Lead the folding motion of the thick origami model under gravity

ABSTRACT Origami have been focused on by the researchers for its foldability and deployability owing to the huge potential in architectural engineering. However, origami was rarely used in practical construction projects as structures due to the lack of systematic investigation of motion control with respect to thickness and gravity. To clarify the lifting process of the origami structure, a preliminary folding test is carried out with a small-scaled single-DOF physical origami model. Folding paths are obtained by the non-contact measurement system and the force-displacement measurement system. The experimental results are confirmed by the numerical simulation of thick origami. The invariant relationship is observed between the folding process and the driving force. Furthermore, the final configuration of the model is found to be strongly related to the location of the suspension points. The methods to realize the complete folding are suggested. Moreover, the multi-unit model is simulated aiming for the application of rapid foldable structures.


Introduction
Origami has been applied in various fields, including aerospace engineering (Morgan, Magleby, and Howell 2016;Mathew, Bharatpatil, Anilchamoli, et al. 2021), bioengineering (Saito, Tsukahara, and Okabe 2016;Rogers et al. 2016), robotics (Shigemune et al. 2016;Peraza-Hernandez et al. 2014), and mechanical engineering (Klett and Middendorf 2016;Evans, Silverberg, and Santangelo 2015;Silverberg et al. 2014;Ye and Ji 2022). In the field of architectural engineering, origami also has potential to expand novel functions (Miura and Pellegrino 2020;Meloni et al. 2021). One of the most striking features is the foldability, which allows origami to perform as a sort of 4D structures. Using this feature, the concepts of deployable origami shelters can be quickly constructed for emergency cases. The applications include shelters (Alegria Mira, Thrall, and De Temmerman 2014;Thrall and Quaglia 2014) as temporary isolation rooms against Covid-19. Another important feature is the unique mechanical properties owing to the 2D-3D origami tessellations. These tessellation patterns lead to improvement in the lightness and strength of origami metamaterials (Schenk and Guest 2013;Xiang, Lu, and You 2020;Liu et al. 2015). Moreover, in terms of aesthetics in architectural design, the 3D unique form of origami offers non-structural functions. They were used as sunshades, canopies, and facades (Grinham et al. 2020;Le-Thanh et al. 2021;. Despite the competitive advantages, origami was rarely used in real movable architecture projects. One of the important problems was the difficulty in controlling the folding process. This issue has been noted as a significant task since Pinero proposed the first modern deployable dome in the 1960s (Pérez-Piñero 1961). Similar problems also occurred in the design of deployable frames (Kawaguchi, Yang, and Sone 2018), linkages (Zhang, Kawaguchi, and Wu 2019a), and movable bridges (Zhang, Kawaguchi, and Wu 2019b;Ario et al. 2013). Therefore, the modeling and design methodologies (Tachi and Hull 2017;Dudte et al. 2016;Cai et al. 2016) improved to deal with engineering factors. For example, elastic deformation was explained by bar and hinge models Filipov et al. 2017;Saito, Tsukahara, and Okabe 2015;Filipov, Tachi, and Paulino 2015). In these studies, nevertheless, the effect of gravity was neglected (Koschitz 2019;Gattas and You 2016), which led to the lack of knowledge on morphing control under gravity. To fill this gap, we proposed a numerical analysis method for simulating the folding process of thick origami model in the gravity environment (Zhang, Kawaguchi, and Wu 2018;Zhang and Kawaguchi 2021;Zhang, Kawaguchi, and Wu 2022).
Though it is a critical task to understand the practical folding process of thick origami in the gravity environment, as far as we know, none of them are systematically validated. An experiment with a physical model is needed to demonstrate the simulated folding process. Furthermore, the relationship between the folding process and the center of gravity was also not clarified in existing numerical studies. Consequently, there is no design consensus on how to define the driving position in the design of morphing control for building engineering. Since these problems are based on rigid folding that ignores elastic deformation, even a small-scaled physical model can reveal the morphing process. Therefore, we investigated in the most straightforward way: the folding test of a simple single-DOF physical model. This paper investigates the folding process of a thick origami model in the gravity environment by means of experiment and numerical analysis. For the first time, the folding paths of thick origami are physically demonstrated under gravity. As the study unit, a scaled physical model of Miura-ori pattern is fabricated and actuated by the suspension cables. Its folding behavior is obtained by non-contact measuring system and force-displacement measurement system. The invariant relationship between the actuating force and the displacement of the center of gravity is observed from the experimental result. Based on the experiment and numerical analysis obtained using methodology in (Zhang and Kawaguchi 2021) as a tool, discussion on the final configurations is carried out. It is found that the final shape can be controlled by means of designing the location of suspension points. This is useful information for the motion design of foldable structures. Moreover, two methods are suggested to realize the completely folded configuration based on the numerical simulation. The model can be compactly folded when the additional horizontal forces are introduced into the folding process. Since the magnitude of the horizontal force can be much less than the suspension force, this approach can be applied into multi-unit models. The discovery illustrated in this paper can be useful in the motion design of practical movable origami structures in the future.

