MSMechanical SciencesMSMech. Sci.2191-916XCopernicus PublicationsGöttingen, Germany10.5194/ms-7-69-2016Towards developing product applications of thick origami using the offset panel techniqueMorganMichael R.LangRobert J.MaglebySpencer P.magleby@byu.eduHowellLarry L.https://orcid.org/0000-0001-8132-8822Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USALang Origami, Alamo, CA 94507, USASpencer P. Magleby (magleby@byu.edu)7March201671697715September201521December20151February2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ms.copernicus.org/articles/7/69/2016/ms-7-69-2016.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/7/69/2016/ms-7-69-2016.pdf
Several methods have been developed to accommodate for the use of thick
materials in origami models which preserve either the model's full range of
motion or its kinematics. The offset panel technique (OPT) preserves both the
range of motion and the kinematics while allowing for a great deal of
flexibility in design. This work explores new possibilities for origami-based
product applications presented by the OPT. Examples are included to
illustrate fundamental capabilities that can be realized with thick materials
such as accommodation of various materials in a design and manipulation of
panel geometry resulting in an increased stiffness and strength. These
capabilities demonstrate the potential of techniques such as the OPT to
further inspire origami-based solutions to engineering problems.
Introduction
In recent years, origami's potential for innovative solutions
facilitated by its complex behaviors, yet simple fabrication methods has
caught the attention of scientists and engineers. Some of the proposed
applications of origami include deployable space applications
, nano-structure fabrication
, robotics , and medical equipment
. While origami's potential can be seen
and even explored using traditional paper origami, many engineering
applications would generally require materials with stiffness and strength.
With increased stiffness and strength, however, comes some degree of
thickness, and as thickness increases folding becomes less and less feasible.
Several formal methods have been developed for accommodating thickness in
origami-based design. These include methods that shift rotational axes to
edges of panels , mount trimmed panels onto
membranes , employ spatial mechanisms at vertices
, taper the edges of the panels , and
replace creases with a rigid link and two axes to allow folding of adjacent
panels . The method used in this paper, the offset panel
technique (OPT) developed by , preserves the kinematics of
an origami model and allows a full range of motion. While the benefits of
these two traits can come at the cost of self-intersection and some
complexity, the technique does allow for a variety of shapes, materials, and
movements to be employed in a thick model based on a given origami pattern.
Previously, a number of basic configuration-related capabilities of the OPT
have been shown by . For example, the technique can
provide designers with flexibility as it accommodates uniform and varying
panel thickness, gaps between panels, and freedom in joint plane placement.
In this paper, the intent of the authors is to move beyond describing the
basic capabilities of the OPT (to accommodate thick panels) to illustrating
how the technique can be used to facilitate the development and engineering
of origami-based devices and products. The examples visually demonstrate that
employing the OPT with various origami patterns can facilitate the design of
mechanisms that use a variety of materials, have links (panels) that are
non-planar shapes, and that can resist loads and/or transfer forces.
Engineers and designers can use the illustrated examples as inspiration for
further designs that use origami patterns.
A comparison of thickness accommodation methods.
MethodKinematicsROMSingle DOFUnfoldsApplication considerationspreservedpreservedflatAxis-shiftNoYesYesYesLimited to selected fold patternsOffset jointYes,NoYesYesvariedLimited to selected fold patternsthicknessMembrane foldsNoYes, if gapsNo, gapsYesDeployed system requiresbetween panelsallow movementtension at edges to keep≥2× thicknessmembranes stretchedTapered panelsYesNoYesYesRequired tapering of panels limitspossible geometry and materialsOffset creaseNoYesNoYesPanels required to be trimmed toavoid self-interference at verticesSpatial linkagesNoYesYesNoFold angles and panel thicknesseslimited by the spatial mechanismOffset panelYesYesYesNoCutouts in panels may be requiredto avoid self-intersectionBackground
As preparation for extending the application of the OPT, we briefly review
three areas. These areas combined lead to an understanding of new ways to
create mechanical products by adapting origami models and motions for thick
materials.
Modeling origami
To take advantage of an origami mechanism's potential for application, it is
useful to have mathematical models of its motion and behaviors. Origami can
be modeled as a web of coupled spherical mechanisms where each vertex is the
center of a spherical mechanism, each panel is a link, and each fold is a
joint (e.g. ). Spherical kinematic mechanisms belong to a subset of
three-dimensional kinematics in which any point on the mechanism is
constrained to be coincident with a spherical surface whose center is the
point of intersection of all joints within the mechanism. The behavior of a
spherical mechanism is defined by the location of the rotational axes. In
other words, if a link's shape or size changes, as long as the rotational
axes have not been altered, the motion will be the same
.
