Rigid Origami Structures with Vacuumatics: GeometricConsiderations
Tomohiro TACHI1, Motoi MASUBUCHI2 and Masaaki IWAMOTO3
1

The University of Tokyo, Tokyo, Japan, tachi@idea.c.u-tokyo.ac.jp
TU Berlin, Berlin, Germany, motoi.masubuchi@mailbox.tu-berlin.de3
Vo Trong Nghia Co. Ltd., Ho Chi Minh City, Vietnam, masaaki.iwamoto@gmail.com
2Summary
We propose a novel type of hybrid structures, in which multi-DOF rigid foldable structures will be
stiffened using Vacuumatics i.e., the structural system of negative pressured double membrane
containing particles. The structure is transformed to a desired 3D configuration following thekinematics of origami when the hinges are plastic and flexible, and then stiffened at dihedral hinges
by strengthening negative pressure. In addition, the negative pressure gives a small moment at thehinge lines, which helps to solve singularity problem of rigid origami at the initial unfolded state.
Keywords: origami; vacuumatics; rigid-foldable structures.

1. Introduction
Rigid foldable structures based on rigid panels and hinges, i.e., rigid origami, are potentiallyapplicable for creating transformable or temporary spaces. The core advantage of rigid origami is in
their geometric scalable mechanisms, i.e., can be scaled up to large scale, and their property offorming continuous surfaces without in-plane deformation. In particular, multi-DOF rigid foldable
structures based on triangle panels have an advantage in its flexibility to follow the change in theenvironment and human activities while the 3D form becomes geometrically stable after fixing theboundary points. One of such example patterns is doubly-expandable transformable shells by Reschand Christiansen [1]. However, such a structure tends to be structurally flexible in reality, even after
it is geometrically pinned at its boundary. For example in the case of [1], numbers of rods (3 foreach concave vertex) are added in order to stabilize the structure; these fixing rods make thestructure not actually transformable or reusable. Therefore, effective temporary stiffening methods
have been required.
In order to effectively solve this problem, we utilize another structural system called Vacuumatics,
which is a negative pressured double membrane containing particles. The structure is initiallyplastic or viscoelastic, but by adding the negative pressure, the friction force between compressed
particles makes the structure stiff. As far as we know, the oldest published concept of an adaptive
structure using Vacuumatics is “LIVING-ROOM” by J. Gilbert in 1971 [2 pp.52-53], preceded bysimple mockup test of a vault in 1970. Vacuumatics concept is researched by several parties in1970s, such as by Frei Otto [3]. It is after decades that the structural principle is paid attention again
when it is rediscovered by people such as Sobek et al. in 2000s [4] [5].With the social backgroundof sustainability, the potential of Vacuumatics as adaptive insulation also attracts some researchers
after the rediscovery. The concept of controlling the stiffness and plasticity of the material by thestrength of vacuum has been studied [6]. One of the unsolved problems of pure Vacuumatics systemis the method for controlling its overall shapes, which is essential in order to use the concept foradaptive structure.
We propose a novel type of hybrid structures, in which multi-DOF rigid foldable structures arestiffened at their hinges using Vacuumatics. The key concept is that the stiffness of hinges iscontrolled by the strength of the vacuum asserted to the structure. The structure is transformed to adesired 3D configuration when the hinges are plastic and flexible, and then stiffened at dihedral
hinges by strengthening negative pressure. In addition, the negative pressure gives a small momentat the hinge lines at the initial construction, helping to remove the instability caused by thesingularity in the unfolded state.
In this paper, we explain this novel concept from the geometric perspective. We will review thekinematics of rigid origami structures to show the singularity in the initial unfolded state and therelation between the 3D form and the 2D boundary configuration.

2. System Overview
2.1

Origami Pattern

Since any rigid origami pattern can be used for the hybrid system, we especially choose a triangle
based regular corrugation pattern of origami composed of both convex and concave vertices. Here,a vertex is convex (or concave) when the solid angle around the vertex is greater (or less) than 2π.
This includes corrugation patterns by Resch [1], origami patterns known as waterbomb corrugation
or “namako” [7], and triangulated version of Miura-ori [8]. It is preferable to include both convex
and concave vertices in order to produce a freeform surface having both positive and negative meancurvatures. For example, the PCCP shell or Yoshimura pattern [8] is not suitable for this purpose
because it comprises only with convex vertices.
Here, we use a variation of waterbomb corrugation based on 36° grids (Figures 1 and 2). This iscompactly flat-foldable and flexible enough in the limited range of folding angle (from 0° to 161°
as described in the following subsection).

