Deployable antenna mechanism based on bottom equilateral triangular pyramid and hexagonal pyramid heterogeneous module and paraboloid profile networking method thereof

Abstract

The invention discloses a deployable antenna mechanism based on bottom equilateral triangular pyramid and hexagonal pyramid heterogeneous modules and a paraboloid profile networking method thereof. The deployable antenna mechanism is formed by combining a plurality of pyramid combination units in a networking mode. Each pyramid combination unit comprises a hexagonal pyramid basic deployable unit and nine triangular pyramid basic deployable units which are arranged around the hexagonal pyramid basic deployable unit and movably connected with the hexagonal pyramid basic deployable unit, and every two adjacent triangular pyramid basic deployable units are movably connected through a foldable connecting rod piece. Parabolic profile networking of the deployable antenna mechanism is realized by adopting a triangular and hexagonal combined graph method, so that the deployable antenna mechanism can keep good profile precision in the unfolding and folding processes, and the adaptability and fault tolerance of fitting a compound profile division unit into a paraboloid are further improved; and the device can better adapt to parabolic profiles with different calibers.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of deployable antenna mechanisms, in particular to a deployable antenna mechanism based on isomorphic modules of equilateral triangular prisms and hexagonal prisms and a parabolic surface networking method thereof. BACKGROUND

[0002] Frame-type deployable antennas have become a research hotspot due to their high folding-to-deploying ratio and large rigidity. The parabolic surface needs to be designed and deployed with high precision through a reasonable surface division method because of its characteristics meeting the needs of sending and receiving electromagnetic wave signals.

[0003] At present, scholars at home and abroad have carried out a series of researches in the field of reflective surface division of deployable antenna mechanisms. Existing deployable antenna mechanisms mainly use hexagonal contour segmentation, three-way grid method, and force density method for surface division and reflector design. These methods have achieved remarkable results in improving structural performance. For example, Chinese patent "Parabolic frame-type deployable antenna mechanism based on shear-type hexagonal prism units" (Patent Application No. CN2023113537339) is composed of multiple shear-type hexagonal prism deployable units. Each unit is connected internally by upper and lower synchronous rods, shear rods, side edges, and bottom rods. The outer center node disc and the inner center node disc cooperate with each other, and the adjacent units are connected at three places through connecting pieces, lower connecting rods, and upper connecting rods, ensuring the compactness and flexibility of the overall structure and improving the overall rigidity of the deployed antenna. However, the research does not use programmatic methods for surface division and modeling, which makes it difficult to improve the accuracy of fitting a composite surface into a parabolic surface and the efficiency of parameterized modeling. SUMMARY

[0004] The first object of the present application is to provide a parabolic surface networking method based on isomorphic modules of equilateral triangular prisms and hexagonal prisms that can achieve accurate design and node fitting of parabolic surfaces.

[0005] The second object of the present application is to provide a deployable antenna mechanism based on isomorphic modules of equilateral triangular prisms and hexagonal prisms with high surface accuracy and strong stability.

[0006] Technical solution: The disclosed parabolic surface networking method based on isomorphic modules of equilateral triangular prisms and hexagonal prisms comprises the following steps:

[0007] S1: Construct a composite surface division unit composed of triangles and hexagons in a two-dimensional plane from the perspective of the top view;

[0008] S2: The two-dimensional planar view of the parabolic surface formed by the deployable antenna mechanism is divided by using the composite profile division unit, to obtain the combined graph with the characteristics of triangle and hexagon combination after division, and a coordinate system xoy is established at the center position of the combined graph;

[0009] S3: The length of the polygon in the composite profile division unit is calculated based on the caliber of the composite profile division unit set in advance, and the coordinates of the center point of the triangle of all composite profile division units in the coordinate system xoy are calculated based on the length of the polygon, to obtain the triangle center point coordinates;

[0010] S4: The rotation coordinate transformation method is used to construct the transformation equation formed by the angle relationship between the nodes based on the triangle and the center point of the triangle, and the two-dimensional vertex coordinates of the triangle are calculated based on the transformation equation and the triangle center point coordinates obtained in step S3;

[0011] S5: Based on the parameter equation of the parabolic surface set in advance , the three-dimensional node coordinates of the triangle node coordinates mapped to the three-dimensional coordinate system xyz are calculated, wherein f refers to the focal length of the parabolic surface; based on the curvature of the parabolic surface, the three-dimensional node coordinates obtained in step S4 are curve fitted and connected, to obtain the parabolic surface of the deployable antenna mechanism in the three-dimensional space;

[0012] S6: Based on the parabolic surface in the three-dimensional space obtained in step S5, the pyramid combination unit composed of three pyramid basic deployable units and six pyramid basic deployable units is constructed, the three-dimensional node coordinates of the triangle obtained refer to the center point of the lower end center node disc of the deployable antenna mechanism, and the length of the connecting line between the adjacent three-dimensional node coordinates refers to the length of the rod of the deployable antenna mechanism.

[0013] Based on the same inventive concept, the application further discloses a deployable antenna mechanism based on a bottom isosceles triangular pyramid and hexagonal pyramid isomodule, which is combined by a plurality of pyramid combination units in a networking manner; the pyramid combination unit comprises one hexagonal pyramid basic deployable unit and nine triangular pyramid basic deployable units arranged around the hexagonal pyramid basic deployable unit and movably connected with the hexagonal pyramid basic deployable unit; two adjacent triangular pyramid basic deployable units are movably connected through a foldable connecting rod; the regular hexagon formed by the orthographic projection of the bottom surface of the triangular pyramid basic deployable unit is a basic regular hexagon, the regular triangle formed by the orthographic projection of the bottom surface of the triangular pyramid basic deployable unit is a basic regular triangle, the side length of the basic regular hexagon is equal to that of the basic regular triangle, the orthographic projection of the bottom surface of the pyramid combination unit forms a combined regular hexagon, and the combined regular hexagon is spliced by the basic regular hexagon at the center and 18 basic regular triangles arranged around the basic regular hexagon, wherein the nine basic regular triangles arranged alternately are formed by the orthographic projection of the bottom surface of the triangular pyramid basic deployable unit, and three of the nine basic regular triangles arranged alternately share edges with the basic regular hexagon.

[0014] Further, the hexagonal pyramid basic deployable unit comprises six lower end center node discs respectively located at six vertexes of the basic regular hexagon, six lower center node disc folding linkage rods respectively located at six bottom edges of the basic regular hexagon, an upper center node disc located directly above the center point of the basic regular hexagon, and six side edges respectively connected between the upper center node disc and the corresponding lower end center node disc through a revolute pair, and the two ends of the lower center node disc folding linkage rod are respectively connected with the corresponding lower end center node disc through a revolute pair; the six lower center node disc folding linkage rods are synchronously folded and link the upper center node disc and the lower end center node disc to move towards the center point, and are folded into a cylindrical structure in which the six lower center node disc folding linkage rods are surrounded outside the six side edges.