Thickness accommodation
Mathematical models have been developed which have been used to predict the
behavior of a given fold pattern and which can generate
crease patterns based on a desired form . These
mathematical models usually assume zero-thickness materials. Due to paper's
relatively thin nature, zero-thickness is a suitable approximation for
traditional origami. However, this approximation breaks down for thicker
origami materials .
An illustration of the concepts of different thickness accommodation
methods. All images, except (e), are from .
(a) The zero-thickness model describes the basic kinematic behavior
of the model. (b) The axis-shift method as demonstrated by
shifts each rotational axis to either the top or bottom of
the thick material. While slightly different conceptually, the method
described by can be illustrated identically.
(c) The membrane folds method by mounts
thick-material facets to a flexible membrane. (d) The tapered panels
method from trims material from the panel edges to maintain
the kinematics. (e) The offset crease technique, described by
, is similar to the membrane folds method, but calls for rigid
material in the gaps between panels. This method was inspired by work done by
(f) The offset panel technique shown by
offsets each panel from a selected joint plane and extends
the rotational axes back to the joint plane.
demonstrates a thickness accommodation method which
uses spatial linkages at the vertices of the origami pattern to achieve its
motion. A single origami vertex is shown that employs such a mechanism. This
illustration is separate because the method cannot be portrayed in the same
two-dimensional form as the other methods. The red dotted lines indicate the
rotational axes. This method was shown to work with degree-4, degree-5, and
degree-6 vertices.
Several techniques have been developed to accommodate thick materials. Each
of the thickness accommodation techniques has its own set of strengths,
weaknesses, and limitations. Figures and
illustrate seven methods, including the OPT, and Table lists
the methods and compares them against four characteristics that will be
important to discussing applications for the OPT. Following is a description
of these characteristics.Kinematics preserved
indicates if the kinematics of the base origami
model (a zero-thickness model) is preserved with the method. While matching
kinematics may not be important for some applications, in many it will be
essential to achieve the same degrees of freedom, the consistency, or the
predictability of a motion identical to the origami model.
ROM preserved
indicates whether or not the range of motion, from
fully folded to fully deployed, is preserved. Many methods do not allow for
full range of motion due to clashes of panels/edges. In some applications
full motion may not be required, but in many there is a need for the folds to
move through the full 180∘.
Single DOF
indicates if the method will result in a single
degree-of-freedom (DOF) system (assuming that the pattern itself has a single
DOF). Many rigid-foldable origami models have one DOF. For many applications,
especially those implementing a deployable application, a single DOF is
desirable.
Unfolds flat
indicates whether or not the thick origami system
resulting from the method will be flat in its fully unfolded shape. Flat is
defined in this case as resulting in a system where the entire upper face of
each panel lies in the same plane. Traditional paper origami unfolds flat by
this definition. Most of the methods do not produce fully flat configurations
as the panels are offset or tapered. The OPT does not result in flat
configurations, but the offset panels (or panel substitutes) can be parallel
to the joint plane.
Application considerations
presents notes on key aspects which
should be considered when creating practical applications with the method.
Depending on the application, the considerations may be impediments or used
as advantages.
In this paper, the applications that are presented take advantage of the
OPT's capabilities to preserve kinematics, allow for full range of motion,
and open with a single DOF. OPT-based systems generally do not fold flat and
have a tiered appearance in the open position. This is used as an advantage
in some of the following examples.
Offset panel technique
One of the major advantages of the OPT is that it maintains both the
kinematic behavior and the full 180∘ of motion as demonstrated by
. The former permits designers to take advantage of the
mathematical models already developed while the latter allows for fully
opened and closed models. The OPT also allows for flexibility in a design.
Since an origami-inspired design is constrained by only the preservation of
the location of the axes and self-intersection, attributes such as varied
panel thickness, spacing between panels, and selectable joint plane placement
are all possible with the OPT (see ).
Fundamental capabilities of the offset panel technique
A major benefit of traditional origami is the simplicity that it offers –
the desired model is a product of a single monolithic piece of paper which
undergoes no process other than folding. While some of this simplicity is
lost in “thick origami”, there are at least two fundamental capabilities
possessed by origami models implemented by the OPT which are not exhibited by
traditional origami. These are the variation of panel geometry and the
material(s) used in a design. We refer to these as fundamental capabilities
because others are either subsets of them or results of combining them.
