The design of a novel two-dimensional deployable mechanism for large-aperture planar antennas

Large-aperture planar antennas are essential in applications like satellite communication and Earth observation. In order to accommodate them within the constrained storage space of rockets during launch, the development of high-folding-ratio mechanisms becomes imperative. However, most of the existing planar antennas rely on one-dimensional deployable mechanisms, limiting their width and impeding the development of planar antennas with larger apertures. The lack of two-dimensional deployable planar antennas is primarily due to their stringent requirements, including the need for flat reflecting surfaces, regular panel shapes to accommodate electronic devices on the panels’ rear sides and minimal inter-panel gaps. In this paper, we introduce a novel two-dimensional deployable mechanism for the development of large-aperture planar antennas, capable of simultaneously meeting these requirements. Firstly, the architecture of the proposed mechanism and its two-step deployment procedure are described; next, the kinematics on the displacement level are conducted; moreover, a case study is presented to demonstrate its folding performance and frequency response. These studies indicate that the antenna thus obtained can achieve efficient two-dimensional folding while offering a reasonable frequency response, which can be a practical solution for the development of large-aperture planar antennas.


Introduction
Planar antennas are essential components in many important applications such as satellite communication, space science, deep space exploration and Earth observation.As the performance of planar antennas is significantly influenced by their aperture size, there is a growing demand for largeaperture planar antennas in recent years.The central challenge in addressing this demand is the development of innovative deployable mechanisms capable of achieving substantial folding ratios, which has attracted the attention of numerous researchers and engineers.As of now, a number of planar antennas have been successfully deployed in space, such as ERS-I, Radarsat-I, SEASAT, and ALOS [1] .Notably, all of them have employed a simple one-dimensional deployable mechanism called " Z-fold", as shown in Figure 1.This mechanism offers several advantages, including architectural simplicity, flat reflecting surfaces, regular panel shapes, and minimal inter-panel gaps.However, since the panel width is preserved during the folding process, their panel width is severely restricted by the diameter of the designated storage space in the rockets.As a result, all their panel width is quite limited, as shown in Table 1.In order to further increase the antenna aperture, many researchers have started investigating two-dimensional deployable mechanisms [2][3][4][5][6] .However, the development of two-dimensional deployable planar antennas faces many challenges due to their stringent requirements, including the IOP Publishing doi:10.1088/1742-6596/2764/1/012007 2 need for flat reflecting surfaces, rectangular panel shapes to accommodate electronic devices on the panels' rear sides and minimal inter-panel gaps.Indeed, many of these designs have uneven reflecting surfaces [4] , non-rectangular-shaped panels [5] , or inevitably introduce inter-panel gaps in order to handle the potential interference of thick panels [6] .In this paper, we introduce a novel two-dimensional deployable mechanism for the development of large-aperture planar antennas.The mechanism description, kinematics analysis, and structure analysis have been conducted, indicating that the proposed mechanism is promising in meeting all the aforementioned requirements while offering good folding performance and reasonable frequency response, making it a potentially practical solution for the development of large-aperture planar antennas.

Design requirements
Firstly, we summarize the requirements that the planar antenna must satisfy:  The antenna system must efficiently fold in two dimensions to fit within the designated storage space of rockets in its stowed state;  The working surfaces of all panels should be on the same plane, with minimal gaps between them;  The panels should ideally be rectangular for accommodating rear-side electronic devices.

The proposed two-dimensional deployable mechanism
As can be seen from Figure 1, the commonly used "Z-fold" scheme meets most of the aforementioned requirements except that the panel width cannot be reduced, which is thus limited by the diameter of the designated storage space within the rockets, preventing us from further increasing its panel width.In order to overcome this limitation while taking advantage of the attractive features of the "Z-fold" mechanism, the following two-step deployment procedure is proposed, as shown in Figure 2: We begin by dividing the antenna (with dimensions l×w m 2 ) into n×2 subpanels, effectively splitting the width dimension into two portions.This allows us to treat the antenna as two separate n×1 "sub-antennas", each of which has n×1 panels that can be folded using the "Z-fold" mechanism, as depicted in Figure 2(a) to Figure 2(b).Moreover, each of these "sub-antennas" is mounted on a plate, which can carry the corresponding "sub-antenna" to undergo a combination of translation (along the Z axis) and rotation for ±90° (about the X axis), ultimately reaching the final stowed state depicted in Figure 2(c).In this way, the width of the antenna will no longer be constrained by the diameter of the designated storage space; instead, its limit will become 2h in this case, with h denoting the height of the designated storage.In this vein, we propose a novel two-step two-dimensional deployable mechanism to realize the deployment process from

