Proceedings of the International Association for Shell and Spatial Structures (IASS)
Symposium 2015, Amsterdam
Future Visions
17 - 20 August 2015, Amsterdam, The Netherlands

Deployable Folded-core Sandwich Panels
Guided by a Generating Surface
Tomohiro TACHI*, Evgueni T. FILIPOVa, Glaucio H. PAULINOab* The University of Tokyo
3-8-1 Komaba, Meguro-Ku, Tokyo 153-8902
tachi@idea.c.u-tokyo.ac.jp
aUniversity of Illinois Urbana-ChampaignbGeorgia Institute of Technology

Abstract
We propose a composite panel that can be assembled in a planar state and can be transformed with aone-DOF kinematic mechanism after the assembly. The proposed structure is comprised of generalized
rigid-foldable tubes. The tubes are assembled such that they are non-trivially compatible with one
another, but still share a desired single-curved surface. Because of the nontrivial assembly, the structureis expected to be flexible only in the desired one-DOF motion, deploying from a flat state to a 3D state,while it is significantly stiffer against other motions. The geometric construction follows the following
procedure; (1) obtain an equivalent origami structure from a generating surface, (2) attach compatible
tubular assemblies on both sides of the surface. This method produces a wide range of rigidly-foldable
composite structures including corrugated surfaces with a flat-foldable compact state.
Keywords: origami, deployable structure, transformable structure, folded-core sandwich

1. Introduction
Origami is a technology that can be used to make lightweight stiff structures or deployable systems.
Folded-core and honeycomb-core sandwich panels are examples of former applications. Folded-corepanels are lightweight stiff composite structures that can be fabricated by folding and assembling thinsheets [1]. These core structures can also be fitted to a single curved surface by changing the folding orcutting patterns of the core [2]. These structures are assembled in a folded 3D state, and once assembled,
they become rigid and non-deformable. The latter type of application for deployable mechanisms canbe realized through a geometric system of rigid origami, where rigid panels are connected by rotational
hinges [3]. Rigid foldable origami tubes and cellular materials have been investigated previously [4,5,6].
Cellular rigid origami are expected to have meta-materialistic properties where the structure is flexible
in one-mode and stiffer in other modes [7].

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

The authors have introduced and explored coupling methods for rigid-foldable tubes [8], that can beused to create flexibly transformable but stiff, lightweight structures. In this paper, we generalize ideasof tube coupling to obtain a variety of designs for sandwich panels that can be deformed only in oneprescribed folding mode. First, we review the basic ideas of tube coupling and folded cores in Section
2. Then, we propose generalized construction process for rigid-foldable tubular assemblies in Sections
3 and 4.

2. Basic Structure
2.1 Corrugated Sandwich Systems
Figure 1 shows a rigid foldable Miura-ori tube, its aligned assembly [4] and the new assembly
introduced in [8]. The stiffness and flexibility of such origami structures can be examined througheigenvalue analysis [9]. In [8], we show a detailed investigation of the different assemblies, and weshow that the new coupling technique can increase the stiffness of the structure while still permitting
for flexible folding and unfolding. The different coupling methods can be repeatedly translated andconnected to form a corrugated sandwich surface (Figure 2). This structure can flat fold in both
directions and is expected to have improved structural properties when in a coupled form. The sandwich
surface consists of a corrugated surface in the middle, which we call generating surface. This surface
is offset in both directions, connected by corrugated walls, and thus forming tubes. The extrusion vector
defining the corrugated walls are different on the top and bottom sides. The prime objective of this study
is to obtain varieties of forms by changing the geometric parameters of the tubes.

Figure 1. Rigid foldable Miura-ori tube (Left) and its assemblies: aligned (Middle) and new (Right) [8].

