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Research Papers

Rigidly Foldable Quadrilateral Meshes From Angle Arrays

[+] Author and Article Information
Robert J. Lang

Lang Origami,
Alamo, CA 94507
e-mail: robert@langorigami.com

Larry Howell

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602
e-mail: lhowell@byu.edu

Manuscript received September 18, 2017; final manuscript received December 17, 2017; published online February 1, 2018. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 10(2), 021004 (Feb 01, 2018) (11 pages) Paper No: JMR-17-1300; doi: 10.1115/1.4038972 History: Received September 18, 2017; Revised December 17, 2017

We present a design technique for generating rigidly foldable quadrilateral meshes (RFQMs), taking as input four arrays of direction angles and fold angles for horizontal and vertical folds. By starting with angles, rather than vertex coordinates, and enforcing the fold-angle multiplier condition at each vertex, it is possible to achieve arbitrarily large and complex panel arrays that flex from unfolded to flatly folded with a single degree-of-freedom (DOF). Furthermore, the design technique is computationally simple, reducing for some cases to a simple linear-programming problem. The resulting mechanisms have applications in architectural facades, furniture design, and more.

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References

Chen, Y. , Peng, R. , and You, Z. , 2015, “ Origami of Thick Panels,” Science, 349(6246), pp. 396–400.
Edmonson, B. J. , Lang , R. J. , Magleby , S. P. , and Howell, L. L. , 2014, “An Offset Panel Technique for Thick Rigidly Foldable Origami,” ASME Paper No. DETC2014-35606.
Evans, T. A. , Lang, R. J. , Magleby, S. P. , and Howell, L. L. , 2015, “ Rigidly Foldable Origami Gadgets and Tessellations,” R. Soc. Open Sci., 2(9), p. 150067. [CrossRef] [PubMed]
Evans, T. A. , Lang, R. J. , Magleby, S. P. , and Howell, L. L. , 2016, “ Rigidly Foldable Origami Twists,” Origami 6: Sixth International Meeting of Origami Science, Mathematics, and Education, American Mathematical Society, Providence, RI, pp. 119–130. [CrossRef]
Lang, R. J. , Magleby, S. , and Howell, L. , 2015, “Single-Degree-of-Freedom Rigidly Foldable Origami Flashers,” ASME Paper No. DETC2015-46961.
Miura, K. , and Tachi, T. , 2010, “ Synthesis of Rigid-Foldable Cylindrical Polyhedra,” Symmetry: Art and Science, pp. 204–213. http://origami.c.u-tokyo.ac.jp/~tachi/cg/FoldableCylinders_miura_tachi_ISISSymmetry2010.pdf
Tachi, T. , 2009, “ Generalization of Rigid Foldable Quadrilateral Mesh Origami,” Symposium of the International Association for Shell and Spatial Structures (IASS), Valencia, Spain, Sept. 28–Oct. 2, pp. 1–8. http://www.tsg.ne.jp/TT/cg/RigidFoldableQuadMeshOrigami_tachi_IASS2009.pdf
Tachi, T. , 2009, “ One-DOF Cylindrical Deployable Structures With Rigid Quadrilateral Panels,” Symposium of the International Association for Shell and Spatial Structures (IASS), Valencia, Spain, Sept. 28–Oct. 2, pp. 2295–2305. http://origami.c.u-tokyo.ac.jp/~tachi/cg/RigidFoldableCylindricalOrigami_tachi_IASS2009.pdf
Tachi, T. , 2010, “ Freeform Rigid-Foldable Structure Using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh,” Advances in Architectural Geometry 2010, Springer-Verlag, Vienna, Austria, pp. 87–102.
Tachi, T. , 2010, “ Geometric Considerations for the Design of Rigid Origami Structures,” International Association for Shell and Spatial Structures Symposium (IASS), Shanghai, China, Nov. 8–12, pp. 458–460.
Tachi, T. , 2011, “ Rigid-Foldable Thick Origami,” Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education, CRC Press, Boca Raton, FL, pp. 253–264. [CrossRef]
Tachi, T. , 2013, “ Composite Rigid-Foldable Curved Origami Structure,” First International Conference on Transformable Architecture (Transformables 2013), Seville, Spain, Sept. 18–20, pp. 18–20.
Tachi, T. , 2018, “Software: Freeform Origami, Origamizer, Rigid Origami Simulator,” Tomohiro Tachi, Tokyo, Japan, accessed Jan. 23, 2018, http://www.tsg.ne.jp/TT/software/
Greenberg, H. C. , Gong, M. L. , Magleby, S. P. , and Howell, L. L. , 2011, “ Identifying Links Between Origami and Compliant Mechanisms,” Mech. Sci., 2(2), pp. 217–225. [CrossRef]
Howell, L. L. , 2001, Compliant Mechanisms, Wiley, Hoboken, NJ.
Yan, C. , Weilin, L. , Junlan, L. , and Zhong, Y. , 2017, “ An Extended Family of Rigidly Foldable Origami Tubes,” ASME J. Mech. Rob., 9(2), p. 021002.
Klett, Y. , and Middendorf, P. , 2016, “ Kinematic Analysis of Congruent Multilayer Tessellations,” ASME J. Mech. Rob., 8(3), p. 034501. [CrossRef]
Qiu, C. , Zhang, K. , and Dai, J. S. , 2016, “ Repelling-Screw Based Force Analysis of Origami Mechanisms,” ASME J. Mech. Rob., 8(3), p. 031001. [CrossRef]
Yasuda, H. , Chen, Z. , and Yang, J. , 2016, “ Multitransformable Leaf-Out Origami With Bistable Behavior,” ASME J. Mech. Rob., 8(3), p. 031013. [CrossRef]
Lang, R. J ., 2017, Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami, CRC Press, Boca Raton, FL.
Justin, J. , 1997, “ Towards a Mathematical Theory of Origami,” Origami Science and Art: Second International Meeting of Origami Science and Scientific Origami, Seian University of Art and Design, Otsu, Japan, pp. 15–30.
Huffman, D. A. , 1976, “ Curvature and Creases: A Primer on Paper,” IEEE Trans. Comput., C-25(10), pp. 1010–1019. [CrossRef]
Thomas, H. , 2006, Project Origami: Activities for Exploring Mathematics, A K Peters, Natick, MA.
Schief, W. K. , Bobenko, A. I. , and Hoffmann, T. , 2008, “ On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces,” Discrete Differential Geometry, Springer, New York, pp. 67–93. [CrossRef]
Miura , K. , and Natori, M. , 1985, “ 2-D Array Experiment on Board a Space Flyer Unit,” Space Sol. Power Rev., 5(4), pp. 345–356.
Barreto , P. T. , 1997, “ Lines Meeting on a Surface: The ‘Mars’ Paperfolding,” Origami Science and Art: Second International Meeting of Origami Science and Scientific Origami, Seian University of Art and Design, Otsu, Japan, pp. 343–359.
Sakoda, J. M. , 1997, “ Hikari-Ori: Reflective Folding,” Origami Science and Art: Second International Meeting of Origami Science and Scientific Origami, Seian University of Art and Design, Otsu, Japan, pp. 333–342.
Hokusan, 2017, “Sanfoot Wood Veneer,” Hoxan Corporation, Chiba, Japan, accessed Jan. 23, 2018, https://www.hoxan.co.jp/english/products/
Lang, R. J. , 2017, “Rigidly Foldable Quad Mesh+Tessellatica,” Robert J. Lang Origami, Alamo, CA, accessed Jan. 23, 2018, http://www.langorigami.com/publication/rigidly-foldable-quadrilateral-meshes-angle-arrays
Ginepro, J. , and Hull, T. C. , 2014, “ Counting Miura-Ori Foldings,” J. Integer Sequences, 17(2), pp. 1–15. http://emis.ams.org/journals/JIS/VOL17/Hull/hull.pdf
Harary, F. , 1994, Graph Theory, Westview Press, Boulder, CO.
Bényi, B. , and Hajnal, P. , 2015, “ Combinatorics of Poly-Bernoulli Numbers,” Stud. Sci. Math. Hung., 52(4), pp. 537–558. http://publicatio.bibl.u-szeged.hu/5940/1/2340311.pdf
Cameron, P. , 2014, “Poly-Bernoulli Numbers,” Peter Cameron, Toowoomba, Australia, accessed Jan. 23, 2018, https://cameroncounts.wordpress.com/2014/01/19/poly-bernoulli-numbers/
Kaneko, M. , 1997, “ Poly-Bernoulli Numbers,” J. Théor. Nombres Bordeaux, 9(1), pp. 221–228. [CrossRef]

