Origami-Inspired Active Structures- A Synthesis and Review

Topical Review

Origami-inspired active structures:a synthesis and review

Edwin A Peraza-Hernandez1,2, Darren J Hartl1,2, Richard J Malak Jr1,3 andDimitris C Lagoudas1,2

1 Texas Institute for Intelligent Materials and Structures, Texas A&M University, College Station, TX77843, USA2Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA3Design Systems Laboratory, Department of Mechanical Engineering, Texas A&M University, CollegeStation, TX 77843, USA

E-mail: [email protected]

Received 10 April 2014, revised 12 May 2014Accepted for publication 13 May 2014Published 11 August 2014

AbstractOrigami, the ancient art of paper folding, has inspired the design of engineering devices andstructures for decades. The underlying principles of origami are very general, which has led toapplications ranging from cardboard containers to deployable space structures. More recently,researchers have become interested in the use of active materials (i.e., those that convert variousforms of energy into mechanical work) to effect the desired folding behavior. When used in asuitable geometry, active materials allow engineers to create self-folding structures. Suchstructures are capable of performing folding and/or unfolding operations without beingkinematically manipulated by external forces or moments. This is advantageous for manyapplications including space systems, underwater robotics, small scale devices, and self-assembling systems. This article is a survey and analysis of prior work on active self-foldingstructures as well as methods and tools available for the design of folding structures in generaland self-folding structures in particular. The goal is to provide researchers and practitioners witha systematic view of the state-of-the-art in this important and evolving area. Unifying structuralprinciples for active self-folding structures are identified and used as a basis for a quantitativeand qualitative comparison of numerous classes of active materials. Design considerationsspecific to folded structures are examined, including the issues of crease pattern identificationand fold kinematics. Although few tools have been created with active materials in mind, manyof them are useful in the overall design process for active self-folding structures. Finally, thearticle concludes with a discussion of open questions for the field of origami-inspiredengineering.

Keywords: active materials, origami, review, morphing structures, design, origami engineering,smart structures

(Some figures may appear in colour only in the online journal)

1. Introduction

Traditionally, the term ‘origami’ has been associated pri-marily with the art of folding paper. The term origami has theJapanese roots ‘ori’ meaning ‘folded’, and ‘kami’ meaning

‘paper’ [1]. Its original purpose was not particularly utilitar-ian, but rather recreational and artistic [2]. Specifically, it wasand remains the art of folding uncut sheets of paper intodecorative and well-defined shapes, either abstract in form orrepresentative of realistic objects (e.g., plants, animals). In

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origami, a goal shape is obtained from an initially planar sheetexclusively through folding operations. For an idealized sheetwith no thickness, a fold is defined as any deformation of thesheet such that the in-surface distance between any two pointsin the sheet is preserved and self-intersection does not result[3]. Stretching and tearing are not permitted; bending is theessential deformation [3]. For a sheet with non-zero thickness,a fold is defined as any deformation of the sheet that preservesa continuous neutral surface (i.e., a surface that neitherstretches nor contracts) and prevents self-intersection.

In the mid-1970s, mathematicians discovered that anendless number of shapes could in theory be created usingtraditional origami (initially planar shape, only folds allowed)[1]. These discoveries enabled new approaches for manu-facturing, assembling, and morphing of devices and structuresbased on origami principles. This is evident in the increasingattention mathematicians, scientists, and engineers have givento origami theories and tools over the past four dec-ades [1, 4–6].

Origami offers engineers novel ways to fabricate,assemble, store, and morph structures. Potential advantagesinclude the capability to compactly store deployable struc-tures (e.g., airbags [7, 8]), the potential for structures to bereconfigurable [9–12], and a reduction in manufacturingcomplexity [13] (reduced part counts and improved assemblyusing collapsible/deployable parts). Furthermore, origami hasbeen demonstrated to be applicable across scales via itsapplications ranging from DNA approaches at the nano-scale[14–17] to deployable structures for space exploration at thevery large scale [18–21]. Other current applications of ori-gami-inspired engineering include: micro-mirrors [22], var-ious space structures (solar panels, solar sail, telescope lenses[23]), batteries and capacitors with improved properties[24–27], robots [28–30], foldable wings and airplanes[31–36], foldcore-based structures for improved mechanicalproperties and impact resistance [37–43], crash boxes andother energy absorption systems [44–47], shelters [48–50],metamaterials [51, 52], microelectromechanical systems(MEMS) [53–57], biomedical devices [58–63], and manyothers [64]. A representative cross-section of these engi-neering applications is shown in figure 1.

For certain origami-inspired engineering applications, itis impractical to externally execute the manipulations neces-sary to produce the folding operations. This is the case at verysmall or very large scales or in remote applications (e.g.,underwater robotics, space structures, invasive biomedicaldevices). In these circumstances, self-folding capabilities areessential. A self-folding structure is one that has the capabilityof folding and/or unfolding to and/or from a desired shapewithout external manipulations. One approach for the devel-opment of self-folding systems is to leverage the use of activematerials agents of fold generation. Active materials arematerials that convert various forms of energy into mechan-ical work [65]. This coupling can be categorized as direct(mechanical response due to field-induced eigenstrains in theactive material) or indirect (mechanical response due to field-induced change in stiffness or other properties). In either case,active materials allow the mechanical work associated with

the folding operations to be obtained via the application of anon-mechanical field (e.g., thermal, electrical, optical, che-mical, etc).

Certain basic origami concepts should be defined prior tothe discussion of active self-folding structures in the sub-sequent sections. In origami, the locations of localized foldson the sheet are formally called creases [66, 67]. The creases,folding directions, folding magnitude, and folding sequencedetermine the ultimate shape of the structure. Typically,creases are defined by their endpoints, formally called ver-tices. Sheet regions bounded by the creases are known asfaces. To determine the fold direction of a crease, a ‘moun-tain-valley’ assignment is typically used. For mountain folds,faces on either side of the crease can be thought of as rotatinginto the page, while for valley folds, they can be thought of asrotating out of it. A crease pattern is a schematic that showsall the creases on a sheet required to fold a structure, typicallywith mountain-valley assignments. These concepts aredepicted figure 2.

Two parameters that describe the magnitude of a fold arethe folding angle and the radius of curvature at the fold line.These parameters are shown schematically in figure 3. In theview that a finite thickness sheet cannot provide sharp foldsand for the purposes of discussion, it is assumed that a finiteregion centered at the fold is bent and has a radius of cur-vature R. The internal fold angle θi is the angle at the inter-section of two line segments stretching collinearly withrespect to the folding faces. The external fold angle θe isdefined as ° θ−180 i.

The objectives of this paper are to provide a survey andclassification of current active self-folding structural approa-ches and design tools that might be applied to them, as well asto provide open questions and challenges in the field of ori-gami-inspired engineering with active materials. Figure 4 is aschematic of the process for developing origami-inspiredactive structures and how it relates to the organization of thispaper. Specifically, the remainder of this paper is organized asfollows: section 2 presents a description of active materialsand describes how they might be used for self-foldingstructures, section 3 presents examples of active self-foldingstructures, section 4 presents a review of design tools forfolding structures, and section 5 presents the conclusions andopen questions for the field of origami-inspired engineering.

2. Self-folding

Self-folding is the capability of a structure to fold and/orunfold without the application of external manipulations. Asmentioned in the preceding section, active materials enablefor self-folding systems since they inherently convert otherforms of energy into mechanical work allowing folding and,in some cases, unfolding operations. Active materials are alsotypically energy dense and geometrically simple when used asactuators [68–70]. Compactness is an important design con-sideration for self-folding systems since flat reference con-figurations are often essential.

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Smart Mater. Struct. 23 (2014) 094001 E A Peraza-Hernandez et al

2.1. Single fold concepts

Concepts for generating individual folds using active mate-rials are provided here while particular examples follow insection 3. The concepts are divided into two categories: hingetype and bending type. Most hinge-type active folds are

associated with one of three local actuator concepts: (i)variable length active rod or spring connected to the two facesjoined by the hinge (figure 5(a)), (ii) active torsional elementat the hinge (figure 5(b)), and (iii) active element with presetfolded shape (figure 5(c)). Throughout this paper these foldconcepts will be referred to as ‘extensional’, ‘torsional’, and‘flexural’, respectively.

Figure 1. Examples of current and potential applications of origami-inspired structures with and without self-folding capability. (b) Reprintedfrom [60], Copyright (2006), with permission from Elsevier. (d) ([28] Reproduced by permission of the American Society of MechanicalEngineers, ASME).

Figure 2. Schematic of a pinwheel crease pattern illustrating variousorigami concepts.

Figure 3. Parameters that define the magnitude of a fold.

3


The extensional concept (figure 5(a)) uses the activematerial in a rod or spring form with its two ends attached tothe faces connected by the hinge and the length of the activeelement controls the rotation of the hinge. The torsionalconcept (figure 5(b)) uses the active material as a torsionalspring or a rod that provides twist at the hinge. The twistangle of the active material thus directly controls the rotationof the hinge. In the flexural concept (figure 5(c)), the activematerial has been manufactured or trained to have a presetfolded configuration but is then deformed to an initially flatconfiguration. Upon application of the activation field, theactive material itself returns to its preset folded configurationand being bonded to the faces of the passive material, inducesthe local hinge to do the same.