Thick origami model
To clarify the folding process in the gravity environment, the well-known Miura-ori pattern is selected as the basic investigation subject. The thickness of the origami model is designed as shown in Figure 1 (Tachi 2011;Hoberman 1988Hoberman , 2010. For simplicity, the thickness is defined as the same value: t 1 ¼ t 2 ¼ t . Thus, from the view in y direction, the hinge lines are located at the z coordinate of 0, À t, and À 2t, respectively (Figure 1 (b)). Since the effect of the thickness is not the key point, the parametric study of the thickness is not discussed in this paper.
A physical model is assembled for folding tests. The topology of the physical model is designed identical to the numerical model introduced in Figure 1. It is a single-DOF physical origami model composed of six wood panels and metal hinges ( Figure 2). All the panels have the same thickness of t ¼ 10:0mm. Four of the hinges are fixed on side A, while five hinges are on side B. The total mass of the model is 705 g. To realize the difference of heights for the hinge lines, an additional thickness of t is designed at the middle hinge. Thus, it is higher than the other four diagonal hinges on side B. To avoid the penetration of physical material, the lines with metal hinges indicate the valley crease lines in the crease pattern.

Experiment setup
The folding process is planned to be actuated by the driving system as illustrated in Figure 3. The steel ramen frame made with L-shaped cross-section is fixed on the ground. Between the two columns of the frame, a white acrylic plate is set as the base to reduce the contact friction between the ground and model. Four suspension cables are connected to the panels through connection joints that are anchored to the panels. On each suspension cable, a turnbuckle is equipped to adjust their lengths separately. The other ends of the suspension cables are joined to the active cable. The active cable runs through the two rope guides that are fixed on the beam of the frame. Therefore, the four suspension cables are synchronized by pulling the active cable. Simultaneously, the origami model is expected to fold as in Figure 4.
It is notable that the driving force F i i ¼ 1; 2; 3; . . . ð Þ does not necessarily to be constant because we controlled the pulling speed by hand at approximately 10 mm/s. At such a low speed, the effect of velocity and acceleration is minimized. Therefore, the folding process can be seen as a sort of quasi-static procedure.
The moving process is tested using the measurement system, including (1) the non-contact measurement system and (2) the force-displacement measurement system.
The non-contact measuring system is based on the 3D measuring method with stereo cameras, which was also used to investigate the deployable truss structures (Wu et al. 2019;Zhang, Wu, and Guan 2016). Before measuring, camera information is collected with the Camera Calibration Toolbox for Matlab, created by Jean-Yves Bouguet (Bouguet 2004). Using the calibrated data, the 2D coordinates of arbitrary target points can be transformed into 3D coordinates ( Figure 5(a)).
The force-displacement measurement system is composed of the laser displacement sensor, the load cell, the logger, and the controlling equipment ( Figure 5(b)). The laser displacement sensor is fixed on the beam of the steel frame, while the load cell connects the suspension cables and the active cable (Figure 4(b)). The laser light is reflected by a flat board fixed on the top of the load cell. Thus, vertical displacement (in the z direction) and tensile force in active cable can be recorded simultaneously.
In the initial state, the model is designed to be fully deployed. To connect the model with the suspension cable, metal rings are anchored into the wood panels. They are divided into six groups and named from T 1 to T 6 according to the symmetry ( Figure 6). These rings can be utilized as the target points in the non-contact measurement system. In particular, the highlighted 6 points are served as the targets of the stereo cameras.

