## Abstract

The hexagonal twist origami pattern has characteristics that made it a candidate for next-generation deployable space arrays. It has a deployed area that is up to 3.3 times larger than the stowed area, has a single-degree-of-freedom which simplifies actuation, it is flat-foldable making flat positions possible in both stowed and deployed positions, and its rigid foldability means that its motion is enabled by rotation about distinct axes without deformation of its panels. Although the pattern shows promise for deployable systems, it cannot be directly applied with thick materials because of the self-intersection of nesting panels. This paper presents the kinematics and mechanical advantages of the hexagonal twist pattern, addresses the self-intersection problem by implementing five different thickness accommodation techniques and provides metrics for comparing thickness accommodation techniques to determine which would be best suited for a given application. The concepts are demonstrated through two applications: a deployable reflectarray antenna and a LiDAR telescope.

## 1 Introduction

The number of collaborative, ride-along efforts being carried out in the space-exploration field has increased [1,2], and payloads that utilize these ride-along options need to be readily stowable to utilize space effectively and efficiently. For power and communication applications, large apertures are required because the size of the aperture directly relates to the efficiency of the system [3]. Origami-based and origami-inspired designs provide paths to achieve highly compact, stowable designs that can be deployed [4,5], usually to large surface areas. This potential is highly beneficial to deployable space arrays because they require compact stowing in a launch payload, but large apertures when deployed.

This work presents the hexagonal twist origami pattern, shown in Fig. 1, as a promising model for deployable space arrays. This work evaluates the characteristics of the hexagonal twist pattern, proposes several approaches for accommodating thickness, describes metrics for evaluating the approaches, and demonstrates the concepts in origami-based reflectarray antenna (RA) and light detection and ranging (LiDAR) designs (see Fig. 2).

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

Deployable systems have been used in many space-exploration efforts because the main focus for design is often the size, weight, and power (SWaP) of the system [6]. However, many of these deployable systems use associated deployment systems, such as booms or trusses to assist in or facilitate the deployment, which requires more volume in the launch vehicle.

Origami has inspired deployable designs that have allowed researchers and designers to realize large area gain from a stowed, compact shape [7,8]. However, even origami proves challenging when attempting to deploy and stop the pattern in a completely flat position, as is required in RA and LiDAR cases. Additionally, most origami fold patterns are developed using zero-thickness models and most engineering materials cannot be approximated as zero-thickness, requiring different techniques to account for material thickness while still providing origami-inspired mechanisms which fold as desired.

## 2 Background

Origami, the ancient art of paper folding, has recently inspired researchers in solving many different design problems. For example, different origami patterns were used as inspiration in each of the following cases. Chen et al. [9] used the Miura-ori, triangulated cylinder, and Waterbomb patterns as case studies for a method to automatically assigning folding directions. de Waal et al. [10] used a graded origami hexagonal honeycomb for energy absorption. Zhang et al. [11] used the Miura-ori pattern in a sandwich structure for energy absorption. Yu et al. [12] used the Miura-ori pattern, but as an actuator. Sargent et al. [13] used a triangulated cylinder pattern to tune the mechanism to the desired bistability. Al-Mansoori et al. [14] showed ways to manufacture the triangulated cylinder pattern and a flasher pattern in carbon fiber composites. Ma et al. [15] determined the folding motion of a tubular waterbomb pattern that could be used as a mechanical metamaterial. Lyu et al. [16] used the square-twist pattern to create mechanical metamaterials. Wang et al. [17] used a square-twist origami pattern to create a reconfigurable antenna. DeFigueiredo et al. [18] used the square-twist, chicken wire, and a modified hexagonal twist pattern to present objects during the actuation of concealing and revealing boxes for high-end consumer products. Many different patterns have proven beneficial for these different design cases. The present work is based on the hexagonal twist pattern [19].

Because most materials used in engineered products have some non-negligible thickness, different techniques must be used to design for these thicknesses, while allowing the pattern to still move as desired [20].

