MATERIALS SCIENCE

Bioinspired spring origamiJakob A. Faber,1 Andres F. Arrieta,2* André R. Studart1*

Origami enables folding of objects into a variety of shapes in arts, engineering, and biologicalsystems. In contrast to well-known paper-folded objects, the wing of the earwig has anexquisite natural folding system that cannot be sufficiently described by current origamimodels. Such an unusual biological system displays incompatible folding patterns, remainsopen by a bistable locking mechanism during flight, and self-folds rapidly without muscularactuation.We show that these notable functionalities arise from the protein-rich jointsof the earwig wing, which work as extensional and rotational springs between facets.Inspired by this biological wing, we establish a spring origami model that broadens thefolding design space of traditional origami and allows for the fabrication of precisely tunable,four-dimensional–printed objects with programmable bioinspired morphing functionalities.

Programmable matter that can self-shape,morph, and actuate through “instructions”embedded into its own material architec-ture iswidespread innature andhasopenedexciting possibilities in robotics, biomedi-

cal technologies, arts, and design (1–3). Origamiis particularly attractive because it allows fold-ing simple, two-dimensional (2D) sheets intocomplex, 3D geometries. This simplicity and ef-fectiveness of folding has inspired mathema-ticians, engineers, and materials scientists toexploit origami (4, 5) as programmable meta-materials (6, 7), reconfigurable structures (8–10),adaptive architectures (11, 12), and soft roboticparts (13, 14).Despite its exciting prospects, major draw-

backs remain when using classic origami princi-ples. Classic origami, or rigid origami, uses two

building blocks: rigid, planar facets of zero thick-ness and distinct, straight-line creases. These ri-gidity assumptions lead, in theory, to a limiteddesign space of possible folding patterns (15).Besides these pattern constraints, a second as-pect that limits functionality is that rigid origamimechanisms have only one degree of freedom,which is not associated with any stiffness duringfolding or unfolding. This lack of stiffness ren-ders the resulting mechanisms purely kinematic(only describingmotions) and thus hindersmanyengineering applications, particularly those in-volving load-bearing functions.Attempts to extend the purely kinematic rigid

origami principles havemainly been pursued byintroducing bending energy in the creases (5, 16).The addition of bending energy in the form ofstiffness to the crease allows for forces and mo-

ments to be linked to the foldingprocess; however,the design space for folding patterns remainsunchanged. To account for additional foldingpatterns and programmability observed in prac-tice on thin sheets, facet bending or twisting hasbeen identified as an extra degree of freedom(17, 18).Despite these efforts, synthetic origami struc-

tures developed so far are still far from reachingthe range of functionalities and design freedomobserved in nature. An impressive natural ex-ample of folding pattern and functionality isthe wings of Dermaptera, an order of insectscommonly known as earwigs (Fig. 1A). The fold-ing ratio (closed/open area) of these highly spe-cializedwings is among the highest in the animalkingdom, with reported values of 1:10 (19) to 1:18(20). This exceptionally high ratio concurrentlyallows for a large wing area during flight and ashort, folded package to navigate the earwig’stight underground habitat (21). There are threevery distinct features separating thewing-foldingmechanism ofDermaptera from the assumptionsof rigid origami. First, the earwig wing employsa pattern (Fig. 1B) that is incompatible with rig-id origami theory because of angle mismatchesand curved creases (22). Second, the wing hasevolved to fully and rapidly self-fold from theopen toward the closed state. The self-folding isachieved without the use of muscles. Instead, it

RESEARCH

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1Complex Materials, Department of Materials, ETH Zürich,8093 Zürich, Switzerland. 2Programmable StructuresLab, School of Mechanical Engineering, Purdue University,West Lafayette, IN 47907, USA.*Corresponding author. Email: andre.studart@mat.ethz.ch (A.R.S.);aarrieta@purdue.edu (A.F.A.)