All concepts can be further improved to allow for foldingin both directions relative to the sheet normal by addingcorresponding antagonistic active components. In the tor-sional concept, this can be achieved by installing two activeelements that generate twist in opposite directions (e.g., [71]).For the extensional and flexural concepts, it can be achievedby pairs of active elements in opposition to each other (e.g.,on opposite sides of the sheet for the extensional concept).However, mechanical restrictions may arise since activationof one active element may subject its associated antagonisticactive element to excessive stress or deformation, hinderingthe folding operation and possibly leading to material failure(e.g., plastic deformation, damage). Another challenge is thatof maintaining both sufficient geometric offset and localized/targeted driving field imposition such that opposing hingemoments do not result.

Other applications and approaches do not assume theexistence of discrete hinge mechanisms but are rather based

on direct local sheet bending caused by the actuation of theactive material. Such concepts are shown in figures 5(d) and(e). Unlike the hinge type fold concepts, the direct bendingapproach (i.e., without hinges) may offer the advantage ofmassive foldability. In other words, folds can occur at anylocation or orientation to which the driving field is applied(unless mechanically restricted). In hinge type concepts, how-ever, folds are clearly restricted to structurally pre-determinedhinge locations. We will refer to the concept of figure 5(d) as‘multi-layer’ and that of figure 5(e) as ‘single layer’.

The multi-layer concept considers self-folding using atwo-layer laminate with one passive layer and one active layer(figure 5(d)). A passive layer generates negligible mechanicalwork compared to the active layer under the application of theactuation inducing field. When such a field (thermal, mag-netic, etc) is applied, the active layer is driven to deform,generally axially, while the passive layer is not. This differ-ence in expansion or contraction between the two layersgenerates localized bending of the sheet. This concept can befurther expanded to allow for folds in both directions relativeto the sheet normal in a manner similar to that employed forhinge-type folds. Specifically, three-layer designs with twoopposing outer layers of active material separated by a pas-sive material can be used. A key challenge for such three-layer designs is to isolate the driving field within only one ofthe two active outer layers. For this purpose, the middle layermay serve in another role as an insulator with respect to thedriving field, preventing the field applied to one active layerfrom reaching the opposing active layer.

The single layer concept considers self-folding viabending without hinges using a sheet with a single activelayer subjected to a graded driving field (figure 5(e)). Such agradient generates a distribution of actuation strain throughthe sheet thickness, causing the sheet to bend. This designallows for folds in both directions relative to the sheetnormal based on the direction of the driving field gradient.However, folding via this approach is generally more diffi-cult as compared to the multi-layer concept since it is notpractical to maintain gradients in some physical fields (e.g.,temperature) at specific locations for a considerable periodof time [72].

It should be noted that folding using active materials isnot restricted to the five concepts presented in this section.Examples of self-folding systems that use other types offolding concepts (e.g., hinged faces with magnetic patchesthat are kinematically manipulated by the direction of themagnetic field without requiring deformation of the magneticpatches [73, 74]) are presented in section 34.

2.2. Active materials for self-folding systems

When considering self-folding from a flat to a deformedconfiguration using active materials, several critical designdrivers should be considered: actuation strain, actuation

Figure 4. Process for developing origami-inspired active structuresas it relates to the organization of this work.

4 Although it is not strictly folding, it should be noted that out-of-planedisplacements and changes in Gaussian curvature can be obtained by theapplication of uniform through thickness but inhomogeneous in-plane strainfields in a sheet [75–77].

4


stress, and the capability of generating and/or manipulatingthe desired field at the chosen scale. This section provides adescription of field/actuation strain/actuation stress relations.

In order to understand how active self-folding systemsare developed, constitutive relations that relate the externallyapplied fields to the obtained actuation strain are needed. Thefollowing expression relates the total strain ε to different

fields [65, 78–81] 5:

ε σ α

ε

= + − +

+ + ∑ − +=

( )( )S d E

d H e

T T

c c , (1)

p

m

i

ni i i ms

0

10

where S is the fourth order compliance tensor, σ is the secondorder stress tensor, α is the second order thermal expansion

Figure 5.Basic active fold concepts. Hinge type: (a) extensional (variable length active rod or spring connected to the two faces), (b) torsional(active torsional element at the hinge), and (c) flexural (active element with preset folded shape). Individual simplified free body diagrams ofthe hinge-face structure and the active element are also shown. Bending type: (d) bilayer consisting of an active and a passive layer, and (e)single layer subjected to graded driving field.

5 This additive decomposition of strain is valid only in the case of smallstrains. We will limit the conversations of this work to such an assumption.

5


tensor, T is the absolute temperature, T0 is the referencetemperature, d p is the third order piezoelectric coefficientstensor, E is the electric field vector, dm is the third orderpiezomagnetic coefficients tensor, H is the magnetic fieldvector, ei is the second order tensor of expansion due toconcentration of i chemical species, ci is the concentration ofthe i chemical species, c i

0 is the reference concentration of thei chemical species, and εms is the second order tensor ofstrains caused by changes in the material micro- and/or nano-structure. This last contribution (εms) may be generated orrecovered due to phase transformation [82], variant reor-ientation [83, 84], change in the crosslinked structure incertain polymers [85, 86], etc. These strains are coupled incertain extent to external variables such as σ , T, H, E, and areoften associated with other internal variables (e.g., phase orvariant fraction).

The total strain ε can clearly be separated into two dis-tinct parts: the strain due to elastic deformation εel and thestrain due to actuation εact. This separation is given byε ε ε= +el act , where the elastic strain is given by ε σ= Sel .Taking this into account, one obtains the following expressionfor the actuation strain:

ε α

ε

= − + +

+ ∑ − +=

( )( ) d E d H

e

T T

c c. (2)

act p m

i

ni i i ms

0

10

It is observed that there are various possible ways tocontrol an active self-folding structure depending on theactive material chosen:

• Use direct coupling by applying fields that induceactuation strain without modifying the material micro-and nano-structure (e.g., modify the electric field E toincrease actuation strain due to piezoelectric effect (d Ep )or modify the temperature to generate thermal expansionor contraction α −T T( )0 ) .

• Use direct coupling by applying fields to modify themicro- and nano-structure of the material (e.g., applytemperature to induce phase transformation in a shapememory alloy (SMA) and generate/recover transforma-tion strains).

• Use indirect coupling and apply fields that modify certainmaterial properties (e.g., use temperature to modify thematerial compliance S and alter the elastic strains.Examples of indirect coupling: [87–90]).

2.2.1. Comparative single fold analysis. To make anassessment of the actuation strain and actuation stress fordifferent active materials, those properties should be mappedinto fold-specific quantitative characteristics (e.g., bending ortorsional moment, folding angle or radius of curvature at thefold). Analytical expressions for the mechanical analysis of asingle fold can be obtained for the torsional and multi-layerconcepts (see figure 5) after making certain simplifications.The influence of secondary material regions or structuralcomponents (e.g., sensors, connectors) that might be essentialin the full physically realized self-folding systems isneglected.

Equations relating actuation strain and actuation stress tovarious self-folding metrics are presented in table 1 where φ,

Figure 6. Active materials performance for the torsional fold concept. The diagram shows typical ranges of actuation strain and actuationstress for different common active materials [65]. The top and right axes represent the normalized hinge rotation and the normalized hingetorque, respectively.

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R, Mt, and Mb are the torsional angle (equal to folding angle θe

for the torsional concept), radius of curvature, torsionalmoment, and bending moment, respectively. Those values aredirectly or inversely proportional to the active materialactuation strain εact and actuation stress σ act. The derivationsfor the relations associated with a single fold via the torsionalconcept and multi-layer concept are presented in appendix Aand appendix B, respectively. The other parameters in theequations shown in table 1 are constants that depend ongeometry and/or material properties. The constants r and Lcorrespond to the cross-section radius and axial length of thetorsional active element at the hinge, respectively. Theconstants h, a1, and w correspond to the total sheet thickness,the thickness of the active layer, and the width of the sheet inthe direction parallel to the fold line, respectively. Thedimensionless constants C1 and C2 are defined as follows

=+ + + +

+( )( ) ( )

( )C

m mn m mn

m

3 1 1 1 ( )

6 1, (3)1

2 2

2

= ++

Cha E

E a E a2

6, (4)2

12

1

1 13

2 23

where m is the ratio of thickness of the active layer to thethickness of the passive layer, n is the ratio of the elasticmodulus of the active layer to the passive layer, a2 is thethickness of the passive layer, and E1 and E2 are the elasticmodulus of the active and the passive layer, respectively.

2.2.2. Active materials assessment. Figures 6 and 7 comparethe actuation strain and actuation stress of common activematerials [65] for the torsional and multi-layer fold concepts,respectively. The bottom axes of both figures 6 and 7represent the actuation strain (related to the stroke of anactuator) and the left axes represent the actuation stress σ act,assuming here that the active material is operating underconstant stress. The top and right axes of figure 6 showmetrics that assess the material performance in the torsionalconcept (according to the results of table 1). The top and rightaxes of figure 7 show quantities that assess the materialperformance as an active layer in the multi-layer concept(table 1). In this plot, it is observable that ceramic-type activematerials (electrostrictive ceramics, magnetostrictiveceramics, and piezoelectric ceramics) may not be suitablefor self-folding systems because they would provide bendingradii that are orders of magnitude below other common active

Table 1. Relations between actuation stress and actuation strain withdifferent fold metrics.

