Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns

Abstract

This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.

1. Introduction

Origami, an ancient art of folding a two-dimensional paper into a three-dimensional structure, has aroused considerable research interests from scientists and engineers in recent years due to some of the unique properties exhibited by the folded structures and consequently has inspired a wide spectrum of state-of-the-art applications ranging from deployable space solar panels [1] to medical stents [2], foldable electronic devices [3–5], lightweight sandwich folded cores [6] and mechanical metamaterials [7–11]. In this context, a fundamental yet challenging issue is origami geometric design, which addresses the problem of how to create a crease pattern that folds into the desired three-dimensional shape. Deriving parametric equations to describe the folding kinematics for a certain crease pattern is a commonly used approach by many researchers [12–14]. However, this approach suffers from vast mathematical complexity and often lacks flexibility and applicability in real circumstances. Lang [15] proposed an origami design method, known as the tree method, which allows one to design hierarchical flat-folded origami silhouettes based on the concept of universal molecule. Tachi [16] developed the first practical method for the construction of true three-dimensional origami structures using edge- and vertex-tucking molecules. Later on, Tachi [17] introduced a method to produce three-dimensional origami structures from generalized Resch patterns that can approximate a given polyhedral surface. Zhou et al. [18] developed a computer-based method, known as the vertex method, which can generate developable three-dimensional origami structures between two singly curved surfaces as well as simulate the unfolding process of the designed folded structure to the corresponding two-dimensional crease pattern. More recently, Dudte et al. [19] presented a numerical algorithm to generate developable origami tessellations that can yield approximations to given surfaces of constant or varying curvature. Despite these developments, designing developable origami tessellations to strictly fit between two doubly curved target surfaces is still intractable.

In this paper, we present a mathematical framework for the generation of rigid-foldable three-dimensional origami tessellations based on trapezoid-mesh crease patterns that can simultaneously fit two doubly curved surfaces with rotational symmetry about a common axis. Specifically, the geometric parameters of the trapezoidal crease pattern and the folding angles of the target three-dimensional state whose outer and inner profiles are prescribed independently are determined by solving a set of mixed geometric and constraint equations. To facilitate the design process, an algorithm to simulate the interim folding sequence of the designed origami tessellation is provided. Furthermore, methods of designing developable and flat-foldable folded rings and stacked folded structures consisting of two layers that can fold independently or in a compatible manner are introduced. The potential applications of the proposed framework include but are not limited to creating deployable roofs or shelters in architecture and doubly curved sandwich structures with folded cores on the aircraft.

The layout of the paper is as follows. First, the folding kinematics of the trapezoidal crease pattern is established, based on which a folding simulation algorithm and the fundamental inverse design theory are developed. Two design examples are provided. Then, the conditions and procedures to design folded ring structures and stacked folded structures are discussed. Finally, a brief summary concludes the paper.

2. Fundamental theory

(a). Folding kinematics of the trapezoidal pattern

Figure 1a illustrates a single degree-4 vertex, where the valley and mountain creases are indicated by the blue and red lines, respectively, sector angles $\mathrm{\angle }AOB$ and $\mathrm{\angle }AOD$ are both equal to φ ($\in [0,\pi /2]$), and creases OA and OC are collinear. The interim folded state of the degree-4 vertex is shown in figure 1b. Denote the dihedral angles along creases OA, OB, OC and OD by θ_A, θ_B, θ_C and θ_D ($\in [0,\pi ]$) and the angles formed by creases OA and OC and creases OB and OD by η_a and η_b, respectively. The following relationships can be obtained:

$$\begin{array}{r}{\theta }_A={\theta }_C,\end{array}$$ 2.1

$$\begin{array}{r}{\theta }_B={\theta }_D,\end{array}$$ 2.2

$$\begin{array}{r}(1+{\cos }^2\phi +{\sin }^2\phi \cos {\theta }_A)(1+{\cos }^2\phi -{\sin }^2\phi \cos {\theta }_B)=4{\cos }^2\phi ,\end{array}$$ 2.3

$$\begin{array}{r}\cos {\eta }_a=\mathrm{si}{\mathrm{n}}^2\phi \cos {\theta }_B-\mathrm{co}{\mathrm{s}}^2\phi \end{array}$$ 2.4

$$\begin{array}{r}\text{and}\cos {\eta }_b=\mathrm{si}{\mathrm{n}}^2\phi \cos {\theta }_A+\mathrm{co}{\mathrm{s}}^2\phi .\end{array}$$ 2.5

Figure 1.

(a) A single degree-4 vertex and (b) The interim folded state of the degree-4 vertex. (Online version in colour.)

The detailed derivations of equations (2.1)–(2.5) are provided in the electronic supplementary material, section A.

