# Computational Modeling Methods for Deployable Structure Based on Alternatively Asymmetric Triangulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Concentric Pleating in Tetragons And Hexagons

#### 2.1. Two Categories of Concentric Pleating

#### 2.2. Parametric Modeling

#### 2.2.1. Geometrical Relationship in Squares

_{i}and b

_{i}(i = 1,2,…) will move on the Y–Z plane and X–Z plane, respectively. Here, the way this paper finds these vertexes will be illustrated with examples of a

_{2}and b

_{2}, as shown in Figure 6. The width of one loop is defined as d, so the length of these creases are

_{1}a

_{2}, while the intersection of the sphere and Y–Z plane is a circle. Similarly, there is a sphere of radius a

_{2}b

_{1}centered at b

_{1}intersecting with the Y–Z plane so that another circle is obtained on this plane, as shown in Figure 7.

_{1}. When there are two different solutions, as shown in Figure 8, the real solution can be easily confirmed because this surface has a saddle-shaped configuration and if a

_{0}a

_{1}is fixed, then a

_{1}must be below the X–Y plane, which means that its y coordinate is a positive number. By this approach, the coordinates of b

_{2}are at the intersection of two circles on the X–Z plane. The two circles are intersections of two different spheres of radii a

_{2}b

_{2}and b

_{1}b

_{2}centered at a

_{2}and b

_{2}, respectively, and this saddle-shaped surface makes this point b

_{2}located above the X–Y plane. Demaine et al. used the mountain-valley assignment to determine the genuine solution and propose an algorithm based on Mathematica’s fully expanded solution for the general case to obtain this structure [10].

_{1}is only decided by the folding angle, which is the basic parameter. When the sub-plate is completed, the whole plate can be constructed by copying the sub-plate itself, as shown in Figure 9.

#### 2.2.2. Geometrical Relationship in Regular Hexagons

_{0}a

_{1}can be assumed as fixed, while for regular hexagons only the mid-point can be regarded as an immobile one, which leads to a new geometric relationship.

_{0}, is permanent, and to obtain the coordinates of the other points there are some rules that can be followed [19].

_{0}is fixed at the original and a

_{i}(i = 1,2,…) is moving within the Y–Z plane. The point b

_{1}is regarded as rotating around the m-axis, the angle between this and the y-axis being 30°. Thus, the folding angle refers to the angle that a

_{0}b

_{1}is rotating around the m-axis and, once the angle is given, the length of a

_{0}b

_{1}can be derived from d and the regular triangle’s geometric properties.

_{i}and b

_{i}are derived from them. This paper assumes that the initial state is when the paper is a flat sheet without deformation, as shown in Figure 11. Once the location of b

_{1}is obtained, a sphere with the radium of a

_{1}b

_{1}, whose center is b

_{1}, will intersect with the Y–Z plane to generate a circle on this plane. Then, a

_{1}is the solution of the intersection of this circle and another one which is centered at a

_{0}with a radius of a

_{0}a

_{1}on this plane. When there are two solutions in the intersection, the genuine solution must follow the rules of a negative Gauss surface, as mentioned in Section 2.2.1. This process is shown in Figure 12.

#### 2.2.3. Parametric Modeling of Deployable Structures Based on Alternating Asymmetric Triangulation

#### 2.3. Analysis of Kinematics Motions

_{1}and μ

_{2}are defined as following to make it clearer:

_{1}refers to the projected area of the top view, A

_{2}is the maximum projected area, H

_{1}is the projected height of the side view, and H

_{2}is the maximum projected height.