Fig. 1. Earwig wing as a natural example of multifunctional program-mable folding. (A) The earwig Forficula auricularia with unfolded wing.[Image reprinted with permission from G.Wizen] (B) The folding patternis incompatible with rigid origami assumptions. [Image adapted fromHaas et al. (19)] (C) Alternating, asymmetric resilin distribution throughoutthe wing drives the self-folding process. The schematic under the

microscopic image indicates the resilin-rich regions (in blue) along thecross-section of the wing. [Microscope image reprinted from (19) withpermission by Elsevier] (D) Bistable central mechanism and anglemismatch. [Image adapted from Haas et al. (19)] (E) The inaccessibleregime is responsible for the bistable function, but not accessible by origamikinematics. [Graph adapted from Haas (23) for a missing angle of 60°]

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is preprogrammed in asymmetrically arranged,prestrained resilin in the joints (19) (Fig. 1C). Final-ly, a bistable snap-through mechanism (Fig. 1D)is incorporated in the structure to keep the un-folded wing in its open state. The open, lockedstate is stable enough to resist the aerodynamicloads during flight. The rigid origamimodel ofthis mechanism describes angular motions nearthe stable states (Fig. 1E), but exhibits a “forbid-den range” (23) or inaccessible regime. Notably,this regime near the snap-through angle deter-mines all functionality of the earwig wing. Thus,although themorphing of the earwig wing arisessolely from the folding pattern and crease design,current origami models are not sufficient to de-scribe its exceptional functionality even if bend-ing facets are considered.Our aim in this study is to identify design

principles of the earwig wing and implementthem in synthetic structures with pronouncedstiffness and fast-morphing programmability.Therefore, we first investigate the earwig wing’sself-folding behavior by finite element analysis(FEA) and derive the underlying principles em-bedded in the design of its natural joints. Wethen simplify the identified joint design usinganalogous mechanical springs. Using the newlyintroduced spring elements, we build a modelof the wing’s core mechanism to quantify its bi-stability and self-folding functions. Finally, thesefunctions are transferred into 4D-printed, syn-thetic folding systems with unmatched and tun-able functionality inspired by the natural example.The self-folding and locking capability of the

earwig wing relies on the presence of resilin-

based joints. Resilin is an elastic biopolymer com-monly linked to energy storage innatural systems,which has been found in symmetrical as well asasymmetrical arrangements in the earwigwing’sjoints (19). This through-thickness distribution ofresilin in the joints determines the spring type: Asymmetric distribution corresponds to an exten-sional spring,whereas an asymmetric distributioncorresponds to a rotational spring. Combinationsof both forms are possible. Whereas rotationalsprings have been discussed elsewhere (5, 16), wenow describe the role and resulting design capa-bilities of extensional springs for extending thepattern design space and generating locking,multistable folding systems.To exploit the effect of the proposed extension-

al springs on the wing’s self-folding behavior, weconducted folding simulations assuming eitherstrict origami or spring-modified origami condi-tions (Fig. 2). Our results show that the geomet-rically incompatible folding pattern of the wingprevents complete folding if the strict conditionsof traditional origami are considered (Fig. 2C).By contrast, full closure of the wing followingthe same pattern as the natural example is ob-served if the joints are treated as extensionaland rotational springs.More details of these FEAsimulations are provided in the supplementarymaterials. To assess the feasibility of this simula-tion, we produced a replica of the wing pattern bymultimaterial 4D printing of stiff polymer facets(acrylonitrile butadiene styrene, ABS) and rubber-like hinges (thermoplastic polyurethane, TPU). Themanually folded package confirms the stackingorder and shape of the simulation results (Fig. 2D).

Despite the complexity of the simulated biolog-ical example and therefore the rather qualitativenature of these results, such proof-of-concept com-putational analysis clearly hints at the crucial roleof membrane extensibility inmaking previouslyincompatible fold designs accessible.The extensibility of joints not only facilitates

the folding of complex patterns, but also pavesthe way to understand and synthetically designbioinspired features that were previously out ofreach. The most notable feature of the earwigwing is its central mid-wing mechanism. It isbistable with two possible states: During folding,it is a classic Miura-Ori pattern with three con-vex and one concave fold. After fully openingand passing through an unstable flat state, it be-comes a concave pyramid (Fig. 3A). In this shape,themid-wingmechanismmaintains thewing inits open flying configuration. Although it consistsof only four facets, this crucial part of the wingsimultaneously exhibits the mentioned featuresof interest; namely, self-folding, self-locking (bi-stability), and imperfect-pattern tolerance. Thus,a fundamental investigation of the interplaybetween membrane (extensional) and bending(rotational) elastic energy stored in folds of themid-wing region provides valuable insights intothe design principles of this morphing structure.To analytically model this four-facet unit cell, weuse simple rotational and extensional springs(Fig. 3). The four rigid facets are interconnectedby rotational and extensional springs at eachjoint (Fig. 3A). The spring constants of possiblematerials and geometries can easily be retrievedfrom the formulas given in the supplementary

