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OrigamiBot-I: A Thread-Actuated Origami Robot for Manipulation and Locomotion Evan Vander Hoff, Donghwa Jeong, and Kiju Lee Abstract— This paper presents OrigamiBot-I, a thread- actuated origami robot, to demonstrate a physical application of an origami design for robotic manipulation and locomotion. The selected design can generate twisting and bending motions by pulling, pushing, or torsional force applied to the origami structure. Thread-based actuation also enables various shapes and motions by using different numbers of threads and routing them through different paths. The kinematics for each twisting and bending motions based on estimated parameters is derived. To evaluate potential use of origami for real-world applications and identify structural weaknesses, preliminary stiffness and durability testing was conducted. For physical demonstrations of robotic manipulation and locomotion, OrigamiBot-I was equipped with four independently-routed threads, where each thread is controlled by a geared DC motor. The robot suc- cessfully demonstrated its simple manipulation and locomotion capabilities. I. I NTRODUCTION The word Origami describes a paper-folding technique that creates 3D structures from 2D paper [1], [2], [3]. Due to the space creation and possibility of miniaturization, origami has been receiving growing attention in the robotics community [4]. In particular, the unique coexisting properties of flexibility and rigidity that the folding patterns provide can bring a breakthrough in robot design. Traditional rigid- body robots have been suited for many situations, but have not fully met the needs for specific situations that may involve unknown or extreme environments where the robot is expected to perform unspecified or a variety of different tasks. Furthermore, paper or thin layers of any foldable material used for building the robot may significantly reduce the weight and the material cost while also being eco- friendly. In the field of robotics, the power of origami structure has been discovered for generating useful functionalities such as actuators [5], [6], springs [7], and printable robotics [4]. More recently, origami design was used for a deformable wheel robot [8] which facilitates quick movement with large wheels and the ability to move through small gaps by folding and reducing the wheel size. Many of the previous robotic designs inspired by origami utilized folding features actuated by Shape Memory Alloys (SMAs). The SMAs are used to extend and contract the structure when a specific amount of heat is applied by a current [9]. Although there are low temperature (70 C) activated SMAs, it can still harm materials that are not thermally resistant, such as paper. In The authors are with the Department of Mechanical and Aerospace Engineering at Case Western Reserve University, Cleveland, OH44106, USA. [email protected] order to utilize SMAs, the origami bodies could be made with special polymers such as polyester, polyether (PEEK), or polytetrafluoroethylene (PTFE) with high thermal resistance [4]. One of the main limitations of origami design in engi- neering applications is the high complexity associated with the folding process. An origami design often requires paper to be folded and bent in a complex sequence, following exact instructions. This process is labor intensive and time consuming when performed by human hands. However, automation of such a process is not an easy engineering problem to solve. Although a robotic system demonstrated autonomous paper-folding, its capabilities were limited to valley and mountain folds which are referred to as “simple” folds [10]. Nevertheless, this system implies that the labor intensive and time consuming folding process that many origami shapes require can be at least partially automated. Origami has brought novel inspiration to other areas of science, engineering, and education. It has been receiving significant interest in mathematics and art as well as other specialized areas including investigation of DNA folding mechanism, design of medical stents, design of crumple zones in automobiles, and building design [11]. In addition, structural engineers deployed origami folding for both the flexibility and the rigidity the folding patterns provide [12]. An origami-inspired approach is also found in shelter system design [13] and novel crash box design that allows for the absorption of energy during vehicle collisions [14]. Many researchers and educators have also pointed out that origami can have further educational benefits and effects [15], [16], [17], [18], [19]. Origami folding requires dexterous hand manipulation that can improve cognitive skills such as hand- eye coordination [20], fine motor skills [21], [22], and work- ing memory skills [23]. Also, origami is highly associated with attention skills, as it requires listening and watching an instructor or reading instructions carefully [15], [16]. In this paper, we present a thread actuated robot, called OrigamiBot-I, based on an origami design called “twisted tower.” To evaluate its potential utilities and applications, stiffness and durability testing of the origami structure were conducted. In addition, two physical demonstrations of OrigamiBot-I were performed. The rest of the paper is organized as follows. Section II describes the design and control scheme of OrigamiBot-I. Section III presents the results from the stiffness and durability testing and physical demonstrations of the robot when used as a manipulator arm and a worm-like crawling robot. 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2014) September 14-18, 2014, Chicago, IL, USA 978-1-4799-6933-3/14/$31.00 ©2014 IEEE 1421

