An over-constrained cross deployable parabolic cylindrical antenna support structure
By designing a constrained cross-type deployable parabolic cylindrical antenna support structure and using a motor-driven traction rope to control the movement of the slider, the problem of high precision and high stability of large spaceborne deployable parabolic cylindrical antennas in complex electromagnetic environments was solved. This resulted in a modular support structure with high rigidity and high strength, suitable for antennas of various apertures.
Patent Information
- Application Number
- CN202310123208.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-02-16
- Publication Date
- 2026-02-24
- Estimated Expiration
- 2043-02-16
AI Technical Summary
Existing technologies are insufficient to meet the high precision, high gain, and high stability requirements of large spaceborne deployable parabolic cylindrical antennas in complex space electromagnetic environments, and my country's research in this field is relatively lagging behind.
Design a constrained cross-type deployable parabolic cylindrical antenna support structure, including an arc-shaped deployable unit, a cylindrical deployable unit, and a motion control device. The traction rope is driven by a motor and a reducer to control the movement of the slider, thereby realizing the deployment and locking of the arc-shaped and cylindrical units. It has modular characteristics.
It achieves a reasonable spatial distribution in the folded state and forms a stable cross-shaped parabolic cylindrical antenna support structure after unfolding, which improves stiffness, strength, unfolding reliability and stability, and is suitable for antenna requirements of different apertures.
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Figure CN116231271B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of aerospace technology, and in particular to an over-constrained cross-type deployable parabolic cylindrical antenna support structure. Background Technology
[0002] With the development of the information age, the demand for large-aperture spaceborne deployable antennas is becoming increasingly urgent in various countries, aiming to achieve the acquisition of low-power signals, transmission of large information capacity, and improvement of remote sensing resolution. Research on large-aperture spaceborne deployable antennas has gradually become a research hotspot in the aerospace field both domestically and internationally. Large-aperture spaceborne deployable antennas have wide applications in important projects such as Earth observation, military reconnaissance, wireless communication, manned spaceflight, and deep space exploration, and are of great significance for improving the communication, reconnaissance, navigation, and resource exploration capabilities of satellites.
[0003] Due to the development of space technology and the increase in the number of satellites, satellite frequency and orbit resources are becoming increasingly scarce and competition is fierce. Furthermore, the space electromagnetic environment is complex, with electromagnetic interference existing between the environment and equipment. Traditional single-beam antennas are insufficient to meet the current demands for information acquisition and transmission in terms of accuracy and gain. Therefore, large aperture, multi-band, high signal-to-noise ratio, high precision, high gain, and high deployment-to-reception ratio have become the development trend for large spaceborne deployable antennas. Deployable parabolic cylindrical antennas possess the strong directivity, high gain, automatic beam scanning capabilities, and ease of multi-band sharing characteristic of parabolic cylindrical antennas. During the launch phase, the deployable parabolic cylindrical antenna is contained within the envelope of the launch vehicle. After entering orbit, it gradually deploys under the combined action of the drive and control systems. Once fully deployed, it forms a stable structure using its structural characteristics and locking mechanism. Research on its configuration design and performance verification has attracted the attention of researchers worldwide, making the study of large spaceborne deployable parabolic cylindrical antennas an important research direction. Europe, the United States, Japan and other countries have successively carried out on-orbit tests and applications and achieved certain results. However, due to the relatively backward research on space deployable structures in the aerospace field in my country, the research on large deployable parabolic cylindrical antennas is still in the stage of theoretical design analysis and verification.
[0004] Against this backdrop, research on the support structure of novel deployable parabolic cylindrical antennas has certain reference value for the design, theoretical analysis, and engineering application of large deployable parabolic cylindrical antennas. At the same time, it also promotes the development of large space deployable antennas, and therefore has important and far-reaching significance. Summary of the Invention
[0005] The present invention aims to provide an over-constrained cross-type deployable parabolic cylindrical antenna support structure, which is intended to provide reference value for the structural design of large deployable parabolic cylindrical antennas.
[0006] To achieve the above objectives, the technical solution of the present invention is as follows: an over-constrained cross-type deployable parabolic cylindrical antenna support structure, comprising multiple arc-shaped deployable units, cylindrical deployable units, and a motion control device.
