Paraboloid division method based on isosceles triangle and regular pentagon heterogeneous module networking and deployable antenna mechanism formed by paraboloid division method

Abstract

The invention discloses a paraboloid division method based on isosceles triangle and regular pentagon heterogeneous module networking and a deployable antenna mechanism formed by the paraboloid division method. The deployable antenna mechanism is formed by combining a plurality of pyramid combination units in a networking mode. Each pyramid combination unit comprises a basic pentagonal pyramid deployable unit and ten basic triangular pyramid deployable units which are arranged on the periphery of the basic pentagonal pyramid deployable unit in a surrounding mode and movably connected with the basic pentagonal pyramid deployable unit, and every two adjacent basic triangular pyramid deployable units are movably connected through foldable connecting rod pieces. According to the invention, the profile of the expandable parabolic reflector antenna is constructed through the combined pattern of the isosceles triangle and the regular pentagon, and the parabolic profile is high in precision; according to the method, conversion from two-dimensional plane division to three-dimensional profile division can be achieved, the bottom center node disc can be conveniently and rapidly determined, the complexity of the shape finding process is reduced, and the matching precision of the curvature and the focal length of the parabolic profile can be improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of deployable antenna mechanisms, and particularly relates to a parabolic surface division method based on isosceles triangle and regular pentagon isomodule networking and a deployable antenna mechanism formed thereby. BACKGROUND

[0002] Parabolic antennas are key components for achieving high gain and directional transmission, playing an important role in satellite communication, space exploration, and radio astronomy. Traditional fixed parabolic antennas are gradually unsuitable for today's strict requirements for weight and space in spacecraft and mobile application environments due to their large volume and weight. To address this challenge, framed deployable antennas have received widespread attention due to their excellent folding ability and compact storage volume. The design of the reflecting surface of such antennas requires precise surface division to achieve modularity and accurate deployment. In-depth research on surface division is crucial for improving the overall performance of the antenna and has a significant driving effect on the progress of space technology and the exploration of space science.

[0003] Deployable antennas mainly use several advanced surface division techniques, including hexagonal contour segmentation, three-way grid design methods, and force density methods, to design and optimize the reflecting surface. The application of these techniques has shown obvious results in improving the overall performance of the antenna structure, such as Chinese patent: Chinese patent "Symmetrical parabolic framed deployable antenna mechanism based on equilateral triangle surface division" (Application number: 2023102855906), which is composed of six identical basic folding and unfolding units connected by a shared central node plate. The reflecting surface of each basic unit is based on a regular hexagon, which is further divided into six identical equilateral triangles. Each equilateral triangle is further divided into N² small equilateral triangles according to the principle of equal division of side length, forming a highly symmetrical parabolic deployable structure. Through the design of shared central node plates and hinges, the folded volume is small, improving the overall stiffness of the antenna after deployment. However, the process of finding nodes at the central node plate and hinge is relatively complex, i.e., the surface division method is relatively complex, and the parabolic surface precision needs to be improved. SUMMARY

[0004] The first object of the present application is to provide a parabolic surface division method based on isosceles triangle and regular pentagon isomodule networking 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 isosceles triangle and regular pentagon isomodule networking that has high surface precision and strong stability.

[0006] Technical solution: The present application discloses a parabolic surface division method based on isosceles triangle and regular pentagon isomodule networking, comprising the following steps,

[0007] S1: Construct a composite profile division unit composed of regular pentagons and two kinds of isosceles triangles in a two-dimensional plane of a top view perspective;

[0008] S2: Divide the two-dimensional planar graph of the parabolic surface formed by the deployable antenna mechanism using the composite profile division unit, obtain the combined graph with the combined characteristics of isosceles triangles and regular pentagons after division, and establish a coordinate system xoy at the center position of the combined graph;

[0009] S3: Calculate the side length of the polygon in the composite profile division unit based on the preset composite profile division unit, and calculate the coordinates of the center point of each isosceles triangle in the coordinate system xoy based on the side length of the polygon, to obtain the isosceles triangle center point coordinates;

[0010] S4: Construct a transformation equation based on the angle relationship between the nodes of the isosceles triangle and the center point of the isosceles triangle using the center point coordinate method, and calculate the isosceles triangle node coordinates based on the transformation equation and the isosceles triangle center point coordinates obtained in step S3;

[0011] S5: Based on the preset parameter equation of the parabolic surface , calculate the three-dimensional node coordinates of the isosceles triangle node coordinates mapped to 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;

[0012] S6: Based on the parabolic surface in three-dimensional space obtained in step S5, construct a pyramid combination unit composed of three pyramid basic deployable units and five pyramid basic deployable units, the three-dimensional node coordinates of the isosceles triangle obtained refer to the center point of the bottom center node plate of the deployable antenna mechanism, and the length of the connection line between 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 an expandable antenna mechanism based on isomerism module networking of isosceles triangle and regular pentagon, which is combined by a plurality of pyramid combination units in a networking manner.

[0014] Further, the five-prism basic deployable unit comprises five bottom center node discs respectively located at five vertices of the basic regular pentagon, five first lower folding linkage rods respectively located at five bottom edges of the basic regular pentagon, a top center node disc located directly above the center of the circumscribed circle of the basic regular pentagon, and five side edges respectively connected between the top center node disc and the corresponding bottom center node disc through a revolute pair; the two ends of the first lower folding linkage rod are respectively connected to the corresponding bottom center node disc through a revolute pair; the five first lower folding linkage rods are synchronously folded and link the top center node disc and the bottom center node disc to move towards the center point, and are folded into a cylindrical structure with the five side edges surrounding the outside of the cylindrical structure.

[0015] Further, the three-prism basic deployable unit comprises three bottom center node discs respectively located at three vertices of the triangle, a second lower folding linkage rod respectively located at the bottom edge of the triangle, two third lower folding linkage rods respectively located at two legs of the triangle, a top center node disc located directly above the center of the circumscribed circle of the triangle, and three side edges respectively connected between the top center node disc and the corresponding bottom center node disc through a revolute pair; the two ends of the second lower folding linkage rod and the third lower folding linkage rod are respectively connected to the corresponding bottom center node disc through a revolute pair; one second lower folding linkage rod and two third lower folding linkage rods are synchronously folded and link the top center node disc and the bottom center node disc to move towards the center point, and are folded into a conical structure with the second lower folding linkage rod and the third lower folding linkage rod surrounding the outside of the three side edges.

