Identifying links between origami and compliant mechanisms

Mech. Sci., 2, 217–225, 2011 www.mech-sci.net/2/217/2011/ doi:10.5194/ms-2-217-2011 © Author(s) 2011. CC Attribution 3.0 License.

Mechanical

Sciences Open Access

Identifying links between origami and compliant mechanisms H. C. Greenberg, M. L. Gong, S. P. Magleby, and L. L. Howell Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA Received: 25 February 2011 – Revised: 26 July 2011 – Accepted: 20 November 2011 – Published: 12 December 2011

Abstract. Origami is the art of folding paper. In the context of engineering, orimimetics is the application

of folding to solve problems. Kinetic origami behavior can be modeled with the pseudo-rigid-body model since the origami are compliant mechanisms. These compliant mechanisms, when having a flat initial state and motion emerging out of the fabrication plane, are classified as lamina emergent mechanisms (LEMs). To demonstrate the feasibility of identifying links between origami and compliant mechanism analysis and design methods, four flat folding paper mechanisms are presented with their corresponding kinematic and graph models. Principles from graph theory are used to abstract the mechanisms to show them as coupled, or inter-connected, mechanisms. It is anticipated that this work lays a foundation for exploring methods for LEM synthesis based on the analogy between flat-folding origami models and linkage assembly. Keywords. Compliant mechanisms, Lamina emergent mechanisms, Origami, Orimimetics

1

Introduction

Origami is the art of folding paper where ori- means fold and -kami means paper. The scope and complexity of origami has exploded in the last twenty years creating many different schools of thought (Demaine et al., 2010). Recently in the fields of science, mathematics and engineering, origami has been used to solve complex problems such as airbag folding, shock absorption (crash box), and deployable telescopic lenses (Cromvik, 2007; Ma and You, 2010; Heller, 2003). Origami design is governed by mathematical laws, which if better understood, could be applied in engineering. Specifically, the authors propose that origami can provide inspiration for synthesis techniques for compliant mechanisms. This paper identifies key concepts that can link origami to mechanism design. Traditionally origami has been static and representational. However, kinetic origami focuses on origami models that have some novel motion. While the term “action origami” is a more generally known term, we introduce the term kinetic origami to refer to models that exhibit a mechanical motion and to differentiate from models such as paper airplanes which are also referred to as action origami. Figure 1 shows two examples, where one is a static origami structure Correspondence to: S. P. Magleby ([email protected]) Published by Copernicus Publications.

and the other is an action origami mechanism called a flasher hat; it is shown in its fabricated and deployed forms. The way an origami model folds also provides insight and another way to classifying origami. The class of rigidly foldable origami focuses on motion where the creases often act as joints and the faces act as links with bending stress only occurring at the creases (Tachi, 2006; Balkcom and Mason, 2008; Hull, 1994; Watanabe and Kawaguchi, 2006). The process of folding origami has been examined using kinematic theory (Dai and Jones, 2002; Balkcom and Mason, 2008; Tachi, 2006; Buchner, 2003). This paper focuses on the motion of a finished origami and not the states in between flat and final states. Since origami relies on the deflection of flexible materials it is a compliant mechanism. The origami mechanisms examined herein are all flat-folding in their final-folded state, and so they are part of a subgroup of compliant mechanisms called lamina emergent mechanisms (LEMs) (Jacobsen et al., 2010). Lamina emergent mechanisms are compliant mechanisms made from planar materials (lamina) with motion that emerges out of the fabrication plane. A productive connection between origami and compliant mechanisms can be developed by drawing upon principles from graph theory to depict the origami mechanisms via simple, planar connected graphs. These graphs can show how the folds and facets interact with each other in their motion. Graphs allow for improved understanding of the interaction

218 amina Emergent

the faces act as links with bending stress only occurring at the creases (Tachi, 2006; Balkcom and Mason, 2008; Hull, H. C. Greenberg et al.: Identifying links between origami and compliant mechanisms 1994; Watanabe and Kawaguchi, 2006).

ri- means fold and lexity of origami ng many different . Recently in the Figure 1. Example of an origami structure (left) and flasher hat mechanism in its fabricated (middle) and deployed form (right). Fig. 1. Example of an origami structure (left) and flasher hat mechering, origami has anism in its fabricated (middle) and deployed form (right). ch as airbag foldto create motion, are referred to as construction creases. In between motion and structure of origami, and can help in loyable telescopic some cases, they coincide with hinge creases (Demaine and understanding how to predict complex motion and develop The process of folding origami has been 2007). examined using O’Rourke, corresponding mechanisms. 10; Heller, 2003). Structural creases. Structural creases are used to define Origami’s rich history and research in design optimization kinematic theory (Dai and Jones, 2002; Balkcom and Mason, cal laws,can which the shape of flaps; they can be hinge creases as well. These provideifan “orimimetic” perspective to LEM design that 2008;of novel Tachi, 2006; Buchner, 2003). This paperinfocuses on and they are not creases are not needed most mechanisms lead to the generation mechanisms. ineering.canSpecifforthe manystates materials. They may/can be substituted or the motion of a finished origami feasible and not in between an provide inspieliminated in various ways (Demaine and O’Rourke, 2007). Objective flat and final states. Crease vs. fold. A fold is an action and a crease is the iant mechanisms. origamiofrelies on the of action. flexible materials Orimimetic design refersSince to the application the concepts of deflection product of that Creases may be folded in one of two folding to mechanism design. It can provide alternative ways ways: mountain or valley. it is a compliant mechanism. The origami mechanisms exto achieve a particular range of motion; compliant mechaMountain crease. A convex crease (Hull, 2002). hereinorigami are allis achieved flat-foldingValley in their state, nisms have a similaramined aim. In addition, crease.final-folded A concave crease (Hull, 2002). from the manipulation of a planar material which motivates its examination as a lamina emergent mechanism. The objective of this paper is to identify links between origami and compliant mechanisms to facilitate the application of origami design literature to compliant mechanism and LEM design. 2

