Folding Angle Regulation by Curved Crease Design

for Self-Assembling Origami Propellers

Shuhei MiyashitaComputer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology32Vassar st. Cambridge, MA, 02139, USAshuheim@csail.mit.edu

Isabella DiDioComputer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology32Vassar st. Cambridge, MA, 02139, USAididio@mit.edu

Ishwarya AnanthabhotlaComputer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology32Vassar st. Cambridge, MA, 02139, USAishwarya@mit.edu

Byoungkwon AnAutodesk ResearchPier 9, San Francisco, CA, 94111, USAkwon.an@autodesk.com

Cynthia SungComputer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology32Vassar st. Cambridge, MA, 02139, USAcrsung@mit.edu

Slava ArabagiBoston Children’s HospitalHarvard University330 Longwood Avenue, Boston, MA, 02115, USAbadeaslava@gmail.com

Daniela RusComputer Science and Artificial Intelligence LaboratoryMassachusetts Institute of Technology32Vassar st. Cambridge, MA, 02139, USArus@csail.mit.edu

1

Abstract

This paper describes a method for manufacturing complex three dimensional curvedstructures by self-folding layered materials. Our main focus is to first show that thematerial can cope with curved crease self-folding and then to utilize the curvature topredict the folding angles. The self-folding process employs uniform heat to induceself-folding of the material and shows the successful generation of several types ofpropellers as a proof of concept. We further show the resulting device is functionalby demonstrating its levitation in the presence of a magnetic field applied remotely.

2

1 Introduction

We are interested in developing rapid manufac-turing of complex structures. To this end, wecombine parameterized designs on origami pat-tern with planar fabrication into a new processfor creating complex three dimensional shapeswith curved surfaces by self-folding. In this pa-per we apply this fabrication method to man-ufacturing several types of propellers. We showexperimentally that the propellers are functionalunits. While the results in this paper focus onpropellers, they can be generalized to other com-plex shapes.

A rapidly increasing demand for manufactur-ing complex, iterative, or fine structures hasdrawn attention toward a fabrication techniquethat combines planar fabrication and origami-inspired transformations [1–3]. Recent innova-tions in origami technique [4, 5] have demon-strated that curved crease folding enables thegeneration of three dimensional geometries un-achievable in traditional prismatic paper fold-ing [4], which uses straight creases alone.

In the curved folding process, it is wellknown that folding also necessitates bending thesheet [4]. Various investigations have been con-ducted focusing on curved crease designs [6], rul-ings (see Section 3 for an explanation of rul-ings) [7, 8], sheet bending [9], formulation of thefolding principle [10–12], or applications pertain-ing to car body design [13].

Self-folding is a recent technique aimed atrapid fabrication of objects by the folding ofmany small and complex creases [14–19]. Toour knowledge, few approaches have attemptedself-folding curved creases [20]. The engineeringchallenges here are (1) to precisely predict thefolding angle of a curved crease, (2) to achieveself-folding of curved creases into a functionalstructure, and (3) to actuate the device after ithas been self-folded, as a way of demonstratingfunctionality.

Under these presented challenges and goals,this paper contributes the following:

1. Simulations for estimating curved creasefolding angles

2. An application of this algorithm toward de-signing propellers

3. An algorithm for computing self-foldingcrease patterns for objects with curved sur-faces, such as propellers

4. A series of self-folding experiments for dif-ferent propellers

5. Levitation experiments of the self-foldedpropellers by remotely applying a rotationalmagnetic field.

2 Outline

The methodology of curved crease self-foldingdescribed in this paper consists of the follow-ing steps. We model and derive a method forpredicting a folding angle of a curved crease(Section 3). We outline the general guidelinesfor making an origami propeller (Section 4.1).We analyze the geometric relationship betweenthe crease and folded propeller structure (Sec-tion 4.2). We develop an algorithm for au-tomatically designing various types of origamipropellers (Section 4.3). We build an electro-magnetic coil system with supporting electron-ics for remote actuation of the propeller (Ap-pendix). We show the experimental resultsof self-folding curved creases (Section 5.1) anddemonstrate self-folding of propellers with dif-ferent crease curvatures (Section 5.2). We inves-tigate the functionality of the folded propeller(Section 5.2). We conclude the study (Section 6).