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OrigamiBot-I: A Thread-Actuated Origami Robot forManipulation and Locomotion

Evan Vander Hoff, Donghwa Jeong, and Kiju Lee

Abstract— This paper presents OrigamiBot-I, a thread-actuated origami robot, to demonstrate a physical applicationof an origami design for robotic manipulation and locomotion.The selected design can generate twisting and bending motionsby pulling, pushing, or torsional force applied to the origamistructure. Thread-based actuation also enables various shapesand motions by using different numbers of threads and routingthem through different paths. The kinematics for each twistingand bending motions based on estimated parameters is derived.To evaluate potential use of origami for real-world applicationsand identify structural weaknesses, preliminary stiffness anddurability testing was conducted. For physical demonstrationsof robotic manipulation and locomotion, OrigamiBot-I wasequipped with four independently-routed threads, where eachthread is controlled by a geared DC motor. The robot suc-cessfully demonstrated its simple manipulation and locomotioncapabilities.

I. INTRODUCTION

The word Origami describes a paper-folding techniquethat creates 3D structures from 2D paper [1], [2], [3].Due to the space creation and possibility of miniaturization,origami has been receiving growing attention in the roboticscommunity [4]. In particular, the unique coexisting propertiesof flexibility and rigidity that the folding patterns providecan bring a breakthrough in robot design. Traditional rigid-body robots have been suited for many situations, but havenot fully met the needs for specific situations that mayinvolve unknown or extreme environments where the robotis expected to perform unspecified or a variety of differenttasks. Furthermore, paper or thin layers of any foldablematerial used for building the robot may significantly reducethe weight and the material cost while also being eco-friendly.

In the field of robotics, the power of origami structure hasbeen discovered for generating useful functionalities suchas actuators [5], [6], springs [7], and printable robotics [4].More recently, origami design was used for a deformablewheel robot [8] which facilitates quick movement with largewheels and the ability to move through small gaps by foldingand reducing the wheel size. Many of the previous roboticdesigns inspired by origami utilized folding features actuatedby Shape Memory Alloys (SMAs). The SMAs are used toextend and contract the structure when a specific amountof heat is applied by a current [9]. Although there arelow temperature (70◦C) activated SMAs, it can still harmmaterials that are not thermally resistant, such as paper. In

The authors are with the Department of Mechanical and AerospaceEngineering at Case Western Reserve University, Cleveland, OH44106,USA. [email protected]

order to utilize SMAs, the origami bodies could be made withspecial polymers such as polyester, polyether (PEEK), orpolytetrafluoroethylene (PTFE) with high thermal resistance[4].

One of the main limitations of origami design in engi-neering applications is the high complexity associated withthe folding process. An origami design often requires paperto be folded and bent in a complex sequence, followingexact instructions. This process is labor intensive and timeconsuming when performed by human hands. However,automation of such a process is not an easy engineeringproblem to solve. Although a robotic system demonstratedautonomous paper-folding, its capabilities were limited tovalley and mountain folds which are referred to as “simple”folds [10]. Nevertheless, this system implies that the laborintensive and time consuming folding process that manyorigami shapes require can be at least partially automated.

Origami has brought novel inspiration to other areas ofscience, engineering, and education. It has been receivingsignificant interest in mathematics and art as well as otherspecialized areas including investigation of DNA foldingmechanism, design of medical stents, design of crumplezones in automobiles, and building design [11]. In addition,structural engineers deployed origami folding for both theflexibility and the rigidity the folding patterns provide [12].An origami-inspired approach is also found in shelter systemdesign [13] and novel crash box design that allows for theabsorption of energy during vehicle collisions [14]. Manyresearchers and educators have also pointed out that origamican have further educational benefits and effects [15], [16],[17], [18], [19]. Origami folding requires dexterous handmanipulation that can improve cognitive skills such as hand-eye coordination [20], fine motor skills [21], [22], and work-ing memory skills [23]. Also, origami is highly associatedwith attention skills, as it requires listening and watching aninstructor or reading instructions carefully [15], [16].

