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Research Papers

Kinematics of Origami Structures With Smooth Folds

[+] Author and Article Information
Edwin A. Peraza Hernandez

Graduate Research Assistant
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: eperaza@tamu.edu

Darren J. Hartl

Research Assistant Professor
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: darren.hartl@tamu.edu

Dimitris C. Lagoudas

Professor
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843;
Department of Materials
Science and Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: lagoudas@tamu.edu

1Corresponding author.

Manuscript received April 24, 2016; final manuscript received July 13, 2016; published online October 11, 2016. Assoc. Editor: Robert J. Wood.

J. Mechanisms Robotics 8(6), 061019 (Oct 11, 2016) (22 pages) Paper No: JMR-16-1115; doi: 10.1115/1.4034299 History: Received April 24, 2016; Revised July 13, 2016

Origami provides both inspiration and potential solutions to the fabrication, assembly, and functionality of various structures and devices. Kinematic modeling of origami-based objects is essential to their analysis and design. Models for rigid origami, in which all planar faces of the sheet are rigid and folds are limited to straight creases having only zeroth-order geometric continuity, are available in the literature. Many of these models include constraints on the fold angles to ensure that any initially closed strip of faces is not torn during folding. However, these previous models are not intended for structures with non-negligible fold thickness or with maximum curvature at the folds restricted by material or structural limitations. Thus, for general structures, creased folds of merely zeroth-order geometric continuity are not appropriate idealizations of structural response, and a new approach is needed. In this work, a novel model analogous to those for rigid origami with creased folds is presented for sheets having realistic folds of nonzero surface area and exhibiting higher-order geometric continuity, here termed smooth folds. The geometry of smooth folds and constraints on their associated shape variables are presented. A numerical implementation of the model allowing for kinematic simulation of sheets having arbitrary fold patterns is also described. Simulation results are provided showing the capability of the model to capture realistic kinematic response of origami sheets with diverse fold patterns.

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Figures

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Fig. 2

Sheet with creased folds in its reference configuration S0 (a) and a current configuration St (b). The planar faces comprising the sheet undergo only rigid deformations.

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Fig. 3

Schematics showing unfolded and folded configurations of a creased fold

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Fig. 4

Schematic showing faces and folds connected to an interior vertex and their associated geometric parameters

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Fig. 1

(a) A conventional origami sheet having creased folds of zeroth-order geometric continuity (G0). (b) A sheet having smooth folds of nonzero surface area and higher-order geometric continuity (Gn).

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Fig. 5

Sheet with smooth folds in its reference configuration S0 (a) and a current configuration St (b)

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Fig. 6

Schematics showing unfolded and folded configurations of a smooth fold, cf. Fig. 3

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Fig. 7

Cross section of a smooth fold. The fold shape variables and the fold-attached coordinate system are shown.

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Fig. 8

Geometric parameters defining F0i

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Fig. 9

Schematic showing faces and smooth folds adjacent to an interior fold intersection and associated geometric parameters

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Fig. 10

(a) Path γj(η) crossing the faces and smooth folds joined to I0j. (b) Vectors wjk and ljk with start-points and end-points corresponding to the points where the path γj(η) crosses the boundary rulings of the smooth folds.

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Fig. 11

Schematics illustrating the transformation associated with folding of the kth smooth fold crossed by γj(η). (a) Reference configuration of the fold. (b) Intermediate step to determine the location of the axes of rotation taking into account the change in the distance between the boundary rulings of the smooth fold in a current configuration. Note that the vector gj k−1 is subtracted from bLjk to account for the previous smooth folds crossed by γj(η). (c) Rotation by (1−ajk)θjk about an axis aligned to mjk and crossing a point with position vector bRjk−gjk. (d) Rotation by ajkθjk about an axis aligned to mjk and a crossing point with position vector bLjk−gj k−1. (e) Resulting configuration of the smooth fold and its adjacent faces.

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Fig. 12

(a) Two equivalent paths γ⌣j(η) connecting the fixed face to P0j. (b) Paths γ⌣j(η), j=1,...,NP, connecting the fixed face to every other face in S0.

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Fig. 13

(a) and (b): Configurations of a sheet having a single interior fold intersection obtained through the guess fold angle increments provided in Eqs. (101) and (102), respectively. (c) Fold angles versus increment number and configurations obtained through the guess fold angle increments provided in Eq. (103).

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Fig. 14

Configurations for origami sheets with vertex coordinates and fold centerlines defined in Fig. 16: (a) sheet with the baseline fold pattern, (b) sheet with a fold pattern generated by modifying the interior vertex coordinates of the baseline fold pattern, and (c) sheet with a fold pattern generated by modifying the boundary vertex coordinates of the baseline fold pattern

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Fig. 15

Graphs showing the vertex coordinates and fold centerlines for the sheets shown in Fig. 14

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Fig. 16

Evolution of fold angles with increment number for the sheets shown in Fig. 14

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Fig. 17

Demonstration of constrained deformation associated with origami sheets having four, five, and six interior fold intersections. Note that configurations of substantial folding are captured without bending or stretching of the faces or tearing of the sheet.

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Fig. 18

Schematic showing two adjacent creased folds (along the vectors mjk and mj k+1) incident to a common interior vertex. The vectors w̃jk and l̃jk are shown.

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