Consider now a crease pattern comprising 12 trapezoidal facets shown in figure 2a, where all vertices locate on five auxiliary dashed lines χ_c1, χ_f1, χ_c2, χ_f2 and χ_c3 that intersect at point O, the angle formed by any two adjacent χ lines equals α₀ and the creases and boundary lines (i.e. the grey lines) extending between any two adjacent χ lines are all parallel to each other. For clarity, we refer herein the creases or boundary lines extending along the χ lines as the radial creases or boundary lines and the rest as the circumferential creases or boundary lines. Any two adjacent circumferential creases or boundary lines are symmetrical about the χ line passing through their common point. Denote the length of line segment $C_{1,k}{C}_{1,k+1}$ by a_ck, where k = 1,2,3, the length of line segment $F_{1,k}{F}_{1,k+1}$ by a_fk, where k = 1,2,3, the length of line segment $C_{1,k}{F}_{1,k}$ by b_k, where k = 1, … , 4, the acute angle formed by line segment C₁₁F₁₁ and line χ_c1 by φ_c, and the acute angle formed by line segment C₁₁F₁₁ and line χ_fk by φ_f. The following simple geometric relationships can be obtained:

$$\begin{array}{rl}& {\phi }_c-{\phi }_f={\alpha }_0,\end{array}$$ 2.6

$$\begin{array}{r}\frac{a_{ck}}{a_{fk}}=\frac{\sin {\phi }_f}{\sin {\phi }_c},k=1,2,3\end{array}$$ 2.7

$$\begin{array}{rl}\mathrm{and}& b_k=b_1+\frac{\sin {\alpha }_0}{\sin {\phi }_f}\sum _{i=1}^{k-1}{a}_{ci},k=2,3,4.\end{array}$$ 2.8

Figure 2.

(a) A trapezoidal crease pattern comprising 12 trapezoidal facets and (b) The interim folded state of the trapezoidal crease pattern. (Online version in colour.)

The interim folded state of the crease pattern is illustrated in figure 2b. Each internal vertex of the crease pattern is a degree-4 vertex shown in figure 1. According to equations (2.1)–(2.3), all of the dihedral angles along the collinear radial creases and all of those along the circumferential creases are equal. Owing to symmetry, the dihedral angles along the creases on line χ_f1 are equal to those along the creases on line χ_f2. Hence, we denote the dihedral angles along the radial creases on lines χ_f1 and χ_f2 by θ_fa, those along the radial creases on line χ_c2 by θ_ca and those along the circumferential creases by θ_b. Applying equation (2.3) to each internal vertex, the following two equations can be obtained:

$$(1+{\cos }^2{\phi }_c+{\sin }^2{\phi }_c\cos {\theta }_{ca})(1+{\cos }^2{\phi }_c-{\sin }^2{\phi }_c\cos {\theta }_b)=4{\cos }^2{\phi }_c$$ 2.9

and

$$(1+{\cos }^2{\phi }_f+{\sin }^2{\phi }_f\cos {\theta }_{fa})(1+{\cos }^2{\phi }_f-{\sin }^2{\phi }_f\cos {\theta }_b)=4{\cos }^2{\phi }_f.$$ 2.10

Note that there are a total of three independent geometric parameters θ_ca, θ_fa and θ_b governed by two kinematics equations. Therefore, the crease pattern shown in figure 2 has single degree-of-freedom (d.f.) of rigid folding motion. Besides, when θ_b becomes zero, both θ_ca and θ_fa reach zero according to equations (2.9) and (2.10), indicating that the crease pattern is flat-foldable. The same conclusions are applicable to a general case with any number of χ lines and any number of vertices along the χ lines.

(b). Motion simulation of the trapezoidal pattern

According to equation (2.4) and the relationships among the dihedral angles discussed above, the angles formed by the radial creases at all internal vertices that locate on the χ_f lines (subsequently referred as the F-type vertices) are equal. So are those formed by the radial creases or boundary lines at all vertices that locate on the χ_c lines (subsequently referred to as the C-type vertices). The similar relationships can be found for the angles formed by the circumferential creases or boundary lines. Hence, we denote the angles formed by the radial creases at the F-type vertices by η_fa, those by the radial creases or boundary lines at the C-type vertices by η_ca and the circumferential counterparts by η_fb and η_cb, respectively. According to equations (2.4) and (2.5), they can be determined as

$$\begin{array}{rl}\cos {\eta }_{ca}& =\mathrm{si}{\mathrm{n}}^2{\phi }_c\cos {\theta }_b-\mathrm{co}{\mathrm{s}}^2{\phi }_c,\end{array}$$ 2.11

$$\begin{array}{rl}\cos {\eta }_{cb}& =\mathrm{si}{\mathrm{n}}^2{\phi }_c\cos {\theta }_{ca}+\mathrm{co}{\mathrm{s}}^2{\phi }_c,\end{array}$$ 2.12

$$\begin{array}{rl}\cos {\eta }_{fa}& =\mathrm{si}{\mathrm{n}}^2{\phi }_f\cos {\theta }_b-\mathrm{co}{\mathrm{s}}^2{\phi }_f\end{array}$$ 2.13

$$\begin{array}{rl}\text{and}\cos {\eta }_{fb}& =\mathrm{si}{\mathrm{n}}^2{\phi }_f\cos {\theta }_{fa}+\mathrm{co}{\mathrm{s}}^2{\phi }_f.\end{array}$$ 2.14

