# Networking Method for the Structure of a Large Modular Deployable Space Antenna

Authors: Dake Tian; Xiaodong Fan; Junwei Zhang; Lu Jin; Rongqiang Liu; Yu Zhang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

A high-precision networking method for a modular deployable antenna with ultramultiple modules is proposed to solve the problem of achieving an accurate formation of the structure of a large-aperture modular deployable space antenna. In accordance with the general requirements for the deployment surface of deployable antennas, the structure composition and deployment principle of a deployable antenna are analyzed, and the topological law of a hexagonal prism modular structure is expounded. On the basis of basic theories of differential geometry, analytical geometry, and robotics, the mathematical model of an envelope circle and that of its center are established. Then, a networking model of structure that includes 61 modules in the fully deployed state is established. Finally, the established model is verified via a numerical simulation method, and the influence law of structural parameters on networking characteristics is analyzed and studied. Simulation results show that the networking method can realize an accurate connection of multiple modules. The networking method for the structure of a modular deployable antenna proposed in this study can provide a reference for research on basic theory of a deployable antenna’s mechanism and related engineering applications.


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## Body

## 1. Introduction

A deployable space antenna is a new type of space structure that has resulted from the rapid development of aerospace technology in the past two or three decades, and it is one of the key payloads of spacecraft [1, 2]. Considering the limited storage space of transport rockets, deployable space antennas must be designed as deployable mechanisms that can be folded into a compact configuration before launch and then gradually deployed and locked into a predetermined shape in orbit. At present, deployable space antennas are widely used in mobile communication, space science, military reconnaissance, navigation, and remote sensing [3–6]. Given their important roles in many major space projects and unique and attractive configuration variations, these antennas have high academic value, engineering demand, and application prospects. Because of the favorable application prospects, deployable antennas have become one of the research frontiers and focuses in the field of aerospace [7, 8].The metal mesh deployable antenna [9–12] is one of the most widely used deployable antenna structures that primarily comprises a flexible metal reflective mesh and a rigid deployable truss. This structure is known for its high storage rate, light weight, and large aperture. As the working surface of the antenna, the metal reflective mesh plays a major role in receiving and transmitting signals; meanwhile, the deployable truss serves as the supporting structure of the antenna that plays a key role in deploying, supporting, and positioning the metal mesh surface [13–15].To increase the coverage area, resolution, and communication distance of satellites, deployable antennas have witnessed large-scale high-precision development in recent years. Shape accuracy is an important index for measuring the working performance of deployable antenna. How to improve the shape accuracy of deployable antennas has become a hot topic in this field. For instance, Di et al. [16] optimized the number of adjustment cables and initial tension by using genetic algorithms, which effectively improved the surface accuracy of deployable antennas. Li et al. [17] proposed two form finding methods, namely, determining the prestress of the cable network and optimizing the prestress distribution of the boundary cable, to ensure a good match between the surface deformation of the reflector and cable tension. Tanaka and Natori [18] proposed two mesh surface shape control methods, namely, the direct method and mode method, and applied them to control the tension and boundary cables. Yang et al. [19] proposed an initial shape design method for the cable net structure based on force density, which realized the initial shape design of the front and rear cable nets by optimizing and improving the uniformity of cable net tension. Wang et al. [20] established the active shape adjustment optimization model of the cable net structure by integrating the piezoelectric ceramic actuator and flexible cable. Given that the flexible mesh surface acts as the direct surface of the antenna and mainly plays the roles of signal reception and transmission, most studies on shape accuracy have focused on flexible structures, such as cable mesh, tension cable, and metal mesh surface. However, only few studies have explored the deployment accuracy of the truss structure, which is another important factor affecting the shape accuracy of deployable antennas.With the rapid development of space technology in recent years and the growing demand for high-speed intelligent communication, the development of large and ultralarge deployable antennas with a deployment aperture of tens or even hundreds of meters has become especially urgent [21, 22]. The modular design provides an inspiration for the high-precision networking of antenna structures. Datashvili [23] proposed a single-layer hoop deployable antenna using a double scissor-type link mechanism that effectively improves the overall stiffness of the deployable antenna. You and Pellegrino [24] proposed a double-layer hoop deployable antenna by using scissor-type link mechanisms to connect the outer and inner hoops. Qi et al. [25] proposed a hoop deployable antenna using the Myard and plane symmetric Bricard linkages. Shi et al. [26] proposed single- and double-layer hoop deployable antennas using slider crank-type units. Han et al. [27] proposed a double-layer hoop deployable antenna using scissor-type link mechanisms. Xu and Guan [28] proposed a deployable truss antenna using tetrahedral units. Chen and Guan [29] proposed a hexagonal prism module driven by a telescopic rod and designed a modular deployable antenna based on this module. Guo et al. [30] proposed a modular deployable antenna based on 3RR-3URU tetrahedral units. Sanchez et al. [31] proposed several modular deployable mechanical networks using derivative configurations of scissor-type link mechanisms. The Japan Aerospace Exploration Agency [32, 33] developed a 14-module deployable antenna for the engineering test satellite ETS-VIII.Although many existing deployable antenna configurations meet the accuracy requirements, existing antennas only have a small number of modules and a short deployment aperture (<20 m). Moreover, only few studies have examined the networking methods of ultramultiple modular deployable antenna structures that comprise dozens of modules. This limitation presents a challenge in promoting the large-scale and ultra-large-scale future development of deployable antennas.To address the development demands of large-scale and high-precision deployable antennas, this paper proposes a high-precision networking method for modular deployable antennas. Compared with the existing networking methods, the proposed method can realize the networking of ultramultiple modules; meet the future demands for deployable antenna on-orbit assembly and deep space detection; and provide a theoretical basis for the structural design, accuracy analysis, form finding, and engineering application of these antennas.The remainder of the paper is organized as follows: in Section2, the structural composition and deployment principle of a deployable antenna are introduced. In Section 3, the topological law of a hexagonal modular structure is studied. Then, the mathematical models of the envelope circle are deduced. Subsequently, the structure networking model is established. In Section 4, the above model is verified and analyzed. In Section 5, conclusions are drawn.


