Effects of Panel Misalignment in a Deployable Origami-Based Optical Array

Deployable origami-based array systems have received increased interest over the past several years, particularly in circumstances where an array must stow compactly and deploy to a large area [1,2]. Suggested areas of application for origami-based systems include solar array concentrators [3], radiative surface heat control [4], origami arrays as actuators [5], energy absorption [6,7], antennas [8,9], automobile airbags [10], biomedical devices [11], and consumer products [12]. They have also been proposed for use as LiDAR telescopes [13].

LiDAR telescopes and other optical arrays need precise placement to function correctly [14]; to achieve such precision, designers have traditionally used segmented mirror arrays. Wang et al. have analyzed the effect of panel misalignment in three degrees-of-freedom (DOFs) on radiation patterns in segmented mirror antennas to develop a tolerance budget [15]. Segmented mirror arrays, however, are heavy, bulky, and require a large number of actuators, which can result in high costs required to place them in orbit [16]. Selecting a telescope design based on a deployable origami array instead of a segmented mirror array could offer many benefits, such as shrinking rocket fairing sizes, increasing aperture sizes, decreasing the number of actuators, and decreasing the cost of launches, which could help to increase the frequency of optical telescope missions [16,17]. Such an increase could lead to higher-resolution images and more space exploration. However, many challenges exist when attempting to use deployable origami patterns as telescopes [18,19]. One challenge, termed deployment, is to align each panel relative to the array’s coordinate frame; another challenge, termed positioning, is to align the entire array relative to the detector.

In certain applications, the misalignment of panels or of the array can severely lower performance [20]; specifically, misalignment can contribute to the difficulty or inability to completely deploy, increased stresses at joint lines, degraded image quality, loss of power and efficiency of the array, and even complete mission failure [21]. Panel misalignment may come about because of machining tolerances, inaccurate assembly, obstructions during deployment, and external factors such as temperature, gravity, or stresses and vibrations experienced during launch [14,22,23]. Array misalignment can be introduced through the same means as well as from the malfunction of positioning mechanisms. Because of the significant effects of misalignment, it is important to characterize and understand it so that it can be mitigated [23].

The objective of this work is to propose a system for characterizing misalignment in deployable origami arrays, analyze the effects of misalignment on performance of a LiDAR telescope, and determine which DOFs are more sensitive to misalignment. This work presents a case study of misalignment in the hexagonal twist origami pattern and its use as a LiDAR telescope, but the results and conclusions can be extended to a variety of systems and applications. The results of this work will provide an approach for designing precision-positioned origami-based arrays and their optical interfaces.

Origami, the ancient art of paper folding, has been used in engineering applications to find unique solutions to modern problems. Many origami patterns can be categorized as flat foldable, meaning they are completely flat once deployed [24]. By Kawasaki’s theorem, flat foldability is determined by alternatively adding and subtracting fold angles around a vertex; if those angles add to zero, the vertex is flat foldable [25]. Furthermore, by Maekawa’s theorem, flat foldability is also confirmed when the absolute value of the number of mountain folds minus the number of valley folds equals two [26]. Origami patterns can also be rigid foldable, meaning they can go from their stowed state to deployed state without any deformation of the panels, i.e., all deformation occurs at the crease line [27]. They can also be characterized by their number of DOFs and deployed-to-stowed-area ratio.

In order for origami patterns, which are traditionally developed in paper and are assumed to have zero-thickness, to be used as engineering mechanisms, a thickness accommodation technique must be applied. Doing so allows the design to fold as expected without needing to employ the zero-thickness assumption. Many thickness accommodation techniques have been developed, and each has different characteristics and effects on the base pattern [28].

Misalignment of panels is the major source of degraded image quality in optical arrays [29]. Telescopes are sensitive to perturbations in the position and orientation of panels in the array because the size of the detector is comparatively small in comparison to the array itself. Most telescopes for high-resolution imaging applications need precision placement of panels on the order of nanometers because the perturbation will directly affect the wavefront error on a 1:1 scale, making alignment of these panels tremendously difficult [30]. However, telescopes used for collecting energy (e.g., LiDAR) have relatively large detectors (in our application, we use a 4 × 4 pixel detector made of square pixels whose side lengths are 64 μm long). Nanometer precision is therefore not needed; several orders of magnitude less suffices.

