Modular Self-Lock Origami: design, modeling, and simulation to improve the performance of a rotational joint

For origami assembly, joint constraints are necessary. Various shapes are created by cutting angles in different origami plates’ positions: any of the origami’s four central angles can be reduced in order to achieve the self-locking property. Although figure 1 demonstrates only one of these, 16 different configurations were considered: four where the central angle was subtracted from both sides of a single fold line (figure 2), and twelve other configurations shown in the supplemental material. Note that only cuts reducing the central angles are considered, as an origami could have various shapes around its edges without compromising its motion zare2021design .

The structure of the grounded plate 1, connected to plate 2, is equivalent to a simple fold (figure 4a). Therefore, the origami’s input angle, $\theta_{1}$, and input moment are the same as the rotational angle and the output moment of a simple fold structure with the same actuator. Different actuators could be used to increase the input angle limits. However, it is not necessary since the output angle changes most for $\theta_{1}$ near zero (figure 4). Both the semi-flat and the MPF state occur at values of $\theta_{1}$ near $90^{\circ}$. Thus, in order to obtain the full range of motion, there must be pouch motors on both sides of plates 1 and 2.

Various motion simulations were implemented using the contact solver of Autodesk Inventor 2019 to find an origami that could maintain a flat configuration between its input plates at its initial state and obtain the maximum rotational movement (optimal performance). Achieving a flat state by the origami’s input plates could offer a more simplified model and better control by the pouch motor’s actuator. The pouch motor will be able to maintain a zero pressure state (no inflation) and provide a known shape and dimensions at the initial state for the modelling.

In all simulations, plate 1 is a grounded link at a fixed position with its outer corner at the origin. Gravity is neglected for simplicity and consistency with the analytical model. The kinematics of the mechanism are simulated by sweeping the input angle between what the “semi-flat” and “maximum possible fold” (MPF) state. The semi-flat state is defined as the state where the absolute sum of all the angles between origami plates is at its maximum. Since the origami models to have zero thickness, they are kinematically capable of achieving fold angles more extreme than a real origami, so the maximum possible fold state is chosen to set the angle $\theta_{4}$ between the input and output plate to the constant MPF angle $\gamma\coloneqq 36.5^{\circ}$.

Figure 2 shows origami types defined by different cutting positions. The green and yellow plates are input and output plates respectively. The “initial state” pictured in figure 2 shows the closest angle that origami’s input plates could get to $180^{\circ}$ (flat form), and the “final state” is the MPF state. The motion of the outer corners of the input and output plate is shown using markers 1 and 2.

In order to obtain an origami which is symmetrical and also capable of reaching a state with fully flat input plates, origami type a is used in the remainder of the paper. If the flat state of the input plates is not considered, all the origami types provide almost the same amount of rotational motions in slightly different directions. This is discussed in the supplemental material (together with an expanded version of figure 2) using an algorithm described previously zare2021design . Additional fold types not pictured were also considered. Certain of these types could also reach a state with flat input plates, but type a is used due to its symmetry, which simplifies modelling.

The kinematics and dynamics of single-vertex rigid origami can be studied by noting that they are mechanically identical to spherical mechanisms bowen2014position ; zare2021design , for which a detailed theory exists chiang1988kinematics . To do this, the origami folds between the plates are treated as revolute joints. Then, since each plate of a single-vertex origami is assumed to be inflexible, the single vertex acts as a fixed centre about which the four plates move as the bars of a spherical four-bar mechanism.

The first quantity of interest is the kinematic relationship between the input angle $\theta_{1}$ and output angle $\theta_{4}$. The kinematics of a spherical four-bar mechanism are determined entirely by the angular lengths $\alpha_{12}$, $\alpha_{23}$, $\alpha_{34}$, and $\alpha_{41}$ of its links. Given these quantities, the input and output angles are related by the following equation chiang1988kinematics :

However, the structure of the Self-Lock Origami allows this equation to be simplified significantly. First, the angles $\alpha_{41}$ and $\alpha_{34}$ are both equal to $90^{\circ}$, which zeros several terms and simplifies others. The symmetry of the system allows the parameter $\alpha\coloneqq\alpha_{12}(=\alpha_{23})$ to be defined, so several more terms can be simplified and combined, resulting in a simple relation between $\theta_{1}$ and $\theta_{4}$.

Besides the relation between the input and output angles, the kinematics of the other angles $\theta_{2}$ and $\theta_{3}$ may also be of interest. Due to the symmetry of the origami, $\theta_{2}=\theta_{4}$, but $\theta_{3}\not=\theta_{1}$. Instead, $\theta_{3}$ is calculated using the following relation yang1964application :

Up to angular equivalence, the kinematic equations 2,3 each have two distinct solution curves, corresponding to the two configurations of the Self-Lock Origami. In the up configuration, $\theta_{4}$ ranges over positive angles from near zero to almost $180^{\circ}$, whereas in the down configuration $\theta_{4}$ ranges over positive angles from near zero to almost $-180^{\circ}$. In computations, the atan2 function is used to ensure that the returned value of $\theta_{4}$ corresponds to the correct configuration, as the single-argument $\tan^{-1}$ would result in a discontinuous curve. Using $\cos^{-1}$ to calculate $\theta_{3}$ returns a single continuous positive curve, so the result is simply multiplied by $-1$ in the down configuration.

