Self‐Folding Method Using a Linkage Mechanism for Origami Structures

Origami is garnering attention in fields such as medical and electronic devices as this approach allows transitioning from a 2D to a 3D structure. Self‐folding method is effective for fabricating origami structures, but conventional strategy of self‐folding by driving all hinges is unsophisticated and thus makes redundancy and unnecessary limitations in fabrication. The behavior of deformation of origami structures can be described as a linkage mechanism, so that the degree of freedom of the origami structure is essentially not equal to the number of hinges. Herein, a self‐folding method is proposed for origami structures such that an entire structure can be folded by driving only a few hinges using the characteristics of force transmission of origami as a linkage mechanism. This proposed self‐folding method allows the selection of the position of the driving hinges and enables the self‐folding of origami structures with the restrictions of the position of the driving hinge. In addition, the method can provide high process compatibility for the fabrication and folding processes of origami devices.


Introduction
Origami structures can be made into various shapes by hinging a flat structure and folding up the flat part. [1,2] The property of seamless transformation from a 2D to a 3D shape, which can be easily applied to various fabrication processes, motivates applications in a wide range of fields, such as medical care, robotics, and electronic devices.  For folding a structure from a flat 2D shape to a 3D shape, self-folding is one of the most effective methods. Self-folding is an attractive designing approach to fabricate complex and fine origami structures because it is a technique to facilitate spontaneous folding of structures by converting external stimuli into mechanical energy by pre-assembling stimulus-responsive active materials in the device. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Several active materials for self-folding have been reported: stimuliresponsive hydrogels on the hinges formed by photolithography, [12][13][14] heat-shrinkable films attached to the entire substrate, [15][16][17][18][19][20][21] shape memory alloys attached to each hinge, [22,23] and solder dispensed on the folded side of the hinge; [24,25] all these methods add active materials to all hinges. If elements are densely mounted on the device surface, or active materials cannot be added to maintain the function of the device surface, the strategy of adding active material to all hinges limits the device fabrication. Therefore, to accomplish self-folding independent of the device structure, it is favorable for the attachment position of the driving material to possess a high degree of freedom. External stimuli that cause a reaction from the active material include heat and changes in acidity. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] However, external stimuli are also applied during device fabrication processes such as photolithography and solder mounting. Therefore, external stimuli applied to devices during the fabrication process induce self-folding at inappropriate times, such as during fabrication processes that can only be applied to flat surfaces. To overcome this problem, it is important to be able to select the timing of the addition of active material.
In general, origami can be analyzed for its deformation using a geometric mechanism called rigid origami, when it is assumed that the flat part is a rigid body and the hinge part is a perfect joint. [2,[26][27][28][29] By simplifying rigid origami, its mode of formation can be reduced to a linkage mechanism, and it can be considered as a structure whose components transmit forces and moments to each other. Although the characteristics of rigid origami are widely used in the design of thick origami structures, [26,30] the use of origami as a force-transmission mechanism is limited.
We propose a self-folding method that minimizes the limitations owing to device design and fabrication through the use of a linkage mechanism to calculate the driving force of self-folding. Using the property that an origami structure can be reduced to a linkage mechanism, it is possible for the entire structure to selffold with the driving force of only some hinges. This can be achieved by transmitting the driving force generated at one hinge to other hinges that do not have a driving force. The origami structure was modeled as a structure consisting of rigid bars and elastic hinges, and the hinge moments required for selffolding to the target shape were derived. Moreover, the proposed self-folding method was experimentally demonstrated by fabricating origami electronic devices. The proposed self-folding method allows the entire device to self-fold even if there are DOI: 10.1002/aisy.202200445 Origami is garnering attention in fields such as medical and electronic devices as this approach allows transitioning from a 2D to a 3D structure. Self-folding method is effective for fabricating origami structures, but conventional strategy of self-folding by driving all hinges is unsophisticated and thus makes redundancy and unnecessary limitations in fabrication. The behavior of deformation of origami structures can be described as a linkage mechanism, so that the degree of freedom of the origami structure is essentially not equal to the number of hinges. Herein, a self-folding method is proposed for origami structures such that an entire structure can be folded by driving only a few hinges using the characteristics of force transmission of origami as a linkage mechanism. This proposed self-folding method allows the selection of the position of the driving hinges and enables the self-folding of origami structures with the restrictions of the position of the driving hinge. In addition, the method can provide high process compatibility for the fabrication and folding processes of origami devices.
hinges without added active material, thus making it possible to design as required even when it is necessary to reduce redundancy in adding active material or there are hinges where it is difficult to add active material. Generally, the attachment of the active material should be done after all fabrication processes that stimulate the active material have been completed to avoid unintended self-folding. The proposed method enables the attachment of active materials at arbitrary timing without depending on the structure of the device due to the increased degree of freedom in the position of the attachment. Thus, the self-folding to the target shape is realized by post-attaching of the active material to some hinges, even if some hinges are inaccessible due to the element mounting or the assembly of the structure.