Faber et al., Science 359, 1386–1391 (2018) 23 March 2018 2 of 6

Fig. 2. Assumptions and FEA simulations describing the self-foldingbehavior of the earwig wing. (A) Strict (rigid) origami model.(B) Proposed interpretation of resilin occurrence as extensional androtational spring elements. (C) Origami-inspired simulation approach

leads to incomplete self-folding, whereas (D) the incorporation of springelements in the origami structure results in complete self-folding ofthe simulated wing (movie S1). A multimaterial printed wing model confirmsthe shape and fold order of the simulated folded package.

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materials. The energy stored in one rotationalspring equals 1

2 cBðDfÞ2 , where cB is the rota-tional spring stiffness and Df is the differencebetween the interfacet angle f and the angleunder which no bending stress occurs, f0;B

(Fig. 3B). Similarly, the extensional spring ener-gy is 1

2 cMðDxÞ2 , where cM is the extensionalspring stiffness andDx is the spring extension xcompared to its stress-free state x0;M . Simpletrigonometry links the spring extension x and

the folding parameter f (see Eq. S3), allowing fora convenient description of all energies in termsof f (Fig. 3C). The extension-free angle obtained,f0;M , is determined by the missing angle be-tween the facets (see Eq. S10).

Faber et al., Science 359, 1386–1391 (2018) 23 March 2018 3 of 6

Fig. 3. Generalized model of bistable four-fold spring origamistructures. (A) Geometry and spring arrangement. Analytical(B) rotational and (C) extensional spring energies and correspondingFEA evaluations. (D) Possible configurations and associated energy land-

scapes of selected four-faced spring origami structures. (E) Influence of theextensional spring stiffness on the energy landscape for b ¼ 5° and definition ofenergy barriers. (F) Design map illustrating the effect of the missing angle andstiffness ratio on the magnitude of the energy barriers EBarr;1 and EBarr;2.

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By computing the rotational and extensionalcontributions of the total energy stored in thejoints of themid-wing region, it is now possibleto establish the energy landscape of the four-facet structure in different configurations (see

assumptions in supplementary materials). Inthe inaccessible regime, themissing anglebmustbe compensated for by membrane extension,giving rise to an energy peak (Fig. 3, C andD). Therotational and extensional energy contributions

can then be combined in endless configurations,depending on the folding patterns, each joint’sstiffness values, and the programming param-eters f0;B and f0;M . Three exemplary config-urations are shown in Fig. 3D. The initially

Faber et al., Science 359, 1386–1391 (2018) 23 March 2018 4 of 6

Fig. 4. Four-dimensional printing and experimental investigation ofspring origami mechanisms. (A) Geometry and material properties.(B) FDM printing in the folded, upright state. (C) Nondevelopable,4D-printed designs using the earwig wing’s central mechanism contours(top) and simplified rectangular facet contours with different missing

angles. (D) Cardan test rig. (E) Measured forces and related energy plots formechanismswith increasingmissing angle b. Model prediction for b = 5° is shownby the dashed green curve. (F) Effect of the missing angle on the energy barriermagnitudes andmodel predictions. (G andH) Fast self-folding after slow internalstimulus (movie S1). (I) A 4D-printed, nondevelopable 3 × 3–panel array.

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mentioned case of a stress-free “pyramid” as thebase state leads to an asymmetrical energy land-scape (Fig. 3D, left). A symmetrical case can bereached by planar assembly of facets, includinga missing angle and using prestretched joint re-gions (Fig. 3D, center). Assembling three convex(f0;B ¼ �p) and one concave fold (f0;B ¼ þp), asin the case of the earwig wing, yields the ex-emplary Miura-Ori configuration, which servesas our base system. These preprogrammed angleslead to a fully folded pattern as the stress-freestate (Fig. 3D, right).For a technical application of the described

