Abstract

Origami patterns have been used in the design of deployable arrays. In engineering applications, paper creases are often replaced with surrogate folds by providing a hinge-like function to enable motion. Overconstraint observed in multivertex origami patterns combined with imperfect manufacturing may cause the resulting mechanisms to bind. The removal of redundant constraints decreases the likelihood of binding, may simplify the overall system, and may decrease the actuation force by reducing friction and other resistance to motion. This paper introduces a visual and iterative approach to eliminating redundant constraints in origami-based mechanisms through joint removal. Several techniques for joint removal are outlined and illustrated to reduce overconstraints in origami arrays.

1 Introduction

Origami, the ancient art of paper folding, has inspired many engineering designs from many different fields. Deployable origami-based mechanisms can stow compactly and deploy into a large configuration. Deployability is achieved through a systematic arrangement of connected panels, or origami arrays. A special category of arrays, origami tessellations, are often used for deployable applications because of their symmetric geometric fold patterns and/or their ability to be repeated linearly or radially.

Deployable origami has inspired designs for areas such as deployable antennas [13], medical devices [46], robotics [7], metamaterials [8], and architecture [9]. Deployable origami arrays have also recently gained extra attention when designing reconfigurable antennas as they are inherently physically reconfigurable [1014].

Designs for these applications generally include mechanisms with many panels and joints to “fold” as origami does. Origami patterns such as the Miura-ori, Tachi–Miura, twist tessellation patterns, and their variations have the behavior of storing compactly and deploying to a large area while also having a single-degree-of-freedom (DoF) [15,16]. Single-DoF mechanisms are attractive for such applications because the entire mechanism can be actuated with only one input, thus reducing the number of actuators and the corresponding cost, weight, and complexity.

Origami can be modeled as connected sets of spherical mechanisms [17], where each vertex corresponds to the center of rotation of an individual spherical mechanism. These spherical mechanisms, each with a loop constraint, are then connected into a single system of mechanisms by sharing links/panels. The combination of constraints between the individual mechanisms often results in redundant constraints, making the system overconstrained. Overconstrained mechanisms are an important topic in mechanism analysis and design. Multivertex origami patterns are overconstrained when at least one hinge is shared between vertices. These redundant constraints have no effect on the intended kinematics of the system when the geometry is assumed to be perfect.

However, when fabricated, imperfect joint axis alignment and placement render the kinematics nonideal. As a result, these systems can exhibit an increase in unintended resistance to motion or can completely bind during motion. Undesirable internal loads can also result from these nonideal conditions. In applications such as deployable space arrays, it is vital that the mechanism’s actuation force be predictable and minimized. Required forces to actuate these mechanisms could be decreased by reducing the number of joints, particularly when compliant joints (such as flexures and flexible membranes) are used. It is also important that the mechanism’s motion be uninhibited throughout deployment, meaning that it does not bind or meet internal resistance to motion. Deployable space arrays must exhibit reliable and predictable motion, ease of actuation, and low unintended resistance to desired motion. Reducing overconstraint in an array by removing redundant joints could lead to improved performance in deployable origami-based space arrays and other multiloop mechanisms.

Removing redundant joints could lead to additional benefits. Many applications that utilize the foldability of deployable origami arrays such as antennas [18] and solar panels [1921] seek to maximize usable area once deployed. Joints used to enable the folding motion of the arrays may take up usable area that could be used for communication [22] or power generation, as well as add to the weight and stored volume. Research has been done to reduce area taken up by hinges and the panel gaps created [23]. The elimination of redundant joints would leave more area to be utilized productively and could result in improved performance for deployable space arrays.

The objective of this paper is to identify overconstraint in origami-based mechanisms and propose techniques to eliminate redundant joints within an array while retaining intended motion.

2 Background

To better understand the techniques that will be discussed, we review the fundamentals of origami-based mechanisms. Spherical mechanisms are explained with their connection to origami patterns. General mobility criteria and overconstraint are defined and discussed.