Fold conceptActuation strain

assessmentActuation stress

assessment

Torsionalconcept

φ ε=rL

2 act πσ=M

r 2t

act

3

Multi-layerconcept ε=R

C h1act

1

σ=M C

whab act2

1

Figure 7. Active materials performance for the multi-layer fold concept. The diagram shows typical ranges of actuation strain and actuationstress for different common active materials [65]. The top and right axes represent the normalized bending radius of curvature and thenormalized bending moment, respectively.

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materials, though they may do so at much higher frequencies[91, 92]. Often, actuation stress limitations may be reduced bythe usage of compliant mechanisms [93–98]. For instance,shape memory polymers (SMPs) [85, 99, 100], which areshown in figures 6 and 7 as having the lowest actuation stresscompared to the other common active materials, have beendemonstrated to be a suitable active material for self-folding(e.g., [101]).

The assessments in table 1 were made using analyticalmodels and they serve as a quantitative first order assessmentof how different materials will perform in self-foldingsystems under certain idealizations (e.g., the influence ofstructural components such as connectors and sensors on thefolding performance is neglected). To provide with furtherinformation regarding the full system integration andcapabilities of active materials within self-folding structures,examples of previously demonstrated systems are presentedin the subsequent section.

3. Active self-folding structures

Here we describe existing examples of self-folding structuresclassified according to the physical field that induces the

folds. The goal is to provide researchers and practitioners witha systematic view of the state-of-the-art in this important andevolving area. Thermal, chemical, optical, electrical, andmagnetic field-induced self-folding structures are presented.tables 2 through 6 summarize the characteristics of the largestclasses of self-folding systems considered. Examples of eachtype are characterized in terms of their inducing field, foldconcept, reversibility of the folds, material system, char-acteristic sheet thickness, and current or potential application.

3.1. Thermally-activated self-folding

The design space for thermally induced self-folding systemsis large in terms of the methods available for localized supplyof heat which include conduction, Joule heating, convection,radiation, etc. Although there are several ways to alter thetemperature field in a structure, the diffusive nature of heatrepresents a design challenge requiring the consideration ofmethods for controlling the spatial distribution of temperatureover time (e.g., by adding thermal insulators to maintain alarge temperature zone concentrated in certain regions of theself-folding system).

At the macro-scale, thermally-induced self-folding hasbeen achieved mostly by the usage of SMAs and SMPs. On

Table 2. Examples of thermally-activated self-folding structures.

Fold concept Reversible Material systemCharacteristic thick-

ness [μm] Application Reference

Flexural Yes SMA 500.0 Potential multi-purposemorphing structures

[102, 103]

Multi-layer and singlelayer

Yes SMA 1000.0 Potential multi-purposemorphing structures

[104–107]

Other Yes SMA 50.0 Stent [60, 108, 109]Other Yes SMA 100.0 Robotics fabrication [110, 111]Extensional Yes SMA 1000.0 Robotics [28, 112]Flexural Yes SMA 1000.0 Robotics [113]Extensional, flexural,and multi-layer

Yes SMA 2000.0 Flexible mobile devices [114]

Multi-layer Yes SMA 100.0 Flexible mobile devices [115]Multi-layer Yes SMA 500.0 Decorative paper

structures[116–118]

Multi-layer Yes SMA and SMP 1000.0 Potential multi-purposemorphing structures

[119]

Flexural Yes SMP 1000.0 Potential multi-purposemorphing structures

[101, 120–122]

Flexural and multi-layer

Yes SMP 500.0 Novel fabricationmethods

[123]

Other Yes SMP 500.0 Potential multi-purposemorphing structure

[124]

Single layer Yes(once)

SMP 250.0 Potential multi-purposemorphing structure

[125]

Multi-layer Yes SMA 25.0 Microgripper [126, 127]Multi-layer Yes Thermoresponsive

polymerNot available (in theorder of 1000.0)

Potential multi-purposemorphing structures

[128]

Multi-layer Yes Polymer-metal bilayer 5.0 Capture/release smallscale devices

[129, 130]

Multi-layer Yes Thermoresponsivepolymer

0.5 Capture/release smallscale devices

[131–133]

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Figure 8. Thermally-induced self-folding structures: (a) sheet with SMA hinges. A flat sheet maneuvers to fold towards an airplane shape[102] (reproduced by permission of the Proceedings of the National Academy of Science of the United States of America, PNAS). (b)Massively foldable SMA-based laminate including worm and rolling locomotion examples and self-folded cube simulations [139] andexperimental demonstrations of the sheet folding towards a ‘bowl shape’ and an ‘S-shape’ [135]. (c) SMP-based self-folding structuresmorphed under uniform heating [121].

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the SMA side, Rus and coworkers [102, 103] developed aconcept of self-folding origami with universal crease patternsthat consisted of a single sheet with repeated triangular tilesconnected by pre-engineered actuated hinges. The repeatedtriangular tiles configuration provides flexibility in the num-ber of shapes into which the sheet can fold. Thin (100 μm)Nitinol foil was used as the actuating hinge. This system wasshown to be successful through demonstrations of the singleplanar sheet folding towards conventional origami shapessuch as a boat or a plane (figure 8(a)). An and Rus provided adesign and programming guide for the development of self-folding sheets of this kind [134]. In their work, they describeand analyze algorithms that generate designs and programsfor the sheet.

Other example of SMA-based self-folding origami is theconcept of massively foldable self-folding sheet developed byHartl, Malak, Lagoudas and others [104, 106, 135, 136]. Theconcept consists of a composite laminate with two outerlayers of SMA separated by a compliant and insulating layer(e.g., an elastomer). The outer layers of the SMA may consistof thin pre-strained SMA films [106, 137] or meshes of pre-strained SMA wires [104, 138]. With this three layer design,the side of the laminate being heated determines the directionof the fold relative to the laminate normal. It was shown viafinite element analysis (FEA) that structures composed of thislaminate are able to morph, raise against gravity, and formthree-dimensional structures as demonstrated in [105].Simulations and experimental demonstrations of this conceptare shown in figure 8(b). More recently, SMA-SMP compo-site laminates have been investigated with this concept tocreate a shape memory composite capable of locking in itsfolded shape upon a heating/cooling cycle [119].

Kuribayashi and coworkers [60, 108] performed thedesign, manufacturing, and characterization process of a self-deployable origami stent graft. This stent is made from asingle foldable SMA foil with mountain and valley folds. Thedeployment of this stent design can be achieved by shapememory effect activated at the body temperature or by mak-ing use of pseudoelasticity. The authors demonstrated that theproposed design successfully deploys as expected. Thisconcept adds to the extensive use of SMAs in the biomedicalfield [140–142].

Robots with SMA-based self-folding components havealso been designed. Rus and coworkers [110] presented anorigami-inspired technique that allows for the application oftwo-dimensional fabrication methods to build three-dimen-sional robotic systems. The laser-machined origami robotsuse only a flat sheet as the base structure for building com-plicated bodies. They showed the fabrication and assemblyprocess of a robot that can undergo worm-like peristalticlocomotion [111]. In such design, NiTi spring actuatorsplaced on the body moved parts of the robot on demand. Leeand coworkers [28] designed a deformable wheel robot withthe ball-shaped waterbomb origami pattern where the sizewas altered by activating an SMA spring actuator. By chan-ging the size of its wheels the robot was able to navigate pathswith various space limitations. Firouzeh and coworkers [113]developed a four-fold sheet-like robot that folds using SMA

actuators. The device also showed promising locomotioncharacteristics.

SMA-based self-folding has also been used for paperanimation. Qi and coworkers [116, 117] utilized SMA wiresto self-fold different shapes such as cranes. Their worksprovide a set of design guidelines and information regardingcircuitry details. The interesting shapes of folding structurescan also be potentially applied to other areas such as flexibledisplay devices [143–146]. For example Roudaut and co-workers [114] and Gomes and coworkers [115] proposed newflexible display designs for mobile devices that can adapt theirshape via SMA actuation driven fold-like operations to createwhat they term shape resolution in mobile devices.

Of course, SMAs are not the only option for thermallyinduced folding. SMPs provide actuation at a higher strain butlower force (see figures 6 and 7). One example of thermally-activated self-folding with SMPs is the work of Demaine,Rus, Wood, and others [101]. They developed a method ofself-folding hinges consisting of SMP, paper, and resistivecircuits. In addition, they created a model for the torqueexerted by such composite hinges, which was experimentallyvalidated. The SMP-based self-folding composites show thecapability of creating complex geometries and locking shapesvia proper sequential folds design. They have studied differ-ent ways to heat the SMP including radiation heating (visiblelight), electrical methods, and uniform heating [120, 121].Examples of uniformly heated SMP-based self-folding com-posites are shown in figure 8(c).

Behl and coworkers have also demonstrated self-foldingusing the shape memory effect in SMPs [124]. In theirdemonstration, the permanent (set) shape of the SMP is thedesired folded configuration while the temporary shape isassociated with the unfolded configuration. Initially at atemperature below the glass transition temperature, the SMPcan be heated above this temperature, morphing from its flattemporary configuration to its folded permanentconfiguration.

Light sources have also been used to heat thermally-activated self-folding devices based on the conversion of lightto heat [72], where the thermal energy is then converted tomechanical energy. Liu and coworkers fabricated self-foldingstructures that employ localized absorption of light in theinfrared spectrum cast over a compositionally homogenoussheet of SMP. The uniform externally applied stimulus (i.e.,unfocused light) generates a focused folding response vialocalized designed light absorption [125]. Their approachuses mass-produced materials without the need for multiplefabrication steps, where the folded regions were defined bythe presence of black ink patterned by a printing process. Thepolymer regions located beneath inked areas heat faster thanthe areas elsewhere and eventually heat beyond the glasstransition temperature of the SMP. After such temperature isexceeded, these local SMP regions relax and the film bends.The original flat configuration can be recovered by heating theentire sheet above the SMP transition temperature.

