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Research
Article submitted to journal

Subject Areas:
structural engineering, mechanicalengineering, mechanics
Keywords:
reconfigurable origami, mechanics ofOrigami tubes withreconfigurable polygonalcross-sections
E. T. Filipov1 , G. H. Paulino1,2 and T. Tachi3
1 Department of Civil and Environmental

Engineering, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA
2 School of Civil and Environmental Engineering,
Georgia Institute of Technology, Atlanta, Georgia
30332, USA
3 Department of General System Studies,
University of Tokyo, 3-8-1 Komaba, Meguro-Ku,
Tokyo 153-8902, Japanorigami tubes, variable cross-section
tubes, programmable structures

Author for correspondence:
G. H. Paulino
e-mail: paulino@gatech.edu

Thin sheets can be assembled into origami tubes tocreate a variety of deployable, reconfigurable, andmechanistically unique three dimensional structures.
We introduce and explore origami tubes withpolygonal, translational symmetric cross-sections,that can reconfigure into numerous geometries. The
tubular structures satisfy the mathematical definitions
for flat and rigid foldability, meaning that they canfully unfold from a flattened state with deformations
occurring only at the fold lines. The tubes do not need
to be straight, and can be constructed to follow a nonlinear curved line when deployed. The cross-section andkinematics of the tubular structures can be reprogrammed
by changing the direction of folding at some folds. We
discuss the variety of tubular structures that can beconceived and we show limitations that govern thegeometric design. We quantify the global stiffness ofthe origami tubes through eigenvalue and structural
analyses and highlight the mechanical characteristicsof these systems. The two-scale nature of the present
work indicates that, from a local viewpoint, the crosssection of the polygonal tubes are reconfigurable
while, from a global viewpoint, deployable tubes ofdesired shapes are achieved. This class of tubes has
potential applications ranging from pipes and microrobotics to deployable architecture in buildings.

c
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1. Introduction

state. (b) Folding sequence of the tube, where the cross-section is reconfigured using the four initially flat panels or switches(n = 4). (c) Four other possible cross-sections into which the tube can be reconfigured.

A variety of origami inspired tubes exist including the Miura-Tachi polyhedron [17,25,26],and variations inspired by the Yoshimura pattern [27]. In this paper, we explore and extend
upon origami tubes that employ the Miura-ori pattern, that were first introduced by Tachi and
Miura [28,29]. We generalize these into a new set of polygonal cross-section tubes that posses thefollowing properties and advantages:
1. Tube cross-sections can take a variety of polygonal shapes.
2. The cross-sections can be made reconfigurable to allow for programmable functionality.
3. A wide variety of new curved tubular forms are possible.

...............................................................

Figure 1. A reconfigurable origami tube with a polygonal cross-section. (a) The tube and cross-section shown at a fully extended2

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Historically, origami has gained popularity in science and engineering because a compactly
stowed or flat system can be folded into a transformable 3D structure with increased functionality.
Proposed applications in the smaller length scales include biomedical devices [1], and micro
robotic assembly [2]. In medium sizes, origami techniques have been used in creating compliant
mechanisms [3], actuators [4], toys, and educational tools [5]. Large origami structures could
be constructed for aerospace [6] and architectural applications [7]. The applications seem to beendless.
More recently, innovation with origami has pivoted on its capability to create programmableand re-programmable systems that can change shape, function, and mechanical properties.
For example Hawkes et al. [8] created a sheet with pre-defined fold lines that can reshapeautonomously into different three dimensional structures. Marras et al. [9] showed that DNA canbe folded to create nano-scale mechanisms with programmable mechanical function. Origami
metamaterials that can be reconfigured, and whose mechanical properties can be tuned andtailored have also become a popular subject of study [10–13].
Thin walled origami tubes have been created by folding thin sheets, but they typicallydiffer from the fundamental definitions of origami. In particular: entire origami tubes are notdevelopable, meaning they cannot be created from a continuous flat sheet; and they require gluing
or some other connectivity for creating the complete tube. Despite the higher complexity of
manufacturing, origami tubes greatly extend the functionality of engineered thin sheet structures.
For example, they can be be used as deployable stents in biomedicine [14], as inflatable structuralbooms for space structures [15,16], or as actuators and bellows [4,17,18]. Origami tubes have a selfconstraining geometry that makes them suitable for energy absorption devices [19–22]. Stacking
and coupling of origami tubes into more complex geometries can lead to stiffening of the systemand enhanced mechanical characteristics [11,13,23,24].

2. Cross-section definitions for polygonal tubes
The popular Miura-ori pattern has inspired the development of rigid foldable origami tubes
discussed in several recent articles [13,23–25,28,29,33]. The cross-sections of these tubes are
symmetric, with the most fundamental tube consisting of two equal symmetric Miura-ori strips
placed opposite from each other. More advanced cross-sections follow isotropic, anisotropic,
or star shaped cylindrical variations [25,28,29]. In this work, we go beyond the previous tube
variations and introduce a translational symmetry method to create a variety of polygonal shaped
tubes. The basic cross-section variations for the polygonal tubes are defined in the Y − Z axis, as
demonstrated by figure 2. For our definition, we divide the geometry of the cross-section into an
upper (U ) and a lower (L) section. The names of these two sections are only representative andtheir location may in fact be side by side as shown later in figures 3 and 4. The two opposing
sections of the tube have to be continuous and can be composed of m ≥ 2 edge groups. The edgegroups are identified by a unique slope angle θ, and denoted by a lower-case letter (a, b, c ...). The
slope angle is taken clockwise from the Z axis of the cross-section, and has the admissible rangeof −180◦ < θ < 180◦ . Each edge group on the upper section can be composed of p ≥ 1 edges, andthe corresponding lower edge group can be composed of q ≥ 1 edges. The length of the ith edge inthe b edge group on the upper (U ) section is denoted as bU i .
To create a valid cross-section, each edge group on the upper section must have acorresponding edge group on the lower section with the same total length and slope angle. Thisdefinition can be written mathematically as:pX
i=1

aU i =

q
X
i=1

aLi ;

p
X
i=1

bU i =

q
X
i=1

bLi

...

p
X
i=1

mU i =

q
X
i=1

mLi ,

(2.1)

...............................................................

