Abstract

Design parameters of the origami flasher pattern can be modified to meet a variety of design objectives for deployable array applications. The focus of this paper is to improve the understanding of design parameters, objectives, and trade-offs of origami flasher pattern configurations. Emphasis is placed on finite-thickness flasher models that would enable engineering applications. The methods presented aim to provide clarity on the effects of tuning flasher parameters based on existing synthesis tools. The results are demonstrated in the design of a flasher-based deployable LiDAR telescope where optimization is used to converge on optimal design parameters and the results are implemented in proof-of-concept hardware.

1 Introduction

Deployable origami-based space arrays improve the potential to achieve large deployed areas to stowed volume ratios. This can help antenna, solar, and optical systems meet critical spatial constraints during launch, while maximizing efficiency gains once deployed. Many origami-based mechanisms could provide adequate solutions to meet this requirement. One pattern, the “flasher”, has received notable attention in recent years due to its potential for favorable stowed-to-deployed size ratios.

The flasher has been implemented into a variety of applications, oftentimes requiring different form factors, deployed and stowed dimensions, and stabilization methods. While several tools and mathematical expressions exist for modifying the base pattern parameters [14], these methods require deep intuition of the resulting pattern behavior. The parameters of the base flasher pattern are herein referred to as: m the rotational order, or number of central hub regular polygon sides, h the height order, r the ring order, and dr the separation between vertices. An in-depth introduction to these parameters can be found in Ref. [5]. A fifth parameter, sf, is herein introduced to denote the scaling factor used to properly scale the pattern to maximize stowing efficiency. This parameter is dependent on stowing constraints. The flasher pattern can be thought of as a template for a family of complex, highly over-constrained spatial mechanisms. It is also sensitive to panel thickness. These and other coupled behaviors mean that selecting and designing a particular flasher configuration can be a cumbersome process.

The early design process for flasher-based deployable systems could be aided by identifying key design parameters and showing their relationships to objectives common to the system performance. It is intended that the heuristics presented in this paper facilitate the design decisions needed to meet case-specific objectives. The design space of flasher configuration possibilities is vast. As such, the methods presented in this paper decode inter-dependencies and present the effects of flasher parameters on design objectives.

As parameters are selected and design objectives are met, the following step involves using these zero-thickness values to convert the system into a thick array suitable for engineering applications. Once deployed, array stability may be an imperative objective to maximize power capabilities. Thus, thick panels provide a means to achieve such objectives by providing a hard stop to prevent motion. Depending on the boundary conditions of the system, increasing thickness can have an influential role in increasing stiffness. Nonetheless, thickness accommodation techniques must be implemented to account for the transition from zero-thickness to finite-thickness and prevent panel interference.

Our primary focus for this development was a finite-thickness deployable array for a LiDAR space telescope. The objective is to demonstrate a deployable telescope that can be stowed inside part of an ESPA-class satellite (∼1 cubic foot) and deploy to 1–2 m in diameter. ESPA is a commercial standard developed to utilize excess rocket launch capacity by mounting secondary payloads below the primary payload and is an acronym for evolved expendable launch vehicle secondary payload adapter (ESPA). An optical telescope has stringent alignment tolerances compared to radio frequency antennas or solar arrays. This requires that we optimize the parameters such as deployed area and stowed volume while assessing the deployed mechanical alignments to evaluate potential optical performance of the deployed system.

This paper presents a novel approach to designing deployable, finite-thickness origami flasher arrays. The concepts are demonstrated in the design of a flasher-based deployable LiDAR telescope.

2 Background

The flasher pattern has been studied and proposed for engineering applications. Guest and Pellegrino [6], and Scheel [7] proposed conceptual flasher patterns, where modifications and experimentations have diversified the possibilities of flasher configurations. Aerospace [810], automotive [11], and medical [12,13] fields have taken advantage of the flasher’s favorable qualities to create systems of varying sizes, materials, and geometries.