[0016] Further, the top center node disc of the five triangular pyramid basic deployable units arranged alternately is connected with the top center node disc of the five pyramid basic deployable unit through the first upper folding linkage rod.

[0017] Further, the connecting rod comprises the second upper folding linkage rod connected between the top center node discs of the two triangular pyramid basic deployable units on the two sides of the same triangle II through the revolute pair, and the third upper folding linkage rod connected between the top center node discs of the two triangular pyramid basic deployable units between the two adjacent triangles II through the revolute pair.

[0018] Further, the folding directions of all the first lower folding linkage rods, the second lower folding linkage rods and the third lower folding linkage rods are consistent, the folding directions of all the first upper folding linkage rods, the second upper folding linkage rods and the third upper folding linkage rods are opposite to the folding direction of the first lower folding linkage rods; the five first lower folding linkage rods, the five second lower folding linkage rods, the twenty third lower folding linkage rods, the five first upper folding linkage rods, the five second upper folding linkage rods and the five third upper folding linkage rods are synchronously folded and link the top center node disc and the bottom center node disc to the center point, and are folded into a cylindrical structure, after the folding, the ten third lower folding linkage rods on the periphery of the triangular pyramid basic deployable unit are located in the outermost layer, the remaining ten third lower folding linkage rods, the five second lower folding linkage rods, the thirty side edges, the five first upper folding linkage rods, the five second upper folding linkage rods and the five third upper folding linkage rods of the triangular pyramid basic deployable unit are located in the middle layer, and the five first lower folding linkage rods and the five side edges of the five pyramid basic deployable unit are located in the innermost layer.

[0019] Further, the pyramid combination unit presents a parabolic surface when fully unfolded, each bottom center node disc falls on the parabolic surface, and the parametric equation of the parabolic surface is as follows:

[0020]

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

[0022] Further, the ratio of the space volume occupied by the pyramid combination unit when fully unfolded to the space volume occupied when fully folded is the folding rate, and the calculation formula of the folding rate 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 reflecting surface disc when the pyramidal unit is fully expanded, d0 is the diameter of the envelope circle of the back frame disc when the pyramidal unit is fully expanded, and h0 is the vertical distance between the reflecting surface disc and the back frame disc 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 reflecting surface disc when the pyramidal unit is fully collapsed, d1 is the diameter of the envelope circle of the back frame disc when the pyramidal unit is fully collapsed, and h0 is the vertical distance between the reflecting surface disc and the back frame disc 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 pentagon of the central combination unit.

[0026] Beneficial Effects: Compared with existing technologies, this invention has the following significant advantages: This invention constructs the surface profile of a deployable parabolic reflector antenna using a combination of isosceles triangles and regular pentagons, ensuring the surface profile accuracy is maintained during deployment and retraction. This enhances the adaptability of the grid cells in forming the parabola and improves the system's fault tolerance. Combining the center point coordinate method with the parabolic surface parametric equations, this invention enables the conversion from two-dimensional planar partitioning to three-dimensional surface partitioning, facilitating rapid determination of the bottom center node disk, reducing the complexity of the shape-finding process, and improving the matching accuracy of the parabolic surface curvature and focal length. The pyramidal combination units of this invention possess symmetrical deployment and seamless splicing capabilities, significantly shortening the on-site assembly time for large antenna reflectors and reducing manual intervention and error accumulation. Furthermore, based on the combination of regular pentagons and isosceles triangles, a final overall regular pentagonal base is formed, resulting in a complete overall structure that maintains stable system performance and significantly enhances system reliability. Attached Figure Description

[0027] Figure 1 This is an orthographic projection of the bottom surface of the pyramidal assembly unit of the present invention;

[0028] Figure 2 For the present invention Figure 1 The labeling diagrams for triangle 1 and triangle 2;

[0029] Figure 3 In the embodiments of the present invention, the numbers ① and ② are respectively. A schematic diagram for calculating the coordinates of the center point of an isosceles triangle;

[0030] Figure 4 In the embodiments of the present invention, numbered ②, A schematic diagram for calculating the coordinates of the center point of an isosceles triangle;

[0031] Figure 5 In the embodiments of the present invention, numbered ③, A schematic diagram for calculating the coordinates of the center point of an isosceles triangle;

[0032] Figure 6 This is a schematic diagram illustrating the calculation of the center point coordinates of isosceles triangles numbered ④ and ⑩ in an embodiment of the present invention;

[0033] Figure 7 This is a schematic diagram illustrating the calculation of the center point coordinates of isosceles triangles numbered ⑤ and ⑨ in an embodiment of the present invention;

[0034] Figure 8 This is a schematic diagram illustrating the calculation of the center point coordinates of isosceles triangles numbered ⑥ and ⑧ in an embodiment of the present invention.

[0035] Figure 9 This is a schematic diagram illustrating the calculation of the center point coordinates of the isosceles triangle numbered ⑦ in an embodiment of the present invention;

[0036] Figure 10 In the embodiments of the present invention, the number is... A schematic diagram for calculating the coordinates of the center point of an isosceles triangle;

[0037] Figure 11 This is a visualization diagram of the surface of a combination unit of isosceles triangles and regular pentagons, generated using a Python program in an embodiment of the present invention.

[0038] Figure 12 This is a schematic diagram of the structure of the basic deployable pentagonal pyramidal unit of the present invention when fully unfolded.

[0039] Figure 13 This is a front view of the pentagonal pyramidal basic expandable unit of the present invention when it is fully collapsed.

[0040] Figure 14 This is a schematic diagram of the structure of the basic deployable triangular pyramid unit of the present invention when fully unfolded.

[0041] Figure 15 This is a front view of the basic deployable triangular pyramidal unit of the present invention when fully collapsed.

[0042] Figure 16 This is a schematic diagram of the pyramidal assembly unit of the present invention when fully unfolded;

[0043] Figure 17 This is a schematic diagram of the pyramidal assembly unit of the present invention when fully collapsed;

[0044] Figure 18 This is a top view of the pyramidal assembly unit of the present invention when fully retracted;

[0045] Figure 19This 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 21 This 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 structure of a pentagonal pyramid basic deployable unit when fully unfolded in an embodiment of the present invention;

[0049] Figure 23 This is a schematic diagram of the structure of a basic deployable triangular pyramid in an embodiment of the present invention when it is fully unfolded;

[0050] Figure 24 This is a diagram showing the diameter of the pyramidal assembly unit of the present invention when it is fully collapsed;

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

[0052] Figure 26 This is a frontal projection of the bottom surface when multiple pyramidal combination units of the present invention are combined. 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 partitioning method based on a heterogeneous network of isosceles triangles and regular pentagons, comprising the following steps:

[0056] S1: Construct a composite surface division unit on a two-dimensional plane from a top-down perspective, consisting of a regular pentagon and two types of isosceles triangles.