Properties 1, 2, 3, 5, and 6 are possible with the new polygonal tube definitions presented herein.
Some of the advantages are motivated by figure 1 that shows a curved tube that can fold in avariety of different cross-sections. The versatility, mechanical characteristics and reconfigurabilityof theses tubes could result in numerous applications as pipelines, architectural structures, robotic
components, bellows, metamaterials, and other reprogrammable systems.
The paper is organized as follows: Section 2 introduces the cross-sections, and Section 3
provides the full three dimensional definition for admissible polygonal tubes. The systemkinematics and reconfigurable characteristics of different tubes are discussed in Section 4. In
Section 5 we extend the tubular definitions to cellular assemblages that can also be reconfigured.
In Section 6 we discuss the elastic modelling and explore the mechanical properties of the tubes
through eigenvalue and structural analyses. Tubes with circular cross-sections are investigated in
Section 7, and Section 8 provides a discussion and concluding remarks. Appendix A explains
folding characteristics of the idealized tubes assuming zero-thickness, Appendix B gives ananalytical comparison between smooth pipes and origami tubes, and Appendix C gives an
outlook for practical implementations and future extensions of the proposed systems.

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4. The tubes are compatible and can be coupled into a variety of assemblages.
5. The mechanical properties of the tubes can be tuned through reconfiguration.
6. Out-of-plane compression stiffness is enhanced similar to corrugated pipe systems.
7. The perimeter of the tubes is continuous, allowing for deployment by inflation and for the
potential capability to carry liquids and gases.
8. Based on idealized zero-thickness kinematics, the tubes are flat foldable meaning that they canfold down to a completely flat state allowing for compact stowage.
9. These systems are rigid foldable, meaning the origami can fold and unfold with deformation
concentrated only along the fold lines (creases), while the panels (facets) remain flat. This
capability could allow the structures to be constructed with panels of finite thickness [30–32],
and to fold in a controlled motion.

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Figure 2. Valid cross-section definitions and basic variations. (a) Six-sided tube cross-section with m = 3 edge groups eachhaving a unique slope angle θ . (b) Folding sequence of a tube created from the cross-section in (a). The cross-section
corresponds to the fully-extended configuration. (c) Six-sided cross-section with the same edge groups as in (a), arranged ina different order. (d) The upper edge group bU is divided in two (p = 2) and rearranged. The corresponding lower edge groupbL can be composed of a single, two, or more corresponding edges with an equal total length (q 6= p).

This property ensures that the cross-section will be closed, thus creating a foldable origami tube
with a continuous un-interrupted circumference. The logic of equation (2.1) can also be thoughof as a sum of two groups (sections) of equal direction vectors (edge groups), segmented (intoedges) and re-arranged to create the cross-section. The re-arrangement of the individual edges
can be performed in any logical manner (e.g. figure 2c), so long as the lower and upper sections
do not intersect. As shown in figure 2d, when an edge group is segmented into several edges,
the number of edges on the upper and lower sections do not need to be the same (i.e. p 6= q). A
non-trivial cross-sections with a negative θ is shown in figure 3 and one with a complex outline isshown in figure 4.

Figure 3. (a) Cross-section with a negative slope angle θ . (b) A tube with that cross-section shown fully extended, and folded to
95% and 10% extension.

22 edges, while the lower section has 29. (b) Dog tube fully extended, and folded to 95% and 10% extension.

3. Three dimensional profile definitions
In this section, we discuss the complete three dimensional definition of the tubes when apreviously defined Y − Z cross-section is used as a basis. The cross-section is projected in
X − Y − Z space, to create a closed continuous tube. The tube definitions assume that the origami
sheets have an infinitesimally small or zero-thickness. In practice, there is a technique that allows
for thickness to be incorporated into the design of rigid foldable tubes [31], however we do nottake these details into account. In Section 3(a) we discuss the basic projection geometries that
preserve the rigid and flat foldability of the polygonal origami tubes. With these definitions thecapability to reconfigure the cross-section is preserved allowing for a programmable system. The projection
discussed in Section 3(b) violates flat foldability conditions, but maintains rigid foldability and theprogrammable characteristics. The projection presented in Section 3(c) is the most geometricallyunrestricted, but it restricts folding for non-square, non-symmetric tubes. The programmablecharacteristics of the tubes are discussed in Section 4, and the folding properties are summarizedin Appendix A.

...............................................................

Figure 4. (a) An admissible cross-section with the shape of a dog, created with six different edge groups. The upper section has

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The fundamental tube that was previously studied [28,33] is a unique case of the generalization
proposed here. The tube is created from only four edges that are symmetric about the Y and Z
axes; that is θB = 180◦ − θA , and the edge lengths are aU 1 = aL1 = bU 1 = bL1 . This tube can be
fully flattened in the X − Y plane and can also be folded into a flat state in the Y − Z plane.
However, as will be shown in Section 4, this most fundamental tube case is not reconfigurable.
To create a reconfigurable tube, the cross-section must have at least three edge groups (m > 2)each with a unique slope angle θ. Although the slope angles can be arbitrary, in our work wedefine reconfigurable cross-sections with one edge group where θ = 90◦ . When this cross-section
is projected in the X − Y plane per Section 3(a), the θ = 90◦ edge group will be completely flat.
As defined, the tube is at a fully extended state (100% extension), because from this state the flatedge group can only fold down. When folding, the θ = 90◦ edges serve as programmable bits
or switches to reconfigure the tube cross-section (see Section 4). A m > 2 cross-section that has noedge group with θ = 90◦ , is not fully extended when initially defined, and the edges with θ closest
to 90◦ serve as the switches.

(a) Admissible projections for rigid and flat foldable polygonal origami

6

tubes

(b) Constant φ = 80◦ projection. (c) Constant φ = 40◦ projection. (d) Constant φ = 60◦ projection, with lengths of segment

i defined as: li = 0.4 + 0.2 ∗ i. (e) Projection with angle variation. Symmetry between the cross-section and projection vector
is preserved in the X-Y plane. (f ) A rigid foldable S-shaped tube constructed by following symmetry rules in (e). All tubes of thisfigure use the cross-section in figure 2a.