Nomenclature

It is helpful to establish terminology to facilitate discussion. Key terms, and their use, are provided below. 2.1

Origami

Origami. Origami is defined as the art of folding paper, and in the context of engineering, it is the use of folding to solve mechanical problems (Demaine and O’Rourke, 2007). Kirigami. Kirigami is defined as the art of folding and cutting paper (Hart, 2007). An example of kirigami is shown in Fig. 2 with the side view showing how creases are made such that the house pops out of the paper. Kinetic kirigami models are often LEMs, especially kirigami pop-up models. Hinge creases. In traditional origami hinge creases define the boundaries between flaps. In the context of mechanisms they are creases along which lies an interface of two planar faces and it is the axis about which both facets rotate (Demaine and O’Rourke, 2007). Construction creases. Creases used to create references in the construction of the mechanism are not directly used Mech. Sci., 2, 217–225, 2011

2.2

Mechanisms

Compliant mechanism. Mechanisms which transfer or transform motion, force, or energy at least in part through the deflection of flexible members are compliant mechanisms (Howell, 2001). Lamina emergent mechanism. Lamina emergent mechanisms (LEMs) are a subset of compliant mechanisms made from planar materials (lamina) with motion that emerges out of the fabrication plane (Jacobsen et al., 2010). They are a subset of compliant mechanisms, in that they use the deflection of flexible members to achieve the desired motion. As a subset of compliant mechanisms, LEMs can provide feasible, repeatable solutions to advance the design and manufacturing of products. The advantages of LEMs include: reducing the number of parts, reducing cost, reducing weight, improving recyclability, increasing precision, and eliminating assembly (Jacobsen et al., 2010). The incorporation and use of LEMs offer many potential advantages in the design of mechanical products. These mechanisms can provide opportunities for more cost-effective, compact, easy to assemble, and modular products. Spherical mechanism. In a spherical mechanism any point in a moving body is confined to move within a spherical surface, and all spherical surfaces of motion are concentric (Chiang, 1988).

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Greenberg et al.: Origami and Compliant Mechanisms H. C. Greenberg et al.: Identifying links between origami and compliant mechanisms

t mechanisms (?). Lamina ms made from rges out of the

219

and compliant pon principles nisms via simcan show how their motion. he interaction d can help in Figure 2. A kirigami house made in cardstock. n and develop

Orimimetic. Orimimetic means the ability to imitate folds. Fig. 2. A kirigami house made in In an engineering design context, it refers to the ability to n optimization use the concept of folding to solve problems. Specifically it M design that refers to the use of folding, either conceptually or literally, in mechanism design. s.

3.1 Origami history cardstock.

The current origami movement began in the early 20th century, though historically origami has its beginning in several countries dating back to the Muromachi period, from 1333– 1573 AD (Demaine and O’Rourke, 2007). Akira Yosihzawa Structural creases. Structural creases are used to define and his origami works have been largely credited for the creapplication of 2.3 Principles from graph theory ative explosion in origami in theThese last century (Demaine and shape of flaps; they can be hinge creases as well. . It canGraph. pro-A graphthe O’Rourke, 2007). Since the 1920’s, origami design has beconsists of points, called vertices, and concreases are not needed in most mechanisms and they are not come increasingly complex and varied. Initially origami was range ofnections, mo- called edges (Marcus, 2008). used to create figures and animals. As the Planar graph.feasible A graph with edges crossing is a planarThey may/can be substituted orart has progressed, forno many materials. m. In addition, the models have become increasingly complex, starting with graph (Marcus, 2008). eliminated in various ways (Demaine a fewO’Rourke, dozen steps to a2007). few hundred or even a thoua planar Facet. ma- The area enclosed by a planar graph is referred to a few toand sand steps. In the last twenty years as a facet. Crease vs. fold. A fold is an action and a crease isnew thebranches of origami mina emergent have begun to be explored by more than artists; a growing Degree. The degree of a vertex is the number of edges that product of that action. Creases may be folded in one of two number of mathematicians, educators, engineers and scienoccurreat that vertex (Marcus, 2008). show the tists have gathered to discuss the applications of origami in Degree sequence. The mountain degree sequence a graph is the ways: orofvalley. isms to facilitheir respective fields (Demaine and O’Rourke, 2007). degrees of each vertex listed in decreasing order (Marcus, Mountain crease. A convex crease (Hull, 2002). Closely related to origami are other forms of paper en2008). e to compliant gineering, such as kirigami, pop-up paper mechanisms, and Regular graph, d-regular graph. A graph is considered Valley crease. concave crease (Hull, 2002). origami architecture (Winder et al., 2009a). regular if all its vertices are of the same A degree. It can be referred to as a d-regular graph, where d is a non-negative integer where each vertex has degree d (Marcus, 2008). 3.2 Origami applications in engineering and science 2.2 A Mechanisms Generically rigid. graph is generically rigid if almost every realization of the graph is rigid. A realization of a Previous work has focused on studying the folding process graph is an assignment of coordinates to joints but without reto map the transition between initial and final states. An exstrictions on edge lengths. Generic realizations avoidMechanisms degenCompliant Mechanism. which transfer orwhich transample is airbag folding designs unfold smoothly from eracies such as having three points collinear, four points contheir flat starting state to their final volume ate discussion. form motion, force,andor energy at least in part through the (Cromvik, 2007). cyclic, or inducing parallel edges (Demaine O’Rourke, The Diffractive Optics Group at Lawrence Livermore Labo2007). . deflection of flexible members ratory are compliant mechanisms has worked on developing a telescope having a 100 m