Result
2.1. Concept and Theory of Self-Folding Method Using Linkage Mechanism Figure 1a shows an origami electronic device fabricated using the self-folding method proposed in this study. First, the fabrication process of the flat device components, except for the active material, is completed. In this step, fabrication processes suitable for processing on a flat surface with stimuli, such as photolithography and element mounting, can be easily applied because the substrate is flat and stimulus-responsive active material is not attached. Subsequently, the active material is attached to specific hinges of the flat device. The active material is attached based on a design theoretically derived from the target origami shape. In this study, a heat-shrinkable polymer was used as the active material for self-folding.
The layer consisting of the heat-shrinkable polymer is called the shrinkable layer and that consisting of the flat device without the active material is called the nonshrinkable layer. The device consisting of the nonshrinkable layer attached with the shrinkable layer is heated. The heat-shrinkable polymer, that is, the shrinkable layer, shrunk by heating and the stress gradient in the thickness direction between the shrinkable and nonshrinkable layers becomes the driving force causing the structure to self-fold. The hinges are driven and the device is self-folded, resulting in a device with a 3D structure. In this example of an origami electronic device, all five hinges folded up even though a shrinkable layer was not attached to the central hinge. This is characteristic of the proposed self-folding method, which takes advantage of the fact that origami is a type of linkage mechanism. By transmitting the driving force of the hinge with the shrinkable layer to the hinges without the shrinkable layer, the entire device can fold itself, even if only a few hinges have attached shrinkable layers. Figure 1b shows a simplified model of the origami electronic device structure reduced to a linkage mechanism. When the driving force is applied only to the hinges, indicated by the solid red lines, the force is transmitted to the hinges-indicated by the broken blue lines-resulting in the folding of all the hinges.
To control self-folding using the linkage mechanism, it is necessary to design the driving force required to shape the object into the target shape and determine the position for attaching the shrinkable layer. The driving force required for each hinge varies, depending on the number of links, hinge characteristics, and the folding angles. To solve this problem, we developed a theory to control the shaping of the self-folding in a structure with N links. As an example, a model of a linkage mechanism with six links is shown in Figure 2. The links are assumed to be rigid bars and the joints are assumed to be elastic hinges, with the length of each link l i and the spring constant of each hinge k i . First, the moment required by each hinge to deform the shape before self-folding (Figure 2a) into the shape after self-folding ( Figure 2b) is calculated. The torque T i generated at each joint when the link mechanism shapes the structure is expressed as a simple product of the spring constant at each joint k i and the deformation angle at each joint Δθ i because an elastic hinge is assumed. However, at the joint where the driving force is generated, the driving force at the joint is the difference of the moment M i that bends the hinge and the torque in the opposite direction proportional to the amount of bending. Therefore, the Figure 1. Concept of the self-folding method using a link mechanism. a) Schematic illustration of origami device applying self-folding method using a linkage mechanism. b) Modeling of origami structure into a linkage mechanism.
www.advancedsciencenews.com www.advintellsyst.com torque generated at each joint after deformation can be expressed as given in Equation (1).
According to Kutzbach-Gruebler's equation, the driving force must be generated at N-3 or more joints to determine the shape uniquely. Each rigid bar is regarded as one link, and the torques and forces generated at each link are shown in Figure 2c. The x-directional component of the force generated at each joint is F ix , and the y-directional component is F iy . The torque balance at each link T total at link i is the sum of the torque generated at the joints at both ends of the rigid bars and the torque generated by the forces in the x-and y-directions at the joints at both ends, and can be expressed as given in Equation (2).
As the linkage mechanism after deformation is in a state of mechanical equilibrium, the forces in the x-and y-directions at all joints are equal, and the balance of torque at each link is zero. Therefore, it is possible to calculate the moment M i that is required at each hinge, where the driving force is generated, by solving the Equation (2) such that the balance of the equation becomes zero. This theory can be applied to micrometer-to centimeter-scale devices, which are typical in the field of selffolding and have sufficiently small mass relative to the driving force of the hinge.