### 2.1 Thickness Accommodation Techniques.

Origami patterns assume a thin material, such as paper and modification is necessary when implemented in thick materials. There are many different thickness accommodation techniques that have been developed for use with origami mechanisms [2123]. Five techniques provide the basis of this study: hinge-shift, tapered panel, offset panel, synchronized offset rolling contact element (SORCE), and split-vertex.

The hinge-shift technique has been implemented in various ways; however, the primary concept is that after the pattern is thickened, the hinges are moved to the valley fold sides of the panels. This technique, though simple, does not work for all patterns [23].

Another early technique for thickness accommodation was proposed by Tachi [24]. This technique is known as “tapered panels” technique. Material is removed from the edges of each thickened panel, creating a taper on each panel, to allow the thick array to fold. However, this technique eliminates the possibility of the pattern folding fully flat.

Edmondson et al. [25] proposed the offset panel technique, which keeps the hinges on the zero-thickness model, preserving the original kinematics, and allows the array to fold flat. However, it moves the panels away from the zero-thickness axis with attachments to these hinges. Depending on the panel thickness, this technique may require much more thickness in the out-of-plane direction.

The split-vertex thickness accommodation technique, proposed by Tolman et al. [19], is similar to the offset hinge technique in that it allows a thickened origami pattern to be folded flat. However, instead of offsetting panels, this technique splits specific creases into two separate creases, creating a six-bar linkage with two spherical centers, and retains single-degree-of-freedom. This also allows the panels to lay in parallel or in the same plane when deployed.

Lang et al. [26] presented a technique that uses rolling joints (cams) with specific shapes that synchronize the motion of the panels, which move at different relative rotational velocities. These joints are termed as SORCE joints.

In this paper, the aforementioned five techniques described will be evaluated as they apply to the hexagonal twist pattern to create deployable space array designs.

## 3 Hexagonal Twist Pattern

The hexagonal twist pattern [19] is composed of three unique panels: a hexagonal panel (1), a pentagonal panel (6), and a triangular panel (6). Figure 3(a) shows the hexagonal twist origami pattern. This pattern is an array of repeating degree-4 vertices consisting of intersections of three mountain folds and one valley fold, or three valley folds and one mountain fold. The sector angles of these vertices are α1 = 90 deg, α2 = 60 deg, α3 = 90 deg, and α4 = 120 deg. Each edge of the hexagon is equal to a, which is also equal to the radius of the circumscribing circle of the central hexagonal panel. The sides of the pentagonal panels that share fold lines with the triangular panels are equal to $3/3a$. These are shown in Fig. 3(b).

Fig. 3
Fig. 3
Close modal

The hexagonal twist pattern is flat-foldable, meaning it will fold flat in the open and closed configurations. Figure 1 shows the folding motion from the flat, closed configuration to the flat, open configuration. The pattern is also rigid-foldable, meaning that the only deformation during the motion of the pattern occurs in the creases. The hexagonal twist is a single-degree-of-freedom pattern, therefore it can be fully actuated with only one input, and if any fold angle is defined, the other three-fold angles are defined throughout the deployment.

The stowed (fully folded) area of the pattern is $33/2a2$. The deployed area is $93/2a2$. This provides a deployed area that is three times the stowed area.

In addition to being rigid-foldable and flat-foldable, the hexagonal twist provides a deployed aperture that approximates a circular aperture. In electromagnetic (EM) and optical applications, circular apertures yield the closest to ideal performance. The close approximation to the circular of the hexagonal twist makes it stand out as an initially good candidate for these applications and is one of the reasons for its selection in this work.