In this paper, we present a thread actuated robot, calledOrigamiBot-I, based on an origami design called “twistedtower.” To evaluate its potential utilities and applications,stiffness and durability testing of the origami structurewere conducted. In addition, two physical demonstrationsof OrigamiBot-I were performed. The rest of the paper isorganized as follows. Section II describes the design andcontrol scheme of OrigamiBot-I. Section III presents theresults from the stiffness and durability testing and physicaldemonstrations of the robot when used as a manipulator armand a worm-like crawling robot.

2014 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS 2014)September 14-18, 2014, Chicago, IL, USA

978-1-4799-6933-3/14/$31.00 ©2014 IEEE 1421

II. ROBOT DESIGN & CONTROL SCHEME

A. Twisted Tower

Fig. 1. Twisted tower: (a) creases to fold an origami segment, (b) foldedsegment, (c) assembled tower when extended, (d) assembled tower whenfully contracted, and (e) assembled tower when bent.

After analyzing many different origami structures, we se-lected the twisted tower, designed by Mihoko Tachibana [24],for physical demonstration of locomotion and manipulation.The twisted tower is made of identical origami segmentsthat are connected in an octagonal pattern and stacked toform a tower. To begin creating the tower, a single pieceof rectangular paper is selected (Fig. 1 (a)). The size of therectangular piece determines the diameter and height of eachoctagon layer and therefore must be determined based onthe size and weight of the actuators, sensors, and associatedcircuits to be embedded in the robot structure. Each octagonpattern is made up of sixteen individual origami segmentsthat have been folded together as shown in Fig. 1 (b). Anynumber of layers can be added to form a tower with a desiredlength and stroke. We selected a 4.25×2.125 inch2 piece ofconstruction paper where the diameter of the circumscribedcircle of the assembled octagon pattern is 3.5 inch. Thicknessof the construction paper is 0.012 inch and the thicknessof the single origami segment is about 0.119 inch when itis fully compressed. Fig. 1 (c) - (e) shows three differentconfigurations of the twisted tower.

B. Kinematics

The kinematics of the twisted tower provide interestingproperties, such as the low bending stiffness and high axialstiffness due to the fact that the paper bends easily butbarely stretch. Therefore, the structure generates “rigid-body-like” motions that allow us to use traditional rigid-bodykinematics with the addition of physical constraints due topaper thickness and structural limitations. We assume thateach top and the bottom plates of an octagon layer arerigid where the possible motions are limited to 1) twistingabout the vertical axis and 2) bending towards each of theeight outer edges of the octagon layer, where workspace isdependent upon the number of layers (Fig. 2). One of theunique properties of the twisted tower is that the horizontaltwisting motion generates vertical displacement. While bothclock-wise and counter-clock-wise rotations result in thesame linear displacement vertically, the orientation of thetop layer is determined by the rotation direction of eachlayer. Bending occurs when one of the eight segment areas

is compressed. Another interesting property of the twistedtower is that the diameter of the structure does not changewhile extended or contracted.

Fig. 3. Single layer origami tower with the assigned body-fixed framewhere h is the link length, θ is the rotation about the z-axis, and l is thedisplacement between two plates measured along the z-axis.

1) Twisting (screw) motion: Fig. 3 shows a single octagonlayer. As shown in the figure, rotation about the z-axisgenerates translation along the same axis. Such motion iscalled a screw motion [25]. The screw motion is generated bytwisting the structure by θ where −π/4+εθ ≤ θ ≤ π/4−εθ

and εθ is a small angle caused by the thickness of the foldedpaper when the top and bottom plates are fully collapsed. Fullcollapse of the top and bottom plate, i.e., θ = −π/4+ εθ

or θ = π/4− εθ , results in a small distance between theplates, εh, where estimated relationship between εθ and εhis εh ≈ (4h/π)εθ . The rigid-body transformation by twistingthe top plate by θ radian is given by

T =

ewθ

(h− 4h

π|θ |)~w

~0T 1

∈ R4×4. (1)

We note that ~w = [0,0,1]T is the unit vector specifying theaxis of rotation, and w is the corresponding skew symmetricmatrix. Eq. (1) can also be written as

T =

cθ −sθ 0 0sθ cθ 0 0

0 0 1 h− 4hπ|θ |

0 0 0 1

(2)

where ‘cθ ’ and ‘sθ ’ denote ‘cosθ ’ and ‘sinθ ,’ respectively.This shows that l = h− 4h

π|θ |.

