Abstract

Linkage origami is one effective approach for addressing stiffness and accommodating panels of finite size in origami models and tessellations. However, successfully implementing linkage origami in tessellations can be challenging. In this work, multiple theorems are presented that provide criteria for designing origami units or cells that can be assembled into arbitrarily large tessellations. The application of these theorems is demonstrated through examples of tessellations in two and three dimensions.

Introduction

An important problem that arises while converting origami from paper-based models to engineered mechanisms is ensuring that they can be made with thick materials. The ability of thick origami to include panels of finite size is essential for making origami feasible to construct and support different loads during use [1]. One technique for achieving this involves using linkage-based origami [2].

Linkage-based origami can be an effective solution to the problems of including thick panels and ensuring high stiffness because of its unique characteristics. While zero-thickness origami is often described using spherical mechanisms [3], linkage-based origami is distinct because it is based on spatial linkages whose hinge axes do not intersect at a common point. Because adjacent creases in such linkages are not co-planar, incorporating thick panels in a design is significantly simplified [4].

This additional thickness also significantly simplifies the process of stiffening origami [5]. In zero-thickness origami, the advantage of including stiff hinges will be diminished by the presence of thin, flexible panels. However, the ability to include thick panels can provide a stiff connection between the panel and the hinges as well as minimize panel deformation.

However, utilizing linkage-based origami in tessellations is complicated by multiple challenges. In addition to managing competing constraints on fold angles throughout the pattern [6], constraints associated with global folding, self-intersection, and panel thickness must be considered [7]. Manufacturing considerations are also important because many models cannot be constructed from single sheets. Because of the lack of a systematic method to address these challenges in thick origami, few thick tessellations have been constructed [8].

In this work, we address some of these challenges by demonstrating an original technique for designing tessellations of linkage-based, origami vertices. Theorems are developed which identify a consistent method for designing arbitrary thick tessellation units. The method is demonstrated on a wide range of linkage combinations to demonstrate its broad applicability. It is shown that chains of these units have simple kinematic behavior that can avoid self-intersection. Techniques for connecting these chains into tessellations in two and three dimensions are also discussed. Solutions to different problems arising during the connection process are also explored. These techniques are demonstrated on multiple models to showcase their effectiveness.

Background

The capability to design origami-based mechanisms is inspiring useful applications [911], including spacecraft solar panels [12,13], packaging [14], efficient airbag stowage [15], shipping containers [16], deployable aerodynamic fairings [17], and optics [18]. In each case, the ability of these mechanisms to change shape and adapt to different design needs is a significant advantage. This advantage derives from enhanced desirable qualities such as reduced size.

Because of these advantages, multiple authors have attempted to address challenges in the construction of thin and linkage-based origami. Chen et al. [4] investigated the relationship between traditional linkage descriptions and origami and showed that origami can be designed based on these linkages. Additional techniques for creating new, linkage-based vertices were developed [19,20] and demonstrated on example mechanisms.

Origami tessellations are another class of mechanisms that has been investigated [21,22]. Processes of joining vertices and cells together to form tessellations in one, two, and three-dimensions as shown in Fig. 1 have been discussed, but further understanding of their fundamentals is needed for general application [3]. Further understanding of tessellations has been achieved by modeling them as networks of linkages. Examples of linkages used in these networks are Altmann [23] and Myard [24] linkages.

Significant effort has also been devoted for finding new thick tessellation models [8]. In these cases, previously developed thickness accommodation techniques have proven useful in adapting zero-thickness tessellations.

Design Process Summarized

The method used to design the regular, linear tessellations shown in Fig. 1 can be summarized using the steps listed below. This systematic nature of the process will facilitate the design of a wide variety of different tessellations of thick origami. These steps are:

2. Convert to origami. Convert the linkage representation to a thick-origami model as described by Chen et al. [4].

3. Reconfigure (Optional). Alter the model geometry using hinge transposes, thickness shifts, and/or vertex split operations as described by Yellowhorse et al. [19].

4. Construct cell. Create a regular cell using a twin-vertex combination if the vertex is not already regular.

• Identify input and output creases. Select the appropriate creases and correct mountain-valley incompatibilities with an adapter.

• Equal input/output angles. Ensure that the input and output dihedral angles are always equal.

5. Choose number of dimensions. Choose the number of dimensions to tessellate.

6. Connect. Join the chosen cells using any combination of connection methods including rigid joints, hinges, mountain-valley adapters, link offsets, or regular cells.

In steps 1–3, the origami cell is constructed as a thick model using techniques discussed previously. Steps 4–6 are unique to this work and guide the construction of a diverse set of original thick tessellations with simplified kinematics. More details on these steps will be given in the following sections.

