Parallel-motion Thick Origami Structure for Robotic Design

Shuai Liu, Huajie Wu, Yang Yang, Michael Yu Wang, Fellow, IEEE

Abstract— Structures with origami design enable objects totransform into various three-dimensional shapes. Traditionallyorigami structures are designed with zero-thickness flat papersheets. However, the thickness and intersection of origamifacets are non-negligible in most cases, uniquely when in-tegrating origami design with robotic design because of themore efficient force transfer between thick plates comparedwith zero-thickness paper-sheets. Meanwhile, the single-layer-paper oriented initial design limited the shape transformationpotential as multiple layer origami structures could conductmore variety of deformation. In this article, we are proposinga general design method of parallel-motion thick origamistructures that could apply in robotic design like a parallel-motion gripper.

I. INTRODUCTIONOrigami is an ancient art of paper folding. Its shape

transforming ability has innovated various engineering de-signs. A typical example is the origami-based deployablearrays integrated with solar panels [1], [2]. Recently, someinnovative designs are proposed for robots [3], wheels [4],actuators [5], [6], gift box[7] etc. The attraction of origamistructures is that it could provide unlimited possibilities ofgeometric transformation and motion with different creasepatterns.

Generally, for robotic design, twisting, bending, and ro-tating motions are all essential. There are some researchworks combining the bending motion of origami with aforceps-like robotic gripper design [8], [9]. A forceps-likegripper is able to work out fine for some small objects,but larger and heavier ones would fall off because of thesmall contacting area of forceps. More complicated origamipatterns are also applied to develop pneumatic actuators[10], [11], [12]. Those pneumatic origami actuators are morecompliant, flexible, and passive; the grippers made fromthose structures are more reliant on the shape adaptive abilityof origami structures. It has the same function as a pneumaticsuction cup or artificial muscles, and therefore the shapeconfiguration cannot be controlled precisely.

Parallel grippers are another type of widely-used engi-neering gripper with higher precision and gripping force

Shuai Liu is with the Department of Mechanical and Aerospace En-gineering, Hong Kong University of Science and Technology. (email:sliubw@connect.ust.hk)

Huajie Wu is with the Shool of Information Science and Tech-nology, University of Science and Technology of China. (email:whj99129@mail.ustc.edu.cn)

Yang Yang is with the Department of Mechanical and Aerospace Engi-neering, Hong Kong University of Science and Technology, Hong Kong andalso with School of Automation, Nanjing University of Information Scienceand Technology, Nanjing 210044. (email: meyang@connect.hku.hk)

Michael Yu Wang is with the Department of Mechanical and AerospaceEngineering, Hong Kong University of Science and Technology. (email:mywang@ust.hk)

because the parallel gripping surface can usually providea larger contacting area between gripper tips and objects.There is already some mature design using split rails toachieve parallel-motion [13]. As those parallel grippers arecomposed of gears and rails, it will break when overloaded.In this work, we want to combine the high control precisionand gripping force of traditional parallel gripper with theflexibility and adaptability of origami structures. There couldbe a lot of application scenarios for parallel origami gripperwhen the compliant and flexible parallel gripper is needed.

Grippers with parallel-grasping capability are more usefulfor various tasks and can be controlled with better motionprecision. Here in this project, we have developed a noveldesign strategy for parallel-motion origami structures, andwe integrated the geometric design with a parallel-motiongripper shown in Fig. 1. The main contribution of this paperis the general idea to take advantage of the parallel motionmechanism, which has always been a fundamental feature oforigami structures.

II. DESIGN METHOD

A. Parallel-motion Design for Zero-thickness Origami PaperSheets

For origami structures, the fundamental motion is rotationbecause all facets are rotating around their connected creases.To make our design strategy simple and easier to deploy, wewill construct our origami pattern based on the widely-usedMiura-ori. Miura-ori is a typical origami folding pattern with1-DOF(Degree of freedom) when all facets are considered tobe rigid, which means the facets can neither bend nor stretch[14]. The mechanical features like bi-stability, geometricrelations between different folding states, and kinematicsof Miura-ori structures have been widely discussed [15],[16], [17]. A unit cell of Miura-ori is shown in Fig. 2. Theunit cell is symmetric about the plane P1P2P3. For activeperiodic Miura-ori patterns, the key parameters are the angleα, which is the angle between two adjacent creases, theinput angle θ, which is the dihedral angle between the initialfacet and the folded facet, and the output angle γ whichis the supplementary angle of 6 P1P2P3. When a Miura-oripatterned origami structure is folding, the relation betweeninput angle θ and output angle γ is given as [18]:

γ = 2k cos−1(cosα√

1− cos2 θ sin2 α) (1)

The variation of output angle γ with the change of inputangle θ and curve angle α is shown in Fig. 3. As α becomeslarger and larger, there will be a jump of output angle γ whenθ changes around zero, which means the acceleration of the

Fig. 1. (a)-(d)Motion of origami gripper under pneumatic actuation,negative pressure for expansion and positive pressure for grasping; (e)-(h)Grasping experiments

moving facets will become extremely large. Therefore toolarge α angle is not suitable for robotic design because ofits jumping output angle. For origami-based gripper design,we suggest choosing an angle of α between 30◦ to 60◦.