#### 2.4. Rigid-Foldable Origami

## 3. Discussion

#### 3.1. The Maximum Foldable Angle of One Single Regular Polygon Using Alternating Asymmetric Triangulation

#### 3.2. The Tessellation of Concentric Pleating Squares

#### 3.3. Suggestions for Configuration

## 4. Conclusions

## Supplementary Materials

Supplementary File 1## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Gantes, C.J. Deployable Structures: Analysis and Design; National Technical University of Athens: Athens, Greece, 2001. [Google Scholar]
- Rogers, J.; Huang, Y.; Schmidt, O.G.; Gracias, D.H. Origami MEMS and NEMS. MRS Bull.
**2016**, 2, 123–129. [Google Scholar] [CrossRef] - Gracias, D.; Cho, J.-H.; Hu, S. A Three Dimensional Self-folding Package (SFP) for Electronics. MRS Proc.
**2010**, 1249, F09. [Google Scholar] [CrossRef] - Vergauwen, A.; De Temmerman, N.; Brancart, S. Design and physical modelling of deployable structures based on curved-line folding. In Proceedings of the 4th International Conference on Mobile, Adaptable and Rapidly Assembled Structures, Ostend, Belgium, 11–13 June 2014; WIT Press: Southampton, UK, 2014. [Google Scholar] [CrossRef]
- Puig, L.; Barton, A.; Rando, N. A review on large deployable structures for astrophysics missions. Acta Astronaut.
**2010**, 67, 12–26. [Google Scholar] [CrossRef] - Naser, M.Z.; Chehab, A.I. Materials and design concepts for space-resilient structures. Prog. Aerosp. Sci.
**2018**, 98, 74–90. [Google Scholar] [CrossRef] - De Temmerman, N.; Roovers, K.; Alegria Mira, L.; Vergauwen, A.; Koumar, A.; Brancart, S.; De Laet, L.; Mollaert, M. Engineering lightweight transformable structures. In Proceedings of the International Conference on Adaptation and Movement in Architecture, Toronto, ON, Canada, 10–11 October 2013. [Google Scholar]
- Gross, D.; Messner, D. The able deployable articulated mast–enabling technology for the shuttle radar topography mission. In Proceedings of the 33rd Aerospace Mechanisms Symposium, Pasadena, CA, USA, 19–21 May 1999. [Google Scholar]
- Van Knippenberg, R. Stable Adaptive Structures. Master’s Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 28 August 2014. [Google Scholar]
- Tachi, T. Geometric considerations for the design of rigid origami structures. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2010, Shanghai, China, 8–12 November 2010. [Google Scholar]
- Tachi, T. Generalization of rigid-foldable quadrilateral-mesh origami. In Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2009, Valencia, Spain, 28 September–2 October 2009. [Google Scholar]
- Miura, K. Method of Packaging and Deployment of Large Membranes in Space; Institute of Space and Astronautical Sciences, 1985; Available online: https://repository.exst.jaxa.jp/dspace/handle/a-is/7293 (accessed on 21 September 2019).
- Sareh, P.; Guest, S.D. Designing Symmetric Derivatives of the Miura-Ori; Advances in Architectural Geometry 2014; Springer: London, UK, 2015; Chapter 15; pp. 233–241. [Google Scholar] [CrossRef]
- Sareh, P.; Guest, S.D. Minimal isomorphic symmetric variations on the Miura fold pattern. In Proceedings of the First Conference Transformables 2013, Seville, Spain, 18–20 September 2013. [Google Scholar]
- Schenk, M.; Guest, S.D. Geometry of Miura-folded metamaterials. Proc. Natl. Acad. Sci. USA
**2013**, 110, 3276–3281. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tachi, T. Designing freeform origami tessellations by generalizing resch’s patterns. J. Mech. Des.
**2013**, 135, 111006. [Google Scholar] [CrossRef] - Resch, R.D. The topological design of sculptural and architectural systems. In Proceedings of the 1973, National Computer Conference and Exposition, New York, NY, USA, 4–8 June 1973; ACM: New York, NY, USA, 1973; pp. 643–650. [Google Scholar] [CrossRef]
- Tachi, T. Origamizing polyhedral surfaces. IEEE Trans. Vis. Comput. Graph.
**2010**, 16, 298–311. [Google Scholar] [CrossRef] [PubMed] - Demaine, E.D.; Demaine, M.L.; Hart, V.; Price, G.N.; Tachi, T. (Non)existence of pleated folds: How paper folds between creases. Gr. Comb.
**2011**, 27, 377–397. [Google Scholar] [CrossRef] - Tachi, T. Rigid Origami Simulator. Available online: http://www.tsg.ne.jp/TT/software/ (accessed on 28 September 2007).
- Demaine, E.D.; Tachi, T. Origamizer: A practical algorithm for folding any polyhedron. In Proceedings of the 33rd International Symposium on Computational Geometry, Brisbane, Australia, 4–7 July 2017; pp. 34:1–34:16. [Google Scholar]
- Tachi, T. Origamizer. Available online: https://tsg.ne.jp/TT/software/Origamizer047.zip (accessed on 29 August 2008).
- Tachi, T. Freeform Origami. Available online: https://tsg.ne.jp/TT/software/FreeformOrigami030.zip (accessed on 16 October 2010).
- Piker, D. Kangaroo. Available online: https://www.grasshopper3d.com/group/kangaroo/ (accessed on 23 May 2015).
- Grasshopper. Available online: https://www.grasshopper3d.com/ (accessed on 23 May 2008).
- Robert McNeel & Associates. Rhinoceros. 2015. Available online: https://www.rhino3d.com/ (accessed on 21 September 2010).
- Fushimi, K.; Fushimi, M. Origami No Kikagaku (Geometry of Origami); Nihon Hyoronsha: Tokyo, Japan, 1979. [Google Scholar]

