Abstract
Origamiinspired engineering design is increasingly used in the development of selffolding structures. The majority of existing selffolding structures either use a bespoke crease pattern to form a single structure, or a universal crease pattern capable of forming numerous structures with multiple folding steps. This paper presents a new approach whereby multiple distinct, rigidfoldable crease patterns are superimposed in the same sheet such that kinematic independence and 1DOF mobility of each individual pattern is preserved. This is enabled by the crosscrease vertex, a special configuration consisting of two pairs of collinear crease lines, which is proven here by means of a kinematic analysis to contain two independent 1DOF rigidfoldable states. This enables many new origamiinspired engineering design possibilities, with two explored in depth: the compact folding of nonflatfoldable structures and sequent folding origami that can transform between multiple states without unfolding.
Introduction
Origamiinspired engineering is to apply origami science and technology to the design of engineering structures or devices with remarkable performance characteristics. Applications have been developed across most engineering disciplines, with examples including space structures^{1,2}, deployable shelters^{3}, sandwich panels^{4,5,6}, metamaterials^{7,8}, medical implants^{9,10}, and automobile components^{11,12}. Many of these applications utilize a rigidfoldable origami pattern, which can fold without twisting or stretching of component panels and thus can be folded from rigid engineering sheet materials^{13}. A substantial body of research into understanding the mathematics and kinematics of rigidfoldable origami patterns has enabled the above engineering explorations^{14,15}. These have shown that conditions for rigidfoldability and the degrees of freedom (DOF) are determined by the geometry of the origami crease pattern, a network of folding lines that are placed in a sheet and intersect at vertices^{16}. Properties such as, the flatfoldability and large areal reduction ratio are also of great importance in practice^{17}. Yet, not all the crease patterns offer a compactly flatfolded configuration and even when such a configuration is achieved, the areal reduction ratio may be limited by the geometric parameters of the pattern design.
Most existing crease patterns are for a solo pattern, for example the Miuraori (rigid with 1DOF^{18}) or the waterbomb tessellation (rigid with multiDOF^{19}). For these, the folding process and the folded shape are clearly predefined by the crease patterns. There are, nevertheless, some exceptions. For example, two sets of patterns have been integrated on printed circuit boards in a Latincross shape without intersection such that it can be folded into a pentahedron or an octahedron, which is controlled by the actuation at each folding line through software and hardware^{20,21}. A ‘universal’ crease pattern has also been investigated that can be dynamically programmed into different shapes by folding a subset of available creases, which is a multiDOF process with multiple folding steps. The order of the folded states has to be followed strictly in order to obtain the desired configurations in which a programmable origami is applied^{22}.
In this paper, we propose a fundamentally new method for generation of multiple 1DOF rigidfoldable configurations from predefined crease patterns. The method leverages the existing knowledge surrounding rigidfoldability to enable crease patterns of different 1DOF rigidfoldable states to be embedded within a single sheet, such that kinematicindependence is preserved. Sheets can thus be created that contain multiple states with different or complementary functionality and substantial potential for application in origamiinspired engineering design.
Results
Superimposed rigidfoldable patterns
Shown in Fig. 1a are two origami patterns. The red pattern, S_{1}, forms a singlecurved surface and the black one, S_{2}, forms a planar surface, both with 1DOF rigidfoldability. S_{1} and S_{2} crease patterns are superimposed on top of each other within the same sheet to form the crease pattern of S_{1,2}, see Fig. 1b. In order to make the folding movement of patterns S_{1} and S_{2} independent, the superimposition cannot be carried out arbitrarily. Many vertices are unchanged from the single pattern to the superimposed one. However, when S_{1} and S_{2} vertices are coincident, a combined vertex is formed such as the eightcrease vertex V_{1}. According to the principles of rigid origami kinematics^{23}, this vertex has a DOF of 5. Similarly, when an S_{1} vertex is intersected by an S_{2} crease line, or vice versa, an intersected vertex is formed such as the sixcrease vertex V_{2}, with a DOF of 3. The combined and intersected vertices thus disrupt the 1DOF rigidfoldability of the initial patterns and thus the global rigidfoldability in the combined pattern S_{1,2} cannot be easily determined or controlled. The third type of new vertex V_{3} appears in S_{1,2} when two straight crease lines, one each from S_{1} and S_{2}, intersect. They are here termed crosscrease vertices. The kinematic effect of this vertex type can be determined as follows.
