Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures
28 September - 2 October 2009, Universidad Politecnica de Valencia, Spain
Alberto DOMINGO and Carlos LAZARO (eds.)

One-DOF Cylindrical Deployable Structures with
Rigid Quadrilateral Panels
Tomohiro TACHI** The University of Tokyo
7-3-1 Hongo , Bunkyo-ku, Tokyo 113-8656, Japanttachi@siggraph.org

Abstract
In this paper, we present a novel cylindrical deployable structure and variations of its design
with the following characteristics:
1. Flat-foldable: The shape flattens into a compact 2D configuration.
2. Rigid-foldable: Each element does not deform throughout the transformation.
3. One-DOF: The mechanism has exactly one degree of freedom.
4. Thick: Facets can be substituted with thick or multilayered panels without introducing
the distortion of elements.

Keywords: origami, deployable structure, flat-foldable, rigid-foldable, isometric transformation

1 Introduction
Cylindrical deployable and collapsible structures composed of two-dimensional elements areuseful in various designs. Such a structure can form a watertight surface that encloses acertain desired volume by the addition of two surfaces at the ends, and the surfaces can becompactly folded down into a two-dimensional state. Several flat-foldable cylindrical deployable structures have been proposed thus far, e.g., by Hoberman [3], Guest and Pellegrino[1], Sogame and Furuya [7], and Nojima [6]. However, all of these existing deployable
tubes are bistable structures whose transformation mechanisms rely on the in-plane elasticdeformation. Since the mechanism of such a structure relies on the material flexibility, theapplications were limited: e.g., an energy absorption device that can be used only once, asproposed by Wu et al. [8], a small-scale medical device by Kuribayashi et al. [4], a deployable membrane structure for use in space proposed by Sogame and Furuya [7].
In this paper, we propose cylindrical deployable structures in which every element of the surface is geometrically free of distortion. This enables the construction of mechanisms usingProceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

stiff materials, and such mechanisms can potentially be applied to designing repeatedly foldable architectural or human-scale structures under gravity. We first introduce the basic unit
structure, and then, we generalize it to obtain isotropic and anisotropic types of structures,
both of which produce one-DOF rigid motions. Furthermore, we present methods for realizing an isotropic type of generalized mechanism with thick panels and piano hinges and forrealizing an anisotropic mechanism with multilayered thin surfaces.

2 Basic Geometry
The basic unit structure is constructed by joining two pieces: a single vertex origami withfour congruent parallelograms and its mirror image (with respect to the zx-plane), as shownin Figure 1(a). The same structure is also known in the field of artistic origami, where Thoki
Yenn designed a model “Flip Flop” by combining two units [9]. The vertex produces a oneDOF transformation mechanism in which the edges on the plane of reflection (zx-plane)
always lie on the plane. This enables the formation of a valid joint structure from two pieces.
The rhombus at each end, which we call the “section rhombus,” is maintained co-planar(parallel to xy-plane) throughout the transformation. This enables the repetition of the units
in the axial direction (z direction), to construct a cylinder of arbitrary length (Figure 2).
φξℓℓ02φ0=π
φ0=7/8π

φ02

yxπ

0.5π

φ0=1/2πazΟ(a) Mechanism of the unit. Note that four edges on the plane ofreflection (dotted lines) are kept on the plane while the section
rhombus (solid lines) is kept coplanar.

0.5ℓ0ℓ0ℓ(b) The relation between ℓ and ϕ

Figure 1: Motion of a single unit.

The structure transforms from the completely unfolded state to the flat-folded state. In theflat-folded state, the distance between two adjacent units is zero. This means that a long
cylindrical structure with multiple units can be compactly folded in the axial direction. We
will now investigate the relationship between the folding motion in the axial direction andthe deformation of the section rhombus. Let ξ and ϕ denote the dihedral angle betweenfacets incident to the edge of the section rhombus (all four dihedral angles are equal) and theexternal angle of the top vertex of the section rhombus, respectively, as shown in Figure 1(a).