[0007] The arc-shaped developable unit includes two linear optical axes, an arc-shaped constraint mechanism, and an asymmetrical cross mechanism. Each linear optical axis is equipped with a limiting nut, and each linear optical axis has a slider at its upper end. The arc-shaped constraint mechanism consists of two first connecting rods rotatably connected at one end, and the other ends of the two first connecting rods are rotatably connected to the corresponding sliders. The asymmetrical cross mechanism includes two upper connecting parts, a first middle hinge, and two lower connecting parts. The first middle hinge consists of two rotatably connected first rotating blocks, and a torsion spring is connected between the two first rotating blocks. The free end of each first rotating block is rotatably connected to a second connecting rod. The four second connecting rods are located in two mutually parallel planes. The two second connecting rods on the upper side are rotatably connected to the corresponding sliders through two upper connecting parts. The length of the two second connecting rods on the upper side is greater than the length of the two second connecting rods on the lower side.
[0008] The deployable cylindrical unit includes two linear optical axes, a cylindrical constraint mechanism, and a symmetrical cross mechanism. Each linear optical axis is equipped with a limiting nut, and each linear optical axis has a slider at its upper end. The cylindrical constraint mechanism includes two third connecting rods and a limiting hinge. One end of each of the two third connecting rods is rotatably connected to the corresponding slider. The limiting hinge is a cross-shaped hinge and is rotatably connected between the two third connecting rods. The symmetrical cross mechanism includes two upper connecting parts, a second middle hinge, and two lower connecting parts. The two upper connecting parts are rotatably connected to the corresponding sliders. The second middle hinge consists of two rotatably connected second rotating blocks. A torsion spring is connected between the two second rotating blocks. A fourth connecting rod is rotatably connected to the free end of each second rotating block. The four fourth connecting rods are located in the same plane. The two fourth connecting rods on the upper side are rotatably connected to the corresponding sliders through two upper connecting parts. The four fourth connecting rods are all the same length.
[0009] The slider in each of the arc-shaped and cylindrical deployable units consists of a housing, a linear bearing, and two elastic retaining rings. The housing has a through hole in the center. The linear bearing is installed inside the housing through two elastic retaining rings that abut against the upper and lower sides of the linear bearing. The housing has grooves that engage with the elastic retaining rings.
[0010] Two adjacent arc-shaped developable units and arc-shaped developable units, as well as two arc-shaped developable units and cylindrical developable units, share a linear optical axis. The lower ends of the linear optical axes of two adjacent arc-shaped developable units and cylindrical developable units are connected to a first hinged disc. Each first hinged disc is rotatably connected to the free ends of two corresponding second connecting rods located on the lower side via a lower connector. The lower ends of the linear optical axes of two adjacent arc-shaped developable units and cylindrical developable units are connected to a second hinged disc. Each second hinged disc is rotatably connected to the free ends of two corresponding fourth connecting rods located on the lower side via a lower connector.
[0011] The motion control device is used to control the slider on any curved inn unit or cylindrical expandable unit to slide up and down on the corresponding linear optical axis.
[0012] Furthermore, the motion control device includes a motor support mounted on a linear optical axis, a motor mounted on the motor support, and a drive spring connecting the motor support and the corresponding slider. The motor support has two fixed pulleys and two fixing screws symmetrically distributed on it. A reducer is connected to the motor, and a winch is connected to the output shaft of the reducer. Two traction ropes are wound on the winch. The corresponding slider has movable pulleys symmetrically distributed on it. The two traction ropes pass through the corresponding fixed pulleys and pass over the corresponding movable pulleys, respectively. The free ends of the two traction ropes are fixed to the corresponding fixing screws, respectively.
[0013] With the above setup, the motor and reducer are used to tighten or release the two traction ropes. During the tightening process, the traction ropes move upward, which drives the movable pulley and the corresponding second slider to move upward. At this time, under the action of the drive spring, the upward speed of the slider will be constrained to a certain extent, thereby realizing the control of the speed of the second slider and improving the stability and reliability of this solution.
[0014] Furthermore, the ratio of the number of arc-shaped developable units to the number of cylindrical developable units is 1:1 or 2:1.
[0015] With the above configuration, the design of this deployable support structure has a high degree of modularity, and can be applied to antennas of different apertures through reasonable combination and matching of deployable units.