Zirbel et al. [5] presented a mathematical model and thickness accommodation techniques to facilitate the transition from the flasher concept to the prototype. The mathematical developments in this work highlight the complexity of the flasher and its vast configuration possibilities.

Lang et al. [14] expanded flasher possibilities by introducing a single-degree-of-freedom cut pattern capable of achieving rigid-foldability and deployment in one motion. Lang furthered flasher design accessibility by creating an open-source tool, Tessellatica [2], where design parameters are input by the user to create fully developed flasher patterns.

Pehrson et al. [15] explored a different approach for achieving flasher foldability and deployment. The design methods proposed to utilize strained joint techniques to allow the pattern to bend at each circumferential fold. This method, however, creates a trade-off by maximizing deployment energy while sacrificing stability once deployed.

Guang and Yang [16] contributed to feasibility of finite-thickness flasher engineered systems by decomposing the coupled deployment motion of the pattern, creating a system that first rotates each sector, followed by a synchronous fold of each sector to create a fully stowed system. A method of tapering was also introduced to create a parabolic system favorable for solar array and antenna applications. Similarly, Kwok [17] developed a framework to simulate finite-thickness origami which considers 1D bending of mountain and valley folds.

Horvath’s investigation of flasher parameters and application to engineered systems provides preliminary guidance on how parameters—particularly dr—affect flasher configurations [18]. The work presented here will expand on Horvath’s presentation by developing flasher systems with non-constant thickness.

Wang and Santer [19] propose a Bayesian and gradient-descent optimization method where optimal doubly-curved finite-thickness flasher patterns are generated to maximize stowage ratio subject to stowed dimension constraints. This work is a stepping stone for further analysis into flasher pattern objective trends and optimization models.

3 Methods

The purpose of this section is to define, analyze, and optimize flasher design parameters to understand their effects on common deployable space array objectives. Designers are often interested in geometric outputs such as the number of total panels, deployed dimensions, and stowed dimensions. The following sections provide context for each design parameter and investigate the effects of varying such parameters. For this study, Tessellatica is used to create flasher patterns and evaluate objectives at each configuration. Trends are presented to aid designers in considering trade-offs.

3.1 Design Parameters.

Designing the flasher pattern requires five design parameters. The first three will be defined as zero-thickness parameters: rotational order, also equal to the number of sides (m), height order (h), and ring order (r). A detailed introduction to these parameters can be found in Zirbel et al. [5].

The fourth design parameter is a thickness parameter, referred to as dr. This parameter is defined as “the desired separation between two nearest-neighbor vertices at the same radial position and the same z-coordinate, normalized to the diameter of the circumcircle of the central polygon” [5]. dr is most closely associated with panel thickness (t). The relationship between the two parameters is defined as
(1)
where
  • i = panel height order

  • h = total height order

  • rcc = central hub circumcircle radius

  • t = projected distance between panel nodes as observed from a top view

Figure 1 provides different views to demonstrate the beginning of a thickness-incorporated flasher on one gore. Further analysis is provided in the following sections.

Fig. 1
A finite-thickness flasher (m = 6, h = 2, r = 2, dr = 0.2) highlighting one gore. Parameters dr, h, and m (which defines rcc) can be varied to tune panel thickness.
Fig. 1
A finite-thickness flasher (m = 6, h = 2, r = 2, dr = 0.2) highlighting one gore. Parameters dr, h, and m (which defines rcc) can be varied to tune panel thickness.
Close modal

Now that base parameters have been established, a fifth parameter is introduced herein referred to as sf—scaling factor. This is considered as an implementation parameter which is only relevant when stowed constraints are established. When a base flasher pattern is designed using the first four parameters, it is assumed that sf = 1. However, the structure may be scaled up or down to fit within the stowed constraints. It is important to note that when the pattern is scaled, each dimension is scaled linearly.

These design parameters form the foundation of the flasher pattern. The following sections provide methods used to clarify the design process for deployable, finite-thickness flasher arrays.