2.1 Origami and Spherical Mechanisms.

In some definitions, origami is an implementation of folds into a single, uncut sheet of paper. The patterns used here are based on origami patterns, and once cut they could also accurately be referred to as kirigami.

A spherical mechanism is a mechanism where all axis of rotational joints point toward and intersect at a specific point in space [2426]. It has also been shown that an origami vertex can be modeled as a spherical mechanism, and many origami patterns are made up of interconnected vertices or spherical mechanisms [17,27,28]. Just as the kinematics of spherical mechanisms are defined by link angles [24], sector angles, denoted by α, define the geometry of an origami vertex. Fold angles, denoted by γ, define the fold angles between relative panels, as shown in Fig. 1. For a symmetric bird’s-foot, it can be shown that due to symmetry, γ1 = −γ3 and γ2 = γ4 [29]. Figure 1 also shows a fold pattern with two connected vertices, denoted by subscripts a and b. For connected vertices, we know that certain fold angles must be the same because panels are shared between vertices. For example, for the degree-4 symmetric bird’s-foot pattern shown in Fig. 1, panels a1 and a4 are the same panels as b2 and b3 respectively, thus γa1 = −γa3 = −γb1 = γb3.

2.2 Mobility of Mechanisms.

Mobility (M), also called the degrees-of-freedom, is used to predict how many independent parameters must be controlled to define the motion of a mechanism. When the mobility is greater than 0 (M > 0), the connected links are “mobile.” When the mobility is less than or equal to 0 (M ≤ 0), the mechanism becomes a structure, being rendered immobile, because there are more constraints than there are degrees-of-freedom. The development of an equation that predicts the mobility of a mechanism has been researched for over 150 years, with many adjustments proposed over the years [30].

A commonly used mobility equation is the traditional Chebychev–Grubler–Kutzbach mobility criterion, defined as
M=λ(NJ1)+i=1jfi
(1)
where M = predicted mobility of mechanism, λ = 6 for spatial constraint space and 3 for planar/spherical constraint space, N = number of links, J = number of joints, and fi = degrees-of-freedom permitted by joint i.
For a spherical/planar mechanism with only revolute and prismatic joints, Eq. (1) simplifies to
M=3(N1)2J
(2)
For a spatial mechanism with only revolute and prismatic joints, Eq. (1) becomes
M=6(N1)5J
(3)

In the application of origami design, a single origami vertex could be modeled as a spherical mechanism where axes of hinges intersect at the vertex [17,27]. The number of degrees-of-freedom in a vertex of degree n is n − 3 [29]. However, when we look at a multivertex origami tessellation as a whole, symmetry and periodicity reduce the overall degrees-of-freedom. In other words, due to special geometry and redundant constraints, the origami mechanism has more degrees-of-freedom than Eq. (1) would predict.

Taking these considerations into account, a few mobility equations for origami-based mechanisms have been proposed.

Tachi et al. proposed that for a general bird’s-foot crease pattern, such as the Miura-ori, the mobility can be defined as
M=B3H+S3k=4Pk(k3)
(4)
where B is the number of edges on the border of the pattern, H is the number of holes in the pattern, S is the number of redundant constraints, and Pk is the number of k-gon facets [9,29].

While traditional mobility criteria would under-predict the mobility of origami mechanisms, this equation takes into account redundant constraints in the system. Despite Eq. (4) including a term for redundant constraints, S, it can be difficult to identify the exact number of redundant constraints within a pattern. Equation (4) is also limited to crease patterns which satisfy the “bird’s-foot condition.” Lang outlined the bird’s-foot condition as a vertex with (1) a set of three creases of one fold assignment, separated sequentially by angles strictly between 0 and π, and (2) one additional crease of the opposite assignment [29].

Proposed methods for predicting the mobility and overconstraint of origami and polyhedral mechanisms have also included screw-based mobility methods [3133], group-theoretic approaches [34], as well as methods based on the Euler–Rodrigues formula [35].

