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Research Papers

Accommodating Thickness in Origami-Based Deployable Arrays1

[+] Author and Article Information
Shannon A. Zirbel

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602

Robert J. Lang

Lang Origami,
Alamo, CA 94507

Brian P. Trease

Jet Propulsion Laboratory,
California Institute of Technology,
Pasadena, CA 91109

Larry L. Howell

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

Submitted to ASME IDETC with Paper No. DETC2013-12348.

2Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 31, 2013; final manuscript received June 17, 2013; published online October 3, 2013. Assoc. Editor: Mary Frecker. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Mech. Des 135(11), 111005 (Oct 03, 2013) (11 pages) Paper No: MD-13-1046; doi: 10.1115/1.4025372 History: Received January 31, 2013; Revised June 17, 2013

The purpose of this work is to develop approaches to accommodate thickness in origami-based deployable arrays with a high ratio of deployed-to-stowed diameter. The origami flasher model serves as a basis for demonstrating the approach. A thickness-accommodating mathematical model is developed to describe the flasher. Practical modifications are presented for the creation of physical models and two options are proposed: allowing the panels to fold along their diagonals or applying a membrane backing with specified widths at fold-lines. The mathematical model and hardware modifications are employed to create several physical models. The results are general and apply to a range of applications. An example is provided by the application that motivated the work: a deployable solar array for space applications. The model is demonstrated in hardware as a 1/20th scale prototype with a ratio of deployed-to-stowed diameter of 9.2 (or 1.25 m deployed outer diameter to 0.136 m stowed outer diameter).

FIGURES IN THIS ARTICLE
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Copyright © 2013 by ASME
Topics: Thickness , Membranes
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References

Tachi, T., 2010, “Geometric Considerations for the Design of Rigid Origami Structures,” Proceedings of IASS Symposium on Spatial Structures—Permanent and Temporary.
Tachi, T., 2011, “Rigid Foldable Thick Origami,” Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education.
Trautz, M., and Kunstler, A., 2009, “Deployable Folded Plate Structures: Folding Patterns Based on 4-Fold-Mechanism Using Stiff Plates,” Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium.
Hoberman, C., 1991, “Reversibly Expandable Structures,” U.S. Patent No. 4,981,732.
Hoberman, C., 1993, “Curved Pleated Sheet Structures,” U.S. Patent No. 5,234,727.
Hoberman, C., 2010, “Folding Structures Made of Thick Hinged Sheets,” U.S. Patent No. 7,794,019.
Faist, K. A., and Wiens, G. J., 2010, “Parametric Study on the Use of Hoberman Mechanisms for Reconfigurable Antenna and Solar Arrays,” Proceedings of IEEE Aerospace Conference, Paper No. #1172.
Miura, K., 1980, “Method of Packaging and Deployment of Large Membranes in Space,” Proceedings of 31st Congress International Astronautical Federation, pp. 1–10.
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.
Sternberg, S., 2010, “Symmetry Issues in Collapsible Origami,” Symmetry: Cult. Sci., 21(4), pp. 345–364.
Guest, S., and Pellegrino, S., 1992, “Inextensional Wrapping of Flat Membranes,” Proceedings of the First International Seminar on Structural Morphology, pp. 203–215.
Nojima, T., 2002, “Origami Modeling of Functional Structures Based on Organic Patterns,” Master's thesis, Graduate School of Kyoto University, Kyoto, Japan.
De Focatiis, D. S., and Guest, S., 2002, “Deployable Membranes Designed From Folding Tree Leaves,” Philos. Trans. R. Soc. London, Ser. A, 360, pp. 227–238. [CrossRef]
Lang, R. J., 1997, Origami in Action, St. Martin's Griffin, New York.
Shafer, J., 2001, Origami to Astonish and Amuse, St. Martin's Griffin, New York.
Shafer, J., 2010, Origami Ooh La La! Action Origami for Performance and Play, CreateSpace Independent Publishing Platform, US.
Chiang, C. H., 1984, “On the Classification of Spherical Four-Bar Linkages,” Mech. Mach. Theory, 19, pp. 283–287. [CrossRef]
NASA Technology Roadmaps, 2012, “TA12: 2.1.3 Flexible Material Systems,” Available at: http://www.nasa.gov/pdf/501625main_TA12-ID_rev6_NRC-wTASR.pdf
NASA Research Announcement NNL12A3001N, 2011, “Game Changing Opportunities in Technology Development,” Available at: http://prod.nais.nasa.gov/cgi-bin/eps/synopsis.cgi?acqid=152634