[0057] During the construction process, the geometric characteristics of regular pentagons and two types of isosceles triangles are followed. A rotational symmetry combination strategy of isosceles triangles and regular pentagons is adopted to ensure that each composite surface division unit is interlocked through a precise splicing method, thereby achieving fine division of the parabolic surface of the deployable antenna mechanism and forming a two-dimensional planar diagram of the parabolic surface with the combined graphic characteristics of isosceles triangles and regular pentagons.

[0058] like Figure 1As shown, a combination of regular pentagons and two types of isosceles triangles is used to divide the parabolic reflector surface of the deployable antenna. On a top-view plane, a regular pentagon and fifteen isosceles triangles are drawn. The isosceles triangles are of two types: triangle 1 (2) and triangle 2 (4). Triangle 1 (2) has a vertex angle of 72° and a base angle of 54°, while triangle 2 (4) has a vertex angle of 36° and a base angle of 72°. The base of triangle 2 (4) shares a side with the regular pentagon; the isosceles sides of triangle 1 (2) and triangle 2 (4) coincide, and the bases of the two triangles 1 (2) are also aligned. Triangles 1 (2) and triangle 2 (4) are closely joined around the regular pentagon. The fifteen isosceles triangles are connected pairwise, with adjacent sides coinciding and sharing. Two triangles 1 (2) sharing a base are placed between two adjacent triangles 2 (4). Finally, one regular pentagon, ten triangles 1 (2), and five triangles 2 (4) are joined to form a larger combined regular pentagon 3, thus creating a composite surface division unit. Using this composite surface division unit as the surface center, a circular array network is formed according to the number of its side lengths, and finally the deployment antenna surface with the combined features of isosceles triangles and regular pentagons is completed on the plane.

[0059] 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 the combined features of isosceles triangles and regular pentagons, and a coordinate system xoy is established at the center of the combined graphic.

[0060] like Figure 2 As shown, each composite surface division unit consisting of a regular pentagon and fifteen isosceles triangles was systematically numbered. The regular pentagon is located at the center of the composite surface division unit, and the remaining fifteen isosceles triangles are arranged around the regular pentagon and are meticulously numbered, designated as number ①~ .

[0061] S3: Calculate the side length of the polygons in the composite surface division unit based on the pre-defined aperture of the composite surface division unit. Based on the side length of the polygons, calculate the coordinates of the center point of each isosceles triangle in the xoy coordinate system in all composite surface division units to obtain the coordinates of the center point of the isosceles triangle. The polygons include regular pentagons and two types of isosceles triangles.

[0062] Let the coordinates of the center point of each isosceles triangle in the composite surface division unit be respectively .

[0063] For the number ①, The coordinates of the center point of the isosceles triangle are obtained by solving the problem. The coordinates of the center point are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, and their x-coordinates are opposite. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, and let K be the side length of the regular pentagon. and points The angular and length relationships with the origin are as follows: Figure 3 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0064] ,

[0065] The angle between the connection between the center point P1 and the origin and the positive x-axis is... The angle between the connection between the center point P1 and the origin and the positive x-axis is... The solution obtained by this formula is and .

[0066] For number ②, The coordinates of the center point of the isosceles triangle are obtained by solving the problem. The coordinates of the center point are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, and their x-coordinates are opposite. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, and B be the distance from the center point of the regular pentagon to the origin. and points The angular and length relationships with the origin are as follows: Figure 4 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0067] ,

[0068] The angle between the base of the isosceles triangle numbered ② and the positive x-axis is . Solving using this formula yields... and .

[0069] For number ③, The coordinates of the center point of the isosceles triangle are obtained by solving the problem. The coordinates of the center point are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, and their x-coordinates are opposite. Let the distance from the center point of the regular pentagon to the origin be B, the side length of the regular pentagon be K, the leg length of the isosceles triangle numbered ③ be D, and the base length of the isosceles triangle numbered ③ be 2E. and points The angular and length relationships with the origin are as follows: Figure 5 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0070] ,

[0071] The distance from the vertex of isosceles triangle ③ to its center point P3 is G; the distance between the point of intersection of isosceles triangle ③ and the regular pentagon and the y-axis is F. Solving using this formula yields... and .

[0072] Solve for the coordinates of the center points of isosceles triangles numbered ④ and ⑩. The coordinates of their center points are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, their x-coordinates are opposite, and both y-coordinates lie on the negative half-axis, so their y-coordinates are negative. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, and point... and points The angular and length relationships with the origin are as follows: Figure 6 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0073] ,

[0074] The solution obtained from this formula is... and .

[0075] Solve for the coordinates of the center points of isosceles triangles numbered ⑤ and ⑨, where the coordinates of the center points are respectively... , Since the two are symmetrical about the y-axis, their y-coordinates are the same, their x-coordinates are opposite, and both y-coordinates are located on the negative half-axis, so their y-coordinates are negative. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, K be the side length of the regular pentagon, and G be the distance from the vertex of the isosceles triangle numbered ⑤ to its center point P5. and points The angular and length relationships with the origin are as follows: Figure 7 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0076] ,

[0077] The isosceles triangle numbered ⑤ has a leg length of 2M, and the perpendicular distance between the center point P5 of the isosceles triangle numbered ⑤ and one of its legs is J. The solution obtained using this formula is... and .

[0078] Solve for the coordinates of the center points of isosceles triangles numbered ⑥ and ⑧. The coordinates of their center points are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, their x-coordinates are opposite, and both y-coordinates are located on the negative half-axis, so their y-coordinates are negative. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, K be the side length of the regular pentagon, and G be the distance from the vertex of the isosceles triangle numbered ⑥ to its center point P6. and points The angular and length relationships with the origin are as follows: Figure 8 As shown. Based on the known conditions, the coordinates... and Solve the following:

[0079] ,

[0080] The perpendicular distance from the center point P6 to the x-axis is H+L, and the perpendicular distance from the center point P6 to the y-axis is... Solving using this formula yields... and .