Yu et al. showed that an adjacency matrix could be used to define the number of degrees-of-freedom in any rigidly foldable origami pattern with multiple vertices [36]. This works very well for traditional origami patterns which include uncut fold patterns. However, once cuts are introduced into the pattern, the adjacency matrix method is limited to single cuts between two vertices and cannot predict DoF of patterns with holes involving external vertices or more than two internal vertices.

2.3 Overconstraint

2.3.1 Overconstrained Mechanisms.

While many adjustments have been made to the original Grubler equation [30], the Chebychev–Grubler–Kutzbach criterion (Eq. (1)) can be shown to predict mobility inaccurately for both single [37,38] and multiloop systems [30,3941]. This inaccuracy is due to overconstraint in the analyzed mechanisms.

Overconstraint is when there exist more constraints than there are degrees-of-freedom in a system [37,38,42,43]. While overconstraint can mean that a linkage becomes immobile, some special geometric conditions, such a symmetry and angular relations, allow motion even when mobility is predicted to be less than one. A constraint that is redundant can be removed without changing the mobility or motion of the mechanism [44,45]. An overconstrained mechanisms is one that has more degrees-of-freedom than is predicted by its mobility equation [37,38,42].

For example, the degrees-of-freedom of the planar linkage system shown in Fig. 2(a) can be calculated using Eq. (1) as M = 1. This is defined as an exactly constrained system, where it is predicted to have a mobility of 1 and is observed to have a mobility of 1.

Now consider the system in Fig. 2(b) where a link is added and is parallel to the other vertical links. The mobility becomes M = 0. Here, the Chebychev–Grubler–Kutzbach criterion (Eq. (1)) predicts it to be an overconstrained system with a mobility of 0. However, due to special geometry, this system still has a mobility of 1 and can be classified as mobile overconstrained due to the redundancy in the constraints. We could continue to add more parallel links to the system in the same manner and we would see the predicted mobility decrease, but the actual observed mobility remains the same.

Similar to this planar example, it has been identified that overconstraint is observed in connected spherical systems, such as origami [46]. Tachi showed that any quadrilateral mesh origami pattern is an overconstrained system because the number of constraints around degree-4 vertices (three for each vertex) exceeds the number of variables (the number of hinges) [46]. However, despite being overconstrained, many quadrilateral mesh patterns are still mobile. Similar to planar link redundancy, as shown in Fig. 2, quadrilateral mesh patterns such as the Miura-ori also have redundant constraints, enabling their motion.

2.3.2 Challenges With Overconstraint.

Overconstrained mechanisms often require mathematically perfect geometries. When analyzed assuming perfect geometry, redundant constraints are observed and motion is allowed. However, without perfect geometry, the previously redundant constraints become conflicting constraints and render the linkage immobile. The conflicting constraints can then limit the mechanism’s motion (by locking or binding), cause choppy motion, induce excessive internal loads, and can lead to fatigue failure [47,48]. For example, Fig. 2(c) shows an overconstrained planar mechanism that is rendered immobile by imperfect joint placement.

Imperfect geometry can be caused by imperfect tolerancing, imperfect hinge placement, and thermal expansion differences. Clearances can be added into hinge designs to allow enough motion for the joints to line up, even when placed imperfectly [49]. Another approach is to introduce compliance so that the system can flex. While clearances and compliance can allow a mechanism to move in the desired way, it also can introduce unwanted motion in other directions, making the motion less predictable.

In many deployable array applications, it may be beneficial to replace an overconstrained system with an exact-constrained system design to eliminate or minimize these problems. Specifically, origami-based deployable designs would benefit from the minimization of redundant constraints and knowing where to remove those constraints.

3 Recognizing Overconstraint

While it may be easy to recognize overconstraint in a planar mechanism, it can be difficult to identify overconstraint in spherical mechanisms. The purpose of this section is to provide the reader techniques that can be used to recognize different overconstraints within an origami pattern.