There are a number of self-folding examples usingthermally-induced actuation at the small scale. Lee and co-workers designed a microgripper that uses Ni–Ti–Cu SMA

10


self-folding to open and close [126, 127]. The microgripper isfabricated by alignment and selective eutectic bonding of twopreprocessed silicon wafers. Deposited SMA films serve asthe outer layers of the microgrippers, acting as actuators.Applications for the microgripper include assembling smallparts for manufacturing, minimally invasive biopsy tissuesampling, remote handling of small particles in extremeenvironments, among others [126]. Clearly the single outerlayer of SMA provides bending actuation in a manner ana-logous to the bilayer design shown in figure 5(d).

The multi-layer concept has also being exploited by otherresearchers for self-folding at the small scales. For instance,Gracias and coworkers fabricated folding structures at themicro-scale able to perform sequential folding via heating ofpre-stressed hinges using lasers [128]. Their hinges werecomposed of Cr/Au-polymer bilayers while their rigid regionswere composed of single Au layers. Upon laser irradiation,the polymer layer softens and the bilayer bends due toexisting pre-stress generated during the bilayer fabricationprocess. Kalaitzidou and coworkers [129, 130] also devel-oped self-folding polymer-metal bilayer structures. Theirbilayer sheets consisted of polydimethysiloxane (PDMS[147, 148]) and gold (Au) layers that had self-folding cap-abilities. The PDMS layer had a thickness of several micro-meters while the thickness of the Au layer was in the order ofnanometers. Upon changes in temperature, the bilayer folds orunfolds due to dissimilar thermal expansion of the two layers.They also created a PDMS-silicon carbide (SiC) bilayer withsimilar behavior to demonstrate that their concept can beapplied using any two materials with dissimilar thermalexpansions. The ability of the bilayers to capture, transportand release different solids was demonstrated indicating theirpotential application as delivery tubes [130]. A similar bilayerapproach was adopted by Ionov and coworkers when fabri-cating polycaprolactone (PCL) - poly-(N-iso-propylacrylamide) (PNIPAM) self-folding polymer bilayers[131, 132]. PCL is hydrophobic (i.e., tending to repel/rejectwater [149]) while the solubility of PNIMAN can be changedreversibly with temperature by going above/below its lowcritical solution temperature [131]. This temperature depen-dent behavior allows PNIPAM to swell or collapse in the

presence of water with changes in temperature. CombiningPNIMAN with the PCL hydrophobic layer allows for tem-perature controlled folding and unfolding of structures createdwith this bilayer.

3.2. Chemically-activated self-folding

Self-folding using chemical stimulus has also been exploredby multiple researchers. Most of these systems are based onthe multi-layer fold concept and utilize the degradation orswelling behavior of certain polymers under the presence ofspecific substances or pH level [150, 151]. Examples of thesesystems are presented in table 3.

Self-folding films sensitive to pH are commonlydesigned using weak polyelectrolytes as active polymers[151]. One example are the microtubes fabricated by Kumarand coworkers [152]. They considered a three-layer laminatesheet of poly-(dimethylsiloxane) (PDMS) / polystyrene (PS)/ poly(4-vinylpyridine) (P4 VP) for self-rolling of micro-tubes. The folding-based mechanism for rolling-up of themicrotubes was based on the different degree of swelling inthe constituting polymer layers. PS demonstrates minimalwater uptake while P4 VP is less hydrophobic and swells inacidic aqueous solutions because of protonation of polymerchains. When a P4 VP layer swells, its volume increasesand, if fixed to a PS layer, will cause the resulting polymerlaminate to fold. In their work, Kumar and coworkers con-verted the PDMS layer into silica after rolling via oxidativepyrolisis. This process allowed them to produce a stiff andthermally and chemically stable microtube. This approachwas also demonstrated to be useful for the fabrication of allmetallic microtubes by using a PDMS–PS polymer bilayeras template by depositing a thin layer of metal onto thebilayer, rolling by the polymer swelling, and then removingthe bilayer [153] where the effects of layer thickness andother parameters on the microtubes fabrication procedurehave been investigated [154]. Micro- and nano-tubes fabri-cated by this method are promising for applicationsincluding nano-syringes for intra-cellular surgery and nano-jet printing [164]. This folding-based fabrication approachshows flexibility in the axial shape of the obtained microtube

Table 3. Examples of chemically-activated self-folding structures.

Fold concept Reversible Material systemCharacteristic thick-

ness (μm) Application Reference

Multi-layer No Hydrogel 0.5 Microtubes fabrication for smallscale devices

[152–155]

Multi-layer No Hydrogel 20.0 Drug release device [156, 157]Multi-layer Yes Hydrogel 50.0 Micro-containers [158]Multi-layer Yes Hydrogel 5.0 Potential multi-purpose morph-

ing structures[159]

Torsional andmulti-layer

Yes Hydrogel 200.0 Potential multi-purpose morph-ing structures

[160]

Multi-layer No Polymerdegradation

Not available (in the orderof 100)

Small scale grippers [161, 162]

Multi-layer No pH responsivepolymer

0.5 Capture/release small scaledevices

[163]

11


(i.e., the tubes do not necessarily have to be straight). Anexample of this is the fabrication of toroidal hollow-coremicrocavities obtained using the polymer bilayer approach[155]. In another work, Shim and coworkers [158] createdrobust microcarriers using hydrogel bilayers that exhibitedreversible folding behavior. The bilayer hydrogel systemconsisted of a layer of poly(2-hydroxyethyl methacrylate-co-acrylic acid), p(HEMA-co-AA), and a layer of poly(2-hydroxyethyl methacrylate), p(HEMA). Planar films com-posed of this bilayer were able to fold towards micro-con-tainers by swelling of the p(HEMA-co-AA) layer at apH of 9.

A similar self-folding bilayer approach was adopted by Heand coworkers in the fabrication of an oral delivery device[156]. The main part of the device consisted of a finger likebilayer composed of pH-sensitive hydrogel based on cross-linked poly(methyacrylic acid), which swells significantlywhen exposed to body fluids, and a second non-swelling layerbased on poly(hydroxyethyl methacrylate). Studies regardingthe degree of folding as a function of the bilayer compositionhave been performed [159]. In such studies, the swelling layerwas prepared with a mixture of poly(ethylene glycol metha-crylate) and poly-(ethylene glycol dimethacrylate). By con-trolling the ratio between the two components of this ‘active’layer, different degrees of folding were achieved. Two differentmicro-scale structures folding using this bilayer are shown infigure 9(a). The bilayer approach was also utilized by Ionovand coworkers to synthesize PCL/polysuccinimide polymerbilayer self-rolled tubes. Both polycaprolactone and poly-succinimide are generally hydrophobic. In physiological buffersurrounding environment, polysuccinimide hydrolyzes formingwater-swellable polyaspartic acid that yields to the rolling ofthe polymer bilayer and the formation of tubes [163].

Jeong and coworkers developed a PDMS—hydrophilicpolyurethane (PU)/2-hydroxyethyl methacrylate (HEMA)bilayer composite that folds when submerged in hexane sol-vent due to the large swelling of the PDMS layer of suchsubstance [160]. This idea allowed them to create pro-grammed two-dimensional sheets that can self-fold into acube, pyramid, and helix forms.

A self-folding gripper that opens and closes by theactuation of polymer hinges was fabricated by Gracias andcoworkers [161, 162]. The actuation of the polymer hingeswas triggered by their sensitivity to the presence of enzymes,where they utilize two different polymer types with twomutually exclusive enzyme sensitivities. The two polymerswere placed at hinges in such a way that bending in oppositedirections is activated given the appropriate stimulus. Whenone polymer is selectively degraded by its associated enzyme,its modulus decreases and the gripper closes. When the otherpolymer is degraded via the action of its own distinct enzyme,its respective hinge bends and the gripper opens. This processis shown in figure 9(b).

3.3. Optically-activated self-folding

Previously in section 3.1, we discussed light driven actuationthat worked on the principle of radiative heating. Here,

however, we consider actuation driven by microstructuralchanges directly caused by light irradiation.

Folding of polymer films with light was investigated byRyu and coworkers [165]. The mechanism used for self-folding in their work was localized photo-induced stressrelaxation. The considered polymer was composed of Pen-taerythritol tetra(3-mercaptopropionate) (PETMP), 2-methy-lene-propane-1, 3-di(thioethylvinylether) (MDTVE), andethylene glycol di(3-mercaptopropionate) (EGDMP) mixedwith photoinitiators. Straining the sheet and subsequentlyirradiating it with light dissociates specific photoinitiators intofree radicals that react with and cleave the MDTVE

Figure 9. Chemically-activated self-folding examples: (a) opticalmicrographs of a cross-shaped microwell and a multiple extensionsmicrowell in their folded and unfolded configurations. All scale barsrepresent 100 μm. Reprinted with permission from [159]. Copyright(2005) American Chemical Society. (b) Optical images andschematic representation of micro-grippers folded by selectivepolymer degradation. Scale bars represent 200 μm. The foldingprocess is also illustrated for a single hinge. Reprinted withpermission from [161]. Copyright (2010) American ChemicalSociety.