[0081] Solve for the coordinates of the center point of isosceles triangle number ⑦. The coordinates of its center point are as follows: The coordinates are located on the negative half of the y-axis, so the x-axis coordinate is 0 and the y-axis coordinate is negative. Let L be the perpendicular distance from the center point of the regular pentagon to the side length, and let the midpoint be... The angular and length relationships with the origin are as follows: Figure 9 As shown. The coordinates are solved using the triangle congruence theorem and other known conditions:

[0082] ,

[0083] The solution obtained from this formula is... .

[0084] For numbered The coordinates of the center point of the isosceles triangle are obtained by solving the problem. The coordinates of the center point are as follows: , Since the two are symmetrical about the y-axis, their y-coordinates are the same, and their x-coordinates are opposite. Let L be the perpendicular distance from the center point of the regular pentagon to its side length, and B be the distance from a vertex of the regular pentagon to the origin. and points The angular and length relationships with the origin are as follows: Figure 10 As shown. The coordinates are solved using the triangle congruence theorem and other known conditions:

[0085] ,

[0086] Center point P 14 The perpendicular distance to the y-axis is N. This can be obtained by solving using this formula. and .

[0087] S4: Construct a transformation equation based on the angular relationship between the nodes of the isosceles triangle and the center point of the isosceles triangle using the center point coordinate method. Calculate the coordinates of the nodes of the isosceles triangle based on the transformation equation and the coordinates of the center point of the isosceles triangle obtained in step S3.

[0088] S5: Parametric equations based on a pre-defined parabolic surface Calculate the coordinates of the isosceles triangle nodes and map them to the three-dimensional coordinate system xyz, where f is 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.

[0089] That is, based on the pre-defined curvature and parametric equations of the parabolic surface. The calculated coordinates of the two-dimensional nodes (isosceles triangle nodes) are substituted into the parametric equations of the parabolic surface to calculate the coordinate height z of each node on the parabolic surface, thereby determining the precise position of the polygonal nodes in three-dimensional space.

[0090] 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 pentagonal pyramid units. The three-dimensional node coordinates of the obtained isosceles triangles refer to the center point of the bottom 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.

[0091] In practical use, Python is used to perform least-squares nonlinear curve fitting on the 3D node coordinate data generated in step S4 to obtain the analytical form of the parabolic surface parameter equations. Subsequently, a spline curve interpolation algorithm is used to complete the smooth connection between discrete vertices, achieving high-precision conformal mapping of the 2D surface segmented by the combined graphics to the unfoldable parabolic reflector. Furthermore, an interactive visualization module is used to perform 3D rendering and curvature pseudo-color display of the segmented surface, enabling efficient retrieval and export of node data.

[0092] like Figure 11As shown, in the Python environment, a parameterized parabolic surface model composed of isosceles triangles and regular pentagons is first established: the parabolic surface parametric equations, the constraint formulas in step S3, and the geometric control parameters are uniformly encapsulated as a structure input. Then, using a hybrid strategy of vectorization and nested for-loops, the center of each isosceles triangular unit is mapped by polar coordinate angles and homogeneous coordinate transformation, iteratively solving for its vertex coordinates in the global coordinate system. Based on plot3, 3D point cloud rendering presents the spatial distribution of each vertex and the corresponding parabolic surface height z in real time, thus completing the topological construction of the combined units in the unfoldable parabolic reflector. Finally, writematrix is ​​used to batch export the node coordinates to standard txt files, achieving lossless persistence of geometric data and downstream interface integration. Based on the rotation coordinate transformation method and the parabolic surface parametric equations, a parameterized model is constructed. Using Python code, the entire process from planar partitioning to 3D surface transformation is automated, reducing the complexity of the shape-finding process and improving the matching accuracy of the parabolic surface curvature and focal length.

[0093] Example 2

[0094] This invention discloses a deployable antenna mechanism based on a network of heterogeneous modules of isosceles triangles and regular pentagons, composed of several pyramidal combination units arranged in a network configuration. Each pyramidal combination unit includes a pentagonal pyramidal basic deployable unit and ten triangular pyramidal basic deployable units arranged around and movably connected to the pentagonal pyramidal basic deployable unit. Adjacent triangular pyramidal basic deployable units are movably connected by foldable connecting rods. Figure 1 As shown, the orthographic projection of the base of the basic developable unit of the pentagonal pyramid forms a basic regular pentagon 1. The orthographic projection of the base of the basic developable unit of the triangular pyramid forms an isosceles triangle, triangle 1-2, with a vertex angle of 72° and a base angle of 54°. The orthographic projection of the base of the pyramid combination unit forms a combined regular pentagon 3. Combined regular pentagon 3 is composed of a basic regular pentagon 1 at the center, ten triangles 1-2 surrounding the basic regular pentagon 1, and five triangles 2-4. Triangles 2-4 are isosceles triangles, and the bases of the five triangles 2-4 coincide with the five sides of the basic regular pentagon 1. The vertex angle of triangles 2-4 is 36° and the base angle is 72°. Between two adjacent triangles 2-4, there are two triangles 1-2, with one leg of triangle 1-2 coinciding with one leg of triangle 2-4, and the bases of the two triangles 1-2 coinciding. The legs of the two triangles 1-2 sharing a side with the same triangle 2-4 are connected to form one side of the combined regular pentagon. Figure 2 As shown, the numbers are ②, ③, ⑤, ⑥, ⑧, ⑨, The isosceles triangle is the orthographic projection of the base of the basic developable unit of the triangular pyramid, that is, triangle 2; labeled ①, ④, ⑦, ⑩, The triangle is 24.

[0095] like Figure 12 and Figure 13 As shown, the basic deployable unit of the pentagonal pyramid includes five base center node disks 5, five first lower folding linkage rods 6, one top center node disk 7, and five side edges 8. The five base center node disks 5 are located at the five vertices of the basic regular pentagon 1, and the five first lower folding linkage rods 6 are located at the five base edges of the basic regular pentagon 1. The two ends of the first lower folding linkage rods 6 are rotatably connected to the corresponding base center node disks 5 through a revolute joint. The included angle between two adjacent first lower folding linkage rods 6 is 108°. The top center node disk 7 is located directly above the center of the circumcircle of the basic pentagonal hexagon 1, and the five side edges 8 are rotatably installed between the top center node disk 7 and the corresponding base center node disks 5 through a revolute joint.