The basic type of projection is further extended by allowing an angle shift to occur, where theprojection angles are not equal throughout (i.e. φ1 6= φ2 6= φ3 ...). Figure 5e shows the projection
where the angle is varied in the X − Y plane. Symmetry is enforced such that the adjacent vertex
angles (α) about the cross-section are kept symmetric. This projection can be used to create anarbitrary geometry in the X − Y plane that is flat and rigid foldable.

...............................................................

Figure 5. (a) Cross-section projection in the X-Y direction using a constant projection angle i.e. φ = φ1 = φ2 = φ3 = 60◦ .

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The first geometric variation for the tubes is to project the the cross-section in the X − Y planewith a constant projection angle as shown in figure 5a-d. The projection is defined by an angle
φ and length l. This projection creates a new cross-section that again lies only in the Y − Z
plane and is parallel with the initial cross-section when looked at from above (X − Y plane). The
corresponding edges of the two cross-sections are connected with thin origami sheets creating asystem of fold lines and panels. A different projection angle φ can be used to create a distinctly
different structure (figure 5b and c). The length of individual projected segments can also bevaried (figure 5d). When the base projection with no length variation is used, all panels areparallelograms and are the same for each cross-section edge. The left vertex angle (α) of eachpanel (internal angle of the parallelogram) can be calculated as αL = arccos(− sin(θ) ∗ cos(φ)).
For other more complex projections discussed herein we leave the geometric derivations to thereader.

(b) Projections for rigid, but non-flat foldable origami tubes

completely flat.

Projection in X − Y space can also be performed without following the symmetry about thecross-section. In figure 6 we show a projection where the projection angles are φ2A 6= φ2B , andthus the adjacent vertex angles are also not symmetric. The system can undergo rigid folding, but
in this case the folding sequence is restricted and the system cannot fold into a completely flatspace (figure 6b and Appendix A).

(c) Extended projections for origami tubes
The final form of projection discussed here is the most general, where the projection is performed
arbitrarily in all three dimensions (X − Y − Z). The vector can be varied in all directions
simultaneously, by using an angle φ to describe the projection vector in X − Y , and γ to vary
the projection vector in Y − Z. Symmetry of the projection is preserved, such that the adjacentvertex angles on opposing sides of a cross-section are equal. This symmetry can be visualized
as mirroring the structure locally, which is shown using transparent planes in figure 7a-c. For the
polygonal cross-section, when this projection is used, the resulting structure is not foldable, and so
it is essentially no longer origami, but a static fully restrained structure (Appendix A). However,
if a simple symmetric cross-section is used, the structure remains rigid and flat foldable. The
structure in figure 7d follows an arbitrary spiral in three dimensional space.

4. Kinematics in reconfiguring polygonal tubes
The folding of the tube can be preformed through an analytical [34–37] or numerical method [38],by changing a fold angle in one vertex, calculating the other angles in the vertex, and cycling
through all of the vertices in the pattern until all fold angles, and the new geometric shape arecalculated. We use the numerical method in [38] to perform the folding. In contrast to more simple
tube structures, the geometry of the tubes presented here can be reconfigured. Figure 8 shows thetwo basic geometry reconfigurations that can be obtained from the simple six-sided origami tube.
The initially flat in Y − Z segments, can be used as switches to change the structural geometry.
A binary system is used to inform the directional change in cross-section and new geometry.
The upper and lower switches are defined as a 0 or a 1 and indicate negative or positive slopechange in the cross-section respectively (valley or mountain fold respectively between the first andsecond panels). The assignment on the upper and lower segments must match to preserve thetranslational symmetry in equation (2.1), thus if the [U: 1] then [L: 1] as well. In figure 9a we
extend these definitions to the eight-sided tube from figure 2d.

...............................................................

φ2B . (b) The folding sequence of the non-symmetric projection shown in top and isometric views. The structure cannot foldrspa.royalsocietypublishing.org Proc R Soc A 0000000

Figure 6. (a) Cross-section projection in the X-Y direction that does not preserve symmetry about the cross-section i.e. φ2A 6=

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the cross-section. Projection of six-sided polygon shown in (a) top, (b) side, and (c) isometric views. This polygonal tube cannot
fold. (d) Folding sequence of four-sided origami tube constructed by projecting along a spiral in three dimensional space. This
tube is rigid and flat foldable.

The eight-sided tube has two switches of equal length on both the upper and lower sections.
The number of positive switches (1s) on the upper section has to correspond to the number of
positive switches on the lower section of the tube. Thus the sum (k) of the L and U switches mustmatch. Figure 9a shows the six possible switch variations for the eight-sided tube. The number ofpossible ways to reconfigure the upper section only, follows a binomial coefficient as:
!
n
n!
=
,
(4.1)
k!(n − k)!
kwhere we have n available switches and we want exactly k of them to be positive. For example
in figure 9a there is only one possible way to reach a total of either k = 2 or k = 0, configurations
I and II respectively. However, there are two possible ways to reach a total of k = 1, i.e. [U: 1 0]
and [U: 0 1]. Because each variation of the upper section can be coupled with a corresponding
lower section with the same polarity sum (k), we need to take the square of these possibilities,
and sum them to find the total number of possible variations for the cross-section. This results inthe central binomial coefficient:
2
n 
X
(2n)!
n!
=
.
(4.2)
k!(n − k)!
(n!)2
k=0

This function gives the total number of unique cross-section variations that can be obtained
when folding a reconfigurable tube with n flat segments or switches. The possible upper
section assignments for a n = 2 and n = 3 tube are shown in figure 9b. The number of possible
upper section variations follow Pascal’s triangle (figure 9c), and the total number of possible
configurations follow the central binomial coefficient (figure 9d). The most basic, four sided tube...............................................................

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Figure 7. Cross-section projection varied in the X-Y-Z directions simultaneously while preserving symmetry of the structure about

9

for the first/last fold (0 = valley, 1 = mountain). The folded cross-sections of the two configurations are not symmetric becauseedge groups a and c in the cross-section definition are not symmetric (figure 2a).

cross-section (e.g. figure 7d) has no switches (n = 0), and thus has only one possible cross-section
configuration. On the other hand, the tube with four symmetric switches (n = 4) shown in figure
1 can reconfigure into 70 distinct cross-sections.