### 3.1 Rigid-Foldable Origami.

The hexagonal twist pattern is rigid-foldable and portions of the design depend on the associated relations. Rigid-foldable patterns allow the desired folding of the pattern through deformation in the creases only. Because no panel deformation is required during folding, relations can be made between the sector angles (α) of the pattern and the fold angles (γ). Figure 4 shows a degree-4 vertex from the hexagonal twist pattern. The relationships between the sector and fold angles, presented by Lang [27] and others, and applied to Fig. 4 are

$γ2=−γ4γ1=γ3$
(1)
and
$tan(12γ2)tan(12γ1)=sin(12(α1+α2))sin(12(α1−α2))$
(2)
where γi is the fold angle between the (i−1) and ith panels and αi represents the sector angle of the ith panel, which is the angle between the creases of that panel. Specifying the sector angles, α, and one fold angle, γ1, and using Eqs. (1) and (2), the entire folding motion of the degree-4 vertex can be determined. Because the hexagonal twist pattern is repeating degree-4 vertices, these relationships can be used to define the motion of every panel in the pattern. Equations (1) and (2), along with the specific parameters of the hexagonal twist pattern were used to determine the rotation, velocity, and acceleration of each crease and are shown in Fig. 5.
Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

The mechanical advantage of a rigid mechanism is defined as the force or torque required at an input link to the force or torque that is realized at the desired output link and can be determined by a balance of the input and output work or energy (W = T11 = T44). Mechanical advantage (MA) can also be quantified as the change in rotation of the output panel due to the change in rotation of the input panel or
$MA=dγ1dγ4=T4T1$
(3)
where i is the change in fold angle or the rotational velocity and Ti is the torque, both in regards to the crease between the (i − 1) and ith panels. The mechanical advantage is dependent on the geometric properties/sector angles of the vertex in an origami pattern. This value can help to quantify how much force or torque will be required to open all portions of the pattern.

The mechanical advantage of the hexagonal twist pattern was determined using the preliminary work on the mechanical advantage from Ref. [28]. From Fig. 6, the mechanical advantage starts at −2, meaning that for an input displacement or torque, double the output displacement or torque is realized. The mechanical advantage is reduced to 0.5 at the flat-unfolded state. This trend shows that the mechanical advantage starts out high and as the folding motion occurs, the mechanical advantage decreases. This allows the determination of how much actuation force is needed throughout the hexagonal twist folding.

Fig. 6
Fig. 6
Close modal

### 3.3 Modified Hexagonal Twist.

If additional deployed area is desired, the hexagonal twist shown in Fig. 3(a) can be modified to increase the deployed area, without causing any panel interference during the folding motion or in the flat, stowed configuration as shown in Ref. [29]. Figure 7(a) shows the modified hexagonal twist pattern with the triangular panels increased to rhombus-shaped panels. This change is shown with the hinge-shift and offset panel technique presented later in this paper. As a note, figures of the other techniques do not show this change, however, all calculated metrics presented in this paper consider this change from a triangular panel to a rhombus panel. However, the general packing efficiency and usable-deployed area trends follow the same shape, with a slight decrease in value, if the triangular pattern is considered.

Fig. 7
Fig. 7
Close modal

The stowed area of this modified pattern is $33/2a2$ and the deployed area is $53a2$, resulting in a deployed area that is 3.33 times the stowed area.

## 4 Thickness Accommodated Hexagonal Twists and Comparison Metrics

Five thickness accommodation techniques were selected as potential candidates to incorporate with the hexagonal twist pattern. These five techniques are briefly discussed later. Metrics are presented to help compare techniques and are demonstrated in making the selection for two different space-borne array systems.

### 4.1 Thickness Accommodated Hexagonal Twists.

Figure 8 shows the hexagonal twist realized in five techniques. Table 1 presents the five techniques with associated metrics. Variables in the table are defined as: t is the desired thickness of the array and w is the width of the joint in the offset panel technique. β1 and β2 are as defined in Ref. [19] and μ is as defined in Ref. [26].

