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Unfolding Polyhedra Method for the Design of Origami Structures with Smooth Folds

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Active Origami

Abstract

In this chapter, we address the method of unfolding polyhedra for origami structures with smooth folds. We develop this method based on the theory of unfolding polyhedra for origami with creased folds studied in Chap. 3. Accordingly, the goal shape is represented as a three-dimensional goal mesh. The objective is to determine the geometry of a planar sheet with smooth folds that can be folded towards a configuration that approximates the goal mesh. We also examine the computational implementation aspects of unfolding polyhedra for origami with smooth folds and provide representative examples.

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Notes

  1. 1.

    Consult Sect. 5.3 for a detailed description of the geometry of smooth folds.

  2. 2.

    Consult Sect. 5.5.2 for details on the determination of the kinematic variables \(\hat {w}_i\) for the cross-section of smooth folds.

References

  1. M. Schenk, A.D. Viquerat, K.A. Seffen, S.D. Guest, Review of inflatable booms for deployable space structures: packing and rigidization. J. Spacecr. Rocket. 51(3), 762–778 (2014)

    Google Scholar 

  2. Y. Shi, F. Zhang, K. Nan, X. Wang, J. Wang, Y. Zhang, Y. Zhang, H. Luan, K.-C. Hwang, Y. Huang, J.A. Rogers, Y. Zhang, Plasticity-induced origami for assembly of three dimensional metallic structures guided by compressive buckling. Extreme Mech. Lett. 11, 105–110 (2017)

    Google Scholar 

  3. Z. Yan, F. Zhang, J. Wang, F. Liu, X. Guo, K. Nan, Q. Lin, M. Gao, D. Xiao, Y. Shi, Y. Qiu, H. Luan, J.H. Kim, Y. Wang, H. Luo, M. Han, Y. Huang, Y. Zhang, J.A. Rogers, Controlled mechanical buckling for origami-inspired construction of 3D microstructures in advanced materials. Adv. Funct. Mater. 26(16), 2629–2639 (2016)

    Google Scholar 

  4. N. Turner, B. Goodwine, M. Sen, A review of origami applications in mechanical engineering. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. (2015). https://doi.org/10.1177/0954406215597713

  5. L.J. Fei, D. Sujan, Origami theory and its applications: a literature review. Int. J. Soc. Hum. Sci. Eng. 7(1), 113–117 (2013)

    Google Scholar 

  6. M. Johnson, Y. Chen, S. Hovet, S. Xu, B. Wood, H. Ren, J. Tokuda, Z. Tse, Fabricating biomedical origami: a state-of-the-art review. Int. J. Comput. Assist. Radiol. Surg. 12(11), 2023–2032 (2017)

    Google Scholar 

  7. Q. Ge, C.K. Dunn, H.J. Qi, M.L. Dunn, Active origami by 4D printing. Smart Mater. Struct. 23(9), 094007 (2014)

    Google Scholar 

  8. H.J. In, S. Kumar, Y. Shao-Horn, G. Barbastathis, Origami fabrication of nanostructured, three-dimensional devices: electrochemical capacitors with carbon electrodes. Appl. Phys. Lett. 88(8), 083104 (2006)

    Google Scholar 

  9. D. Han, S. Pal, J. Nangreave, Z. Deng, Y. Liu, H. Yan, DNA origami with complex curvatures in three-dimensional space. Science 332(6027), 342–346 (2011)

    Google Scholar 

  10. C.D. Santangelo, Extreme mechanics: self-folding origami. Annu. Rev. Condens. Matter Phys. 8, 165–183 (2017)

    Google Scholar 

  11. E.A. Peraza Hernandez, D.J. Hartl, R.J. Malak Jr, D.C. Lagoudas, Origami-inspired active structures: a synthesis and review. Smart Mater. Struct. 23(9), 094001 (2014)

    Google Scholar 

  12. M.B. Pinson, M. Stern, A.C. Ferrero, T.A. Witten, E. Chen, A. Murugan, Self-folding origami at any energy scale. Nat. Commun. 8, 15477 (2017)

    Google Scholar 

  13. Z. Song, T. Ma, R. Tang, Q. Cheng, X. Wang, D. Krishnaraju, R. Panat, C.K. Chan, H. Yu, H. Jiang, Origami lithium-ion batteries. Nat. Commun. 5, 3140 (2014)

    Google Scholar 

  14. I. Shitanda, S. Kato, S. Tsujimura, Y. Hoshi, M. Itagaki, Screen-printed, paper-based, array-type, origami biofuel cell. Chem. Lett. 46, 726–728 (2017)

    Google Scholar 

  15. J.P. Rojas, D. Conchouso, A. Arevalo, D. Singh, I.G. Foulds, M.M. Hussain, Paper-based origami flexible and foldable thermoelectric nanogenerator. Nano Energy 31, 296–301 (2017)

    Google Scholar 

  16. J.T. Early, R. Hyde, R.L. Baron, Twenty-meter space telescope based on diffractive Fresnel lens, in Proceedings of the SPIE’s 48th Annual Meeting, Optical Science and Technology (International Society for Optics and Photonics, San Diego, 2004), pp. 148–156

    Google Scholar 

  17. L. Xu, T.C. Shyu, N.A. Kotov, Origami and kirigami nanocomposites. ACS Nano 11(8), 7587–7599 (2017)

    Google Scholar 

  18. G. Stoychev, M.J. Razavi, X. Wang, L. Ionov, 4D origami by smart embroidery. Macromol. Rapid Commun. 38(18), 1700213 (2017)