3 Curved Crease Folding

This section investigates a basic theory of curvedcrease folding. We numerically analyze the re-lationship between shapes of various curvaturesdrawn on a two-dimensional crease pattern andthe resulting three-dimensional folded shapesand curvature. Here, we employ the superellipseas an example of general curvature. A character-istic of the Superellipse is that it can representa sector of major shapes such as squares, circles,or triangles, continuously being transformed bychanging one parameter.

3

3.1 The Model

The curves we investigate are superellipses de-scribed on an x-y plane centered at x = π

2 withradius κ = π

2 :∣∣∣∣x− κ

κ

∣∣∣∣n +∣∣∣yκ

∣∣∣n = 1. (1)

The superellipse has a unique characteristicsuch that, by changing a single parameter, n,it changes curvature continuously from a square(n → ∞) to a circle (n = 2), to a triangle(n = 1), and to a “star” (n < 1), while main-taining the width and the height (see Fig. 1 (a)).This is an ideal characteristic for our study sincevarious curvatures can be compared in a singleparametric space.

The positive half of the superellipse equationis

y = κ ·(1−

∣∣∣∣x− κ

κ

∣∣∣∣n) 1n

. (2)

We depict the curves of n = 1.5, n = 2, andn = 4 in Fig. 1 (a), and show the illustration ofthe folded plane in Fig. 1 (b) with an example inthe side picture.

The folding angle α is the angle between thetwo sides of the curve after it is folded, and whosechange with respect to x can be described as [11]

dα

dx=

tan(α(x))

2 · dxds

K2D(x)

·(cotβL(x)− cotβR(x)), (3)

where s is the arc length of the curve, which is afunction of x:

s(x) =

∫ x

0

√1 +

(dy

dx

)2

dx, (4)

K2D is the curvature of the flat curve defined as

K2D =

∣∣∣ d2ydx2

∣∣∣(1 + ( dydx)

2) 3

2

, (5)

and β· (· ∈ [R,L]) is defined as the angle be-tween the tangent line to the curve and the rul-ings of the surface that are made from the fold(Fig. 1 (b)).

Taking the derivative and reciprocal in Eq. (4)yields

dx

ds=

1dsdx

=1√

1 +(dydx

)2 . (6)

Rulings (or ruling lines) are straight lines thatdefine a space curve by sweeping along the sur-face (shown as the red dot lines in Fig. 1 (b)and Fig. 8 (b)). Hypothesizing that both sur-faces formed by the crease form cylindrical cur-vatures when folded into three dimension, β· arecalculated using vertical rulings (Fig. 1 (a)).

With vertical rulings, the angle βR can becomputed as

βR = tan−1

(1dydx

), (7)

and βL is simply

βL = π − tan−1

(1dydx

). (8)

The derivative of the superellipse equation is

dy

dx= −sgn

[x− κ

y

]·∣∣∣∣x− κ

y

∣∣∣∣n−1

. (9)

From Eq. (9) we obtain the second derivativeof y as

d2y

dx2= (1− n)

∣∣∣∣x− κ

y

∣∣∣∣n−2

·(−x+ κ

y2· dydx

+1

y

)−∣∣∣∣x− κ

y

∣∣∣∣n−1

· 2δ(x− κ

y

),

(10)

where δ is the Dirac delta function.

3.2 Simulation Results

Fig. 2 shows the simulation results of Eq. (3) fordifferent surface conditions, which was run for nvalues of 1.5, 2, and 4. As the stiffness of thesheet material affects the bendable curvature ofthe surface and thus affects α, to run the simula-tion, we chose α at the end of the creases (α| dy

dx=1

;

termed αend) by supposing that the entire crease

4

consists of two straight lines divided from themiddle of the curve to the edges (an exampleshown as the green line in Fig. 1 (a)). This way,when the surface is fully folded flat, αend = Γ−γ(Γ and γ were obtained by taking the inverse tan-gent of the ratio of the horizontal and verticalradii of the curve), and when the surface foldsup to π/2, αend = (Γ− γ + π)/2, premising thatthe actual value falls between them. This condi-tion was selected by referring to our pre-testedmanual folding experiments and the self-foldingexperiments appearing in the following sections.