Fig. 8
Fig. 8
Close modal
Table 1

Comparison of thickness accommodated hexagonal twists

Hinge-shiftOffset panelTapered panelSORCESplit-vertex
Design parameter choicest, at, a, wt, a, δ1t, at, a, β1, β2
Number of hinges1518181833
Width of hingesN/AwN/A$3tπμor3tπ$N/A
Kinematics retainedNoYesYesYesNo
Folded thickness4t4t$t2(1+cosδ1)+32asinδ1$4t4t
Packing efficiencySee Fig. 9 for trends over range of $ta$
Usable-deployed areaSee Fig. 10 for trends over range of $ta$
Obstructions presentNoneOne-sideOne-sideOne-sideNone
Hinge-shiftOffset panelTapered panelSORCESplit-vertex
Design parameter choicest, at, a, wt, a, δ1t, at, a, β1, β2
Number of hinges1518181833
Width of hingesN/AwN/A$3tπμor3tπ$N/A
Kinematics retainedNoYesYesYesNo
Folded thickness4t4t$t2(1+cosδ1)+32asinδ1$4t4t
Packing efficiencySee Fig. 9 for trends over range of $ta$
Usable-deployed areaSee Fig. 10 for trends over range of $ta$
Obstructions presentNoneOne-sideOne-sideOne-sideNone

Specific to the hinge-shifted technique, creating a thickened hexagonal twist requires three of the hinges to be cut to allow folding. These cut hinges are the hinges that connect the hexagonal panel to the pentagonal pattern those folds on the top of the pattern to allow the array to fold. These hinges are shown in Fig. 7(b) by small dashed lines.

Specific to the tapered panel technique, the designer is required to choose the amount of folding that will be achievable, with more folding requiring more tapering of the panels. In Fig. 8(b), a fold angle, γ1, was determined to be 150 deg, making $δ1=15deg$. Selecting this value and using the relations of rigid-foldable origami allows the other fold angles and the taper angles to be determined by Eqs. (1) and (2) and
$δ2=π−γ22δ3=π−γ32δ4=π−γ42$
(4)

### 4.2 Design Metrics.

Metrics were determined using SWaP criteria, packing efficiency, and usable-deployed area. Other metrics considered were the number of hinges present, preservation of the original kinematics, and if the thickness accommodation technique presented obstructions (such as from raised panels or hinges). Because the packing efficiency and the usable-deployed area depend on the thickness (t) and side length (a) of the array, the following two sections discuss the trends in these metrics.

#### 4.2.1 Packing Efficiency.

The volume of each thickened pattern and the smallest cuboid in which the design could fit (bounding box) were calculated. The ratio of these two values constitutes how well the thickened pattern uses the required stowing space and will be referred to as the packing efficiency. The trends are shown in Fig. 9. From the figure, it is observed that the offset panel technique provides better packing efficiency at low thickness-to-side-length ratios (t/a) while for higher ratios, the highly tapered panel technique provides better packing efficiency. As a note, the volume of the SORCE pattern was determined by multiplying the usable area by the thickness (neglecting the volume of the joints) and therefore underestimates the actual volume of the SORCE design, however, the packing efficiency will probably not exceed the packing efficiency of the split-vertex pattern.

Fig. 9
Fig. 9
Close modal

#### 4.2.2 Usable-Deployed Area Ratio.

The ratio of the area of the deployed thickened pattern to the area of the deployed origami pattern is referred to as the usable-deployed area ratio. Figure 10 shows the usable-deployed area ratio for t/a from 0 to 1. As may be expected since split panels are being added, the split-vertex has the most usable area. The tapered panel technique has the next most usable area, followed by the hinges shift and offset panel techniques, with the SORCE technique having the least usable area. This is because, for the offset panel hexagon, only three offsets are occurring on the hexagonal panel and only three of the pentagonal panels, while there are SORCE joints on every panel of the SORCE hexagonal array.

Fig. 10
Fig. 10
Close modal

For the tapered panel technique, only the lower bound line shows the area that would be contained in the plane that cuts the whole design. Therefore, all the other portion of the shaded area shows the usable area if all the tapered portions could be used (in cases where the same or parallel planes are not needed), such as in the case of a solar array.

## 5 Demonstration of Hexagonal Twist in Space-Borne Arrays

Two case studies are presented to illustrate the characteristics of the hexagonal twist pattern for use in space applications and demonstrate the concepts described earlier. The first presents the hexagonal twist as the mechanism to create a deployable RA for CubeSat missions. The second case study presents the hexagonal twist pattern being implemented as a LiDAR telescope. These applications present design cases that have some requirements in common (SWaP) but also have differing requirements. The RA reflects EM signals to a receiver (or horn), to be processed. The LiDAR telescope focuses light that is passing through onto a sensor.