Panel and joint materials
With the option of using non-paper materials, it becomes clear that virtually
any solid material can be utilized in designs that incorporate folding.
Designers of origami-inspired products now have the ability to accommodate
materials that are, for example, transparent/opaque, conductive/insulative,
lubricative/abrasive, adhesive, stiff, modifiable, expansive, electrically
charged, absorbent, or reflective.
The unfolding of an origami-based kinetic sculpture which employs
the OPT. The mechanism is made of panels with a thickness of one inch. This
was part of an exhibit in BYU's Museum of Art.
Additionally, not only can a model be made of virtually any solid material,
since thick origami panels are fabricated individually in most cases, the
individual panels of a model can be assigned different materials. For
example, Fig. shows a kinetic sculpture based on an
origami fold pattern. In the design of this model, a lightweight material was
desired for the moving panels, while strength was needed in the supporting
ground panel. A lightweight foam board was chosen to be used for the moving
panels and MDF was used for the ground panel. The different sculpture
materials can best be seen from the back, as shown in Fig. .
In the kinetic sculpture, the grounded panel is made of MDF while
all others are made of a foam board.
A schematic of the pattern used in the design of the kinetic
sculpture. (a) The square twist crease pattern. (b) A side
view of the stacked panels and joint plane (represented by the red line).
(c) The panel stack with offsets. (d) The unfolded panels
including the offsets for the OPT model.
A foldable circuit board with one panel constructed from metal. The
thickness of the panel is approximately 0.063 inches.
This kinetic sculpture is based on the square twist fold pattern shown in
Fig. a. Figure b illustrates the closed
stack of panels (looking down at the sculpture from the top). In this design,
the panels all have an equal thickness of one inch and, while interference
has been accounted for, the panel profiles have not been altered.
Figure c shows the same panel stack but includes the offsets
which have the same color as the panels to which they are rigidly attached.
The offsets are shown to demonstrate the preservation of the axes' locations
which in the closed configuration all lie on a single plane (referred to as
the joint plane) indicated by the dotted red line.
The next example shows the possibilty of a foldable circuit board which uses
the OPT. This is shown in Fig. . All but one panel is made from
a PCB substrate, while the exception is made of a metal plate intended to act
as the ground layer. While simpler crease patterns (i.e., a tesselated
tri-fold pattern) could offer the similar stowed-to-deployed area ratios,
this pattern facilitates connections between each panel and its neighbors and
a single DOF.
A square twist pattern, as illustrated in Fig. a, was used
as a base for this circuit board. The illustration in
Fig. b shows the stacked panels from the side and
Fig. c shows the same panel stack with the offsets.
A schematic of (a) the pattern used in the design of the
foldable circuit board. (b) A side view of the panel stack and the
joint plane. (c) The stacked panels with offsets. (d) The
unfolded pattern with the offsets. (b) and (c) are 2X
scale.
Panel geometry
One of the most applicable and inspiring capabilities of thick origami (the
OPT in particular) is the freedom a designer has to alter the geometry of the
panels. Once it is determined that an origami-inspired design will employ the
OPT, the only limitations are the preservation of the rotational axes and the
prevention of self-intersection. Panel material can be added and/or removed
to add structure, give form, achieve motion, etc. Combining this idea with
that of using various materials, not only is it possible to alter the shape
of a model's panel material, but panels can be replaced entirely by
mechanical parts, electrical parts, displays, wheels, optical devices, solar
panels, molds, etc. As long as axes are not moved and self-intersection is
avoided, the model will maintain the kinematic behavior and the range of
motion of the source model.
To demonstrate the manipulation of panel geometry, a model of a folding
sphere is shown in Fig. . From the closed configuration of a
basic OPT model with uniform thickness, material is added to each panel in
such a way that the model takes on the shape of a sphere. This is done
without affecting the rotational axes and with care taken to ensure no
self-intersection.
A version of the square twist as shown in Fig. a formed
the origami base of this foldable sphere. Figure b
and c show the panel stack alone and with the offsets,
respectively, and Fig. d shows the panel stack with
offsets and the material added to form a sphere.
A kinematic simulation model of how panel geometry can vary, in this
case to form a sphere.
A schematic of the pattern used in the design of the foldable
sphere. (a) The crease pattern. (b) The panel stack and
joint plane. (c) The panel stack with offsets. (d) The
offsets shown with the open model. (e) The panel stack with offsets
and modified panel geometry.
As another example of non-constant panel geometry, see the engineer's toolbox
in Fig. . This model explores the utility of adding and
removing material by carving out material from the panels to form
compartments for supplies and adding material to panels to form the walls of
the box. The box's opening motion is shown in Fig. .