2.2.1
The deployable mechanism for the first dimension This deployable mechanism, as depicted in Figure 3, is responsible for the deployment of the antenna system from the state shown in Figure 2(c) to that in Figure 2(b).Since the two "sub-antennas" remain rigidly attached to their respective plates during this step, only the two plates carrying them are illustrated here for clarity.Notably, joint O translates along the line of symmetry from top to bottom during this process, while the two plates rotate 90° (or -90°) simultaneously.To this end, the antenna's width dimension aligns with its designated direction in the working state.To achieve this trajectory, a planar five-bar mechanism (1-2-3-4-5) is designed, where Rod 2 and Rod 3 are the aforementioned plates while 5 is the satellite, as shown in Figure 3(c).Apparently, two motors are needed to actuate this mechanism, which can be installed, e.g., at the hinges between Rod 1 and Rod 5, and those between Rod 4 and Rod 5, as shown in Figure 3(c); alternatively, a single motor together with a pair of identical gears can be used to drive both joints, as they only need to rotate with the same amplitude but in opposite directions [7] .Moreover, locking mechanisms, to be introduced in Section 2.2.2, are installed between each pair of adjacent panels along the width dimension (respectively sitting on plates 2 or 3), which will be triggered when reaching the state shown in Figure 3(c).

2.2.2
The deployable mechanism for the second dimension Now we are ready to deploy the second dimension, corresponding to the state transitions from that shown in Figure 2(b) to that shown in Figure 2(a), and eventually, to the deployed state.To achieve this, two identical sets of deployable mechanisms based on the "Z-fold" scheme are designed for the two "sub-antennas", which are installed on their respective plates.Moreover, for each "sub-antenna", each pair of its adjacent panels is supported by the widely-used deployable mechanism with the quadrangular pyramid configuration (Figure 4(a)) [1] , which has one degree of freedom (Figures 4(b), (c) and (d) for its stowed, intermediate and deployed states).Furthermore, all the "quadrangular pyramids" are actuated simultaneously by a chain of scissors mechanism, which reaches a singular configuration at the final state to enhance the stiffness of the supporting mechanism.This scheme is adopted due to the various attractive features of the "Z-fold" mechanism, which has been validated by its numerous successful applications.Due to the locking mechanisms mentioned in Section 2.2.1,only one motor is needed for the deployment of the second dimension.

The two-step deployment procedure of the two-dimensional deployable mechanism
As stated earlier, the deployment process of the two-dimensional deployable mechanism can be divided into two steps, which are illustrated in Figure 6.Moreover, these two steps are completely uncoupled; as a result, their kinematics analysis can be conducted separately, which are discussed in Sections 3.2 and 3.3, respectively.Figure 6.The deployment process of the two-dimensional deployable mechanism.

Kinematics analysis of the deployable mechanism for the first dimension
As stated in Section 2.2.1, the deployable mechanism for the first dimension is realized by a planar linkage of two degrees of freedom, where the two joints at the hinges between Rod 1 and Rod 5, and that between Rod 4 and Rod 5 can be chosen to be actuated.Moreover, since the trajectories of the left and right "sub-antennas" are required to be always symmetric, their input joint variables must always be equal to each other but of opposite signs.Hence, we only look at the left "sub-antenna" for clarity.Next, we introduce the following notations according to Figure 3: The base frame ℱ O is established at point O, with its x-axis and y-axis respectively parallel to the width dimension and the length dimensions of the antenna panel, as shown in Figure 3(a).Moreover,  ,  ,  and  denote the joints between neighboring rods, while the two rectangles     and    O represent the two plates that carry the two "sub-antennas", which are rigidly attached to Rods 2 and 3, respectively.Furthermore,  represents the distance from points  to the y-axis, while  is the distance from  to   ; finally,  and  denote the length and width of the "sub-antenna", respectively.To this end, the lengths of Rods 1 and 2, denoted as  and  , can be determined via the following relation, namely, deployable mechanism for the first dimension can be readily established using the closed-loop vector method, namely, ) Where α i denotes the rotation angle of the relevant rods (i=1,2), while d is the displacement of the joint O along the y-axis; α 1 also represents the rotation angle of the actuator, as shown in Figure 3.