The simulated result showed that the creasefor n = 4 folded the most acutely while thatfor n = 1.5 folded the least in the middle ofthe creases (called αmiddle) This trend is moreclearly displayed in Fig. 3 with the change offolding angle αmiddle with respect to n. Thecrease folded more as n increased, irrespective ofαend, although the tendency was more apparentfor smaller values of αend. This result suggeststhat when we plan to induce a difference in fold-ing angles in real fabrication, it would be bet-ter to aim for smaller folding angles in order toachieve clear differentiation in their folding an-gles over different curved creases. We also verifythis trend in later experiments (see Section 5.1).

Another noticeable trend is that α increasedtoward the center of the crease when αend > π

2 ,while it decreased when αend < π

2 This trendwas observed with other conditions of αend inthe simulation, and is confirmed with manuallyfolded models.

See the folding angles from experiments in Sec-tion 5.1.

4 Origami Propeller

This section builds further upon the establishedtheoretical implications of curved crease foldingto present the actual designs used in fabricating3- and 4-blade propellers based on the origamidesigns introduced by Mitani [21] and the ge-ometric rationale for having designed them inthis manner. We first show a few largely well-known design guidelines on propeller design de-rived from the theories of aerodynamics. We

then analyze the geometric relationship betweena crease pattern and a three dimensional pro-peller so that the design parameters such aswidths and lengths of blades can be dynamicallyoptimized and reflected in the creases.

4.1 Design Guidelines

From helicopter aerodynamics, the thrust force,Fth, produced by a propeller consisting of multi-ple rotor blades is

Fth = Ct ρAω2 r2

∼ Ct ρω2 r4, (11)

where A is the disk area swept by the blades, Ct

is the aerodynamic thrust coefficient intrinsic tothe blade profile, ω is the angular velocity, andr is the length of the blade [22]. The Reynoldsnumber, Re, for the aerodynamic flow of a rotorin water is:

Re =c ω r

ν, (12)

where c is the mean chord length of the blade(approximated at 1 cm), ω is the angular veloc-ity (approximated at 30Hz · 2π), and ν is thekinematic viscosity. Approximating c as 1 cm,ω ∼ 30Hz · 2π, r ∼ 1 cm, and ν = 106m2/s,being the kinematic viscosity of water at 20◦C,we obtain a Re ∼ 19000. Given the high Renumber and the fact that each blade of the ro-tor will be passing in the aerodynamic wake ofthe blade ahead of it (in hovering mode) the flowis expected to be turbulent. Thus a larger an-gle of attack of 45◦ for the rotor blades is cho-sen over the smaller pre-stall angles of attack tomaximize the lift coefficient curve. Although in-creasing drag, this regime features a broader liftcoefficient maxima thus allowing variability inangles of attack with little effect on lift.

Furthermore, the following facts on propellerengineering have been used as general designguidelines:

1. The lift force is proportional to the square ofboth angular velocity and disk radius sweptby the blades, r, implying that scaling downof the disk area needs to be compensated bya proportional increase in rotational veloc-ity, which greatly increases required power.

5

2. Longer blades provide larger lift forces atthe expense of added weight.

3. A 45◦ angle of attack is chosen for each rotorblade per the above discussion.

4. A spanwise twist along the rotor blade thatincreases the angle of attack at the root anddecreases it at the tip compensates for theincreasing incoming stream velocity alongthe blade, thus allowing for a more uniformlift force profile [23]).

5. Increasing the number of blades mostlyserves the purpose of reducing vibratoryloads, since the power requirement increasesproportionally with blade count.

4.2 Blade Geometry

The crease pattern of the 4-blade propeller is ro-tationally symmetric and is composed of fourstraight and four curved lines, as shown inFig. 5(a). The center point is O, and a curved

line has two sections: a curved section OI anda straight section IK. The curve can be one ofmany different types of a superellipse sector anddoes not have to intersect at position I. In thisinstance, a circular sector (n = 2) is used forthe curve, and the angle of incidence is set to be IKJ 45◦.