## 2. Structure Composition and Deployment Principle

### 2.1. Structure of Deployable Antenna

Earth’s organisms have undergone billions of years of superiority and inferiority, and the biological prototypes of many animals and plants in nature can provide lessons and inspiration for the study of modular deployable antennas. Honeybees can use their own secreted wax to produce a hexagonal prism modular honeycomb with the least material, the largest synthesis space, and the strongest structure (Figure1). On the basis of the honeycomb morphology, a hexagonal prism modular deployable antenna configuration scheme with a hexagonal module section is proposed in the current study. The overall structure scheme is shown in Figure 2.Figure 1
Honeycomb in nature.Figure 2
Structure of a modular deployable antenna.The modular deployable antenna consists of several hexagonal prism modules with the same or similar structures [34]. Each module consists of two major components: (1) the mesh surface and cable network and (2) the truss structure (Figure 3).Figure 3
Structure of a module.The mesh surface and cable network is the working part of the antenna. It consists of several flexible mesh surfaces, including the front net, the mesh surface, the tie cable, and the rear net. The front net is used to connect the mesh surface. The mesh surface is the working surface of the deployable antenna. It is woven with a gold-plated molybdenum wire, used for satellite signal reception and transmission, and it exhibits a paraboloid shape after deployment. The tie cable is used to connect the front and rear nets; it mostly plays a role in precisely adjusting the shape of the mesh surface. The rear net is the base of the mesh surface and cable network. It provides the carrier for the connection and tensioning of the front net, mesh surface, and tie cable. The truss structure is the skeleton of the antenna; it plays roles in deploying, supporting, and positioning the mesh surface and cable network, providing sufficient stiffness and precision. The truss structure primarily consists of the deployable structure and the crossing cable. The deployable structure is the core of the truss structure; it is a multilink mechanism consisting of six radially arranged rib units with stowing and deploying functions. The crossing cable is installed on the outer beam of the deployable structure; it is used to strengthen the stiffness of the mechanism after deployment.In accordance with the relative position of the antenna reflector and the feed, parabolic antennas can be divided into two types: normal feed and offset feed [35]. The paraboloid is cut with a cylindrical surface coaxial to the rotating paraboloid, and the cut surface is called the normal feed antenna. The rotating paraboloid is cut with a cylindrical surface that has a different axis from that of the rotating paraboloid, and the cut surface is called the offset feed antenna (Figure 4).Figure 4
Types of parabolic antenna. (a) Normal feed. (b) Offset feed.
(a)(b)The work surface of the normal feed antenna is partly blocked by the feed; thus, the efficiency of the normal feed antenna is lower than that of the offset feed antenna when the antenna aperture, operating frequency, and manufacturing accuracy are the same. Therefore, many satellite antennas currently use the offset feed type. Simultaneously, considering that the working surface is parabolic, the idea of fitting the parabolic surface with a spherical surface of a consistent curvature composed of the support structure is used to implement the network model study to improve the consistency and generality of the structure of each module and reduce development cost [36]. The relationship between the truss structure and the working surface is illustrated in Figure 5.Figure 5
Relationship between the truss structure and the working surface.


### 2.2. Deployment Principle

The rib unit is the smallest deployable unit of the truss structure. It primarily consists of a central beam, a slider, a support beam, an upper beam, an outer beam, a lower beam, a small diagonal bar, a large diagonal beam, and a host spring. Its structure is shown in Figure6. The host spring is the driving source of the mechanism, and the motor is the deployment speed control device.Figure 6
Structures of a rib unit. (a) Stowed. (b) Deployed.
(a)(b)When the rib unit is stowed, the host spring is in the compressed energy storage state, and the slider is at the bottom of the central beam. When the mechanism is deployed, the motor slowly releases the rope, and then, the host spring drives the slider to move upward. Meanwhile, the support beam rotates and drives the rest of the beams to deploy. The axes of the small and large diagonal bars coincide after complete unfolding, and the mechanism is in the “dead point” position. The entire mechanism is locked, and the rib unit becomes a stable structure. In accordance with the structure composition and deployment principle introduced earlier, a set of single-module truss structure prototype is developed in the current study, as shown in Figure7.Figure 7
Prototypes of single module. (a) Stowed. (b) Deployed.
(a)(b)


## 2.1. Structure of Deployable Antenna

Earth’s organisms have undergone billions of years of superiority and inferiority, and the biological prototypes of many animals and plants in nature can provide lessons and inspiration for the study of modular deployable antennas. Honeybees can use their own secreted wax to produce a hexagonal prism modular honeycomb with the least material, the largest synthesis space, and the strongest structure (Figure1). On the basis of the honeycomb morphology, a hexagonal prism modular deployable antenna configuration scheme with a hexagonal module section is proposed in the current study. The overall structure scheme is shown in Figure 2.Figure 1
Honeycomb in nature.Figure 2
Structure of a modular deployable antenna.The modular deployable antenna consists of several hexagonal prism modules with the same or similar structures [34]. Each module consists of two major components: (1) the mesh surface and cable network and (2) the truss structure (Figure 3).Figure 3
Structure of a module.The mesh surface and cable network is the working part of the antenna. It consists of several flexible mesh surfaces, including the front net, the mesh surface, the tie cable, and the rear net. The front net is used to connect the mesh surface. The mesh surface is the working surface of the deployable antenna. It is woven with a gold-plated molybdenum wire, used for satellite signal reception and transmission, and it exhibits a paraboloid shape after deployment. The tie cable is used to connect the front and rear nets; it mostly plays a role in precisely adjusting the shape of the mesh surface. The rear net is the base of the mesh surface and cable network. It provides the carrier for the connection and tensioning of the front net, mesh surface, and tie cable. The truss structure is the skeleton of the antenna; it plays roles in deploying, supporting, and positioning the mesh surface and cable network, providing sufficient stiffness and precision. The truss structure primarily consists of the deployable structure and the crossing cable. The deployable structure is the core of the truss structure; it is a multilink mechanism consisting of six radially arranged rib units with stowing and deploying functions. The crossing cable is installed on the outer beam of the deployable structure; it is used to strengthen the stiffness of the mechanism after deployment.In accordance with the relative position of the antenna reflector and the feed, parabolic antennas can be divided into two types: normal feed and offset feed [35]. The paraboloid is cut with a cylindrical surface coaxial to the rotating paraboloid, and the cut surface is called the normal feed antenna. The rotating paraboloid is cut with a cylindrical surface that has a different axis from that of the rotating paraboloid, and the cut surface is called the offset feed antenna (Figure 4).Figure 4
Types of parabolic antenna. (a) Normal feed. (b) Offset feed.
(a)(b)The work surface of the normal feed antenna is partly blocked by the feed; thus, the efficiency of the normal feed antenna is lower than that of the offset feed antenna when the antenna aperture, operating frequency, and manufacturing accuracy are the same. Therefore, many satellite antennas currently use the offset feed type. Simultaneously, considering that the working surface is parabolic, the idea of fitting the parabolic surface with a spherical surface of a consistent curvature composed of the support structure is used to implement the network model study to improve the consistency and generality of the structure of each module and reduce development cost [36]. The relationship between the truss structure and the working surface is illustrated in Figure 5.Figure 5
Relationship between the truss structure and the working surface.