The same square twist pattern as used with the kinetic sculpture was used for
the toolbox, but with a different crease assignment as shown in
Fig. a. To facilitate the closing of the box, a small space
was added between the panels as demonstrated by . This
spacing is shown in Fig. b and the same panel stack is
shown in Fig. c, but with offsets.
An electrical engineer's toolbox created from an origami crease
pattern and the OPT in (a) the unfolded postion and (b) the
folded position. The panel thickness in this model is
0.5 inches.
A demonstration of the motion of the
toolbox.
A schematic of the pattern used in the design of the toolbox.
(a) The crease pattern. (b) The stacked panels and the
joint plane (in red). (c) The stacked panels with added offsets.
(d) The unfolded pattern including the offsets.
This origami-based table supports a significant amount of weight.
Unlike the other examples in this paper, the table is designed to unfold to
an intermediate position that does not correspond to the zero-thickness
model's fully unfolded position. The panels in this design are
0.75 inches.
Resultant capabilities – stiffness and strength
Having discussed the use of various materials and geometry, it is now
befitting to examine an example of a capability that is a result of
simultaneously varying materials and geometry. We will explore the use of
materials commonly used in engineering combined with the necessary geometry
to develop designs which can support loads and apply forces. Based on the
design specifications of an origami-inspired product, a fold pattern can be
selected which would give the desired behavior – a particular motion,
mechanical advantage, shape, or unfolded/folded size ratio. After the desired
pattern has been determined, materials with sufficient stiffness and strength
to perform the desired task can be selected and any geometrical modifications
necessary can be made.
For example, an origami-inspired table is shown in Fig. .
In the folded configuration, the table is compact. In the unfolded
configuration, the table is much larger and, based on the kinematics and the
materials of the model, can support a significant amount of weight. The
table's unfolding motion is demonstrated in Fig. . The table
uses the same square twist pattern as the kinetic sculpture and also has the
same panel stacking configuration (see Fig. ).
An origami-inspired table is shown through its opening
motion.
In the next example, we consider a lift. By using a single reverse fold, a
substantial mechanical advantage is possible. In this case, the mechanical
advantage is approximately 20 at the open state (top-left in
Fig. ) and gradually decreases to 5 at its highest point
(top-right in Fig. ). To withstand the stresses that accompany
the loads and large mechanical advantage of this example, MDF was the
selected material.
The lift employs a reverse fold which is a single vertex origami pattern as
shown in Fig. a. Figure b shows the
stack of the panels and Fig. c shows that same stack with
the offsets and altered geometry. To create the offsets for this model, the
entire inner faces of the long green panels were extended inward to the joint
plane while leaving clearance for the small blue panels. This is equivalent
to assigning different thicknesses to the panels as shown possible by
.
The origami-inspired lift mechanism shown through its motion as it
lifts the black weight. The panels of the lift are
0.75 inches.
The schematic of the reverse fold pattern used in the design of the
lift. The dotted red line represents the joint plane. (a) The fold
pattern. (b) A side view of the panel stack. (c) The panel
stack with offsets which in this design are large and cover the majority of
the inside face of each green panel. (d) The open pattern with the
offsets.
Conclusions
Origami's simple fabrication methods, infinite possibilities,
and predictability provide it potential to emerge as a source of inspiration
for many innovative designs. While paper origami models are useful for
quickly visualizing and prototyping origami-inspired products, paper is often
insufficient as a material for a finished product. However, the use of other
materials in such origami-inspired designs often presents a handful of major
difficulties including the folding of these materials and the interference
issues that accompany it. This paper has illustrated how designs based on
origami patterns can be realized using non-paper materials in a variety of
shapes and configurations. The examples show that common engineering-related
product objectives can be achieved with origami-based mechanisms. The
techniques shown in the examples can be applied to a wide variety of origami
patterns.
Several methods for accommodating thickness have been developed, each with
its own capabilities and limitations. While the examples in this paper use
the offset panel technique, the capabilities of accommodation of various
materials, manipulation of panel geometry, and strength and stiffness can be
applied to other thickness accommodation methods. These ideas add new and
exciting possibilities for origami-inspired design and facilitate the
development of creative solutions to real-world problems.
Acknowledgement
This material is based on work supported by the National Science Foundation
and Air Force Office of Scientific Research under NSF Grant
EFRI-ODISSEI-1240417.
Edited by: G. Hao
Reviewed by: two anonymous referees
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