Additionally, to increase aperture size, optical telescopes are often composed of multiple panels; while this approach is generally beneficial, it makes alignment more difficult. Even when each optical element is only out of phase by tens or hundreds of wavelengths, the aberration and image shift caused by the accumulated effect of all optical elements’ misalignments can significantly degrade image quality [31]; in severe cases, this combined effect can result in shifting the image spot outside of the detector. Compensation is often employed to account for this misalignment, but to do so requires knowing the exact misalignment of each panel, which can be difficult [32,33].

In this section, we describe the models used in this work and illustrate the several types of potential misalignment in deployable origami arrays. We also detail the simulation process used to quantify the effects of misalignment in the hexagonal twist LiDAR telescope.

The hexagonal twist origami pattern has been proposed as both a reflect array antenna and LiDAR telescope [13]. The hexagonal twist has several unique properties that make it a promising candidate for a LiDAR space telescope, including simplicity, symmetry, and a single DOF. Three different models of the hexagonal twist pattern are used in this paper.

The first model is the paper model, as shown in Fig. 1, and is assumed to have zero-thickness. The pattern is constructed from one equal-sided hexagon surrounded by six identical right pentagons and six identical equilateral triangles, and it has a deployed-to-stowed-area ratio of three. Each interior vertex of the hexagonal twist array is a four-link spherical mechanism whose center lies at the intersection point of crease lines. The pattern therefore contains six coupled spherical mechanisms, resulting in a single DOF spatial mechanism, which facilitates panel alignment and can reduce the number of actuators needed for deployment. Every vertex in the array is flat foldable and rigid foldable; the entire array is also flat foldable and rigid foldable [27]. A flat foldable array simplifies the design of the optical interface, and rigid foldability prohibits panel deformation during deployment, which protects the embedded optics from stretch and compression.

The second model is the optical model, as shown in Fig. 2(a). This model is nearly the same as the paper model, the slight difference being that the size of the array is specified and that a small amount of area is removed to account for rigid frames that surround each panel for enclosing the optical membrane. Removing this area gives an accurate estimate of the available optical area in the final model. This model is used only for optical simulations; no thickness accommodation is applied. The optical model’s hexagon circumscribes a 0.5-m-diameter circle, while the larger outer hexagon (the shape of the array when deployed) inscribes a 1-m-diameter circle. Each panel has 25.4-mm-thick frames around its inside edges; the thick frames and crease lines are assumed to be rigid.

The third model is the engineering model, as shown in Figs. 3 and 4. It is manufactured using rigid materials at full scale. This model is produced by applying the split vertex thickness accommodation to the paper model [34]. This thickness accommodation is promising for use in antennas and telescopes because it increases optical area, retains a single DOF, and improves stability. This model is also designed to accommodate an optical membrane. Optical membranes have been used in deployable structures as reflectors, lenses, and more [35–37]. One approach to integrate the optical components, as shown in Fig. 5, is to use a membrane with optical features that is held between a top and bottom part of the panel. The engineering model is fully defined by five independent variables, whose final values are listed in Table 1. The array has 33 joint lines and is made from 22 two-part panels. The overall thickness of the array is 38.1 mm when deployed. The membrane is placed inside the frame at 6.35 mm from the top face of the array to avoid damage to the optical features through self-contact when stowed. The middle area of each panel is removed so that light can pass through the optical membrane. The optical membrane is approximately 1.52 mm thick, so, it is approximated to be zero-thickness, and therefore does not affect the kinematics of the array.

These models and results can be used to inform the design of actuation and positioning mechanisms used to deploy and align the system.

We provide a general practice for characterizing misalignment in flat foldable, deployable origami arrays. Because many deployable optical mechanisms are not meant to be functional in the stowed state (Fig. 5(a)), misalignment in this state has no effect on the performance of the array; simulating misalignment in this state is therefore not needed. Misalignment can enter the system during the deployment process and its effects can be further perpetuated into the deployed state, resulting in a poorly aligned final state. We perform all misalignment simulations on the array when in its deployed state.