3.2 Number of Panels.

The total number of panels is a direct result of m, h, and r combinations, as described by Eq. (2). This pattern characteristic is important when considering array designs because a practicality limit may exist on how many panels can be used for the array application. Different applications may present different limits. In most cases, it is favorable to minimize the number of panels (np).
(2)
From Eq. (2), we see r as bearing the most influence. Trends are further explored in Fig. 2. We visually see the effect of varying sides, rings, and height order. Increasing rings causes the greatest influence on the number of panels. As a designer, this should be considered when deciding which m-h-r combination will be most suitable.
Fig. 2
Number of panels as a function of m, h, and r. m is represented on the x-axis, and r and h are represented by colors and symbols, respectively. Connecting lines are used to improve visualization of trends.
Fig. 2
Number of panels as a function of m, h, and r. m is represented on the x-axis, and r and h are represented by colors and symbols, respectively. Connecting lines are used to improve visualization of trends.
Close modal

3.3 Deployed and Stowed States.

Figure 3 provides visual context for the design objectives of interest associated with the flasher pattern. Here, we see a scaled representation of how the deployed state is able to achieve a large diameter from a small stowed state.

Fig. 3
(a) Stowed height, diameter, and isometric views of a 4-2-2-0.1 (m-h-r-dr) pattern and (b) deployed incircle diameter of the same pattern. The central hub dimensions are held constant from stowed to deployed.
Fig. 3
(a) Stowed height, diameter, and isometric views of a 4-2-2-0.1 (m-h-r-dr) pattern and (b) deployed incircle diameter of the same pattern. The central hub dimensions are held constant from stowed to deployed.
Close modal

It is desirable to non-dimensionalize the design objectives and consider the flasher in terms of ratios between deployed and stowed dimensions. For the following analysis, the ratios of interest are defined as follows:

  • ψ = deployed diameter to stowed diameter

  • κ = deployed diameter to stowed height

  • ζ = stowed height to stowed diameter

In most deployable space array applications, it is often desirable to maximize ψ and κ to achieve the largest deployed diameter while minimizing stowed diameter and height. It is also often a requirement that the array must stow within a particular form factor. The objective is set to target a specific ζ.

Figures 4, 5, and 6 present non-dimensionalized trends associated with varying the base flasher parameters m, h, r, dr. Individual design parameters are varied on each side of each graph. Shades are used to present objective value results, as a ratio of the central hub incircle diameter, herein assumed to equal 1 unit. dr is incremented across the bottom horizontal axis from 0 to 0.1 in 0.01 steps. Rotational order is discretely varied across the left vertical axis from 3 to 7 sides. Height order ranges discretely from 1 to 3 on the top horizontal axis, and ring order varies discretely from 1 to 2. The ranges used in this analysis provide sufficient intuition on design trends and can be extended to higher-order values. For this analysis, sf is assumed to equal 1 due to the general approach being presented; constraints are not established. This analysis was conducted using Tessellatica; objectives were produced at each parameter combination.

Fig. 4
Deployed diameter to stowed diameter trends. Assuming sf = 1.
Fig. 4
Deployed diameter to stowed diameter trends. Assuming sf = 1.
Close modal
Fig. 5
Deployed diameter to stowed height trends. Assuming sf = 1.
Fig. 5
Deployed diameter to stowed height trends. Assuming sf = 1.
Close modal
Fig. 6
Stowed height to stowed diameter trends. Assuming sf = 1.
Fig. 6
Stowed height to stowed diameter trends. Assuming sf = 1.
Close modal

3.3.1 Analysis.

To better understand the trends observed in Figs. 4, 5, and 6, a simple example will be demonstrated using the analysis shown in Fig. 7. We will herein assume sf = 1.

Fig. 7
(a)–(d) Visual representations showing the effects of varying each parameter level on each objective while holding three of the four parameters constant across each graph. In (a), dr is discretely varied from 0 to 0.2, while m, h, and r are held constant. The pattern m = 4, h = 2, r = 2, and dr = 0.1 is held as the base pattern across each graph.
Fig. 7
(a)–(d) Visual representations showing the effects of varying each parameter level on each objective while holding three of the four parameters constant across each graph. In (a), dr is discretely varied from 0 to 0.2, while m, h, and r are held constant. The pattern m = 4, h = 2, r = 2, and dr = 0.1 is held as the base pattern across each graph.
Close modal

Figure 7 shows a decomposed and decoupled analysis of the effects of changing one design parameter individually. Each of the four graphs presents the 4-2-2-0.1 pattern and analyze the corresponding parameter at two other levels.