12


functionalities along the polymer backbone [165]. Suchevents irreversibly rearrange the network connectivity andmacroscopically result in stress relaxation that generates thefolds. The authors were able to create folded arcs and a closedcube-shaped box using this self-folding approach. Furtherinformation about optically induced deformation in polymerfilms can be found elsewhere [166, 167]. Polymers are not theonly option when considering optical actuation. For instance,Zanardi Ocampo and coworkers fabricated optically-actuatedmicro-mirrors based on laminates with strained GaAs-basedalloys. Actuation of such laminates was achieved by laserirradiation [22, 56].

3.4. Electrically-activated self-folding

With the recent advanced in electronics and MEMS tech-nologies, limitations for the generation and manipulation ofelectric fields across scales have diminished. These advances

along with the extensive research in dielectric elastomers[168, 169] and other electro-active polymers [170] have madepossible for the development of electrically-activated self-folding structures. Examples of these structures are providedin table 5.

On the large scale, White and coworkers [29] demon-strated the feasibility of implementing dielectric elastomeractuators for a bending component that can be applied tomodular robotics. They present the design and experimentalanalysis of a dielectric elastomer actuator that consists of twolayers of dielectric elastomer separated from each other. Itwas noted that improvements could be made with respect tothe bending performance by optimizing the geometricalparameters of the device and/or adding more dielectric elas-tomer layers. A different design for a dielectric elastomeremployed for self-folding structural actuation was investi-gated by Frecker and coworkers [73, 171, 172]. Theydeveloped a bending actuator that consisted of three primary

Table 4. Example of optically-activated self-folding structures.

Foldconcept Reversible Material system Characteristic thickness (μm) Application Reference

Singlelayer

No Photo-responsivepolymer

Not available (in the order of1000.0)

Potential multi-purpose morphingstructures

[165]

Multi-layer Yes GaAs-based alloys 1.0 Micro-mirrors [22, 56]

Figure 10. Electrically and magnetically-activated self-folding structures: (a) a self-folding sheet with dielectric elastomer actuator under noapplied electrical field (left) and with applied electrical field (right) [73] (reproduced by permission of the American Society of MechanicalEngineers, ASME). (b) Photo series of a biomorphic origami robot showing the robot translating using accordion-like movement activatedvia electrically-induced folds [173] (Reproduced by permission of IOP Publishing. All rights reserved). (c) Magnetically induced folding of aPDMS-MAE box under zero magnetic field (left) and under non-zero magnetic field (right) [73] (reproduced by permission of the AmericanSociety of Mechanical Engineers, ASME).

13


layers: the active dielectric elastomer, a passive substrate(scotch tape), and compliant electrodes (carbon grease). Thebending samples configured in this manner were able togenerate a sample length normalized tip displacement ofslightly above 1, demonstrating the promising qualities of thisas a self-folding concept. Figure 10(a) shows the foldingbehavior of the dielectric elastomer-based bending sample.Self-folding with dielectric elastomers based on the exten-sional concept (figure 5(a)) were developed by Roudaut andcoworkers for flexible mobile device displays (previouslydiscussed in section 3.1). Their concept considered bendingof a flexible mobile device via contraction and expansion of alinear dielectric elastomer actuator connected to two sides ofthe device [114].

More complex deformations were demonstrated byOkuzaki and coworkers, who created a biomorphic robotfabricated by a folding a conducting polymer film [173]. Thefolding actuation was generated by electrically-inducedchanges in the elastic modulus of a conducting humido-sensitive polymer film. Those changes of elastic moduluswere due to absorption and desorption of water vapor mole-cules depending on the imposed electrical field. Theydemonstrated the feasibility of the concept by fabricatingdifferent prototypes such as an origami robot that movesrectilinearly with a caterpillar-like motion. Such motion wasachieved by repeated expansion and contraction of itsaccordion-like body via folding/unfolding of electricallysensitive Polypyrrole (PPy [176, 177]) (figure 10(b)).

At the small scale, Inganás and Lundstróm utilizedconducting polymers as actuators within electrically-drivenpolymer-gold folding bilayers [174, 175]. They used con-ducting PPy as the active layer. The bilayer is submerged inNa+DBS−electrolyte solution (where DBS represents dode-cylbenzenesulfonate). When a negative voltage potential isapplied, the PPy film fills with Na+ and is expanded to its

original size, thus the flattens. When a positive voltagepotential is applied, the volume of the PPy layer reduces andthe bilayer bends.

3.5. Magnetically-activated self-folding

Magneto-active elastomers (MAE [178]) have been investi-gated as actuators for self-folding materials by von Locketteand coworkers [73, 74]. The MAE materials were fabricatedby mixing 30% (by weight) barium ferrite (BaM) particlesinto a silicone rubber matrix. The passive substrate consistedsimply of a silicone RTV compound. The MAE-based self-folding composite was able to bend with a sample lengthnormalized tip displacement of slightly above 0.5. Compo-sites consisting of PDMS sheets with MAE patches were ableto achieve locomotion under application a oscillating mag-netic field. In addition, a cross-shaped PDMS sheet with fourMAE patches on its sides that raised under the application ofmagnetic field was shown to fold into a box (figure 10(c)).However, it should be noted that in this particular example,the MAE patches were used only for their ability to bemagnetized rather than for any magnetic to mechanicalenergy conversion.

The subsequent section provides a survey of anotherlevel in the design process of a self-folding structure, whichentails crease pattern design and fold planning.

4. Crease pattern/fold kinematics design

Previously in section 2, we addressed the design of individualfolds as well as the material selection for active self-foldingsystems. This section is concerned with the design of creasepatterns and the analysis of folding maneuvers to achievecertain shapes or functionalities, irrespective of the active

Table 5. Examples of electrically-induced self-folding structures.

Foldconcept Reversible Material system

Characteristic thick-ness (μm) Application Reference

Other Yes Dielectricelastomer

1000.0 Robotics [29]

Multi-layer Yes Dielectricelastomer

350.0 Potential multi-purpose morphingstructures

[73, 171, 172]

Other Yes Electroactivepolymer

20.0 Robotics / locomotion [173]

Extensional Yes Dielectricelastomer

2000.0 Flexible mobile devices [114]

Multi-layer Yes Electroactivepolymer

0.5 Small scale tools [174, 175]

Table 6. Example of magnetic-induced self-folding structures.

Foldconcept Reversible Material system

Characteristic thick-ness (μm) Application Reference

Other Yes Magneto-activeelastomer

3000.0 Potential multi-purpose morphingstructures

[73, 74]

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material or other mechanism being used to effect the foldingbehavior.

To realize a useful active self-folding structure, designersmust consider several issues beyond the choice of activematerial and folding concept. The motivation for a designproblem is to fulfill one or more desired functions. Forexample, designers might achieve a storage/deploymentfunctionality for satellite solar panels using a folding struc-ture. Important considerations are to determine what is thefinal folded shape (or shapes, in the case of a reconfigurablestructure), to identify a crease pattern that can achieve thedesired shape(s), and a sequence of folding actions that resultsin the desired shape(s). Designers must achieve all of thissubject to local and system-level failure criteria and require-ments for interfacing the structure with other parts of thesystem (e.g., mating points). Table 7 is a summary of thisdesign problem.

Although many origami design approaches have theirlimits with respect to actual material folding behavior (e.g.,they commonly neglect thickness or assume an infinitelysmall fold radius), they provide useful options for addressingthe design of folded structures in general. While this survey isfar from complete, we do attempt to address the options mostsuitable for the design of self-folding systems with a focus onthe most recent works.

4.1. Software developments

Designers need tools and methods tailored to each part of thedesign problem of self-folding structures (table 7). The pre-vious sections of this article have focused primarily ondomain knowledge about fold concepts and the various activematerials available for achieving self-folding behavior.Existing computer aided design and FEA tools can be useddirectly or extended for use in the analysis and design ofactive folds. However, the challenges of identifying creasepatterns and fold sequencing require unique tools and meth-ods. This section is a review of the prior work on origamiengineering tools in support of these design activities.

One of the most well known software packages for foldpattern design is Robert Langʼs TreeMaker [179]. Langdescribed the theoretical foundation on which TreeMaker isbased: the tree method [179]. This method allows for thedesign of an origami base, which is defined as a non-stretching transformation of the sheet into 3-space such thatall facets remain flat. The base can be fully defined by thelocation of creases, their angles, and the orientation andlocation of each facet [179]. The base is then folded andshaped into an origami model. TreeMaker determines a creasepattern that results in the desired base via optimizationmethods. An example of the application of this algorithm isshown in figure 11(a), where the tree graph of a bug is showntogether with the computed crease pattern generated byTreeMaker. Further information and advances of this algo-rithm can be found in the literature [180, 181].

Another approach for crease pattern design is to directlyimpose the exact final shape that the sheet should take afterfolding. Demaine and coworkers [182] developed a

mathematical framework for folding a sheet into a thin stripthat is essentially ‘wrapped’ around the desired shape. In theirwork, they showed that one can obtain a crease pattern for aparticular flat silhouette or for a three-dimensional polyhedralsurface. They proposed different algorithms for wrapping thedesired shape with the sheet, these include one that uses anysheet area arbitrarily close to the surface area of the desiredshape, another that maximizes the width of the sheet stripsubject to certain constraints, etc [183].