We model the pouch motor as non-extensible with zero bending stiffness. If fully inflated while detached from the origami, it would take a cylindrical form with height $D$, but it is affixed to the origami in its flat state as shown in figure 3, resulting in a rectangular shape with width $L_{0}$ and height $D$. This length is split between plates 1 and 2, so we also define $L_{p}\coloneqq L_{0}/2$. The dimensions are chosen to fit the available space on the cut input plates, as shown in figure 3.

The moments that have been applied to (input moment) or generated by (output moment) the origami are modelled under the assumption of constant air pressures on the system’s pouch motor. The moments could be calculated using the origami’s input angle. A range of $-90^{\circ}$ to $90^{\circ}$ degrees has been selected for the input angle, $\theta_{1}$, due to the physical limitation of the pouch motor to produce even near $90^{\circ}$ degrees rotational angle. Both upward and downward movements can be created by adding two pouch motors on the origami’s input plates: one on top for $\theta_{1}:(0^{\circ},90^{\circ})$ range of motion, and another one on the bottom for $\theta_{1}:(-90^{\circ},0^{\circ})$ range of motion.

To estimate the input moment applied to a simple fold (a hinged structure), Niiyama et al. developed equation 4 niiyama2015pouch . The Self-lock origami’s input plates have a structure similar to a simple fold, so this equation can be used to calculate its input moment. Based on the proposed origami structure, the equation relies on the origami’s input angle $\theta_{1}$ and $\alpha_{12}$. Given the fixed pressure $P$ as well as the width $D$, half-length $L_{p}$, and central angle $S$ of the pouch, the input moment can be calculated as follows:

Here $S$ is derived from the assumption that the surface of the pouch acts as a section of a cylinder with constant curvature. The central angle is the angle subtended by the arc of this surface, which depends on $\theta_{1}$ and can be approximated well for $\theta_{1}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ as follows sun2015self :

$M_{\text{input}}$ and $S$ also depend implicitly on the dimensions of the rectangular uninflated pouch. All four plates of the origami were originally squares of side length $m=25mm$, but the reduction of central angles to $\alpha$ on the input plates reduces the space available for the pouch, as illustrated in figure 3. The length of the top side is reduced by a length $L_{1}=m\cot\alpha$, and the misalignment of the right angles of the plates and the pouch means that the top right corner of the pouch touches the side of the plate at one point a distance $n=(m-L_{1})\cot\alpha$ from its corner. The pouch length $L_{p}=L_{0}/2$ and width $D$ are then computed geometrically as follows:

Finally, the mechanical advantage of a generic spherical four-bar mechanism has been shown to be calculated as follows (adapted to a different naming convention for central angles) yang1965static :

The rotational manipulator shown in figure 6 consists of two origami units, connected by a welded joint between one of the output plates of the first origami and one of the input plates of the second origami. Connecting them on more than one plate could jeopardise their mobility; the connection method described here leaves each origami unit with a single degree of freedom. There is a possibility of a collision between the first origami’s output plates and the second origami’s input plates while the manipulators move. To avoid this, the plates are cut at a slight angle in those problematic areas (figure 6Ab). Cutting them does not affect the manipulator’s movements because the plates’ central angles are unaffected. The motion of the manipulator is given in terms of the position of a designated end effector on one corner of the second origami’s output plate, indicated in figure 6A with the red dot labelled “marker”.

Both of the origami making up the manipulator are constructed in the downwards configuration. Figure 6A gives the coordinate system of manipulators and shows the curling motion in 3D. Figure 6B shows the planar projection of this movement onto the $x$-$z$, $y$-$z$, and $x$-$y$ planes. This manipulator is designed to produce a simple rotation about the $x$ axis, so the end effector does not move in this direction. However, due to the change in the geometry, the end effector moves further from the $x=0mm$ line with decreasing $\alpha$. The greater the reduction of the central angles (i.e. the further $\alpha$ is from $90^{\circ}$), the shorter the range of motion on the y-axis. The $89^{\circ}$ and $85^{\circ}$ degrees manipulators have similar and very close rotation around $x=0mm$ due to their very small central angle cuts. Due to defining the common MPF state at output angle $\gamma\coloneqq 36.5^{\circ}$ for all manipulator models, their final states and positions in all plots are the same.

The five manipulators with different values of $\alpha$ follow the overall trajectory but with a different initial state, so they all have the same range of motion on the $z$-axis. The larger their central angle cuts are, the bigger the inaccessible configuration area near their flat state. Therefore, their semi-flat configuration (initial state) starts farther away in the range of motion. The movements of different rotational manipulator models in the 3D space are demonstrated in figure S8 in the supplementary material.