### 5.1 Deployable Antenna for CubeSat Applications

#### 5.1.1 Reflectarray Antenna Background and Requirements.

RAs are antennas that are flat and use metallic patches to direct the electromagnetic waves to a receiver (or horn) [3]. This setup is shown in Fig. 11. The gain, or how well a signal is directed in a specific desired direction [30], of a reflectarray is dependent on the size and efficiency of the reflectarray designed. The larger the aperture the higher the possible gain. [3]. The required flatness of these types of antennas is between 1/20th and 1/40th of the desired wavelength that is to be sent or received [31,32]. The efficiency of the RA is dependent on many characteristics, with one of the most important factors being the shape. A circular aperture is generally the best shape for high efficiency [3]. Additionally, to increase the ability to transport the array to space, it would be beneficial if the reflectarray were stowable and fit on the side of a CubeSat.

Fig. 11
Fig. 11
Close modal

Furthermore, for reflectarray systems on CubeSats, the reflectarray should be as low profile as possible so the reflectarray can reside on the side of the CubeSat in the space between the CubeSat and the CubeSat deployer. This thickness can vary but a standard thickness on the side of the CubeSat is 6.5 mm. The folded reflectarray must be less than or equal to this 6.5 mm thickness.

#### 5.1.2 Selected Techniques and Array Design.

The hinge-shift and the offset panel thickness accommodated hexagonal twists were selected to be used as the mechanism of the folding reflectarray. These techniques were selected because of the high packing efficiency at a small thickness-to-side-length ratio (see Fig. 9).

The hexagonal reflectarray thickness was selected based on the folded thickness of each of these techniques, which for both is 4t. If the thickened hexagonal twist pattern is used as the substructure on which EM panels are placed, then t = 0.813 mm. If only EM panels are used, then the panels can be 1.625 mm thick.

The folded array needs to fit on the side of a CubeSat, providing design information to determine the side length, a, and subsequently all other parameters of the pattern. These values are shown in Table 2, where the width of the hinges for the offset panel hexagon was assumed to be w = 2 mm.

Table 2

The side length of the reflectarray hexagonal twists. The figures show the area the array will take and the remaining usable areas (green boxes) of different unit CubeSats for other components

CubeSat Size
Accommodation technique1U2U3U6U
Hinge-shiftFigure
amax50 mm57.74 mm57.74 mm115.47 mm
ARemaining1339.8 mm28452 mm218,452 mm213,812 mm2
Offset panelFigure
amax48.27 mm56 mm56 mm113.74 mm
ARemaining1339.4 mm28453 mm218,453 mm213,811 mm2
CubeSat Size
Accommodation technique1U2U3U6U
Hinge-shiftFigure
amax50 mm57.74 mm57.74 mm115.47 mm
ARemaining1339.8 mm28452 mm218,452 mm213,812 mm2
Offset panelFigure
amax48.27 mm56 mm56 mm113.74 mm
ARemaining1339.4 mm28453 mm218,453 mm213,811 mm2

#### 5.1.3 Other Considerations.

Other considerations in the design include the hinge design, controlling the flatness, and actuation and grounding.

##### 5.1.3.1 Hinge design.

For the hinge-shifted hexagonal reflectarray, three of the hinges were removed to allow for folding. These hinges are shown in Fig. 7(b). This can be done and still retain the single-degree-of-freedom. The same three hinges were removed in the offset panel hexagonal reflectarray but this was done to increase the usable area in the offset panel reflectarray. Brown et al. [33] showed that this technique was possible without changing the kinematics of the mechanism.

The hinges selected in both reflectarray designs were membrane hinges. Membrane hinges were selected due to their low profile and high range of motion. The hinge-shift hexagonal twist used polyimide tape for the hinges on the top (above the EM panels) and bottom layers where the hinges are located. For the offset panel hexagonal twist, the membrane hinges were placed on the substructure, and then the EM panels were laid on top.