3.1 Geometric Overconstraint.

An origami pattern containing multiple vertices can be visualized as a system of connected spherical mechanisms, with each vertex within the pattern being the center of its own spherical mechanism. Redundant constraints can be found within multiloop origami patterns. When individual spherical mechanisms are joined, loop constraints are created and redundant constraints are added. Generally, the number of redundant constraints is defined as the difference between the observed mobility and the mobility predicted by the mobility equation [37,38,42,43]. For rigid origami, the mobility criterion can be simplified to
M=J3V
(5)
where J is the number of internal joints, and V is the number of internal vertices [36]. When special geometry exists, such as symmetry and periodicity, the observed mobility will be larger than that predicted by Eq. (5).

The difference in the observed and predicted mobilities indicates the number of redundant fold angle constraints within the pattern. For example, consider a degree-4 vertex within an origami pattern with a negative predicted mobility. A degree-4 vertex requires one fold angle be constrained to define the position of all panels in the vertex. Loop constraints from adjacent vertices may define more than one fold angle within the degree-4 vertex. Since only one fold angle constraint is required, the redundant constraint may be removed. These are the redundant constraints identified using Eqs. (4), (5), and the adjacency matrix method [36].

As an example, consider the Tachi–Miura pattern shown in Fig. 3. It can be shown using multiple mobility criterion and techniques that the predicted overconstraint (S) in the origami pattern is S = 2 (see Table 1). Mobilities listed in Table 1 are predicted using Eqs. (4), (5), and the adjacency method as presented in Ref. [36].

3.2 Intervertex Spatial Overconstraint.

Additional overconstraints can be identified apart from those given by Eqs. (4), (5), and the adjacency matrix method [36]. Consider the example of the two-vertex origami pattern shown in Fig. 4 taken from a larger origami tessellation. The adjacency matrix method [36] and Eq. (4) predict a mobility of 1, with no redundant constraints. Equation (2) predicts a mobility of M = 1, giving no indication of overconstraint.

However, when we analyze this as a spatial mechanism using Eq. (3) the mobility is predicted to be M = −5, implying some overconstraint. Removing one joint from the mechanism, Eq. (3) results in a mobility of 0. This implies that while we have removed one joint, the mechanism is still overconstrained. Due to panel rigidity and the remaining intact joints, the split edges cannot move relative to each other [50]. This resulting mechanism can be classified as a symmetric double-spherical 6-bar mechanism, which has been shown to be an overconstrained mechanism of mobility 1 [38,43].

While Eqs. (4), (5), and the adjacency matrix method [36] indicated no overconstraint, one hinge between the two vertices is redundant. Due to panel rigidity and the remaining intact joints, the split edges cannot move relative to each other [50].

3.2.1 Visual Representation.

It can be difficult to identify this intervertex spatial overconstraint. The remainder of this section introduces a visual representation of spherical mechanism constraints in a planar mechanism analog to visually identify this overconstraint.

Consider two adjacent vertices within an origami pattern that are rigidly connected by two shared panels, as shown in Fig. 4. For this example, we will use two symmetric degree-4 vertices.

Now consider a special spherical mechanism case where the point at which the axes intersect is infinity. The axes are now parallel to each other, turning the spherical mechanism into a planar mechanism, similar to that done by Wiener [51]. The ratio between sector angles may be represented by a ratio between link lengths. The spherical mechanism in Fig. 5(a) can be visualized as a planar mechanism shown in (b). However, we must conserve the angular constraints exhibited in spherical mechanisms. Because γa1 = −γa3 = −γb1 = γb3, we know that the angles between links must follow the same relation.

Maintaining the pin joints and angle relations, the system is overconstrained. If all hinges are placed in their correct locations, motion of the mechanism is not limited. However, consider if the center joint were slightly off-center as shown in Fig. 6. Physically this could be due to poor manufacturing, poor tolerances, thermal expansion, etc. The previously redundant constraints of the center joint now conflict with the angular constraints, and the mechanism becomes a structure. This can be resolved by removing the redundant constraint, the center pin joint. With the removal of the center joint constraint, the mechanism is still fully constrained.