## 2.2. Deployment Principle

The rib unit is the smallest deployable unit of the truss structure. It primarily consists of a central beam, a slider, a support beam, an upper beam, an outer beam, a lower beam, a small diagonal bar, a large diagonal beam, and a host spring. Its structure is shown in Figure6. The host spring is the driving source of the mechanism, and the motor is the deployment speed control device.Figure 6
Structures of a rib unit. (a) Stowed. (b) Deployed.
(a)(b)When the rib unit is stowed, the host spring is in the compressed energy storage state, and the slider is at the bottom of the central beam. When the mechanism is deployed, the motor slowly releases the rope, and then, the host spring drives the slider to move upward. Meanwhile, the support beam rotates and drives the rest of the beams to deploy. The axes of the small and large diagonal bars coincide after complete unfolding, and the mechanism is in the “dead point” position. The entire mechanism is locked, and the rib unit becomes a stable structure. In accordance with the structure composition and deployment principle introduced earlier, a set of single-module truss structure prototype is developed in the current study, as shown in Figure7.Figure 7
Prototypes of single module. (a) Stowed. (b) Deployed.
(a)(b)


## 3. Structure Networking Modeling of a Deployable Antenna

The shape accuracy of deployable antenna is usually judged according to the Root Mean Square Error (RMSE) theory. RMSE, also known as standard error, refers to the root mean square of the deviation between the measured value and its theoretical value. The smaller the RMSE, the higher the surface accuracy. The measured value can be measured by experiment, while the theoretical value needs to be calculated by the networking model of deployable antenna.To build a networking model for the structure of a hexagonal prism modular deployable antenna, the coordinates of the key points on each module in the global coordinate system should be found within the spherical space. In accordance with the structural characteristics of the modules, the current work applies appropriate simplifications to the structure, extracts its key features, and proposes a relatively fast and convenient envelope circle equivalent method for building the networking model. The modeling processes of the structure network are as follows: first, the topological rules of the hexagonal prism modular structure are analyzed. Second, the mathematical model of the envelope circle is established based on basic theory of robotics. Third, the 3D coordinates of the intersection point (key point) of the envelope circles are calculated, and the structure networking model is established.


### 3.1. Laws of Module Topology

The hexagonal prism modular structure has an evident topological law. After careful observation and study of the structure, two topological laws are identified: hierarchical topology and toroidal array.In hierarchical topology, the central module is the center (i.e., the first layer), stacked layer by layer from the inside out, as shown in Figure8(a). The relationship between the number of modules per layer gi and the number of layers i can be expressed as follows:
(1)gi=1,6i−1,i=1,i≥2.Figure 8
Module topology laws. (a) Hierarchical topology. (b) Toroidal array.
(a)(b)In toroidal array, the entire hexagonal prism modular structure is formed by several typical modules via a 60° toroidal array, as shown in Figure8(b). In each layer, the relationship between the number of typical modules fi and the number of layers i can be written as
(2)fi=1,i−1,i=1,i≥2.The relationship between the total number of typical moduleshi and the number of layers i can be obtained as follows:
(3)hi=∑k=1ifk.


### 3.2. Definition of the Envelope Circle Equivalent Method

Each module is divided into upper and lower surfaces in accordance with the position of key points in space. Each module has six key points each on the upper and lower surfaces. The key points on the upper surface areA1–A6, and the key points on the lower surface are A7–A12 (Figure 9). The normal distance between the upper and lower surfaces is the length of the outer beam; therefore, as long as the coordinates of the key points on one surface are known, the coordinates of the key points on the other surface can be calculated. The current study presents the upper surface as an example for illustration.Figure 9
Envelope circle equivalent process.From the structural composition, each module consists of six identical rib units, and then, six key pointsA1–A6 on the upper surface are in the same circle, which is called the envelope circle of the module. Point S1 is the center of the envelope circle, and r is the radius of the envelope circle (Figure 9). Then, the support structure of each hexagonal prism module can be simplified as an envelope circle. That is, each hexagonal prism module can be equivalent to an envelope circle, and the problem of finding the coordinates of key points can be equivalent to that of finding the intersection point between envelope circles. In the current study, the method for simplifying the hexagonal prism module with an envelope circle and obtaining the coordinates of the key points of each module is called the envelope circle equivalent method. When the hexagonal prism module becomes equivalent to the envelope circle, the topological law of the envelope circle is the same as that of the hexagonal prism module. Assuming that the radius of the spherical surface constructed by several hexagonal prism modules is R and point O is the spherical center, a global coordinate system O is established at the spherical center. The research flow of multimodule networking is illustrated in Figure 10.Figure 10
Research flow of multimodule networking.


### 3.3. Mathematical Model of the Envelope Circle

In space, a space circle is the intersection of a sphere and a plane. The point normal equation of a plane can be determined by a point on the plane and a normal vector of the plane. Assuming that the space rectangular coordinate system is built at the spherical centerO, then the equation of any envelope circle can be expressed as
(4)x2+y2+z2=R2,−xSnx−xSn−ySny−ySn−zSnz−zSn=0,where Sn is the coordinate of a circle n, and its column vector is xSnySnzSnT. xSn,ySn,zSn correspond to the x, y, and z components of Sn, respectively; and R is the spherical radius.In addition, onlySn is an unknown variable in Equation (4). Thus, the problem of finding the equation of the envelope circle n can be transformed into the problem of finding the circle center coordinate Sn of the envelope circle n.


### 3.4. Mathematical Model of the Envelope Circle Center

Figure11 is a top-down schematic of the spatial rectangular coordinate system in Figure 10 (not a strict top-down view, but only a schematic illustrating the position relationship), and the envelope circle in Figure 11 is the envelope circle that corresponds to the typical module presented in Figure 8.Figure 11
Typical envelope circle.In accordance with the module’s topological rules, the centers of envelope circles 1, 2, 4, 7, and 11 are on planey=x⋅tanπ/6; the centers of envelope circles 3 and 8 are on plane y=0; and the centers of envelope circles 5, 6, 9, and 10 are symmetrical around plane xoz. That is, in envelope circles 1–11, the center of a class of envelope circles is on a plane. The center of another type of envelope circle, although not in a special position, is symmetrical with respect to some planes.On the basis of the analysis of the two aforementioned special position relationships of the envelope circle center in space, two calculation methods of the circle center coordinates are proposed for each type of position relationship.