Kinematic analysis of the deployable mechanism for the second dimension
Since all the hinge axes in the deployable mechanism for the second dimension are parallel to the y-axis shown in Figure 2, this mechanism can be simplified to a planar mechanism in a plane normal to the hinge axes, as depicted in Figure 7.Moreover, using the same approach that has been adopted by Wang et al [1] , its kinematics model can be established using the vector equations associated with the three individual closed-loops, namely loop-1 (O 0 BCD 1 O 1 ), loop-2 (O 1 D 1 H 1 I 1 ), and loop-3 (O 1 D 1 G 1 O 2 E 1 ), leading to the following relations: Where l i (i=1, 2…,11) denotes the lengths of the relevant rods, while θ i denotes the rotation angle of rod i, (i=2…,11), all illustrated in Figure 7.To this end, all the relevant angles can be determined during

Case study
In this section, a case study is provided for verification, where an antenna with the dimension 16 m×10 m ×6 cm is used for demonstration.Using the technique proposed in this paper, the antenna panel is divided into 10×2 subpanels; moreover, the parameters related to the supporting mechanisms are given in Table 2.The supporting mechanisms attached to the rear sides of the panes will also take some space along the thickness dimension of the panel panes in the stowed state, which also needs to be been considered.Eventually, it is found that the antenna system can be folded to a stowed state with the dimension of 5 m×3.2 m ×1 m, which can be transported by existing rockets.4100 Next, in order to verify the kinematics models and the frequency responses of the proposed antenna, a case study is conducted based on the set of feasible architecture parameters shown in Table 2. Figure 8 illustrates the trajectories of the relevant angles and displacement during the deployment process of the first dimension, with solid lines representing those generated by the kinematics models and markers representing those generated by the SolidWorks simulation module.Similarly, the trajectories of the relevant angles during the deployment process of the second dimension are illustrated in Figure 9.As can be seen, all these trajectories consistently show a match between their simulation values (S_v) and theoretical values (T_v).3.Moreover, binding constraints are added between the antenna panels and rods, while fixed constraints are adopted at the plates, which connect the satellite and the antenna system.Thereafter, the finite element model meshed, with the number of nodes and the number of elements being 19965 and 7718, respectively.Finally, simulation results indicate that its first natural frequency is equal to 0.85984 Hz, with its first-order vibration mode shown in Figure 10(b).This natural frequency is comparable to the data shown in Table 1, which can be further increased upon conducting an optimization procedure.
Table 3. Material properties of each component in model.

Conclusion
In this paper, a novel two-dimensional deployable mechanism is proposed for the development of largeaperture planar antennas, capable of simultaneously meeting stringent requirements such as flat reflecting surfaces, regular panel shapes and minimal inter-panel gaps.The mechanism architecture, its two-step deployment procedure and the kinematics analysis are described successively, thereby confirming the feasibility of the proposed deployable mechanism.Moreover, the case study indicates that the proposed antenna can achieve efficient two-dimensional folding while offering a reasonable frequency response.Notably, the width of the antenna is increased from the state-of-art dimension (roughly 3 m) to 10 m, with the potential for further expansion when necessary.This presents a practical solution for the development of large-aperture planar antennas.

Figure 3 .
Figure 3.The deployment process of the deployable mechanism for the first dimension.

Figure 4 .
Figure 4.The deployment process of the deployable mechanism for the second dimension.Finally, to improve the structural stiffness and frequency response of the antenna, various springloaded locking mechanisms have been designed and installed at specified locations (Figures 5(a)-(d)).The locking mechanism in Figure5(d) is triggered at the end of the deployment of the first dimension, securing the panels of the two "sub-antennas" to ensure their simultaneous deployment in the second dimension.In contrast, the other three types are triggered at the final stage of deployment.

Figure 5 .
Figure 5.The locking mechanisms to enhance the stiffness and frequency response.
, according to Figure3(a), the kinematic model of the .1088/1742-6596/2764/1/012007 6 the deployment process, which provides a theoretical foundation for the development of the control strategies.

Figure 7 .
Figure 7. Simplified planar closed-loop mechanism of the deployable mechanism for the second dimension.

Figure 8 . 7 Figure 9 .
Figure 8. Angle and displacement trajectories for the deployable mechanism for the first dimension, acquired through the kinematics model, and Solidworks motion simulation.

Figure 10 .
Figure 10.Finite element model and the first-order vibration mode of the antenna.

Table 2 .
The parameters of the supporting mechanism.