##### 5.1.3.2 Flatness.

The hinge-shift technique uses the inherent hard stop interactions between panels as a means to control the deployed surface flatness. To increase the flatness and stability of the array with the offset panel technique, and because the panels are offset in the deployed state, the top pentagonal panel (when the pattern is folded) was modified to cover the exposed portion of the bottom pentagonal panels. In addition to filling this portion of unused volume, it also produces a larger portion of the panel for the rhombus panels to lie on in the deployed state, thus increasing the stability. This addition is shown in Fig. 12.

Fig. 12
Fig. 12
Close modal
##### 5.1.3.3 Actuation and grounding.

Both thickened reflectarray designs allow for the antenna to be connected (or grounded) to the CubeSat by either the central hexagonal panel, an outer pentagonal pattern, or through the connection of the reflectarray to the CubeSat by a mechanism that rotates the array off the CubeSat, as shown in Ref. [29]. Actuation of the pattern can be accomplished by using a membrane material that stores strain energy when the antenna is folded. If pin-joints are used, then a pin-joint with a torsional spring could be used to actuate the pattern also. These options reduce the need for external actuation or power requirements from the CubeSat.

The EM performance was determined for the two reflectarray designs in Ref. [29]. Prototypes of both reflectarray designs are shown in Fig. 13.

Fig. 13
Fig. 13
Close modal

### 5.2 Deployable LiDAR Telescope

#### 5.2.1 LiDAR Telescope Background and Requirements.

LiDAR uses lasers or light to determine the range of objects and thus has been used as a technology to scan Earth’s surfaces as well as the atmosphere. LiDAR, similar to reflectarrays, benefits from having receivers that have large apertures [34,35] and suggests the need for a deployable system. Figure 14 shows the basic principle of the LiDAR telescope redirecting the light through a specially designed membrane to a sensor to be processed.

Fig. 14
Fig. 14
Close modal
##### 5.2.1.1 Requirements.

Because it also uses EM principles, specifically optics, many of the requirements remain the same for the LiDAR telescope as for the RA. The array must achieve a required flatness for the optical membrane, which focuses the incoming light, to perform properly and a circular aperture provides higher efficiencies. However, other requirements include the need for light to be able to pass through the array unobstructed and unshadowed. Additionally, specific requirements for this project were that the array must fit into a 660 mm × 559 mm × 305 mm (26 in. × 22 in. × 12 in.) volume when folded and the array should deploy to have an area of approximately 1 m2. An optical membrane will be used to focus the light. This membrane must be protected and should be located on the same plane throughout the design.

#### 5.2.2 Selected Techniques and Array Design.

The split-vertex hexagonal twist was chosen as the array design for the LiDAR telescope. This design is shown in Fig. 15, where
$b=33ac=a−2d1tan(β1−π2)d=32a−36(a−2d1tan(β1−π2))e=33(a−2d1tan(β1−π2))d1=2tsec(μ1−π2)$
(5)
and μ1 is based on the selected β1. The choice of the split-vertex hexagonal twist was made based on the following design decisions:
1. The split-vertex provides a consistent plane throughout the pattern that can contain the optical membrane.

2. Larger stowing volume dimensions allow larger t/a and a larger area gain is realized using the split-vertex technique.

3. The split-vertex does not have joints that protrude upwards from panels, alleviating the potential of joints damaging the membrane.

4. All joints can be placed on the top plane of the pattern, simplifying manufacture.

5. The split-vertex hexagon twist has many joints where panel interference occurs when open to the flat state. This benefits the relative flatness of the pattern.

Fig. 15
Fig. 15
Close modal

The triangular panels of the hexagonal twist were used in this design because the usable optical area is determined from the inscribed circle of the deployed split-vertex hexagonal twist. The added area of the rhombus panels did not increase the diameter of the inscribed circle. Therefore, it was not necessary to use the rhombus panels, which added complexity to the design without contributing to the optical performance of the LiDAR telescope.

The following discusses how the requirements were met.