For a tower consisting of N layers, the composite trans-formation is given by

T N0 =

ewθ1,··· ,N

(Nh− 4h

πΣN

i=1|θi|)~w

~0T 1

∈ R4x4 (3)

where θ1,··· ,N = ∑Ni=1 θi. For our physical prototype, h = 1.35

[inch] and εθ = 0.113 [radian].2) Bending motion: A bending motion in a single octagon

layer occurs when one of the eight edges on the top platecollapses towards the bottom plate. As shown in Fig. 4, thetransformation from the bottom plate to the top plate in asingle octagon layer, where the frames are assigned at the

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Fig. 2. Three different configurations of the twisted tower by applying (a) linear contraction by twisting two layers in the counter-clock-wise direction;(b) linear contraction by twisting the lower layer in the clock-wise direction and the upper layer in the counter-clock-wise direction; and (c) bending bycollapsing one of eight sides of the octagon layer.

Fig. 4. Schematics of a single layer twisted tower when bending isoccurred.

center of each plate where the z-axis pointing upward, isgiven by

T =

cθ −sθ 0 0

cφsθ cφcθ −sφ r− (r+ l)sφ

−sφsθ −sφcθ cφ lcφ + rsφ

0 0 0 1

(4)

where r is the radius of the circumscribed circle of theoctagon layer, l is the displacement between two platesmeasured along the z0-axis, φ is the bending angle, and θ

is the twisting angle resulted by bending. Using that θ = 0when φ = 0 and |θ | ≈ θmax when φ = φmax, the relationshipbetween θ and φ can be approximated by

θ ≈ θmax

φmaxφ

where φmax is the maximum bending angle that a singlelayer of twisted tower can generate and θmax is the resultingtorsional angle. The relationship between l and φ can be alsoestimated by

l ≈ h− (h− εh)

φmaxφ .

Our physical prototype measures r = 1.75, θmax = 0.672, andφmax = 0.323 where 0≤ φ ≤ φmax.

3) Singularities: We assume that each layer can generateeither pure twisting or pure bending motion. For the layergenerating twisting motion, the singularity occurs when θ =0 or θ =±(π/4− εθ ) where the structure is fully stretchedor fully compressed by twisting. For bending motion, thesingularity also occurs when φ = 0 or φ = φmax where thestructure is fully stretched or fully bent.

C. Thread-Actuated Hardware Design

Fig. 5. Examples of three different routing in the threads.

The selected origami structure allows for the generationof screw and bending motions by selectively twisting andsqueezing the structure. To achieve these motions, we con-sidered a thread-based actuation system that achieves thedesired motions by routing the threads via different paths.Fig. 5 shows some examples. Fig. 5 (a) shows linear motionachieved by routing two pieces of thread diagonally andthe other two linearly, (b) shows torsional motion achievedby routing the thread diagonally, and (c) shows alternativetorsional motion by routing the thread linearly.

For experiments, we used four motors that control fourlinearly routed threads independently (Fig. 5 (c)). We se-lected thread-based actuation for OrigamiBot-I over ShapeMemory Alloys (SMAs) which are commonly used for suchapplications. In order to use the SMAs, thermal resistant ma-terials are needed for hardware to endure the heat generatedby SMAs, which is typically more than 70◦ Celsius. We usedpaper because it’s recyclable, cheap, easily obtainable, andeasily handcrafted. Therefore, SMAs were excluded from ourconsideration due to flammability. Four DC motors generatestrokes by winding and releasing the thread, which is betterthan the small percentage shrinking rate of the SMAs.

D. Control Circuit Design

OrigamiBot-I can generate simple oscillatory motions byusing a low-complexity, memoryless control module for

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Fig. 6. Memoryless control circuit for OrigamiBot-I.

forward locomotion without preprogrammed logic in embed-ded memory or a processor. The circuit only contains lowcomplexity electronic components such as diodes, transistors,resistors, and a timer (Fig.6). For oscillatory motion, 555timers are used with stable mode generating continuouspulses. One timer controls the top and bottom motor whichmake forward locomotion, and the other timer sets thefrequency of the oscillation of right and left side motors,which allows the robot to bend to the side. The pulse widthis 50% for both timer and each time period is modulatedmanually by potentiometers. To detect a wall or an obstaclewhile navigating, two photo sensors are installed and controlthe right and/or left side motor. Two serially connected 3.7V(7.4V) lithium polymer batteries power the robot. To achievehigher level of manipulability and programmability, we alsobuilt an Arduino-based control circuit.