We define frames for each panel to aid in defining misalignment in the deployed state. The origin of the coordinate frame for each panel is placed at the midpoint of its primary hinge line. The x axis runs parallel to the joint line, and the y axis is perpendicular to the joint line. The z axis is normal to the face of all adjacent panels when they are in the completely deployed and flat position. Figure 5(b) shows the axes and panel B in their perfectly aligned position and orientation.

There are two categories of misalignment perturbations: local and global. Local perturbations are perturbations applied to a single panel other than the central hexagon which affect the position or orientation of the panel relative to central hexagon. There are six types of local perturbations: translation along each panel’s x, y, and z axes and rotation about each panel’s x, y, and z axes. These local misalignments are defined in Figs. 6 and 7. Global perturbations treat the array as a single unit and are applied to the array’s position and orientation. Five global perturbations are simulated in this paper: translation along the x, y, and z axes and rotation about the x and y axes of global coordinate frame (GF) in Fig. 2(a). Rotation along the z axis (i.e., clocking of the array) has no fundamental influence on the energy on the detector due to the rotational symmetry of the array, so it is not considered further in this paper.

For this work, a panel is considered perfectly aligned when its position and orientation relative to the central panel of the array are as prescribed by the design. The array is considered perfectly aligned when its position and orientation relative to the detector are as prescribed by the design.

The simulation environment includes the deployable panel assembly (DPA), collimating lens assembly, spatial light modulator (SLM), imaging lens assembly, and detector, as seen in Fig. 8. The SLM is used as a compensator to either adjust the phase function of the optical path and therefore adjust the orientation or balance the aberration caused by panel misalignment. It is placed in the collimated optical path and is conjugative to the DPA to maximize its compensation ability. All simulations are performed on the optical model.

The purpose of the simulation was to evaluate the effect of the mechanical misalignment of deployable panels on the energy level on the detector and to check the compensation ability of the SLM. We performed the simulation based on our theoretical optical model shown in Figs. 2(a) and 8. We selected panel P1 for the analysis of local perturbations because panels P1–P6 are identical and cover much larger area than the six triangle panels.

For each perturbation, we began by simulating the power output using the aligned nominal model to get a baseline power output value. A predetermined perturbation along a specific DOF was then applied either locally (to panel P1) or globally (to the DPA). The power output was again simulated and recorded. This process was repeated for increasing amounts of misalignment.

The SLM compensation was then simulated using the optimizer in Zemax Optical Studio optical design and analysis software. For compensation, we chose the energy level on the detector to be a merit function, and the optimizer performed a search in the n-dimension parameter space of the phase function of the SLM. The optimizer finds a local optimum for the SLM phase function that produces maximum energy on the detector. The post-compensation energy level on the detector was recorded.

The baseline power output of the hexagonal twist LiDAR array was calculated assuming perfect alignment of panels to serve as a control for the simulations. Assuming a total of 1 W energy, the total energy on the detector with no misalignment present was 0.6697 W. The results are shown in Fig. 9 and listed in the “Nominal power output” column in Table 2. Local perturbations were applied to panel P1 while keeping all other panels in their aligned positions, and global perturbations were applied to the DPA. After misalignment perturbations were applied, the energy received on the detector from the affected panel(s) was reduced. The energy received on the detector before and after the SLM compensation is listed in Table 2. The results of all the simulations are shown in Fig. 10.

For global misalignments to the DPA, more power loss comes from decenter X, decenter Y, tilt, and tip DOFs. Both piston and clocking DOFs are robust to misalignment. In this case, clocking does not produce any loss of power because of the rotational symmetry of the DPA. For local misalignments, decenter X, decenter Y, and clocking DOFs are more sensitive to misalignment. Tilt and tip DOFs are moderately sensitive; the notable exception is piston misalignment, which causes little power loss compared to the other DOFs.

