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Research Papers

Thick Rigidly Foldable Origami Mechanisms Based on Synchronized Offset Rolling Contact Elements

[+] Author and Article Information
Robert J. Lang

Lang Origami,
Alamo, CA 94507
e-mail: robert@langorigami.com

Todd Nelson

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602
e-mail: toddgn@gmail.com

Spencer Magleby

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

Larry Howell

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

Manuscript received October 10, 2016; final manuscript received December 22, 2016; published online March 9, 2017. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 9(2), 021013 (Mar 09, 2017) (17 pages) Paper No: JMR-16-1300; doi: 10.1115/1.4035686 History: Received October 10, 2016; Revised December 22, 2016

We present a general technique for achieving kinematic single degree of freedom (1DOF) origami-based mechanisms with thick rigid panels using synchronized offset rolling contact elements (SORCEs). We present general design analysis for planar and 3D relative motions between panels and show physically realized examples. The technique overcomes many of the limitations of previous approaches for thick rigidly foldable mechanisms.

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References

Figures

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Fig. 1

A thick degree-4 vertex with sector angles α1α4 and fold angles γ1γ4. Left: the unfolded state, for which all panels are coplanar with zero offset. Right: the flat-folded state, for which the four panels should be offset from their zero-thickness position.

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Fig. 2

Offset panels in a degree-4 vertex. Each panel is offset perpendicularly to its zero-thickness facet by an amount zi(t), where t parameterizes the state of folding.

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Fig. 3

Three positions of a variable-offset joint, viewed in a plane perpendicular to the axis of rotation: (a) t = 0, unfolded, (b) intermediate t; the rotations and offsets are characterized by (γ(t),zl(t),zr(t)), and (c) fully flat-folded. Both panels are offset relative to their zero-thickness facets, whose positions are indicated by the dotted lines in all three subfigures.

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Fig. 4

Two positions of a pure revolute joint, viewed in a plane perpendicular to the axis of rotation: (a) t = 0, unfolded, (b) intermediate t; the rotation angle is (γ(t); the offsets are zl(t)=zr(t)=0. The zero-thickness facets' positions are indicated by the dotted lines.

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Fig. 5

Geometry of a circular rolling contact between two contacts with circular cross section of radius r at a fold angle ofγ(t)

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Fig. 6

Geometry of a general rolling contact between two surfaces: (a) at a fold angle γ(0)=0, (b) at an intermediate fold angle of γ(t) with t > 0, and (c) geometric relations that relate the elevation functions zl(t) and zr(t) to the vector q(t)

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Fig. 7

Surface functions and panel positions for the four joints at four different values of the fold parameter t. Left to right: t = 0 (unfolded), 0.333, 0.667, and t = 1.0 (flat folded). Top row: joint corresponding to fold γ1. Second row: γ2. Third row: γ3. Bottom row: γ4. The zero-thickness facets and fold axis are indicated by black dotted lines and dot, respectively.

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Fig. 8

Convexity functions (signed curvatures) for the four joints. Left: for left surfaces sl,i. Right: for right surfaces sr,i.

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Fig. 9

Design of the optimized rolling-contact surfaces for a degree-4 vertex. Top left: the fold angles versus t (note that γ2 and γ4 overlap). Top right: the offset functions (note that z1 and z4 overlap, as do z2 and z3). Bottom row: the four convexity functions for left and right surfaces.

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Fig. 10

Design of the optimized rolling-contact surfaces for a degree-4 vertex. Panels and surfaces for the four joints at t = 0. Top: γ1 and γ2. Bottom: γ3 and γ4. The zero-thickness facets are indicated by black dotted lines; the fold axis is indicated by the heavy black dot in the middle of each image.

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Fig. 11

Three-dimensional printed rolling core joints with custom rolling surfaces

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Fig. 12

Degree-4 vertex constructed with 3D printed panels and rolling joints

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Fig. 13

Schematic of two facets Fi and Fj undergoing planar motion forming rolling contact between surfaces si,j(t) and sj,i(t)

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Fig. 14

Geometry of the zero-thickness reference with an offset d(t) between the two halves of the fold

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Fig. 15

Splitting a symmetric bird's-foot vertex. Left: the crease pattern (folded state at t = 0). Right: the split zero-reference surface in a partially folded state (t > 0).

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Fig. 16

Surface functions and panel positions for the four joints at four different values of the fold parameter t with linear offsets. Left to right: t = 0 (unfolded), 0.333, 0.667, and t = 1.0 (flat folded). Top row: joint corresponding to fold γ1. Second row: γ2, third row: γ3, and bottom row: γ4. The zero-thickness facets and fold axis are indicated by black dotted lines and dot, respectively.

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Fig. 17

Surface functions and panel positions for the four joints at four different values of the fold parameter t with a quadratic offset. Left to right: t = 0 (unfolded), 0.333, 0.667, and t = 1.0 (flat folded). Top row: joint corresponding to fold γ1. Second row: γ2, third row: γ3, and bottom row: γ4. The zero-thickness facets and fold axis are indicated by black dotted lines and dot, respectively.

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Fig. 18

Surface functions and panel positions for the four joints at four different values of the fold parameter t with three circular-CORE joints. Left to right: t = 0 (unfolded), 0.333, 0.667, and t = 1.0 (flat folded). Top row: joint corresponding to fold γ1. Second row: γ2, third row: γ3, and bottom row: γ4. The zero-thickness facets and fold axis are indicated by black dotted lines and dot, respectively.

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Fig. 19

Bird's-foot vertex constructed from 3D printing panels and rolling joints

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Fig. 20

Configuration of two interacting panels Fi and Fj undergoing relative Eucliean motion

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