[0096] When the basic expandable unit of the pentagonal pyramid is folded up, the five side edges 8 fold inward into the parabolic surface, and the five first lower folding linkage rods 6 fold simultaneously and link the top center node disk 7 and the bottom center node disk 5 to move towards the center point, folding into a cylindrical structure surrounded by the five first lower folding linkage rods 6 around the five side edges 8.

[0097] like Figure 14 and Figure 15 As shown, the basic deployable unit of the triangular pyramid includes three base center node disks 5, one second lower folding linkage rod 9, two third lower folding linkage rods 10, one top center node disk 7, and three side edges 8. The three base center node disks 5 are located at the three vertices of triangle 2. The second lower folding linkage rod 9 is located at the base of triangle 2, and its two ends are rotatably connected to the corresponding base center node disks 5 via revolute joints. The two third lower folding linkage rods 10 are located at the two sides of triangle 2, and their two ends are rotatably connected to the corresponding base center node disks 5 via revolute joints. The top center node disk 7 is located directly above the center of the circumcircle of triangle 2. The three side edges 8 are rotatably connected between the top center node disk 7 and the corresponding base center node disks 5 via revolute joints. The included angle between the two third lower folding linkage rods 10 of the basic deployable unit of the triangular pyramid is 72°, and the included angle between the second lower folding linkage rod 9 and the third lower folding linkage rod 10 of the basic deployable unit of the triangular pyramid is 54°.

[0098] When the basic deployable unit of the triangular pyramid is folded up, the three lateral edges 8 fold inwards towards the parabolic surface. A second lower folding linkage 9 and two third lower folding linkages 10 fold simultaneously and, in conjunction with the top center node disk 7 and the bottom center node disk 5, move towards the center point, forming a frustum structure with the second lower folding linkage 9 and the third lower folding linkages 10 surrounding the three lateral edges 8. The length of the second lower folding linkage 9 is greater than the length of the third lower folding linkage 10. When the basic deployable unit of the triangular pyramid is folded up, the vertex of the second lower folding linkage 9 is higher than the vertex of the third lower folding linkage 10.

[0099] like Figure 16 to 18 As shown, the side edges 8 of the basic deployable triangular pyramid unit and the basic deployable pentagonal pyramid unit are of the same length. Therefore, when the deployable antenna mechanism is fully deployed and fully retracted, the top center node disks 7 of both the basic deployable triangular pyramid unit and the basic deployable pentagonal pyramid unit are at the same height. The top center node disks 7 of the five alternating basic deployable triangular pyramid units and the top center node disk 7 of the basic deployable pentagonal pyramid unit are all connected by a first upper folding linkage rod 11 via a swivel joint. The first upper folding linkage rod 11 not only improves the stability of the deployable antenna mechanism, but also makes the deployment and retraction process of the deployable antenna mechanism smoother and the overall coordination better.

[0100] The connecting rods include five second upper folding linkage rods 12 and five third upper folding linkage rods 13. The two triangular pyramidal deployable basic units located on either side of the same triangle 4 are rotatably connected by the second upper folding linkage rods 12 via a revolute joint; as shown... Figure 2 As shown, the top center node disks 7 of the two triangular pyramidal developable basic units corresponding to the two triangles 2 (labeled ⑥ and ⑧) are rotatably connected by a second upper folding linkage rod 12 via a revolute joint. The top center node disks 7 of the two triangular pyramidal developable basic units located between two adjacent triangles 4 are rotatably connected by a third upper folding linkage rod 13 via a revolute joint; as shown... Figure 2 As shown, the top center node disks 7 of the two triangular pyramidal developable basic units corresponding to the two triangles 2 (labeled ⑧ and ⑨) are rotatably connected by a third upper folding linkage rod 13 via a rotary joint. Furthermore, the two triangular pyramidal developable basic units located between the two adjacent triangles 4 are mirror-symmetrical.

[0101] The length of the first upper folding linkage 11 is greater than the length of the second upper folding linkage 12, and the length of the second upper folding linkage 12 is greater than the length of the five third upper folding linkages 13.

[0102] All the first lower folding linkages 6, the second lower folding linkages 9, and the third lower folding linkages 10 fold in the same direction, while all the first upper folding linkages 11, the second upper folding linkages 12, and the third upper folding linkages 13 fold in the opposite direction to the first lower folding linkage 6.

[0103] When the deployable antenna mechanism retracts, the thirty-five side edges 8 retract into the parabolic surface. The five first lower folding linkage rods 6, the five second lower folding linkage rods 9, the twenty third lower folding linkage rods 10, the five first upper folding linkage rods 11, the five second upper folding linkage rods 12, and the five third upper folding linkage rods 13 fold synchronously and link the top center node disk 7 and the bottom center node disk 5 to move towards the center point and retract into a cylindrical structure. After retraction, the ten third lower folding linkage rods 10 on the periphery of the triangular pyramid deployable basic unit are located in the outermost layer. The remaining ten third lower folding linkage rods 10 and five second lower folding linkage rods 9 of the triangular pyramid deployable basic unit, the thirty side edges 8 of the triangular pyramid basic deployable unit, the five first upper folding linkage rods 11, the five second upper folding linkage rods 12, and the five third upper folding linkage rods 13 are located in the middle layer. The five first lower folding linkage rods 6 and five side edges 8 of the pentaangular pyramid basic deployable unit are located in the innermost layer. After being folded together, all the top center node disks 7 are at the same height.

[0104] Taking a pyramidal composite unit as an example, the pyramidal composite unit consists of a pentagonal pyramidal basic expandable unit and... Figure 2 The numbers are ②, ③, ⑤, ⑥, ⑧, ⑨. The basic unit of a pyramid is formed by combining ten identical triangular pyramids created from isosceles triangles. For example... Figure 16 As shown, the pyramidal assembly unit includes ten externally located bottom center node disks, five internally located bottom center node disks, ten externally located top center node disks, one internally located top center node disk, five first lower folding linkage rods, five second lower folding linkage rods, twenty third lower folding linkage rods, five first upper folding linkage rods, five second upper folding linkage rods, five third upper folding linkage rods, and thirty-five equal-length side edges.