Figures

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Fig. 1

Conventions for labeling and measuring fold angles. γi is a valley fold; γj is a mountain fold. Left: in a crease pattern. Right: in the corresponding folded form.

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Fig. 2

Schematic of a D4V with sector angles α1α4 and fold angles γ1γ4. Left: crease pattern. Right: folded form.

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Fig. 3

Indexing scheme for the RFQM mesh. Vertices are in black; horizontal edges in gray; vertical edges in black.

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Fig. 4

Direction angles at a vertex

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Fig. 5

Direction angles and fold angles around a single vertex

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Fig. 6

Schematic of vertex construction. The gray-shaded lines show creases whose vertices are constructed relative to one another from their intervening edges.

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Fig. 7

A crease pattern, designed from the angle arrays ϕl=(30 deg,20 deg), γl=(25 deg,−35 deg), ϕb=(80 deg,100 deg), and γb=(−85 deg,95 deg)

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Fig. 8

The folded form of the crease pattern of Fig. 7, with fold angles transformed by Eq. (25). Top left: m = 0.1. Top right: m = 1.0. Bottom left: m = 2.0. Bottom right: m = 5.0.

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Fig. 9

Two DLFF-RFQMs that reproduce known rigidly foldable patterns. Top: the Miura-ori. Bottom: Barreto's Mars.

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Fig. 10

A DLFF-RFQM with the same direction angles and fold-angle magnitudes as Barreto's Mars but different fold-angle signs

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Fig. 11

Two DLFF-RFQMs in which zero-length edges give rise to degree-6 vertices within the pattern

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Fig. 12

A set of direction and fold angles for which it is not possible to find a kinematic mechanism using rectangular paper, but for which allowing nonrectangular paper provides a solution

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Fig. 13

Top: fold pattern with only rectangular paper and minimum-edge-length constraints. Bottom: the same pattern but with all four corner facets constrained to lie within the same plane.

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Fig. 14

Left: crease pattern for a rigidly foldable tubular form. Right: the folded form.

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Fig. 15

Left: a vertex with minor fold angle approaching zero. Right: a vertex with major-fold angle approaching π.

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Fig. 16

Top: a schematic rectangular-boundary configuration. Bottom: the same pattern embedded within a larger unconstrained-boundary configuration.

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Fig. 17

Left: crease pattern. Right: folded form, implemented in folded wood veneer.

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Fig. 18

Left: crease pattern. Right: folded form, implemented in folded wood veneer.

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Fig. 19

Schematic of major-direction assignment for an example crease pattern with nh=4,nv=3. Left: the crease pattern. Arrow points from major fold to minor fold. Right: the oriented complete bipartite graph K(3, 2) that encodes the major-direction assignment.

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