To derive the positions of the vertices, we define a cylindrical coordinate system (figure 2b) whose z-axis is perpendicular to the plane defined by the circumferential boundary lines passing through C_i,1 and F_j,1, where i = 1,2,3 and j = 1,2, r-axis passes through C₁₁ and origin O has equal distances to C_i,1, i = 1,2,3. Denote the projected lengths of line segments C₁₁C₁₂, C₁₂C₁₃ and C₁₃C₁₄ on the z-axis by h₁, h₂ and h₃, respectively. They can be determined as

$$h_k=a_{ck}\cos \left(\frac{{\eta }_{ca}}{2}\right),k=1,2,3.$$ 2.15

The sector angle α, defined by the angle formed by line segments OF_k,1 and OC_k,1 or OC_k+1,1, k = 1,2, is given by

$$\alpha =\frac{{\eta }_{cb}-{\eta }_{fb}}{2}.$$ 2.16

Finally, for a general trapezoidal crease pattern containing m × n C-type vertices and (m − 1) × n) F-type vertices, the coordinates of the C-type vertices are obtained as

$$\begin{array}{rl}{r}_{i,j}^c& =\frac{b_j\sin \hspace{0.17em}({\eta }_{fb}/2)}{\sin \alpha },i=1,\dots ,m,\hspace{0.17em}j=1,\dots ,n,\end{array}$$ 2.17

$$\begin{array}{rl}{\theta }_{i,j}^c& =2(i-1)\alpha ,i=1,\dots ,m,\hspace{0.17em}j=1,\dots ,n,\end{array}$$ 2.18

$$\begin{array}{rl}{z}_{i,1}^c& =0,i=1,\dots ,m\end{array}$$ 2.19

$$\begin{array}{rl}\mathrm{and}{z}_{i,j}^c& =\sum _{k=1}^{j-1}{(-1)}^k{h}_k,i=1,\dots ,m,\hspace{0.17em}j=2,\dots ,n,\end{array}$$ 2.20

and those of the F-type vertices as

$$\begin{array}{rl}{r}_{i,j}^f& =\frac{b_j\sin \hspace{0.17em}({\eta }_{cb}/2)}{\sin \alpha },i=1,\dots ,m-1,\hspace{0.17em}j=1,\dots ,n,\end{array}$$ 2.21

$$\begin{array}{rl}{\theta }_{i,j}^f& =(2i-1)\alpha ,i=1,\dots ,m-1,\hspace{0.17em}j=1,\dots ,n\end{array}$$ 2.22

$$\begin{array}{rl}\mathrm{and}{z}_{i,j}^f& =z_{i,j}^c,i=1,\dots ,m-1,\hspace{0.17em}j=1,\dots ,n.\end{array}$$ 2.23

The algorithm to simulate the folding motion of the trapezoidal pattern is as follows. First, select one of the three dihedral angles as the control parameter. Then, solve the other two dihedral angles through equations (2.9) and (2.10). Next, η_ca, η_fa, η_cb and η_fb are determined through equations (2.11)–(2.14). Finally, the coordinates of all the vertices are obtained using equations (2.17)–(2.23), with which the current interim folded state is determined.

(c). Inverse design algorithm

The folded structure of the trapezoidal pattern, as illustrated in figure 2b, has rotational symmetry about the z-axis. Hence, projecting all the vertices and creases onto the r–z plane yields a superimposed two-dimensional pattern shown in figure 3. It is noted that the outer and inner cross-sectional profiles of the folded structure are determined by the F-type vertices with odd radial indices, i.e. F_i,j, i = 1, … , m − 1, j = odd and the C-type vertices with even radial indices, i.e. C_i,j, i = 1, … ,m, j = even, respectively. Given two non-intersecting doubly curved surfaces with rotational symmetry about the z-axis whose cross-section equations in the r–z plane are given by z = f_out(r) and z = f_in(r), respectively, in order for the folded structure to fit into the interspace between these surfaces, the following constraints need to be satisfied for arbitrary i:

$${\tilde{z}}_{i,2j-1}^f=f_{\mathrm{out}}(r_{i,2j-1}^f),j=1,\dots ,\left[\frac{n+1}{2}\right]$$ 2.24

and

$${\tilde{z}}_{i,2j}^c=f_{\mathrm{in}}(r_{i,2j}^c),\hspace{0.17em}j=1,\dots ,\left[\frac{n}{2}\right],$$ 2.25

where [ζ] means taking the integer part of a real number ζ, and ${\tilde{z}}_{i,j}^c$ and ${\tilde{z}}_{i,j}^f$ are the z-coordinates of the C-type and F-type vertices of the folded structure with an offset distance h₀ along the z-axis, respectively, i.e.

$${\tilde{z}}_{i,j}^c={\tilde{z}}_{i,j}^f=z_{i,j}^c+h_0,i=1,\dots ,m,\hspace{0.17em}j=1,\dots ,n.$$ 2.26

Figure 3.