Fig. 5 (b) shows the folded propeller. Eachof the corners A,B,C,D in Fig. 5(a) becomesan end A′, B′, C ′, D′ of the blades in Fig. 5 (b),respectively. Each of the points E,F,G,H inFig. 5(a),which represent the maximum ampli-tude of the circular sector curve, folds to thepoints E′, F ′, G′,H ′ of the blades in Fig. 5 (b),respectively.

Each point O, I, L in Fig. 5(a) is placed onpoint O′, I ′, L′ of the blades in Fig. 5 (b), respec-tively.

1. Blade Length. Let a denote the distancebetween O and I, b denote the distance be-tween I and J , and t denote the shortestdistance between F and OI, where O′ is thecenter point and OJB is 90◦. In the foldedpropeller, the point I ′ is under the pointO′ at the center. Since IKJ is 45◦, thelength of BK is a and the length of JK is

b. The length of each blade is the equal toBJ , which is a+ b (apothem).

2. Blade Width. The width of the blade is t+bbecause the distance between E′′ and centerline (I ′L′) is t, and the distance between I ′

and B′K ′ is b, where E′′ is the projectedpoint of E′ to the bottom (the plane of△I ′L′B′) and point K ′ is point K in foldedstatus.

3. We look next at the outer circle formed by�E′F ′G′H ′. Since E′F ′ of the folded pro-peller is 2t, the radius of the outer circle is2t√2.

The crease pattern of the 3-blade propeller isshown in Fig. 6(a). The design is similar toFig. 5(a) in that all of the curvature lines arecomprised of both a straight line and a curvedline and they are pointed symmetrically towardthe center O. As with the 4-blade propeller, eachcurved line is composed of two parts: a curvedsection OG and a straight section GI. The curvecan be any line of the superellipse. We select 30◦

for the angle GIH.Fig. 6 (b) shows the folded 3-blade propeller

seen from the top. Each corner A,B,C inFig. 6(a) becomes each end A′, B′, C ′ of theblades in Fig. 6(b), respectively. Each pointD,E, F in Fig. 6(a) is placed on each of thepoints D′, E′, F ′ of the blades in Fig. 6(b), re-spectively. Each point O,G, J in Fig. 6(a) isplaced on each of the points O′, G′, J ′ of theblades in Fig. 6(b), respectively.

1. Blade Length. Let O denote the centerpoint, and let OHB be 90◦. Let a denotethe distance between O and G, b denote thedistance between G and H, and t denote theshortest distance between E and OG. In thefolded propeller, point E′ and the correlat-ing points on the other curved lines meet atthe center. Since GIH is 30◦, the lengthof BI is

√3a and the length of IH is

√3b.

The length of each blade is the equal to BHand is

√3(a+ b).

2. Blade Width. If we project the propeller tothe bottom, the angle between the projected

6

lines of D′E′ and G′O′ is 120◦. The widthof the blade is t√

3+ b because the distance

between E′ and the center line is t√3, and

the distance between J and CH is b.

3. We look next at the outer circle formed by△D′E′F ′. Since E′F ′ of the folded propelleris 2t, the radius of the outer circle is 2t√

3.

4.3 Auto Generation of a Self-foldingCrease Pattern and the Sheet

The fabrication of the propeller that was usedin the study followed a protocol established pre-viously by [16], used in [19], and is briefly out-lined here. The developed self-folding sheet hasa three layer structure, wherein a heat-sensitivecontraction sheet is sandwiched between tworigid structural layers (Fig. 4). When heat isapplied to the structure, the middle contractionlayer shrinks. As a result, the entire structurefolds in the direction that opens as a gap in thesheet.

We developed a Matlab program for auto-generation of multi-blade propeller crease pat-terns that allows users to vary the parame-ters of the design for the purpose of optimiza-tion. A graphic interface displays the front andback designs and allows users to adjust the gapwidths along the folds (Fig. 7). A sample auto-generated pattern for a 4-blade propeller withcircular curvature is shown in Fig. 7 (a). User-controlled parameters include: (1) the numberof blades, (2) the type of curvature, either n =1.5, 2, 4, or sinusoidal, (3) the apothem of theregular polygon, (4) the length of the curvedcrease, (5) the amplitude of the curved crease,(6) the incident angle of the curved crease, and(7) the gap width.