diameter lens that could collapse to fit into a space vehicle (Howell, 2001). having a 4 m diameter and 10 m length. Lang used origami a design that wouldmechafit and maintain the integrity Lamina Emergent Mechanism.to identify Lamina emergent These sections describe the relevant history and current apof the surface when deployed into space (Heller, 2003; Wu nisms (LEMs) are a subset of compliant mechanisms made plications of origami. and You, 2010; Wei and Dai, 2009). Origami has also been ng paper, and applied to the crash box emerges of a car to improve from planar materials (lamina) with motion that out energy absorbtion lding to solve of the fabrication plane (Jacobsen et al., 2010). They are a www.mech-sci.net/2/217/2011/ Sci., 2, 217–225, 2011 , 2007). subset of compliant mechanisms, in that they use theMech. deflecf folding and tion of flexible members to achieve the desired motion. As a 3

Origami background

et al., 2009a). 4

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H. C. Greenberg et al.: Identifying links between origami and compliant mechanisms

Greenberg et al.: Origami and Compliant Mechanisms

For a model of an animal, efficiency is often defined by the ratio of the size of the final model to the initial sheet. Some branches of origami that are of special relevance to LEMs are flat-folding origami, which is origami that folds to a flat state, and tessellations or tilings, which could be useful for LEM applications in arrays. The repetition of basic folds in tessellations leads to the development of an origami mechanism that expands andofcontracts capturing Figure 4. A rigid-foldable version the squarethereby twist origami con- the motionfrom thatpolypropylene would be required forpaper. an array-type structure. structed sheet and The polypropylene shows that it1 behaves rigidly, where bendingcorollaries is only allowed at the Table lists examples of origami in each of the categories. The potential for origami insights LEM deFig. 4. folds. Asix common double slit kirigami modeled andin its correspondsign can be seen by mapping existing applications of origami ing PRBM. to the six application LEMs suggested by Alconfiguration of paper is categories determinedfor by internal spring forces Figure A common double representation and conits Fig. 3. A3.rigid-foldable versionslitofkirigami the square twist origami brechtsen et al. (Albrechtsen et al., 2010). as well as external forces and constraints” (Balkcom and Macorresponding PRBM both shown in the The crease of a folded structed from polypropylene sheet placed and paper. polypropylene son, 2008). Thus, the origami can be viewed as a compliant sheet of paper. shows that it behaves rigidly, where bending is only allowed at the Origami 4.3 as a Lamina Emergent Mechanism mechanism which Applications can be modeled using the pseudo-rigidfolds. Table 1. Origami by LEM Technology Category body model. in a low speed collision (Ma and You, 2010). Miura explored focusing mechanisms on origami in a fabricated statefrom we only exBecauseByorigami are made lamina mateApplication Class Origami Corollaries asmethods well as of external (Balkcom andsimMa- amine LEM foldingforces maps and that constraints” could be unfolded with the the hinge creases that contribute to the mechanism’s rials it follows that they are LEMs. Traditionally, origami son, origami can be viewed as he a compliant ple 2008). pull of aThus, cornerthe (Miura, 2002). In the 1980’s invented structure and motion, not the folds that were used to conDisposable mechanisms Packaging designsstruct areit.judged by their efficiency and accuracy in terms mechanism which can which be modeled the pseudo-rigidthe Miura-ori pattern, is usedusing as a basis for folding Space sails solarmodel. arrays (Miura and Natori, 1985). In 2005, Mahadevan body of material Novel usageArray andmechanisms number of folds (Lang and Hull, 2005) Telescopes published findings that thisinsame pattern exists foldBy focusing on origami a fabricated stateinweleaf only ex- Pseudo-rigid-body model ing, wings, and flower 2005). Scalable mechanisms Origami at nano-level amine the hinge creasespetals that (Mahadevan contribute toand theRica, mechanism’s The pseudo-rigid-body model (PRBM) uses rigid-body comCellular origami structure and motion, not the folds that were used to conponents, rigid links and springs, that have equivalent forcestruct it. Surprising Motion mechanisms Pop-up books 4 Origami mechanisms deflection characteristics to model the deflection of flexible members (Howell, 2001). Figure 3 shows a common Shock-absorbing mechanisms Crash box 4.2.1 Pseudo-Rigid-Body Model or origami mechanisms This paper considers kinetic origami kirigami double slit and its corresponding PRBM (Winder as opposed to origami structures. We will use the terms kiDeployable mechanisms Airbags The pseudo-rigid-body model (PRBM) uses rigid-body com- et al., 2009a). Stents netic origami and origami