The cross-sectional view in the r–z plane of the folded state in figure 2b, in which all the vertices and creases are projected onto the same r–z plane. (Online version in colour.)

Furthermore, given that the total height of the folded structure in the z-direction is H_z and the circumferential span is θ_s ($\in [0,2\pi ]$), the following constraints hold for arbitrary i:

$$H_z=\{\begin{array}{ll}{\tilde{z}}_{i,1}^c-{\tilde{z}}_{i,n}^c,& \mathrm{if}\hspace{0.17em}n=\mathrm{even}\\ {\tilde{z}}_{i,1}^c-{\tilde{z}}_{i,n-1}^c,& \mathrm{if}\hspace{0.17em}n=\mathrm{odd},\end{array}$$ 2.27

and

$${\theta }_s={\theta }_{m,j}^c-{\theta }_{1,j}^c=2(m-1)\alpha .$$ 2.28

To design a folded structure, α can be determined independently from equation (2.28) where it is assumed that m and n are design inputs. Then, equations (2.24), (2.25) and (2.27) along with equations (2.9)–(2.14) and (2.16) can be solved together for parameters φ_c, φ_f, a_ci, b₁, θ_ca, θ_fa, θ_b, η_ca, η_cb, η_fa, η_fb and h₀, where i = 1, … , n − 1. Note that the total number of equations and unknowns are counted as n + 8 and n + 10, respectively. Hence, two out of the n + 10 parameters need to be chosen as the control parameters so that the rest can be uniquely determined by the equations. In other words, the design d.f. is two. Once these parameters are solved, the rest of the geometric parameters of the plane crease pattern can be determined using equations (2.6)–(2.8) and the resulting folded structure determined by equations (2.17)–(2.23).

(d). Examples

To demonstrate the inverse design process discussed above, two examples are provided below. In the first example, the outer and inner cross-sectional profiles are given by

$$z=f_{\mathrm{out}}(r)=0.0015r^2+450$$ 2.29

and

$$z=f_{\mathrm{in}}(r)=0.0021r^2+440,$$ 2.30

respectively. The design inputs m, n, H_z and θ_s are taken as 16, 21, 100 mm and 3π/2 rad, respectively. h₀ and φ_c are chosen as the control parameters whose values are taken as 445 mm and π/3 rad, respectively. The calculated parameters are listed in table 1. The resulting folded geometry and the corresponding two-dimensional crease pattern are shown in figure 4. An animation of the folding motion of this example is provided in the electronic supplementary material, movie S1.

Table 1.

The calculated parameters of the first example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α	π/20 rad	ac1	16.143	ac11	20.160
φf	0.895 rad	ac2	10.175	ac12	5.361
θca	1.746 rad	ac3	17.031	ac13	20.733
θfa	1.523 rad	ac4	9.208	ac14	4.418
θb	1.076 rad	ac5	17.889	ac15	21.128
ηca	1.464 rad	ac6	8.241	ac16	3.493
ηcb	1.451 rad	ac7	18.708	ac17	21.227
ηfa	1.673 rad	ac8	7.276	ac18	2.597
ηfb	1.137 rad	ac9	19.473	ac19	20.789
b1	13.665 mm	ac10	6.315	ac20	1.747

Figure 4.

(a) The resulting target folded structure of the first example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern and (d) a paper model of the designed folded structure. (Online version in colour.)

The outer and inner cross-sectional profiles of the second example are determined as

$$z=f_{\mathrm{out}}(r)=n_1-\sqrt{n_1^2-\frac{n_1^2}{m_1^2}{(r-m_1+40)}^2}$$ 2.31

and

$$z=f_{\mathrm{in}}(r)=n_1-\sqrt{n_2^2-\frac{n_2^2}{m_2^2}{(r-m_1+40)}^2},$$ 2.32

respectively, where m₁, n₁, m₂ and n₂ are 580, 300, 575, 278, respectively. The design inputs m, n, H_z and θ_s are the same as the first example. Again, h₀ and φ_c are taken as the control parameters and their values are taken as 140 mm and π/3 rad, respectively. The calculated parameters and resulting folded structure are shown in table 2 and figure 5, respectively. An animation of the folding motion is provided in the electronic supplementary material, movie S2. It is noted that in both examples the designed folded structures strictly fit between the given outer and inner profiles (figures 4b and 5b), indicating the validity of the proposed design theory (figure 6).

Table 2.

The calculated parameters of the second example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α	π/20 rad	ac1	49.445	ac11	17.827
φf	0.901 rad	ac2	5.833	ac12	7.440
θca	2.056 rad	ac3	27.089	ac13	17.602
θfa	1.856 rad	ac4	5.929	ac14	8.078
θb	1.385 rad	ac5	22.030	ac15	17.677
ηca	1.682 rad	ac6	6.159	ac16	8.847
ηcb	1.671 rad	ac7	19.685	ac17	18.004
ηfa	1.846 rad	ac8	6.490	ac18	9.778
ηfb	1.357 rad	ac9	18.452	ac19	18.566
b1	14.719 mm	ac10	6.915	ac20	10.910

Figure 5.