    Google Scholar 

  19. E.D. Demaine, J. O’Rourke, Geometric Folding Algorithms (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  20. T.A. Evans, R.J. Lang, S.P. Magleby, L.L. Howell, Rigidly foldable origami gadgets and tessellations. R. Soc. Open Sci. 2(9), 150067 (2015)

    Google Scholar 

  21. X. Zhou, H. Wang, Z. You, Design of three-dimensional origami structures based on a vertex approach. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471(2181), 20150407 (2015)

    Google Scholar 

  22. W. Liu, K. Tai, Optimal design of flat patterns for 3D folded structures by unfolding with topological validation. Comput Aided Des. 39(10), 898–913 (2007)

    Google Scholar 

  23. S. Bouzette, F. Buekenhout, E. Dony, A. Gottcheiner, A theory of nets for polyhedra and polytopes related to incidence geometries. Des. Codes Cryptogr. 10(2), 115–136 (1997)

    Google Scholar 

  24. W. Schlickenrieder, Nets of polyhedra. Master’s thesis, Technische Universität Berlin, 1997

    Google Scholar 

  25. M. Bern, E.D. Demaine, D. Eppstein, E. Kuo, A. Mantler, J. Snoeyink, Ununfoldable polyhedra with convex faces. Comput. Geom. 24(2), 51–62 (2003)

    Google Scholar 

  26. B. Grünbaum, Nets of polyhedra II. Geombinatorics 1(3), 5–10 (1991)

    Google Scholar 

  27. M. Damian, R. Flatland, J. O’Rourke, Epsilon-unfolding orthogonal polyhedra. Graph. Comb. 23(1), 179–194 (2007)

    Google Scholar 

  28. S. Takahashi, H.-Y. Wu, S.H. Saw, C.-C. Lin, H.-C. Yen, Optimized topological surgery for unfolding 3D meshes. Comput. Graph. Forum 30(7), 2077–2086 (2011)

    Google Scholar 

  29. E.D. Demaine, A. Lubiw, A generalization of the source unfolding of convex polyhedra, in Spanish Meeting on Computational Geometry (Springer, Berlin, 2011), pp. 185–199

    Google Scholar 

  30. Z. Xi, Y. Kim, Y.J. Kim, J.-M. Lien, Learning to segment and unfold polyhedral mesh from failures. Comput. Graph. 58, 139–149 (2016)

    Google Scholar 

  31. M.A. Alam, I. Streinu, Star-unfolding polygons, in International Workshop on Automated Deduction in Geometry (Springer, Berlin, 2014), pp. 1–20

    Google Scholar 

  32. V. Chandru, R. Hariharan, N.M. Krishnakumar, Short-cuts on star, source and planar unfoldings, in International Conference on Foundations of Software Technology and Theoretical Computer Science (Springer, Berlin, 2004), pp. 174–185

    Google Scholar 

  33. S. Kiazyk, A. Lubiw, Star unfolding from a geodesic curve. Discret. Comput. Geom. 56(4), 1018–1036 (2016)

    Google Scholar 

  34. J. Itoh, J. O’Rourke, C. Vîlcu, Star unfolding convex polyhedra via quasigeodesic loops. Discret. Comput. Geom. 44(1), 35–54 (2010)

    Google Scholar 

  35. Y. Lee, M. Cho, Self-folding structure using light-absorption of polystyrene sheet, in Proceedings of SPIE Smart Structures and Materials+Nondestructive Evaluation and Health Monitoring (International Society for Optics and Photonics, San Diego, 2017), p. 101650L

    Google Scholar 

  36. H. He, J. Guan, J.L. Lee, An oral delivery device based on self-folding hydrogels. J. Control. Release 110(2), 339–346 (2006)

    Article  Google Scholar 

  37. J. Guan, H. He, D.J. Hansford, L.J. Lee, Self-folding of three-dimensional hydrogel microstructures. J. Phys. Chem. B 109(49), 23134–23137 (2005)

    Article  Google Scholar 

  38. G. Stoychev, N. Puretskiy, L. Ionov, Self-folding all-polymer thermoresponsive microcapsules. Soft Matter 7(7), 3277–3279 (2011)

    Article  Google Scholar 

  39. C. Yoon, R. Xiao, J. Park, J. Cha, T.D. Nguyen, D.H. Gracias, Functional stimuli responsive hydrogel devices by self-folding. Smart Mater. Struct. 23(9), 094008 (2014)

    Google Scholar 

  40. S. Chen, J. Li, L. Fang, Z. Zhu, S.H. Kang, Simple triple-state polymer actuators with controllable folding characteristics. Appl. Phys. Lett. 110(13), 133506 (2017)

    Google Scholar 

  41. S. Pandey, M. Ewing, A. Kunas, N. Nguyen, D.H. Gracias, G. Menon, Algorithmic design of self-folding polyhedra. Proc. Natl. Acad. Sci. 108(50), 19885–19890 (2011)

    Article  Google Scholar 

  42. Y. Liu, J.K. Boyles, J. Genzer, M.D. Dickey, Self-folding of polymer sheets using local light absorption. Soft Matter 8(6), 1764–1769 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Texas A&M University, College Station, TX, USA

    Edwin A. Peraza Hernandez, Darren J. Hartl & Dimitris C. Lagoudas

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Peraza Hernandez, E.A., Hartl, D.J., Lagoudas, D.C. (2019). Unfolding Polyhedra Method for the Design of Origami Structures with Smooth Folds. In: Active Origami. Springer, Cham. https://doi.org/10.1007/978-3-319-91866-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-91866-2_6

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91865-5

  • Online ISBN: 978-3-319-91866-2

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