[0015] Further, the triangular pyramid basic deployable unit comprises three lower end center node discs respectively located at three vertexes of the basic regular triangle, three lower center node disc folding linkage rods respectively located at three bottom edges of the basic regular triangle, an upper center node disc located directly above the center point of the basic regular triangle, and three side edges respectively connected between the upper center node disc and the corresponding lower end center node disc through a revolute pair, and the two ends of the lower center node disc folding linkage rod are respectively connected with the corresponding lower end center node disc through a revolute pair; the three lower center node disc folding linkage rods are synchronously folded and link the upper center node disc and the lower end center node disc to move towards the center point, and are folded into a cylindrical prism structure in which the three lower center node disc folding linkage rods are surrounded outside the three side edges.

[0016] Furthermore, the connecting rods include nine upper center node disk folding linkage rods and another six lower center node disk folding linkage rods. The upper center node disk folding linkage rods are rotatably connected between the upper center node disks of two adjacent triangular pyramidal deployable basic units through a rotary joint, and the upper center node disk folding linkage rods are longer than the lower center node disk folding linkage rods. The nine basic equilateral triangles include three basic equilateral triangles A that share sides with the basic regular hexagon and six basic equilateral triangles B that share angles with the basic regular hexagon. The six lower center node disk folding linkage rods are rotatably connected between the lower center node disks at the outer periphery of two adjacent basic equilateral triangles A and B through a rotary joint.

[0017] Furthermore, all the lower center node disk folding linkage rods fold in the same direction, while the upper center node disk folding linkage rods fold in the opposite direction to the lower center node disk folding linkage rods. The nine upper center node disk folding linkage rods and the thirty-six lower center node disk folding linkage rods fold synchronously and link the upper center node disks and the lower center node disks to move towards the center point, forming a cylindrical structure in which the twelve lower center node disk folding linkage rods on the base edge of the combined regular hexagon are located on the outermost layer, the remaining twenty-four lower center node disk folding linkage rods and the twenty-seven side edges of the basic deployable triangular pyramid unit are located in the middle layer, and the six side edges of the basic deployable hexagon unit are located in the innermost layer.

[0018] Furthermore, the included angle between each side edge of the basic developable unit of the triangular pyramid and the lower central node disk connected thereto is w, w∈[45°,55°]; the included angle between each side edge of the basic developable unit of the hexagonal pyramid and the lower central node disk connected thereto is θ, θ∈[40°,50°].

[0019] Furthermore, when the pyramidal assembly unit is fully unfolded, it exhibits a parabolic surface, with each lower central node disk resting on the parabolic surface, and the parametric equation of the parabolic surface is as follows:

[0020]

[0021] Where x, y, and z form a three-dimensional coordinate system, and f is the focal length of the parabolic surface.

[0022] Furthermore, the ratio of the spatial volume occupied by the pyramidal assembly unit when fully expanded to when fully collapsed is called the collapse ratio, and the formula for calculating the collapse ratio is as follows:

[0023]

[0024] Where V0 is the volume of the pyramidal unit when fully expanded, D0 is the diameter of the envelope circle of the central node disk of the reflecting surface when the pyramidal unit is fully expanded, d0 is the diameter of the envelope circle of the central node disk of the back frame when the pyramidal unit is fully expanded, and h0 is the vertical distance between the central node disk of the reflecting surface and the central node disk of the back frame when the pyramidal unit is fully expanded; V1 is the volume of the pyramidal unit when fully collapsed, D1 is the diameter of the envelope circle of the central node disk of the reflecting surface when the pyramidal unit is fully collapsed, d1 is the diameter of the envelope circle of the central node disk of the back frame when the pyramidal unit is fully collapsed, and h1 is the vertical distance between the central node disk of the reflecting surface and the central node disk of the back frame when the pyramidal unit is fully collapsed.

[0025] Furthermore, among the multiple pyramidal combination units, one pyramidal combination unit is selected to form a central combination unit, and the remaining pyramidal combination units are arranged in a circular grid according to the side length of the combination regular hexagon of the central combination unit.

[0026] Beneficial Effects: Compared with existing technologies, this invention has the following significant advantages: This invention uses a combination of triangles and hexagons to achieve parabolic surface networking of deployable antenna mechanisms, enabling the deployable antenna mechanism to maintain good surface accuracy during deployment and retraction, further improving the adaptability and fault tolerance of the composite surface partitioning unit fitting into a parabolic surface. Based on the rotation coordinate transformation method and the parametric equations of parabolic surfaces, this invention can achieve the conversion from two-dimensional planar partitioning to three-dimensional surface partitioning, with high computational efficiency during the partitioning process. This not only reduces the complexity of the shape-finding process but also helps improve the matching accuracy of the curvature and focal length of the parabolic surface. By combining the development law of polyhedra, this invention forms pyramidal combination units with symmetry and tight splicing characteristics of combined graphics. Rapid expansion is achieved through circular array networking, which can better adapt to parabolic surfaces of different apertures. Attached Figure Description

[0027] Figure 1 This is a flowchart of the method of the present invention;

[0028] Figure 2 This is a plan view of the composite surface division unit of the method of the present invention;

[0029] Figure 3 This is a plan view of the combined array network of multiple composite surface partitioning units in the method of the present invention;

[0030] Figure 4 This is a diagram showing the corresponding angles of two types of triangles, one placed face-up and the other inverted, in the method of this invention.

[0031] Figure 5 This is a visualization of the surface of the composite surface combination unit in an embodiment of the present invention;

[0032] Figure 6 This is a visualization diagram of the surface of a network of multiple composite surface combination units in an embodiment of the present invention.

[0033] Figure 7 This is an orthographic projection of the bottom surface of the pyramidal assembly unit of the present invention;

[0034] Figure 8 For the present invention Figure 7 A diagram showing the labeling of a basic equilateral triangle;

[0035] Figure 9 This is a frontal projection of the bottom surface of the multiple pyramidal combination units of the present invention when they are combined.

[0036] Figure 10 This is a top view of the fully unfolded basic deployable unit of the triangular pyramid of the present invention.

[0037] Figure 11 This is a schematic diagram of the structure of the basic deployable triangular pyramid unit of the present invention when fully unfolded.

[0038] Figure 12 This is a front view of the basic deployable triangular pyramidal unit of the present invention when fully collapsed.

[0039] Figure 13 This is a top view of the basic deployable triangular pyramid unit of the present invention when fully collapsed.

[0040] Figure 14 This is a top view of the hexagonal pyramid basic deployable unit of the present invention when fully unfolded.

[0041] Figure 15 This is a schematic diagram of the structure of the basic deployable hexagonal pyramidal unit of the present invention when fully unfolded.

[0042] Figure 16 This is a front view of the basic expandable hexagonal pyramidal unit of the present invention when fully collapsed.

[0043] Figure 17 This is a top view of the hexagonal pyramidal basic expandable unit of the present invention when it is fully collapsed.