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

Figure 2. A transformable corrugated sandwich surface. Red indicates the generating surface. In thedesign phase, we construct the top and bottom offset structures based on the given middle surface.
2.2 Problem Description
In this paper, we consider the following inverse problem: given a generating surface as a general singlycurved polyhedral surface, construct rigidly foldable sandwich assembly with different extrusion
directions on the top and bottom sides (Figure 3). So, we decide to explore the design variations using
the generating surface as the guiding geometry. Here, the generating surface is given by a quadrilateral
strip, which is not necessarily constrained to cylinders. From fine subdivision, we can naturally extend
the geometric construction to smooth developable surfaces including general cylinders, cones, andtangent surfaces.
Constructing the structure only on one side is relatively straightforward. The structure is an assemblyof aligned tubes, which is obtained by sweeping extruding the parallelogram cross-section grid along
an arbitrary guiding polyline on the generating surface (Figure 2). The sweeping is sequential: first,
section grid is extruded along a segment of the guiding polyline by sufficient length; then the extruded
surface is trimmed by a plane passing though the fold line of generating surface; the new section forms
another parallelogram grid, so we continue the extrusion along the next segment of the guiding polyline.
Since the cross-section remains a parallelogram grid when folded, these aligned tubular structurespossess a one-DOF mechanism.
However, combining top and bottom structures is a non-trivial problem. If we separately obtain top andbottom structures, we will end up with two separate one-DOF structures that cannot synchronize once
they start to fold. This is because the folding motions of the shared generating surfaces can be different
in general. We need to explore geometric constraints that make the top and bottom mechanisms
compatible with each other. One obvious example of compatible mechanisms are ones sharing the same
extrusion vectors on the top and bottom, and this would create an aligned type assembly in both
directions. However, we would like to use the new type of coupling, because of the improved structural

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

properties discussed in [8] . We need to explore non-trivial intrinsic symmetry in order to explore thesespecial cases.

Figure 3. A transformable structures obtained from a given generating surface.
2.3. Equivalent Origami Structures
In order to simplify the geometric constraints, we extract the fundamental elements from the mechanism
comprising of the generating surface and two distinct wall surfaces (Figure 4). By connecting portionsof these surfaces, we construct an equivalent origami mechanism composed of 3
quadrilateralpanels, mechanically equivalent to the original structure. Because the equivalent origami gives different
extrusion vectors for the top and bottom, we can easily construct the whole cellular structure from it.
Therefore, the design problem of rigid foldable sandwich structures is reduced to obtaining a rigid
foldable equivalent origami structure with 3
quadrilateral panels. Here, note that the equivalent
origami structure does not necessarily have to be developable in order to form a mechanism. If it isdevelopable, the sandwich structure has a completely flat state; fabrication is especially easy in such astate because we can use flat sheets and assemble in a flat state.

Figure 4. Left four columns: Extraction of equivalent origami from structures in Figure 2. Right column:
Equivalent origami of model in Figure 3. Top row: equivalent surface composed of 3
quadrilateral
panels. Middle row: extracted surfaces. Bottom row: generating surface and top and bottom tubes.

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

3. Cylindrical Generating Surface with Mirror Symmetric Walls
Equivalent origami structures can be constructed when the generating surface is cylindrical, i.e., thefold lines are parallel. In this case, we can construct a mirror plane perpendicular to the surface.
Arbitrarily constructed wall surface of one side can be mirror reflected to construct the wall on the otherside. Figure 5 shows an example design of mirror-reflected construction and generated sandwich
structures.

Figure 5. Cylindrical surface with mirror symmetric wall surfaces.
In this example, the structure does not have a flat state.

4. General Generating Surface with Bi-Directionally Flat-Foldable Structures
A family of equivalent origami structure can be obtained using bi-directionally flat-foldable
quadrivalent mesh structure, i.e., a polyhedral surface each vertex of which comprises of four corners
with angles , , , or , ,,, in this order. If this type of structure has non-flat
configuration, it has a mechanism where the tangent of half fold angles (supplementary angle of dihedral
angle) of all fold lines are always proportional to each other [10]. The motion of generating surface ispredefined by this constraints independent of other parameters of the attached structures. Top andbottom wall structures fold in a compatible motion if they share a common generating surface. Becauseof bidirectional flat foldability, the structure can fold into two extreme flat states. It can fold to adeveloped state, and to a completely flat-folded state if we ignore collision.
We start from the generating surface and obtain a wall structure. Figure 6 shows the process ofconstruction. First, the generated surface is given as a sequence of planes , , … , . Then, a wall
can be sequentially constructed such that each vertex is bi-directionally flatsurface , , … ,surrounded by, ,,foldable. The construction is as follows. Consider a vertex,, respectively, in counterclockwise order. According tocomprising corner angles , ,by tan and that between[10], if we denote the tangent of the half fold angle between andandby tan , the fold angles around the vertex in counterclockwise order istanwheretan