(a) The resulting target folded structure of the second example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern and (d) a paper model of the designed folded structure. (Online version in colour.)

Figure 6.

Explanation of the developable and flat-foldable condition of the ring design. (Online version in colour.)

3. Discussion

(a). Developable and flat-foldable ring design

For some practical applications, it is desirable to design a folded structure that can not only fit between the given surfaces but at the same time form a closed ring by itself. In other words, it is desired that the opposite edges in the θ-direction of the folded structure meet each other, which can be achieved by making θ_s in equation (2.28) equal 2π, i.e.

$$(m-1)\alpha =\pi .$$ 3.1

If it is further required that the designed folded structure satisfying equation (3.1) is both developable and flat-foldable for the ease of assembly, storage or transportation, the following conditions must be satisfied:

$${\phi }_c>\frac{\pi }{4},\hspace{0.17em}{\phi }_c+{\phi }_f>\frac{\pi }{2}\hspace{0.17em}$$ 3.2

and

$$\cos {\theta }_b=\frac{\sin 2{\phi }_f(1+{\cos }^2{\phi }_c)-\sin 2{\phi }_c(1+{\cos }^2{\phi }_f)}{\sin 2{\phi }_f{\sin }^2{\phi }_c-\sin 2{\phi }_c{\sin }^2{\phi }_f}.$$ 3.3

The detailed derivations of these conditions are provided in the electronic supplementary material, section B. A physical illustration of how these conditions work is provided in figure 6, where the solid, dashed and dotted lines correspond to three cases in which φ_c = 70° and φ_f = 60°, φ_c = 50° and φ_f = 40°, and φ_c = 30° and φ_f = 20°, respectively. It is shown that there exists a maximum value for α, denoted by α_max, only when inequality (3.2) is strictly satisfied, i.e. the solid line. Under this circumstance, when α of the designed folded structure locates either left or right to the peak, the folded structure is either flat-foldable but not developable or the other way round due to the penetration of the opposite edges in the θ-direction during folding or unfolding which is physically prohibited. Only when α of the designed folded structure is equal to α_max can it be both developable and flat-foldable. Equation (3.3) ensures that θ_b of the designed structure corresponds to the peak position of α on the curve. If inequality (3.2) is not satisfied (e.g. the dashed and dotted lines), α increases monotonically with θ_b, indicating that the designed folded structure is always flat-foldable but not developable.

The algorithm to design a developable and flat-foldable self-closed folded ring is as follows. First, α is determined through equation (3.1). Then, equations (2.9)–(2.14), (2.16), (2.24), (2.25), (2.27) and (3.3) are solved together to obtain parameters φ_c, φ_f, a_ci, b₁, θ_ca, θ_fa, θ_b, η_ca, η_cb, η_fa, η_fb and h₀, where i = 1, … , n − 1. Note that the total number of equations and unknowns are n + 9 and n + 10, respectively. Hence, the design d.f. is now reduced to one. Owing to inequality (3.2), it is convenient to choose φ_c as the control parameter which is set to be larger than π/4 rad. Once the other parameters are solved, one ought to check back if inequality (3.2) is satisfied. Alternatively, one may specify both φ_c and h₀ and solve the remaining parameters without equation (3.3) in the first step. Next, keep h₀ unchanged as the sole control parameter and solve the rest of the parameters including φ_c with equation (3.3) included. In this second step, the specified value for φ_c and the solutions of the other parameters obtained in the first step can be used as the initial guesses for solutions of the nonlinear equation system. Usually, as long as m is large enough and φ_c is set to be well above π/4 rad, inequality (3.2) can always be satisfied.

A design example is given here in which m = n = 21, H_z = 100 mm, θ_s = 2π rad, and the outer and inner cross-sectional profiles of the target surfaces are

$$z=f_{\mathrm{out}}(r)=450\sqrt{1-\frac{r^2}{{580}^2}}$$ 3.4

and

$$z=f_{\mathrm{in}}(r)=440\sqrt{1-\frac{r^2}{{480}^2}},$$ 3.5

respectively. First, we use h₀ and φ_c as the control parameters whose values are taken as 445 mm and π/3 rad, respectively, and the rest of the parameters are solved using the inverse design algorithm discussed in §2c. Then, h₀ is taken as the sole control parameter, and the remaining parameters including φ_c are solved with equation (3.3). The obtained parameters are listed in table 3. The graphic results are shown in figure 7. An animation of the folding motion of the structure is provided in the electronic supplementary material, movie S3. Note that the designed folded structure forms a complete ring (figure 7a) and strictly fits between the inner and outer profiles (figure 7b). Moreover, the total circumferential span of the folded structure is always smaller than 2π during the entire folding range (figure 7d), indicating that the designed folded structure is both developable and flat-foldable.

Table 3.