mechanism interchangeably. ponents, rigid links and springs, that have equivalent forcedeflection characteristics to model the deflection of flexi- 4.3 Origami as a lamina emergent mechanism 4.1members Kinematic modeling of origami ble (Howell, 2001). Figuremechanisms 4 shows a common Because origami mechanisms are made from lamina matekirigami slit and corresponding PRBM (Winder Origami double mechanisms areitsmodeled with hinge creases as rials follows Origami that they to areMechanisms LEMs. Traditionally, origami 5 itLinking with Graphs etjoints al., 2009a). and facets as links (Winder et al., 2009b). Origami designs are judged by their efficiency and accuracy in terms that is rigid-foldable is a “piecewise linear origami that is ofPrinciples material usage number of folds (Lang fromand graph theory can be usedand as Hull, a tool2005). to undercontinuously transformable without the deformation of each For a model an animal, efficiency often defined by the stand how of origami functions as a ismechanism (Dobrjanskyj facet” (Tachi, 2010). Rigid origami is also defined as having ratio the size of the1967). final model the model initial sheet. andof Freudenstein, Graphstocan both origami and regions of the paper between crease lines that do not need A branch of origami that is of special relevance LEMs mechanisms on an abstract level and show theirtosimilarities. to bend or twist in the folding process (i.e., the facet could isThe flat-folding origami, which is origami that folds to flat abstraction of origami mechanisms to graphsa demonbe replaced with sheet metal and hinge creases replaced by state, and tessellations or tilings, which could be useful for strates how the origami functions as a mechanism. Origami hinges and it would still fold up) (Hull, 1994; Tachi, 2006). LEM applications in arrays. The repetition of basic folds in mechanisms can be reduced to graphs which show how they Figure 4 shows an example of rigid origami made by modtessellations leads to the development of an origami mechabehave also. Applications of this are seen in Dai’s carton eling the links of a square twist, a common origami unit for nism that expands and contracts thereby capturing the motion folding research where graph theory is applied to show an tessellations, converted to a polypropylene compliant mechthat would be required for an array-type structure. equivalent mechanism (Dai and Jones, 2002). anism. Table 1 lists examples of origami corollaries in each of the In a graphThe of apotential kinetic for origami model folds become six categories. origami insights in LEM de- line Fig. 4. A common double slit kirigami modeled and its correspondsegments (edges) and links become nodes (vertices). sign can be seen by mapping existing applications of origamiFrom ing 4.2PRBM. Origami and compliant mechanisms such graphs it can be seen that origami mechanisms to the six application categories for LEMs (Albrechtsen et may al., be thought of as interconnected linkages, with each loop repOrigami mechanisms are compliant mechanisms; their mo2010). resenting a linkage system. It is important to note that in tion is a result of the deflection of the material. As a mate4.3 Origami as a Lamina Emergent Mechanism graph theory the shape of an edge or the position of the verrial undergoes deformation, the resulting stored strain energy tices does not matter since graphs depict connections (MarBecause mechanisms are made fromnotes lamina gives riseorigami to an internal spring force. Balkcom that mate“the cus, 2008). rials it follows that they are LEMs. Traditionally, origami designs are judged by their efficiency and accuracy in terms The crease pattern can be related to the graph of the Mech. Sci., 2, 217–225, 2011 www.mech-sci.net/2/217/2011/ of material usage and number of folds (Lang and Hull, 2005). origami mechanism in some cases where all creases are hinge

creases which aid in the folding of the origami but do not contribute to the motion and therefore are not considered in H. C. Greenberg et al.: Identifying links between origami compliant corresponding mechanisms 221 theand origami’s graph. Table 1. Origami Applications by LEM Technology Category.

5

LEM Application Class

Origami Corollaries

Disposable mechanisms

Packaging

Novel Array mechanisms

Space sails Telescopes

Scalable mechanisms

Origami at nano-level Cellular origami

Surprising Motion mechanisms

Pop-up books

Shock-absorbing mechanisms

Crash box

Deployable mechanisms

Airbags Stents

Linking origami to mechanisms with graphs

5. The crease pattern for the square twist shown in Fig. 4 is Fig. 5. Figure The crease pattern for the square twist shown in Figure 3 is shown with its corresponding graph overlaid in black. In the crease shown with its corresponding graph overlaid the crease pattern the red corresponds to mountain folds in andblack. blue to In valley pattern folds. the red corresponds to mountain folds and blue to valley folds.