#### 3.4.1. Point-Plane Method

The point-plane method is a technique for obtaining the envelope circle center coordinates in accordance with two known conditions. These conditions are as follows: (a) the coordinates of a point on the envelope circle are known, and (b) the center of the circle of the envelope circle is known on a plane and the equation of the plane is known. The point-plane method is suitable for the case in which the center of the circle of the envelope circle is on a known plane.Given pointS2, the circle center of envelope circle 2 is presented as an example in Figure 11. Assume that the coordinates of point A1 are known and point S2 is on the plane. Then, point S2 satisfies the following equations:
(5)xA1−xS22+yA1−yS22+zA1−zS22=r2,xS22+yS22+zS22+r2=R2,yS2=xS2tanπ/6,where r is the radius of the envelope circle and R is the radius of the sphere surface. Equation (5) has two solutions: one is the coordinate of point S1, and the other is the coordinate of point S2.


#### 3.4.2. Two-Point Method

The two-point method is a technique for obtaining the coordinates of envelope circle centers in accordance with a known condition. The known condition is that the coordinates of two noncoincident points on the envelope circle are known. The two-point method is suitable for cases in which the circle center of the envelope circle is not located in a special position.Given pointS5, the circle center of envelope circle 5 is presented as an example in Figure 11. Assume that the coordinates of points D3 and C1 are known. Then, the coordinates of point S5 can be expressed by
(6)xC1−xS52+yC1−yS52+zC1−zS52=r2,xD3−xS52+yD3−yS52+zD3−zS52=r2,xS52+yS52+zS52+r2=R2.Equation (6) has two solutions: one is the coordinate of point S3, and the other is the coordinate of point S5.The point-plane and the two-point methods are used to solve the coordinates of the circle center of the 11 envelope circles in Figure11. The coordinates of the circle center of envelope circles 1, 2, 3, 4, 7, 8, and 11 can be calculated using the point-plane method. Meanwhile, the coordinates of the circle center of envelope circles 5 and 9 can be calculated using the two-point method. The coordinates of the circle center of envelope circles 6 and 10 can be obtained in accordance with the symmetrical relation (the circle center of envelope circles 5, 6, 9, and 10 are symmetrical with respect to plane xoz). The detailed calculation flow is depicted in Figure 12.Figure 12
Calculation process of the coordinates of the circle center of a typical envelope circle.From Figures9 and 10, the coordinates of points S1 and A1 are as follows:
(7)S1=000−R2−r2T,A1=r0−R2−r2T.Thus, in accordance with the known conditions: (a) the coordinates of pointA1 and (b) point S2 are on plane y=x⋅tanπ/6. The coordinates of point S2 can be calculated using the point-plane method, as shown in Figure 11.PointB1 is the intersection of envelope circle 2 and plane y=0. Hence, the coordinates of point B1 can be calculated as
(8)x2+y2+z2=R2,y=0,xS2x−xS2+yS2y−yS2+zS2z−zS2=0.Equation (8) has two solutions: one for point A1 and the other for point B1.Therefore, in accordance with the known conditions: (a) the coordinates of pointB1 and (b) point S3 on plane y=0, and the coordinate of point S3 can be solved using the point-plane method, as shown in Figure 12. The calculation process of points S4, S7, S8, and S11 is similar to that of points S2–S3. The calculation process of point S5 is presented in Equation (6) by using the two-point method. The calculation process of point S9 is similar to that of point S5.In addition, the symmetric relation of the plane in Figure12 (the circle center of envelope circles 5, 6, 9, and 10 are xoz symmetrical with respect to the plane) indicates that the coordinates of points S6 and S10 can be determined using the coordinates of points S5 and S9, as follows:
(9)S6=xS5−yS5zS5T,S10=xS9−yS9zS9T.Therefore, the circle center coordinatesS1–S11 of typical envelope circles can be calculated in accordance with the calculation process of the circle center coordinates shown in Figure 12.The envelope is numbered in accordance with the topological rule of the module in Figure8, as shown in Figure 13.Figure 13
Envelope circle number.In accordance with the module’s topological rules, the circle center coordinates of the other envelope circles can be determined by the circle center coordinates of typical envelope circles (envelope circles 1–11).(10)Sn=Rotz,π3Sn−10n=12,13,⋯,21,Rotz,2π3Sn−20n=22,23,⋯,31,Rotz,3π3Sn−30n=32,33,⋯,41,Rotz,4π3Sn−40n=42,43,⋯,51,Rotz,5π3Sn−50n=52,53,⋯,61,where n is the envelope circle number, and its range of value is from 1 to 61. Rotz,θ is the coordinate transformation matrix that rotates around the z-axis.
(11)Rotz,θ=cosθ−sinθ0sinθcosθ0001.Equation (10) can be used to calculate the circle center coordinates of the remaining 50 envelope circles. The mathematical model of all the envelope circles can be determined by substituting the circle center coordinates of all the envelope circles into Equation (4).


### 3.5. Structure Networking Model

In the aforementioned envelope circle model, the intersection point of adjacent envelope circles is the connection point of the corresponding hexagon prism module, and the coordinate of the intersection point of the envelope circle is the coordinate of the corresponding key point of the hexagon prism module. If envelope circlesp and q intersect and the points of intersection are W1 and W2, then the coordinates of points W1 and W2 can be solved using the equation of two space circles:
(12)x2+y2+z2=R2,xSpx−xSp+ySpy−ySp+zSpz−zSp=0,xSqx−xSq+ySqy−ySq+zSqz−zSq=0,where Sp and Sq are the circle center coordinates of envelope circles p and q, respectively; and the values of p and q range from 1 to 61. Equation (12) has two solutions that correspond to the coordinates of points W1 and W2.The spatial geometric coordinates of all the key points can be solved in accordance with this method, and the networking model of a modular deployable antenna can be established.


## 3.1. Laws of Module Topology

The hexagonal prism modular structure has an evident topological law. After careful observation and study of the structure, two topological laws are identified: hierarchical topology and toroidal array.In hierarchical topology, the central module is the center (i.e., the first layer), stacked layer by layer from the inside out, as shown in Figure8(a). The relationship between the number of modules per layer gi and the number of layers i can be expressed as follows:
(1)gi=1,6i−1,i=1,i≥2.Figure 8
Module topology laws. (a) Hierarchical topology. (b) Toroidal array.
(a)(b)In toroidal array, the entire hexagonal prism modular structure is formed by several typical modules via a 60° toroidal array, as shown in Figure8(b). In each layer, the relationship between the number of typical modules fi and the number of layers i can be written as
(2)fi=1,i−1,i=1,i≥2.The relationship between the total number of typical moduleshi and the number of layers i can be obtained as follows:
(3)hi=∑k=1ifk.