Finally, holes for the guiding pole, which wasemployed to stabilize the posture of a propellerduring the levitation experiments, was added ineach blade. When folding is complete, the holeappears as the conjugation of half-folded circles.

In the fabrication process, the generatedcrease pattern was printed onto a rigid sheet ma-terial using a laser cutter machine. After the ex-cessive components of the pattern were removed,

the construction sheet (polyvinylchloride; PVC)was placed between the front and the back formsof the crease pattern. These two layers werethen laminated upon one another, sandwichingthe contraction sheet. Lastly, the entire struc-ture was subjected to uniform thermal applica-tion in a heated oven, thus self-folding from atwo-dimensional crease pattern into a three di-mensional propeller.

5 Results

This section presents the results from experi-ments on self-folding and on propeller levitation.

5.1 Single Curved Crease Self-folding

We fabricated three types of self-folding sheets ofcurved creases that we modeled in Section 3 andcompared the folds to the simulated values of α.The self-folding experiments were performed onwater in an oven (Cuisinart TOB-100) by settingthe temperature to about 110◦C. The water waspre-warmed to approximately the deformationtemperature of PVC (∼ 50◦C) before the place-ment of a self-folding sheet. The sheets werefolded on water to provide uniform heating (see[24]). In addition, folding on water helps reducefriction between the propeller and the ground.Under the set temperature of 110◦C, successfulself-folding of the curved crease was observed(Fig. 8 (a)). Once folding started, the processmaintained the speed of folding for a while be-fore it slowed down and converged to the finalangle.

The folded sheets are displayed in Fig. 8 (b).In our measurements, the folding angles at themiddle (αmiddle) are 2.0 rad (113◦) for n = 1.5crease, 1.9 rad (107◦) for n = 2 crease, and1.1 rad (61◦) for n = 4 crease, showing that themost acutely folded crease was from n = 4 andthe least folded from n = 1.5 (see the overlaidplots in Fig. 2). This trend was predicted inthe simulation and also supports our intuition;when n = 4, the crease has almost a straightline in the middle. Approximating the curvesas a straight crease should result in yielding avery small folding angle α. Conversely, when n

7

is smaller, n = 1.5, the crease can be approxi-mated as two straight lines going from the edgesintersecting in the middle. In this case, the fold-ing angle reaches to π

2 given the surface can befolded fully flat.

The surface of the n = 1.5 curve inducedthe largest bend at the middle of the surface,whereas the n = 4 curve showed at the edge(compare the indications of ruling in Fig. 8 (b)).The influence of the stiffness of the self-foldingsheet, which hindered the bending of the sur-face and was not counted in the model, canbe recognized in the experimental results; withn = 1.5 and n = 2.0, the αmiddle show smaller(more acute) values than expected by the model,whereas the influence can be seen on αend withthe n = 4 curve, in which smaller αend was ob-served (αend = 1.96 rad (112◦) for n = 1.5 crease,1.76 rad (101◦) for n = 2 crease, and 1.21 rad(70◦) for n = 4 crease). This result implies apotential to improve the model by reflecting thestiffness of the material in rulings.

5.2 Propeller Self-Folding

We show the snapshots of self-folding with then = 2 (circular) curve propeller design in Fig. 9.

Self-folding took about 3 minutes, from whendeformation first began to when the sheet suc-cessfully achieved the targeted propeller shape.The self-folding successfully proceeded with 180◦

foldings along straight lines, resulting in about a0.91 rad (52.2◦) angle of attack (Fig. 10 (a)). Incontrast, self-folded propellers of curved creasesn = 1.5 showed a low attack angle of 0.86 rad(49.4◦), while creases n = 4 showed a high at-tack angle of 1.08 rad (61.7◦) (Fig. 10 (b)).