vertices (links) on the boundaries separating four segments

Any(joints) simple planar connected graph with four segments is representative of a four-bar linkage. Principles from graph theory can be used as a tool to underconnecting four vertices that is a 2-regular graph represents stand how origami functions as a mechanism (Dobrjanskyj a four-bar mechanism. The graphs depict the degree of in6 Paper mechanism examples and Freudenstein, 1967). Graphs can model both origami and terconnection for each linkage system. This is shown in how mechanisms on an abstract level and show their similarities. Four paper mechanisms are shown in Table 2 with their each facet relates to its neighbouring facets. Each facet in a The abstraction of origami mechanisms to graphs demoncorresponding kinematic and graphical representations. For strates how the origami functions as a mechanism. Origami graph represents linkage having the same of links simplification,a all four paper mechanisms are number flat-folding mechanisms can be reduced to graphs which show how they mechanisms and are rigid-foldable. as nodes. For example, a facet (enclosed region) having four behave also. Applications of this are seen in Dai’s carton The kinematic representations for separating each mechanism exam(links) on the boundaries four segments folding research where graph theory is applied to showvertices an ine a portion of the larger mechanism and its motion. When equivalent mechanism (Dai and Jones, 2002). (joints)theispaper representative a four-bar mechanisms of have a flat initiallinkage. state and they can In a graph of a kinetic origami model folds become line be represented with kinematics, by definition they can be resegments (edges) and links become nodes (vertices). From alized as a LEM. In addition, the graphs representing each such graphs it can be seen that origami mechanisms may be paperMechanism mechanism show more abstractly the mechanism and 6 Paper Examples thought of as interconnected linkages, with each loop rephow they are coupled. The graphs also indicate the type of resenting a linkage system. It is important to note that in fold, where mountain folds are shown in red and valley folds graph theory the shape of an edge or the position of the verFour paper mechanisms are shown in Table 2 with their are in blue. tices does not matter since graphs depict connections (MarThe four-bar double slit mechanism (Table 2, row 1) uses corresponding kinematic and graphical representations. For cus, 2008). the PRBM to show how each crease can be modeled as a joint simplification, all four paper mechanisms are flat-folding The crease pattern can be related to the graph of the with a torsional spring. Its graph is a 2-regular graph. It has mechanisms andfolds, are rigid-foldable. origami mechanism in some cases where all creases are hinge three valley and one mountain fold as indicated by the creases and the origami can be flattened via actuation. In thisThe colors kinematic representations for each mechanism of edges in the graph. Each link has only two joints examcase the graph and crease pattern are considered dual graphs. that define its motion. ine a portion of the larger mechanism and its motion. When Figure 5 shows the square twist with its crease pattern and the 45-degree-fold twisting mechanism, also known the paperFormechanisms have a flat initial state and they can corresponding graph overlaid. However, for the general case, as the pop-up spinner card (Table 2, row 2) each center be represented kinematics, by definition they can the crease pattern may include structural and construction fold, alongwith the spine of the mechanism, is a 45 degree fold be recreases which aid in the folding of the origami but do alized not (O’Rourke, 2011). The kinematic shows only each as a LEM. In addition, therepresentation graphs representing contribute to the motion and therefore are not consideredpaper in four links along the spine. The 45-degree-fold twisting and mechanism show more abstractly the mechanism the origami’s corresponding graph. mechanism is a chain of four-bar mechanisms where those how they are coupled. The graphs also indicate the type of links not on the ends share two links between each neighAny simple planar connected graph with four segments fold, where foldsand areeach shown redis and valley folds bouringmountain linkage system, insidein link constrained connecting four vertices that is a 2-regular graph represents by three joints with one of those joints shared between each a four-bar mechanism. The graphs depict the degree of are in- in blue. linkage system. terconnection for each linkage system. This is shown in howThe neighbouring four-bar double slit mechanism (Table 2, row 1) uses each facet relates to its neighbouring facets. Each facet in a The square twist mechanism (Table 2, row 3) is a sethe PRBM to show how eachmechanisms crease can be modeled as a joint ries of coupled spherical which collapse onto graph represents a linkage having the same number of links with a one torsional graph is a square 2-regular as nodes. For example, a facet (enclosed region) having four another. spring. Figure 4 Its shows the basic twist, graph. and the It has www.mech-sci.net/2/217/2011/

three valley folds, and one mountain fold as indicated by the colors of edges in the graph. Mech. EachSci., link2,has only 2011 two joints 217–225, that define its motion.

222


Table 2. Table of Mechanisms. Paper Mechanism

Kinematic Representation

Graph

1. Four-Bar Double Slit

2. 45-degree-fold Twisting Mechanism

3. Square Twist

4. Water Bomb Base Tessellation

Mech. Sci., 2, 217–225, 2011



kinematic representation shows one of the four-bar linkages seen at the corner of the twisting square platform. It may be better to express the square twist as inter-connected four-bar spherical mechanisms, which may be seen in the graph representation. The outer corner links each connect to only two joints. There are two neighbouring links on each end that are connected to three joints and are shared between two linkage systems. Of the 7 nodes of degree four, 3 are shared between four linkage systems and 4 are shared between three linkage systems. Lastly, the kinematic representation for the water bomb base tessellation is shown in Table 2, row 4. As a graph the water bomb base tessellation is a 3-regular graph. The water bomb base tessellation contains six-bar linkage systems that are interconnected. Each link is shared between three linkage systems via three joints. This is also the basic tessellation used in the origami stent design which allows for the mechanism to compact radially as well as lengthwise (Kuribayashi et al., 2006). 7

The examples in Table 2 examine a single four-bar mechanism, four-bars connected in series, four bars in series and parallel, and a six-bar spherical mechanism. The use of graph theory helps bridge the fields of origami and mechanisms. Visually the graphs show how the origami models are interconnected spherical mechanisms. The following sections will introduce concepts for knowing whether a graph can be realized as a flat-folding mechanism. 7.1