Tachi presented a practical method for ‘origamizing’objects [184, 185] (i.e., obtaining the crease pattern that foldsa single sheet of material into a given polyhedral surfacewithout any cut). It is based on the idea of using folds in aninitially planar sheet to create flaps that are ‘tucked’ (hidden)to form the desired three-dimensional shape. The Origamizersoftware solves the inverse problem of providing a creasepattern from the input, which may be an arbitrary polyhedralsurface. In his work Tachi investigated the conditionsrequired for constructing a valid crease pattern.

Tachi has also contributed methods and software forcreating freeform foldable structures that can approximatecurved surfaces [186–188]. In such processes, curved bodiessuch as disks and cylinders can be obtained from collapsiblevariations of rigid-foldable surfaces. One of the proposedtechniques consists of obtaining a freeform variation of rigid-foldable and bidirectionally flat-foldable disk surfaces, whichis a combination of generalized Miura-Ori [189] and eggboxpatterns [190]. A software that provides these capabilities hasbeen created. In addition to folding pattern generation, it alsoallows for the visualization of the folding/unfolding processfor a given model. In a more recent work [191], Tachi pro-posed a method to produce origami tessellations from givenpolyhedral surfaces. The method consists of first separatingfaces of the desired shape and then inserting folded partsbetween them. The obtained configuration is then modified bysatisfying geometric constraints of developability6, foldingangle limitations, and self-intersections. Such method is notonly applicable to paper but also sheets that do not allow 180°degree folding. In another work, Tachi developed a software

Table 7. High-level definition of the design problem for active self-folding structures.

Given Desired design functionality

Find 1. Folded shape(s) that achieve desiredfunctionality

2. Crease pattern that can produce said shape(s)3. Sequence in which to execute folds (foldkinematics)

4. Appropriate active fold concept(s) (section 2.1)and detailed geometry

5. Appropriate active material(s) (sections 2.2 and3) and passive structure materials

Subjectto

Failure criteria and interface requirements

6 A surface is developable if it has zero Gaussian curvature everywhere[192, 193].

15


for interactive simulation of origami based on rigid origami[194]. The simulation program can generate the continuousprocess of folding a flat sheet into a 3D shape by calculatingthe configuration from the crease pattern. The configuration ofthe sheet is then determined by the crease angles and thetrajectory is calculated by projecting the crease angles into theconstrained configuration space.

Ishida and coworkers developed a method using con-formal transformations to design crease patterns of circularmembranes that can be wrapped up compactly [195]. Themethod follows an analogous procedure for the generation offluid flow streamline profiles using velocity potentials andstreamline functions. In their work, similar potential andstreamline functions are used to generate the fold lines profilein circular shells. The graphical analogy between fluid flowsand origami pattern is shown in figure 11(b) where it isobserved that unidirectional flow is analogous to a creasepattern for a cylindrical shell while flow with a vortex fila-ment is analogous to a crease pattern for a conical shell. Themethod enables the design of complex deployable structuressystematically and efficiently from simple structures, con-trolling angles among fold lines. In their work they demon-strate the feasibility and applicability of their algorithm bycreating plastic film and stainless steel plate folded models.

An origami computational interface called Eos [196–199]that permits a user to fold sheets virtually as if they werefolding paper by hand has been developed by Ida and co-workers. The software allows for visualization and interactionof origami constructions. Eos provides two methods for fold-ing: the mathematical fold and the artistic fold. The formerfolding method is based on axiomatic definitions of origamifolds [196, 200, 201]. Such method requires fair knowledge ofnomenclature and coding definitions for different foldingoptions. The artistic fold method is more user-friendly andallows one to specify relatively straightforward folding featuressuch as whether fold is a valley or a mountain, which lines areto be folded, and to what angle they should be folded. Thismethod is preferred for the creation of artistic origami.

Akleman and coworkers developed a method to unfold agiven convex polygonal shape into a one-piece planar (i.e.,developable) surface [105]. The algorithm first triangulates(i.e., divides into triangles) the initial shape to guarantee thatall faces of the shape are planar. A ‘dual mesh’ is then con-structed from the triangulated shape such that every facebecomes a vertex and every original vertex becomes a face[202]. A dual graph consisting of the edges and vertices of thedual mesh is then created along with an associated spanningtree that includes all vertices from the dual graph. Everyvertex of the dual graph is then two-dimensionally thickenedinto a triangle and every edge of the spanning tree is two-dimensionally thickened into a developable quadrilateral. Thespanning tree is then unfolded into two-dimensional planarshapes [203]. The algorithm has been extended such that eachtwo-dimensional planar shape can be further subdivided intosmaller quadrilaterals and triangles for input into finite ele-ment software, which allows full mechanical analysis of thefolding (or self-folding) process. An example of this proce-dure for the creation of a single-panel unfolding for a cube is

Figure 11. Examples of folding pattern/sheet design tools.(a) Illustration of the tree algorithm using one of Meguroʼs bugs.(Not all creases are shown.) [181] (Springer and The MathematicalIntelligencer, 27, 2005, page 93, Origami design secrets: mathema-tical methods for an ancient art, Robert J. Lang, figure 2, © 2005Springer Science+Business Media, Inc. is given to the publication inwhich the material was originally published, by adding this figure.With kind permission from Springer Science and Business Media).(b) Conformal transformation on fluid flows and analogy to origamipatterns: (left) uniform flow with analogous crease pattern for acylindrical shell, and (right) flow with a vortex filament with itsanalogous crease pattern for conical shell [195] (Reproduced bypermission of the American Society of Mechanical EngineersASME). (c) Procedure to obtain a single-panel unfolding from acube: initial mesh, spanning tree, unfolding, and refinement for finiteelement analysis (Reprinted from [105], Copyright (2013), withpermission from Elsevier).

16


shown in figure 11(c). The feasibility of the algorithm wastested by considering the complete unfolding/re-folding pro-cess associated with a dome and a cube shapes.

Fuchi and coworkers [204] proposed a folding patterndesign method based on topology optimization [205]. In theirapproach, the existence and type of fold lines are assumed;this given configuration is known as the ‘ground structure’.The folding angles are the design variables. For a givenground structure and set of folding angles, the three-dimen-sional geometry of the folded sheet can be computed usingorigami approaches similar to those described previously inthis section. A topology optimization method is then used inan iterative effort to find the optimal set of folding angles that,combined with the ground structure, provide the desired fol-ded geometry.

4.2. Studies of folding and the behavior of folded structures

One main consideration when designing a crease pattern forconventional origami designs is flat-foldability. A flat-fold-able origami structure must be able fold into a shape that has

dihedral angles of zero between pairs of adjacent faces. Theimposition of flat foldability constraints onto the design oforigami structures ensures that such structures can be becreated from an initially flat sheet or, conversely, be deployedtoward a fully flat configuration. It has been shown [206] thatcrease patterns created under this constraint can be applied forthe design of physically realized sheets that have non-zerothickness and may not provide perfectly flat folds. Accordingto the previous works of Demaine and OʼRourke [66],Schneider [207], and Poma [208], global flat-foldability mustsatisfy the following conditions:

(i) Local flat-foldability: considering a single vertex, it hasbeen shown that is flat-foldable if the alternating anglessurrounding the vertex sum to 180° . This is known as theKawasakiʼs theorem [209–211]. In general, a creasepattern is everywhere locally foldable if there is amountain-valley assignment so that each vertex locallyfolds flat [183].

(ii) 2-colorable: this condition refers to the binary designa-tion assigned to the faces (finite areas) bounded by the

Figure 12. Examples of folding kinematics studies. (a) In-plane Poissonʼs ratio of Miura-ori folded sheet for different folding states. Theresults show that Miura-ori folded sheets have metamaterial characteristics independent of the sheet material [52]. The figure was obtainedcourtesy of Simon Guest (reproduced by permission of the Proceedings of the National Academy of Science of the United States of AmericaPNAS). (b) Fold planning [221, 222] examples: motion paths to fold an icosahedron and an sphere from single panels (courtesy of NancyAmato and her research group at TAMU). (c) Fold designs of membrane-panel model for thickness accommodation. The rigid panels aredepicted in green while the flexible membranes are represented as red lines; 180° mountain fold, 180° valley fold, and 60° valley fold areshown. Based on the works of [223].

17


creases (see figure 2) referred to as ‘colors’. To ensureglobal flat foldability no neighboring faces should sharethe same color.

(iii) Able to stack all faces flat without penetration or collisionbetween faces during the folding process.

The kinematics of both folding and the mechanicalresponse of folded structures is also an extensive area of studyand is relevant to active self-folding systems, especially interms of their potential applicability. For example, Schenkand coworkers studied the kinematics of Miura-ori foldingpatterns [52] and showed that a folded structure folded in thispattern provides a negative Poissonʼs ratio for in-planedeformations and a positive Poissonʼs ratio for out-of-planebending. An example of these results are provided infigure 12(a) where the negative in-plane Poissonʼs ratios in aMiura-ori pattern for different folding stages and patterngeometries are shown. In addition, they also considered afolded structure based on a stacking of individual foldedlayers taking into account kinematic compatibility betweenlayers. They showed that such multilayer folded structure isable to fold and unfold uniformly and can be designed to lockits motion in a specific configuration.