Compensation can be beneficial for regaining power lost to misalignment. An example of compensation at the local level is shown in Fig. 11. In this application, compensation can effectively remedy misalignments in the local clocking, local piston, global tilt, global tip, and global piston DOFs. Compensation is moderately efficient at making up for misalignments in the global decenter X and global decenter Y DOFs. It fails to effectively correct misalignments in the local tilt, local tip, local decenter X, and local decenter Y DOFs.

An understanding of which DOFs are more severely affected by misalignment can enable the design of deployable optical arrays with increased performance and can focus resources to areas of the highest impact. Specifically, these results can direct the design of deployment systems, which primarily affect local misalignments, as well as positioning systems, which primarily affect global misalignments. By examining which DOFs cause the most power loss and which cannot be effectively compensated for, we conclude that local decenter X and local decenter Y misalignments (Figs. 6(a) and 6(c)) are the translational DOFs most sensitive to misalignment. For rotational DOFs, we conclude that local tilt and tip (Figs. 7(a) and 7(b)) are the most sensitive to misalignment. Therefore, the alignment of these DOFs should be prioritized.

Based on these conclusions, local misalignments are generally more severely affected than their global counterparts. This would suggest that it is more important to ensure excellent local alignment of the array before focusing on global misalignment.

Considering the results of Fig. 10 and assuming a maximum percent power loss before compensation of about 10%, the global alignment tolerance must be near 0.05 deg and 0.05 mm. Pinpointing a tolerance limit for the local misalignments is more difficult because different panels may have different misalignments, and those misalignments could be coupled, either positively or negatively.

In most other rigid foldable arrays, the decenter X and decenter Y misalignments would likely appear as in-plane misalignment (in-plane when the array is in the deployed configuration). This observation may be applied to other arrays by designing misalignment-reducing surrogate folds, which are not dependent on a specific pattern. For example, many rigid foldable arrays are assembled by attaching rigid panels together using surrogate folds (hinges) inspired by compliant mechanisms. Previous work has been done to investigate the properties and improve the design of compliant hinges [38,39]. Many of these hinges, though generally accepted as having several improvements beyond traditional hinges, such as reduced cost, better manufacturability, and lower part count, can introduce undesirable parasitic motion, leading to greater misalignment [40]. Therefore, the results of this work could be applied to other arrays by designing compliant surrogate folds which mitigate misalignment in sensitive DOFs and then implementing those hinges onto the array.

In this work, we performed an optical simulation on the hexagonal twist pattern to analyze the effect of panel and array misalignment on energy output of a LiDAR telescope. We set forth a method for characterizing misalignment in deployable origami-based arrays. The results of the simulation show that the local decenter X and Y, and local tilt and tip misalignments are the most sensitive to misalignment and for which compensation is the least effective, for translational and rotational DOF, respectively. It was previously unknown which misalignment DOFs were the most sensitive; with this new knowledge, designers and engineers will be equipped with the necessary tools to design mechanisms that mitigate misalignment in these DOFs. Additionally, we specify a tolerance level that can give a target for acceptable alignment of the array. We also prototype a flat, deployable, LiDAR telescope in rigid materials for use in space applications and anticipate that such a design will help reduce both cost and rocket fairing sizes, thus enabling larger deployed systems to be launched in smaller payloads.

In practice, reducing misalignment can take many forms, including intentional selection of hardware to minimize misalignment-susceptible DOF, improving machining tolerances, accounting for external factors like temperature through approximating the thermal expansion, robust testing of deployment mechanisms, or investigation into and development of precision positioning systems. The results presented here could apply to other origami arrays that are rigid foldable, flat foldable, and have a single DOF. The results of this work are intended to inform the engineering designs of systems that can mitigate the negative effects of misalignment and help dedicate efforts and resources to the primary sources of performance reduction.

Areas of future work in addition to those listed above may include performing a similar analysis on other types of origami arrays, including radial, modular, tessellation, or truss structures [17], investigating the positive or negative effects of pattern selection and thickness accommodation techniques on the alignment of panels in a deployed array, or simulating the coupling of several local misalignments.

This paper is based on work supported by NASA Earth Science Technology Office through contract 22003-20-041. Thanks to Brandon Sargent and Jeff Niven for early concepts leading to this paper and articulating the misalignment characterizations.

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

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