Mobility

Given any generic graph we can determine whether it is generically rigid or flexible and subsequently realizable as a mechanism or a structure, where a mechanism would be classified as generically flexible and a structure as generically rigid. A generic graph can be realized as a mechanism in two dimensions when e < 2n − 3

(1)

or in three dimensions when e < 3n − 6

(2)

where n is the number of nodes and e is the number of edges in the graph (Demaine and O’Rourke, 2007). When e = 2n − 3

(3)

the graph is generically rigid. 7.2

For a four-bar mechanism to be assembled it is required to meet the criteria of


(5)

Theorem 2 (Maekawa [Kasahara and Takahama], Justin). In a flat-foldable single-vertex mountain-valley pattern defined by angles θ1 + θ2 + ... + θn = 360◦ , the number of mountains and the number of valleys differ by ±2 (Demaine and O’Rourke, 2007). Theorem 2 is generally know as Maekawa’s Theorem. It is essentially a condition for assembling a four-bar closed kinematic chain. Theorem 3 (Kawasaki, Justin). If an angle θi is a strict local minimum (i.e., θi−1 > θi < θi+1 ), then the two creases bounding angle θi have an opposite mountain-valley assignment in any flat-foldable mountain-valley pattern (Demaine and O’Rourke, 2007). This theorem is analogous to the conditions defining the classes for spherical four-bar LEMs where the shortest link has to connect to two links having opposite direction assignment. An additional criterion is that a sheet can never penetrate a fold. To determine this it is required to have the geometry specified as well as the angles. While criteria exist for determining the local flatfoldability (for a single vertex crease pattern), determining global flat-foldability is np-hard (non-deterministic polynomial-time hard). However we will explore the flatfoldability of the square twist to better understand these rules for assembly. Square twist

Assembly

l < s+ p+q

where l refers to the length of the longest link, s to the length of the shortest link and p and q the lengths of the other links. Likewise for an origami to be folded flat it must follow certain criteria. In determining flat-foldability it is assumed models can be folded perfectly, ignoring folding error, that paper has zero thickness, and that our creases have no width (Hull, 2002). The following criteria were developed for determining local flat-foldability, where local means about a single vertex. Theorem 1 (Kawasaki, Justin, Hull). A single-vertex crease pattern defined by angles θ1 +θ2 +...+θn = 360◦ is flatfoldable if and only if n [the number of creases] is even and the sum of the odd angles θ2i+1 is equal to the sum of the even angles θ2i , or equivalently either sum is equal to 180◦ (Demaine and O’Rourke, 2007). This theorem is generally known as Kawasaki’s Theorem, though also discovered independently by Justin in 1989. It’s analogy in mechanisms is the condition for a changepoint mechanism where s+l = p+q

Discussion

223

(4)

The square twist crease pattern, shown in Fig. 6, has 212 possible assignments and of those only 16 are flat-foldable (Hull, 2003). Figure 7 shows the 16 valid assignments for the square twist. It should be noted that only 6 of these 16 are unique. Mech. Sci., 2, 217–225, 2011

uired to

The square twist crease pattern, shown in Figure 6, has 2

(Hull, 2003). Figure 7and shows the 16 assignments possible assignments of those onlyvalid 16 are flat-foldablefor (4) the(Hull, square twist. It should be noted that only 6 of these 2003). Figure 7 shows the 16 valid assignments for16 unique. (4) arethe square twist. It should be noted that only 6 of these 16 224 H. C. Greenberg et al.: Identifying links between origami and compliant mechanisms ngth are unique. Greenberg et al.: Origami and Compliant Mechanisms enks. length ercerlinks. 1/2 generic graph is ow 1/2 it is believed th thecerme the whether it repre rror, relationship betw g error, 1/4 have the application o 1/4 es have velcompliant mech develeans Acknowledgement

l means

rtex e-vertex o o 0360 is is even is even f the m of the o o to 180 180

rin Wada for his d ported by the Nati 0800606. Any op tions expressed in necessarily reflect

Fig. 6. 6. Crease pattern for twist. Fig. Crease pattern forsquare square twist. Figure 6. Crease pattern for square twist.