Tessellations

Because a significant amount of variation in tessellation design is attributable to variations in cell construction and connections between cells, methods for exploring these variations will be discussed. In this paper, a tessellation cell is the unique, repeating unit that forms the tessellation. This process will allow designers to take advantage of the variety of different folding motions that the resulting tessellations can provide.

To simplify the design process, the large number of constraints in these models are used to develop guidelines that can efficiently direct its progress. These constraints can be divided into categories that allow tessellation design to be decomposed into the following sub-problems:

1. Cell construction/selection

2. Connection in 1D

3. Connection in 2D

4. Connection in 3D

Methods of addressing each of these aspects of tessellation construction can then be developed.

Combined with previously published techniques for generating and transforming linkage-based origami vertices [19,20], a systematic method for designing novel tessellation cells can be generated possessing unique properties. Techniques for connecting cells in tessellations can also be demonstrated given certain assumptions about the regularity of the cell kinematics. Together, the methods discussed in this work will help overcome the significant obstacles associated with developing new, thick cells.

Regular Tessellations.

Generally, tessellations can assume a wide range of geometry and incorporate significant curvature in their design [25]. While this can be advantageous in certain circumstances, ensuring that the cells of a tessellation are evenly spaced and are arranged in straight lines, as in Fig. 1, can also provide significant benefits. These benefits include part uniformity, improved metamaterial uniformity, and simplified kinematics. In this paper, such a tessellation is defined as regular because it does not curve at a global level. One way to create this regularity involves ensuring that connections between neighboring cells are always parallel.

Illustrations of regular tessellations in one, two, and three dimensions are shown in Fig. 1. Figure 1(a) shows an individual cell with its one-dimensional tessellation in Fig. 1(b). Figure 1(c) shows a two-dimensional tessellation created by joining multiple chains laterally. This tessellation represents one of a limited number of two-dimensional, regular tilings [26]. Finally, tessellations in three dimensions can be created by joining two-dimensional tessellations as shown in Fig. 1(d). In all three cases, the same cell is used and the resulting tessellation has no curvature. The tessellation regularity must also be retained over all folded states of the cell to ensure that the cell maintains a degree-of-freedom in motion.

For the tessellation method shown in Fig. 1, design criteria can be described that permit new cells to be designed and connected. Defining these criteria requires multiple definitions of linkage-based vertex behavior.

Definition 1
Let L be a linkage-based vertex if and only if it can be described by n links connected sequentially and joined at the ends to form a loop. Such a vertex will be described by
$I=∏i=0nTi$
(1)
where ∏ specifies a right product and Ti is a homogeneous transformation describing the ith link. Here, Ti is written in the Denavit–Hartenburg notation as
$Ti=(Rd01)$
(2)
where R is a 3 × 3 rotation matrix, d is a 3D translation vector, and 0 is a 3D zero vector.

An example of a linkage-based origami vertex is the Bennett linkage shown in Fig. 2. This vertex is represented by four distinct Ti where each describes the transformation from frame Si to frame Si+1. Although the word vertex is often used when describing zero-thickness origami, it will be used here to describe thick origami because of the close relationship between these mechanisms.

In addition to defining the specific geometry and features of a vertex, it will also be necessary to define a connection process for joining these vertices into cells and tessellations. This connection process consists of joining two creases of similar mountain-valley assignment as shown in Fig. 3(a) where link 1 is fixed relative to link 2 and link 3 relative to 4. The crease is still allowed to fold. The model in Fig. 3(a) was created by connecting the dotted or output crease to a solid or input crease of the second vertex as indicated by Fig. 3(b). A different model would result if the output crease was changed as in Fig. 3(c). The spatial relationship between these different connection choices can be defined as follows:

Definition 2
Let the creases ia and ib of the linkage-based vertex in Fig. 4 be input and output creases of the ith origami vertex if and only if their orientations are related by a homogeneous transformation Hi. If the frames Sia and Sib are fixed to the links described by Ta and Tb from Definition 1, then a point xib in frame Sib can be expressed in frame Sia by the relation
$(xia1)=H(xib1)$
(3)

This definition is useful because it establishes a notation for describing the transformation produced by a vertex with a given input and output crease as shown in Fig. 4. Because this description applies to any kinematic configuration of the linkage-based vertex, the transform H is a function of the linkage fold angles. This notation can now be used to define properties of vertex combinations and criteria for ensuring that these combinations tessellate efficiently.