[0016] The basic deployable module can be composed of 2 curved deployable units and 2 cylindrical deployable units combined with a slider and a linear optical axis, with a ratio of 1:1; or it can be composed of 4 curved deployable units and 2 cylindrical deployable units combined with a slider and a linear optical axis, with a ratio of 2:1.
[0017] If other methods are used to increase the number of deployable units in the assembly module, this will reduce the stiffness and strength of the deployable support structure, as well as the reliability and stability of the deployment process.
[0018] Compared with existing technologies, the beneficial effects of this solution are:
[0019] 1. When folded up, this design forms a regular cuboid, which facilitates reasonable spatial distribution during transport. At the same time, multiple arc-shaped deployable units and cylindrical deployable units can be unfolded in the horizontal and vertical directions respectively to form the support structure of the cross-shaped parabolic cylindrical antenna.
[0020] 2. This solution features modular design and assembly, which reduces design and manufacturing complexity and shortens development time. Under constraints such as mass and volume, this solution can produce a deployable support structure for parabolic cylindrical antennas with high rigidity and strength, as well as high deployment reliability and stability, to meet the needs of various missions. Attached Figure Description
[0021] Figure 1 This is a schematic diagram of the retracted state of the over-constrained cross-type deployable parabolic cylindrical antenna support structure of the present invention;
[0022] Figure 2 This is a schematic diagram of the unfolded state of the over-constrained cross-type deployable parabolic cylindrical antenna support structure of the present invention;
[0023] Figure 3 This is a schematic diagram of the structure of the arc-shaped developable unit in this embodiment;
[0024] Figure 4 This is a side view of the retracted state of the arc-shaped expandable unit in this embodiment;
[0025] Figure 5 This is a schematic diagram of the arc-shaped constraint mechanism in this embodiment;
[0026] Figure 6 This is a schematic diagram of the asymmetric cross mechanism in this embodiment;
[0027] Figure 7 This is a schematic diagram of the structure of the first hinge in this embodiment;
[0028] Figure 8 This is a schematic diagram of the cylindrical deployable unit in this embodiment;
[0029] Figure 9 This is a side view of the cylindrical expandable unit in the collapsed state in this embodiment.
[0030] Figure 10 This is a schematic diagram of the cylindrical constraint mechanism in this embodiment;
[0031] Figure 11 This is a schematic diagram of the symmetrical cross mechanism in this embodiment;
[0032] Figure 12This is a schematic diagram of the second hinge in this embodiment;
[0033] Figure 13 This is a schematic diagram of the structure of the second rotating block in this embodiment;
[0034] Figure 14 This is a schematic diagram of the slider structure in this embodiment;
[0035] Figure 15 This is a cross-sectional view of the slider in this embodiment;
[0036] Figure 16 This is a schematic diagram of the motion control device in this embodiment;
[0037] Figure 17 This is a schematic diagram of the structure of the first hinged flower plate in this embodiment;
[0038] Figure 18 This is a schematic diagram of the structure of the second hinged flower plate in this embodiment. Detailed Implementation
[0039] The present invention will be further described in detail below through specific embodiments:
[0040] The reference numerals in the accompanying drawings of the instruction manual include: linear optical axis 1, limiting nut 2, housing 3, linear bearing 4, elastic retaining ring 5, first connecting rod 6, upper connecting piece 7, lower connecting piece 8, first rotating block 9, pin shaft 10, second connecting rod 11, third connecting rod 12, fixed block 13, slide rod 14, second rotating block 15, rotating groove 16, torsion spring 17, fourth connecting rod 18, limiting block 19, first hinge plate 20, second hinge plate 21, motor support 22, motor 23, reducer 24, drive spring 25, fixed pulley 26, winch 27, traction rope 28, movable pulley 29.
[0041] Example
[0042] As attached Figure 1-18 As shown, a constrained, cross-shaped deployable parabolic cylindrical antenna support structure includes multiple arc-shaped deployable units, cylindrical deployable units, and motion control devices. In this embodiment, the ratio of arc-shaped deployable units to cylindrical deployable units is 2:1, but a 1:1 ratio can also be used. The number of motion control devices needs to be determined based on the number of arc-shaped and cylindrical deployable units. In this embodiment, the minimum number of arc-shaped and cylindrical deployable units in the support structure includes four arc-shaped deployable units and two cylindrical deployable units. Therefore, the motion control devices are located on the diagonal of the deployed support structure, and the number of motion control devices is two.
