One-DOF Superimposed Rigid Origami with Multiple States

Origami-inspired engineering design is increasingly used in the development of self-folding structures. The majority of existing self-folding 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, rigid-foldable crease patterns are superimposed in the same sheet such that kinematic independence and 1-DOF mobility of each individual pattern is preserved. This is enabled by the cross-crease 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 1-DOF rigid-foldable states. This enables many new origami-inspired engineering design possibilities, with two explored in depth: the compact folding of non-flat-foldable structures and sequent folding origami that can transform between multiple states without unfolding.

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 eight-crease 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 six-crease vertex V 2 , with a DOF of 3. The combined and intersected vertices thus disrupt the 1-DOF rigid-foldability of the initial patterns and thus the global rigid-foldability 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 cross-crease 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 z-axis, 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 right-hand 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 , (b) Patterns S 1 and S 2 are superimposed in the same sheet to form a pattern S 1,2 , in which V 1 is the eight-crease combined vertex, V 2 is the six-crease intersected vertex, and V 3 is the four-crease cross-crease vertex. (c) Unit patterns of S 1 and S 2. (d) Superimposed units with dimensional shift. (e) Overhanging regions of S 2 translated to within S 1 . (f) 2 by 2 small S 2 units superimposed within a single S 1 unit. (g) A superimposed pattern S 1,2 with only additional crosscrease vertices and its prototype with two independent 1-DOF rigid-foldable states, in which superscripts U, M, D and P represents the fully unfolded, intermediate folding, deployed and fully packed states, respectively. 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 closed-loop 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 Under this framework, the rotation of the crease lines can be obtained. Yet, here our attention is on the cross-crease vertex, which can be considered kinematically as a four-crease 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 2 4 1 3 which indicates that the rigid origami pattern in the cross-crease vertex is in general 1-DOF, 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 cross-crease vertices only, that is it has no intersected or combined vertices. The actuation of S 1 will constrain all cross-crease 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 cross-crease vertices are added in the combined pattern, kinematic independence between states and 1-DOF mobility are preserved. To ensure all new intersections in superimposed origami patterns are cross-crease vertices, we can consider the superimposition method of the pattern units. For the arc-Miura pattern, S 1 in Fig. 1a 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 non-flat-foldable structures. There are a very large number of 1-DOF rigid-foldable patterns that can be combined within a single sheet using the methods described above. Rather than simply combining known 1-DOF 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 1-DOF distributed frame accordion shelter 25 is shown in Fig. 3a, with nearly flat-folded 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 1-DOF Miura-ori shown in Fig. 3b, has an in-plane folding motion that can never self-intersect during folding and reaches a compact flat-folded 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 1-DOF rigid-foldability, that is only cross-crease 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 zero-thickness panels, with a comprehensive kinematic synthesis for rigid origami of thick panel presented in ref. 26. With a change from zero to non-zero 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 thick-panel multi-state 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 zero-thickness case, cross-crease vertices in the thick-panel model are able to maintain 1-DOF 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 Miura-ori embedded patterns can be obtained by following the superimposition process outlined previously. However, a modified process enables between-state 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 Miura-ori pattern boundary, as shown in Fig. 4d. A Miura-ori pattern with sector angle ϕ, shown in red, can thus be superimposed onto the folded square twist configuration and subsequently permit a second-stage of folding into the Miura-ori pattern. The between-state 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   due to the projection of the Miura-ori 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 Miura-ori 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 Miura-ori, as shown in Fig. 4e and SI Video 5. The Miura-ori 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 Miura-ori 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 Miura-ori 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 2 cos 12 (8) m i and the areal folding ratio of the indirect case 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 square-twist 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 And the total areal folding ratio of the second folding sequence is 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 γ = . 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 1-DOF rigid-foldable origami patterns within one single sheet in such a way that only cross-crease vertices are generated. The particular kinematic properties of cross-crease vertices mean that the folding of one motion path suppresses the other, and vice versa. The Scientific RepoRts | 6:36883 | DOI: 10.1038/srep36883 superimposed pattern with cross-crease vertices thus has preserved 1-DOF 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 self-folding and origami-inspired 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 non-flat foldable shelter embedded with a compact packaged state. States with radically different functionality can also be combined to create multi-tool sheets capable of performing a broad range of functions. Finally, as the superposition method is purely enabled by the special cross-crease vertex, the principles of thick-panel and 1-DOF rigid origami design remain applicable and so the method can be applied across a range of material types and application domains.