Figure 1: Left: A variational waterbomb corrugation pattern. Right: close-up of the pattern (thick:mountain fold, thin: valley fold). A vertex with 4 (or 2) mountain folds is convex (or concave resp.).

Figure 2: Sequence of unfolding

2.2

Hinge Structure

Flexible hinges allow the continuous rigid plates to be transformed to a desired configuration;
however they must be stiffened to hold it against the self-weight and imposed loading. Thisrequirement is essential in the folding process as well as in the final folded state. Our proposedsolution is that the flexible hinge is stiffened by given negative pressure.
2.2.1 Layout
Rigid origami structures can be built with double layered thick composite panels. In order to avoid
the collision of panels, we offset the panel outline as proposed in [9]. The width of offset isdetermined by maximum folding angle and the thickness, in the example design, the folding angle
can change from 0°to 169°, with the ratio of thickness to the length (longest part) of exterior andinterior panel are approximately 0.009 and 0.006 respectively.
This produces gaps between the panels on one of its sides where we fill in aggregate particles
packed in breathable material. Then the whole structure is packed in an airtight membrane forming
a single air chamber (Figures 3 and 4)
2.2.2 Angular stiffening
The structural principle of vacuum is the difference of the air pressure between inner and outer
enclosed volume. Smaller inner air pressure act on the aggregate particle as compression forcewhich is balanced with the tension force loaded on the enclosing membrane. By the compression
force the aggregate particles are deformed and their maximum contact between each of them isachieved. Under this prestressed status the stiffened angular hinge can be effective as supporting
structures of the global system and could work until the compression stress remains equal to or
larger than the tensile stress caused by loading.
Another interesting aspect of Vacuumatics structure is in its plastic behavior that the particles
reconfigure themselves to be in another equilibrium state. Therefore, we can produce a plastic hinge
whose strength is controlled by the strength of negative pressure. Asserting appropriate amount of(smaller) vacuums keeps the whole structure in a plastic state and assists the process of folding by
removing excess force asserted to the control points.
2.2.3 Small additional moment
To allow folding the rigid plates in one direction, the packing particle is set on only one side (valleyfolding side) of the plates. Consequently the negative pressure gives a small moment at the hingelines. Certainly the resulting moment would be regarded as an unfavorable effect in the structural
point of view, however this can be helpful to solve singularity problem of rigid origami at the initial
unfolded state.

Figure 3: Hinge Design. Left: particles are placed in between panels. Middle: Self-folding moment
is induced by the vacuum. Right: The vacuum keeps the hinge stiff.

Figure 4: Hinge Design
2.3

Construction Process

The prospective construction process is as follows (Figure 5):(1) Prefabrication:
Layout the membranes, panels, particles, and fabrics in a plane to form the composite structure.
Then, the negative pressure is added to the structure, which induces the small folding moment.
This makes the structure non-singular, and the whole folding motion can be actuated byconstraining the boundary points. Fold the structure to a compact state so that it can be carried
to the site.
(2) On-site
deployment:
Unfold the structure and transform it by controlling the configuration of the boundary vertices.
The folding process is performed under an appropriate amount of vacuum to make the structure
plastically deformable. Strengthen the vacuum when the structure is fixed to the final state. The
structure can be re-used by removing the vacuum and re-folding it to the compact state.

Figure 5: Idea of the construction process

3. Kinematics
3.1

BasicsThe kinetic behavior of rigid origami based on triangle panels is generally described using unstable
truss model. Here we denote the numbers of vertices, vertices on the boundary, and the edges by V,
V0, E. The configuration of the structure is represented by the coordinate 3V-vector⋯⋯, which are constrained by its edge lengths

(E-vector) being unchanged from the original state. When 3
0 , the infinitesimaltransformation of the structure can be represented by the Jacobian matrix (E×3V matrix) of thelength vector.
Δ1

A nontrivial solution of Equation (1) gives the transformation mode, and the number of degrees of
freedom of the structure (including rigid transformation of the whole structure) is the dimension ofthe solution space, which is3, where S is the number of degenerate constraints or
the number of self-equilibrium state. In the case of triangular mesh homeomorphic to a disk, thenumbers of elements are related by Euler’s equation, which yields
3 S. By adding
sufficient number of support conditions, e.g., pinning vertices to the ground, the structure becomesstable (0).
3.2Singularity at a Flat StateThe initial unfolded state of a rigid origami structure has at least V singular conditions. This isbecause the elements of the 1 mod 3 -st columns of the Jacobean matrix are all 0, i.e., ⁄0, where. For a developable origami surface, this degeneracy holds
only when every fold is unfolded to a plane. As a result, the valid configuration space comprises
3-dimensional cells sharing one point (where everything is unfolded). This means that weneed to choose the right transformation mode at the singular flat state in order to reach a desiredstate (Figure 6). The bending moment at each edge resulting from the vacuum produces the initialtransformation mode that follows correct mountain and valley assignment and also convexity andconcavity of vertices.