## 3.2. Definition of the Envelope Circle Equivalent Method

Each module is divided into upper and lower surfaces in accordance with the position of key points in space. Each module has six key points each on the upper and lower surfaces. The key points on the upper surface areA1–A6, and the key points on the lower surface are A7–A12 (Figure 9). The normal distance between the upper and lower surfaces is the length of the outer beam; therefore, as long as the coordinates of the key points on one surface are known, the coordinates of the key points on the other surface can be calculated. The current study presents the upper surface as an example for illustration.Figure 9
Envelope circle equivalent process.From the structural composition, each module consists of six identical rib units, and then, six key pointsA1–A6 on the upper surface are in the same circle, which is called the envelope circle of the module. Point S1 is the center of the envelope circle, and r is the radius of the envelope circle (Figure 9). Then, the support structure of each hexagonal prism module can be simplified as an envelope circle. That is, each hexagonal prism module can be equivalent to an envelope circle, and the problem of finding the coordinates of key points can be equivalent to that of finding the intersection point between envelope circles. In the current study, the method for simplifying the hexagonal prism module with an envelope circle and obtaining the coordinates of the key points of each module is called the envelope circle equivalent method. When the hexagonal prism module becomes equivalent to the envelope circle, the topological law of the envelope circle is the same as that of the hexagonal prism module. Assuming that the radius of the spherical surface constructed by several hexagonal prism modules is R and point O is the spherical center, a global coordinate system O is established at the spherical center. The research flow of multimodule networking is illustrated in Figure 10.Figure 10
Research flow of multimodule networking.


## 3.3. Mathematical Model of the Envelope Circle

In space, a space circle is the intersection of a sphere and a plane. The point normal equation of a plane can be determined by a point on the plane and a normal vector of the plane. Assuming that the space rectangular coordinate system is built at the spherical centerO, then the equation of any envelope circle can be expressed as
(4)x2+y2+z2=R2,−xSnx−xSn−ySny−ySn−zSnz−zSn=0,where Sn is the coordinate of a circle n, and its column vector is xSnySnzSnT. xSn,ySn,zSn correspond to the x, y, and z components of Sn, respectively; and R is the spherical radius.In addition, onlySn is an unknown variable in Equation (4). Thus, the problem of finding the equation of the envelope circle n can be transformed into the problem of finding the circle center coordinate Sn of the envelope circle n.


## 3.4. Mathematical Model of the Envelope Circle Center

Figure11 is a top-down schematic of the spatial rectangular coordinate system in Figure 10 (not a strict top-down view, but only a schematic illustrating the position relationship), and the envelope circle in Figure 11 is the envelope circle that corresponds to the typical module presented in Figure 8.Figure 11
Typical envelope circle.In accordance with the module’s topological rules, the centers of envelope circles 1, 2, 4, 7, and 11 are on planey=x⋅tanπ/6; the centers of envelope circles 3 and 8 are on plane y=0; and the centers of envelope circles 5, 6, 9, and 10 are symmetrical around plane xoz. That is, in envelope circles 1–11, the center of a class of envelope circles is on a plane. The center of another type of envelope circle, although not in a special position, is symmetrical with respect to some planes.On the basis of the analysis of the two aforementioned special position relationships of the envelope circle center in space, two calculation methods of the circle center coordinates are proposed for each type of position relationship.


### 3.4.1. Point-Plane Method

The point-plane method is a technique for obtaining the envelope circle center coordinates in accordance with two known conditions. These conditions are as follows: (a) the coordinates of a point on the envelope circle are known, and (b) the center of the circle of the envelope circle is known on a plane and the equation of the plane is known. The point-plane method is suitable for the case in which the center of the circle of the envelope circle is on a known plane.Given pointS2, the circle center of envelope circle 2 is presented as an example in Figure 11. Assume that the coordinates of point A1 are known and point S2 is on the plane. Then, point S2 satisfies the following equations:
(5)xA1−xS22+yA1−yS22+zA1−zS22=r2,xS22+yS22+zS22+r2=R2,yS2=xS2tanπ/6,where r is the radius of the envelope circle and R is the radius of the sphere surface. Equation (5) has two solutions: one is the coordinate of point S1, and the other is the coordinate of point S2.


### 3.4.2. Two-Point Method

The two-point method is a technique for obtaining the coordinates of envelope circle centers in accordance with a known condition. The known condition is that the coordinates of two noncoincident points on the envelope circle are known. The two-point method is suitable for cases in which the circle center of the envelope circle is not located in a special position.Given pointS5, the circle center of envelope circle 5 is presented as an example in Figure 11. Assume that the coordinates of points D3 and C1 are known. Then, the coordinates of point S5 can be expressed by
(6)xC1−xS52+yC1−yS52+zC1−zS52=r2,xD3−xS52+yD3−yS52+zD3−zS52=r2,xS52+yS52+zS52+r2=R2.Equation (6) has two solutions: one is the coordinate of point S3, and the other is the coordinate of point S5.The point-plane and the two-point methods are used to solve the coordinates of the circle center of the 11 envelope circles in Figure11. The coordinates of the circle center of envelope circles 1, 2, 3, 4, 7, 8, and 11 can be calculated using the point-plane method. Meanwhile, the coordinates of the circle center of envelope circles 5 and 9 can be calculated using the two-point method. The coordinates of the circle center of envelope circles 6 and 10 can be obtained in accordance with the symmetrical relation (the circle center of envelope circles 5, 6, 9, and 10 are symmetrical with respect to plane xoz). The detailed calculation flow is depicted in Figure 12.Figure 12
Calculation process of the coordinates of the circle center of a typical envelope circle.From Figures9 and 10, the coordinates of points S1 and A1 are as follows:
(7)S1=000−R2−r2T,A1=r0−R2−r2T.Thus, in accordance with the known conditions: (a) the coordinates of pointA1 and (b) point S2 are on plane y=x⋅tanπ/6. The coordinates of point S2 can be calculated using the point-plane method, as shown in Figure 11.PointB1 is the intersection of envelope circle 2 and plane y=0. Hence, the coordinates of point B1 can be calculated as
(8)x2+y2+z2=R2,y=0,xS2x−xS2+yS2y−yS2+zS2z−zS2=0.Equation (8) has two solutions: one for point A1 and the other for point B1.Therefore, in accordance with the known conditions: (a) the coordinates of pointB1 and (b) point S3 on plane y=0, and the coordinate of point S3 can be solved using the point-plane method, as shown in Figure 12. The calculation process of points S4, S7, S8, and S11 is similar to that of points S2–S3. The calculation process of point S5 is presented in Equation (6) by using the two-point method. The calculation process of point S9 is similar to that of point S5.In addition, the symmetric relation of the plane in Figure12 (the circle center of envelope circles 5, 6, 9, and 10 are xoz symmetrical with respect to the plane) indicates that the coordinates of points S6 and S10 can be determined using the coordinates of points S5 and S9, as follows:
(9)S6=xS5−yS5zS5T,S10=xS9−yS9zS9T.Therefore, the circle center coordinatesS1–S11 of typical envelope circles can be calculated in accordance with the calculation process of the circle center coordinates shown in Figure 12.The envelope is numbered in accordance with the topological rule of the module in Figure8, as shown in Figure 13.Figure 13
Envelope circle number.In accordance with the module’s topological rules, the circle center coordinates of the other envelope circles can be determined by the circle center coordinates of typical envelope circles (envelope circles 1–11).(10)Sn=Rotz,π3Sn−10n=12,13,⋯,21,Rotz,2π3Sn−20n=22,23,⋯,31,Rotz,3π3Sn−30n=32,33,⋯,41,Rotz,4π3Sn−40n=42,43,⋯,51,Rotz,5π3Sn−50n=52,53,⋯,61,where n is the envelope circle number, and its range of value is from 1 to 61. Rotz,θ is the coordinate transformation matrix that rotates around the z-axis.
(11)Rotz,θ=cosθ−sinθ0sinθcosθ0001.Equation (10) can be used to calculate the circle center coordinates of the remaining 50 envelope circles. The mathematical model of all the envelope circles can be determined by substituting the circle center coordinates of all the envelope circles into Equation (4).


