Research Papers

Nonrigidly Foldability Analysis of Kresling Cylindrical Origami

[+] Author and Article Information
Cai Jianguo

Key Laboratory of C & PC Structures
of Ministry of Education,
National Prestress Engineering Research Center,
Southeast University,
Si Pai Lou 2#,
Nanjing 210096, China
e-mails: j.cai@seu.edu.cn;

Liu Yangqing

School of Civil Engineering,
Southeast University,
Nanjing 210096, China
e-mail: iamliuyangqing163@163.com

Ma Ruijun

School of Civil Engineering,
Southeast University,
Nanjing 210096, China
e-mail: 1175455218@qq.com

Feng Jian

School of Civil Engineering,
Southeast University,
Nanjing 210096, China
e-mail: fengjian@seu.edu.cn

Zhou Ya

Wuxi Architectural Design &
Research Institute Co. Ltd.,
Wuxi 214001, Jiangsu, China
e-mail: zhouya5166@126.com

1Corresponding author.

Manuscript received August 1, 2016; final manuscript received April 26, 2017; published online June 14, 2017. Assoc. Editor: Guimin Chen.

J. Mechanisms Robotics 9(4), 041018 (Jun 14, 2017) (10 pages) Paper No: JMR-16-1224; doi: 10.1115/1.4036738 History: Received August 01, 2016; Revised April 26, 2017

Rigid origami is seen as a fundamental model in many self-folding machines. A key issue in designing origami is the rigid/nonrigid foldability. The kinematic and foldability of Kresling origami, which is based on an origami pattern of the vertex with six creases, are studied in this paper. The movement of the single-vertex is first discussed. Based on the quaternion method, the loop-closure equation of the vertex with six creases is obtained. Then, the multitransformable behavior of the single vertex is investigated. Furthermore, the rigid foldability of origami patterns with multivertex is investigated with an improved dual quaternion method, which is based on studying the folding angle and the coordinates of all vertices. It can be found that the Kresling cylinder is not rigidly foldable.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