In rigid origami, facets and crease lines are equivalent to rigid panels and revolute joints, respectively. As all crease lines meet at vertices, a rigid origami pattern with a single vertex is kinematically equivalent to a spherical linkage and a pattern with multiple vertices is equivalent to the assembly or network of a number of spherical linkages.
Therefore, the kinematic properties of rigid origami can be obtained by analyzing the corresponding spherical linkages and networks with standard kinematic theory. Here, the Denavit and Hartenberg (DH) matrix method^{24} is applied, see Fig. 2a. The axes of four revolute joints (or crease lines) are z_{i}. The DH coordinates are then setup on each joint i along zaxis, where axis x_{i} is commonly normal to z_{i} and z_{i−1}, and axis y_{i} is normal to x_{i} and z_{i} following the righthand rule. Thus the kinematic geometric parameters are defined as a_{(i−1)i}, the distance between axes z_{i−1} and z_{i}, positive along x_{i} (a_{(i−1)i} = 0 for spherical linkages), and α_{(i−1)i}, the angle between axes z_{i−1} and z_{i}, positive along x_{i}. The kinematic variable θ_{i} is defined as the rotation between two panels joined by the crease or revolute joint z_{i}. For a closedloop spherical linkage, shown in Fig. 2b, the necessary and sufficient mobility condition is obtained when the product of the transformation matrices equals the unit matrix, that is
in which, Q_{i(i+1)} is the transformation matrix from the ith coordinate system on joint i to the (i+1)th coordinate system on joint (i+1), i.e.,
Under this framework, the rotation of the crease lines can be obtained. Yet, here our attention is on the crosscrease vertex, which can be considered kinematically as a fourcrease vertex in which alternate pairs of crease lines are collinear and of the same polarity. Its corresponding linkage form is shown in Fig. 2c. The kinematic geometric parameters of this linkage are
where both crease lines 1 and 3 are both either mountain folds (0 < θ < π) or valley folds (−π < θ < 0), as are crease lines 2 and 4. Substituting (3) into the closure condition (1), we can get the kinematic closure equation of the linkage in Fig. 2c as
or
which indicates that the rigid origami pattern in the crosscrease vertex is in general 1DOF, but in the unfolded configuration θ_{1} = θ_{2} = θ_{3} = θ_{4} = 0, a kinematic bifurcation exists with two possible moving paths, see Fig. 2d,e. However, this bifurcation behavior will not disturb the folding movement of the origami pattern and indeed can be utilized for practical benefit in superimposed origami patterns. For instance, a superimposed pattern is shown in Fig. 1g which has crosscrease vertices only, that is it has no intersected or combined vertices. The actuation of S_{1} will constrain all crosscrease vertices to one of two paths given in equations (4) while the other path is held inactive. Pattern S_{1,2} is able to independently fold between each state S_{1} or S_{2} as shown in the prototype. We can therefore conclude that if origami patterns are superimposed such that only crosscrease vertices are added in the combined pattern, kinematic independence between states and 1DOF mobility are preserved.
To ensure all new intersections in superimposed origami patterns are crosscrease vertices, we can consider the superimposition method of the pattern units. For the arcMiura pattern, S_{1} in Fig. 1a, there are 2 by 3 units, one of which is shown as red in Fig. 1c with dimensions a_{m} and b_{m}. For the double corrugated pattern, S_{2} in Fig. 1a, there are 2 by 2 units, one of which is shown as black in Fig. 1c with dimensions a_{d} and b_{d}. If units are superimposed as shown in Fig. 1d, it can easily be seen that no combined or intersected vertices are present in the superposed pattern and there are only cross crease vertices added. The overhanging regions of one state can be shifted in both directions to ensure there are no combined or intersected vertices generated when units are tessellated, shown in Fig. 1e. Then, if the dimensions of both units satisfy a_{m} = ka_{d} (or a_{d} = ka_{m}) and b_{m} = lb_{d}, (or b_{d} = lb_{m}) where k and l are positive integers, tessellation of the superimposed units can be carried out to generate a larger origami pattern with only crosscrease vertices present. The examples k = l = 1 in Fig. 1e and k = l = 2 in Fig. 1f can both be tessellated into a larger origami pattern such as that given in Fig. 1g and SI Video 1.
Compact folding of nonflatfoldable structures
There are a very large number of 1DOF rigidfoldable patterns that can be combined within a single sheet using the methods described above. Rather than simply combining known 1DOF states, it is more useful to consider how secondary patterns can be combined with a primary pattern to develop applications with extended functionality. This is discussed in the context of specific examples as follows.