[0044] Figure 18 This is a schematic diagram of the pyramidal assembly unit of the present invention when fully unfolded;

[0045] Figure 19 This is a top view of the pyramidal assembly unit when fully unfolded in an embodiment of the present invention;

[0046] Figure 20 This is a schematic diagram of the pyramidal assembly unit when fully unfolded in an embodiment of the present invention. Figure 1 ;

[0047] Figure 21This is a schematic diagram of the pyramidal assembly unit when fully unfolded in an embodiment of the present invention. Figure 2 ;

[0048] Figure 22 This is a schematic diagram of the pyramidal assembly unit of the present invention when fully collapsed;

[0049] Figure 23 This is a top view of the pyramidal assembly unit of the present invention when fully retracted;

[0050] Figure 24 This is a schematic diagram of the fully unfolded pyramidal assembly unit of the present invention, which has a parabolic shape.

[0051] Figure 25 This is a diagram showing the diameter of the pyramidal assembly unit of the present invention when it is fully collapsed;

[0052] Figure 26 This is a diagram showing the diameter of the pyramidal assembly unit of the present invention when fully expanded. Detailed Implementation

[0053] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0054] Example 1

[0055] This invention discloses a parabolic surface networking method based on heterogeneous modules of equilateral triangular pyramids and hexagonal pyramids on the base, such as... Figure 1 As shown, it includes the following steps:

[0056] S1: Construct a composite surface division unit composed of triangles and hexagons on a two-dimensional plane from a top-down perspective.

[0057] The combination of triangles and hexagons employs a rotational symmetry combination strategy to ensure that each composite surface division unit is precisely spliced ​​together to achieve fine division of the parabolic surface of the deployable antenna mechanism, forming a two-dimensional planar diagram of a parabolic surface with the graphic characteristics of a combination of triangles and hexagons.

[0058] like Figure 2 As shown, a two-dimensional planar diagram of a parabolic surface is divided using a composite surface division unit composed of triangles and hexagons. On the top-view plane, a hexagon and eighteen triangles are drawn, each with a side length identical to that of the hexagon, tightly joined around it. Three triangles share one side with the hexagon. These triangles are connected pairwise, with adjacent sides overlapping and sharing, ultimately forming a larger hexagonal composite unit composed of triangles and hexagons—the composite surface division unit.

[0059] The method of combining triangles and hexagons can realize the deployment of parabolic antenna surfaces, which can maintain good surface accuracy during the deployment and retraction of the antenna, and further improve the adaptability and fault tolerance of the network unit to fit the parabolic surface.

[0060] S2: The two-dimensional planar diagram of the parabolic surface formed by the deployable antenna mechanism is divided using composite surface division units to obtain a combined graphic with triangular and hexagonal features, and a coordinate system xoy is established at the center of the combined graphic.

[0061] like Figure 3 As shown, taking the composite surface division unit O as the center of the parabolic surface, a circular array is formed according to the number of side lengths of the composite surface division unit O. Finally, a two-dimensional planar diagram of the parabolic surface with a combination of triangles and hexagons is completed on the plane, thus obtaining the combined graphic. A coordinate system xoy is established with the center point of the composite surface division unit O as the origin. The remaining composite surface division units, arranged in a circular array around the center point of the composite surface division unit O, are sequentially labeled A, B, C, D, E, and F in a counter-clockwise direction. Furthermore, the eighteen triangles within each composite surface division unit are meticulously numbered, following the same counter-clockwise order. The triangles in composite surface division units O, A, B, C, D, E, and F are numbered o1~o18, a1~a18, b1~b18, c1~c18, d1~d18, e1~e18, and f1~f18, respectively; and the coordinates of the center points of the triangles in composite surface division unit O are respectively... The coordinates of the center points of the triangles in the composite surface division unit A are respectively The coordinates of the center points of the triangles in the composite surface division unit B are respectively The coordinates of the center points of the triangles in the composite surface division unit C are respectively The coordinates of the center points of the triangles in the composite surface division unit D are respectively The coordinates of the center points of the triangles in the composite surface division unit E are respectively The coordinates of the center points of the triangles in the composite surface division unit F are respectively .

[0062] S3: Calculate the side length of the polygons in the composite surface division unit based on the pre-defined diameter of the composite surface division unit, and calculate the coordinates of the center point of the triangle in the xoy coordinate system based on the side length of the polygon, thus obtaining the coordinates of the triangle center point.

[0063] The aperture, curvature, and focal length of the parabolic surface are set according to the actual situation. Based on the aperture of the composite surface unit, the side lengths of the triangles and hexagons in the composite surface unit can be calculated. Based on the calculated side lengths of the triangles and hexagons and the position of each composite surface unit on the xoy coordinate system, the coordinates of the center points of all triangles in all composite surface units are calculated, that is, the coordinates of the triangle center points are obtained, which is the calculation. , , , , , , .

[0064] S4: Construct a transformation equation based on the angular relationship between the nodes of the triangle and the center point of the triangle using the rotation coordinate transformation method. Calculate the coordinates of the triangle nodes based on the transformation equation and the coordinates of the triangle center point obtained in step S3; and calculate the parametric equations of the parabolic surface based on the pre-defined parabolic surface. Calculate the three-dimensional node coordinates of the triangle node in the three-dimensional coordinate system xyz, where f refers to the focal length of the parabolic surface.

[0065] like Figure 3 and 4 As shown, the triangles in the composite surface division unit have only two positional relationships: one is an equilateral triangle placed face-up, such as those numbered o1, o3, o5, o7, etc.; the other is an equilateral triangle inverted from the face-up triangle, such as those numbered o2, o4, o6, o8, etc. Therefore, it can be inferred that the angular relationships corresponding to all triangles in the composite surface of the deployable antenna mechanism are also only these two. Figure 7 As shown, establish a center point coordinate system based on the center point coordinates of the two types of triangles, one placed face-up and the other inverted. The geometric relationships of the figure are used to calculate the line connecting each node of the triangle to its center coordinates. Angle measure of the positive axis At the same time, the radius of the circumcircle of the triangle can be calculated based on the side lengths. .

[0066] Construct the coordinate matrix of the center point of the triangle in the composite surface partitioning unit. and two types of angle matrices :

[0067] (1)

[0068] (2)

[0069] in The interior angles of an equilateral triangle are used to establish the transformation equations relating the center point of the polygon to the angles using the rotation coordinate transformation method, based on the formulas (1) and (2) above, as shown in equation (3):

[0070] (3)

[0071] In the formula, This represents the x-value of the i-th coordinate in matrix P. This represents the y-value of the i-th coordinate in matrix P. This represents the included angle matrix corresponding to the i-th coordinate in matrix P. Let be the radius of the circumcircle of the triangle.

[0072] Substitute the coordinates of all the triangle center points obtained in step S3 into the established formula (3), and calculate the node coordinates of each triangle through coordinate transformation. Simultaneously, the coordinates of the center points of all triangles in the obtained composite figure and the coordinates of the transformed triangle nodes are substituted into the pre-defined parabolic surface parametric equations. In this process, the coordinate height z of each triangle vertex on the parabolic surface is calculated, and the two-dimensional triangle vertex coordinates are converted into three-dimensional triangle top coordinates.