## 3.4.1. Point-Plane Method

The point-plane method is a technique for obtaining the envelope circle center coordinates in accordance with two known conditions. These conditions are as follows: (a) the coordinates of a point on the envelope circle are known, and (b) the center of the circle of the envelope circle is known on a plane and the equation of the plane is known. The point-plane method is suitable for the case in which the center of the circle of the envelope circle is on a known plane.Given pointS2, the circle center of envelope circle 2 is presented as an example in Figure 11. Assume that the coordinates of point A1 are known and point S2 is on the plane. Then, point S2 satisfies the following equations:
(5)xA1−xS22+yA1−yS22+zA1−zS22=r2,xS22+yS22+zS22+r2=R2,yS2=xS2tanπ/6,where r is the radius of the envelope circle and R is the radius of the sphere surface. Equation (5) has two solutions: one is the coordinate of point S1, and the other is the coordinate of point S2.


## 3.4.2. Two-Point Method

The two-point method is a technique for obtaining the coordinates of envelope circle centers in accordance with a known condition. The known condition is that the coordinates of two noncoincident points on the envelope circle are known. The two-point method is suitable for cases in which the circle center of the envelope circle is not located in a special position.Given pointS5, the circle center of envelope circle 5 is presented as an example in Figure 11. Assume that the coordinates of points D3 and C1 are known. Then, the coordinates of point S5 can be expressed by
(6)xC1−xS52+yC1−yS52+zC1−zS52=r2,xD3−xS52+yD3−yS52+zD3−zS52=r2,xS52+yS52+zS52+r2=R2.Equation (6) has two solutions: one is the coordinate of point S3, and the other is the coordinate of point S5.The point-plane and the two-point methods are used to solve the coordinates of the circle center of the 11 envelope circles in Figure11. The coordinates of the circle center of envelope circles 1, 2, 3, 4, 7, 8, and 11 can be calculated using the point-plane method. Meanwhile, the coordinates of the circle center of envelope circles 5 and 9 can be calculated using the two-point method. The coordinates of the circle center of envelope circles 6 and 10 can be obtained in accordance with the symmetrical relation (the circle center of envelope circles 5, 6, 9, and 10 are symmetrical with respect to plane xoz). The detailed calculation flow is depicted in Figure 12.Figure 12
Calculation process of the coordinates of the circle center of a typical envelope circle.From Figures9 and 10, the coordinates of points S1 and A1 are as follows:
(7)S1=000−R2−r2T,A1=r0−R2−r2T.Thus, in accordance with the known conditions: (a) the coordinates of pointA1 and (b) point S2 are on plane y=x⋅tanπ/6. The coordinates of point S2 can be calculated using the point-plane method, as shown in Figure 11.PointB1 is the intersection of envelope circle 2 and plane y=0. Hence, the coordinates of point B1 can be calculated as
(8)x2+y2+z2=R2,y=0,xS2x−xS2+yS2y−yS2+zS2z−zS2=0.Equation (8) has two solutions: one for point A1 and the other for point B1.Therefore, in accordance with the known conditions: (a) the coordinates of pointB1 and (b) point S3 on plane y=0, and the coordinate of point S3 can be solved using the point-plane method, as shown in Figure 12. The calculation process of points S4, S7, S8, and S11 is similar to that of points S2–S3. The calculation process of point S5 is presented in Equation (6) by using the two-point method. The calculation process of point S9 is similar to that of point S5.In addition, the symmetric relation of the plane in Figure12 (the circle center of envelope circles 5, 6, 9, and 10 are xoz symmetrical with respect to the plane) indicates that the coordinates of points S6 and S10 can be determined using the coordinates of points S5 and S9, as follows:
(9)S6=xS5−yS5zS5T,S10=xS9−yS9zS9T.Therefore, the circle center coordinatesS1–S11 of typical envelope circles can be calculated in accordance with the calculation process of the circle center coordinates shown in Figure 12.The envelope is numbered in accordance with the topological rule of the module in Figure8, as shown in Figure 13.Figure 13
Envelope circle number.In accordance with the module’s topological rules, the circle center coordinates of the other envelope circles can be determined by the circle center coordinates of typical envelope circles (envelope circles 1–11).(10)Sn=Rotz,π3Sn−10n=12,13,⋯,21,Rotz,2π3Sn−20n=22,23,⋯,31,Rotz,3π3Sn−30n=32,33,⋯,41,Rotz,4π3Sn−40n=42,43,⋯,51,Rotz,5π3Sn−50n=52,53,⋯,61,where n is the envelope circle number, and its range of value is from 1 to 61. Rotz,θ is the coordinate transformation matrix that rotates around the z-axis.
(11)Rotz,θ=cosθ−sinθ0sinθcosθ0001.Equation (10) can be used to calculate the circle center coordinates of the remaining 50 envelope circles. The mathematical model of all the envelope circles can be determined by substituting the circle center coordinates of all the envelope circles into Equation (4).