Results of the folding test
The folding test of the thick physical origami model is conducted using the driving system introduced in Section 2.1. During the folding process, the moving paths of the target nodes and the driving force are measured by the measurement system illustrated in Section 2.2. In this section, the suspension cables are attached to the rings of Group 1 (Figure 6).

Folding process
The folding motion is realized by pulling the free end of the active cable. The origami model is gradually lifted by the four suspension cables. The actuation continues until the model slightly leaves the base plate. Thus, the final configuration obtained in this study is the equilibrium state by only suspending force and gravity. This folding process is recorded by the two cameras (Figure 7). It is  clear that the folding process stopped even though it is not completely folded. To study the consistency in the folding test, three folding tests are conducted under the same experiment protocol. All of them have the same final stable configuration.
The displacements of the 3D coordinates of the target points are illustrated in Figure 8. The horizontal axis refers to the time during the folding test, whereas the vertical axis indicates the coordinates of the target points.
In x direction (Figure 8(a)), the six target points are separated in the initial state. When the model starts to fold, T 4 ; T 5 , and T 6 decrease, while T 1 ; T 2 , and T 3 keep the initial value. A slight peak occurs around Step 30, which is obvious for T 1 . In y direction (Figure 8(b)), the value of T 1 and T 6 is not strictly constant to 0 as expected. The reason for this inconsistency might be the rigid translational and rotational motion in the x-y plane.
In the z direction (Figure 8(c)), the initial and last values of the z coordinate are approximately 350 mm. Interestingly, even x and y directions show inconsistency, the three paths of z direction reach the same value in the end. This means that even the rigid motion occurs, the final configurations are still the same for the  three different tests. Therefore, the inconsistency in the x and y direction has no significant impact on the results in z direction.

Driving force in folding process
To clarify the mechanical performance during folding process, the physical model is investigated using the force-displacement measurement system. The displacements and tensile forces are obtained simultaneously by the laser displacement sensor and the load cell, respectively. Again, three tests are carried out to eliminate the unexpected error. However, the three displacement-time curves do not perfectly overlap (Figure 9(a)) owing to the difference in driving velocity. For the same reason, the three curves describing force-time relationship are not consistent although the folding process was conducted with the same experimental settings (Figure 9(b)).
To clarify the reason for the gaps of the three curves, we extract the displacement and force data and plot as Figure 10. When we set the displacements and the tensile force as the horizontal and vertical axes, respectively, the 3 curves almost overlap. This invariant relationship means that the individual tests match to each other if the influence of "time" is eliminated from the results.
Although the folding paths are different and independent, the equilibrium relationship maintains high consistency throughout the folding process. The result proves again that in this quasi-static experiment, the morphing velocity and acceleration will not affect the test result. Moreover, since the error comes from the human error due to the hand pulling speed, it can be avoided in the future research by means of actuators with constant pulling velocity.  (c) in z direction The folding process illustrated in Figure 10 can be divided into 3 stages: Stage 1, the driving force increases without movement. The model keeps the completely deployed configuration, while the gravity is balanced by the sum of the suspension force and the reaction force from the base plate. In stage 2, the model folds smoothly with a relatively gentle gradient until the displacement reaches approximately 305 mm. In stage 3, the force increases rapidly and the displacement stops at 353 mm, with a force of 6.86 N that is equal to the gravity of the physical model. In the practical construction of a single-DOF origami structure, the driving force could be controlled similarly if the force-displacement relationship is previously designed. The same principle can be a significant hint for the motion control of the complicated deployment process.