[0073] S5: Curvature and Parametric Equations Based on Parabolic Surfaces The three-dimensional node coordinates obtained in step S4 are curve-fitted and connected to obtain the parabolic surface of the deployable antenna mechanism in three-dimensional space.

[0074] In practical use, Python is used to perform curve fitting and connection processing on the three-dimensional node coordinate data generated in step S4, thereby achieving accurate conversion from two-dimensional surfaces based on combined graphics to deployable antenna parabolic surfaces, and performing visualization of the divided surfaces and efficient data extraction.

[0075] The code for creating parabolic surfaces by dividing a composite shape using triangles and hexagons was written in Python. This included defining the aperture of the composite surface dividing unit, the curvature of the parabolic surface, and the parametric equations of the parabolic surface. The transformation equation from step S4 is also input, and a for loop is used to convert the center coordinates of each triangle into nodal coordinates according to angular relationships and coordinate transformation methods. Simultaneously, using the software's plot3 function, the nodal coordinates of each triangle and its height z on the parabolic surface are visualized, thus completing the construction of the composite parabolic surface partitioning unit for the deployable antenna based on a combination of triangles and hexagons, and the expansion unit of multiple composite surface partitioning units, such as... Figure 5 and 6As shown. In actual use, the Python `writematrix` function is used to save the coordinates of each node obtained from the programmatic surface division calculation as a txt file.

[0076] A parameterized model is constructed based on the rotation coordinate transformation method and the parametric equations of the parabola. The entire process from planar division to 3D surface transformation is automated using Python code, which can reduce the complexity of the shape finding process and improve the matching accuracy of the parabolic surface curvature and focal length.

[0077] S6: Based on the parabolic surface in three-dimensional space obtained in step S5, construct a pyramid combination unit formed by combining basic deployable triangular pyramid units and basic deployable hexagonal pyramid units. The three-dimensional node coordinates of the obtained triangles refer to the center point of the lower center node disk of the deployable antenna mechanism, and the length of the line connecting adjacent three-dimensional node coordinates refers to the rod length of the deployable antenna mechanism.

[0078] Following the shape rules of the combined graphic division, multiple pyramidal combination units are arranged in an orderly manner according to the layout principle of the circular array, and finally a deployable antenna mechanism with a parabolic surface based on triangular pyramids and hexagonal pyramids is formed.

[0079] This embodiment combines the unfolding rules of polyhedra to form a modular pyramidal combination unit with symmetry and tight splicing characteristics of combined graphics. It can be rapidly expanded through circumferential array networking, and can better adapt to parabolic reflective surfaces of different diameters, that is, it can just adapt to parabolic surfaces of different diameters.

[0080] This embodiment, based on deployable antenna mechanisms of triangular and hexagonal pyramids and their parabolic surface networking method, forms a pyramidal composite unit deployable antenna mechanism. This mechanism not only breaks the traditional division method of a single basic unit, ensuring the consistency of the antenna surface during deployment and retraction, but also features a modular pyramidal composite unit with symmetry and tightly spliced ​​composite graphics. It can be rapidly expanded through circumferential array networking to adapt to different shapes and larger sizes of reflective surfaces, improving the deployability and stability of the mechanism and meeting the high surface accuracy reflection requirements of high-performance antennas.

[0081] Example 2

[0082] This invention discloses a deployable antenna mechanism based on heterogeneous modules of equilateral triangular pyramids and hexagonal pyramids on the base. The deployable antenna mechanism is composed of several pyramidal combination units arranged in a network manner, such as... Figure 18 As shown, the pyramidal assembly unit includes a hexagonal pyramidal basic deployable unit and nine triangular pyramidal basic deployable units arranged around and movably connected to the hexagonal pyramidal basic deployable unit. Adjacent triangular pyramidal basic deployable units are movably connected by foldable connecting rods. Figure 7As shown, the orthographic projection of the base of the basic developable triangular pyramid unit forms a basic regular hexagon 1. The orthographic projection of the base of the basic developable triangular pyramid unit forms a basic equilateral triangle 2, and the side lengths of the basic regular hexagon 1 and the basic equilateral triangle 2 are equal. The orthographic projection of the base of the pyramid combination unit forms a combined regular hexagon 3, which is composed of the basic regular hexagon 1 at the center and 18 identical basic equilateral triangles 2 surrounding it. Nine of these 18 basic equilateral triangles 2 are formed by the orthographic projection of the base of the basic developable triangular pyramid unit, and three of these nine alternating basic equilateral triangles 2 share sides with the basic regular hexagon 1. Figure 8 As shown, the nine basic equilateral triangles 2 with odd numbers are formed by the orthographic projection of the base of the basic developable unit of the triangular pyramid, namely, those labeled ①, ③, ⑤, ⑦, ⑨, , The basic equilateral triangle is the orthographic projection of the base of the basic expandable unit.

[0083] like Figure 10 to Figure 13 As shown, the basic deployable unit of the triangular pyramid includes three lower center node disks 4, three lower center node disk folding linkage rods 5, one upper center node disk 6, and three side edges 7. The three lower center node disks 4 are located at the three vertices of the basic equilateral triangle 2, and the three lower center node disk folding linkage rods 5 are located at the three bases of the basic equilateral triangle 2. The two ends of the lower center node disk folding linkage rods 5 are rotatably connected to the corresponding lower center node disks 4 through revolute joints. The upper center node disk 6 is located directly above the center point of the basic equilateral triangle 2. The height of the upper center node disk 6 from the center point of the basic equilateral triangle 2 is set according to actual needs. The three side edges 7 are arranged around the upper center node disk 6 and serve to support the upper center node disk 6. The tops of the three side edges 7 are rotatably connected to different positions of the upper center node disk 6 through revolute joints, and the bottoms of the three side edges 7 are rotatably connected to the corresponding three lower center node disks 4 through revolute joints. The three side edges 7 constitute the three side edges of the triangular pyramid. Furthermore, the included angle between each side edge 7 and the lower center node disk 4 connected to it is w, where w ∈ [45°, 55°]. Each lower center node disk folding linkage rod 5 can be folded inward, and the included angle between two adjacent lower center node disk folding linkage rods 5 is 60°.

[0084] When the basic expandable unit of the triangular pyramid is fully retracted, the three side edges 7 retract into the parabolic surface, and the three lower center node disk folding linkage rods 5 also fold into the parabolic surface at the same time. After retraction, the three lower center node disk folding linkage rods 5 are at the outermost layer, and the three side edges 7 are at the innermost layer. The three side edges 7 and the three lower center node disk folding linkage rods 5 simultaneously drive the upper center node disk 6 and the three lower center node disks 4 to move towards the center point, and finally retract into a cylindrical frustum structure.