[0043] The arc-shaped deployable unit includes two linear optical axes 1, an arc-shaped constraint mechanism, and an asymmetrical cross mechanism. Each linear optical axis 1 has a limiting nut 2 in its middle, which controls the distance the slider moves, thus controlling the specific shape and size of the unfolded arc-shaped deployable unit. Each linear optical axis 1 has a slider slidably connected to its upper end. The slider consists of a housing 3, a linear bearing 4, and two elastic retaining rings 5. The housing 3 has a through hole in its center, with two spaced grooves inside. The linear bearing 4 is sleeved on the linear optical axis 1 and positioned between the two elastic retaining rings 5, which engage in the grooves and abut against the upper and lower sides of the linear bearing 4. The arc-shaped constraint mechanism consists of two first connecting rods 6 rotatably connected at one end. The ends of the two first connecting rods 6 that are close to each other are hinged together, and the two ends that are far apart are rotatably connected to the corresponding sliders. The asymmetrical cross mechanism includes two upper connectors 7, a first middle hinge, and two lower connectors 8. The first middle hinge consists of two rotatably connected first rotating blocks 9. A pin 10 passes through the middle of the two first rotating blocks 9. The pin 10 is covered by a torsion spring 17 connected between the two first rotating blocks 9. A slot for placing the torsion spring 17 is provided at the connection point of the two first rotating blocks 9. The free end of each first rotating block 9 is rotatably connected to a second connecting rod 11. The four second connecting rods 11 are located in two mutually parallel planes. The two second connecting rods 11 on the upper side are rotatably connected to the corresponding sliders through the two upper connectors 7. The length of the two second connecting rods 11 on the upper side is slightly greater than the length of the two second connecting rods 11 on the lower side.
[0044] The cylindrical deployable unit includes two linear optical axes 1, a cylindrical constraint mechanism, and a symmetrical cross mechanism. Each linear optical axis 1 is equipped with a limiting nut 2, which controls the distance the slider moves, thereby controlling the specific shape and size of the unfolded curved deployable unit. Each linear optical axis 1 has a slider at its upper end, consisting of a housing 3, a linear bearing 4, and two elastic retaining rings 5. The housing 3 has a through hole in its center, with two spaced grooves inside. The linear bearing 4 is fitted onto the linear optical axis 1 and positioned between the two elastic retaining rings 5, which engage within the grooves and abut against the upper and lower sides of the linear bearing 4. The cylindrical constraint mechanism includes two third links 12 and a limiting hinge. The ends of the two third links 12 that are far apart from each other are rotatably connected to corresponding sliders. The limiting hinge is a cross-shaped hinge, which includes two fixed blocks 13 and an L-shaped slide rod 14. The cross-sectional shape of the two fixed blocks 13 is U-shaped. The two fixed blocks 13 are respectively connected to the ends of the two third links 12 that are close to each other. Each fixed block 13 has a sliding groove on both its front and rear sides. The two ends of the slide rod 14 are each provided with a sliding shaft. The two sliding shafts are slidably connected in the corresponding sliding grooves, thereby realizing the rotational connection of the ends of the two third links 12 that are close to each other by means of the slide rod 14 and the sliding shaft. The limiting hinge avoids the problem of return stroke when the cylindrical constraint mechanism is unfolded to 180°. The symmetrical cross mechanism includes two upper connectors 7, a second middle hinge, and two lower connectors 8. The two upper connectors 7 are rotatably connected to their corresponding sliders. The second middle hinge consists of two rotatably connected second rotating blocks 15. The two second rotating blocks 15 have mutually cooperating rotating grooves 16. The middle part of each second rotating block 15 is rotatably connected to the rotating groove 16 of the other second rotating block 15. A pin 10 passes through the middle of the two second rotating blocks 15. The pin 10 is covered with a torsion spring 17 connected between the two first rotating blocks 9. A placement groove for the torsion spring 17 is provided at the connection point of the two second rotating blocks 15. The free end of each second rotating block 15 is rotatably connected to a fourth link 18. The four fourth links 18 are located in the same plane. The two fourth links 18 located on the upper side are rotatably connected to their corresponding sliders through the two upper connectors 7. The lengths of the four fourth links 18 are all the same.