Definition 3
In Fig. 5, let L1, L2, L3, … be a set of origami vertices according to Definition 1 and S1a, S2a, …, Sna and S1b, S2b, …, Snb be the input and output creases of these vertices according to Definition 2. This set forms an origami cell C if and only if a chain can be formed by connecting output Sib to input crease Si+1,a and a point x1a in frame S1a can be related to a point xnb in frame Snb by
$(x1a1)=∏i=1nHi(xnb1)$
(4)
where ∏ specifies a right product and Hi is the transformation between input and output frames for the ith vertex.

One way of ensuring that a tessellation remains regular is by guaranteeing that the panels at the end of a one-dimensional chain are always parallel to the panels at its input. This can be done by requiring that the frames at the beginning and end of the chain are not rotated relative to each other.

Definition 4
An origami cell according to Definition 3 is regular if and only if
$∏i=1nHi=(Id01)$
(5)
where ∏ specifies the right product, I is the identity matrix, and d is an arbitrary vector.

Here, it should also be noted that this relationship must hold as the degrees-of-freedoms in vertices i through n are varied.

Using the definition for regular cells, it is possible to identify classes of vertices and their combinations which are guaranteed to form regular cells. This is advantageous because it will help designers quickly construct different regular cells without requiring significant experimentation.

One way to categorize techniques for constructing regular cells is based on the number of vertices in the cell. Here, we consider methods for constructing cells with one, two, and three or more vertices. Methods for connection are based on whether the connections are being created in one, two, or three dimensions.

Single-Vertex Regular Cells.

From Definition 4, a cell (or vertex, in this case) is only regular if there is no rotation between the input and output frames. This is only true if the input and output creases are parallel throughout the folding process. An example of such a case can be found when a split-vertex transformation [19,20] is applied to a single fold as shown in Fig. 6. The fold initially behaves as a two-link chain as in Fig. 6(a) but becomes a six-link mechanism after transformation, as shown in Fig. 6(b). The split-vertex transformation is valid for any real, offset link geometry, therefore many different geometries are possible. Because this vertex is inherently regular, no further effort is needed before connection.

Twin-Vertex Regular Cells.

When a cell consists of two vertices, tessellation design becomes significantly more complicated. Criteria for designing regular, twin-vertex origami cells can be developed if we consider different combinations of two origami vertices. The specific combinations that will be attempted were found using the two-dimensional diagram shown in Fig. 7(a). In this image, the bend introduced by transformation H is opposed by an equal and opposite bend, ensuring that the outermost segments always remain parallel. This suggests that the vertex arrangement shown in Fig. 7(c) should also be capable of forming a regular origami cell. If we describe the transformations H for each vertex as shown in Fig. 7(a) with the frames S1a, S1b, S2b, S2a, and S3, then a relationship between the orientation of S1a and S3 can be found.

Theorem 1
Let two vertices L1 and L2 according to Definition 2 be connected at the output crease of L1 and the input crease of L2. If these vertices are described by homogeneous transformations H1 and H2, then this vertex pair is regular if and only if
$R1MyR2My=I$
(6)
where My is a rotation matrix of π radians about the y-axis, R2 is the rotation matrix component of H2 describing L2, and R1 is the rotation matrix component of H1 describing L1 according to Definition 2.
Proof
Let the input and output creases of the two vertices shown in Fig. 7 be related by H1 and H2, respectively. If xi is a point in frame Si, then
$(x1a1)=(R1d101)(x1b1)$
(7)
$(x2a1)=(R2d201)(x2b1)$
(8)
If frames S1b and S2a and S2b and S3 are rigidly connected by the transformations
$(x1b1)=(My001)(x2a1)$
(9)
$(x2b1)=(My001)(x31)$
(10)
where My describes a rotation of $180deg$ about the y-axis, then x1b, x2a, and x2b can be eliminated from Eqs. (7) and (8), giving
$(x1a1)=(R1MyR2MyR1Myd2+d101)(x31)$
(11)
If we require that this pair of vertices be a regular origami cell according to Definition 4, then
$(R1MyR2MyR1Myd2+d101)=(Id01)$
(12)
and
$R1MyR2My=I$
(13)
The reverse of this proof can also be shown. If we consider a twin vertex where Eq. (13) is true, then this pair is also described by Eq. (11). Substituting Eq. (13) into Eq. (11) yields
$(x1a1)=(IR1Myd2+d101)(x31)=(Id01)(x31)$
(14)
Therefore, if a vertex pair is connected and Eq. (13) applies, then the resulting transformation is regular. Consequently, a paring of two vertices L1 and L2 is regular if and only if R1MyR2My = I.
One solution to Eq. (13) occurs when the vertices L1 and L2 are identical and L2 is upside-down relative to L1. In this case,
$R1MyR1My=I$
(15)
This translates to a requirement that C = MyR1 must be involutory (CC = I). A matrix is involutory if and only if $12(C+I)$ is idempotent. This requires that
$12(C+I)=12(C+I)12(C+I)=14(MyR1MyR1+2MyR1+I)=14(I+2MyR1+I)=12(MyR1+I)$
(16)
Consequently, Eq. (6) is always satisfied by the geometric assumption that vertices L1 and L2 are identical. However, to also be kinematically regular over the range of mechanism motion, this relationship must be true for all mechanism configurations. This requirement can also be satisfied by requiring that they two vertices be in the same kinematic state. For origami vertices, this can be accomplished by coupling equal fold angles.