unit cell, it is necessary to understand andmodelthe relation between design parameters and re-sulting morphing and mechanical properties.Applying the basic building blocks from Fig. 3,A to D, we computed design maps of two kinds:First, Fig. 3E shows the bending,membrane, andtotal stored energies over f for the self-foldingMiura-Ori configuration, displaying a constantmissing angleb. A lowmembrane stiffness ratio(cM/cB) leads to monostable systems resultingfrom the lack of an energy barrier that could giverise to two distinct mechanical states. These sys-tems are bending-dominated and will return totheir preprogrammed shape immediately. Forstiffness ratios of cMcB > 0:5� 104, a barrier de-velops in the energy landscape, leading to thetwo minima that characterize a bistable system.This threshold frommono- to bistability changeswith the missing angle, which increases the en-ergy barrier and therefore shifts the overall be-havior toward more pronounced bistability. Thecombined effect of both stiffness valuescM andcB,as well as the missing angle b, on the magnitudeof the energy barriers EBarr;1 and EBarr;2 is shownin the designmap depicted in Fig. 3F. EBarr scalesproportionally with cM and cB as long as theirratio is constant (see Eqs. S1 and S2). For eachcMcB

ratio, there is a critical missing angle bc thatmakes the folding pattern bistable, and vice versa(Fig. 3F). The parameter space revealed by thesemaps allows us to determine appropriate varia-bles to design folding behavior, stable states, andtuned energy barriers in spring origami systems.Such features control the direction and order offolding, the obtainable geometries, and the self-

locking strength. Furthermore, the derivatives ofthe energy plots allow one to directly obtain fold-ing moments and forces, which are relevant forthe design of load-bearing functions in both stablestates (see Eqs. S1 and S5). As is the case in theearwig wing, this unit cell can serve as an em-bedded control unit for much larger folding pat-terns and complex geometries. In the followingsection, we discuss how this Miura-Ori configu-ration can be exploited in exemplary syntheticsystems to achieve fast folding and lockingmech-anisms with minimal actuation inspired by theearwig wing.We transferred the biological design principles

extracted from the earwig wing into a function-al synthetic folding system that can be directlymanufactured by 4D printing using a conven-tional additive manufacturing process (Fig. 4).The base configuration was a fourfold structureof 100mmby 100mmunfolded edge lengthwith1.2-mm-thick facets made of a stiff component[polylactic acid (PLA) or ABS] and joints madefrom an elastomeric component (TPU) with athickness of 1.0mm(Fig. 4A).Usingmultimaterialfused deposition modeling (FDM, Fig. 4B), wecan overcome several limitations of conventionalorigami-type folding approaches. First, printingallows us to preprogram the folding pattern in anelegant way: Instead of using selective shrinkingor swelling of a bilayer architecture to inducebending (14, 24, 25), we printed the samples inthe fully folded configuration. This allows us toset f0;B ¼ Tp in the desired combination. Ifbistability is simultaneously required, simplefolding techniques also fail. Any missing angleb ≠ 0 renders the fold pattern nondevelopable,which means not foldable from a flat sheet.Our folded-printing approach allows us to pro-gram f0;M by directly implementing the missingangle b in the geometry. Figure 4C demonstratesthe relationship between the printed and theunfolded geometry with the missing angle branging from 0° to 40°.To illustrate the distinctive folding and me-

chanical functionalities of the printed spring ori-gami, we measured the force required to keepthe structure at a constant interfacet angle nearand within the inaccessible region (Fig. 1E). The

energy curves obtained through integration ofthe measured forces feature the two-well land-scapes characteristic of bistable systems (Fig. 4E,blue curves). The experimental curves and theearlier model predictions match quantitativelyvery well the dependence of the stored energy onthe folding angle (Fig. 4E, dashed line). Themag-nitude of the energy barrier observed in thesecurves rises in a progressive manner with in-creasing missing angles b and shows the pre-dicted bistability threshold behavior (Fig. 4F).Small missing angles lead to the monostable sys-tems predicted by our model (compare Fig. 3E),whereas bistability arises above the expectedthreshold.Our ability to tune the energy barrier between

bistable states using simple geometrical and ma-terial properties (Fig. 3F) enables the design andfabrication of spring origami structures that canundergo fast morphing, triggered by an environ-mental stimulus. As opposed to other commonbiological and synthetic morphing systems thatreact to external stimuli (26–28), the elastic en-ergy stored in spring origami results in very fastintrinsic folding of the structure. Tracking themo-tion of a Miura-Ori structure during self-foldingallowed us to quantify the dynamics of this fastmorphing process (Fig. 4, G and H). The snap-through between the two stable states occurs in80 ms, much faster than conventional diffusion-drivenmechanisms. This is equivalent to the fastactuation mechanism used by the Venus flytrapor the underwater suction trapUtricularia, whichalso convert a slow stimulus to a rapidmovementby bistability concepts (29, 30).We also demonstrate the scalability of our

