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Research Papers

Self-Foldability of Rigid Origami

[+] Author and Article Information
Tomohiro Tachi

Department of General Systems Studies,
Graduate School of Arts and Sciences,
The University of Tokyo,
Tokyo 153-8902, Japan
e-mail: tachi@idea.c.u-tokyo.ac.jp

Thomas C. Hull

Department of Mathematics,
College of Arts and Sciences,
Western New England University,
Springfield, MA 01119
e-mail: thull@wne.edu

1Corresponding author.

Manuscript received October 11, 2016; final manuscript received December 10, 2016; published online March 9, 2017. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 9(2), 021008 (Mar 09, 2017) (9 pages) Paper No: JMR-16-1301; doi: 10.1115/1.4035558 History: Received October 11, 2016; Revised December 10, 2016

When actuating a rigid origami mechanism by applying moments at the crease lines, we often confront the bifurcation problem where it is not possible to predict the way the model will fold when it is in a flat state. In this paper, we develop a mathematical model of self-folding and propose the concept of self-foldability of rigid origami when a set of moments, which we call a driving force, are applied. In particular, we desire to design a driving force such that a given crease pattern can uniquely self-fold to a desired mode without getting caught in a bifurcation. We provide necessary conditions for self-foldability that serve as tools to analyze and design self-foldable crease patterns. Using these tools, we analyze the unique self-foldability of several fundamental patterns and demonstrate the usefulness of the proposed model for mechanical design.

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References

Hawkes, E. , An, B. , Benbernou, N. M. , Tanaka, H. , Kim, S. , Demaine, E. D. , Rus, D. , and Wood, R. J. , 2010, “ Programmable Matter by Folding,” Proc. Natl. Acad. Sci. U. S. A., 107(28), pp. 12441–12445. [CrossRef] [PubMed]
Felton, S. , Tolley, M. , Demaine, E. , Rus, D. , and Wood, R. , 2014, “ A Method for Building Self-Folding Machines,” Science, 345(6197), pp. 644–646. [CrossRef] [PubMed]
Na, J.-H. , Evans, A. A. , Bae, J. , Chiappelli, M. C. , Santangelo, C. D. , Lang, R. J. , Hull, T. C. , and Hayward, R. C. , 2015, “ Programming Reversibly Self-Folding Origami With Micropatterned Photo-Crosslinkable Polymer Trilayers,” Adv. Mater., 27(1), pp. 79–85. [CrossRef] [PubMed]
Abel, Z. , Cantarella, J. , Demaine, E. D. , Eppstein, D. , Hull, T. C. , Ku, J. S. , Lang, R. J. , and Tachi, T. , 2016, “ Rigid Origami Vertices: Conditions and Forcing Sets,” J. Comput. Geom., 7(1), pp. 171–184.
Akitaya, H. , Cheung, K. , Demaine, E. D. , Horiyama, T. , Hull, T. C. , Ku, J. , and Tachi, T. , 2016, “ Rigid Foldability With Optional Creases is NP-Hard,” (Unpublished).
Hanna, B. H. , Magleby, S. P. , Lang, R. J. , and Howell, L. L. , 2015, “ Force-Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms,” ASME J. Appl. Mech., 82(8), p. 081001. [CrossRef]
Waitukaitis, S. , Menaut, R. , Chen, B. G. , and van Hecke, M. , 2015, “ Origami Multistability: From Single Vertices to Metasheets,” Phys. Rev. Lett., 114(5), p. 055503. [CrossRef] [PubMed]
Silverberg, J. L. , Evans, A. A. , McLeod, L. , Hayward, R. C. , Hull, T. , Santangelo, C. D. , and Cohen, I. , 2014, “ Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials,” Science, 345(6197), pp. 647–650. [CrossRef] [PubMed]
Kawasaki, T. , 1997, “ R(γ) = I,” Second International Meeting of Origami Science and Scientific Origami, pp. 31–40.
Belcastro, S. , and Hull, T. , 2002, “ A Mathematical Model for Non-Flat Origami,” Origami3: 3rd International Meeting of Origami Mathematics, Science, and Education, pp. 39–51.
Tachi, T. , 2016, “ Rigid Folding of Periodic Triangulated Origami Tessellations,” Origami6: Sixth International Meeting on Origami Science, Mathematics, and Education, K. Miura , T. Kawasaki , T. Tachi , R. Uehara , P. Wang-Iverson , and R. J. Lang , eds., pp. 97–108.
Tachi, T. , 2010, “ Freeform Rigid-Foldable Structure Using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh,” Advances Architectural Geometry 2010, Springer, Vienna, Austria, pp. 87–102.
Tachi, T. , 2009, “ Generalization of Rigid-Foldable Quadrilateral-Mesh Origami,” J. Int. Assoc. Shell Spatial Struct., 50(3), pp. 173–179.
Evans, T. A. , Lang, R. J. , Magleby, S. P. , and Howell, L. L. , 2016, “ Rigidly Foldable Origami Twists,” Origami6: Sixth International Meeting on Origami Science, Mathematicsw, and Education, K. Miura , T. Kawasaki , T. Tachi , R. Uehara , P. Wang-Iverson , and R. J. Lang , eds., pp. 119–130.

Figures

Grahic Jump Location
Fig. 1

A vertex with three mountains and three valleys. This can pop up or pop down even with the same MV assignment. Note that pop-up state has sharper mountains and pop-down state has shaper valleys.

Grahic Jump Location
Fig. 2

Tangent vectors of well-behaved continuous rigid folding and valid tangents

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Fig. 3

Two folding modes of a flat-foldable vertex with four creases. The fold angles of opposite creases have the same magnitude.

Grahic Jump Location
Fig. 4

Visualization of the configuration space of a flat foldable vertex with sector angles α = π/4 and β = π/2. Note that 4D parameter space is projected along ρ0 to 3D space formed by ρ1, ρ2, and ρ3. The configuration space is the union of paths of mode 1 and 2. Each path lies on a plane perpendicular to each other.

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Fig. 5

A model with two flat-foldable vertex having four different folding modes. Left: original crease pattern with edge numbering. Middle: mountain and valley assignments. Right: folded forms.

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Fig. 6

A regular square twist with six different folding modes. Essentially two modes, one has four rotational variations, and the other has two rotational variations.

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Fig. 7

An irregular “twist” with two different folding modes. Here, p = 1/2 in the crease pattern.

Grahic Jump Location
Fig. 8

The configuration space of a degree-6 vertex with threefold symmetry described by solid and dashed curves. At the flat state, the force along (2+3,1) should be chosen to uniquely fold toward mode 1+.

Grahic Jump Location
Fig. 9

The configuration space remapped using the tangents of quarter angles. ρT is the actual target that we desire, from which off-configuration target ρT′ is derived. A rotational spring potential energy toward ρT′ (Eq. (17)), illustrated by gray area, will uniquely drive from any point on the configuration space to ρT along the configuration space. Note that there is no bifurcation at (0, 0) because energy increases in both modes 2+ and 2−.

Grahic Jump Location
Fig. 10

Crease line position in the folded state

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Fig. 11

Napier's analogies

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