**Figure 1.**Examples of deployable structures tht are used: (

**a**) Optical images showing experimentally realized packaging of cantilever sensors and magnetic field sensitive strain gauges in polyhedral geometries [3]; (

**b**) conceptual design of a facade with adaptive shading elements based on curved-line folding [7]; (

**c**) the ADAM mast deployed and canister [8]; (

**d**) expandable base (courtesy of NASA) [6].

**Figure 2.**Square concentric pleating origami and the crease pattern: (

**a**) The crease pattern; (

**b**) the origami model.

**Figure 3.**Foldable triangulations of the hyperbolic paraboloid crease pattern: (

**a**) Asymmetric triangulation; (

**b**) alternating asymmetric triangulation.

**Figure 8.**Using the intersections of circles on the different co-ordinate planes to find a

_{2}and b

_{2}: (

**a**) Intersections of circles on the Y–Z plane; (

**b**) intersections of circles on the X–Z plane.

**Figure 10.**A regular hexagon using alternating asymmetric triangulation. (

**a**) The creases of a regular hexagon, in which the full lines represent the mountains and the dotted lines represent the valleys in this folding pattern. The green lines are used to triangulate the trapezoids; (

**b**) a sub-plate of a regular hexagon in the X–Y plane.

**Figure 15.**Demonstration of the use of Kangaroo in Grasshopper to model the folding motion of a pattern using alternating asymmetric triangulation.

**Figure 16.**The parametric model of squares and regular hexagons with alternating asymmetric triangulation obtained in the digital environment of Rhinoceros: (

**a**) 3-D models of squares; (

**b**) 3-D models of regular hexagons.

**Figure 18.**The trends of μ

_{1}and μ

_{2}in squares and regular hexagons: (

**a**) The trend of μ

_{1}in squares; (

**b**) the trend of μ

_{2}in squares; (

**c**) the trend of μ

_{1}in regular hexagons; (

**d**) the trend of μ

_{2}in regular hexagons.

**Figure 19.**The perspective observed in Rhinoceros. (

**a**) The folding of a regular hexagon when the folding angle reaches its maximum foldable angle; (

**b**) the perspective of the hexagon when the angle is above the maximum foldable angle.

**Figure 20.**Tessellation of squares with alternating asymmetric triangulation: (

**a**) Top view; (

**b**) front view.

**Figure 21.**Reverse buckling of squares with alternating asymmetric triangulation: (

**a**) No buckling; (

**b**) critical state; (

**c**) buckling.

Type of Polygons | The Central Point | Points on 1st Closed Loop | Points on 2nd Closed Loop | Points on 3rd Closed Loop | … | Points on Edges |
---|---|---|---|---|---|---|

Regular tetragon | 4 | 4 or 6 | 4 or 8 | 4 or 8 | 4 or 8 | 3 or 5 |

Regular hexagon | 6 | 4 or 6 | 4 or 8 | 4 or 8 | 4 or 8 | 3 or 5 |

Regular octagon | 8 | 4 or 6 | 4 or 8 | 4 or 8 | 4 or 8 | 3 or 5 |

N-regular polygon | n | 4 or 6 | 4 or 8 | 4 or 8 | 4 or 8 | 3 or 5 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Q.; Yu, D.; Li, X.; Fan, F.
Computational Modeling Methods for Deployable Structure Based on Alternatively Asymmetric Triangulation. *Symmetry* **2019**, *11*, 1278.
https://doi.org/10.3390/sym11101278

**AMA Style**

Zhang Q, Yu D, Li X, Fan F.
Computational Modeling Methods for Deployable Structure Based on Alternatively Asymmetric Triangulation. *Symmetry*. 2019; 11(10):1278.
https://doi.org/10.3390/sym11101278

**Chicago/Turabian Style**

Zhang, Qingwen, Danmin Yu, Xinye Li, and Feng Fan.
2019. "Computational Modeling Methods for Deployable Structure Based on Alternatively Asymmetric Triangulation" *Symmetry* 11, no. 10: 1278.
https://doi.org/10.3390/sym11101278