III. EXPERIMENTS

Fig. 7. Loading and unloading durability testing.

We conducted preliminary experiments to evaluate thepotential of OrigamiBot-I as a robotic manipulator or amobile robot. Because the main body is made with paperand the spring-like behavior generated by the origami designis an essential property for achieving desired motions, wefirst tested stiffness and durability of the structure. Physicaldemonstrations of OrigamiBot-I acting as a manipulator armand a worm-like crawling robot are also presented.

A. Stiffness and Durability

Fig. 8. Performance result of 1000 cycles of loading and unloading origamistructure. The color bar in the right side shows the number of cycles of theloading and unloading up to 1000 cycles.

The origami tower was tested by an Inston servo-hydraulictesting machine (model 8501) in order to estimate the stiff-ness and force capacity, and dynamic characteristics, suchas fatigue over time, (see Fig. 7). 1000 sinusoidal cycleswere modeled with 1Hz frequency and 1.5 inch amplitude (3inches stroke). A 250 lb capacity force transducer is used forthe test. As shown in the Fig. 8, the performance results tellus the origami structure made of paper, functions well as aspring until 2 inches stroke out of 6 inches long. The averagestiffness for 1-inch contraction to 3-inch contraction is 8.61lb/inch for loading and 5.81 lb/inch for the first cycle. Thishysteresis indicates that the energy is dissipated by frictionor heat during loading and unloading.

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

Cycles

Sti

ffn

ess

(lb

/in

)

Loading

Unloading

Fig. 9. Stiffness changes induced by fatigue.

Another observation is the fatigue of the structure byrepetitive motion. Fig. 9 shows that the stiffness changesgradually by the time but not significantly indicating there

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will be limited life-time using the paper based origami robot.We used paper rather than other materials due to low cost, ob-tainability, and disposability. However, recyclable polymerssuch as PETE or high-density polyethylene (HDPE) can beconsidered for the next version of OrigamiBot.

B. Manipulator Arm

Fig. 10. Video captured images of bending motion (top) and loading test(bottom).

TABLE IDEFLECTION ANGLE (α ) PER LOADED WEIGHT.

Weight (lb) 1.0 1.5 2.0 2.5 3.0α (degree) 3 6 8 12 N/A

An interesting application of OrigamiBot-I is to use it as amanipulator arm (Fig. 10). The Arduino-based controller wasused to generate manipulating motions. Using two actuators,the manipulator arm can demonstrate linear extension andcontraction and swing from the left to the right or vice versa.With additional actuators, the manipulator arm is capableof reaching specific location within the workspace. Thistype of movement can be beneficial in a situation wherea flexible arm is needed to move objects from one placeto another. By adding more segment to OrigamiBot-I, themanipulator arm could reach a larger area as well as allowingobstacle avoidance using the flexible structure. To test theamount of load that the robot can hold without damagingthe origami structure, 1lb, 1.5lb, 2lb, 2.5lb, and 3.0lb of loadwere attached to the robot’s end-effector (Fig. 10 (bottom)).Table I shows the deflection angle (α) caused by each weight.Although the origami structure was damaged with 3lb ofload, the result shows that the structure is fairly durable formanipulating with weight attached.

C. Worm-like Crawling Robot

1) Forward locomotion: In order to make the robot moveforward with oscillation, mass-distribution and friction wereconsidered. OrigamiBot-I generates a symmetric flexion andextension motion by winding and releasing the thread viarotaries. Unlike the majority of the worm-like crawlingrobots which employ peristaltic wave motion for locomotion,we used a composite of two different friction material-made

Fig. 11. Forward movement in top view (Oscillation frequency is 0.25Hz).