Figures

Grahic Jump Location
Fig. 1

The four-sided flasher model is shown in paper, with the crease pattern overlaid. Mountain folds are indicated with solid red lines; valley folds are indicated with dashed blue lines.

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Fig. 2

Two approaches for enabling origami with thickness, from Tachi [2]: (a) The axis of rotation for the fold is shifted from the midline of the material to the material surface. (b) Material is removed near the fold line. In both (a) and (b), the dashed line represents the zero-thickness model.

Grahic Jump Location
Fig. 3

Crease pattern for the Miura-ori fold, based on Refs. [8,9]

Grahic Jump Location
Fig. 4

Comparison of (a) ideal and (b) thickness-accommodating winding membrane from Guest and Pellegrino [11]

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Fig. 5

The six-sided flasher model. (a) Model assuming zero-thickness. (b) Crease pattern for the model assuming finite thickness, with additional diagonal folds included to enable rigid-foldability.

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Fig. 6

The parameters for defining the mathematical model are (a) rotational order, m, which is the number of sectors around the central polygon, (b) height order, h, which is the number of axial bends between the bottom and top of the folded cylinder, and (c) number of rings, r

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Fig. 7

A single labeled sector of crease pattern (left) and folded form (right), showing how the vertices are indexed. In the crease pattern, the central polygon is outlined in light grey. For this sector, m = 6, h = 2, r = 2.

Grahic Jump Location
Fig. 8

A thickness-accommodating flasher (m = 6, h = 2, r = 2, δ = 0.1). Top row, l–r: crease pattern, single sector; folded form, single sector; folded form, side view. Bottom row, l–r: full crease pattern; full folded form; top view, folded form.

Grahic Jump Location
Fig. 9

The six-sided flasher (m = 6, h = 3, r = 1) is rigid-foldable when additional fold-lines are added to the model along the diagonals of each panel. (a) The model in its deployed configuration. (b) The model in its stowed configuration. The valley folds are marked by the bold black lines.

Grahic Jump Location
Fig. 10

The gap width was adjusted at fold-lines to enable rigid folding of the model on a membrane backing. In this model, m = 6, h = 3, r = 1.

Grahic Jump Location
Fig. 11

The folds of the membrane model are illustrated with the panels in blue and the membrane as the dashed red line. The gap sizes defined here are lower limits. (a) The 180 deg mountain folds require minimal gap between panels because the membrane folds back on itself. (b) The 180 deg valley folds require a minimum of twice the thickness of the panels because the membrane folds around two panels. (c) The 60 deg valley folds require a minimum of one thickness because of the triangle formed by the panel edges.

Grahic Jump Location
Fig. 12

A rigid-foldable six-sided flasher (m = 6, h = 2, r = 1). The thickness of the panels governs the final size of the stowed model. (a) Deployed configuration; the orange path marks the number of panels on each side of the stowed model. (b) The rigid panels are taped on the side corresponding to the direction of the fold (mountain or valley). (c) Stowed configuration.

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Fig. 13

A computer-generated model of the solar array as it might work on a spacecraft

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Fig. 14

1/20th scale model of the deployable hex-flasher (m = 6, h = 4, r = 2, δ = 0.01). The deployed-to-stowed diametral ratio of the model is 9.2.

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