Using a similar approach while further addressing morephysical considerations, Qiu and coworkers performed akinematic analysis of origami carton-type packages[212, 213]. They addressed design issues of origami macro-scale packaging and presented mathematical models to pre-dict their folding characteristics. They idealized creases astorsional springs with arbitrary stiffness during folding.Experiments were performed to validate their models. Suchwork has potential value in origami packaging design andelastic type origami elements in robotics.

The kinematics of origami-inspired robotic linkages wereaddressed by several authors. Qin and Dai studied an eight barrobotic mechanism where folded panels acted as links and thefold creases themselves acted as revolute joints [214]. Theyexplored the configuration space of this mechanism underdifferent geometric constraints. Moses and coworkers inves-tigated the kinematics of origami-inspired rotors consisting ofa snake-like strip connected to two fixed platforms at its endsand having a rotating body in the middle [215]. The entiremechanism can be folded from a single initially planar sheet,and the body at the middle of the strip rotates only due tofolding maneuvers of the strip. Potential applications of thisidea include rotating propellers for micro underwater or fluidimmersed robots and high mobility wheel legs for crawlingvehicles [215].

Researchers have also explored origami models that arefolded so that in their final folded/deployed state they exhibitmotion [216, 217]. The term action origami [218], has beenused to describe such focus. Bowen and coworkers[216, 219, 220] studied a large number of action origamimodels and proposed a classification scheme for them. Theclassification is based in the idealization of vertices of inter-secting folding lines as spherical mechanisms. In this form-alism, each vertex is considered to lie at the center of animagined sphere and the motion of the folds connected to

each vertex are tracked by considering their projections ontothat the sphere. The classification takes into account char-acteristics of the folding pattern such as whether the fold linesform a loop, the linearity of the fold line chains, whether theyare periodic, and other aspects.

With several examples in nature of origami-like pro-cesses [224–229], bioinspiration has also provided ideas forfold designs [230]. For instance, De Focatiis and Guest [231]presented a folding pattern design of deployable structuresinspired by a model of deploying tree leaves. They investi-gated the effects of combining several corrugated leaf patternsto produce deployable surfaces, such as solar sails, solarpanels, and antennas. Nature inspired approaches for creasepattern analysis are also being developed by McAdams andcoworkers [232, 233] where they consider pixelated multi-cellular representations to define origami structures and acrease generation method based on the analogy between acrease pattern and ice-craks on a frozen lake surface. The firstmethod consists of defining a crease pattern via a set of dis-tributed cells. The cells define the crease pattern by their color(each face contains a collection of cells with the same color).A crease restoration algorithm to extract the equivalent creasepattern from the multicellular representation was proposed.The second method is based on the analogy between ice-cracks on a frozen lake surface where each crack is equivalentto a crease and each forking point to a vertex [233]. To formthe creases and vertices in an ‘ice-cracking’-like origamicrease pattern, a vertex is picked as the starting location andthe rest of the creases and vertices are grown in an analogousmanner to crack growth and fork formation in ice. A geneticalgorithm [234, 235] encodes the geometric information offorming the creases and vertices according to the develop-ment sequence based on the ‘ice-cracking’ process which canbe adapted to accelerate the emergence of optimal designoutcomes through the evolutionary design process.

One issue not addressed in the studies and developmentsmentioned up to this point is that of fold planning, whichassesses the physical feasibility of folding operations incomplex forms. Amato and coworkers investigated motionplanning algorithms [221, 222, 236, 237] that are applicableto the fold planning problem. Their motion planning algo-rithm can obtain, as input, the geometry of the sheet in bothits folded and unfolded shapes and can generate a continuouspath through the state space that will allow the sheet to movefrom one configuration to the other without self-intersectingitself. The motion planner is based on Probabilistic Road-Maps [238] and has been improved through the years toincrease computational efficiency. Two examples of suc-cessfully determined folding paths are shown in figure 12(b)and includes examples are folding paths for an icosahedronand a sphere.

The majority of what has been described so far in thissection is relevant for folding of only very thin sheets.However, the examples described in section 3 largely addressself-folding of ‘thick’ sheets. Fortunately, there are a numberof works addressing the kinematics and fold design pertainingto thick sheets. For example, Guest and Pellegrino [239]developed a methodology for wrapping thin membranes

18


around a central hub. They begin with the assumption of zerothickness for the generation of a folding pattern and thenshow a way to correct the proposed folding pattern toaccommodate for a sheet of finite thickness. Tachi proposed amethod for creating a 3D structure composed of finite thick-ness panels. The kinematic behavior of an associated zero-thickness model of the same structure is preserved byembedding said zero-thickness mechanism within a set offinite thickness panels [240]. Thus, the proposed mechanismfolds in the same way of the base zero-thickness modelthough with some limitations such as that the final foldedstate of the structure cannot have coplanar panels (e.g., thepanels must remain within a non-zero internal folding angleand/or a finite distance between them). Zirbel and coworkers[223, 241, 242] addressed the accommodation of thickness inorigami-based deployable arrays, motivated by the need tofold thick rigid panels that cannot bend during folding ordeployment (e.g., solar panels made of brittle materials). Intheir work they present a mathematical model for the mod-ification of folding patterns to accommodate material thick-ness in the context of the design, modeling, and testing of adeployable system. Examples of panel-membrane hingedesigns for common folds are shown in figure 12(c). Hingedesigns for 180° mountain folds, 180° valley folds, and 60°valley folds are shown. They demonstrate the applicability oftheir model using a 1/20 scale prototype of a deployable solararray for space applications. Their physical constructionincluded gap widths at the folding lines (not present in zerothickness origami models) to accommodate for thickness.

5. Conclusions

Although the art of origami is ancient, the science and tech-nology associated with origami-inspired engineering is newand developing rapidly. As identified in this review, smartmaterials can play a significant role in the realization of self-folding origami-inspired structures. Researchers havedemonstrated self-folding behavior in many active materialsystems with inducing fields that include thermal, chemical,optical, electrical, and magnetic. Several combinations ofmaterials, geometry, and inducing field are feasible, yieldingan array of design options. Considerations such as whetherfolds must be reversible, restrictions on sheet thickness,boundary conditions, availability of a particular inducingfield, and the overall folded structure concept are among thedrivers for design decisions. Evaluation of common activematerials and self-folding structures for various fold conceptsconfirms that there is no dominant active material ormechanism for self-folding applications and that the materialand mechanism selection are application-dependent (e.g.,dependent on the desired folding radii or angle, surroundingenvironment).

Mechanically, one can use smart materials to producefolding along a hinge or to approximate folds through a small-radius bending action in a composite laminate structure orsingle-layer active material using a graded field. Concepts inthe latter category are less similar to creases in paper, but can

be similar enough to benefit from origami design principles.The lack of pre-engineered hinge locations in bending foldconcepts also makes them more adaptable in principle.However, concepts with pre-engineered hinge locations canproduce folds that are more idealized and their more restrictedstructure can simplify aspects of the design process such asfold planning.

A number of design tools exist for origami, many ofwhich are useful for designing self-folding structures. Mostassume a desired folded form is known and thus address oneof the two origami subproblems: determining where foldsshould go and determining the order in which folds should bemade. Early investigations into the mathematics of origamiaddressed basic questions such as whether a particular shapeis foldable and relied on strong idealizations such as having afolding sheet of zero thickness. Newer research into topicssuch as folding with thickness, the kinematics of origamistructures, and bio-inspired folded structures is yieldingresults that will help bridge the gap between idealized origamiand the reality of smart self-folding structures.

Although the research on origami principles and foldingstructures has been extensive, multiple challenges and openquestions remain. Three questions are of particular impor-tance in the context of smart self-folding systems:

• Increased understanding of the mechanics of folding inthick structures toward improved flat foldability: eventhough multiple theories and software exist for the designand analysis of zero thickness folding structures (seesection 4), very few exist for the development of foldingstructures with non-zero thickness. Thick origami foldingis a challenging problem and has to be addressed for thedevelopment of applications that require compact storagein non-planar configurations (e.g., stacking of faces) andconventional origami structures where the thickness ofthe sheet is not several orders of magnitude below thecharacteristic length of the sheet.

• Function-driven design of self-folding structures: in bothits artistic goals and mathematical theory, the theme oforigami has traditionally captured form and geometricconsiderations. For example multiple software productsexist for the determination of crease patterns and/orfolding sequences that will allow a sheet to be mappedinto a certain goal three-dimensional shape. However,engineering applications of origami principles are moreoften motivated and constrained by function rather thanform. Only a few examples have been addressed such asthe determination of folded shapes for crash boxapplications [44–46], and folded shapes towards simpleobjectives such as maximizing self-folded structuralheight under gravity [206]. However, these limitedexamples apply to specific sheet materials and constraintsand lack the generality desired of an engineering tool todetermine ideal crease patterns/folding sequences for theoptimal performance of specific functions. Perhapsincreased efforts by engineering design researchers, suchas those who draw inspiration from the functions of

19


folding in nature [233, 243–245], will help to expandthese capabilities.

• Development of remote field-induced reconfigurablefoldability: every one of the active material-based foldingor fold-like structures described herein relies on theimposition, local or global, of a single given physicalfield (e.g., thermal, magnetic, etc). Sheets are pre-engineered to respond in a single particular manner tosuch field application. Therefore, reconfigurability (i.e.,from one morphed/folded shape to another) under globalfields becomes difficult, this may prevent the level oftruly reconfigurable self-folding structures at the smallscale or other situations where local field impositionbecomes unfeasible. Thus, reconfigurability under global(remote) fields would be an important advancement. Thiscould be accomplished by designing for multi-fieldactuation (e.g., thermal and electric [246], thermal andmagnetic [247]) or by allowing discrete folding regionsto be tuned so as to respond to changes in a single globalfield type (e.g., as in changes in magnetic fieldfrequency). Multi-field self-folding is being addressedby some researchers in both large [73] and small scales[54] but there is still much to be done in this area.