##### 5.2.2.1 Protect optical membrane.

When the panels fold over themselves in the split-vertex hexagonal twist, the top faces of the panels contact each other. To protect the optical membrane from damage, the optical membrane was inset into the panel and off the zero-thickness line so that the membranes on different panels would not touch each other.

Additionally, this design allowed for a sandwiched attachment approach, where the panels were cut into two layers and the membrane was sandwiched between them. This alleviates the concerns of relying strictly on adhesives to attach the optical membrane to the top surface, as many of these compounds have a high affinity to outgas in space, causing the membrane adhesion to be lost. Because of this, the membrane was attached to the panels using both adhesion and compressive forces between two halves of the panels.

##### 5.2.2.2 Area optimization.

The panel frame width, w1, and thickness, t, together determine the structural stiffness of the split-vertex hexagonal twist. But, as w1 increases, the optical membrane area decreases. The system was optimized so the array would have the most area possible and still fit in the specified box. The optimization, led to the following parameters for the array: a = 300.3 mm, t = 38.1 mm, and w1 = 12.7 mm. β1 was chosen to equal β2.

##### 5.2.2.3 Hinge design.

The hinge design of the hexagonal LiDAR telescope uses spring loaded piano hinges at several hinge locations, shown in Fig. 15(c) by the solid lines. Since the pattern has only a single-degree-of-freedom, any sort of rigid hinge used would have to be precisely placed. Any significant misalignment in multiple hinge lines would cause the seizing of the array. Membrane hinges, shown in Fig. 15(c) as dashed lines, were used at other joints to provide a slight amount of compliance to allow the array to open, even if the piano hinges were misaligned.

##### 5.2.2.4 Flatness.

The split-vertex hexagonal LiDAR telescope has inherent hard stops at creases 1, 2, 4, 5, and 6. Additionally, compliant latches and magnets are used in the panels at creases 1, 4, 5, and 6 to provide added stability and flatness to the deployed pattern.

##### 5.2.2.5 Actuation.

The driving hinges used torsional springs to initiate and continue the motion of the panels throughout the actuation. The split-vertex hexagonal pattern has different kinematics because of the added joints. Figure 16 shows the angular position, velocity, and acceleration of each unique joint angle given the initial conditions and assuming a constant base panel input of 1 deg/s and the joint angles labeled in Fig. 15.

Fig. 16
Fig. 16
Close modal

Some techniques may be more beneficial in actuating the pattern to the desired flat position. Mechanisms that use other forms of actuation other than just rotation may be helpful to actuate this and other patterns, while avoiding the issue of low mechanical advantage near (and at) the flat state.

A prototype of the LiDAR telescope design is shown in Fig. 17.

Fig. 17
Fig. 17
Close modal

## 6 Conclusions

The hexagonal twist origami pattern has characteristics that make it a candidate for space-borne arrays. As shown in the cases presented earlier, design requirements influence thickness accommodation technique selection. These requirements also lead to trade-offs between the mechanical and electromagnetic or optics requirements. This work demonstrates that the hexagonal twist provides a pattern that not only provides desired foldability and motion, but also provides designs that can incorporate materials that can be used to create internal actuation, stabilize the systems in the flat state, and provide a viable solution to applications such as RAs and LiDAR telescopes.

While basic metrics are provided within this work, additional metrics could be determined to help compare thickness accommodation techniques. Many of the concepts presented within this work are applied to the hexagonal twist, but the metrics and methods shown in this paper can also be applied to other patterns. The concepts were demonstrated through applications to deployable RAs and LiDAR telescopes, but are general enough to apply to other space-borne and earthbound applications.

## Acknowledgment

This paper is based on work supported by the Air Force Office of Scientific Research grant FA9550-19-1-0290 through Florida International University, the NASA Earth Science Technology Office through contract 22,003-20-041, and the Utah NASA Space Grant Consortium.

The technical input of Dr. Mark Stephen of NASA Goddard Space Flight Center and Jeffrey Niven is gratefully acknowledged. The authors would also like to thank Aliya Mittelman for helping in creating Fig. 2.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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