We further attempted two types of propellerself-foldings from the circular curve based on thecrease designs in Fig. 7(b),(c). Fig. 11 showsthe self-folded 3-blade propeller (on left) and5-blade propeller (on right) for the verificationthat our parameterized design generates validself-foldable propeller patterns. In the 3-blademodel, compared to the 4-blade propeller, thenumber of blades are fewer and the length of thecreases of straight lines is longer compared tothe curved creases; thus, the design caused wider

folding angles along the curved creases. The5-blade model shows opposite attributes whenfolded. The folding angles of curved creases showsmaller values compared to the 3-blade model.Currently, attaching paired magnets to odd num-ber blade propellers is difficult for balancing pro-pose, although we propose to utilize a diametri-cally magnetized hollow cylindrical magnet forfuture work.

5.3 Performance of the Propeller

To demonstrate its functionality as a self-foldedstructure, we levitated the self-folded 4-bladepropeller inside water by remotely actuating itwith a magnetic field. For this purpose, twopairs of two cylindrical magnets (axially magne-tized, �3.27mm×H1.62mm, K&J magnet) werehorizontally attached onto the tip of two oppo-site blades pointing in the same directions (seeFig. 10 (b)). The two coupled magnets keep thepositions on a blade by pinching it from bothsides. To obtain a rotational magnetic field alongthe horizontal direction, we powered two-pairedcoil sets switching alternatively, accelerating therotational speed of the magnetic field from 20Hzto around 40Hz (see Fig. 12 in Appendix: Re-mote Magnetic Actuation). Fig. 12(b) showsthe height of levitation over time. The aver-age heights of levitation for 15 s were 11.77mmfor n = 1.5, 14.69mm for n = 2, and 13.02mmfor n = 4, which corresponds to 1.01, 1.26, and1.11 body lengths, respectively, showing that then = 2 propeller shows the best levitation level.While in motion, the propeller iteratively experi-enced levitation and step out resulting from thelevitation, changing the height repeatedly. Stepout occurred as the propeller moved afar fromthe coils by levitation and thus received a weakermagnetic field.

A unique behavior was from the propeller ofn = 1.5, where levitation proceeded slowly com-pared to the other propellers due to the shallowerangle of attack. As a result, it showed a ratherlong duration for levitation before stepping out,appearing in the smooth trajectory in the fig-ure. Despite the environment being underwater,the experiment shows the functional motion as

8

a propeller, which was generated by self-foldingfrom a sheet structure.

6 Conclusion

This study shows a method of rapid prototyp-ing of 3D curved structures based on a self-folding technique. We explore design and mod-eling approaches for regulating folding anglesby changing the curvature of creases and ap-plied this to the fabrication of propeller blades.Our results demonstrating self-folding propellerssupported by mathematical estimation, auto-mated crease generation, and self-folding mate-rials shows promise for the automation of fab-ricating complex three dimensional structuresthrough a folding process of layered intelligentsheet materials.

Acknowledgment

Part of the experimental setup on the electro-magnetic system was developed in the Nano-Robotics Laboratory at the Carnegie MellonUniversity. We thank Metin Sitti, Eric Diller,Hideyuki Tsukagoshi, Kenichi Yuasa, and Tomo-hiro Hatakeyama for their support.

We thank Anna Leonard for her assistance inmeasuring the coil characterization, and PaigeStuder and Marvin Ludersdorfer for their assis-tance in the self-folding experiments.

Support for this work has been provided par-tially by National Science Foundation grants1240383 and 1138967, and the Swiss NationalScience Foundation Fellowship Grant PBZHP2-133472.

Appendix: Remote MagneticActuation

A setup consisting of four electromagneticsolenoid coils was employed to apply a rotationalmagnetic field to the propeller. The magneticcoils consist of copper wire wound on squarepillar-shaped ferrous cores of cross-sectional sidelengths 2D. An xyz coordinate set is defined for

each coil, such that the origin lies at the cen-troid of each coil, the x − y plane is parallel tothe surface of the coil, and z is normal to thesurface, as illustrated in Fig. 13. A small mag-net a distance from the coil can be regarded as amagnetic dipole moment m. Assuming that themagnet’s shape can be approximated as a spher-ical shape of radius a, m can be described withthe saturation magnetization M sat as

m =4

3πa3M sat, (13)

where Msat is intrinsically given by the materialof the magnet.