Comparison with numerical simulation
To confirm the results tested with the designed motion, the folding process obtained from the physically thick physical origami model is compared with the numerical analysis. Numerical simulation is conducted using generalized inverse theory (Zhang, Kawaguchi, and Wu 2022) based on the movable frame model (Figure 11). This kinematic calculation method describes the origami frame model and folds them quasi-statically by introducing the external loads. Gravity of the panels is introduced as the equivalent nodal forces directly onto the numerical model. For each panel, the magnitude of "gravity" is equal to the value of the "weight". The equivalent nodal force of an  arbitrary node has the same magnitude as the weight of the occupying volume. Since this methodology has been introduced in (Zhang and Kawaguchi 2021), the detailed equations will not be repeated in this paper. The actuators are also defined as the driving forces in the vertical direction.
To clarify the detailed folding behavior, the z coordinate of the center of gravity is calculated from the obtained nodal coordinates ( Figure 12). Again, the same coordinate system is used as shown in Figure 6. The initial value is close to 10 mm, which indicates that the model is in fully deployed state. The center of gravity increases during the folding procedure and finally stops at around 228 mm. This value is almost equal to that of the numerical result as illustrated by the dashed line. Thus, the tested data of the physical model are confirmed by calculated results.
Nevertheless, the horizontal axis of Figure 12 has different physical meanings between the test and the numerical simulation. This leads to difficulty in a straightforward comparison. An alternative way is to present the moving paths of the tested nodes and the results analyzed in the same 3D space (Figure 13). The measured data of the moving paths are plotted as the symbols, whereas the calculated results are illustrated as the curves. Despite the inconsistency in x and y directions owing to the rigid motion, the moving paths matched well in the z direction.

Configurations in the final state
The above investigation demonstrates that both the experiment and numerical simulation reach the semideployed final state. This stable configuration is equilibrated by the suspension force and gravity. The base plate no longer offers resistance force. Owing to the symmetricity (or singularity) of the origami model, the basic moment equilibrium equation equals to the force equilibrium equation. For example, the force equilibrium equation equivalents to the moment equilibrium equation in the case illustrated in Figure 14. The suspension force T can be divided into T x ; T y and T z corresponding to the global coordinate system. However, the horizontal terms T x and T y is eliminated in the derivation of the equations. Thus, the result shows that only T z contributes to the moment equilibrium equations. As a result, the final configuration cannot be explicitly determined by the traditional mechanical approaches using equilibrium equations.

Equivalent nodal forces for gravity
Driving forces as actuator  Since the final configuration of the model is related to the horizontal term of the suspension force and the thickness of the material, an alternative way to investigate the motion control and the final configuration is to analyze the folding behaviors corresponding to the actuating force. In this paper, the actuating force is introduced by the suspension cables. Therefore, the location of the suspension cable is discussed here. Comparison tests are carried out with loading points from Cases 1 to 6. The suspension points are determined along the ridge crease at the rings from Groups 1 to 6, respectively ( Figure 15). In cases 1, 2, and 3, the folding behavior is observed. The models stop at different final configurations. On the other hand, the Cases 4, 5, and 6 do not fold at all. The models are kept in flat shape while lifted. Therefore, we plot the displacements of the center of gravity in the z direction for only Cases 1, 2, and 3. The final value of z has a maximum value of 228 mm, 199 mm, and 170 mm in Cases 1, 2, and 3, respectively.
The final configurations are different with respect to the location of the suspension points. When the suspension points are set at the highest location, the structure is likely to be deployed to the final shape as Case 1. Otherwise, when the suspension points are slightly lower. The final configuration is also folded but in a slightly deployed condition as Cases 2 and 3. However, if the suspension points are in the gray zone, which is outside the centroids  of the peripheral panels, the folding process does not occur.