From (1), it is obvious that in a Miura origami structurecomposed of two unit cells with curved angle α1 and α2

respectively, once α1 = α1 we will get γ1 = γ1. Fig. 4(a)shows a paper sheet folded into two Miura-ori cells, whichwe call it first-order parallel-motion Miura-ori. The boundarylines of the upper and bottom facets will keep parallel in thewhole folding process. From this, we can extract the unitstructure for parallel-motion origami, in Fig. 4(a), the upperand bottom beams are heading for reverse directions whenfolding. However, to make full use of the parallel motionmechanism, we want two parallel surfaces orienting to thesame center point so we can integrate the mechanism withgripper design. Traditionally, origami patterns are designedin one sheet of paper by simply adding mountain and valleycreases, and it usually results in the design in Fig. 4(a), herewe solve this problem by reverse extending the bottom facetsshown in Fig. 4(b). This transformation will create a dual-layer origami structure when the input angle θ = 0◦. Dual-layer means the whole structure cannot be unfolded into one

flat paper sheet, when the input angle is 0, it is composedof two stacked layers, which is shown in Fig. 5

This reverse extending strategy could apply to Miura-oristructure with more unit cells, in Fig. 4(c)and(d), a parallel-motion origami structure with 4 Miura-ori cells is shown,which we call it second-order parallel-motion Miura-ori asit can be treated as two first-order Miura-ori combined. Theupper and bottom blue lines will keep parallel as long asα1 = α2, β1 = β2, because γ1 + γ2 + γ3 + γ4 = 360◦ whenprevious conditions are satisfied.

Unlike the single-unit Miura-ori structure, which has onlyone possibility for parallel-motion, there could be differentchoices for multiple Miura-ori structures when we want toget parallel-motion tips. Fig. 4(d)(e)(f) showed three differentways to get those parallel-motion facets.

Fig. 2. (a)Miura-ori unit cell; (b)Thick Miura-ori panels

Fig. 3. Relations between θ and γ

B. Panel Thickness and Hinge Effects

Previous design and discussion are all based on ideallyrigid and zero-thickness origami structure. In real engi-neering situations, the thickness of origami facets is non-negligible. There are various discussions and methods onthick origami structure hinge design [19], [20]. In this article,we are using the hinge-shift technique to transform theMiura-ori unit to thick origami structures shown in Fig. 2(b).In zero-thickness Miura-ori unit, the four hinges will allpoint to P2 and will stay in the same plane when inputangle is zero, while in thick Miura origami structure, thefour hinges will locate on two parallel planes and construct

Fig. 4. Construction of parallel motion tips for Miura-ori structures:(a)(b)Extending the bottom strips inversely to create parallel-motion tipswhich towards inside; (c)(d)(e)(f)Three different ways to make the endmosttips heading inside.

a spatial 4R linkage also known as Bennett linkage [21]. Thethick origami structure will have only 1 DOF. Assume thethickness of the panel is t then the thickness for the stagebetween facets C and D is 2t.

After we get the unit thick Miura-ori structure, we canapply it to parallel-motion design. As shown in Fig. 5(a),the simplest parallel-motion structure is composed of onedual-layer tip and one thick Miura-ori cell (Fig. 5(b)).

Similarly, the second-order parallel-motion thick origamistructure could be designed (Fig. 6). For an ideal rigidorigami model, the number of hinges will not influence themechanical property of the whole structure. However, fororigami structure designed with living hinges, there will bestress concentration in hinge areas, and it will acceleratehinge failure. Generally, the durability of hinges mainlydepends on the hinge material and geometric parametersof hinges. In our design, we will use 3D-printed thermo-plastic elastomer(TPE) for longer hinges and nylon-6 forshort hinges. As the material is determined, the geometricparameters of hinges are the main issue we need to concern.

A significant difference between first and second-orderparallel motion Miura-ori is the second-order unit canachieve the same travel distance with two smaller curveangles α2 and β2 compared to the curve angle of first-orderunit α1. If the width for the origami panels is fixed, then asmaller curve angle will result in a longer hinge length. Thisfeature of the second-order origami unit can decrease thestress concentration in hinges. However, we cannot reducethe stress concentration further by keeping adding hingesbecause force and displacement cannot fully pass livinghinges, so the increased number of hinges will decrease theprecision of geometric constraints between origami panels.So here we only tried second-order origami unit for the

sake of compromise between motion precision and structurestiffness and durability.