The z-directed magnetic flux density Bz cen-tered on the z axis at position z is

Bz =µ0I

4π

2D2

(D2 + z2)32

. (14)

The gradient along the z axis is then

−∂Bz

∂z∝ 2z

(D2 + z2)52

. (15)

When D ≪ z, using Taylor series, the forcethat the magnet experiences is proportional to

−∂Bz

∂z∼ 2z(

1 + 52D2 z2

) . (16)

The four coils were evenly spaced around thecentral vertical axis and tilted at 45◦ from thehorizontal. The stage was set on the point wherethe z axes of all the coils intersect. This config-uration allows the generation of any arbitrarymagnetic field vector at the stage via the su-perposition of individual fields of each coil (seeFig. 13). In addition, a quasi-uniform field isguaranteed with arbitrary strength along theGx − Gy plane and a non-uniform field alongthe Gz-direction. Because the magnetic fieldstrength is stronger at positions closer to thecoils, the propeller can experience magnetic stepout as it levitates higher and enters a relativelyweaker magnetic field region.

The force F and the torque τ that the mag-net experiences in a magnetic flux density B aregiven as

F = (m · ∇)B, (17)

τ = m×B, (18)

9

where B is the globally created superimposedmagnetic flux density of four coils. The torquereaches the maximum when the relative anglebetween the magnet and the applied field reaches90◦.

In real measurements, the coil exerted mag-netic fields of 7.0mT, 5.7mT, and 19.7mT at acurrent flow of 4A measured at the center of thesurface at respective distances of z = 0mm (onthe surface), z = 10mm, and z = 40mm. Theamount and duration of the current to the coilswere driven by motordrivers (SyRen10), whichwere manually controlled through serial commu-nication with a PC via ArduinoMega.

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[12] Yao Z, Bowick M, Ma X, SknepnekR. 2013, Planar sheets meet negative-curvature liquid interfaces. EurophysicsLetters. 101(4):44007.

[13] Kilian M, Flory S, Chen Z, Mitra NJ, Shef-fer A, Pottmann H. 2008, Curved Folding.ACM Transactions on Graphics. 27(3):#75,1–9.

[14] Hawkes E, An B, Benbernou NM, TanakaH, Kim S, Demaine ED, et al. 2010, Pro-grammable matter by folding. Proceed-ings of the National Academy of Sciences.107(28):12441–12445.

[15] Felton SM, Tolley MT, Shin B, Onal CD,Demaine ED, Rus D, et al. 2013, Self-folding with shape memory composites. SoftMatter. 9:7688–7694.

[16] Miyashita S, Onal CD, Rus D. 2013,Self-pop-up Cylindrical Structure by GlobalHeating. In: IEEE/RSJ International Con-ference on Intelligent Robots and Systems(IROS), pp. 4065–4071.

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[17] Yasu K, Inami M. 2012, POPAPY: instantpaper craft made up in a microwave oven.In the 9th International Conference on Ad-vances in Computer Entertainment.

[18] Tolley MT, Felton SM, Miyashita S, AukesD, Rus D, Wood RJ. Self-folding origami:shape memory composites activated by uni-form heating. The IOP Journal SmartMaterials and Structures. doi:10.1088/0964-1726/23/9/094006.

[19] Miyashita S, Meeker L, Goldi M, Kawa-hara Y, Rus D. 2014, Self-Folding Print-able Elastic Electric Devices: Resistor, Ca-pacitor, and Inductor. In: IEEE Interna-tional Conference on Robotics and Automa-tion (ICRA), pp. 1446–1453.

[20] Guberan C, 2012, Available from: http:

//vimeo.com/39914902.

[21] Mitani J. 2009, Fushigina Kyutai RittaiOrigami Tokyo: Futami shobo.

[22] Leishman JG, editor. 2006, Principles ofHelicopter Aerodynamics. Cambridge Uni-versity Press, 2nd ed.

[23] Keys C, Tarzanin F, McHugh F. 1987, Ef-fect of Twist on Helicopter Performance andVibratory Loads. In 13th European rotor-craft forum.

[24] Miyashita S, Meeker L, Goldi M, TolleyMT, Wood RJ, Rus D. Self-Folding Minia-ture Elastic Electric Device. The IOPJournal Smart Materials and Structures.doi:10.1088/0964-1726/23/9/094005.