Figure 4 shows how ψ increases as h and r increase and m and dr decrease. Taking a look at Fig. 7(d), we visually inspect how increasing h increases deployed diameter at a faster rate than stowed diameter. While both are increasing, it is evident that deployed diameter is growing at a faster rate as a function of increasing h. Hence, the trend observed in Fig. 4 by increasing h is established.

Next, we inspect the effects of increasing r. From both Figs. 4 and 7(b), the ratio of deployed to stowed diameter (ψ) demonstrates apparent increase. The rate of increase for deployed diameter exceeds to the rate of increase for stowed diameter.

ψ is seen to increase as m decreases in Fig. 4. From Fig. 7(c), the largest ratio between deployed and stowed diameters is indeed seen at the smallest rotational order. As m increases, this ratio decreases as a result of deployed diameter’s decrease rate overpowering the decrease rate of stowed diameter.

The last analysis for ψ is seen in the effects of varying dr. Figure 4 shows ψ increasing by decreasing dr. Observing Fig. 7(a), the following values can be extracted:

  • dr = 0, ψ ≈ 6.4

  • dr = 0.1, ψ ≈ 3.8

  • dr = 0.2, ψ ≈ 3.0

As such, we see that by increasing dr, ψ decreases. The same process can be used to understand the trends of Figs. 5 and 6.

4 Case Study

4.1 Requirements.

For this case study, as part of a research collaboration with NASA, specific requirements are established for a deployable LiDAR telescope. The main objective is to maximize deployed optical area of the array, while constrained to a 0.3048 m × 0.5588 m × 0.6604 m (12 in. × 22 in. × 26 in.) prismatic volume for launch. The flasher was selected as a potential candidate for developing a deployable frame structure due to its approximately circular form factor when deployed and it’s potential for large optical areas as measured by its high packing efficiency—the ratio of the deployed surface area to the stowed volume. Designing a finite-thickness flasher array involves finding an appropriate thickness accommodation technique capable of adapting and folding rigid, thick panels.

4.2 Design Selection.

Now that an understanding of parameter effects on deployed diameter and stowed dimensions has been established, a conclusive decision can be made on which pattern will meet the objectives of the telescope application given the constraints.

The coupled nature of these objectives and parameters suggests an optimization approach may be best suited for this analysis. Utilizing Tessellatica as a function evaluator, a gradient-free binary-encoded genetic algorithm optimization routine was employed. This method was chosen due to its robustness and compatibility with discrete (m, h, r), and continuous (dr, sf) variables. However, due to the computational expense of evaluating Tessellatica, a stepwise regression surrogate model was implemented to reduce computation time.

The surrogate model was constructed using a statistical software package where a third order stepwise fit was implemented to create a regression model. Minimizing the Bayesian information criterion number was used for model selection. While inaccuracies exist as a result of using a heuristic gradient-free optimization routine, the results remained consistent across various iterations and were sufficient for this analysis.

As a result, the following optimization statement was implemented:
(3)

The objective is to maximize deployed diameter (to align with optimization convention, the objective is written as a minimization statement). The first two constraints state that the resultant pattern stowed dimensions (height and diameter) must be less than or equal to the established constraints. For this case, the limiting dimensions were defined by Hb = 12 and Db = 22. Upper and lower bounds define the practicality limits of the flasher pattern parameters. Rotational order must be between 3 and 7, height and ring order between 1 and 2, and dr shall not be smaller than 0.1 due to thickness requirements. Lastly, scaling factor should be greater than 0 to maximize volumetric efficiency.