A 1DOF distributed frame accordion shelter^{25} is shown in Fig. 3a, with nearly flatfolded frame elements separated by spacer panels. In a deployed state S_{1}^{D}, the frame elements give the shelter a high structural stiffness but prevent the shelter from reaching a compact packaged state. This is a critical weakness for a deployable structure, which are almost always required to be packaged for transport. A second pattern, the 1DOF Miuraori shown in Fig. 3b, has an inplane folding motion that can never selfintersect during folding and reaches a compact flatfolded state S_{2}^{P} with a cuboid volume boundary.
Pattern superposition will allow the packaged configuration of S_{2} to resolve the packaging limitation of S_{1}. One such combination S_{1,2} is shown in Fig. 3c which has necessary conditions for independent 1DOF rigidfoldability, that is only crosscrease vertices are introduced during pattern superposition. Design flexibility is introduced in the superimposed pattern that warrants further inspection. S_{1} is specified with seven independent control parameters: sector angle φ, side lengths a and b, spacer plate width w_{p}, folded edge angle η_{A}, and number of tessellated units along x and y axes, M and N, respectively. S_{2} is similarly specified but without the spacer plate width parameter and so has six independent control parameters. Assuming S_{1} is the base pattern, design flexibility in specification of S_{2} is mostly preserved, with four of six parameters remaining free after the two tessellation unit length constraints are applied.
A full scale prototype of the crease pattern shown in Fig. 3c and SI Video 2 was constructed from a 2400 mm × 3600 mm × 3 mm extruded corrugated polypropylene sheet material. A 2 mm wide heated roller was used to compress the sheet along crease lines and the folded states were reached with manual folding by three people. As predicted by the kinematic analysis, the bifurcation in the unfolded state did not interrupt the folding motion of either state once the creases of a particular state were actuated. Good correspondence and independent foldability are seen between predicted and prototype folded forms at S_{1}^{D} and S_{2}^{P}.
Transition to structural applications of origami geometry also necessitates the consideration of thick, rather than zerothickness panels, with a comprehensive kinematic synthesis for rigid origami of thick panel presented in ref. 26. With a change from zero to nonzero sheet thickness, the physical polarity of crease lines is determined and so it is much easier to restore origami to a completely unfolded state and thus transition between states. A thickpanel multistate accordion shelter was constructed as shown in Fig. 3d and SI Video 3, with the superimposed crease pattern generated from methods described above. Similar to the zerothickness case, crosscrease vertices in the thickpanel model are able to maintain 1DOF mobility for each embedded pattern and the prototype exhibits a smooth transition between both independent states.
Sequent folding of superimposed patterns
Although previous superimposed patterns can realize transformation between embedded patterns, it can be seen that all shown state transformations were via the unfolded configuration, that is the independent mobility followed the process of ‘fold S_{1} – unfold – fold S_{2}’. However there exist certain special superimposed patterns that can directly transform between states, that is a ‘fold S_{1} – fold S_{2}’ transformation that skips the unfolded configuration.
A superimposed pattern with square twist and Miuraori embedded patterns can be obtained by following the superimposition process outlined previously. However, a modified process enables betweenstate transformation. A square twist pattern with sector angle θ and square length c is shown in Fig. 4a. It has a modified boundary condition with side lengths a and b, and sector angle φ, shown in black, such that when folded, its folded configuration corresponds exactly with a Miuraori pattern boundary, as shown in Fig. 4d. A Miuraori pattern with sector angle φ, shown in red, can thus be superimposed onto the folded square twist configuration and subsequently permit a secondstage of folding into the Miuraori pattern. The betweenstate transformation enables a much more compact final package. The areal folding ratio of the first folding step being the folded state of square twist pattern is
and that of Miuraori pattern is
So the total areal folding ratio of superimposed pattern is
For the superimposed pattern in Fig. 4a, with θ = 45°, c = 27.0 mm, φ = 70°, a = 92.2 mm, b = 82.8 mm, the areal folding ratio of two folding steps is γ_{s} = 0.30 and γ_{m} = 0.16, respectively. And the total areal folding ratio γ_{s+m} is 0.05. Referring back to the superimposed unfolded state, additional Miuraori pattern crease lines are created due to the projection of the Miuraori pattern ‘through’ the folded square twist panels. A prototype of the superimposed pattern and its full folding sequence are shown in Fig. 4d and SI Video 4.