5. Cellular extensions for reconfigurable origami tubes
The projection technique for creating polygonal tubes can be extended to creating cellularassemblages that have similar geometric characteristics. When the translational symmetry is usedin the cross-section(s) and an admissible projection is followed to construct the three dimensional
structure, the folding and reconfigurable characteristics remain similar to before. In figure 10 weshow two assemblages that use a constant angle projection, although it is possible to use moreadvanced curved projections as well. The cross-section in figure 10a is created by discretizing
the cross-section into smaller sections. All of the internal cross-sections, as well as the global
external cross-section, follow the translational symmetric rules in equation (2.1). This assemblage
can still be reconfigured as shown in figure 8. In figure 10b we combine four tubes together,
two of which have reconfigurable cross-sections. This assemblage can now be reconfigured
into four different cross-sections, with configurations III and IV being rotationally symmetric.
A variety of new assemblages can be constructed using these ideas, however the initial crosssections cannot have overlapping components, and the kinematics of reconfigurations should be
carefully analysed. When multiple tubes and cross-sections are reconfigured, it may be possiblefor different components to experience interaction or contact, and some of the reconfigurations
may be obstructed.
Polygonal origami assemblages can be further enhanced by using different projection angles
and projection directions, for the different tubes within the assemblage [13,39]. The mostinteresting case is a zipper coupling of tubes, where a positive projection φ is used for one tube anda negative φ for another. These zipper assemblages can be used to create significantly stiffer thinsheet origami structures. The assemblages can be generalized in numerous ways, but they also
limit some of the projection directions that can be used to create the system [39]. Furthermore, it
is possible to introduce techniques for locking the origami configuration into a sandwich-like...............................................................

the single flat segment (n = 1). A cross-section schematic with positive or negative slopes is used to inform the fold assignmentrspa.royalsocietypublishing.org Proc R Soc A 0000000

Figure 8. The six-sided tube can be folded into two different configurations by changing the polarity of folds (valley or mountain) on

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Figure 9. Variations in reconfiguring polygonal tubes. (a) An eight-sided tube with two equal length segments (switches) canreconfigure into six unique configurations. The three dimensional models are shown at 10% and 95% extension. (b) Possible
switch variations for the upper section only, with n = 2 and n = 3 switches. The variable k corresponds to the sum of the of switch
assignments. (c) A Pascal’s triangle shows the number of variations for the upper section of the tube only. This is the binomial
coefficient with n representing the rows and k the columns. (d) The total number of possible cross-section configurations. Thisis equivalent to the central binomial coefficient.

structure [11,40]. These additions can enhance the structural rigidity of the systems, but can
restrict the deployment and reconfigurable kinematics of the polygonal tubes. Future research
can explore the numerous assemblage variations proposed and determine useful methods forenhancing the mechanical characteristics of the structures.

6. Elastic behaviour of polygonal tubes
In this section, we explore the global mechanical characteristics of the tubes with a finite-element(FE) analysis software (Abaqus [41]). Each of the origami panels is discretized with 8 × 8 shellelements and the folds are modelled using rotational hinges as shown in figure 11. The model uses
standard S4 general purpose shell elements with finite membrane strains that are appropriate
for the small deformation analyses of the thin sheet origami structures. We model the eightsided reconfigurable tube from figure 2d. The cross-section edges for the upper section haveslopes of [θa , θb , θc ] = [30, 90, 125]◦ , and lengths of [bU 1 , aU 1 , bU 2 , cU 1 ] = [0.5, 0.7, 0.5, 1] cm. The
tube is ten segments long, and is created with constant projection of φ = 60◦ and l = 1 cm. The
configuration of the structure is defined based on the idealized zero-thickness rigid kinematics,
however, to define the stiffness of the structure we assign a thickness of 0.1 mm which translates toroughly L/t ≈ 50 − 100. The model does not however account for detailed effects of the thickness
such as intersection that may occur when we attempt to fold an origami with finite thickness.

11

configurations similar to before. (b) Assemblage consisting of two six-sided and two four-sided tubes together. This structure
can reconfigure into four states.

Figure 11. A portion of the eight-sided reconfigurable tube with the corresponding FE discretization. The inset shows the localized
zero-length connectivity at the fold lines.

Other model parameters are defined as Young’s modulus E = 5 GPa, Poisson’s ratio ν = 0.33,
and density ρ = 650 kg/cm3 . In reality the behaviour and stiffness of the fold lines can dependon the material and fabrication used to make the origami. Here, we assume linear elastic folds
where the stiffness for a rotation of ρ radians is specified as Kρ = 0.0164 N*cm/rad per one
centimetre of fold line. The fold lines are assumed to be more flexible in bending than the panels,
and thus Kρ is specified to be one tenth (1/10) the bending stiffness of an origami panel with adiagonal length of one centimetre. The dimensions and units used here are chosen arbitrarily
but within a realistic range to give qualitative insight to the origami behaviour. Quantitative
results for engineered origami systems could be obtained using known dimensions and material
properties. The analytical model captures the elastic behaviours of origami type structures: 1)
panels stretching and shearing, 2) panels bending, and 3) bending along prescribed fold lines.
We have evaluated the mesh convergence for the tube when it is loaded as a cantilever laterin this section. The 8 × 8 shell mesh approximates displacements within 4% of a significantly
finer mesh discretized with 32 × 32shell elements per panel. In this paper, we use elastic andsmall displacement approximations for all analyses. Future research will be needed to understandlocalized behaviours in origami structures, as well as the large displacement behaviours which
could be of significant importance.

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(a) The basic six-sided polygon has four smaller parallelogram tubes inserted within. This assemblage has two possible

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Figure 10. Cross-sections, isometric folding sequence and possible reconfigured cross-sections of cellular origami assemblages.

the extension of the tube in configuration I. (b) Corresponding eigen-modes at 75% extension. (c) Eigenvalues seven through ten
presented for each of the six possible geometric reconfigurations of the tube.