## 3.5. Structure Networking Model

In the aforementioned envelope circle model, the intersection point of adjacent envelope circles is the connection point of the corresponding hexagon prism module, and the coordinate of the intersection point of the envelope circle is the coordinate of the corresponding key point of the hexagon prism module. If envelope circlesp and q intersect and the points of intersection are W1 and W2, then the coordinates of points W1 and W2 can be solved using the equation of two space circles:
(12)x2+y2+z2=R2,xSpx−xSp+ySpy−ySp+zSpz−zSp=0,xSqx−xSq+ySqy−ySq+zSqz−zSq=0,where Sp and Sq are the circle center coordinates of envelope circles p and q, respectively; and the values of p and q range from 1 to 61. Equation (12) has two solutions that correspond to the coordinates of points W1 and W2.The spatial geometric coordinates of all the key points can be solved in accordance with this method, and the networking model of a modular deployable antenna can be established.


## 4. Model Verification and Analysis

To verify the correctness of the envelope circle model and networking model, a numerical simulation method is used in the current work. Then, the effects of structure parameters, such as deployment aperture and number and size of modules, on networking characteristics are analyzed and studied.


### 4.1. Verification of Envelope Circle Model

Assume that the parameters areR=4701 mm and r=600 mm. These parameters are used for the succeeding analysis. Then, 61 envelope circles can be simulated with the established envelope circle mathematical model shown in Figure 14.Figure 14
Envelope circle model.The analysis of Figure14 shows the following:
(1)
Envelope circles can be expanded in accordance with the expected topological rules. The numbers of first-, second-, third-, fourth-, and fifth-layer envelopes are 1, 6, 12, 18, and 24, respectively. This finding indicates that the relationship between the number of modules in each layer and the number of layers is correct; i.e., Equation (1) is correct. That is, the layered topology of modules is correct(2)
A total of 61 envelopes in 5 layers agree well with the spherical surface. This finding shows that the envelope circle model established using Equations (4) and (10) is correct and that the array topology of the module is correct(3)
Any adjacent three envelope circles have only one intersection. In the first four layers, six intersections occur on each envelope circle, and the distribution of intersections is relatively uniform. This finding indicates that constructing a spherical surface with an envelope circle idea while retaining the same structural parameters of the rib unit is feasible, and the envelope circle model proposed in the current study is correct


### 4.2. Verification of Networking Model

By using the same structural parameters as those in Section4.1, the coordinates of the module connection points can be calculated using Equation (12), and several modules can be installed at predetermined positions similar to “building blocks.” Then, a modular deployable antenna structure networking model can be obtained (Figure 15).Figure 15
Structural networking model of a modular deployable antenna.As shown in Figure15, several hexagonal prism modules achieve the goal of accurate connection and spherical construction. This result demonstrates that the networking model is correct.


### 4.3. Analysis of Module Limit Topology Layers

In the process of forming a spherical network, a certain constraint is imposed among the number of modules, size of modules, and antenna aperture. When the antenna aperture is fixed, the larger the single module, the fewer modules are required. When the module is smaller, the number of required modules is higher. However, regardless of the number of modules, a spherical network should ensure that no self-shielding structure occurs; that is, the expansion of modules cannot exceed the hemisphere; otherwise, signal transmission is affected. Thus, the number of topological layers of a module has a limit, and this rule is analyzed as follows. Figure16 depicts the top view of Figure 14. As shown in Figure 16, the size of the envelope circle model in the y-axis direction is larger than that in the x-axis direction; hence, the envelope circle in the y-axis direction will reach the hemisphere first.Figure 16
Top view of the envelope circle model.Focus on the five envelope circles in the positivey-axis direction of Figure 16. The midpoints of the chord intersecting envelope circles 1 and 12 are M1, and the midpoints of the chord intersecting envelope circles 12 and 14 are M2, as shown in Figure 17.Figure 17
Positional relationships of several envelope circles: (a)xoy view and (b) yoz view.
(a)(b)In accordance with the geometric relationship illustrated in Figure17, five angle parameters can be obtained as follows:
(13)ϕ2=ϕ1=arctanrcosπ/6R2−r2,ϕ4=ϕ3=arctanLS12M2R2−r2,ϕ6=ϕ5=arctanLS14M3R2−r2,ϕ8=ϕ7=arctanLS17M4R2−r2,ϕ9=arctanLS21M5R2−r2.The structure cannot exceed the hemispherical surface; thus, the following equation can be expressed as(14)2ϕ1+2ϕ3+2ϕ5+2ϕ7+ϕ9≤π2.When Equation (13) is substituted into Equation (14), the following relationship can be obtained:
(15)2arctanrcosπ/6R2−r2+2arctanLS12M2R2−r2+2arctanLS14M3R2−r2+2arctanLS17M4R2−r2+arctanLS21M5R2−r2≤π2.As shown in Figure16, the length of intersecting chords becomes shorter as the number of modules increases. Thus, the following equation can be determined:
(16)rcosπ6≤LS12M2≤LS14M3≤LS17M4≤LS21M5.Therefore,(17)arctanrcosπ/6R2−r2≤arctanLS12M2R2−r2≤arctanLS14M3R2−r2≤arctanLS17M4R2−r2≤arctanLS21M5R2−r2.The following equation can be obtained by substituting Equation (17) into Equation (15):
(18)9arctanrcosπ/6R2−r2≤π2.The result of Equation (18) can be extended from five to more layers.
(19)2i−1arctanrcosπ/6R2−r2≤π2,where i represents the number of layers of a modular structure.Equation (19) is simplified as follows:
(20)i≤π+2arctanrcosπ/6/R2−r24arctanrcosπ/6/R2−r2.Equation (20) denotes that a spherical surface with radius R is constructed using a hexagonal prism module with radius r of the envelope circle. The maximum number of layers that can be topological in the case of a spherical surface is less than or equal to that in the case of a hemispherical surface.When parametersR=4701 mm and r=600 mm are substituted into Equation (20), the limit layer number is seven when the spherical surface is less than or equal to the hemispherical surface. To verify the correctness of Equation (20), an envelope circle model with eight layers is established using a numerical simulation method, as shown in Figure 18.Figure 18
Envelope circle model with eight layers (including 169 envelope circles): (a) 3D view, (b)xoy view, and (c) xoz view.
(a)(b)(c)As illustrated in Figure18, the partial envelope circle crosses the hemispherical surface when the number of layers is eight and not seven. The limit layer number is seven, and Equation (20) is correct.