Fig. 12. Directional passive flaps.

passive flaps (Fig. 12). Conventional flap based locomotionuses an anchoring mechanism [4]. However, the anchoringmechanism of the current design does not guarantee loco-motion on a slippery surface such as glass or fine plastic.OrigamiBot-I adopts flexible and directional passive flapsso that locomotion can be achieved to increase directionalfriction. One side of the passive flaps has low-friction andthe other side with high friction such as rubber. For thesimplicity, we attached coated tape on one side of the rubberto reduce the friction. Forward locomotion with directionalpassive flaps was tested as shown in the video-capturedimages in Fig. 11. OrigamiBot-I moves at the speed of0.25 inch/sec. We observed no backing motion since thedirectional passive flaps prohibit backward movement duringlocomotion. In this experiment, the robot carries the load thatweights about 0.3 lb including batteries, circuits, sensors, andfour DC geared motors.

Fig. 13. Obstacle navigation with sequential control scheme.

2) Obstacle navigation by turning motion: For obstaclenavigation, a dome-shape cover is added around the controlmodule to easily turn around an arbitrarily shaped obstaclewithout advanced control (Fig. 13). To move forward, the

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bottom motor winds up and releases the wire to makecontract and release the structure with cyclic motion. Turningto the left or right motion involves holding period in thetop motor while left or right motor starts oscillating (Fig.13). This sequential control scheme enables OrigamiBot-Ito the left or right allowing it to move around obstacles inits path. On average, utilizing the tail flicking movement,OrigamiBot-I can turn at 0.5◦ per second to either the rightor the left. The total weight of the load that OrigamiBot-Icarries in this experiment was about 0.76 lb. The asymmetricweight distribution due to circuits and dome weight wasnot significant factor for the performance compared to thedirectional passive flaps.

IV. DISCUSSION AND FUTURE WORK

In this paper, we presented the design and development ofa thread-actuated origami robot made with paper. In order togenerate locomotion, we designed simple directional passiveflaps which allow directional friction to move the robotdynamically. Stiffness and fatigue tests were conducted, re-vealing the elastic zone and life-time of the origami structure.For a preliminary evaluation of OrigamiBot-I, manipula-tive motions and locomotion (i.e., forward locomotion andobstacle navigation) were demonstrated. Further evaluationmay include object manipulation using the OrigamiBot-Iarm, sensor integration, and investigation on different threadrouting and control schemes.

The proposed twisted tower structure shows unique prop-erties: 1) horizontal twisting motion applied to the structurecreates linear displacement vertically; and 2) the diameterof the structure does not change when it is contractedor stretched. These properties have been fully utilized inour physical system on demonstrating its manipulation andlocomotion capabilities. We are also developing a wheel-based robot with reconfigurable origami wheels where eachwheel is made by OrigamiBot-I. The second property of thetwisted tower will maintain the wheel diameter and thereforekeep the distance between the robot’s center of mass and theground as well as the speed of the robot. This can be a greatbenefit in controlling wheel-based locomotion.

The deformation of paper or paper-based origami struc-tures are dependent upon environmental conditions, as wellas time and loading patterns. For example, humidity ortemperature of the robot’s surroundings will affect the vis-coelastic properties of the origami structure, which couldchange its behaviors. Nevertheless, a paper-based origamirobot has benefits such as low-cost, fast and easy manufac-turing by hand craft, and eco-friendliness. Potential issues onthe limited life-time due to abrasion as well as disposabilityor recyclability can be overcome by considering possiblematerial selections such as disposable, abrasion resistantpolymer or thermal resistant polymer. Not only for thesebenefits, folding and assembly skills require dexterous handmanipulation as well as cognitive skills so it can be used fortraining and improving these skills. Therefore, OrigamiBot-Ican be adopted as an educational kit for children.

ACKNOWLEDGMENT

This work was partially supported by the National ScienceFoundation under Grant No. 1109270.

REFERENCES

[1] T. Tachi, “3D Origami Design based on Tucking Molecule,” in Proc.4th International Conference on Origami in Science, Mathematics, andEducation, 2006.

[2] E. Hawkes at al., “Programmable Matter by Folding,, PNAS, Vol.107(28), pp. 12441-12445, 2010.

[3] P. Jackson, “Folding Techniques for Designers: From Sheet to Form,”Pap/Cdr. Laurence King Publishers, 2011.