[0105] like Figure 19 As shown, an internally located top center node disk refers to the top center node disk 001 at the vertex of the basic developable element of the pentagonal pyramid. The ten externally located top center node disks are respectively located at... Figure 2 The Chinese number is ⑨, ⑧, ⑥, ⑤, ③, ②, At the vertices of the pentagonal pyramid basic developable unit formed by isosceles triangles, top center node disks 002, 003, 004, 005, 006, 007, 008, 009, 010, and 011 are respectively set. Five internal bottom center node disks are located at the bottom edge nodes of the pentagonal pyramid basic developable unit, and are respectively set as bottom center node disks 012, 013, 014, 015, and 016. The ten outer bottom center node disks are located at the nodes on the outer periphery of the bottom edge of the basic developable unit of the triangular pyramid, and are set in clockwise order as bottom center node disk 017, bottom center node disk 018, bottom center node disk 019, bottom center node disk 020, bottom center node disk 021, bottom center node disk 022, bottom center node disk 023, bottom center node disk 024, bottom center node disk 025, and bottom center node disk 026.

[0106] like Figure 20As shown, the five first lower folding linkages are located at the five base edges of the basic expandable unit of the pentagonal pyramid, and are respectively set as lower folding linkages 027, 028, 029, 030, and 031. The two ends of the lower folding linkage 027 are rotatably connected to the bottom center node disk 016 and the bottom center node disk 012 via a rotary joint, the two ends of the lower folding linkage 028 are rotatably connected to the bottom center node disk 012 and the bottom center node disk 013 via a rotary joint, the two ends of the lower folding linkage 029 are rotatably connected to the bottom center node disk 013 and the bottom center node disk 014 via a rotary joint, the two ends of the lower folding linkage 030 are rotatably connected to the bottom center node disk 014 and the bottom center node disk 015 via a rotary joint, and the two ends of the lower folding linkage 031 are rotatably connected to the bottom center node disk 015 and the bottom center node disk 016 via a rotary joint. Five second lower folding linkages are located at the base of five triangles 2, and are respectively set as lower folding linkages 052, 053, 054, 055, and 056. The two ends of lower folding linkage 052 are rotatably connected to the bottom center node disk 012 and the bottom center node disk 019 through a rotary joint. The two ends of lower folding linkage 053 are rotatably connected to the bottom center node disk 013 and the bottom center node disk 021 through a rotary joint. The two ends of lower folding linkage 054 are rotatably connected to the bottom center node disk 014 and the bottom center node disk 023 through a rotary joint. The two ends of lower folding linkage 055 are rotatably connected to the bottom center node disk 015 and the bottom center node disk 025 through a rotary joint. The two ends of lower folding linkage 056 are rotatably connected to the bottom center node disk 016 and the bottom center node disk 017 through a rotary joint.

[0107] Twenty third-downward folding linkages are located at the waists of ten triangles (2), and are respectively designated as downward folding linkages 032 to 051. The two ends of downward folding linkage 032 are rotatably connected to the bottom center node disks 017 and 018 via revolute joints, the two ends of downward folding linkage 033 are rotatably connected to the bottom center node disks 016 and 018 via revolute joints, the two ends of downward folding linkage 034 are rotatably connected to the bottom center node disks 012 and 018 via revolute joints, and the two ends of downward folding linkage 035 are rotatably connected to the bottom center node disks 018 and 019 via revolute joints. The two ends of rod 036 are rotatably connected to bottom center node disks 019 and 020 respectively via revolute joints; the two ends of lower folding linkage rod 037 are rotatably connected to bottom center node disks 012 and 020 respectively via revolute joints; the two ends of lower folding linkage rod 038 are rotatably connected to bottom center node disks 013 and 020 respectively via revolute joints; the two ends of lower folding linkage rod 039 are rotatably connected to bottom center node disks 020 and 021 respectively via revolute joints; the two ends of lower folding linkage rod 040 are rotatably connected to bottom center node disks 021 and 022 respectively via revolute joints; and the two ends of lower folding linkage rod 041 are connected to bottom center node disks 021 and 022 respectively via revolute joints. The two ends of the lower folding linkage rod 042 are rotatably connected to the bottom center node disks 013 and 022 respectively via revolute joints. The two ends of the lower folding linkage rod 043 are rotatably connected to the bottom center node disks 022 and 023 respectively via revolute joints. The two ends of the lower folding linkage rod 044 are rotatably connected to the bottom center node disks 023 and 024 respectively via revolute joints. The two ends of the lower folding linkage rod 045 are rotatably connected to the bottom center node disks 014 and 024 respectively via revolute joints. The two ends of the lower folding linkage rod 046 are rotatably connected to the bottom center node disks 015 and 022 respectively via revolute joints. The center node disk 024 is rotatably connected. The two ends of the lower folding linkage rod 047 are rotatably connected to the bottom center node disk 024 and the bottom center node disk 025 respectively through a rotary joint. The two ends of the lower folding linkage rod 048 are rotatably connected to the bottom center node disk 025 and the bottom center node disk 026 respectively through a rotary joint. The two ends of the lower folding linkage rod 049 are rotatably connected to the bottom center node disk 015 and the bottom center node disk 026 respectively through a rotary joint. The two ends of the lower folding linkage rod 050 are rotatably connected to the bottom center node disk 016 and the bottom center node disk 026 respectively through a rotary joint. The two ends of the lower folding linkage rod 051 are rotatably connected to the bottom center node disk 026 and the bottom center node disk 017 respectively through a rotary joint.

[0108] like Figure 21As shown, the five first upper folding linkage rods are respectively set as upper folding linkage rod 067, upper folding linkage rod 068, upper folding linkage rod 069, upper folding linkage rod 070, and upper folding linkage rod 071. The top ends of the upper folding linkage rods 067-71 are rotatably connected to the top center node disk 001 through a rotary joint. The bottom ends of the upper folding linkage rods 067, 068, 069, 070, and 071 are rotatably connected to the top center node disks 003, 005, 007, 009, and 011 through rotary joints, respectively. The five second upper folding linkages are respectively set as upper folding linkage 057, upper folding linkage 058, upper folding linkage 059, upper folding linkage 060, and upper folding linkage 061. The two ends of upper folding linkage 057 are rotatably connected to the top center node disk 003 and the top center node disk 004 respectively through a rotary joint. The two ends of upper folding linkage 058 are rotatably connected to the top center node disk 005 and the top center node disk 006 respectively through a rotary joint. The two ends of upper folding linkage 059 are rotatably connected to the top center node disk 007 and the top center node disk 008 respectively through a rotary joint. The two ends of upper folding linkage 060 are rotatably connected to the top center node disk 009 and the top center node disk 010 respectively through a rotary joint. The two ends of upper folding linkage 061 are rotatably connected to the top center node disk 011 and the top center node disk 002 respectively through a rotary joint. The five third upper-upward folding linkages are respectively set as upper folding linkage 062, upper folding linkage 063, upper folding linkage 064, upper folding linkage 065, and upper folding linkage 066. The two ends of upper folding linkage 062 are rotatably connected to the top center node disk 002 and the top center node disk 003 respectively through a rotary joint. The two ends of upper folding linkage 063 are rotatably connected to the top center node disk 004 and the top center node disk 005 respectively through a rotary joint. The two ends of upper folding linkage 064 are rotatably connected to the top center node disk 006 and the top center node disk 007 respectively through a rotary joint. The two ends of upper folding linkage 065 are rotatably connected to the top center node disk 008 and the top center node disk 009 respectively through a rotary joint. The two ends of upper folding linkage 066 are rotatably connected to the top center node disk 010 and the top center node disk 011 respectively through a rotary joint.