[0045] In each of the aforementioned arc-shaped and cylindrical deployable units, a limiting block 19 is provided at the upper end of the linear optical axis 1 without a motion control device. Furthermore, in this embodiment, all first links 6, second links 11, third links 12, and fourth links 18 are made of carbon fiber.
[0046] Two adjacent arc-shaped developable units and arc-shaped developable units, as well as between arc-shaped developable units and cylindrical developable units, share a linear optical axis 1. The lower ends of the linear optical axis 1 of two adjacent arc-shaped developable units and cylindrical developable units are connected to a first hinged disc 20. Each first hinged disc 20 is rotatably connected to the free ends of two corresponding second connecting rods 11 located on the lower side via a lower connecting piece 8. The lower ends of the linear optical axis 1 of two adjacent arc-shaped developable units and cylindrical developable units are connected to a second hinged disc 21. Each second hinged disc 21 is rotatably connected to the free ends of two corresponding fourth connecting rods 18 located on the lower side via a lower connecting piece 8. The first hinged disc 20 and the second hinged disc 21 differ only in the position and number of hinge points. The first hinged disc 20 has two hinge points, distributed on the left and right sides of the first hinged disc 20. The second hinged disc 21 has four hinge points, with one hinge point on each of the front, back, left, and right sides of the second hinged disc 21.
[0047] The motion control device is used to control the slider on any cylindrical deployable unit to slide up and down on the corresponding linear optical axis 1. The motion control device includes a motor support 22 mounted on the linear optical axis 1, a motor 23 bolted to the motor support 22, and a drive spring 25 connecting the motor support 22 and the corresponding slider. The motor support 22 is fixedly mounted on the top of the corresponding linear optical axis 1. Through slots are provided on both the left and right sides of the motor support 22. A fixed pulley 26 and a fixing screw located outside the fixed pulley 26 are rotatably connected in each through slot. The output shaft of motor 23 is connected to reducer 24, which is fixedly connected to motor support 22 via a fixing plate. A winch 27 is connected to the output shaft of reducer 24, and a fixing frame fixedly connected to motor support 22 is rotatably connected to the other side of winch 27. Two spaced traction slots are provided on winch 27, and a traction rope 28 is wound in each traction slot. Movable pulleys 29 are symmetrically distributed on the left and right sides of the corresponding slider (the slider on the side without motion control device does not have a movable pulley 29, but the other structures are the same). The two traction ropes 28 pass through the corresponding fixed pulley 26 and pass over the corresponding movable pulley 29, respectively. The free ends of the two traction ropes 28 are fixed to the corresponding fixing screws.
[0048] The working process of this plan:
[0049] When the support structure is in the retracted state, it forms a cuboid, with the sliders located at the upper end of the corresponding linear optical axis 1, and the drive spring 25 in a compressed state.
[0050] When the support structure needs to be deployed, all the motors 23 in the motion control devices are activated simultaneously. At this time, each motor 23 drives the winch 27 to rotate through the reducer 24. When the winch 27 releases the traction rope 28, the traction rope 28 drives the slider downward through the fixed pulley 26 and the movable pulley 29 under the elastic force of the drive spring 25. After the slider moves, the linear optical axis 1 of the arc-shaped deployable unit and the cylindrical deployable unit are mutually deployed through the asymmetrical cross mechanism and arc-shaped constraint mechanism in the arc-shaped deployable unit, and the symmetrical cross mechanism and cylindrical constraint mechanism in the cylindrical deployable unit. When the slider moves to the limit nut 2, the angle between the first link 6 of the arc-shaped constraint mechanism and the third link 12 of the cylindrical constraint mechanism reaches 180°. The slider is pressed against the limit nut 2 by the drive spring 25, and the arc-shaped deployable unit and the cylindrical deployable unit are fully deployed, and the mechanism self-locks at the same time, and the support structure is fully deployed.