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

These two angles are related as follows:
tanϕ
ϕ0
ξ
cot= cos ,222(1)

where ϕ0 (0 < ϕ0 < π) is the angle ϕ when the model is flat-folded. The distance ℓ betweenadjacent units is written in terms of ϕ as√
ϕϕ0ℓ = 1 − tan2 cot2
ℓ0 ,(2)22
where ℓ0 is the distance in the unfolded state. As shown in Figure 1(b), ϕ0 is the key parameter that controls the relation between two motions. When ϕ0 is close to π a strongly
non-linear motion approximated by a series of two separate motions is produced. We assumed that ϕ0 ̸= π, because for ϕ0 = π, the folding motions of the units are completelyindependent of each other.
The unit occupies a volume of a2 ℓ sin ϕ, where a indicates the length of each side of therhombus. Theand flat-folded states, and it attains the max√volume is zero in the unfolded
1+cos ϕ012imum value. The maximum volume increases as ϕ0ℓaatcos
ϕ=0ϕ
2 021+2 cot

2approaches π.

Figure 2: A rigid-foldable cylindrical structure with ϕ0 close to π (ϕ0 = 168◦ ).

3 Generalizations
Rigid-foldability of the proposed structure can be interpreted in two different ways, each ofwhich results in a different method of generalization: “isotropic generalization” or “anisotropicProceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structuresgeneralization.”
3.1 Isotropic Generalization
The first interpretation is that the unit is composed of mechanical joints, which we call “folds”
and “elbows.” The joints are constructed by reflecting a two-facet strip with a V-shaped
section with respect to a plane perpendicular to the bisector plane of the dihedral angle;
the fold is constructed by ray reflection, while the elbow is constructed by image reflection(Figure 3(a)). We call these vertices isotropic because in these special cases of vertices, theabsolute values of dihedral angles for incidence on opposite edges are equal.
We model the unit of a cylinder by connecting these joints to form a closed linkage. The
bisector planes of dihedral angles are set on the common section plane. The polygonal linkagechanges its shape within the plane according to the common dihedral angle ξ. This generally
prevents a unit from being rigid-foldable, i.e., changing the ξ of a cylindrical shape can resultin its edges tearing, as shown in Figure 4. Our goal is to ensure that the loops are always validfor any value of ξ.
ξφφφφ

Foldπ

π

φ0φ0Elbow

ξFoldElbow
ξ(a) Fold and Elbow

ξ

Ο

0.5ππ

ξΟ

0.5ππ

ξ

(b) Relationship between ξ and ϕ

Figure 3: Fold and Elbow

Figure 4: Unfolding (Left) or folding (Right) results in breaking the valid loop.
A joint enables a one-DOF mechanism that converts motion in the axial direction to motionin the section plane, which are represented by angles ξ and ϕ, respectively (Figure 3(a)). In

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

the same manner as Equation (1), the conversion from ξ to ϕ can be determined in terms of aparameter ϕ0 , the angle ϕ in the flat-folded state (Figure 3(b)).
)(ϕ0ξfold(3)ϕ (ϕ0 , ξ) = 2 arctan cos tan
22{()
2 arctan sec 2ξ tan ϕ20if ξ ̸= πϕelbow (ϕ0 , ξ) =(4)πif ξ = π.
We can construct a valid loop by first constructing a half-loop linkage whose end edges
remain parallel for any ξ and then connecting the linkage with its copy rotated by 180 degrees.
The half loop must satisfy the following equation for any ξ (0 ≤ ξ ≤ π):∑∑
ϕelbow (ϕ0 , ξ) = π.
(5)ϕfold (ϕ0 , ξ) +all elbows

all folds

In order to obtain such a half-loop, we use the following three combinations that make theirsums constant:ϕfold (ϕ0 , ξ) + ϕfold (−ϕ0 , ξ) ≡ 0,(6)ϕelbow (ϕ0 , ξ) + ϕelbow (−ϕ0 , ξ) ≡ 0,(7)(π − ϕ0 , ξ) ≡ π.

(8)

ϕfoldelbow(ϕ0 , ξ) + ϕ

These three pairs can be seen as the fundamental pairs of joints required to construct a validlinkage. Connecting these pairs of joints in an arbitrary ordering such that the sum of the angles is π results in a valid half-loop linkage. The sequence in the basic structure is constructedusing Equation (8) and can be represented as follows:f(ϕ0 )e(π − ϕ0 ),where e(ϕ0 ) and f(ϕ0 ) represent an elbow and a fold, respectively. Figure 5 shows an exampleof the isotropic generalization realized by using this method. The sequence of joints in thisexample (half-loop from the bottom facet to the top facet in the counterclockwise direction)is,f(7π3π 3π
3π 3π π7π 3π
3π 3π
3π 7π)f(− )f( )f(− )f( )e( )f(− )f( )f(− )f( )f(− )f( ).
8444488444483π
In this example, the substructures based on Miura-ori, f(− 3π4 )f( 4 ), are inserted in order tomake the model expand in the radial directions as well as in the axial direction.