If the two vertices are at two different folded states, then the cell regularity is not certain. An example of this is shown in Fig. 8. This linkage was constructed with sector angles of 2π/3, π/2, π/6, π/6, and π/2 and is based on a model by Chen et al. [4]. In this specific model, the input crease and output crease have different dihedral angles. This forces the second vertex in the chain to have a different folded state from the previous vertex and consequently a different R. To avoid this problem, input and output creases must be chosen so that all the mechanism dihedral angles are equal throughout the folding process. The Myard linkage in Fig. 8 is symmetric about the bisector of the large blue link and so its fold angles on either side are always equal. Although inverting one vertex and connecting using these creases may not work because the incompatibility between mountain and valley creases, a transition mechanism can be constructed to allow this. Such a transition device is shown in Fig. 9(a) and the resulting 1D chain is shown in Fig. 9(b).

Similar relationships can be developed for a derivative configuration made possible by a hinge transpose [19]. This configuration is shown in Fig. 10(a) and is obtained by inverting the output crease of one of the vertices as shown in Fig. 11. The crease supporting frame S1 in Fig. 11 is transposed, resulting in the mechanism shown at right in Fig. 11.

Theorem 2
Let two vertices L1 and L2 according to Definition 2 be connected at the output crease of L1 and the output crease of L2. If these vertices are described by homogeneous transformations H1 and H2′ = H2My, then this vertex pair is regular if and only if
$R2=R1$
(17)
where R2 is the rotation matrix component of H2 describing L2, and R1 is the rotation matrix component of H1 describing L1 according to Definition 2.
Proof
For a general vertex transformation H, a transpose of the output crease results in a new matrix H′ given by H′ = HMy. This allows us to describe the configuration in Fig. 10(a) with
$(x1a1)=(R1d101)(x1b1)$
(18)
$(x2a1)=(R2d201)(My001)(x2b1)=(R2Myd201)(x2b1)$
(19)
$(x2b1)=(My001)(x1b1)$
(20)
where x1a, x1b, x2a, and x2b represent points in frames S1a, S1b, S2a, and S2b. Solving these equations and eliminating x1b and x2b yields
$(x2a1)=(R2R1T−R2R1Td1+d201)(x1a1)$
(21)
If this cell is regular according to Definition 4, then we also know that
$R2R1T=I$
(22)
This can only be true if R2 = R1 because $R1T=R1−1$ for orthonormal rotation matrices. This indicates that the specific configuration shown in Fig. 7 can only be a regular cell if both vertices are identical before the transpose was applied.
The reverse of this proof can also be quickly demonstrated. For a general twin vertex shown in Fig. 10(a), the cell transformation is given by Eq. (21). If Eq. (22) is true, then substituting Eq. (22) into Eq. (21) yields
$(x2a1)=(I−R2R1Td1+d201)(x1a1)=(Id01)(x1a1)$
(23)
Therefore, for a cell subject to Eq. (22) and composed of two vertices as shown in Fig. 10(a), it must be true that the vertex pair is regular.

Large Regular Cells.

In cases where more than two vertices are desired in a regular cell, multiple options exist for the design. In addition to varying vertex type, different combinations of single- and twin-vertex regular cells can also be combined as shown in Fig. 12. Because each of the cells is regular, it is logical to assume that the entire chain is also regular. When inconsistencies in mountain-valley assignment exist, they can be corrected using adapters as in Fig. 9(a).

Together, the relationships in Eqs. (12), (13), (21), and (22) establish basic requirements for designing regular origami cells and identify a simple technique for constructing a diverse set of tessellations. For single- and twin-vertex cells, different transformations of identical vertices were shown to be a method for constructing regular tessellations as described. For cells containing more vertices, combinations of single- and twin-vertex cells can be used.

Connection Methods.