The calculated parameters of the first developable and flat-foldable ring design example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α	π/20 rad	ac1	15.615	ac11	21.695
φf	0.997 rad	ac2	11.924	ac12	11.435
θca	1.997 rad	ac3	16.403	ac13	24.095
θfa	1.768 rad	ac4	11.761	ac14	11.450
θb	1.170 rad	ac5	17.343	ac15	27.495
ηca	1.433 rad	ac6	11.629	ac16	11.521
ηcb	1.728 rad	ac7	18.482	ac17	32.838
ηfa	1.591 rad	ac8	11.530	ac18	11.675
ηfb	1.414 rad	ac9	19.893	ac19	43.161
b1	17.738 mm	ac10	11.464	ac20	11.982

Figure 7.

(a) The resulting target folded structure of the first developable and flat-foldable ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern; (d) the θ_s versus θ_b curve where the blue circle indicates the designed state and (e) a paper model of the designed folded structure. (Online version in colour.)

Consider another example in which the prescribed outer and inner surfaces have negative and positive Gaussian curvatures, respectively, whose cross-sectional profiles are determined as

$$z=f_{\mathrm{out}}(r)=300-\sqrt{{300}^2-\frac{{300}^2}{{580}^2}{(r-540)}^2}$$ 3.6

and

$$z=f_{\mathrm{in}}(r)=-0.0038{(r+80)}^2+190,$$ 3.7

respectively. The design inputs m, n, H_z and θ_s are the same as the previous example, and the control parameter h₀ is taken as 160 mm. The obtained parameters are summarized in table 4, the graphic results are illustrated in figure 8, and an animation of the folding motion of the structure is provided in the electronic supplementary material, movie S4. Again, the designed ring structure successfully satisfies both the prescribed geometries (figure 8b) and the developable and flat-foldable requirement (figure 8d).

Table 4.

The calculated parameters of the second developable and flat-foldable ring design example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α	π/20 rad	ac1	15.215	ac11	11.286
φf	1.254 rad	ac2	6.164	ac12	5.687
θca	1.795 rad	ac3	12.009	ac13	13.410
θfa	1.505 rad	ac4	5.193	ac14	6.998
θb	0.567 rad	ac5	10.521	ac15	17.420
ηca	0.732 rad	ac6	4.768	ac16	9.456
ηcb	1.728 rad	ac7	10.011	ac17	25.392
ηfa	0.844 rad	ac8	4.714	ac18	14.464
ηfb	1.414 rad	ac9	10.254	ac19	44.019
b1	5.560 mm	ac10	4.997	ac20	26.790

Figure 8.

(a) The resulting target folded structure of the second developable and flat-foldable ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern; (d) the θ_s versus θ_b curve where the blue circle indicates the designed state and (e) a paper model of the designed folded structure. (Online version in colour.)

(b). Stacked design

The design algorithm discussed above can be generalized to design stacked folded structures consisting of multiple folded layers. For simplicity, only the two-layer configuration is discussed here, as shown in figure 9. In the sequel, we refer to the bottom and top layers as layer 1 and 2, respectively. In each individual layer, equations (2.6)–(2.23) still hold except that equation (2.20) in layer 2 is modified as

$$z_{i,j}^c=\sum _{k=1}^{j-1}{(-1)}^{k+1}{h}_k,i=1,\dots ,m,\hspace{0.17em}j=2,\dots ,n,$$ 3.8

due to the reversed mountain-valley assignment for the creases in layer 2. It is noted that the outer and inner profiles of the folded structure are determined by the F-type vertices with even radial indices in the top layer and the C-type vertices with even radial indices in the bottom layer, respectively. Therefore, given two non-intersecting surfaces with rotational symmetry about the z-axis whose cross-section equations in the r–z plane are given by z = f_out(r) and z = f_in(r), respectively, the following constraints need to be satisfied for arbitrary i:

$${\tilde{z}}_{i,2j}^{f,2}=f_{\mathrm{out}}(r_{i,2j}^{\hspace{0.17em}f,2}),j=1,\dots ,\left[\frac{n}{2}\right]$$ 3.9

and

$${\tilde{z}}_{i,2j}^{c,1}=f_{\mathrm{in}}(r_{i,2j}^{c,1}),j=1,\dots ,\left[\frac{n}{2}\right].$$ 3.10

where the superscripts 1 and 2 denote the layer numbers, and

$${\tilde{z}}_{i,j}^{c,1}={\tilde{z}}_{i,j}^{f,1}=z_{i,j}^{c,1}+h_0^1,i=1,\dots ,m,\hspace{0.17em}j=1,\dots ,n$$ 3.11

and

$${\tilde{z}}_{i,j}^{c,2}={\tilde{z}}_{i,j}^{f,2}=z_{i,j}^{c,2}+h_0^2,i=1,\dots ,m,\hspace{0.17em}j=1,\dots ,n.$$ 3.12

Figure 9.

The cross-sectional profile of a two-layer configuration viewed in the r–z plane. (Online version in colour.)