[0085] like Figure 14 to Figure 17 As shown, the basic deployable unit of the hexagonal pyramid includes six lower center node disks 4, six lower center node disk folding linkage rods 5, one upper center node disk 6, and six side edges 7. The six lower center node disks 4 are located at the six vertices of the basic regular hexagon 1, and the six lower center node disk folding linkage rods 5 are located at the six base edges of the basic regular hexagon 1. The two ends of the lower center node disk folding linkage rods 5 are rotatably connected to the corresponding lower center node disks 4 through revolute joints. The upper center node disk 6 is located directly above the center point of the basic regular hexagon 1. The height of the upper center node disk 6 from the center point of the basic regular hexagon 1 is set according to actual needs. The six side edges 7 are arranged around the upper center node disk 6 and serve to support the upper center node disk 6. The tops of the six side edges 7 are rotatably connected to different positions of the upper center node disk 6 through revolute joints, and the bottoms of the six side edges 7 are rotatably connected to the corresponding six lower center node disks 4 through revolute joints. The six side edges 7 constitute the six side edges of the hexagonal pyramid. Furthermore, the included angle between each side edge 7 and the lower center node disk 4 connected to it is θ, where θ∈[40°,50°]. Each lower center node disk folding linkage rod 5 can be folded inward, and the included angle between two adjacent lower center node disk folding linkage rods 5 is 120°.

[0086] When the basic expandable unit of the hexagonal pyramid is fully collapsed, the six lateral edges 7 of the hexagonal pyramid collapse inward into the parabolic surface, and the six lower central node disk folding linkage rods 5 also collapse inward into the parabolic surface at the same time. After collapse, the six lower central node disk folding linkage rods 5 are at the outermost layer, and the six lateral edges 7 are at the innermost layer. The six lateral edges 7 and the six lower central node disk folding linkage rods 5 simultaneously drive the upper central node disk 6 and the six lower central node disks 4 to move towards the center point, and finally collapse into a cylindrical structure.

[0087] like Figure 18 As shown, the connecting rods include nine upper center node disk folding linkage rods 8 and another six lower center node disk folding linkage rods 5. The upper center node disk folding linkage rods 8 are rotatably connected between the upper center node disks 6 of two adjacent triangular pyramidal deployable basic units through a rotary joint; as shown... Figure 2 and Figure 18 As shown, the nine basic equilateral triangles 2 include three basic equilateral triangles A that share the same side as the basic regular hexagon 1 and six basic equilateral triangles B that share the same angle as the basic regular hexagon 1. The six lower center node disk folding linkage rods 5 are respectively connected to the lower center node disks 4 at the outer periphery of two adjacent basic equilateral triangles A and B through a rotary joint.

[0088] like Figure 22 and 23As shown, the folding directions of all lower center node disk folding linkage rods 5 are consistent, while the folding directions of the upper center node disk folding linkage rods 8 are opposite to those of the lower center node disk folding linkage rods 5. The nine upper center node disk folding linkage rods 8 and the thirty-six lower center node disk folding linkage rods 5 fold synchronously and link the upper center node disk 6 and the lower center node disk 4 towards the center point, forming a cylindrical structure in which the twelve lower center node disk folding linkage rods 5 on the bottom edge of the combined regular hexagon 3 are located on the outermost layer, the remaining twenty-four lower center node disk folding linkage rods 5 and the twenty-seven side edges 7 of the basic deployable triangular pyramid are located in the middle layer, and the six side edges 7 of the basic deployable hexagonal pyramid are located in the innermost layer. After folding, all the upper center node disks 6 are at the same height.

[0089] Taking a pyramidal composite unit as an example, the pyramidal composite unit consists of a basic expandable hexagonal pyramidal unit and... Figure 8 The basic form of a triangular pyramid is composed of nine identical triangular pyramids whose center point coordinates are odd-numbered. For example... Figure 18 As shown, the pyramidal assembly unit includes twelve external lower center node disks, six internal lower center node disks, nine external upper center node disks and one internal upper center node disk, thirty-three equal-length side edges, thirty-six equal-length lower center node disk folding linkages, and nine equal-length upper center node disk folding linkages. Figure 19 As shown, the inner upper central node disk refers to the upper central node disk 001 of the basic developable hexagonal pyramidal element, and the nine outer upper central node disks are... Figure 8 The center points are numbered odd: ①, ③, ⑤, ⑦, ⑨. Each basic developable triangular pyramid has an upper central node disk, numbered from smallest to largest as follows: outer upper central node disk 2002, outer upper central node disk 3003, outer upper central node disk 4004, outer upper central node disk 5005, outer upper central node disk 6006, outer upper central node disk 7007, outer upper central node disk 8008, outer upper central node disk 9009, and outer upper central node disk 10010. Six lower central node disks located inside are the lower central node disks corresponding to the basic developable triangular pyramid units, numbered as inner lower central node disk 1011, inner lower central node disk 2012, inner lower central node disk 3013, inner lower central node disk 4014, inner lower central node disk 5015, and inner lower central node disk 6016. Twelve lower central node disks located outside are the lower central node disks corresponding to nine identical basic developable triangular pyramid units, numbered in counter-clockwise order. Set as the following external lower center node disks: 017, 018, 019, 020, 021, 022, 023, 024, 025, 026, 027, and 028.

[0090] like Figure 20 As shown, the thirty-six equal-length lower center node disk folding linkages are composed of the base sides of a basic regular hexagon 1 and eighteen identical basic equilateral triangles 2. The six inner lower center node disk folding linkages are respectively the lower center node disk folding linkages 17, 18, 19, 20, 21, and 22 of the hexagonal pyramid basic deployable unit. Figure 19 and Figure 20As shown, the two ends of the lower center node disk folding linkage rod 17 are connected to the inner lower center node disk one 011 and the inner lower center node disk two 012. The two ends of the lower center node disk folding linkage rod 18 are connected to the inner lower center node disk two 012 and the inner lower center node disk three 013. The two ends of the lower center node disk folding linkage rod 19 are connected to the inner lower center node disk three 013 and the inner lower center node disk four 014. The two ends of the lower center node disk folding linkage rod 20 are connected to the inner lower center node disk four 014 and the inner lower center node disk five 015. The two ends of the lower center node disk folding linkage rod 21 are connected to the inner lower center node disk five 015 and the inner lower center node disk six 016. The two ends of the lower center node disk folding linkage rod 22 are connected to the inner lower center node disk six 016 and the inner lower center node disk one 011.The twelve lower center node disks located on the outermost lower center node disks are connected end-to-end by twelve lower center node disk folding linkage rods, which are respectively set as lower center node disk folding linkage rods 51 to 62. These twelve lower center node disk folding linkage rods are located on the outer perimeter, and lower center node disk folding linkage rods 53, 54, 57, 58, 61, and 62 are connecting rods; the lower center node disk folding linkage rods are folding linkage rods. The two ends of linkage rod 51 are connected to the outer lower center node disk 1 017 and outer lower center node disk 2 018. The two ends of the lower center node disk folding linkage rod 52 are connected to the outer lower center node disk 2 018 and outer lower center node disk 3 019. The two ends of the lower center node disk folding linkage rod 53 are connected to the outer lower center node disk 3 019 and outer lower center node disk 4 020. The two ends of the lower center node disk folding linkage rod 54 are connected to the outer lower center node disk 4 020 and outer lower center node disk 5 021. The two ends of the folding linkage rod 55 are connected to the outer lower center node disk 5021 and the outer lower center node disk 6022. The two ends of the lower center node disk folding linkage rod 56 are connected to the outer lower center node disk 6022 and the outer lower center node disk 7023. The two ends of the lower center node disk folding linkage rod 57 are connected to the outer lower center node disk 7023 and the outer lower center node disk 8024. The two ends of the lower center node disk folding linkage rod 58 are connected to the outer lower center node disk 8024 and the outer lower center node disk 9025. The two ends of the folding linkage rod 59 are connected to the outer lower center node disk 9 025 and the outer lower center node disk 10 026. The two ends of the lower center node disk folding linkage rod 60 are connected to the outer lower center node disk 10 026 and the outer lower center node disk 11 027. The two ends of the lower center node disk folding linkage rod 61 are connected to the outer lower center node disk 11 027 and the outer lower center node disk 12 028. The two ends of the lower center node disk folding linkage rod 62 are connected to the outer lower center node disk 12 028 and the outer lower center node disk 1 017.