The results from the optimization routine show that a 4-2-2-0.2225 flasher pattern with sf = 4.203 is capable of maximizing deployed diameter while meeting the constraints. The resultant deployed diameter measured 1.59 m. Figure 8 demonstrates the results of the genetic algorithm output from matlab. After 57 generations, convergence was achieved with 36 individuals landing on this configuration. Furthermore, Fig. 9 demonstrates the trade-offs between maximizing ψ and targeting a specific stowed form-factor ζ. From this analysis, we see that if the space allotted for the array to stow during launch is wide and short, the deployed diameter will not be much larger than the stowed diameter. However, if the stowage space is tall and narrow in diameter, the array exhibits a large deployed to stowed diameter ratio ψ.

Fig. 8
Convergence of genetic algorithm optimization routine used to maximize deployed diameter. Optimal results show maximum deployed diameter of 1.59.
Fig. 8
Convergence of genetic algorithm optimization routine used to maximize deployed diameter. Optimal results show maximum deployed diameter of 1.59.
Close modal
Fig. 9
A plot showing trade-offs between ψ and ζ. The Pareto front is shown for this design space.
Fig. 9
A plot showing trade-offs between ψ and ζ. The Pareto front is shown for this design space.
Close modal

4.3 Finite-Thickness Design.

Once pattern parameters are selected, the next step involves accommodating for thickness of the panels.

4.3.1 Thickness Accommodation.

Accommodating the flasher for finite-thickness panels is a difficult task. The pattern exhibits degree-4, degree-5, and degree-6 vertex interactions. Many panels occupy the same plane in the zero-thickness model, which is impossible to do with thickened material because they would have to occupy the exact same volume. Theoretically, one could “nest” them inside each other, but because none of the panels are uniform shape or size, and because the constraints did not allow for anything to deform or penetrate the center of the panels (so an optical membrane could be included), following the zero-thickness model exactly is impossible, and thickness accommodation techniques need to be considered.

Several of the thickness accommodation techniques reviewed by Lang et al. [20] were evaluated, but because the optical membrane needed to be on the same plane throughout the entire flasher when deployed and be protected from scratching the other panels, some techniques were not feasible. Techniques such as the hinge shift technique were eliminated because the membrane would not align on a single plane, and techniques like the offset panel technique were not selected because they consume too much volume and are difficult to implement in non-axially-symmetric patterns.

4.3.2 Proposed Thickness Accommodation Technique.

A modified tapered-panel technique was selected because of its ability to accommodate for thickness in multiple directions at once, while maintaining the flat plane for the optical membrane. The folded state does require a greater stowed volume than the zero-thickness model, but no zero-thickness fold lines occupy the same plane and the additional volume around the fold lines allows for thickness to be added to the panels, as shown in Fig. 10. The use of the pattern alteration presented by Lang et al. [20] presented an approach to the flasher pattern, offsetting the zero-thickness fold lines and providing space for non-zero-thickness tapered panels. The frames were tapered and chamfered to allow the pattern to fold within the anticipated folded configuration. The panels also had the center removed to allow for optical area, creating a border frame-like structure for each panel. In this work, a taper means a cut across the entire panel, and a chamfer is a smaller cut along the edge. These methods are discussed in more detail in the following sections.

Fig. 10
A 2D demonstration of the tapering process, as used in the prototypes
Fig. 10
A 2D demonstration of the tapering process, as used in the prototypes
Close modal
Taper.

The tapering process, as shown in Fig. 10, involves thickening both sides of the zero-thickness line, then removing material to allow the mountain and valley folds to come together without interference. The tapering method shown in this figure is only demonstrated in one dimension, but the flasher pattern requires a more complex tapering process, demonstrated on a triangular-shaped flasher panel in Fig. 11. This same process of removing thickness in each of the fold directions allows the entire flasher to fold along the original zero-thickness line, preserving the original motion of the zero-thickness origami pattern.