Figure 6: Unfavorable (Left) and valid (Right) transformation modes.
3.3

3D state and Constraints

In a generic half folded state, the degrees of freedom is given by
3. This implies thatwe can fix the whole configuration by only fixing a part of the boundary points, for example
pinning at least ⁄3 1 vertices. In Figure 7, we are pinning nearly every other boundary points
to the ground, which results in an statically indeterminate stable structure with about ⁄6 excess
constraints.
Therefore, the whole 3D configuration is dynamically controlled by the configuration of the pinned
vertices. Because of the excess constraints, the configuration cannot be determined arbitrarily. Here,
we show an interactive approach for formfinding the valid configuration of pinned vertices. We use
the unstable truss model in Equation 1 and add geometric constraints (Figure 7),
1.

FoldAngle

Constraints:

foreveryfoldangle,| |180°.

This condition comes from the design of hinges for treating the thickness.
the example design.

11° is chosen in

2.

Ground Constraints: for every vertex to be pinned on the ground plane, i.e.,0,
is a reference point on the ground, e.g.,andwhere is the normal of the plane andfor a horizontal plane. Since our method is based on iteration, this is adaptable to anarbitrary nonlinear terrain surface.

3.

User
DefinedConstraints:
In this example, several boundary points are constrained to be on a vertical plane, derived froman existing building.

In this model, the pinned vertices are allowed to slide on the ground and only fixed in one
dimension. This forms an underdetermined system, in which an infinitesimal valid motion isrepresented byΔΔ,

2

whererepresents the above mentioned constraints. Here, inequality conditions are treated asequality conditions if and only if they are not satisfied (penalty function). The minimal normsolution of this equation from an arbitrary estimation Δ is given using Moore-Penrose generalized.
inverseΔΔ,

3

We use Freeform Origami [10] for interactively giving Δ while numerically solving theconstraints. We have checked that the transformation path exists between the unfolded state, thecompactly folded state, and the states shown in Figure 8, while keeping the fold angle constraints
and ground constraints. The continuous transformation sweeps the pinned vertices to draw planarpaths. Moving the pinned vertices along the paths actuates the 3D configuration of the model in thephysical model.

Figure 7: Geometric Constraints.

Figure 8: Variational configurations controlled by the boundary shapes.

Figure 9: Illustrative image of the example design with gourd shape.

4. Conclusion and Future works
We proposed a novel concept of hybrid structure based on multi-DOF rigid origami and
Vacuumatics hinges in order to achieve an adaptive freeform surface whose 3D geometry andstructural stiffness are controllable. By controlling the stiffness of the hinges, the stiffness of thestructures can be effectively changed. The moment at the hinges produced by the vacuum removes
the singularity in the unfolded state and makes the folding of the pattern controllable by the positionof the boundary vertices.
In this paper, we only worked on the concept and the geometric behavior. We would like to develop
the study to verify the effectiveness of the concept though experimental and analytical approaches.
The patterns can be

5. Acknowledgement
This research is supported by JST PRESTO.

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Herzog T., Minke G., Eggers H., Pneumatische Konstruktionen. Bauten aus Membranen undLuft, Hatje Cantz Verlag, 1986.

[3]

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[4]

Haase, Lemaitre, Schmidt, Sobek, Vacuumatics – Deflated Forms of Construction, 2007
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[6]

Huijben, F., Herwijnen, F. van, Nijsse, R.,”Vacuumatics 3DFormwork Systems: Customised
Free-Form Solidification”, in Proceedings of International Conference on Textile
Composites and Inflatable Structures, Stuttgart, 2009.

[7]

Fujimoto S., Souzousei wo kaihatsu suru rittai origami (in Japanese). Hyougo-ken Gakkou
Kouseikai Tamba Shibu, 1976.

[8]

Miura, K., “Proposition of pseudo-cylindrical concave polyhedral shells”, in Proceedings ofIASS Symposium on Folded Plates and Prismatic Structures (1970).

[9]

Tachi, T., "Geoemtric Considerations for the Design of Rigid Origami Structures", inProceedings of IASS Symposium 2010, Shanghai, China, pp. 771-782, 2010.

[10]

Tachi T. “Freeform Origami”, http://www.tsg.ne.jp/TT/software/, 2010.