References

heorem, rem, 89. It’sIt’s gepoint oint

Albrechtsen, N., nology Push P tunities and Po Fig. 8. The 6 possible unique square twist crease patterns and their Figure 8. The 6 possible unique square twist crease patterns and nisms, in: Proc corresponding graphs given Theorem 2. neering Techni their corresponding graphs given Theorem 2. 2010. Balkcom, D. J. a ternational Jou //dx.doi.org/10 Buchner, T.: Kin tured Origami T nology, 2003. Chiang, C. H.: K University Pres Cromvik, C.: Num and Origami, M ogy and G¨oteb Fig. 9. Folded origami models corresponding to the 6 possible Dai, J. S. and Jo square twists corresponding to those shown in Figure 8. carton folds in equivalent., Pr neers – Part C – flat it does not mean that it is kinetic. Since all folds here 959–970, doi:1 are simple folds and they are non-locking folds this allows Demaine, D. E., D us to actuate the square twist to have the twisting motion to and Gould, V.: Demaine, E. and go from a folded state (half the size of the original square) Figure 9. Folded origami models corresponding to the 6 possible Cambridge Un back to the original square. square twists corresponding to those shown in Fig. 8. Dobrjanskyj, L. a theory to the st 8 Conclusions neering for Ind sembled. As such it is anticipated that the abstraction toHart, G.: Modular kinetic origami the and ability kirigamito models were used to Heller, A.: A Gia graphGraphs theoryofwill improve synthesize LEMs. show how the of origami mechanisms arepaper related.mech- nology Review In addition, by fields examining theand graphs of some 12–18, 2003. We propose that some of the literature from the origami field anisms it is understood that their motion is achieved becauseHowell, L. L.: Co can be applied to mechanisms and vice versa. This was Hull, T.: On the m they are a system of coupled, or interconnected, shown by comparing the criteria for flat-foldabilitymechanisms. to some Graphconditions theory required includesfortheories formechanisms identifying whether a antium, p. 215, changepoint to be asHull, T.: The Com sembled. such itoris flexible anticipated the abstractionAs to such generic graph As is rigid in that n-dimensions. Third Internati theorythat will given improveany the ability to synthesize LEMs. it is graph believed generic graph we can know and Education, In addition, by examining the graphs of some paper mechwhether it represents a structure or mechanism. Showing theHull, T.: Countin anisms it is understood that their motion is achieved because Ars Combinato relationship and mechanisms will facilitateJacobsen, J. O., W they are abetween system of origami coupled, or interconnected, mechanisms. the application ofincludes origamitheories designfor principles the design of Lamina emerge Graph theory identifyinginwhether a

Fig. 8. The 6 possible unique square twist crease patterns and their corresponding graphs given Theorem 2.

(5)

(5)

hama],

ma], ley patpat- of mber er of emaine aine

Fig. 7. The 16 possible square twists given Theorem 3.

rem. It Fig. Figure 7. 16 Thepossible 16 possible square twists given Theorem 7. The square twists given Theorem 3.3. Next by applying Theorem 2 we can determine the remainm.closed It ing crease designations which2are in Figure the 8. remainosed Next by applying Theorem weshown can determine a strict ing crease It is easier to useTheorem thewhich graph (onshown thedetermine right) as opposed to Next bydesignations applying 2are we can the remainin Figure 8. creases the pattern for visualizing how Theorem 2 is applied. ing crease designations which are shown in Fig. 8. trict It is easier to use the graph (on the right) as opposed to assignNote eachto subgraph for one ofthetheright) four as four-bars has It isthat easier use graph (on opposed to ases the crease pattern for the visualizing how Theorem 2 is applied. emaine either 3 mountain and 1 valley folds or 1 mountaing and 3 the crease pattern for visualizing how Theorem 2 is applied. ignNote thatfolds. each The subgraph for onemodels of the are fourshown four-bars has9. Folded origami models corresponding to the 6 possible Fig. valley corresponding in FigNote3that each subgraph for one of the four four-bars and has 3 aine either mountain and 1 valley folds or 1 mountaing square twists corresponding to those shown in Figure 8. ure 9. 3 mountain and 1 valley folds or 1 mountain and ing the either 3 folds. Themodels corresponding models are shown in Figest link valley All six paper in Figure 9 are flat-folded versions valley folds. Corresponding paper representations of each ureof9.the square twist which are kinetic origami. An important the assignare shown in Fig. 9. link All six paper models in Figure 9 are flat-folded distinction to make is that while an origami can be versions folded All six paper models in Fig. 9 are flat-folded versions flatof it does not mean that it is kinetic. Since all folds here ignof the square twist which are kinetic origami. An important the square twist which are kinetic origami. An important are simple folds and they are non-locking folds this allows distinction an origami origamican canbebefolded folded distinctiontotomake make isis that that while while an flat it does not mean that it is kinetic. Since all folds us hereto actuate the square twist to have the twisting motion to are simple folds and they are non-locking folds this allows go from a folded state (half the size of the original square) us to actuate the square twist to have the twisting motion to back to the original square. go from a folded state (half the size of the original square) back to the original square.

8 8

compliant mechanisms and LEMs. Conclusions

Conclusions Acknowledgements. The authors would like to acknowledge

GraphsLarrin of kinetic origami and kirigami models Wada for his drawings. This material is basedwere upon used to Graphs of kinetic origami and kirigami models were used to work the supported by of the origami National Science under Grant show how fields and Foundation mechanisms are related. show how the fields of origami and mechanisms are related. No. CMMI-0800606. Any opinions, findings, and conclusions or We propose that some of the literature from the origami We field propose that some of the literature the field recommendations expressed in this material arefrom those of the origami authors can be applied to mechanisms and vice versa. This can was be and do not necessarily reflect the views the National applied to mechanisms andofvice versa.Science This was shown by comparing the criteria for flat-foldability to some Foundation. by comparing the criteria for flat-foldability to some conditions required for changepoint mechanisms to beshown asconditions required for changepoint mechanisms to be assembled. As such it is anticipated that the abstraction to Mech. Sci., 2, 217–225, 2011 www.mech-sci.net/2/217/2011/ graph theory will improve the ability to synthesize LEMs. In addition, by examining the graphs of some paper mech-

H. C. Greenberg et al.: Identifying links between origami and compliant mechanisms Edited by: J. Herder Reviewed by: H. Stachel and another anonymous referee