Fig. 11
The tapering process as applied to a single panel of the flasher. The dashed line shows the modifications from one image to the next. The last cut would be considered a chamfer, but the previous cuts cross the entire panel, and are considered tapers.
Fig. 11
The tapering process as applied to a single panel of the flasher. The dashed line shows the modifications from one image to the next. The last cut would be considered a chamfer, but the previous cuts cross the entire panel, and are considered tapers.
Close modal
Chamfer.

Tapers allow the mountain and valley folds to preserve motion along the radial folds, but in order to allow panels to wrap around each other in the circumferential directions, additional material must be removed to avoid panel interference and to expose the zero-thickness plane at those locations to maintain the intended zero-thickness motion. Figure 12 demonstrates how the chamfers appear in both the unfolded and folded states. The application of these chamfers can be seen in Fig. 13, as they are visible on the unfolded state, and help the panels wrap around each other in the folded state.

Fig. 12
A demonstration of a chamfer along two adjacent panels
Fig. 12
A demonstration of a chamfer along two adjacent panels
Close modal
Fig. 13
A single section (or gore) of the flasher is demonstrated with the tapered panel approach. Chamfers can be seen on the circumferential folds, while tapers are used on all folds.
Fig. 13
A single section (or gore) of the flasher is demonstrated with the tapered panel approach. Chamfers can be seen on the circumferential folds, while tapers are used on all folds.
Close modal

4.3.3 Final Design.

Figures 14, 15, and 16 present the final 4-2-2-0.2225 thickness accommodated prototype. The panels were manufactured using wood and assembled using compliant spring hinges for deployment ease.

Fig. 14
The deployed finite-thickness prototype, with people shown for scale
Fig. 14
The deployed finite-thickness prototype, with people shown for scale
Close modal
Fig. 15
The stowed finite-thickness prototype, viewed from above to illustrate compactness. Tapered sections can also be seen.
Fig. 15
The stowed finite-thickness prototype, viewed from above to illustrate compactness. Tapered sections can also be seen.
Close modal
Fig. 16
An isometric view of the stowed prototype, illustrating the complexity of the nesting
Fig. 16
An isometric view of the stowed prototype, illustrating the complexity of the nesting
Close modal

The images in Figs. 14, 15, and 16 show the prototype that was created according to project specifications. Figure 14 demonstrates the entire flasher prototype, whereas Fig. 15 demonstrates the compactness of the folding. Figure 16 shows an isometric view where the frames are visible through the sides, illustrating the difficulty of the nesting process. There were some issues involved with the intermediate stages between stowed and deployed states with this prototype, but it was able to successfully fit in the volume allotted and reaches the aperture specified, which was the objective of this prototype.

4.4 Next Steps.

The next steps for the physical flasher model focus on intermediate stages of deployment, including developing more detailed and analytical models that account more fully for thickness accommodation and energy storage in the springs.

Additional compliance in the system would enable the proper motion because the panels are so inherently stiff that they add stress to the hinges. Because of this, models will need to be developed to determine the ideal locations for this compliance.

Features that ensure precise final deployed position would be valuable for array applications. With compliance being added, the inherent hard stops may not be sufficient, but stability could be achieved after the deployment process is accomplished by using magnets, latches, or locking mechanisms to lock the panels into place.

5 Conclusion

The methods presented in this paper contribute to the understanding and development of the flasher pattern. The resulting increased ability to model and design flasher-based arrays will further enable engineering applications. The contributions of this paper are two-fold: (1) demonstrate analytical and visual optimization methods used to understand the effects of each flasher parameter on common space array objectives, and (2) introduce a method for accommodating thickness of flasher panels relying on finite-thickness and non-negligible material. A case study of a design of a flasher-based deployable LiDAR telescope was used to demonstrate the methods presented.

Acknowledgment

This work was funded by the Utah NASA Space Grant Consortium and by the NASA Earth Science Technology Office through contract 22003-20-041. The authors would like to acknowledge Jaxon Jones, Andy Avila, and Jared Hunter of the Compliant Mechanisms Research Group at BYU for prototyping assistance, and Joseph Seamons of BYU for aiding in optimization model development.

Conflict of Interest

There are no conflicts of interest.

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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