For the eight-sided reconfigurable tube the seventh eigen-mode follows the kinematic folding
and unfolding of the structure (figure 12a-b). The seventh mode has the least energy indicatingthat it is easiest to deform the structure by following the prescribed folding sequence. The
eight mode is a squeezing mode, where one end of the tube is folding and the other end is
unfolding. By changing the geometry or through tube coupling described in Section 5 it could bepossible to substantially increase the band-gap λ8 − λ7 , creating a structure that is easy to deploy,...............................................................

Figure 12. Eigenvalue analyses of eight-sided reconfigurable tube with two switches presented in figure 9. (a) Eigenvalue versus

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We use an eigenvalue analysis to explore the global mechanical properties of the reconfigurable
origami tube. These analyses can be used to determine how flexible or stiff the structure would
be for bending, twisting or other deformations. We obtain the eigenvalues λi and corresponding
eigen-modes vi of the structure using a linear dynamics system of equations Kvi = λi Mvi where(K) is the stiffness matrix and (M) is the mass matrix of the structure. The eigenvalues are arranged
in an incremental order (i) and represent the excitation frequencies that would deform thestructure into the corresponding eigen-mode. The eigenvalues are also proportional to the totalenergy of each eigen-mode (elastic strain energy and kinetic energy), meaning that eigen-modes
with higher eigenvalues require more energy to achieve the given deformation. The structure isanalysed with no boundary conditions, thus the first six eigenvalues are zero, with eigen-modes
representing rigid body motion of the structure in three-dimensional space (three displacement
and three rotational modes). We start considering the seventh and subsequent eigen-modes of thestructure.

configuration I at 95% extension. (b) The stiffness of different tube configurations in the three Cartesian directions with respect tothe extension. (c) The tube stiffness for different loading directions in the Y − Z plane represented as a radial plot. The tubes
are at an extension of 95%.

In figure 13 we present a cantilever analysis of the eight-sided tube in different configurations.
One end of the cantilever is fixed and a small uniformly distributed load (summing to a totalof 0.001 N, e.g. FX = 0.001 N) is applied on the other end. We perform static, linear elastic,
small displacement analyses of the structures, with the main objective of exploring the globalbahaviours and anisotropy of the tubes. The system displacements (∆X , ∆Y , ∆Z ) are calculatedusing the equation F = K∆, where F is a vector of forces. Subsequently, the system stiffness
is calculated as KX = FX /δX , where δX is the mean X direction displacement of the loaded
nodes. A squeezing type deformation occurs for some of the loaded cases, and this is believed toresult in lower stiffness than if the origami was engaged in stretching and shearing. Different
cross-section configurations, can have drastically varying stiffness characteristics, with up to
an order of magnitude between different cross-sections (figure 13b). Typically, configurations I
and V are the stiffest while configurations II and IV are the most flexible. We also show thestiffness perpendicular to the X axis, as a radial plot in figure 13c. The I and V configurations
have large oval plots, meaning they have relatively higher stiffness in most directions. Each...............................................................

Figure 13. Structural cantilever analyses of eight-sided tube. (a) Representative deformed shapes scaled ×1000 for the tube in

13

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but is substantially stiffer for other deformations. The ninth mode of the structure is anothermanifestation of squeezing with the centre unfolding and the ends folding . The tenth mode is alocalized mode, where the panels at the end of the tube fold. The ninth and tenth eigenvalues aresubstantially higher, meaning the structure is stiffer for these and other types of deformations.
Because the geometry of the system changes, the magnitudes of the eigenvalues also changewith respect to the extension of the system. Extension here is defined as a percentage of the fully
extended length. When the structure is at 0% extension it is completely folded down, while at
a 100% extension the switches flatten and the system can be reconfigured. The eigenvalues forrigid folding and squeezing remain essentially the same regardless of the folded configuration,although there are some small differences in magnitude. However, the ninth and tenth modeare greatly affected by the different folding configurations (figure 12c). This is because the crosssectional geometry has a higher influence in determining the more complex localized and global
bending modes.

We explore a uniform circular pipe (made from a thin sheet) experiencing uniform out-of-planeloading and compare it with similar origami tubes. Figure 14a shows the pipe that is L = 10 cm
long, loaded in space with symmetric out-of-plane distributed load equal to F/L. In this section,
we use a total force of F = 0.001 N, and we assume linear elastic, small displacement behaviours.
The radius of the pipe is r = 2 cm, and all other parameters (i.e. t and E) are the same as for theorigami analysis in Section 6. An analytical solution for this problem is found using Castigliano’s
theorem where the pipe is simplified to a two dimensional bending of a thin beam (Appendix B).
The total diametric deflection (δd ), coaxial with the applied load, is found to be
2 12F r3π−
= 0.00286 cm.
(7.1)
δd =
4
π ELt3Subsequently, we preform similar analyses on the origami tubes with the same parameters,
and dimensions defined to match the pipe as closely as possible. All cross-sectional edge lengths
are defined as 2πr/NEdge where NEdge are the total number of edges on the circular tube. As
such, the tube perimeter is the same as the analytical case. The edges are arranged in a symmetricfashion so that the cross-section becomes a regular polygon (figure 14b). Three cases with NEdge =6, 10, and 14 are used, such that there is a single flat segment in the initial configuration, meaningthat the initial configuration is the fully deployed state. The number of panels in the X direction
is chosen as 6, 8 and 12 for the three cases respectively, so that the structure is symmetric and thepanels are approximately square. The projection angle defining the three dimensional shape isvaried, and a consistent projection length is used so that the origami tube is L = 10 cm long in the
fully deployed (same as initial) configuration. We perform a static analysis by loading the vertices
on the top flat segment with a downward force, such that the edge vertices carry half the load of
the internal vertices (grey versus black triangles in figure 14b). The loads are defined such that the
total applied load sums to F = 0.001 N. The bottom vertices of the tubes are restrained in the Z
direction, representing a symmetric loading similar to figure 14a.
We use static, linear elastic, small displacement analyses to evaluate the mechanical propertiesof the origami tubes. Scaled deformed shapes of NEdge = 10 tubes with two different projection
angles are shown in figure 14c-d. The tube with φ = 65◦ is much stiffer and has an irregulardeformed shape where panels bend and stretch. The tube with φ = 85◦ has a more regulardeformed shape, similar to what we would expect from a thin pipe, and in this case, deformation
occurs primarily by bending along the longitudinal fold lines. Stiffness with respect to theprojection angle φ, of the tubes with different NEdge is shown in figure 14e. The origami
stiffness is calculated as in Section 6, and the analytical stiffness solution for the circular pipe iscalculated as F/δd . Similar to the deformed shapes, tubes with lower projection angles have lower
displacement and are stiffer, while tubes with a projection angle close to 90◦ are more flexible
because they permit folding along the longitudinally oriented fold lines. The origami tubes withprojection angles between φ = 45◦ − 75◦ are stiffer than the analytical solution for a circular pipe.
This behaviour is similar to that of corrugated pipes and sheets [42]. Corrugated pipes have ahigher stiffness for out-of-plane loadings, which makes them suitable for many applications suchas culverts. Tubes with more edges e.g. NEdge = 14 have more fold lines along their cross-section
perimeter, making them more flexible. The results are verified with physical models (figure 14f-h).
The stiffness of the fold lines RF P factor, does not influence the deflection significantly for cases...............................................................