[0091] Eighteen lower center node disk folding linkage rods located between the inner and outer perimeters are used to connect the twelve lower center node disks located on the outside and the six lower center node disks located on the inside. These are respectively configured as lower center node disk folding linkage rods 63 to 80. Specifically, lower center node disk folding linkage rod 63 connects at both ends to the inner lower center node disk 1 011 and the outer lower center node disk 1 017; lower center node disk folding linkage rod 64 connects at both ends to the inner lower center node disk 1 011 and the outer lower center node disk 2 018; lower center node disk folding linkage rod 65 connects at both ends to the inner lower center node disk 2 012 and the outer lower center node disk 2 018; and lower center node disk folding linkage rods 6... 6. Both ends of the lower center node plate are connected to the inner lower center node plate 2012 and the outer lower center node plate 3019. The two ends of the lower center node plate folding linkage rod 67 are connected to the inner lower center node plate 2012 and the outer lower center node plate 4020. The two ends of the lower center node plate folding linkage rod 68 are connected to the inner lower center node plate 3013 and the outer lower center node plate 4020. The two ends of the lower center node plate folding linkage rod 69 are connected to the inner lower center node plate 3013 and the outer lower center node plate 5021. The two ends of the lower center node plate folding linkage rod 70 are connected to the inner lower center node plate 3013 and the outer lower center node plate 6022. The two ends of the lower center node plate folding linkage rod 71 are connected to the inner lower center node plate... The lower center node disk 4014 is connected to the outer lower center node disk 6022. The two ends of the lower center node disk folding linkage rod 72 are connected to the inner lower center node disk 4014 and the outer lower center node disk 7023. The two ends of the lower center node disk folding linkage rod 73 are connected to the inner lower center node disk 4014 and the outer lower center node disk 8024. The two ends of the lower center node disk folding linkage rod 74 are connected to the inner lower center node disk 5015 and the outer lower center node disk 8024. The two ends of the lower center node disk folding linkage rod 75 are connected to the inner lower center node disk 5015 and the outer lower center node disk 9025. The two ends of the lower center node disk folding linkage rod 76 are connected to the inner lower center node disk 5015 and the outer lower center node disk 9025. The lower center node disk folding linkage rod 77 is connected to the inner lower center node disk 6 016 and the outer lower center node disk 10 026 at both ends. The lower center node disk folding linkage rod 78 is connected to the inner lower center node disk 6 016 and the outer lower center node disk 11 027 at both ends. The lower center node disk folding linkage rod 79 is connected to the inner lower center node disk 6 016 and the outer lower center node disk 12 028 at both ends. The lower center node disk folding linkage rod 80 is connected to the inner lower center node disk 1 011 and the outer lower center node disk 12 028 at both ends. Moreover, the angle between each pair of the eighteen lower center node disk folding linkage rods located in the middle of the inner and outer perimeters is 60°, and they can be folded into the parabolic surface.

[0092] like Figure 21As shown, the thirty-three equal-length lateral edges are composed of six lateral edge rods of a basic developable hexagonal pyramid unit and twenty-seven lateral edge rods of nine identical basic developable triangular pyramid units. The six lateral edges of the basic developable hexagonal pyramid unit are uniformly designated as lateral edge rods A, and the twenty-seven lateral edges of the nine identical basic developable triangular pyramid units are uniformly designated as lateral edge rods B. The upper end of each lateral edge rod A is connected to the inner upper center node disk 001, and the lower end of each lateral edge rod A is connected to the inner lower center node disks 011, 012, 013, 014, 015, and 016 respectively. The angle between each lateral edge rod A and the inner lower center node disk is [not specified]. θ is 41° for all three sides. The twenty-seven side bars B are grouped into three groups of three, arranged counter-clockwise from group one to group nine. The upper ends of the side bars B in groups one to nine are connected to the outer upper center node disks 1-002, 2-003, 3-004, 4-005, 5-006, 6-007, 7-008, 8-009, and 9-010 respectively. The lower ends of the three side bars B in the first group are connected to the outer lower center node disks 1-017, 2-018, and 1-011 respectively. The lower ends of the three side bars B in the second group are connected to the outer lower center node disks 1-017, 2-018, and 1-011 respectively. The ends of the three side ribs B in the third group are respectively connected to the outer lower center node disk 2018, the outer lower center node disk 3019, and the inner lower center node disk 2012. The lower ends of the three side ribs B in the fourth group are respectively connected to the outer lower center node disk 4020, the inner lower center node disk 2012, and the inner lower center node disk 3013. The lower ends of the three side ribs B in the fifth group are respectively connected to the outer lower center node disk 5021, the outer lower center node disk 6022, and the inner lower center node disk 3013. The lower ends of the three side ribs B in the fifth group are respectively connected to the outer lower center node disk 6022, the outer lower center node disk 7023, and the inner lower center node disk 4014. The lower ends of the three side ribs B in the sixth group are respectively connected to the outer lower center node disk 4012. The seventh group of three side ribs B are connected to the lower center nodes of the outer and inner lower center nodes, respectively, namely, the outer lower center node disk 9 025, the outer lower center node disk 10 026, and the inner lower center node disk 5 015. The eighth group of three side ribs B are connected to the lower center nodes of the outer and inner lower center nodes, respectively, namely, the outer lower center node disk 10 026, the outer lower center node disk 11 027, and the inner lower center node disk 6 016. The ninth group of three side ribs B are connected to the lower center nodes of the outer and inner lower center nodes, respectively, namely, the outer lower center node disk 12 028, the inner lower center node disk 6 016, and the inner lower center node disk 1 011. The angle w between each side rib B and its corresponding inner or outer lower center node disk is 49°.