2tan,tan⋯, ,,,, tan, tan
222tan , and is the fold coefficient of vertex defined by(1)Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

tantanIf we assume that the vertex structure atknown. By solving equation (2), we obtain

2
2

tan
tan2,(2)2

is solved, then, parameters,

and tan

are

as

1(3)
tan .
2 12
This determines the configuration of with parameter. Therefore, after choosing initial angles
andtan , the rest of the wall structure is obtained sequentially. In the same manner, we canconstruct the structure on the opposite side with a different parameter ofand with different or thesame parameter . The resulting sandwich structure is shown in Figure 7.
tanFigure 6. Sequential construction of structures from generating surface. Solid and dashed lines
indicate mountain and valley folds with negative and positive fold angles, respectively.

Figure 7. Generated sandwich structure.

5. Conclusion
We have shown a method for obtaining rigidly foldable sandwich structures that follow given singly
curved surfaces. The proposed structures are composed of the assembly of aligned rigid foldable tubeson top and bottom, where the compatibility of the top and bottom mechanisms are satisfied by making
an equivalent origami structure rigidly foldable. We have shown a family of structures that follow

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam
Future Visions

general cylindrical surfaces, and a family of bi-directionally flat-foldable structures that follow generic
single curved surfaces, i.e., general planar quad strips or a developable surfaces. The latter family allows
for high degree of freedom of design, while it has potentially useful property that the structures can befabricated in a flat state. The thin sheet becomes a thick stiff structure as the surface bends into a desired
single-curved surface. Potential applications of the proposed structures include deployable vault
structures (Figure 8) and deployable sandwich surface patches that can cover doubly curved structures.

Figure 8. Model of deployable vault structure.

Acknowledgement
The first author is supported by Japan Science and Technology Agency (JST) Presto program.

References[1]

Miura K., “Zeta-Core Sandwich―Its Concept and Realization”. Institute of Space andAeronautical Science Report, 1972; 37; 137-164.
[2] Klett Y. and Drechsler K., “Designing Technical Tessellations”. Origami5, 2010, 305-321[3] Tachi T., “Geoemtric Considerations for the Design of Rigid Origami Structures”, in Proceedingsof IASS Symposium 2010, 2010, 771-782.
[4] Tachi T., 2011. “One-DOF rigid foldable structures from space curves”. In Proceedings of theIABSE-IASS Symposium, 2011.
[5] Tachi T. and Miura K., “Rigid-Foldable Cylinders and Cells”, Journal of the International
Association for Shell and Spatial Structures, 2012; 53; 217-226.
[6] Schenk M. and Guest S.D., “Geometry of Miura-Folded Meta-Materials”; Proceedings of theNational Academy of Sciences (PNAS), 2013; 110; 3276-3281[7] Cheung K, Tachi T, Calisch S, Miura K, “Origami Interleaved Tube Cellular Materials”, Smart
Materials and Structures, 2014; 23, 094012 (10pp)[8] Filipov E., Tachi T., and Paulino G., “Origami tubes and cellular assemblages coupled into stiff
yet reconfigurable metamaterials”. Under review.
[9] Schenk M., Guest S.D., “Folded Textured Sheets”, In proceedings of IASS 2009 Symposium, 2009[10] Tachi T., “Freeform Rigid-Foldable Structure using Bidirectionally Flat-Foldable Planar
Quadrilateral Mesh”. Advances in Architectural Geometry 2010; 2010, 87-102.