7. Cylindrical origami tubes

14rspa.royalsocietypublishing.org Proc R Soc A 0000000

of the cross-sections also has a different direction (in Y − Z) where it has a lower or higher
stiffness. This phenomena indicates that the reconfigurable tubes have a highly adjustableanisotropy when used as cantilevers. The behaviours observed in this section show that the crosssection geometry can have a significant influence on the mechanical properties of the system.
Thus, the reconfigurable polygonal tubes can be used to create highly tunable and adaptive
structural systems. Detailed research is needed in this area to determine the influence of different
cross-section geometries, as well as the tunability achieved from each reconfiguration.

15...............................................................

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Figure 14. Out-of-plane compression on a pipe. (a) Problem definition and analytical approximations (Appendix B). (b) Origami
tubes with NEdge = 6, 10, and 14. The origami tube cross-sections are overlayed with a r = 2 cm circle. The loading isonly shown for the NEdge = 6 tube. (c) A φ = 65◦ tube, and (d) a φ = 85◦ tube with NEdge = 10. The top (X-Y) view isshown as a reference and the lower views show the deformed shapes. The deformed shapes are scaled ×10000 for the stiffer
φ = 65◦ tube and ×200 for the more flexible φ = 85◦ tube. (e) The out-of-plane stiffness of tubes versus the projection angle
φ. (f-h) Physical models of a uniform sheet, φ = 65◦ , and φ = 85◦ tubes respectively, loaded out-of-plane with 400 grams. The
φ = 85◦ tube is only loaded with one 100 gram weight due to the much larger deformation.

with lower projection angle φ < 75◦ . However, for higher φ the fold lines are the primary locationof deflections, and thus their stiffness greatly affects the tube stiffness.

8. Concluding remarks
We introduce a new category of origami tubes that have reconfigurable polygonal cross-sections.
The tubes are rigid and flat foldable, and have a continuous perimeter. The cross-sections of thetubes can be a wide variety of convex or non-convex polygonal shapes that follow translational
symmetry. Projection is used to define the three dimensional shape of the tube, but non-admissible(e.g. non-symmetric) projections, may limit the flat and rigid foldability of the system. The crosssection geometry can contain any number of n switches that can be used like binary bits toCompeting interests statement. We have no competing interests.
Authors’ contributions. ETF, GHP and TT designed the research, conceived the geometric designs andcomputational models, interpreted the results, and wrote the paper. All authors gave final approval for
publication.
Funding statement. This research was partially supported by the National Science Foundation (NSF) grant
CMMI 1538830. The authors also acknowledge support from the NSF Graduate Research Fellowship; Japan
Society for the Promotion of Science Fellowship; Raymond Allen Jones Chair at the Georgia Institute ofTechnology; and Japan Science and Technology Agency Presto program.
Ethics statement. This work did not involve any human data.

Appendix A. Foldability of origami tubes
In this appendix we verify the developability, flat foldability and initial rigid foldability using theapproach introduced by Tachi [43]. We assume that the origami panels have an infinitesimallysmall or zero-thickness to satisfy the mathematical definitions. The origami tubes defined by
Sections 2 and 3 contain a total number of nvert internal vertices where four fold lines meet, anda number of npanel four-sided panels. The folding characteristics of the origami can be exploredby performing the following vector calculations for the vertices and panels:
"
#
4
X
dev
αk,i
= 0,
(8.1)
c
= 2π −
k=1

cflat =

" 4
X

nvert ×1#(−1)k αk,i

k=1

= 0,

(8.2)

nvert ×1

 
cplanar = ρj npanel ×1 = 0,

(8.3)

where αk,i represents the k-th vertex angle in the i-th vertex, and ρj represents the dihedral angle
between the normals of two triangles that together create the j-th panel of the tube. When cdev = 0
for all vertices, then the origami is developable, meaning it can be created from a single flat pieceof material. The origami tubes presented here have mostly non-developable vertices, and thus
they cannot be folded from a single flat piece of material. However, some of the vertices may be
developable and thus a portion of the tube may be constructed from an initially flat sheet (e.g.

...............................................................

Data accessibility statement. The work does not have any experimental data.

16

rspa.royalsocietypublishing.org Proc R Soc A 0000000

program the geometric reconfiguration of the cross-section. We show that the total number ofpossible cross-section variations for a tube follow the central binomial coefficient of n. A cellularcross-section or coupling of multiple tubes can be used to create a new variety of assemblages
that enhance the functionality and reconfigurable properties of the tubes.
In addition to the geometric variations and reconfigurable kinematics, this paper also explores
some mechanical properties of the polygonal tubes. We show that the tubes have only one
flexible mode for kinematic deployment for which the stiffness is not significantly influenced byreconfiguring the cross-section. On the other hand, the cross-section configuration can influenceother deformation modes and the out-of-plane stiffness of the tubes. This property can be used
to make tunable structures that can change their mechanical properties. If the origami tubes
are used as circular pipes, they can be designed to have a high out-of-plane stiffness similar
to that of corrugated pipes. An appendix proposes future research directions on applications,fabrication, and non-linear deformations, all of which will enhance the practicality, functionality
and capability of the reconfigurable tubes. We envision that the physical attributes, versatility,and programmable characteristics of the polygonal origami tubes will enable solutions of varying
scale in science and engineering.