[0093] Nine equal-length upper center node disk folding linkage rods are used to connect nine external upper center node disks located on the outside. They are respectively set as upper center node disk folding linkage rod one 41, upper center node disk folding linkage rod two 42, upper center node disk folding linkage rod three 43, upper center node disk folding linkage rod four 44, upper center node disk folding linkage rod five 45, upper center node disk folding linkage rod six 46, upper center node disk folding linkage rod seven 47, upper center node disk folding linkage rod eight 48, and upper center node disk folding linkage rod nine 49. The upper center node disk folding linkage rod 41 is connected at both ends to the outer upper center node disk 1 002 and outer upper center node disk 2 003; the upper center node disk folding linkage rod 42 is connected at both ends to the outer upper center node disk 2 003 and outer upper center node disk 3 004; the upper center node disk folding linkage rod 43 is connected at both ends to the outer upper center node disk 3 004 and outer upper center node disk 4 005; the upper center node disk folding linkage rod 44 is connected at both ends to the outer upper center node disk 4 005 and outer upper center node disk 5 006; and the upper center node disk folding linkage rod 45 is connected at both ends to the outer upper center node disk 5 006 and outer upper center node disk 6 000. The upper center node disk folding linkage rod 46 is connected to the outer upper center node disk 6 007 and outer upper center node disk 7 008 at both ends. The upper center node disk folding linkage rod 47 is connected to the outer upper center node disk 7 008 and outer upper center node disk 8 009 at both ends. The upper center node disk folding linkage rod 48 is connected to the outer upper center node disk 8 009 and outer upper center node disk 9 010 at both ends. The upper center node disk folding linkage rod 49 is connected to the outer upper center node disk 9 010 and outer upper center node disk 1 002 at both ends. Each upper center node disk folding linkage rod can be folded towards the back of the parabola to assist the pyramid module assembly unit in retraction.

[0094] like Figure 22 and Figure 23As shown, when the pyramidal assembly unit is fully collapsed, the connections of each component are as follows: the side rods B of the nine identical triangular pyramidal basic unfoldable units, along with the twelve outer lower central node disk folding linkage rods 51-62 and the eighteen middle lower central node disk folding linkage rods 63-80, simultaneously fold inwards into the parabolic surface; the six side rods A of the hexagonal pyramidal basic unfoldable unit, along with the six inner lower central node disk folding linkage rods 17-22, also simultaneously fold inwards into the parabolic surface; after collapse, each lower central node disk folding linkage rod is in a folded state, with the twelve outer lower central node disk folding linkage rods at the outermost layer, the eighteen middle lower central node disk folding linkage rods, the six inner lower central node disk folding linkage rods, and the side rods B at the middle layer, with the lower central node disk folding linkage rods surrounding the side rods B, and the six side rods A at the innermost layer; the nine upper central node disk folding linkage rods 41-49 are also in a folded state, with the folding direction opposite to that of the lower central node disk folding linkage rods. During the folding process, each side rib, lower center node disk folding linkage rod, and top center node disk folding linkage rod simultaneously drive the connected upper center node disk and lower center node disk to move towards the center point, folding into a cylindrical structure. The twelve external lower center node disks 017~028 and the nine external upper center node disks 002~010 are in the outermost circle, while the six internal lower center node disks 011~016 and one internal upper center node disk 001 of the basic expandable hexagonal pyramid unit are in the innermost circle, ultimately achieving a fully folded pyramidal combination unit.

[0095] like Figure 24 As shown, when the pyramidal composite element is fully unfolded, it presents a parabolic surface. Each lower central node disk 4 rests on the parabolic surface, and the parametric equation of the parabolic surface is as follows:

[0096]

[0097] Where x, y, and z form a three-dimensional coordinate system, and f is the focal length of the parabolic surface. In practical applications, parabolic reflector deployable antenna mechanisms of arbitrary size and curvature can be formed by changing the aperture of the parabolic surface and the focal length in the parametric equation. The aperture of the parabolic surface determines the side lengths of the basic triangle of the triangular pyramid basic deployable unit and the basic regular hexagon of the hexagonal pyramid basic deployable unit in the pyramidal assembly unit. In practical applications, the side lengths of the basic triangle and basic regular hexagon are determined according to the required aperture size of the parabolic surface.

[0098] like Figure 25 and 26 As shown, the ratio of the spatial volume occupied by a pyramidal composite unit when it is fully unfolded to when it is fully collapsed is called the collapse ratio, and the formula for calculating the collapse ratio is as follows:

[0099]

[0100] Where V0 is the volume of the pyramidal unit when fully expanded, D0 is the diameter of the envelope circle of the central node disk of the reflecting surface when the pyramidal unit is fully expanded, d0 is the diameter of the envelope circle of the central node disk of the back frame when the pyramidal unit is fully expanded, and h0 is the vertical distance between the central node disk of the reflecting surface and the central node disk of the back frame when the pyramidal unit is fully expanded; V1 is the volume of the pyramidal unit when fully collapsed, D1 is the diameter of the envelope circle of the central node disk of the reflecting surface when the pyramidal unit is fully collapsed, d1 is the diameter of the envelope circle of the central node disk of the back frame when the pyramidal unit is fully collapsed, and h1 is the vertical distance between the central node disk of the reflecting surface and the central node disk of the back frame when the pyramidal unit is fully collapsed.

[0101] In this embodiment, the diameter of the pyramidal assembly unit is 1600mm, and the equation of the parabola is... and focal length The pyramidal composite unit in its fully collapsed state , , The pyramidal assembly unit in its fully unfolded state , , After the convergence rate calculation formula The calculated ratio of the space volume occupied by the pyramidal composite unit in its fully expanded and fully collapsed states is 89.76.

[0102] Multiple pyramidal unit combinations are used. One of these pyramidal unit combinations is selected to form the central unit, and the remaining pyramidal unit combinations are arranged in a circular grid according to the side length of the central unit's hexagonal shape (3). For example... Figure 9 As shown, pyramid combination unit O is set as the central combination unit, and pyramid combination units A, B, C, D, E, and F are arranged in a circular array network with pyramid combination unit O as the center.

Claims

1. A method for networking parabolic surfaces based on isomorphic modules of base equilateral triangular and hexagonal pyramids, characterized by: The method comprises the following steps: S1: constructing a composite profile division unit composed of triangles and hexagons in a two-dimensional plane of a top view perspective; S2: dividing a two-dimensional planar view of a parabolic surface formed by the deployable antenna mechanism using the composite profile division unit to obtain a combined pattern with triangular and hexagonal combination characteristics after division, and establishing a coordinate system xoy at the center position of the combined pattern; S3: calculating the side length of the polygon in the composite profile division unit based on the caliber of the composite profile division unit set in advance, calculating the coordinates of the center point of the triangle in the coordinate system xoy based on the side length of the polygon, and obtaining the triangle center point coordinates; S4: constructing a transformation equation formed by the angle relationship between the nodes of the triangle and the center point of the triangle using the rotation coordinate transformation method, and calculating the triangle node coordinates based on the transformation equation and the triangle center point coordinates obtained in step S3; S5: Parametric equations based on a pre-defined parabolic surface Calculate the coordinates of the triangle nodes and map them to the three-dimensional node coordinates on the three-dimensional coordinate system xyz, where f refers to the focal length of the parabolic surface; based on the curvature of the parabolic surface, perform curve fitting and connection on the three-dimensional node coordinates obtained in step S4 to obtain the parabolic surface of the deployable antenna mechanism in three-dimensional space. S6: based on the parabolic surface in the three-dimensional space obtained in step S5, constructing a pyramid combination unit composed of three-pyramid basic deployable units and six-pyramid basic deployable units, and obtaining the three-dimensional node coordinates of the triangle, which are the center points of the lower end center node disc of the deployable antenna mechanism, and the length of the connecting line between the adjacent three-dimensional node coordinates is the length of the rod of the deployable antenna mechanism.