3.2 Anisotropic Generalization
The proposed structure is a collection of parallelogram strips —in each strip, it is ensured
that the edges between facets are parallel. We can construct a general valid cylindrical structure based on the parallelogram strips in the following manner. First, draw an arbitrary closed
polyline with two-fold rotational symmetry, i.e., a zonogon, on the section plane; the zonogon
represents a shape of the section. We call this zonogon a “section zonogon.” Then, extrude

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

Figure 5: An isotropic generalization with radial expansive motion.
the section zonogon along an arbitrary vector at an angle of ξ20 to the section plane in orderto from a closed parallelogram strip. Without loss of generality, we assume that this vector,
which we call the(“extrusion vector,” is)parallel to the zx plane; the extrusion vector is represented as l0 = ℓ cos ξ20 , 0, −ℓ sin ξ20 , where ℓ is the length of the vector. Next, connect
the strip to its mirror image with respect to the extruded section plane. Thus, we obtain thesingle unit.
The valid motion of the structure is produced as follows. We pick up half of the parallelogram strip and fold it along its edges. It is ensured that the edges, i.e., the extruded vertices,remain parallel after( the folding. The directionof the edges is represented as a rotated ex)
trusion vector l = ℓ cos 2ξ , 0, −ℓ sin 2ξ . On the other hand, the section zonogon must lie
on the section plane throughout the transformation. These constraints determine the orientation of each facet, and thus, they determine the orientation of each side of the section
zonogon, independently with respect to the rotation of adjacent facets. Assume that a sideof the section zonogon forms an angle θ with the x axis. Thus, it is represented as a vector r = (r cos θ, r sin θ, 0), where r is the length of the segment (Figure 6). The vector
r is rotated such that the angle between the edge and the extrusion vector is constant, i.e.,l · r = const. Hence, we obtain the following relation:cos θ =cos ξ20
cos 2ξ

cos θ0 ,where θ0 = θ|ξ=ξ0 . The sign of θ is determined by the sign of θ0 . If θ0 = 0, the facet canbe rotated in both directions. Now, we have obtained the folded half-strip. The other half ofProceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

the strip is folded in the same manner and the corresponding end edges of the half strips meet
because they are parallel and equal in length.

sectionzonogo
nr

yzr0x

θ

θ0

ξ0−
ξ−2 2Figure 6: Rotation of a parallelogram.
The generalized structure is collapsible: the flat-folded state corresponds to the case where
the half-strip is completely flattened. However, unlike the isotropic generalized structures,the completely unfolded state does not commonly exist. This is because the extrusion ofan arbitrary polyline produces anisotropic vertices, in which the two dihedral angles of theopposite edges are not equal. The unfolding motion from the flat-folded state stops when anyof the sides become horizontal i.e., cos θ = 1. Figure 7 shows an example of the generalizedform.

4 Realization of Thick Surface
For the structure described in the previous sections, it was assumed that the thickness of thematerial is zero. However, in a real engineering application, the surface must be realizedusing a material with non-negligible thickness. In particular, in an architectural context, itis necessary to design structures with rigid panels or multilayered surfaces with a desired
thickness. We propose two methods to realize the desired thickness without the loss of rigidfoldability: a method in which facets and foldlines are replaced with thick panels and piano
hinges, respectively, and a method that enables thick multilayed surfaces composed of surfaces of negligible thickness. The former can be applied to the isotropic generalization andthe latter to the anisotropic generalization.
4.1 Thick Panels
The basic structure (Section 2) and its isotropic generalization (Section 3.1) can be realized
by using thick panels and rotating hinges (Figure 8). In order to avoid the interference thatoccurs when folding thick panels, we shift each axis of rotation to the side of the mountainfold, as shown in Figure 9(a). The structures of the vertices are changed in order to accountfor the point of intersection of the shifted axes, which are no longer coincident. In general,a vertex with non-concurrent incident axes yields 6 constraints, as opposed to a concurrent