(a) Constant angle projection in X − Y . (b) Projection in X − Y with symmetry enforced. (c) Projection in X − Y without
preserving symmetry. (d) Simultaneous projection in X − Y − Z . Low eigenvalues correspond to a soft, rigid folding mode of
the origami.

(8.3) indicates that all panels are planar or flat for a given configuration. The dihedral angle (ρj )can be calculated using the four nodes on the corners of the panel, and will always equal 0 at theinitial projected configurations defined using Section 3. Thus all tubes satisfy cplanar = 0, however,this is only a necessary condition for rigid foldability and is not sufficient. For rigid foldability,
folding along fold lines should permit the structure to transition between states while cplanar = 0
is continuously satisfied. Analytical derivations of the kinematics and geometric characteristics
of foldability (including rigid foldability) have been previously discussed [34–37], however thesetend to be cumbersome for verifying the rigid foldability of complex origami systems. A morestraightforward method to check rigid foldability, is to perform the eigenvalue analyses describedin Section 6 with the fold stiffness (Kρ ) substantially reduced (e.g. to 10−7 ) representing fold lines
with no stiffness. In these analyses, the seventh and possibly higher eigenvalues will be near zero,
indicating a rigid folding motion where a kinematic transition is permitted by folding along thefold lines. Figure 15 shows the eigenvalues and eigen-modes for the basic origami assemblies
studied in this paper. All cases except symmetric X − Y − Z projection have a λ7 that is low(≈ 10−2 ), indicating a rigid folding motion. For the X − Y − Z projection case, λ7 is of muchhigher order indicating that bending of the panels must occur to deform the structure, and thatthe tube does not have a rigid folding mode. For the structures in figure 15a-c, λ8 is substantiallyhigher than λ7 indicating that only one rigid folding motion exists, these systems can be classified
as one degree of freedom for rigid folding.

...............................................................

Figure 15. Schematic (top row), seventh mode (middle row) , and eighth mode (bottom row), of basic projection definitions.

17rspa.royalsocietypublishing.org Proc R Soc A 0000000

the single four-sided tube can be constructed from two flat sheets [28]). When cflat = 0, then allvertices of the origami are flat foldable meaning that, they can fold down to a flat 2 dimensional
state. The definitions in Section 3(a) and 3(c) intentionally ensure symmetry when preforminga projection of the cross-section, thus they ensure that all vertices are flat foldable. However, in
Section 3(b) where symmetry is not preserved, we loose the flat foldability (cflat 6= 0). Equation

18

IV and is symmetric to V; and Mode nine corresponds to configuration VI and is symmetric to III. Mode ten is the squeezing mode.

In figure 16 we show the eigenvalue and eigen-modes of the eight-sided tube with two
switches (n = 2). Curiously, for this case, the system has three soft modes where the rigid folding
can occur, and the tenth mode is the first to engage the origami panels in bending. These rigid
folding modes each correspond to one of the system configurations shown in figure 9, and eachone of them also has a symmetric inverse that corresponds to another system configuration.
These results indicate that the system has three non-symmetric degrees of freedom for rigid
folding. However, once the structure enters one of the folding configurations (extension < 100%),
it behaves like a one degree of freedom system, where it only has a single flexible mode for rigid
folding (figure 12). This phenomena of the eight-sided tube is similar to a flat sheet that canenter numerous different folding patterns when initially folded. Future, research could investigate
differences in rigid folding configurations, the symmetric inverse eigen-modes, and the varying
programmability possible with the polygonal tubes.

Appendix B. Analytical solution for a pipe loaded out-of-plane
The exact analytical solution for the out-of-plane bending of a pipe can be calculated using
Castigliano’s theorem where we simplify the problem to a two dimensional bending of a thincurved beam. The theorem states that the displacement δq at the point where a load Q is applied
can be found byZl
Mx ∂Mx
∂U
=
dx,
(8.4)
δq =
∂Q
0 EI ∂Q
where U is the elastic strain energy, Mx is the bending moment, I is the area moment of inertia,
and x is the distance along the beam. By using symmetry, we only consider a quadrant of thepipe’s cross-section which is loaded with a force V = F/2 (figure 14a). The idealized thin beam
has a width equal to the length of the pipe L (X direction), and a depth of t in the bending axis
(perpendicular to X), resulting in the area moment of inertia I = Lt3 /12. A point along the beam
is defined as a function of the angle θ, and the bending moment (Mθ ) and the partial derivatives
are calculated asMθ = V r sin θ − M0 ,∂Mθ= r sin θ,
∂V

∂Mθ
= −1.
∂M0

Using the theorem we can now calculate
Zl
Z
π
 r
Mx ∂Mx
1 π/2
δM0 =
dx =
(V r sin θ − M0 ) ∗ (−1) ∗ rdθ =M0 − V r
,
EI 0
2
EI
0 EI ∂M0
Zl
Z
π
 r2
Mx ∂Mx
1 π/2
δV =
dx =
(V r sin θ − M0 ) ∗ r sin θ ∗ rdθ =V r − M0
.
EI 0
4
EI
0 EI ∂V

(8.5)

(8.6)
(8.7)

By enforcing symmetry, the rotation at the unrestrained end of the beam will be M0 = 0, and using
equation (8.6) we find that M0 = 2V r/π. Substituting M0 into equation (8.7), the total diametric...............................................................

seven corresponds to configuration II and is the symmetric inverse to configuration I; Mode eight is corresponds to configurationrspa.royalsocietypublishing.org Proc R Soc A 0000000

Figure 16. The seventh to tenth eigenvalues and eigen-modes of the eight-sided tube when it is at a fully extended state. Mode

19

If we wish to find the total diametric deflection perpendicular with the applied load, we can use
a fictitious load H applied horizontally at the free end of the curved beam, and use the same
methodology to find




2
2
1 V r3
1 12F r3
−
=
−
.
(8.9)
2δH = 2π
2 EI
π
2 ELt3

Appendix C. Practical considerations and future extensions forreconfigurable origami tubes
In this appendix we propose future research on the reconfigurable tubes to explore (a) practicalapplications, (b) considerations for physical fabrication, and (c) non-linear behaviours that can
extended capabilities. This section is meant to inform and motivate future research, rather than toprovide a holistic discussion on the different topics.