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]
Chen, Y. , Peng, R. , and You, Z. , 2015, “ Origami of Thick Panels,” Science, 349(6246), pp. 396–400. [CrossRef] [PubMed]
Yasuda, H. , Chen, Z. , and Yang, J. , 2016, “ Multitransformable Leaf-Out Origami With Bistable Behavior,” ASME J. Mech. Rob., 8(3), p. 031013. [CrossRef]
Lee, T. U. , and Gattas, J. M. , 2016, “ Geometric Design and Construction of Structurally Stabilized Accordion Shelters,” ASME J. Mech. Rob., 8(3), p. 031009. [CrossRef]
Ma, J. , and You, Z. , 2014, “ Energy Absorption of Thin-Walled Square Tubes With a Prefolded Origami Pattern—Part I: Geometry and Numerical Simulation,” ASME J. Appl. Mech., 81(1), p. 011003. [CrossRef]
Cai, J. G. , Xu, Y. X. , and Feng, J. , 2013, “ Geometric Analysis of a Foldable Barrel Vault With Origami,” ASME J. Mech. Des., 135(11), p. 114501.
Miura, K. , 1980, “ Method of Packaging and Deployment of Large Membranes in Space,” 31st Congress of International Astronautics Federation (IAF-80-A31), Tokyo, Japan, Sept. 21–28, pp. 1–10.
Demaine, E. D. , and O'Rourke, J. , 2005, A Survey of Folding and Unfolding in Computational Geometry, Vol. 52, Mathematical Sciences Research Institute Publications, Berkeley, CA, pp. 167–211.
Demaine, E. D. , and Demaine, M. L. , 1997, “ Computing Extreme Origami Bases,” Technical, School of Computer Science, University of Waterloo, Waterloo, ON, Canada, Report No. CS-97-22.
Streinu, I. , and Whiteley, W. , 2005, “ Single-Vertex Origami and Spherical Expansive Motions,” Lecture Notes in Computer Science, Vol. 3742, Springer, Berlin, pp. 161–173.
Watanabe, N. , and Kawaguchi, K. , 2006, “ The Method for Judging Rigid Foldability,” Origami 4th International Conference on Origami in Science, Mathematics, and Education, Pasadena, CA, Sept. 8–10, pp. 165–174.
Dai, J. S. , and Jones, J. R. , 1999, “ Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds,” ASME J. Mech. Des., 121(3), pp. 375–382. [CrossRef]
Wei, G. W. , and Dai, J. S. , 2013, “ Origami-Inspired Integrated Planar-Spherical Overconstrained Mechanisms,” ASME J. Mech. Des., 136(5), p. 051003. [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]
Zhang, K. , Qiu, C. , and Dai, J. S. , 2016, “ An Extensible Continuum Robot With Integrated Origami Parallel Modules,” ASME J. Mech. Rob., 8(3), p. 031010. [CrossRef]
Wu, W. , and You, Z. , 2010, “ Modelling Rigid Origami With Quaternions and Dual Quaternions,” Proc. R. Soc. A: Math., Phys. Eng. Sci., 466(2119), pp. 2155–2174. [CrossRef]
Yoshimura, Y. , 1951, “ On the Mechanism of Buckling of a Circular Cylindrical Shell Under Axial Compression and Bending,” Reports of the Institute of Science and Technology, University of Tokyo, Tokyo, Japan (English Translation: Technical Memorandum 1390 of the National Advisory Committee for Aeronautics, Washington DC, 1955).
Hunt, G. W. , Lord, G. J. , and Peletier, M. A. , 2003, “ Cylindrical Shell Buckling: A Characterization of Localization and Periodicity,” Discrete Continuous Dyn. Syst. B, 3(4), pp. 505–518. [CrossRef]
Guest, S. D. , and Pellegrino, S. , 1994, “ The Folding of Triangulated Cylinders—Part I: Geometric Considerations,” ASME J. Appl. Mech., 61(4), pp. 773–777. [CrossRef]
Kresling, B. , 1996, “ Plant ‘Design’: Mechanical Simulations of Growth Patterns and Bionics,” Biomimetics, 3(3), pp. 105–222.
Kresling, B. , 2008, “ Natural Twist Buckling in Shells: From the Hawkmoth's Bellows to the Deployable Kresling-Pattern and Cylindrical Miura-ori,” 6th International Conference on Computation of Shell and Spatial Structures, J. F. Abel and J. Robert Cooke , eds., IASS-IACM, Spanning Nano to Mega, Ithaca, NY, May 28–31, pp. 1–4.
Cai, J. , Deng, X. , Zhang, Y. , Feng, J. , and Zhou, Y. , 2016, “ Folding Behavior of a Foldable Prismatic Mast With Kresling Origami Pattern,” ASME J. Mech. Rob., 8(3), p. 031004. [CrossRef]
Tachi, T. , 2009, “ Generalization of Rigid Foldable Quadrilateral Mesh Origami,” International Association for Shell and Spatial Structures Symposium (IASS), Madrid, Spain, Sept. 28–Oct. 2, pp. 2287–2294.
Tachi, T. , 2010, “ Freeform Rigid-Foldable Structure Using Bidirectionally Flat-Foldable Planar Quadrilateral Mesh,” Advances in Architectural Geometry 2010, Springer, Vienna, Austria, pp. 87–102.
Yasuda, H. , Yein, T. , Tachi, T. , Miura, K. , and Taya, M. , 2013, “ Folding Behaviour of Tachi-Miura Polyhedron Bellows,” Proc. R. Soc. A, 469(2159), p. 20130351. [CrossRef]
Nojima, T. , 2002, “ Modelling of Folding Patterns in Flat Membranes and Cylinder by Origami,” JSME Int. J. Ser. C, 45(1), pp. 364–370. [CrossRef]
Nojima, T. , 2007, “ Origami Modeling of Functional Structures Based on Organic Patterns,” VIPSI Conference, Tokyo, Japan, May 31–June 6, pp. 1–21.
Liu, S. C. , Lv, W. L. , Chen, Y. , and Lu, G. X. , 2016, “ Deployable Prismatic Structures With Rigid Origami Patterns,” ASME J. Mech. Rob., 8(3), p. 031002. [CrossRef]
Cai, J. G. , Zhang, Y. T. , Xu, Y. X. , Zhou, Y. , and Feng, J. , 2016, “ The Foldability of Cylindrical Foldable Structures Based on Rigid Origami,” ASME J. Mech. Des., 138(3), p. 031401. [CrossRef]
Kuipers, J. B. , 2002, “ Quaternions and Rotation Sequences: A Primer With Applications to Orbits,” Aerospace and Virtual Reality, Princeton University Press, Princeton, NJ.
Dai, J. S. , 2015, “ Euler-Rodrigues Formula Variations, Quaternion Conjugation and Intrinsic Connections,” Mech. Mach. Theory, 92, pp. 144–152. [CrossRef]
Cai, J. G. , Ma, R. J. , Feng, J. , Zhou, Y. , and Deng, X. W. , 2016, “ Foldability Analysis of Cylindrical Origami Structures,” Advances in Reconfigurable Mechanisms and Robots II, Springer, Berlin, pp. 143–153.
Chen, G. , Zhang, S. , and Li, G. , 2013, “ Multistable Behaviors of Compliant Sarrus Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021005. [CrossRef]
Ma, F. , and Chen, G. , 2015, “ Modeling Large Planar Deflections of Flexible Beams in Compliant Mechanisms Using Chained Beam-Constraint-Model,” ASME J. Mech. Rob., 8(2), p. 021018. [CrossRef]
Zhou, W. , Chen, Y. , Peng, B. , Yang, H. , Yu, H. , Liu, H. , and He, X. , 2014, “ Air Damping Analysis in Comb Microaccelerometer,” Adv. Mech. Eng., 6, p. 373172. [CrossRef]
Zhou, W. , Chen, L. , Yu, H. , Peng, B. , and Chen, Y. , 2016, “ Sensitivity Jump of Micro Accelerometer Induced by Micro-Fabrication Defects of Micro Folded Beams,” Meas. Sci. Rev., 16(4), pp. 228–234. [CrossRef]


Grahic Jump Location
Fig. 1

Kresling origami pattern

Grahic Jump Location
Fig. 2

One vertex with six creases

Grahic Jump Location
Fig. 3

The rotation of a vector

Grahic Jump Location
Fig. 4

The first four steps of the rotation of the edge and normal vectors: (a) the first step, (b) the second step, (c) the third step, and (d) the fourth step

Grahic Jump Location
Fig. 5

Output folding angles t6 versus input folding angles t3

Grahic Jump Location
Fig. 6

Output folding angles t4, t5, t6 versus input folding angles t1 and t3: (a) t4, (b) t5, and (c) t6

Grahic Jump Location
Fig. 7

A typical configuration in which every crease can fold

Grahic Jump Location
Fig. 13

The origami pattern of a cylinder

Grahic Jump Location
Fig. 14

Crease vectors in vertex P1

Grahic Jump Location
Fig. 15

Vertex coordinate transfer along a closed path

Grahic Jump Location
Fig. 16

Coordinate changes of the last vertex



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In