The translational manipulator shown in figure 7 is constructed from four separate origami units alternating downward and upward configurations. This is necessary in order to enable the “zigzag” state depicted in figure 7Aa. To obtain the closest possible motion to a linear translational movement, the manipulator’s geometry is required to be symmetrical. Figure 7Ac shows the location of these connections and cuts.

Similar to the rotational manipulator, the origamis are connected on only one of their plates to retain the full number of degrees of freedom, and some perimeter cuts are made to avoid overlapping between the origamis’ movements. This manipulator is the only case where plate length deviates from the previously fixed 25mm, as indicated in figure 7Ab. The first and last plates have length 25mm, but the lengths of other plates were derived geometrically. The MPF angle $\gamma\coloneqq 36.5^{\circ}$ and the desired height $d=25mm$ of the origami in the MPF state are used to find the plate lengths $f=d\cot\gamma$ and $q=d\sec\gamma$.

Figure 7A shows the initial and final states of the $89^{\circ}$ manipulator’s translational movement in different views. All origami units start in a semi-flat state and must be actuated simultaneously in order to achieve the depicted straight-line motion. The first and the last joint are folded until reaching a $90^{\circ}$ output angle. The second and third joints’ final configurations are MPF. Therefore, as for the rotational manipulator, varying $\alpha$ affects the initial but not the final position.

The two-dimensional plots of the manipulators’ movements using a marker and defined reference frame are presented in figure 7C. The main translational movement occurs along the $y$ axis, but the end effector does also move somewhat in the $z$ axis. In the $y$-$z$ plane depicted in figure 7Ba, the $89^{\circ}$ manipulator stays closest to the $z=0mm$ line. The farther the origami models get from $\alpha=90^{\circ}$, the further their semi-flat state gets from flat. As a result, the range of motion in the $z$ axis increases and the $y$ axis decreases, and their movements changes from a translational to a somewhat more rotational motion. Among all the translational manipulators, the $89^{\circ}$ model produces the closest to ideal translational movement.

Both manipulators could be combined or their constituent origami units could be actuated in other permutations in order to obtain a combination of translational and rotational motions which could be used for complex tasks. Using the origami design in a modular structure could provide the opportunity to create and mimic traditional robots’ complex movements. This could be useful in exploration, where various motions are required in a limited space. This section explores the possibility of a modular manipulator which combines Self-Lock Origami units in an arbitrary way.

A modular manipulator can be built up from Self-Lock Origami units using two different basic construction cells, depicted in figure 8A. The first consists of two origami connected directly to each other with welded joints and small cutouts to avoid self-collision as discussed previously. The second connects two origami via a rigid “bounding plate” interposed between them. This enables connecting the origami plates at different angles to change or increase the manipulator’s workspace. Bounding plates could be designed in different shapes to help the origami robots to explore different locations, but in the present work only considers a square plate equal in size to the origami units. In the binding phase, constraints are defined for the movements in 3 directions or welded joints to create configurations 1 and 2. Then, the last origami joint is connected to a base plate which serves as the ground link.

An example of such a modular manipulator made of $89^{\circ}$ degrees origami joints is presented in figure 8. By defining different rotational motions on the origami joints, the manipulator could take various shapes in the simulation. Figure 8B shows the manipulator’s compact and semi-flat states with two $89^{\circ}$ origami connected at a right angle using a square bounding plate. In the semi-flat state, none of the origami have been activated, and the manipulator is in the closest possible configuration to the flat state. In the compact state, all joints have been activated and folded to the MPF configuration to minimise the overall size of the manipulator. Figure 8Ba additionally indicates the location of the end effector and the coordinate system.

The plots in figure 8C present various possible trajectories which could be taken by this modular manipulator in each of the two-dimensional projections. Each trajectory corresponds to a different sequence of joint activation, indicated in the legend as a sequence of joint numbers. Each joint in the sequence folds from the semi-flat to MPF state, with each fold stopping early if a collision occurs. For instance, activated joints 1-2 refers to folding of joint 1 and then folding joint 2 without overlapping joint 1. The joint activation orders 1-2-3 and 1-2-3-4 have similar motion since after joints 1, 2, and 3 are activated, joint 4 does not have much space for folding.

Across the possible joint folding orders shown, the marker exhibits a large range of motion and a variety of different directions of motion in all three planes. If only the joints on one side of the bounding plate are activated, the end effector trajectory is a a straight line. The joint combination 1-2 generates straight-line movement in both the $x$-$y$ and $x$-$z$ planes, whereas the joint combination 3-4 produced straight-line motion in the $y$-$z$ plane.

One of the advantages of using origami in modular manipulators is the ability to convert them into structures with small volumes due to the joints’ geometries. Manipulators can achieve this by moving their origami into MPF or semi-flat states in different directions. Also, depending on the application, some of the manipulator’s joints can be excluded in order to produce different manipulators. This can even be enforced temporarily, without modifying the manipulator, by simply folding the joints using their actuators and holding them in that state.