Applied to the original two-vertex origami pattern, it is observed that the joint connecting the two vertices can be removed, resulting in the pattern shown in Fig. 7. This joint is removed while retaining the original single-degree-of-freedom motion.

Figure 8 compares this process of identifying redundant constraints between planar mechanisms and spherical mechanisms found in origami.

Although this type of overconstraint is not captured in many mobility criteria, it can be found in multi-internal-vertex origami patterns.

4 Techniques to Reduce Overconstraint

This section will introduce various techniques that may be used to reduce geometric overconstraint and intervertex spacial overconstraint. Due to simplicity, techniques for the intervertex spatial overconstraint will be explained and illustrated. Then various methods for reducing geometric constraint will be introduced, illustrated, and discussed.

4.1 Intervertex Spatial Overconstraint.

Because the constraint imposed by the hinge between the two vertices is redundant, the joint can be removed while remaining fully constrained. In addition, removing the joint does not change the mobility due to panel rigidity and angle constraints between vertices. Because the panels in the vertices are rigid and other joints are still in place, edges of the panel where the joint was removed are still kinematically defined and cannot move relative to each other. Figure 9 shows a general case of two internal vertices (A and B) where the joint between them can be removed.

Yu et al. calculated the mobility of a “ring pattern with six creases,” six panels, and two local spherical centers from the dimension of the null space of the Jacobian matrix [52,53], showing it has a mobility of 1 [36]. Essentially, this pattern can be seen as two adjacent degree-4 vertices with their shared fold removed. While this was shown with two degree-4 vertices, this can be extended to vertices of other degrees. Removing the joint between two vertices of any degree does not increase its observed mobility [50].

While not used for the purpose of removing redundant constraints, this single internal joint removal has been shown to work in previous research. Lang et al. showed that the center joint is unnecessary in a prototype for a new thickness accommodation technique and its removal can simplify geometry [54]. Single internal joint removal has also been shown to enable motion in a hinge-shifted thick hexagon twist [55].

This can be extended and multiple joints within a large origami array can be removed together. By analyzing each pair of adjacent internal vertices within the origami pattern, joints between vertices can be removed as described earlier. Note that analysis of each pair of internal vertices can be done independently of any surrounding previous cuts. In other words, a joint between two vertices can be removed as long as (1) the cut does not touch the edge of the pattern (which would require an external vertex) and (2) cuts do not touch each other (as two cuts touching each other would no longer be a cut of length one).

With these conditions, it can be shown that the number of single internal joints that can be removed, R, is
R=V2
(6)
where V is defined as the number of internal vertices and brackets denote the use of the ”floor” function.
Using Euler’s polyhedral formula [56] to solve for the number of internal vertices [9,31,57], and putting into terms of Eqs. (2) and (3), it can be shown that
R=V2=J+1N2=M3M66
(7)
where M3 is the mobility estimate from Eq. (2) and M6 is the mobility estimate from Eq. (3). Note that the relation V = J + 1 − N can also be obtained by combining Eqs. (2) and (5).

An example of this is shown for the Miura-ori pattern and a hexagon twist pattern in Fig. 10. The mobility of these patterns both before and after the joint removal is one. Since any joints can be removed while following these conditions, there may be many permutations of cuts that can be made for a single pattern. For example, Fig. 10(a) shows two combinations of cut joints for a Miura-ori pattern.

Prototypes of a thickened Miura-ori and hexagon patterns, diagrammed in Fig. 10, were made to illustrate and support the mobility predictions earlier. Joints were removed as shown in Fig. 10, and a portion of each panel adjacent to each cut hinge was removed to visually emphasize the hinge removal. The mobility is illustrated in Fig. 11 with the prototypes placed into closed, intermediate, and open configurations.

4.2 Geometric Overconstraint.

Equation (7) calculates the minimum number of cuts that may be made within a pattern without increasing its mobility. The following section outlines techniques that may be used to increase the number of removed joints from a pattern.

4.2.1 Joints Adjacent to Edge.

While single joints between two vertices can be removed without increasing mobility, if a joint adjacent to the edge is removed, the mobility prediction increases.