Fig. 5. First-order parallel-motion thick origami structure

Fig. 6. Second-order parallel-motion thick origami structure

Fig. 7. Dimensions for motion analysis

III. MOTION ANALYSIS

Because of the symmetry in our parallel-motion structure,the motion analysis can be simplified as a simple 3D rotationproblem. Shown in Fig. 13(a)(b), for a single-vertex Miura-ori unit, P1 is the initial position of a randomly selectedpoint P on the surface, P2 is the current position of P atany folding state. Assume the coordinate of P1 is alreadyknown, the coordinate of P2 can be calculated through twosequential rotating transformations around hinge 1 and hinge

2. For any given point P at an initial state, its coordinate atany time could be easily calculated by multiplying differentrotation matrix around certain hinges:

Pt = RnRn−1...R1P0 (2)

Where Rn is the rotation matrix for spatial rotation transformalong arbitrary axis and here it refers to hinge n.

The geometric dimensions for motion analysis are shownin Fig. 7. For Fig. 7(a), H1 = 50mm,H2 = 100mm,H3 =30mm. For Fig. 7(b), H1 = 50mm,H2 = 30mm,H3 =100mm,H4 = 25mm,H5 = 15mm. For both structures thebeam width d = 15mm, panel thickness t = 2.5mm.

The motion analysis results are shown in Fig. 12. Apoint P on the origami tips is picked for demonstration.Obviously, from the figure, we can tell how the curve anglewill influence the motion of the origami structure. As thecurve angle changes, the path of P is extending, but thedirection barely changes (Fig. 12(a)(d)). From the X and Ycoordinates change, we find that the acceleration is changingrapidly due to the increase of curve angle. All these resultsindicate that we need to find a suitable curve angle to balancetravel distance and control capability.

Fig. 8. Rotation axis and actuation surface of thick origami units

IV. EXPERIMENTS

The final design and motion simulation of the gripper isbased on the geometry shown in Fig. 13(c)(d). Two first-orderthick Miura-ori structures are connected through five flatconnecting panels. This design is to increase the gripper stiff-ness in the tangential direction and provide two horizontalpanels for the further upgrade; for example, soft tips can beinstalled through the reserved screw holes. The final gripperis made by 3D-printing with nylon-6 filament. Those hingeswhich may encounter stress concentration are extended toprevent hinge failure. The gripper is actuated through apneumatic rotating actuator, which is also 3D-printed withTPU filament. The actuator can provide a rotating motionthrough an axis that is offset from the centerline, whichis perfectly compatible with thick origami motion (Fig. 8).This kind of rotation actuator could actually also consideredas origami structures [22], [23]. The actuation and graspingexperiments of the origami gripper are shown in Fig. 1. Theparallel motion worked out nicely, and the gripper can holdan aluminum block, which weighs up to 500g.

Fig. 9. Pneumatic actuator for origami actuation

Fig. 10. FEM simulation results of hinge effects

Fig. 11. Motion and grasping experiments of second-order origami gripper

Fig. 12. Motion analysis of first and second-order parallel-motion thick origami structure

Fig. 13. Design and motion of first-order origami gripper

The second-order gripper is also fabricated through 3D-printing. The difference is the body of the second-orderone is printed with PLA and hinges with TPU filamentsseparately. As shown in Fig. 11, The motion of the second-order gripper is nicely identical to the simulation. Fivegrasping tests are shown in Fig. 11(e)-(i). The box that thegripper holds in Fig. 11(i) weighs up to 1.5kg.

An interesting phenomenon we found during our experi-ments is hinge thickness will deeply influence the parallelismof the structure. If the hinge is too thin, it will easilybreak, and no parallel motion will exist. On the other side,if the hinge is too thick, the large stress to deform thehinge will decrease the parallelism. As shown in Fig. 10,the FEM(Finite Element Method) simulation results indicate

that when hinge has a thickness of 0.8mm, the resulted tipbending angle 4ϕ will be apparent.

V. CONCLUSIONSOrigami structures have great potential in developing var-

ious robotic actuators. It could be either hard thick origamipanels composed of rigid plates and bearings or soft elas-tomer sheets or even in between them. The parallel-motionmechanism demonstrated in this article could be applied toboth high precision mechanical structure or compatible andcompliant metamaterials. We have proposed two differenttypes of parallel-motion mechanisms and conducted thegrasping experiments for both. The future work of thisproject will mainly focus on the improvement of fabricationand modeling. For the fabrication part, produce hinges with

lower thickness while higher stretching stiffness will beconsidered. For the modeling, we will try to propose asimplified model based on FEM and pseudo-rigid bodymodel to model the motion and elastic deformation insidethe whole structure.

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