Active material technologies have already enabled a widerange of engineering applications, many of them accuratelydescribed as being based on ‘smart structures’. As the com-munity that champions such solutions progresses, it is likelythat researchers may find great advantage in looking beyondtraditional design approaches and permitting inspiration fromalternative sources, such as the art and theory of origami. Ashas been discussed, creating recoverable folds in complexmaterial systems is more difficult than creasing paper; indu-cing localized self-folding is more challenging still. Appli-cation of past discoveries and current tools toward futuremorphing structures challenges, however, will allow engi-neers to address the open issues posed and offer new tech-nological solutions to the world at large.

Acknowledgments

This work is supported by the National Science Foundationunder grant EFRI-1240483. Any opinions, findings, andconclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect theviews of the National Science Foundation.

Appendix A. Analysis of folding using an activetorsional element

For the analysis of the torsional self-folding concept(figure 5(b)), the active torsional element is isolated from theother bodies (hinge and connected faces are not modeled). Aboundary value problem of the individual torsional element isconsidered by modeling the interactions of the element withother bodies in terms of applied moments. The active

torsional element is represented as a rod of circular cross-section with radius r and length L. The rod is composed of anactive material and is initially unstressed. Under the uniformapplication of the inducing field, the rod twists and, if con-strained to some degree at its ends, is subjected to equal andopposite end moments Mt. Such constraints are associatedwith the resistance of the sheet to the fold-like operation (e.g.,caused by friction at the hinge, inertial forces, interactions ofthe sheet with other bodies, the effects of gravity, etc). Theinitial and final configurations of the torsional element areshown in figure A1.

The centroidal axis of the bar coincides with the x1-axis.By the rotational symmetry of the problem, it is reasonable toassume that the motion of each cross-sectional region withthickness xd 1 is restricted to a rigid body rotation about the x1

-axis [248]. From equilibrium, in the absence of body forces,it can be found that the local twist angle θ varies linearly withx1; therefore, θ θ= = ′xd d constant1 . Under such displace-ment assumption, the only non-zero strains, which are due toelastic strains (denoted by superscript ‘el’) and mechanism-agnostic actuation strains (denoted by superscript ‘act’) aregiven by:

ε ε θ ε ε θ+ = − ′ + = ′x x12

,12

. (A.1)el act el act12 12 3 13 13 2

The corresponding non-zero stress components are:

⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠σ θ ε σ θ ε= − ′ − = ′ −G x G x

12

,12

, (A.2)act act12 3 12 13 2 13

where G is the shear modulus of the active material bar,assumed to be constant. To obtain bounds for an actuationstress and strain assessment, two different cases are explored.To relate actuation strain to the folding performance of thetorsional element, it is assumed that the magnitude of theactuation shear strain far exceeds that of the elastic shearstrain due to the choice of active material and/or theassumption of relatively small resistive moments at the foldσ σ= =( )012 13 . Considering either equation (A.1) or (A.2)under this assumption:

ε θ ε θ= − ′ = ′x x12

,12

. (A.3)act act12 3 13 2

The magnitude of the maximum principal actuation strainεact at any point in the bar under the strain state given inequation (A.3) is:

ε θ= ′ +( )x x12

. (A.4)act22

32 1 2

The maximum value of εact will occur when

+ =( )x x r22

32 . Therefore, the limit of the change in angle

per unit length the bar axis θ′ multiplied by r is given by

θ ε′ =r 2 . (A.5)act

The total twist angle φ (that is expected to translate into afolding angle) is obtained by integrating θ′ along the entire

20


length of the bar

∫ ∫φ θ θ θ= ′ = ′ = ′l l Ld d . (A.6)L L

0 0

By substituting equation (A.5) into (A.6), the followingdimension-normalized relation between εact and the total twistangle φ is obtained

φ ε=rL

2 . (A.7)act

To obtain a bound for the assessment of actuation stress,the case in which the sheet is fully constrained from folding isconsidered. In this case θ θ= ′ =x( ) 01 . Usingequations (A.1) and (A.2) the following is obtained:

σ ε ε σ ε ε= − = = − =( ) ( )G G G G, . (A.8)act el act el12 12 12 13 13 13

The preceeding equation corresponds to the elasticsolution of the problem. The relations between the stressesand the applied moments are given as follows [248]:

σ σ= − =M x

I

M x

I, , (A.9)t

p

t

p12

313

2

where Ip is the polar second moment of the circular cross-

section ( π=I r 2p4 ). The maximum normal or shear stress

due to actuation σ act occurs at the boundary of the bar and arerelated to the reaction moment Mt by [248]:

πσ=M

r 2. (A.10)t

act

3

Equations (A.7) and (A.10) provide an assessment of thefolding characteristics of concept (b) (twist angle, obtainedmoment) as function of actuation stress and strain of theactive material.

Appendix B. Analysis of folding using an active/passive bilayer

For the analysis of the multi-layer self-folding concept(figure 5(d)), bending of an active-passive bilayer sheet isconsidered. The base analytical solution for this case isobtained from Timoshenko [249]. In 1925, Timoshenkoanalyzed bending a bilayer sheet due to unequal thermalexpansion. The bilayer sheet is depicted in figure B1. Thesheet of total thickness h has a layer of material I withthickness a1 and a layer of material II with thickness a2 so that

+ =a a h1 2 . The width of sheet is denoted as w.Material I has an elastic modulus and thermal expansion

coefficient of E1 and α1, respectively. Similarly, material II hasan elastic modulus and thermal expansion coefficient of E2

and α2, respectively. When the bilayer film is uniformly heatedfrom temperature T0 to temperature T, it bends to a config-uration with radius of curvature R due to unequal thermalexpansion of the constituent layers (i.e., when α α≠1 2). Therelationship between the obtained radius of curvature R andthe geometric parameters and material properties of the filmwas found by Timoshenko [249] to be

α α=

− − ++ + + +( )( )

( ) ( )( ) ( )R

T T m

h m mn m mn

1 6 ( ) 1

3 1 1 1 ( ). (B.1)2 1 0

2

2 2

where =m a a1 2 and =n E E1 2. Here we consider material Ias the active material while material II is the passive material.In equation (B.1) the eigenstrain is assumed to be due tothermal expansion of the constituent layers; however, the endstate of such a bilayer beam would be the same regardless ofthe origin of the eigenstrain (neglecting out-of-plane effects).Thus we use the following simple substitution

α ε α≈ ≡ − −( )T T0, , (B.2)act2 1 0

where εact is the actuation strain. The first expression inequation (B.2) emerges from the assumption that the passivematerial provides negligible strain under the applied field. Thesecond expression implies that the active layer is contractingwith a strain of εact. When substituting the expressions inequation (B.2) into equation (B.1), the following is obtained7

ε=

++ + + +( )( )

( )( ) ( )R

m

h m mn m mn

1 6 1

3 1 1 1 ( ). (B.3)

act 2

2 2

The radius of curvature normalized with the film thick-ness is then given as follows

ε=

+ + + ++

( )( ) ( )( )

Rh

m mn m mn

m

3 1 1 1 ( )

6 1. (B.4)

act

2 2

2

To simplify the preceding equation, the constant C1,defined in equation (3), is substituted into equation (B.4) and

Figure A1. Local twisting of an active circular bar under theapplication of a locally uniform field.

7 Assuming expansion of the active layer will change the sign of R (meaningthat the final configuration is convex down) but the magnitude of R will beunaffected.

21


the following is obtained

ε=RC h

1. (B.5)act

1

Equation (B.5) provides a non-dimensional analyticalexpression for the assessment of the folding behavior of abilayer film with a layer of active material and a layer ofpassive material. Clearly, as actuation strain increases, theradius of curvature of the bent sheet decreases.

For an assessment of actuation stress σ act, the maximumstress experienced (σmax) by the active layer is found. Anexpression for the maximum stress was also found byTimoshenko [249] and it consists of two parts: the stress dueto axial force and the stress due to bending. It is given by

σ σ= = +Pwa

a E

R2, (B.6)act max

1

1 1

where P is the total axial force in the active layer. It can beshown [249] that the total bending moment Mb relates to Pand R as follows:

= = + =+( )

PM

hR

E I E I

M

w E a E a

M

2,

12, (B.7)b

b b

1 1 2 2 1 13

2 23

where I1 and I2 are the moments of inertia of the cross-sectionsof layer I and II, respectively. The term +E I E I1 1 2 2 representsthe effective bending stiffness of the bilayer. The moments ofinertia were substituted in equation (B.7) using the result

=I wa 12i i3 where =i 1, 2. Substituting the results from

equation (B.7) into equation (B.6) the following is obtained

⎛⎝⎜

⎞⎠⎟σ = +

+M

wha

ha E

E a E a2

6. (B.8)act b

1

12

1

1 13

2 23

To simplify the form of equation (B.8), the constant C2,defined in equation (4), is substituted into equation (B.8) andthe following relation between actuation stress and bendingmoment is obtained:

σ = M C

wha. (B.9)act b 2

1

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Origami-Inspired Active Structures- A Synthesis and Review

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