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

Figure 7: Folding motion in an example of anisotropic generalization. Only the top facets areflat in the most unfolded state (Left).
origami vertex with 3 constraints; this means that a general degree-4 vertex can become anoverconstrained structure.
In order to realize a valid mechanism without overconstraints, we utilize special rigid body
joints that replace the folds and the elbows. There are three types of mechanisms, as shownin Figure 9(b): one for the fold and two for the concave and convex elbows. A fold joint isreplaced by panels with two thicknesses; in order to achieve valid motion, the thickness of theconnection part in the fold joint is set as a half of the thickness in the other parts. The sameidea is used by Hoberman [2], who realizes deployment mechanisms based on thick origami.
A concave elbow (an elbow with a negative gauss area) and a convex elbow (an elbow with apositive gauss area) can be replaced with panels of constant thickness.
4.2 Multilayering
An anisotropic structure based on parallelogram strips can be realized using multilayered
surfaces. As discussed in Section 3.2, each edge of the zonogon is uniquely assigned avalid orientation for a given folding angle 2ξ of the extrusion vector. This enables a connectedcylindrical structure extruded from multiple zonogons sharing a vertex and edges to transform
without being separated. Hence, a composite structure of rigid-foldable and flat-foldable
multilayered cylindrical structures is obtained by tessellating the original section zonogon
with multiple zonogonal tiles (Figure 10). We can control the composition of the layers bychanging the size of tiles or by subdividing the parallelograms.
This multilayering method also yields various designs with arbitrary composite sections. Figure 11 shows an example of a flat-foldable and rigid-foldable multilayered single curved sur-

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

Figure 8: Folding motion of the structure with thick panels. The axes of rotation lie on theedges of the thick panels.

(a) The shifting of theaxes.

(b) Rigid body structure with non-concurrent axes replacing the joints. Left:
Fold joint. Middle: Concave elbow. Right: Convex elbow.

Figure 9: Realization with thick panels and hinges.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

Figure 10: Multilayered cylinders. The section is composed of zonogons.

Figure 11: The folding motion of a multilayered surface along a given two-dimensional curve.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia
Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures

face. The structure is automatically constructed such that the section is formed by connected
parallelograms along a given two-dimensional curve. In order to make the flat-folded shapecompact, we use an evenly spaced zigzag polyline. We have implemented a parametric designof this type of structure using Grasshopper [5].

5 Conclusion
We have presented two types of general flat-foldable cylindrical deployable structures composed of rigid quadrilateral panels. The isotropic generalization is based on combinations ofappropriate joint structures, and the anisotropic generalization is based on the folding motionof parallelogram strips. We have shown that isotropic general structures can be realized byusing thick panels and hinges and anisotropic structures can be realized by using multilayered
surfaces.

Acknowledgement
This research is supported by a Grant-in-Aid for JSPS Fellows funded by Japan Society for
the Promotion of Science.

References[1] Guest, S. D. and Pellegrino, S., The folding of triangulated cylinders, Part I: Geometricconsiderations, ASME Journal of Applied Mechanics, 1994, Vol.61, 773–777[2] Hoberman, C., Reversibly expandable three-dimensional structure. United States Patent
No. 4,780,344, 1988
[3] Hoberman, C., Curved pleated sheet structures. United States Patent No. 5,234,727, 1993
[4] Kuribayashi, K., Tsuchiya, K., You, Z., Tomus, D., Umemoto, M., Ito, T. and Sasaki,
M., Self-deployable origami stent grafts as a biomedical application of ni-rich tini shapememory alloy foil, Materials Science and Engineering A, 2006, Vol.419, 131–137[5] McNeel, grasshopper - generative modeling for rhino, http://grasshopper.
rhino3d.com/[6] Nojima, T., Origami modeling of functional structures based on organic patterns,
2007, Presentation Manuscript at VIPSI Tokyo impact.kuaero.kyoto-u.ac.
jp/pdf/Origami.pdf[7] Sogame, A. and Furuya, H., Conceptual study on cylindrical deployable space structures,
In IUTAM-IASS Symposium on Deployable Structures: Theory and Applications (2000),
pp. 383–392[8] Wu, Z., Hagiwara, I. and Tao, X., Optimisation of crush characteristics of the cylindricalorigami structure, International Journal of Vehicle Design, 2007, Vol.43, 66–81[9] Yenn, T., Flip flop, http://erikdemaine.org/thok/flipflop.html