Control of the Driving Force Generated on the Hinge
To control the shape of a structure that self-folds using a linkage mechanism, the moment M i generated at the hinge must be controlled. In general, the moment M in an elastic hinge is proportional to the folding angle θ of the hinge. Hence, if the folding angle of the hinge can be controlled, the moment generated in the hinge can be controlled. In this study, as a method to control the folding of the hinge, the area of the attached shrinkable layer was varied. The design of the test specimens and the cut pattern of the hinge are shown in Figure 3a. The material used for the flat part of the substrate must have sufficient bending rigidity to allow the structure to be modeled into a linkage mechanism. Therefore, materials with high strength, such as thick paper or substrates for electronic devices, are suitable, while materials that bend or stretch easily, such as thin elastomer film, are not suitable. The test pieces were fabricated by attaching a 12 μm www.advancedsciencenews.com www.advintellsyst.com thick heat-shrinkable film to a 33 μm thick polyimide copper substrate using double-sided adhesive tape. The hinge created the slits in an alternating pattern using a UV laser beam machine to locally reduce the bending rigidity. [21] This structure is expected to be applied to electronic devices because electrical connections can be maintained across the hinge. The width of the hinge was 3 mm, and the slit had a width of 0.1 mm in the shorter side direction of the beam structure. The vertical length of the hinge of the attached shrinkable layer, L s , was varied from 1 to 8 mm to change the attached area of the shrinkable layer. The test specimens of each hinge were heated in an oven at 75°C for 1 min to shrink the heat-shrinkable film, and the hinge self-folded. The hinges after self-folding at each length of the shrinkable layer are shown in Figure 3b, showing that the folding angle θ of the hinges increases as L s increases. The relationship between the folding angle of each test specimen and the length of the shrinkable layer L s , shown in Figure 3c, indicates a positive correlation. The graph plots the average value, with the maximum and minimum values shown as error bars. It was confirmed that there is a nonlinear relationship between the shrinkable layer length L s and the folding angle θ. A second-order approximate curve of the experimental values was used as the formula for the relationship between L s and θ, θ = f(L s ), to calculate the theoretical shrinkable layer length, L s , required for each hinge.

Experimental Verification of the Self-Folding Method
To verify the theory of the self-folding method using a linkage mechanism, two types of origami electronic devices were fabricated ( Figure S1, Supporting Information). First, a device with light-emitting diodes (LEDs) mounted on a structure that can be modeled as a simple 4-bar linkage model, as shown in Figure 4, was used for the verification. Figure 4a shows the fabrication process of the device, which was as follows. The substrate was processed; the LEDs were placed, following which mounting by heating was performed. Next, a shrink layer was attached from the backside, and the entire structure self-folded by heating again ( Figure S2, Movie S1, Supporting Information). A model of the 4-bar linkage model of this device is shown in Figure 4b. Theoretically, a 4-bar linkage has one degree of freedom, and hence, the shape of the structure is uniquely determined by determining the angle and moment of one or more of the hinges. This origami electronic device was designed so that only hinge 3 was folded in the valley-fold direction and the other four hinges were folded in the mountain-fold direction when the LED-mounted side was on the top. Therefore, by attaching shrinkable layers to the four hinges, except hinge 3, a driving force in the mountain-fold direction was generated, and this force was transmitted to the central hinge, which can be folded up into a valley-fold direction. The device fabricated by these processes is shown in Figure 4c. In this study, the driving force of hinges 1, 2, 4, and 5 was used, and the length of the shrinkable layer was the same for all four hinges. The designed fold angles θ T were 30°, 45°, and 60°, and the length of the attached shrinkable layer was calculated using Equation (2) and the relationship between L s and θ, θ = f(L s ). The cause of the error from the design value is considered to be the deviation from the real device and the model, such as the bending of the flat plate part, which was assumed to be rigid. These results prove that the theory of self-folding using the linkage mechanism can be used for origami devices. Next, the theory was verified using the origami device with a structure that can be modeled in the 6-bar linkage model. This device was based on a recent study of thermoelectric generators (TEGs) using origami structures for the substrate to make it www.advancedsciencenews.com www.advintellsyst.com flexible (origami-TEG) (8,9). In this study, the origami structure of the origami-TEG was fabricated by self-folding ( Figure S2, Movie S2, Supporting Information). Figure 5a shows the fabrication process of one unit of thermoelectric generators. The fabrication process was the same as in Figure 4a, the only difference being that the shrinkable layers were attached to the two outer surfaces of the upper and bottom substrates. The origami-TEG is a 6-bar linkage model, as shown in Figure 5b, when modeled as a 2D linkage mechanism, so the degree of freedom is 3 when there are no constraints on the hinges. Therefore, the number of driving hinges required to determine the shape is three, and the length of the shrinkable layer to be attached to each hinge is calculated using Equation (2) and the relationship between L s and θ, θ = f(L s ). The results of self-folding devices are shown in Figure 5c, which are designed with a substrate fold-up angle of 20°at the wiring areas of hinges 1, 3, 4, and 6. Although the shrinkable layer is attached at different positions and has different lengths on the upper and bottom substrates, the device is folded uniformly, indicating that the origami-TEG self-folds into the target shape. The relationship between the designed and measured folding angles for each hinge is shown in Figure 5d. Upon measuring the current-voltage (I-V ) characteristics, the origami-TEG showed a maximum output power of 0.43 μW with a heat source temperature of 70°C ( Figure S3, Supporting Information), confirming that the device works properly as a thermoelectric generator.