Yellowhorse et al. showed that this can be used to make immobile rigid folding systems mobile [50]. It was shown that for a group of n vertices in an origami crease pattern, if one crease connecting a vertex to the edge of the pattern is removed, the mobility of the system increases by 2 [50].

Using this technique, we can remove redundant hinges in an overconstrained system. The location of the removed creases is very important to maintain 1-DoF.

When removing joints between two exterior panels, each panel must have at least two additional joints to fully constrain the panels. In other words, any exterior panel with only two connecting joints is not a candidate for joint removal. Any removal of joints would leave the panel with only one joint constraint, allowing the split edges to move relative to each other. An example of this is illustrated in Fig. 12. Location A is not a candidate for removal because that would result in an under-defined panel (highlighted) meaning that the panel can move separate from the rest of the 1-DoF system. Location B is a candidate for single joint removal because the separated panels (highlighted) are still fully defined through other intact joints.

The predicted mobility and predicted overconstraint for this pattern are shown listed in Table 2. For a 3 × 4 Miura-ori pattern, mobility is predicted to be 1 and it has two redundant constraints. After cutting the joint at location B, it can be calculated that the mobility remains at 1, but the overconstraint becomes 0. These mobility predictions can be made through modeling using multiple mobility criterion (see Table 2).

Notice that even with the pattern being cut at location “B” and the overconstraint prediction at 0, additional joints can be removed as explained in Sec. 4.1.

A thickened Miura-ori prototype of the pattern in Fig. 12 was 3D printed from polylactic acid (PLA) with joints made from spinnaker tape. Only joint location “B” was removed and was emphasized by removing sections of adjacent panels. The prototype is shown in Fig. 13. The observed mobility of the mechanism remained unchanged due to the removed joint.

4.2.2 Removing Multiple Joints From Around an Internal Panel.

Yellowhorse et al. also showed that multiple creases could be split around a single internal panel as a method to increase mobility of a nonrigid foldable pattern [50]. In a similar manner, we can remove consecutive creases around an internal panel to remove overconstraint while maintaining 1-DoF.

Consider the hexagon twist pattern shown in Fig. 14(a). Uncut, the mobility is predicted to be 1 with 1 redundant constraint. This overconstraint may be removed by making two consecutive cuts around the center panel, as shown in Fig. 14(a). It can be difficult to verify the mobility of an entire mechanism without considering sections of the pattern separately. For example, in a single pattern, half of it might be overconstrained while the other may have more than one degree-of-freedom. However, the overall mobility prediction would not reflect the actual mobility of the mechanism accurately. As such, sections of patterns may need to be analyzed separately for local mobility.

For this hexagon twist origami pattern, sections can be analyzed separately as shown in Fig. 14(b). The pattern is split by creating a section with panels adjacent to the multijoint cut. The second section is made from the remainder of the pattern and the connecting panels shared with the first section. For this pattern, it is split into two sections, A and B, each including the shared panels (see Fig. 14).

Predicted mobilities and overconstraints for the overall uncut hexagon pattern and each separate section are reported in Table 3. We can see that section A is underconstrained, and section B is exactly constrained. When combined into one hexagon pattern, the 1-DoF motion of section B outputs the needed 2 DoF section A requires, resulting in the pattern shown in Fig. 14(c) with an overall mobility of 1. In other words, the 2-DoF section is driven by the 1-DoF section for an overall mobility of 1.

For this pattern, we cannot cut more than two consecutive cuts around the center panel or it would result in an overall mobility greater than 1. However, it is possible to cut more than two consecutive panels around a single internal panel, as long as the center panel remains defined (with at least two noncolinear hinges), and the separated sections result in an overall mobility of 1.

A prototype was fabricated to illustrate the predicted overall mobility of 1. The prototype was made from 3D printed PLA, with joints made from spinnaker tape as a membrane hinge [58]. Because the model has a finite thickness, a thickness accommodation technique is required to enable the motion. For this prototype, the offset panel thickness accommodation technique was used [59]. The assembled prototype can be seen in Fig. 15. Removed joints are emphasized by large holes in the pattern. While the mobility is 1, it was observed that the mechanism has a bifurcation point midway through the motion.