[4] C. D. Onal, R. J. Wood, and D. Rus, “Towards Printable Robotics:Origami-Inspired Planar Fabrication of Three-Dimensional Mecha-nisms, IEEE ICRA, pp. 4608-4613, 2011.

[5] H. Okuzaki at al., “A Biomorphic Origami Actuator Fabricated byFolding a Conducting Paper, 4th World Congress on Biomimetics,Artificial Muscles and Nano-Bio, 127, 2008.

[6] R. V. Martinez at al., “Elastomeric Origami: Programmable Paper-Elastomer Composites as Pneumatic Actuators, Advanced FunctionalMaterials, Vol. 22, Iss. 7, pp. 1376-1384, 2012.

[7] C. C. Min and H. Suzuki, “Geometrical Properties of Paper Spring,41th CIRP Conference on Manufacturing Systems, pp. 159-162, 2008.

[8] D. Lee at al., “Deformable Wheel Robot Based on Origami Structure,IEEE ICRA, pp. 5612- 5617, 2013.

[9] J. Koh and K. Cho, “Omegabot: Biomimetic Inchworm Robot UsingSMA Coil Actuator and Smart Composite Microstructures (SCM),IEEE ROBIO, pp. 1154-1158, 2009.

[10] D. J. Balkcom, and M. T. Mason, “Robotic Origami Folding,”IJRR,Vol. 27, no. 5 613-627, 2008.

[11] W. Wu and Z. You, “Modeling Rigid Origami with Quternions andDual Quaternions, Proceedings of the Royal Society A: Mathematical,Physical & Engineering Sciences, 2010.

[12] M. Schenk and S. D. Guest, “Origami Folding: A Structural Engineer-ing Approach, Origami 5: International Meeting of Origami Science,Mathematics, and Education, Taylor and Francis Group, 2011.

[13] N. De. Temmerman at al., “Design and Analysis of a Foldable MobileShelter System, Int. J. Space Struct., 22(3), pp.161-168, 2007.

[14] J. Ma and Z. You, “The Origami Crash Box, Origami 5: Fifth Inter-national Meeting of Origami Science, Mathematics, and Education,Taylor and Francis Group, pp. 277-290, 2011.

[15] G. Levenson, “The educational benefits of origami,http://home.earthlink.net/ robertcubie/origami/edu.html.

[16] S. Cakmak, M. Isiksal, and Y. Koc, “Investigating Effect of Origami-Based Instruction on Elementary Students Spatial Skills and Percep-tions, The Journal of Educational research, Vol. 107, Iss. 1, 2014.

[17] L. Coad, “Paper folding in the middle school classroom and beyond,Australian Mathematics Teacher, 62(1), 6-13.

[18] J. Justin, “Towards a mathematical theory of origami, in Proc. the 2ndInternational Meeting of Origami Science and Scientific Origami, pp.1529, 1994.

[19] K. Chen, “Math in motion: origami math for students who are deafand hard of hearing, Journal of Deaf Studies and Deaf Education,11(2), 262-266.

[20] C. Lee and C. Chui, “Training and Measuring the Hand-Eye Coordi-nation Capability of Mentally Ill Patients,” Advanced in TherapeuticEngineering, pp. 45- 81, 2012.

[21] M. L. Fiol, N. Dasquens, and M. Prat, “Student Teachers IntroduceOrigami in Kindergarten and Primary Schools: Froebel Revisited,”Origami 5: Fifth International Meeting of Origami Science, Math-ematics, and Education, Taylor and Francis Group, 2011.

[22] A. Wong, N. Marcus, P. Ayres, L. Smith, G. A. Cooper, F. Paas, andJ. Sweller, “Instructional animations can be superior to statics whenlearning human motor skills, Computers in Human Behavior, Vol. 25,Iss. 2, pp. 339-347, 2009.

[23] L. R. Novick and D. L. Morse, “Folding a fish, making a mushroom:The role of diagrams in executing assembly procedures,” Memory andCognition, Vol. 28, Iss. 7, pp. 1242-1256, 2000.

[24] http://origamimaniacs.blogspot.com/2012/02/twisted-tower-by-mihoko-tachibana.html.

[25] R. M. Murry, X. Li, S.S. Sastry, “A Mathematical Introduction toRobotic Manipulation,” CRC Press.

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