[0109] The thirty-five equal-length lateral edges are composed of five lateral edges of the basic developable unit of a pentagonal pyramid and thirty lateral edges of the basic developable unit of a triangular pyramid. For example... Figure 22As shown, the five lateral edges of the basic developable unit of the pentagonal pyramid are designated as lateral edges 072, 073, 074, 075, and 076. The tops of lateral edges 072-076 are rotatably connected to the top center node disk 001 via a revolute joint. The bottoms of lateral edges 072, 073, 074, 075, and 076 are rotatably connected to the bottom center node disks 012, 013, 014, 015, and 016 via revolute joints, respectively. Figure 23 As shown, with Figure 2 Taking the isosceles triangle with number ⑨ as an example, the three lateral edges of the basic developable triangular pyramid are set as lateral edge 077, lateral edge 078, and lateral edge 079, respectively. The tops of lateral edges 077 to 079 are rotatably connected to the top center node disk 014 through a swivel joint. The bottoms of lateral edges 077, 078, and 079 are rotatably connected to the bottom center node disk 014, the bottom center node disk 023, and the bottom center node disk 023, respectively, through a swivel joint.

[0110] like Figure 17 and Figure 18As shown, when the pyramidal assembly unit is fully collapsed, the connections of each component are as follows: the lower folding linkage rods 032~051 and 052~056 of the basic deployable triangular pyramidal unit simultaneously fold inwards towards the parabolic surface and converge; the thirty side edges of the ten basic deployable triangular pyramidal units also converge inwards simultaneously. Similarly, the lower folding linkage rods 027~031 of the basic deployable pentagonal pyramidal unit simultaneously fold inwards towards the parabolic surface and converge; the side edges 072~076 of the basic deployable pentagonal pyramidal unit also converge inwards simultaneously. During collapse, each folding linkage rod and side edge simultaneously drives the connected central node disk towards the center point of the pyramidal assembly unit, collapsing into a cylindrical structure. The bottom center node disks 017-026 and the top center node disks 002-011 are located on the outermost ring, while the bottom center node disks 012-016 and one top center node disk 001 are located on the innermost ring. The lower folding linkage rods 032-051 and 052-056, along with the thirty side edges of the basic deployable triangular pyramid unit, jointly drive the outer bottom center node disks 017-026 and the top center node disks 002-011 to move towards the center of the pyramid combination unit. After folding, each lower folding linkage rod is in a folded state, with the ten outermost lower folding linkage rods (lower folding linkage rods 032, 035, 036, 039, 040, 043, 044, 047, 048, 049, 040, 040, 041, 042, 043, 044, 045, 046, 047, 048, 049, 040, 042, 043, 044, 045, 046, 047, 048, 049, 040, 040, 041, 042 ...9, 040, 041, 042, 049, 040, 041, 042, 049, 040, 041, 8.051) is located in the outermost layer. The fifteen lower folding linkages (lower folding linkages 033, 034, 037, 038, 041, 042, 045, 046, 049, 050, lower folding linkages 052~056), the fifteen upper folding linkages (upper folding linkages 057~071), and the thirty side edges of the basic deployable unit of the triangular pyramid are located in the middle layer. The lower folding linkages 027~031 and the five side edges (side edges 072~076) of the basic deployable unit of the pentagonal pyramid are located in the innermost layer. The fifteen upper folding linkages (upper folding linkages 057~071) are also in a folded state, and the folding direction is opposite to that of the lower folding linkages. In the collapsed pyramidal assembly unit, the bottom center node disks 017~026 and the top center node disks 002~011 are on the outermost ring, while the bottom center node disks 012~016 and the top center node disk 001 are on the innermost ring, thus achieving the fully collapsed state of the pyramidal module assembly unit.

[0111] like Figure 24 As shown, It is the diameter of the envelope circle of the reflective disc when the pyramidal assembly unit is fully collapsed. It is the diameter of the envelope circle of the back structure disc when the pyramidal assembly unit is fully closed. It is the maximum vertical distance between the reflecting surface disc and the back disc when the pyramidal assembly unit is fully closed; such as Figure 25 As shown, It is the diameter of the envelope circle of the reflective disc when the pyramidal assembly unit is fully unfolded. It is the diameter of the envelope circle of the back structure disc when the pyramidal assembly unit is fully unfolded. This is the maximum vertical distance between the reflecting and rear facets of the pyramidal unit when it is fully extended. The ratio of the volume occupied by the pyramidal unit in its fully extended state to its fully collapsed state is called the collapse ratio. The calculation formula is:

[0112]

[0113] 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 52.38.

[0114] Multiple pyramidal unit combinations are used. One pyramidal unit is selected to form the central unit, and the remaining pyramidal units are arranged in a circular grid according to the side length of the central unit's regular pentagon 3. For example... Figure 26 As shown, pyramid combination unit O is set as the central combination unit, and pyramid combination units A, B, C, D, and E are arranged in a circular array network with pyramid combination unit O as the center.

