形状记忆合金（SMA）制成的弧形条带，安装在每个单元的中心顶点上以进行重新配置，以及整个结构上的柔性气密皮肤和一个用于在每个配置中部署骨架的泵（文本S4）。图 4B 显示了 SMA 条带在一个单元中的排列，其中一对 SMA 条安装在中心顶点的两侧，因此总共有六个 SMA 条带。加热 1 可能会迫使顶点反转，导致单元从 ⊕ 配置切换到 ⊖配置，反之亦然（电影 S3）。与传统的执行器不同，SMA 条带不提供连续控制。相反，它只提供了一个冲动的时刻，让装置从 ⊕ 卡到⊖，因此在设计上要简单得多。一旦骨架进入特定配置，例如⊖-⊕-⊖，SMA致动器产生的力就可以被移除而不引起配置变化，因为每个配置在机械上都是稳定的。然而，它的形状可以通过改变袋子中的压力来进一步改变。由于肢体有三个单元，因此它总共有八种配置（2³）。 图 4C 显示了这些配置。

The robotic limb has many potential applications. For instance, when the limb is in ⊖-⊕-⊕ configuration, it can lift a weight (movie S4). To act as a gripper, it needs first to reconfigure from ⊖-⊕-⊖ to ⊕-⊕-⊕ configuration through SMA strips and then it grabs an object through its shape change activated by withdrawing air from the airbag. The entire process is demonstrated in both Fig. 4D and movie S5. In comparison with existing artificial limbs with a single operation mode, this multistable pneumatic-driven limb provides more operational programmability. It achieves, for instance, lifting and gripping modes in one assembly. Moreover, the gripper can also easily cope with objects of various shapes and weights.
机器人肢体有许多潜在的应用。例如，当肢体处于 ⊖-⊕-⊕ 配置时，它可以举起重物（电影 S4）。要充当抓手，它首先需要通过 SMA 条从 ⊖-⊕-⊖ 重新配置为 ⊕-⊕-⊕配置，然后通过从安全气囊中抽出空气来激活其形状变化来抓取物体。整个过程在图4D和视频S5中都有所演示。与现有的单一操作模式的假肢相比，这种多稳态气动驱动的假肢提供了更多的操作可编程性。例如，它在一个组件中实现了提升和抓取模式。此外，夹持器还可以轻松应对各种形状和重量的物体。

Not only could the bistable units be placed in series to form an artificial limb, it could also be extended laterally to create reconfigurable and programmable mechanical metamaterials to achieve desirable features such as negative Poisson’s ratio, variable stiffness, and shape transformations.
不仅可以将双稳态单元串联放置以形成假肢，还可以横向扩展以创建可重构和可编程的机械超材料，以实现所需的特性，例如负泊松比、可变刚度和形状变换。

Figure 5A shows an origami metamaterial built using 0.12-mm-thick cards, whose pattern consists of three columns of units laterally. In the longitudinal direction, these columns of units are arranged similarly to that of the multistable limb. The total number of columns is seven, and thus, there are 21 units. The origami pattern is given in Fig. 5B. In its original state, the metamaterial takes the shape of ⊕, ⊖, ⊕, ⊖, ⊕, ⊖, and ⊕. Note that unit within the same column behave exactly the same, i.e., they are either ⊕ or ⊖, otherwise, the mountain and valley rules would be violated. It has therefore a total of 128 stable states (2⁷) (movie S6) among which five symmetrical stable configurations of the same metamaterial are displayed in Fig. 5A. The interchangeable configurations could be reached once external stimuli are applied when the origami is close to its flat shape. We have investigated the Poisson’s ratios of the metamaterial, which are defined as functions of the folding ratio in the y direction (text S5). We placed an acrylic plate on top of the metamaterial and two weights were placed on top of the plate to impose a compressive force in the y direction. The Poisson’s ratio, ν_xy and ν_zy were then obtained, which are plotted in Fig. 5 (C and D). ν_xy was always negative and monotonically decreasing with the increase of the folding ratio, while ν_zy could be either positive or negative according to the folding ratio.
图 5A 显示了使用 0.12 毫米厚的卡片构建的折纸超材料，其图案由横向的三列单元组成。在纵向上，这些单元柱的排列类似于多稳肢的排列。列总数为 7 列，因此有 21 个单元。折纸图案如图 5B 所示。在其原始状态下，超材料呈现⊕、⊖、⊕、⊖、⊕、⊖和⊕的形状。请注意，同一列中的单元的行为完全相同，即它们要么是⊕的，要么是⊖的，否则将违反山脉和山谷规则。因此，它总共有128个稳定态（^2,7）（电影S6），其中同一超材料的5个对称稳定构型如图5A所示。当折纸接近其平面形状时，一旦应用外部刺激，就可以达到可互换的配置。我们已经研究了超材料的泊松比，它被定义为 y 方向（文本 S5）的折叠比的函数。我们在超材料顶部放置了一块亚克力板，并在板顶部放置了两个重物，以在 y 方向上施加压缩力。然后得到泊松比 ν_xy 和 ν_zy，如图 5 所示（C 和 D）。ν_xy始终为负，且随着折叠率的增加呈单调减小趋势，而ν_zy根据折叠率的不同，可以是正的，也可以是负的。

The number of distinct configurations of a reprogrammable mechanical metamaterial depends on the number of units in series. It is C(n, 2) where n is the number of the units. Figure 5E shows a single structure made from 11 units in four configurations out of 55 possible ones. Each column shows the deployable sequence of the structure in a particular configuration. Instead of using solely symmetrical units where α = β (see Fig. 1A), unsymmetrical ones could also be included where α ≠ β. An example is shown in Fig. 5F with its four configurations amongst many others. Use of unsymmetrical units can lead to spiral profile, avoiding collision between first and final units in a chain of units.
可重编程机械超材料的不同配置数量取决于串联单元的数量。它是 C（n， 2），其中 n 是单位的数量。图 5E 显示了由 11 个单元组成的单一结构，这些单元在 55 种可能的配置中有四种配置。每列都显示特定配置中结构的可部署顺序。在α = β的地方，而不是只使用对称的单位（见图1A），也可以在α ≠ β的地方包括不对称的单位。图 5F 中显示了一个示例，其中包含四种配置以及许多其他配置。使用不对称单元会导致螺旋形轮廓，从而避免单元链中的第一个单元和最终单元之间的碰撞。