In addition to removing two consecutive joints, an additional single internal joint between two vertices (as explained in Sec. 4.1) may be removed opposite from the current cut without changing the mobility.

4.2.3 Removing Joints Traversing Different Panels.

It was shown earlier that it is possible to maintain 1-DoF while removing consecutive joints around a single panel. Similarly, this can also be done while removing several consecutive joints traversing different panels. To maintain 1-DoF while making these cuts, certain conditions must be met.

4.2.3 Connected 1-DoF Sections.

Joints may be removed such that it creates multiple separate single-degree-of-freedom mechanisms, while still maintaining enough connections to ensure a mobility of 1 overall.

Beatini et al. showed that entire panels may be removed from Miura-ori patterns to remove overconstraint while maintaining a mobility of 1 [60]. Building on that work, instead of removing entire panels to remove redundant constraints, one could remove the joint constraints between panels using similar rules. A comparison and example of this is shown in Fig. 16. If the width of rows/columns of the “excessive faces” is defined as W, it could also be analyzed with W = 0. The mechanism would still maintain a mobility of 1, but the previously removed panels become removed hinge constraints. This can be done because separate sections of mobility 1 are connected with enough hinge constraints to define the motion with only one overall input.

A similar example using the Miura-ori is shown in Fig. 17. Notice that this system can be analyzed as multiple smaller mechanisms, each with a single-degree-of-freedom. Sufficient connectivity constraints ensure an overall mobility of 1. Beatini et al. suggested several connectivity guidelines when connecting multiple Miura-ori patterns [60].

This is not limited to the Miura-ori pattern but can be applied when connecting multiple 1-DoF patterns into a single 1-DoF system.

4.2.3 End-to-End Chains.

When cuts are made such that a section is not 1-DoF, special considerations must be made to ensure an overall mobility of 1.

An end-to-end chain is a sequence of panels where each panel is only connected to two other panels using only two joints. It may be difficult to calculate an overall mobility of a mechanism with an end-to-end chain because part of the pattern may be underconstrained, while an adjacent section is overconstrained. For example, consider the example of the Miura-ori in Fig. 18. The same number of joints is removed in both patterns; however, one guarantees an overall mobility of 1 using the methods from the earlier section and the other creates an end-to-end chain. The Grubler criterion calculates both of these to have a mobility of 1, however due to special geometry, pattern (b) has a larger mobility. We get this error in the mobility prediction because pattern (b) has a section (the end-to-end chain) with multiple degrees-of-freedom connected with a largely overconstrained section (uncut section). This is not captured in the mobility criterion.

It may be necessary to analyze sections separately to accurately predict the mobility and overconstraint. One example of this was shown in Fig. 14.

It is required for an end-to-end chain to have a predicted mobility of 0 to not add an additional degree to the system. For a spherical/planar mechanism with only revolute joints, using Eq. (2) it can be shown that to not increase the mobility (M = 0), N = 3, and J = 3. Thus, the end-to-end chain length limit for spherical/planar mechanisms is 2.

For a spatial mechanism with only revolute joints, we use Eq. (3). In order to not increase the mobility of the mechanism, the end-to-end chain mobility must be 0. It can be shown that for M = 0, N = 6, and J = 6, meaning the maximum spatial end-to-end chain length is 5.

It must also be required that a chain must not have more than three continually parallel joints, whether consecutive or not. Having four continually parallel joints would construct a planar 4-bar, even if nonparallel joints are between these joints. It is important to note that this restriction also applies to four spherical joints if the hinges continually intersected at a point. This would result in motion about the spherical center.

Figure 19 shows various end-to-end chains that may result from pattern cutting. Many of these would introduce an additional degree-of-freedom, while (a) and (e) would not.