Considerations for complete folding
Concerning the practical use in architectural engineering, a compactly folded configuration has advantages in convenience in construction and transportation. Therefore, how to completely fold the model becomes an important point to discuss. From Section 4.2, the final configuration is relevant to the distance between the suspension points and the x-axis. This can also be confirmed using the numerical simulation. The three cases discussed in Figure 15 are demonstrated by the curves describing the numerical results ( Figure 16). The final values (illustrated by the curve symbols) are the same as the tested data. To realize the completely folded shape, two are suggested in this paper.
(1) Method 1: When the suspension point is assumed to be on the inner side of the panel, the folding path reaches a fully folded configuration (dashed line). This is an important feature of the origami fabricated by thick materials,  which has not been found before. Thus, one feasible way to fully fold the model is to adjust the suspension points inside the panel. However, because of the thickness of the material, it might be difficult to create enough space for the suspension points. Compared to the detail of Case 1, the two suspension cables overlap since the y coordinates are 0. Thus, the four cables appear to be only two cables. The drawback is that the details of the structures will be much more complicated. For practical use, it is also necessary to solve the connection method between the suspension cables and the panels.
(2) Method 2: A more feasible way is to introduce an additional horizontal force F H for the final configuration, for example, Case 1. Thus, the folding process can be divided into two phases. In phase 1, only the suspension force is introduced. The model reaches a stable configuration at Step 141. Then, the horizontal forces are introduced, whereas the vertical forces keep the constant value in phase 2. Finally, the model is compactly folded rapidly. The advantage of this method is that the magnitude of the additional horizontal force F H can be less than 5% of the suspension force in Case 1. This leads to convenience in construction if low-powered hydraulic jacks can be used in the folding and deployment process. In addition, this advantage results in the feasibility of folding the multi-unit models since the required horizontal force is low. It is expected to be useful in the construction technique of rapidly foldable large-span architectures.
Taking the model with four units as an example, the folding process is illustrated as Figure 17. Owing to the single-DOF feature, the model can be actuated by two sets of suspension cables. On the one hand, the folding process can be realized by increasing the tensile force of suspension cables, referring to Figure 10. After reaching the final state of phase 1, the model can be completely folded with horizontal forces, whose process is indicated by phase 2. Inversely, by decreasing the tensile force of suspension cables, the model can be deployed from the compact shape into a flat shape.
The above knowledge suggests that the selection of an appropriate suspension location is important. Nevertheless, the motion can be designed and controlled towards the expected configuration by introducing additional horizontal forces. Although this study is limited to the scaled physical model, the approach shows potential impact for the application of morphing structures in architectural engineering.

Conclusions
Through the experiment, numerical analysis, and discussion on folding process of the single-DOF origami model, the following results are concluded.
(1) The folding test of the physical model is conducted. The measured results show consistency with the data obtained from the numerical calculation. The folding paths are confirmed to be trackable in both experimentally and numerically.
(2) The relationship between the actuation force and the folding path is found to be invariant in the quasi-static folding process. (3) The final configuration is found to be strongly related to the location of centroids and the suspension points, regardless of the folding velocity or acceleration. (4) Two methods to realize the completely folded configuration are presented. As the more feasible one, the method with additional horizontal force is also useful for multi-unit cases.
Though the measured data are based on quasi-static processes ignoring the influence of the velocity and acceleration, it showed consistency to the numerical calculation. Since the practical construction process is also planned to be carried out slowly, the features discovered in this paper are helpful for understanding the principle of motion control. This study shows potential and gives inspiration for the motion design for the practical construction of origami structures in the future.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by JSPS KAKENHI Grant Number JP19K23547 and 21K14286.