4D-printing approach beyond four-facet systemsby printing larger arrays (Fig. 4I), or by using asingle spring origami element to control morecomplex structures that fold via conventionalmechanisms. To illustrate this possibility, wefabricated a spring origami gripper that actuateswith a low-energy input (Fig. 5 and movie S2).The gripper consists of a passive fold pattern(white) responsible for the kinematic grippingmovement and a central, 4D-printed spring ori-gami element (gray) that defines the energy land-scape of the folding system. By designing the

Faber et al., Science 359, 1386–1391 (2018) 23 March 2018 5 of 6

Fig. 5. Spring origami gripper (movie S2). (A) Fold pattern. Gray facets andcomputer-assisted design model: preprogrammed control cell. White facets:

passive part of mechanism. (B) Stable state 1: open. (C) Stable state 2: closed.The closed state exerts force on the gripped object without constant actuation.

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materials and geometrical parameters (Fig. 5A)of this cell—namely, missing angle, stiffness val-ues, and strain-free fold angles—the inherent en-ergy landscape of the entire gripper mechanismcan be programmed. Our spring origami grippereventually displays the bistable and fast self-folding functionalities of the earwig wing. Theprogrammed stable states (Fig. 5, B and C) andenergy landscape allows the spring origami struc-ture to exert a force on the gripped objectwithoutthe need for constant external actuation. This con-trasts with purely passive rigid origami mecha-nisms, which would yield to any applied loadowing to the absence of an energy landscape orstable states. The spring origami gripper canthus remain in both the open (Fig. 5B) or closedpositions (Fig. 5C) and lift objects of its own bodyweight.Origami structures featuring extensional and

rotational joints inspired by the earwig wingshow unusual self-locking, fast-morphing, andgeometry-tolerant folding patterns that are notallowed in conventional origami theory. The pos-sibility of 4D printing 3D objects whose morph-ing and mechanical behavior are programmedwith the material architecture brings us closerto the design strategies underlying the exquisitedynamics of biological self-shaping structures.The ample design space provided by the pro-posed spring origami systems can potentially beused to fabricate biomedical deviceswith patient-specific morphing features, collapsible portabledisplays, soft robots, or deployable spacecraftmodules.

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ACKNOWLEDGMENTS

We especially thank K. Masania and S. Gantenbein for help with the3D-printing experiments, M. Binelli for the gripper suggestion,E. Jeoffroy for help with photos and movies, and all colleagues forfruitful discussions. Funding: We are grateful for financial supportfrom ETH Zürich, Purdue University, European Office of AerospaceResearch and Development (grant FA9550-16-1-0007), and theAir Force Office of Scientific Research (grant FA9550-17-1-0074).Authors contributions: All authors designed the research.The analytical, numerical, and experimental work was designed,performed, and evaluated by J.A.F. The manuscript was preparedby J.A.F., A.R.S., and A.F.A. All authors discussed and criticallyassessed the results and their implications, and revised themanuscript at all stages. Competing interests: The authorsdeclare no competing financial interests. Data and materialsavailability: All data are available in the manuscript orsupplementary materials.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/359/6382/1386/suppl/DC1Materials and MethodsFigs. S1 and S2Table S1Movies S1 to S3Python Code of Spring Origami ModelRaw Data of Mechanical TestingRaw Data of Image Analysis3D-Printing Files

25 August 2017; accepted 30 January 201810.1126/science.aap7753

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Bioinspired spring origamiJakob A. Faber, Andres F. Arrieta and André R. Studart

DOI: 10.1126/science.aap7753 (6382), 1386-1391.359Science 

, this issue p. 1386Sciencepassive self-folding behavior.deformations and variable stiffness. Prestretching generated energetically bistable origami patterns that exhibitedusing origami and can alter its shape for flight. The authors replicated this ability by using a membrane that allows for

studied the wing of an earwig, which can fold in ways not possibleet al.cannot be created using standard folds. Faber Origami involves folding two-dimensional sheets into complex three-dimensional objects. However, some shapes

More than just simple folding

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