These restrictions enable us to cut joints traversing different panels. An example is shown in Fig. 20 using the Tachi–Miura pattern. The end-to-end chain created by each cut is a spatial linkage of length 5 and does not increase the overall mobility. Note that both chains include only three parallel hinges and do not increase the mobility.

5 Combination of Techniques

Each of the earlier joint removal techniques can be combined and used together in a single pattern so long as applicable restrictions described earlier are followed. With these techniques as options, a designer may apply one or multiple of them, in any order, to modify a base pattern.

As a general sequence, a designer would choose the potential joints to remove using one of the earlier techniques and verify that the corresponding stated conditions are satisfied. After verifying the cuts do not increase mobility, the designer could then continue to remove additional joints using the same or another technique. The choice of joints at each step is up to the designer as well as the design requirements for the application. For example, consider an application where panels of an origami mechanism intersect with needed actuators. In this case, the designer could choose to remove adjacent joints to make room for the actuators, while maintaining a single-degree-of-freedom.

Figure 21 shows an example of a Tachi–Miura with several joints removed using multiple joint removal techniques. Equation (7) estimates a minimum of 14 cuts may be removed. By using a combination of techniques, 18 joints were removed in Fig. 21. Both the uncut and cut versions of this pattern have a mobility of 1. However, the uncut pattern has an overconstraint prediction of 19, and the uncut has a reduced overconstraint of 3. This illustrates that multiple of the joint removal techniques explained in this paper can be used within one origami mechanism to reduce overconstraint.

6 Discussion

It has been shown that many of these techniques can be used together and can include many permutations of cut-joint patterns for the same origami pattern. Kinematically, each cut pattern is the same, being that the mobility and motion are similar so long as cuts are only reducing the degree of overconstraint and not changing the degrees-of-freedom. Different patterns may have varying degrees of remaining overconstraint.

While the techniques shown ensure a mobility of 1 and kinematic properties are similar, it is likely that other properties such as stiffness, actuation forces, and stability may be different for varying permutations. Although this paper does not seek to analyze which pattern may be better than others when acted upon by a force, it presents techniques a designer could use to identify redundant hinges that may be removed without affecting the kinematics. In engineering practice, it would be important to consider each of these performance behaviors when choosing a cut pattern. Based on different functional objectives, one permutation might be beneficial over another. Possible methods for analyzing forces onto origami-based mechanisms have been proposed [6163]. These performance requirements would inform the placement, pattern, and amount of removed joints within the design.

While these methods work to locally identify candidate joints for joint removal, analysis can get complex when removing joints in a large origami pattern. In addition, while these methods can be used to maintain a mobility of 1, they do not inform anything about singularities, bifurcation, and multifurcation points. For example, while the example shown in Fig. 14 had a mobility of 1, it was found to have a bifurcation point midway through its range of motion. Such behaviors would be important to consider when designing an origami-based mechanism.

These techniques have been shown to reduce the overconstraint of zero thickness origami fold patterns. While many of these can be directly applied to thickened versions of these patterns, some considerations may need to be made when choosing thickness accommodation techniques. For example, when using the split vertex thickness accommodation technique [64], additional folds are introduced. These introduced folds may also be reanalyzed for possible hinge removal.

This paper has shown how to remove redundant constraints in origami mechanisms without changing its mobility and kinematic behavior. While the nonredundant joints must be left intact, the otherwise removed redundant joints could be replaced for other benefits such as stored strain energy to assist in actuation.

7 Conclusion

Single-degree-of-freedom origami patterns have proved to be useful in the design of deployable arrays. However, due to an excess of constraints, many origami patterns are overconstrained. Overconstraint introduces many problems to mechanisms when combined with imperfect manufacturing and rigid panels.

This paper gives designers a visual and iterative tool that they can use to find alternative joint patterns in 1-DoF systems. The reduction of redundant constraints reduces manufacturing cost, reduces overall pattern stiffness, and reduces the likelihood of binding due to conflicting constraints.

Acknowledgment

This paper is based on work supported by the Air Force Office of Scientific Research (FA9550-19-1-0290) through Florida International University.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

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