Discussion and Conclusion
In this study, we proposed a self-folding method to fold an entire origami structure by driving a few hinges, which is applicable to the fabrication of devices with an origami structure. This method allows for a higher degree of design freedom in self-folding, which is an effective approach for fabricating origami devices. Theoretical moments of each hinge necessary for self-folding from a certain shape to the target shape were calculated, and the driving force was controlled by the size of the active material (heat-shrinkable film). This control approach has the potential to be applied to self-folding with other driving materials. Selffolding methods using stimulus-responsive hydrogels, shape memory alloys, solders, magnetic materials, etc., can change the folding force at each hinge by changing the state of the components that generate the driving force. Therefore, it is expected that proposed driving force control method can be applied by experimentally clarifying the correlation between state parameters such as thickness, area, and volume of the added active material and the folding angle of the hinge. Based on this theory, we fabricated a device with an origami structure consisting of multiple links.
In the LED-mounted device, the top surface of the substrate was densely mounted with elements, making it difficult to attach a shrinkable layer. In the general self-folding method, the driving force is the difference in stress generated at the hinge due to the www.advancedsciencenews.com www.advintellsyst.com volume change of the active material in response to stimulation. Therefore, two strategies are generally used to control mountain and valley folding: 1) the active material is placed inside the substrate in advance and the hinge structure is used properly; 2) the position where the active material is attached is reversed vertically. With these strategies, it is difficult to self-fold origami structures with both mountain and valley folds by attaching the drive material only on one side of the outer part, indicating that this research extends the fabrication range of self-folding. In the fabrication of the origami-TEG, the upper and bottom substrates self-fold into a uniform-bellows fold, as designed, by attaching a shrinkable layer from the outside of the device after the elements were mounted. This demonstrates that even in a linkage mechanism, which generally has more than one degree of freedom, the proposed method enables self-folding to the target shape by controlling the driving force of the minimum number of hinges required to determine the shape. The proposed self-folding method can be applied to the fabrication of any device with an origami structure, making it easier to fabricate devices with self-folding. Although, in this study, a structure that can be modeled as a 2D linkage mechanism was selected as the device used to control the folding angle, the proposed approach can also be applied to an origami structure that can be modeled as a 3D linkage mechanism ( Figure S4, Supporting Information), and establishing a control method for hinge moments with a 3D structure can realize a self-folding system with high versatility and expansibility.

Experimental Section
Fabrication of Origami Devices: The materials used for the test specimens and origami devices were 1) a polyolefin film (Taiyo Electric Industry Co., HS-2520) with a thickness of 12 μm as the heat-shrinkable polymer for the shrinkable layer, 2) a film with a thickness of 8 μm copper deposited on a 25 μm thick polyimide as the nonshrinkable layer (Toray Advanced Materials Korea, Metaloyal), 3) a double-sided adhesive tape (NEION Film Coating Corp., NE-NCP3) with a thickness of 3 μm as the adhesive layer that laminates the shrinkable layer and the nonshrinkable layer. The hinge was cut using a UV laser machine (Osada Photonics International, OLMUV-355-5A-K). The width of the hinge was 3 mm, and the width of the cutting in the short direction of the beam structure was 0.1 mm. Chip LEDs (Samsung Semiconductor, Inc., SPMWHT541MD5WARMS2) and thermoelectric elements (Toshima Manufacturing, p-type: Bi 0.3 Sb 1.7 Te 3 , n-type: Bi 2 Te 3 , 2 Â 2 Â 1 mm) were mounted on a substrate coated with cream solder (Sunhayato Corp., SMXH05) and heated to 220°C on a hot plate. The shrinkable layer and double-sided tape were attached to a weak adhesive sheet and were subsequently laser-cut at a power sufficient to cut only the shrinkable layer and the double-sided tape, removing the unnecessary portions and transferring any remaining patterns onto the device. The fabricated device was placed on a tray made of aluminum foil and heated in an oven (Yamato Scientific Co., DK300) to self-fold.
Device Demonstration Setup: The LED-mounted origami device was supplied with a power supply (TEXIO Technology Co., PA18-3B) and was confirmed to emit light for several minutes. The thermoelectric generator with the origami structure on the substrate was placed on a Peltier temperature controller (VICS, VTH1.8K-70S), and the I-V characteristics were obtained by sweeping the voltage with a source meter (Keithley Instruments, 2614B). The characteristics were measured at 23°C under natural convection.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.