Claims

1. A parabolic partitioning method based on a heterogeneous network of isosceles triangles and regular pentagons, characterized in that: The method comprises the following steps, S1: constructing a composite profile division unit composed of regular pentagons and two kinds of isosceles triangles 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 by using the composite profile division unit to obtain a combined pattern with the characteristics of isosceles triangles and regular pentagons 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, and calculating the coordinates of the center point of each isosceles triangle in all composite profile division units in the coordinate system xoy based on the side length of the polygon, to obtain the isosceles triangle center point coordinates; S4: constructing a transformation equation formed by the angle relationship between the nodes based on the isosceles triangle and the center point of the isosceles triangle by using the rotation coordinate transformation method, and calculating the isosceles triangle node coordinates based on the transformation equation and the isosceles triangle center point coordinates obtained in step S3; S5: Parametric equations based on a pre-defined parabolic surface Calculate the coordinates of the isosceles 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 five-pyramid basic deployable units, the three-dimensional node coordinates of the isosceles triangle obtained are the center point of the bottom 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. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of 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 five-pyramid basic deployable unit and ten three-pyramid basic deployable units arranged around the outer periphery of the five-pyramid basic deployable unit and movably connected thereto, and two adjacent three-pyramid basic deployable units are movably connected through a foldable connecting rod; the regular pentagon formed by the orthographic projection of the bottom surface of the five-pyramid basic deployable unit is a basic regular pentagon (1), the isosceles triangle formed by the orthographic projection of the bottom surface of the three-pyramid basic deployable unit is triangle one (2), and the orthographic projection of the bottom surface of the pyramid combination unit forms a combined regular pentagon (3); the combined regular pentagon (3) is composed of one basic regular pentagon (1) located at the center and ten triangle ones (2) and five triangle twos (4) arranged around the outer periphery of the basic regular pentagon (1), the triangle two (4) is an isosceles triangle, and the bottom edges of the five triangle twos (4) coincide with the five edges of the basic regular pentagon (1); there are two triangle ones (2) between two adjacent triangle twos (4), one of the waists of the triangle one (2) coincides with one of the waists of the triangle two (4), the bottom edges of the two triangle ones (2) coincide, and the waists of the two triangle ones (2) connected with the same triangle two (4) form one edge of the combined regular pentagon.

3. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 2, characterized in that: The five-prism basic developable unit comprises five bottom center node discs (5) respectively located at five vertices of the basic regular pentagon (1), five first lower folding linkage rods (6) respectively located at five bottom sides of the basic regular pentagon (1), a top center node disc (7) located directly above the center of the circumscribed circle of the basic regular pentagon (1), and five side edges (8) respectively connected between the top center node disc (7) and the corresponding bottom center node disc (5) through revolute pairs, and two ends of the first lower folding linkage rod (6) are respectively connected with the corresponding bottom center node disc (5) through revolute pairs; the five first lower folding linkage rods (6) are synchronously folded and link the top center node disc (7) and the bottom center node disc (5) to move towards the center point, and are folded into a cylindrical structure with the five side edges (8) surrounding the outer periphery of the cylindrical structure.

4. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 3, characterized in that: The three-prism basic developable unit comprises three bottom center node discs (5) respectively located at three vertices of the triangle one (2), a second lower folding linkage rod (9) respectively located at a bottom side of the triangle one (2), two third lower folding linkage rods (10) respectively located at two waists of the triangle one (2), a top center node disc (7) located directly above the center of the circumscribed circle of the triangle one (2), and three side edges (8) respectively connected between the top center node disc (7) and the corresponding bottom center node disc (5) through revolute pairs, and two ends of the second lower folding linkage rod (9) and the third lower folding linkage rod (10) are respectively connected with the corresponding bottom center node disc (5) through revolute pairs; the second lower folding linkage rod (9) and the two third lower folding linkage rods (10) are synchronously folded and link the top center node disc (7) and the bottom center node disc (5) to move towards the center point, and are folded into a conical frustum structure with the second lower folding linkage rod (9) and the third lower folding linkage rods (10) surrounding the outer periphery of the three side edges (8).

5. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 4, characterized in that: The top center node disc (7) of each of the five three-prism basic developable units arranged alternately is connected with the top center node disc (7) of the five-prism basic developable unit through a first upper folding linkage rod (11).

6. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 5, characterized in that: The connecting rod comprises a second upper folding linkage rod (12) connected between the top center node discs (7) of the two three-prism basic developable units located on both sides of the same triangle two (4) through a revolute pair, and a third upper folding linkage rod (13) connected between the top center node discs (7) of the two three-prism basic developable units located between adjacent two triangle two (4) through a revolute pair.

7. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 6, characterized in that: The folding directions of all the first lower folding link rods (6), the second lower folding link rods (9) and the third lower folding link rods (10) are consistent, and the folding directions of all the first upper folding link rods (11), the second upper folding link rods (12) and the third upper folding link rods (13) are opposite to the folding direction of the first lower folding link rods (6); the five first lower folding link rods (6), the five second lower folding link rods (9), the twenty third lower folding link rods (10), the five first upper folding link rods (11), the five second upper folding link rods (12) and the five third upper folding link rods (13) are synchronously folded and link the top center node disc (7) and the bottom center node disc (5) to the center point, and are folded into a cylindrical structure, after being folded, the ten third lower folding link rods (10) on the outer periphery of the three pyramid unfoldable basic units are located in the outermost layer, the remaining ten third lower folding link rods (10), the five second lower folding link rods (9), the thirty side edges (8), the five first upper folding link rods (11), the five second upper folding link rods (12) and the five third upper folding link rods (13) of the three pyramid unfoldable basic units are located in the middle layer, and the five first lower folding link rods (6) and the five side edges (8) of the five pyramid unfoldable basic units are located in the innermost layer.

8. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 4, characterized in that: The pyramid combination unit is in a parabolic surface when being fully unfolded, each bottom center node disc (5) is 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 isosceles triangle and regular pentagon isomerism module networking of claim 2, characterized in that: The ratio of the space volume of the pyramid combination unit when being fully unfolded to the space volume of the pyramid combination unit when being fully 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 being fully unfolded, D0 is the diameter of the envelope circle of the reflecting surface disc when the pyramid combination unit is fully unfolded, d0 is the diameter of the envelope circle of the back frame disc when the pyramid combination unit is fully unfolded, h0 is the vertical distance between the reflecting surface disc and the back frame disc when the pyramid combination unit is fully unfolded; V1 is the volume of the pyramid combination unit when being fully folded, D1 is the diameter of the envelope circle of the reflecting surface disc when the pyramid combination unit is fully folded, d1 is the diameter of the envelope circle of the back frame disc when the pyramid combination unit is fully folded, and h0 is the vertical distance between the reflecting surface disc and the back frame disc when the pyramid combination unit is fully folded.

10. The deployable antenna mechanism based on isosceles triangle and regular pentagon isomerism module networking of claim 2, characterized in that: A plurality of the pyramid combination units, one of which is selected to form a center combination unit, and the remaining pyramid combination units are arranged in a circle according to the number of the side length of the regular pentagon (3) of the center combination unit.

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