[0051] The above are merely embodiments of the present invention, and common knowledge such as specific structures and / or characteristics in the solutions are not described in detail here. It should be noted that those skilled in the art can make various modifications and improvements without departing from the structure of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.
Claims
1. A support structure for an over-constrained, cross-type deployable parabolic cylindrical antenna, characterized in that: It includes multiple curved deployable units, cylindrical deployable units, and motion control devices. The arc-shaped developable unit includes two linear optical axes, an arc-shaped constraint mechanism, and an asymmetrical cross mechanism. Each linear optical axis is equipped with a limiting nut, and each linear optical axis has a slider at its upper end. The arc-shaped constraint mechanism consists of two first connecting rods rotatably connected at one end, and the other ends of the two first connecting rods are rotatably connected to the corresponding sliders. The asymmetrical cross mechanism includes two upper connecting parts, a first middle hinge, and two lower connecting parts. The first middle hinge consists of two rotatably connected first rotating blocks, and a torsion spring is connected between the two first rotating blocks. The free end of each first rotating block is rotatably connected to a second connecting rod. The four second connecting rods are located in two mutually parallel planes. The two second connecting rods on the upper side are rotatably connected to the corresponding sliders through two upper connecting parts. The length of the two second connecting rods on the upper side is greater than the length of the two second connecting rods on the lower side. The deployable cylindrical unit includes two linear optical axes, a cylindrical constraint mechanism, and a symmetrical cross mechanism. Each linear optical axis is equipped with a limiting nut, and each linear optical axis has a slider at its upper end. The cylindrical constraint mechanism includes two third connecting rods and a limiting hinge. One end of each of the two third connecting rods is rotatably connected to the corresponding slider. The limiting hinge is a cross-shaped hinge and is rotatably connected between the two third connecting rods. The symmetrical cross mechanism includes two upper connecting parts, a second middle hinge, and two lower connecting parts. The two upper connecting parts are rotatably connected to the corresponding sliders. The second middle hinge consists of two rotatably connected second rotating blocks. A torsion spring is connected between the two second rotating blocks. A fourth connecting rod is rotatably connected to the free end of each second rotating block. The four fourth connecting rods are located in the same plane. The two fourth connecting rods on the upper side are rotatably connected to the corresponding sliders through two upper connecting parts. The four fourth connecting rods are all the same length. The slider in each of the arc-shaped and cylindrical deployable units consists of a housing, a linear bearing, and two elastic retaining rings. The housing has a through hole in the center. The linear bearing is installed inside the housing through two elastic retaining rings that abut against the upper and lower sides of the linear bearing. The housing has grooves that engage with the elastic retaining rings. Two adjacent arc-shaped developable units and arc-shaped developable units, as well as two arc-shaped developable units and cylindrical developable units, share a linear optical axis. The lower ends of the linear optical axes of two adjacent arc-shaped developable units and cylindrical developable units are connected to a first hinged disc. Each first hinged disc is rotatably connected to the free ends of two corresponding second connecting rods located on the lower side via a lower connector. The lower ends of the linear optical axes of two adjacent arc-shaped developable units and cylindrical developable units are connected to a second hinged disc. Each second hinged disc is rotatably connected to the free ends of two corresponding fourth connecting rods located on the lower side via a lower connector. The motion control device is used to control the slider on any arc-shaped or cylindrical developable unit to slide up and down on the corresponding linear optical axis.
2. The over-constrained cross-type deployable parabolic cylindrical antenna support structure according to claim 1, characterized in that: The motion control device includes a motor support mounted on a linear optical axis, a motor mounted on the motor support, and a drive spring connecting the motor support and the corresponding slider. Two fixed pulleys and two fixing screws are symmetrically distributed on the motor support. A reducer is connected to the motor, and a winch is connected to the output shaft of the reducer. Two traction ropes are wound on the winch. Movable pulleys are symmetrically distributed on the corresponding sliders. The two traction ropes pass through the corresponding fixed pulleys and pass over the corresponding movable pulleys, respectively. The free ends of the two traction ropes are fixed to the corresponding fixing screws, respectively.
3. The over-constrained cross-type deployable parabolic cylindrical antenna support structure according to claim 1, characterized in that: The ratio of the number of arc-shaped developable units to the number of cylindrical developable units is 1:1 or 2:1.
Citation Information
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