Figure 4b shows a Miuraori pattern, with sector angle φ and side lengths a and b, superimposed with a double corrugated unit, with sector angle β, parallelogram angle α and side lengths c and d. There are two permissible folding sequences, each of which ends with a fully folded Miuraori, as shown in Fig. 4e and SI Video 5. The Miuraori can be folded directly in a single step, or indirectly in two steps, with the first step being the folded state of the double corrugated pattern. For the indirect case, certain Miuraori crease lines reverse their polarity in the second folding step, shown in blue, due to the overlap of crease lines in the double corrugated state. In order to ensure the overlap of crease lines in Miuraori pattern after folding the double corrugated pattern, the crossed crease lines must be perpendicular to each other. The overlapped crease lines reduce the effective side length of the indirect Miura configuration and so the final packaged size is much smaller in the indirect case. The areal folding ratio of the direct case is
and the areal folding ratio of the indirect case is
with
for the first doublecorrugated folding step and
for the second Miuraori folding step. For the superimposed pattern in Fig. 4b, with φ = 70°, a = 102.5 mm, b = 77.9 mm, α = 70°, β = 28°, c = 35.4 mm, d = 33.0 mm, the areal folding ratio of the first folding sequence is and the total areal folding ratio of the second folding sequence is .
Finally, Fig. 4c shows a superimposed pattern with two embedded square twist patterns. Each square twist pattern has identical sector angle θ and square length c, but with different side length, a for black and 2a for red. It again has two permissible folding sequences, each of which has two steps, either folding the red squaretwist pattern first then the black one second, or vice versa, as shown in Fig. 4f and SI Video 6. Unlike the previous example though, both sequences fold into the same final square, despite the folded configuration of the first folding step of two sequences being different. The total areal folding ratio of the first folding sequence is
with
And the total areal folding ratio of the second folding sequence is
with
For the superimposed pattern in Fig. 4c, with θ = 45°, a = 145.3 mm, c = 41.0 mm, the total areal folding ratio of the first folding sequence is , with for the first folding step and for the second. The total areal folding ratio of the second sequence is also , with for the first folding step and for the second. The folding sequence is able to be determined by controlling the order with which the blue crease lines fold, shown circled in Fig. 4f. For example, if the blue crease lines which are collinear with black lines remain collinear during the first folding step, then the black square twist pattern is folded first and the red twist pattern second, and vice versa.
Discussion
The superimposition method presented in the paper combines two 1DOF rigidfoldable origami patterns within one single sheet in such a way that only crosscrease vertices are generated. The particular kinematic properties of crosscrease vertices mean that the folding of one motion path suppresses the other, and vice versa. The superimposed pattern with crosscrease vertices thus has preserved 1DOF mobility and is able to fold independently between each embedded state. Sequent folding between origami patterns is also shown to be possible and enables folding between different states without the requirement of transition through an unfolded state.
The ability to have two or more independent objects coexistent within a single sheet enables many new types of selffolding and origamiinspired engineering design possibilities. States with complementary functionality can be combined to give a device extended performance capabilities, as was seen in the example of a nonflat foldable shelter embedded with a compact packaged state. States with radically different functionality can also be combined to create multitool sheets capable of performing a broad range of functions. Finally, as the superposition method is purely enabled by the special crosscrease vertex, the principles of thickpanel and 1DOF rigid origami design remain applicable and so the method can be applied across a range of material types and application domains.
Additional Information
How to cite this article: Liu, X. et al. OneDOF Superimposed Rigid Origami with Multiple States. Sci. Rep. 6, 36883; doi: 10.1038/srep36883 (2016).
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Acknowledgements
Y.C. would like to thank the financial support from the National Natural Science Foundation of China (Projects No. 51422506, No. 51275334 and No. 51290293) and the Ministry of Science and Technology of China (Project 2014DFA70710). J.M.G. is grateful for the support received from, the UQ Confucius Institute, the Australian Research Council under the Discovery Early Career Researcher funding scheme (DE160100289), and to undergraduate research students who assisted with construction of early prototypes.
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School of Mechanical Engineering, Tianjin University, Tianjin 300072, China
 Xiang Liu
 & Yan Chen
School of Civil Engineering, University of Queensland, St Lucia, QLD 4072, Australia
 Joseph M. Gattas
Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China
 Yan Chen
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Contributions
Y.C. and J.M.G. proposed this research, supervised, and wrote the paper. X.L. carried out research, modelling, and prepared figures and Videos.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Yan Chen.
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