On the interface between the two layers, the C- and F-type vertices with odd radial indices in layer 2 should coincide with those in layer 1, respectively. Under this requirement, the following relationships can be obtained according to figure 2b:

$$\begin{array}{rl}{\eta }_{cb}^1& ={\eta }_{cb}^2,\end{array}$$ 3.13

$$\begin{array}{rl}{\eta }_{fb}^1& ={\eta }_{fb}^2\end{array}$$ 3.14

$$\begin{array}{rl}\text{and}{\alpha }^1& ={\alpha }^2.\end{array}$$ 3.15

Owing to equation (2.16), two of the three equations are independent. Equation (3.15) ensures that the position constraints in the θ-direction for both C- and F-type vertices on the interface are satisfied. For the C-type vertices, the position constraints on the r and z-directions are given by

$$r_{i,2j-1}^{c,1}=r_{i,2j-1}^{c,2},j=1,\dots ,\left[\frac{n+1}{2}\right]$$ 3.16

$${\tilde{z}}_{i,2j-1}^{c,1}={\tilde{z}}_{i,2j-1}^{c,2},j=1,\dots ,\left[\frac{n+1}{2}\right].$$ 3.17

Substituting equations (2.17), (3.14) and (3.15) into equation (3.16) yields

$$b_j^1=b_j^2,j=\mathrm{odd}.$$ 3.18

Considering equations (2.21), (3.13), (3.15) and (3.18) together and due to equation (2.23), the constraints in the r and z-directions for the F-type vertices on the interface are automatically satisfied. Furthermore, given the total height of the stacked folded structure in the z-direction H_z and the circumferential span θ_s ($\in [0,2\pi ]$), the following constraints hold for arbitrary i:

$$H_z=\{\begin{array}{ll}{\tilde{z}}_{i,2}^{c,2}-{\tilde{z}}_{i,n}^{c,1},& \mathrm{if}\hspace{0.17em}n=\mathrm{even}\\ {\tilde{z}}_{i,1}^{c,2}-{\tilde{z}}_{i,n-1}^{c,1},& \mathrm{if}\hspace{0.17em}n=\mathrm{odd}\end{array}$$ 3.19

and

$${\theta }_s=2(m-1){\alpha }^1=2(m-1){\alpha }^2.$$ 3.20

To design a stacked folded structure, α¹ and α² can be determined directly from equation (3.20). Then, equations (2.9)–(2.14) and (2.16) for both layers along with constraint equations (3.9), (3.10) and (3.13), (3.16), (3.17) and (3.19) can be solved together for parameters ${\phi }_c^k$, ${\phi }_f^k$, $a_{ci}^k$, $b_1^k$, ${\theta }_{ca}^k$, ${\theta }_{fa}^k$, ${\theta }_b^k$, ${\eta }_{ca}^k$, ${\eta }_{cb}^k$, ${\eta }_{fa}^k$, ${\eta }_{fb}^k$ and $h_0^k$, where k = 1,2 and i = 1, … , n − 1. Note that the total numbers of equations and unknowns in the equation system are 2(n + 8) and 2(n + 10), respectively. Therefore, the design d.f. for the two-layer configuration is four.

If a stacked folded ring structure is to be designed, one needs simply to set θ_s in equation (3.20) to 2π. If it is further required that each layer in the structure is both flat-foldable and developable, conditions given in inequality (3.2) and equation (3.3) should be satisfied by each layer. Adding equation (3.3) for each layer into the overall equation system of the stacked structure reduces the total design d.f. from four to two. The selection of control parameters is similar to the single-layered ring design discussed in §3a.

For certain applications, it is desirable that both layers of the designed stacked folded structure can fold or unfold together in a compatible manner. To achieve this, the following condition needs to be satisfied:

$${\phi }_c^1={\phi }_c^2.$$ 3.21

The detailed derivation of the above equation is provided in the electronic supplementary material, section C. In this context, when a developable and flat-foldable stacked ring structure is to be designed, due to the compatible folding condition enforced by equation (3.21), the entire structure is developable and flat-foldable as long as inequality (3.2) and equation (3.3) are satisfied by any one of the constituent layers. Therefore, the total design d.f. remains two. As a rule of thumb, $h_0^1$ and ${\phi }_c^1$ are the robust choices for the control parameters.

We finalize this section with a design example for a developable, flat-foldable and compatibly foldable stacked ring structure, where m = 19, n = 21, H_z = 117 mm, θ_s = 2π rad, and the outer and inner cross-sectional profiles of the target surfaces are given by

$$z=f_{\mathrm{out}}(r)=480\sqrt{1-\frac{r^2}{{650}^2}}$$ 3.22

and

$$z=f_{\mathrm{in}}(r)=460\sqrt{1-\frac{r^2}{{550}^2}},$$ 3.23

respectively. The control parameters are $h_0^1$ and ${\phi }_c^1$ and they are taken as 470 mm and π/3 rad, respectively. The calculated parameters are listed in tables 5 and 6. The graphic results are plotted in figure 10. Animations of the folding motion of the three-dimensional structure and the cross-sectional profile change are provided in the electronic supplementary material, movies S5 and movie S6, respectively. The results suggest that the designed stacked ring structure strictly fits between the inner and outer profiles (figure 10b) and is developable and flat-foldable (figure 10e) and the two layers do not separate throughout the folding motion (electronic supplementary material, movie S6).