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s

45°

45°

45°

(a)

(b)fold

x

y

n=4

n=2

n=1.5

region used in the propellers

rulings

rulings

fold

superellipse (crease)

middle

end

R

L

R

L

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Figure 1: Curved crease folding. (a) Superellipse curves of n = 1.5, n = 2, and n = 4. The rulingsand corresponding β· are defined. (b) The folding angle α over arc lengths s.

12

experimentally measured

middle (cf. Fig. 8(b))

0

0.5

1

1.5

2

2.5

3

α [r

ad]

0.5 1 1.5 2 2.5 30

X

α [d

eg]

30

60

90

120

150

0

180

end= -

end=( - /2

αend =( - + )/2n=4αend= - n=4

αend =( - + )/2n=1.5αend = - n=1.5

αend =( - + )/2n=2αend = - n=2

n=1.5 (experiment)

n=2 (experiment)

n=4 (experiment)

end

(cf. Fig. 8(b))

Figure 2: Simulated folding angles α, for n = 1.5, 2, and 4, with different conditions of αend.Experimental results of αend and αmiddle are superimposed.

13

setting αend = ( - + )/2

1.5 2 3 4 5 6 7 8

0

n

30

60

90

120

150

180

[d

eg

]α

mid

dle

0

0.5

1.5

2.5

3α

mid

dle

[r

ad

]

2

1

setting αend = -

experimentally measured

αmiddle (cf. Fig. 8(b))

Figure 3: Change of αmiddle over n for αend = (Γ− γ+π)/2 (upper curve) and αend = Γ− γ (lowercurve). As a general trend, the larger n becomes, the more acutely it folds.

14

contraction layer

(PVC; middle)

structural layer

(top and bottom)

self-fold

gap (crease)

slit

front

back

heat

Figure 4: Three layer structure for the heat sensitive self-folding method [16].

15

a b

BA

CD

EH

FG

O I J

K bt

H’G’

E’F’

O’, I’ (bo!om)A’

D’

B’

C’

L’

(a) (b)

L

valley foldmountain fold

Figure 5: 4-blade origami propeller. (a) The crease pattern. (b) Folded propeller in top view.

16

O Ga b

B

A

C

D

EF H

I

b

D’F’

E’

B’

C’

A’

O’, G’J’

(a) (b)

t

J(bo om)

valley foldmountain fold

Figure 6: 3-blade origami propeller. (a) The crease pattern. (b) Folded propeller in top view.

17

(a)

(b) (c)

Figure 7: Automated self-folding crease pattern generation. (a) The user interface. (b) Front andback crease pattern of a 3-blade propeller. (c) Crease pattern of a 5-blade propeller. In a curvedcrease, a superellipse and a straight line are connected smoothly at an inclination of 45◦.

18

75s 216s0s

1cm 1cm

n=1.5 (2.0rad) n=2 (1.9rad)

rulings

n=4 (1.1rad)

(a)

(b)

Figure 8: Self-folded curved creases with different curvature patterns. (a) The snapshots of then = 2 model while self-folding. (b) Self-folded curved creases (n = 1.5, 2, 4 from left to right,respectively).

19

0s

68s 118s

33s 98s

164s

Figure 9: Self-folding 4-blade propellers (n = 2 model). The whole process was completed in about3 minutes.

20

1cm

(a) n=2

angle of attack

magnets(b) n=1.5 n=4n=2

1cm

Figure 10: Self-folded 4-blade propellers. (a) The angled view. (b) Comparison of angle of attackof n = 1.5, n = 2, and n = 4 propellers.

21

1cm

5 wing propeller3 wing propeller

Figure 11: Self-folded 3-blade propeller (left) and 5-blade propeller (right).

22

20mm

0.6s 6.2s 9.2s 12.4s 15.0s

guiding pole(a)

(b)

ground level

0 5 10 150

10

20

30

Time [s]

Levitation [m

m]

n=1.5 n=2 n=4

Figure 12: Attained levitation of origami propellers. (a) Snapshots of levitation from n = 2propeller, (b) Height of levitation over time.

23

propeller

pole

electromagnetic coils

x,y

z

x, yG G

zG

stage

2D

Figure 13: The developed electromagnetic coil system.

24