For a given set of origami cells, numerous options exist for assembling them into tessellations. Potential options include connection through

1. Rigid joints

2. Hinges

5. Regular cells

Connection through rigid joints involves fixing adjacent links in neighboring vertices and is shown in the single-vertex tessellation in Fig. 13(a) and the twin-vertex tessellations in Figs. 13(c) and 13(e).

Hinges are required to create other varieties of tessellations and higher-dimensional tessellations. Figures 13(b), 13(d), and 13(f) show two-dimensional tessellations created using lateral hinge connections. Table 1 provides additional details on the construction of these models.

Connection using hinges is also possible in all three dimensions because of the large number of features which remain parallel during the motion of regular tessellations. This is shown in Figs. 14(a) and 14(b). Figure 14(a) shows a 2D tessellation of a single-vertex cell with offset panels connected by hinges protruding from its surface. Figure 14(b) shows the resulting 3D model when the extended panels are connected to the underside of a second tessellation layer using hinges.

An interesting observation is that zipper tubes [27,28] belong to this category of tessellations. Specifically, many examples are a type of 3D tessellation which can be regular or irregular and whose cells are connected through hinges or link offsets. An example of such tubes is shown in Fig. 15. Figures 15(a) and 15(b) show the cell and its tessellations while Fig. 15(c) shows a single tube. A 2D tessellation of tubes is shown in Fig. 15(d). Because tubes in the references are zero-thickness tubes, the models must be adjusted to accommodate this difference. One approach involves connecting adjacent 2D layers with the gray offset links.

Lastly, regular cells can also function as bridges joining cells into tessellations. This is inherently true for one-dimensional tessellations. An example of a two-dimensional tessellation using regular cells is shown in Fig. 16. Figure 16(b) shows the one-dimensional chain composed of two single vertices sharing an input-to-input connection and Fig. 16(c) shows the cells that connect these chains laterally into a 2D tessellation. A larger view of the tessellation with four chains and two cells in each chain is shown in Fig. 16(a). The resulting tessellation can be extended infinitely either vertically or horizontally although the pattern will include holes.

Models of several two-dimensional tessellations were created to demonstrate the model kinematics and folding motion. These tessellations are shown in Fig. 17. Figures 17(a), 17(c), and 17(e) show twin-vertex tessellations where the first two use Bennett linkages and the last uses Bricard linkages. All three are geometrically different and are based on the models in Figs. 13(c), 13(e), and 13(h). Figures 17(b), 17(d), and 17(f) show their folded configurations.

Conclusion

Including thick panels in origami-inspired mechanisms and enhancing their stiffness are often needs to be associated with their design. This work has developed methods for designing linkage-based origami tessellations that help achieve these objectives.

This effort has resulted in multiple important results. First, theorems were developed which provide criteria for designing origami cells suitable for tessellation. Specific requirements for constructing combinations of thick vertices that can be joined into regular tessellations were described. Proofs for these theorems guarantee that the resulting cells will be regular as defined previously. This is important because it ensures that the tessellation will not curl upon itself and produce additional self-intersection.

Techniques for assembling these cells into tessellations of multiple dimensions were introduced. Common connection techniques were illustrated and then demonstrated using multiple example cases. Together, these results significantly augment existing capability to consistently design tessellations of thick origami in one, two, and three dimensions. Although challenges in the design of stiff, thick tessellations remain, these results facilitate implementations of origami in engineering applications.

Acknowledgment

This work was supported by a NASA Office of the Chief Technologist’s Space Technology Research Fellowship and by the National Science Foundation and the Air Force Office of Scientific Research under NSF Grant EFRI-ODISSEI-1240417. The authors would also like to acknowledge the help of Bethany Parkinson and Bridget Beatson in creating physical prototypes.

Nomenclature

• d =

3 × 1 translation vector

•
• x =

3D column vector describing a point

•
• L =

origami vertex with input and output creases

•
• C =

arbitrary 3 × 3 rotation matrix

•
• H =

4 × 4 homogeneous transformation between frames attached to the input and output creases of a vertex

•
• I =

identity matrix

•
• R =

3 × 3 rotation matrix

•
• T =

4 × 4 homogeneous transformation matrix describing a mechanism link

•
• 0 =

1 × 3 zero vector

•
• Sia =

frame attached to the input crease of the ith vertex

•
• Sib =

frame attached to the output crease of the ith vertex

•
• My =

3 × 3 rotation matrix encoding 180 deg rotation about the y-axis

•
• Mz =

3 × 3 rotation matrix encoding 180 deg rotation about the z-axis

Appendix

Here, we list figures giving dimensions for some of the models presented in these papers. These dimensions refer to the figures mentioned in Table 1 and give the measurements of the parts used (Figs. 1821).

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