(a) Practical applications
The polygonal cross-section origami tubes discussed in this paper open up a variety of
applications in science and engineering. The continuous perimeter of the cross-sections could
enable the tubes to be used in fluid flow applications. More traditional applications would involve
primarily using these tubular origami as deployable pipe-like [4,15,16] or bellow systems [17,18].
These could have wide and varied applications including deployable pipes for construction,
biomedical devices, or inflatable space structure components. The new projection definitions
introduced in Section 3, provide a new capability where the origami tubes can follow a curved
profile when deployed, versus the straight profile of previously introduced tubes. An instancetaking advantage of this benefit, would be constructing a ventilation system, where the entire
origami tube is deployed to carry air through a congested area, rather than connecting multiple
straight and curved pipe segments. The properties studied in Section 7, show added benefits
where the polygonal tubes have more stiffness for out-of-plane loading, than a conventional
pipe with a constant cross-section. This property could allow for the deployable constructionof culverts, or other pipes that need to carry large loads.

Figure 17. Potential application of origami tubes used as a robotic arm with reconfigurable components. The cross-section shownon the bottom reduces in area and could be used as a gripper when the tube is retracting.

The programmable capability of the tube cross-sections offers novel applications where thestructure can morph and adapt. The tubes can have an adaptable volume, surface properties,...............................................................

(8.8)

rspa.royalsocietypublishing.org Proc R Soc A 0000000

deflection coaxial with the applied load is found to be




π
π
2 V r3
2 12F r3
2δV = 2
−
=
−
.
4
π EI
4
π ELt3

Figure 18. (a) Strips of panels cut out from flat sheets can be used to construct the three dimensional, non-developable tube.
Dashed lines indicate fold lines, and the tabs at the sides of the sheets can be used to attach sheets together. (b) Physical modelof a six-sided polygonal tube that forms a star when fully deployed (c) Physical model of the reconfigurable origami from figure 1
is shown in different configurations. The tabs for attachment are visible on the bottom.

There is currently a tremendous amount of research aimed at making origami feasible forreal world applications. The geometric origami design, fabrication methods, materials, anddeployment mechanisms, all depend on the scale and function of the origami system. For small
applications, origami can be 3D printed with living hinges [44]. More simply however, it ispossible to cut out the origami from a flat sheet and fold the system along perforated or etchedfold lines. As a proof of concept, we have fabricated several small (≈ 30 cm) paper models (figures
14, 18, and 19) to highlight the capabilities of the reconfigurable polygonal tubes. All models are
manufactured from 160 g/m2 paper that has an approximate thickness of 0.25 mm. Panel heights
and widths vary from 1 to 3 cm, thus maintaining a relatively high length/thickness ratio thatis typical for origami. The folds are created by perforating the paper with 0.5 mm cuts spaced
evenly at 1 mm. Because the tubes are not developable, we cut out a flat sheet for each of thecross-section edges, and use tabs to adhere the multiple sheets together (figure 18a). This or a...............................................................

(b) Design and fabrication20

rspa.royalsocietypublishing.org Proc R Soc A 0000000

mechanical characteristics and more, simply through reconfiguring the polygonal cross-section.
For example, components placed inside aircraft wing could be used to change the lift and dragproperties of the wing for different stages of flight [6]. The variable stiffness properties of theorigami tubes discussed in Section 6 could allow for new devices in aerospace, mechanical, andcivil engineering. Robotic components, such as the deployable and reconfigurable arm in figure
17 could be designed to simultaneously fulfil multiple functions. A griper can be used with thereconfigurable cross-section, while the cellular divisions could add stiffness and carry electricalwiring, pneumatic tubes, or other utilities (e.g. similar to multi-functional dental tools). Althoughthese applications are still far from reality, they offer many potential advancements from current
day engineering approaches.

Figure 19. Localized distortion in the six-sided origami tube (left) can bring about new non-linear behaviours similar to those of
bendable drinking straws (right). (a) Unfolding of the structures in the prescribed straight direction. (b) A single transition point
indicated by a T is introduced in the origami tube. At this point a panel of the reconfigurable segment bends across its diagonal,
allowing for a change in configuration to occur in the middle of the tube. (c) Multiple transition points lead to a global curvature
over the length of the tube.

Most research on origami, as well as the body of this paper, take advantage of only the rigid
and prescribed folding mechanisms of the system. However, some recent findings have shownthat there exists a wide range of origami deformations where bending in the panels is encouraged[12,46]. These deformations could be substantially more complex than the rigid kinematics, andcould correspond to highly non-linear behaviours of the thin sheet origami. In figure 19b we show
localized bending that occurs on one of the switch panels of a polygonal tube with six edges. This
allows the tube to have different cross-section configurations at different locations of the tube, i.e.
Configuration I: below the transition point T and Configuration II: above it. The tube is initiallyconstructed straight with 30 constant angle projections, but with the transition point there is a shift
in the direction that the tube follows. Although each transition points causes a localized change...............................................................

(c) Non-linear deformations and extensions

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similar methodology would need to be used for manufacturing the polygonal origami out offlat sheets. When extending origami to the medium scales it is possible to use layered composites
where a flexible sheet that allows folding is sandwiched between more rigid panels [2,8,45]. Large
origami structures could be constructed by using thickened panels interconnected by hinges
rather than fold lines. For various applications in the real world the finite thickness of origami
sheets begins to affect the system behaviour, and the idealized zero-thickness assumptions are
no longer valid. Current research aims to account for thickness in kinematics and manufacturing
in order to prevent self-intersection while minimizing the size of the stowed structure [30–32].
To make the reconfigurable polygonal tubes reliable and cost effective for industrial applications
more innovation will still be needed. In particular, research should explore: materials and systems
to allow multiple folding/unfolding cycles; rapid fabrication methods; mechanisms to facilitate
deployment; and incorporating thickness into the tube design. The programmable switches of thepolygonal tubes may also require new methods for rapid or remote actuation and reconfiguration.

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