2. An expandable antenna mechanism based on the isomodule of base equilateral triangular pyramid and hexagonal pyramid according to claim 1, characterized in that: The deployable antenna mechanism is composed of a plurality of pyramid combination units in a networking manner; the pyramid combination unit comprises one six-pyramid basic deployable unit and nine three-pyramid basic deployable units arranged around the six-pyramid basic deployable unit and movably connected with the six-pyramid basic deployable unit, and the adjacent two three-pyramid basic deployable units are movably connected through a foldable connecting rod; the orthographic projection of the bottom surface of the three-pyramid basic deployable unit forms a basic regular hexagon (1), the orthographic projection of the bottom surface of the three-pyramid basic deployable unit forms a basic regular triangle (2), and the side length of the basic regular hexagon (1) is equal to that of the basic regular triangle (2), the orthographic projection of the bottom surface of the pyramid combination unit forms a combined regular hexagon (3), and the combined regular hexagon (3) is formed by the basic regular hexagon (1) at the center position and 18 identical basic regular triangles (2) arranged around the outer periphery, the nine basic regular triangles (2) arranged alternately in the 18 basic regular triangles (2) are formed by the orthographic projection of the bottom surface of the three-pyramid basic deployable unit, and three of the nine basic regular triangles (2) arranged alternately share an edge with the basic regular hexagon (1).

3. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 2, wherein: The hexagonal pyramid basic developable unit comprises six lower end central node discs (4) respectively located at six vertexes of the basic regular hexagon (1), six lower central node disc folding linkage rods (5) respectively located at six bottom sides of the basic regular hexagon (1), an upper central node disc (6) located directly above the center point of the basic regular hexagon (1), and six side edges (7) respectively connected between the upper central node disc (6) and the corresponding lower end central node disc (4) through revolute pairs, and two ends of the lower central node disc folding linkage rod (5) are respectively connected with the corresponding lower end central node disc through revolute pairs; the six lower central node disc folding linkage rods (5) are synchronously folded and link the upper central node disc (6) and the lower end central node disc (4) to move towards the center point, and are folded into a cylindrical structure with the six lower central node disc folding linkage rods (5) surrounding the outer periphery of the six side edges (7).

4. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 3, wherein: The triangular pyramid basic developable unit comprises three lower end central node discs (4) respectively located at three vertexes of the basic regular triangle (2), three lower central node disc folding linkage rods (5) respectively located at three bottom sides of the basic regular triangle (2), an upper central node disc (6) located directly above the center point of the basic regular triangle (2), and three side edges (7) respectively connected between the upper central node disc (6) and the corresponding lower end central node disc (4) through revolute pairs, and two ends of the lower central node disc folding linkage rod (5) are respectively connected with the corresponding lower end central node disc (4) through revolute pairs; the three lower central node disc folding linkage rods (5) are synchronously folded and link the upper central node disc (6) and the lower end central node disc (4) to move towards the center point, and are folded into a cylindrical prism structure with the three lower central node disc folding linkage rods (5) surrounding the outer periphery of the three side edges (7).

5. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 4, wherein: The connecting rod comprises nine upper central node disc folding linkage rods (8) and six lower central node disc folding linkage rods (5), the upper central node disc folding linkage rod (8) is connected between the upper central node discs (6) of two adjacent triangular pyramid developable basic units through a revolute pair, and the upper central node disc folding linkage rod (8) is longer than the lower central node disc folding linkage rod (5); the nine basic regular triangles (2) comprise three basic regular triangles A sharing a side with the basic regular hexagon (1) and six basic regular triangles B sharing a corner with the basic regular hexagon (1), and the six lower central node disc folding linkage rods (5) are connected between the lower end central node discs (4) at the outer peripheral edges of the adjacent basic regular triangle A and basic regular triangle B through revolute pairs.

6. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 5, wherein: All the lower central node plate folding linkage rods (5) have the same folding direction, and the folding direction of the upper central node plate folding linkage rods (8) is opposite to that of the lower central node plate folding linkage rods (5); the nine upper central node plate folding linkage rods (8) and the thirty-six lower central node plate folding linkage rods (5) are synchronously folded and linked to fold the upper central node plate (6) and the lower end central node plate (4) to the center point, and the twelve lower central node plate folding linkage rods (5) on the bottom edge of the combined regular hexagon (3) are located in the outermost layer, the remaining twenty-four lower central node plate folding linkage rods (5) and the twenty-seven side edges (7) of the triangular pyramid basic developable unit are located in the middle layer, and the six side edges (7) of the hexagonal pyramid basic developable unit are located in the innermost layer, forming a cylindrical structure.

7. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomerization module of claim 4, characterized in that: The angle between each side edge (7) of the triangular pyramid basic developable unit and the lower end central node plate (4) connected thereto is w, and w∈[45°, 55°]; the angle between each side edge (7) of the hexagonal pyramid basic developable unit and the lower end central node plate (4) connected thereto is θ, and θ∈[40°, 50°].

8. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomerization module of claim 4, characterized in that: When the pyramid combination unit is completely unfolded, it presents a parabolic surface, and each bottom central node plate (4) falls on the parabolic surface, and the parametric equation of the parabolic surface is as follows: Where x, y and z form a three-dimensional coordinate system, and f is the focal length of the parabolic surface.

9. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 2, wherein: The ratio of the space volume occupied by the pyramid combination unit when it is completely unfolded to the space volume occupied by the pyramid combination unit when it is completely folded is the folding rate, and the calculation formula of the folding rate is as follows: Where V0 is the volume of the pyramid combination unit when it is completely unfolded, D0 is the diameter of the envelope circle of the reflecting surface central node plate when the pyramid combination unit is completely unfolded, d0 is the diameter of the envelope circle of the back frame central node plate when the pyramid combination unit is completely unfolded, h0 is the vertical distance between the reflecting surface central node plate and the back frame central node plate when the pyramid combination unit is completely unfolded; V1 is the volume of the pyramid combination unit when it is completely folded, D1 is the diameter of the envelope circle of the reflecting surface central node plate when the pyramid combination unit is completely folded, d1 is the diameter of the envelope circle of the back frame central node plate when the pyramid combination unit is completely folded, and h1 is the vertical distance between the reflecting surface central node plate and the back frame central node plate when the pyramid combination unit is completely folded.

10. The deployable antenna mechanism based on the base isosceles triangular pyramid and hexagonal pyramid isomodule of claim 2, wherein: A plurality of the pyramid combination units, one of which is selected to form a central combination unit, and the remaining pyramid combination units are arranged in a circular array according to the length of the side of the combined regular hexagon (3) of the central combination unit.

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