References Albrechtsen, N., Magleby, S. P., and Howell, L. L.: Using Technology Push Product Development to Discover Enabled Opportunities and Potential Applications for Lamina Emergent Mechanisms, in: Proceedings of the ASME International Design Engineering Technical Conferences, DETC2010–28 531, ASME, 2010. Balkcom, D. J. and Mason, M. T.: Robotic origami folding, Int. J. Robot. Res., 27, 613–627, doi:10.1177/0278364908090235, compendex, 2008. Buchner, T.: Kinematics of 3D Folding Structures for Nanostructured Origami TM, Ph.D. thesis, Massachusetts Institute of Technology, 2003. Chiang, C. H.: Kinematics of Spherical Mechanisms, Cambridge University Press, 1988. Cromvik, C.: Numerical Folding of Airbags Based on Optimization and Origami, Master’s thesis, Chalmers University of Technology and G¨oteborg University, Sweden, 2007. Dai, J. S. and Jones, J. R.: Kinematics and mobility analysis of carton folds in packing manipulation based on the mechanism equivalent, Proceedings of the Institution of Mechanical Engineers – Part C – J. Mech. Eng. Sci., 216, 959–970, doi:10.1243/095440602760400931, 2002. Demaine, D. E., Dinh, G., Crimp, V. F. L., Golan, M., Hull, D. T., and Gould, V.: Independent Lens: Between the Folds, 2010. Demaine, E. and O’Rourke, J.: Geometric Folding Algorithms, Cambridge University Press, 2007. Dobrjanskyj, L. and Freudenstein, F.: Some applications of graph theory to the structural analysis of mechanisms, J. Eng. Ind., 89, 153–158, 1967. Hart, G.: Modular Kirigami, San Sebastian Spain, 1–8, 2007. Heller, A.: A Giant Leap for Space Telescopes, Science and Technology Review by Lawrence Livermore National Laboratory, 12–18, 2003. Howell, L. L.: Compliant Mechanisms, Wiley-Interscience, 2001. Hull, T.: On the mathematics of flat origamis, Congressus numerantium, p. 215, 1994. Hull, T.: The Combinatorics of Flat Folds: A Survey, in: Origami3: Third International Meeting of Origami Science, Mathematics, and Education, A K Peters, Ltd., 29–38, 2002. Hull, T.: Counting Mountain-Valley Assignments for Flat Folds, Ars Combinatoria, 67, 175–178, 2003. Jacobsen, J. O., Winder, B. G., Howell, L. L., and Magleby, S. P.: Lamina emergent mechanisms and their basic elements, Journal of Mechanisms and Robotics, 2, 1–9, doi:10.1115/1.4000523, compendex, 2010. Kuribayashi, K., Tsuchiya, K., You, Z., Tomus, D., Umemoto, M., Ito, T., and Sasaki, M.: Self-Deployable Origami Stent Grafts as a Biomedical Application of Ni-rich TiNi Shape Memory Alloy Foil, Mat. Sci. Eng. A-Struct., 419, 131–137, 2006.


225

Lang, R. J. and Hull, T. C.: Origami design secrets: mathematical methods for an ancient art, The Mathematical Intelligencer, 27, 92–95, doi:10.1007/BF02985811, 2005. Ma, J. and You, Z.: The Origami Crash Box, in: Origami 5: the 5th International Conference on Origami in Science Mathematics and Education, 2010. Mahadevan, L. and Rica, S.: Self-Organized Origami, Science, 307, 1740, doi:10.1126/science.1105169, 2005. Marcus, D. A.: Graph theory: a problem oriented approach, MAA, 2008. Miura, K.: The Application of Origami Science to Map and Atlas Design, in: Origami3: Third International Meeting of Origami Science, Mathematics, and Education, edited by: Hull, T., A K Peters, Ltd., 2002. Miura, K. and Natori, M.: 2-D Array Experiment on Board a Space Flyer Unit, Space Power, 5, 345–356, 1985. O’Rourke, J.: How To Fold It, Cambridge University Press, 2011. Tachi, T.: Simulation of Rigid Origami, in: Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education, edited by: Lang, R., A K Peters, Ltd., 175–187, 2006. Tachi, T.: Rigid-Foldable Thick Origami, in: Origami 5: the 5th International Conference on Origami in Science Mathematics and Education, 2010. Watanabe, N. and Kawaguchi, K.-I.: The Method for Judging Rigid Foldability, in: Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education, edited by: Lang, R., 165–174, 2006. Wei, G. and Dai, J.: Geometry and kinematic analysis of an origami-evolved mechanism based on artmimetics, in: Reconfigurable Mechanisms and Robots, 2009, ReMAR 2009. ASME/IFToMM International Conference on, 450–455, 2009. Winder, B. G., Magleby, S. P., and Howell, L. L.: A study of joints suitable for lamina emergent mechanisms, in: 2008 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC 2008, 3–6 August 2008, vol. 2 of 2008 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC 2008, ASME, New York City, NY, United states, compendex, 339–349, 2009a. Winder, B. G., Magleby, S. P., and Howell, L. L.: Kinematic representations of pop-up paper mechanisms, Journal of Mechanisms and Robotics, 1, 1–10, doi:10.1115/1.3046128, compendex, 2009b. Wu, W. and You, Z.: Modelling rigid origami with quaternions and dual quaternions, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 466, 2155–2174, doi:10.1098/rspa.2009.0625, 2010.

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