Table 5.

The calculated parameters of layer 1 of the stacked ring design example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α1	π/18 rad	$a_{c1}^1$	23.519	$a_{c11}^1$	17.790
${\phi }_f^1$	0.884 rad	$a_{c2}^1$	9.870	$a_{c12}^1$	12.922
${\theta }_{ca}^1$	2.171 rad	$a_{c3}^1$	13.869	$a_{c13}^1$	34.827
${\theta }_{fa}^1$	1.963 rad	$a_{c4}^1$	16.682	$a_{c14}^1$	6.678
${\theta }_b^1$	1.518 rad	$a_{c5}^1$	26.082	$a_{c15}^1$	22.041
${\eta }_{ca}^1$	1.783 rad	$a_{c6}^1$	8.843	$a_{c16}^1$	10.810
${\eta }_{cb}^1$	1.745 rad	$a_{c7}^1$	15.488	$a_{c17}^1$	48.055
${\eta }_{fa}^1$	1.950 rad	$a_{c8}^1$	14.839	$a_{c18}^1$	5.385
${\eta }_{fb}^1$	1.396 rad	$a_{c9}^1$	29.460	$a_{c19}^1$	41.048
$b_1^1$	16.422 mm	$a_{c10}^1$	7.795	$a_{c20}^1$	8.089

Table 6.

The calculated parameters of layer 2 of the stacked ring design example.

parameter	value	parameter	value (mm)	parameter	value (mm)
α2	π/18 rad	$a_{c1}^2$	9.870	$a_{c11}^2$	12.922
$h_0^2$	470 mm	$a_{c2}^2$	23.519	$a_{c12}^2$	17.790
${\phi }_c^2$	π/3 rad	$a_{c3}^2$	16.682	$a_{c13}^2$	6.678
${\phi }_f^2$	0.884 rad	$a_{c4}^2$	13.869	$a_{c14}^2$	34.827
${\theta }_{ca}^2$	2.171 rad	$a_{c5}^2$	8.843	$a_{c15}^2$	10.810
${\theta }_{fa}^2$	1.963 rad	$a_{c6}^2$	26.082	$a_{c16}^2$	22.041
${\theta }_b^2$	1.518 rad	$a_{c7}^2$	14.839	$a_{c17}^2$	5.385
${\eta }_{ca}^2$	1.783 rad	$a_{c8}^2$	15.488	$a_{c18}^2$	48.055
${\eta }_{cb}^2$	1.745 rad	$a_{c9}^2$	7.795	$a_{c19}^2$	8.089
${\eta }_{fa}^2$	1.950 rad	$a_{c10}^2$	29.460	$a_{c20}^2$	41.048
${\eta }_{fb}^2$	1.396 rad
$b_1^2$	16.422 mm

Figure 10.

(a) The resulting target folded structure of the stacked ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles and the black and blue lines indicate layers 1 and 2, respectively; (c) the corresponding two-dimensional crease pattern of layer 1; (d) the corresponding two-dimensional crease pattern of layer 2; (e) the θ_s versus θ_b curve where the blue circle indicates the designed state. (Online version in colour.)

4. Summary and final remarks

In this paper, a mathematical framework for the design of rigid-foldable doubly curved origami tessellations that can fit between two doubly curved target surfaces with rotational symmetry about a common axis has been established. Under the framework, an algorithm to simulate the folding motion of the designed origami structure is provided, and specific conditions for the design of doubly curved folded ring structures that are developable and flat-foldable and stacked folded structures whose constituent layers can fold independently or in a compatible manner are identified. The validity and versatility of the proposed framework were demonstrated by several design examples and paper models. This study paves the way towards various potential engineering applications. For example, the developable and flat-foldable ring design with 1-d.f. rigid folding motion are particularly useful for designing retractable domes or portable shelters requiring minimum driving systems. When manufactured with composite materials, such as carbon fibre-reinforced polymer and Kevlar prepregs, the doubly curved folded structures can help guide the development of lightweight folded cores for doubly curved sandwich structures on the aircraft, such as fairing and fuselage. Moreover, the stacked folded structures with compatibly foldable layers may lead to new doubly curved tunable metamaterial designs with intriguing properties. The main limitation of the current work is that the designed folded structures can only fit between two doubly curved surfaces with rotational symmetry about a common axis. Chopping off the parallel condition for the creases and boundary lines extending between any two adjacent χ lines in figure 2a and introducing proper diagonal creases to the general quadrilateral facets is a potential direction to overcome this limitation, which will be considered in our future work.

Data accessibility

The electronic supplementary material supporting this article are available via https://dx.doi.org/10.6084/m9.figshare.c.3729916.

Authors' contributions

K.S. derived some of the equations, wrote computer programs for the examples and made the physical models; X.Z. designed the study, derived some of the equations and wrote the paper; S.Z. derived some of the equations; H.W. coordinated the study and commented on the paper; Z.Y. conceived of the study and commented on the paper. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

X.Z. is funded by National Science Foundation of China (grant no. 51408357).