## 4.1. Verification of Envelope Circle Model

Assume that the parameters areR=4701 mm and r=600 mm. These parameters are used for the succeeding analysis. Then, 61 envelope circles can be simulated with the established envelope circle mathematical model shown in Figure 14.Figure 14
Envelope circle model.The analysis of Figure14 shows the following:
(1)
Envelope circles can be expanded in accordance with the expected topological rules. The numbers of first-, second-, third-, fourth-, and fifth-layer envelopes are 1, 6, 12, 18, and 24, respectively. This finding indicates that the relationship between the number of modules in each layer and the number of layers is correct; i.e., Equation (1) is correct. That is, the layered topology of modules is correct(2)
A total of 61 envelopes in 5 layers agree well with the spherical surface. This finding shows that the envelope circle model established using Equations (4) and (10) is correct and that the array topology of the module is correct(3)
Any adjacent three envelope circles have only one intersection. In the first four layers, six intersections occur on each envelope circle, and the distribution of intersections is relatively uniform. This finding indicates that constructing a spherical surface with an envelope circle idea while retaining the same structural parameters of the rib unit is feasible, and the envelope circle model proposed in the current study is correct


## 4.2. Verification of Networking Model

By using the same structural parameters as those in Section4.1, the coordinates of the module connection points can be calculated using Equation (12), and several modules can be installed at predetermined positions similar to “building blocks.” Then, a modular deployable antenna structure networking model can be obtained (Figure 15).Figure 15
Structural networking model of a modular deployable antenna.As shown in Figure15, several hexagonal prism modules achieve the goal of accurate connection and spherical construction. This result demonstrates that the networking model is correct.


## 4.3. Analysis of Module Limit Topology Layers

In the process of forming a spherical network, a certain constraint is imposed among the number of modules, size of modules, and antenna aperture. When the antenna aperture is fixed, the larger the single module, the fewer modules are required. When the module is smaller, the number of required modules is higher. However, regardless of the number of modules, a spherical network should ensure that no self-shielding structure occurs; that is, the expansion of modules cannot exceed the hemisphere; otherwise, signal transmission is affected. Thus, the number of topological layers of a module has a limit, and this rule is analyzed as follows. Figure16 depicts the top view of Figure 14. As shown in Figure 16, the size of the envelope circle model in the y-axis direction is larger than that in the x-axis direction; hence, the envelope circle in the y-axis direction will reach the hemisphere first.Figure 16
Top view of the envelope circle model.Focus on the five envelope circles in the positivey-axis direction of Figure 16. The midpoints of the chord intersecting envelope circles 1 and 12 are M1, and the midpoints of the chord intersecting envelope circles 12 and 14 are M2, as shown in Figure 17.Figure 17
Positional relationships of several envelope circles: (a)xoy view and (b) yoz view.
(a)(b)In accordance with the geometric relationship illustrated in Figure17, five angle parameters can be obtained as follows:
(13)ϕ2=ϕ1=arctanrcosπ/6R2−r2,ϕ4=ϕ3=arctanLS12M2R2−r2,ϕ6=ϕ5=arctanLS14M3R2−r2,ϕ8=ϕ7=arctanLS17M4R2−r2,ϕ9=arctanLS21M5R2−r2.The structure cannot exceed the hemispherical surface; thus, the following equation can be expressed as(14)2ϕ1+2ϕ3+2ϕ5+2ϕ7+ϕ9≤π2.When Equation (13) is substituted into Equation (14), the following relationship can be obtained:
(15)2arctanrcosπ/6R2−r2+2arctanLS12M2R2−r2+2arctanLS14M3R2−r2+2arctanLS17M4R2−r2+arctanLS21M5R2−r2≤π2.As shown in Figure16, the length of intersecting chords becomes shorter as the number of modules increases. Thus, the following equation can be determined:
(16)rcosπ6≤LS12M2≤LS14M3≤LS17M4≤LS21M5.Therefore,(17)arctanrcosπ/6R2−r2≤arctanLS12M2R2−r2≤arctanLS14M3R2−r2≤arctanLS17M4R2−r2≤arctanLS21M5R2−r2.The following equation can be obtained by substituting Equation (17) into Equation (15):
(18)9arctanrcosπ/6R2−r2≤π2.The result of Equation (18) can be extended from five to more layers.
(19)2i−1arctanrcosπ/6R2−r2≤π2,where i represents the number of layers of a modular structure.Equation (19) is simplified as follows:
(20)i≤π+2arctanrcosπ/6/R2−r24arctanrcosπ/6/R2−r2.Equation (20) denotes that a spherical surface with radius R is constructed using a hexagonal prism module with radius r of the envelope circle. The maximum number of layers that can be topological in the case of a spherical surface is less than or equal to that in the case of a hemispherical surface.When parametersR=4701 mm and r=600 mm are substituted into Equation (20), the limit layer number is seven when the spherical surface is less than or equal to the hemispherical surface. To verify the correctness of Equation (20), an envelope circle model with eight layers is established using a numerical simulation method, as shown in Figure 18.Figure 18
Envelope circle model with eight layers (including 169 envelope circles): (a) 3D view, (b)xoy view, and (c) xoz view.
(a)(b)(c)As illustrated in Figure18, the partial envelope circle crosses the hemispherical surface when the number of layers is eight and not seven. The limit layer number is seven, and Equation (20) is correct.


## 5. Conclusions

This study proposes a networking model for the structure of a hexagonal prism modular deployable space antenna. This model is verified using a numerical simulation method, and the effects of the parameters on networking characteristics are simulated and analyzed. The major conclusions are as follows:(1)
By studying the hexagonal prism honeycomb structure, the expansion rule of a hexagonal prism modular structure is clarified, and the results provide the necessary basis for establishing the networking model of the deployable antenna structure(2)
Simplifying the structure with an envelope circle idea is feasible. The networking model built using this method can form a spherical surface network. This method can also accurately connect at least dozens of modules and provide a theoretical basis for the configuration design and kinematics modeling of modular deployable antennas with ultramultiple modules(3)
The proposed networking method can accurately calculate the spatial coordinates of key nodes on the module and provide data support for investigating the surface accuracy of the truss structure and has important guiding significance for the detailed design and engineering application of parabolic deployable antenna or parabolic cylindrical deployable antenna. In the future, more in-depth research on the shape accuracy of deployable mechanism can be carried out to provide technical support for accelerating the engineering application of modular deployable antenna(4)
Strong constraints are imposed among the number of topological layers, the radius of the fitted sphere, and the radius of the envelope circle. The extremum of the number of topological layers is based on the premise that no self-shielding occurs in the structure. This analysis provides a theoretical basis for the combination design of the number and size of modules, the deployment aperture, and other parameters



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*Source: 4583374-2022-09-26.xml*