Method for calculating optimal matching paraboloid of radio telescope under action of structural deformation

Abstract

The invention relates to the field of radio telescope precision calculation, and discloses a method for calculating an optimal matching paraboloid of a radio telescope under the action of structural deformation, which comprises the following steps of: 1, designing an original paraboloid equation of an OXYZ coordinate system and an optimal matching paraboloid equation of a # imgabs0 # coordinate system; 2, according to the displacement # imgabs1 # of the vertex, the rotation angle # imgabs2 # of the focal axis and the variable quantity # imgabs3 # of the focal length, an equation of the optimal matching paraboloid under the original design paraboloid is obtained; 3, calculating node displacement influence matrixes Mx, My and Mz of the optimal matching paraboloid, and calculating undetermined parameters of the optimal matching paraboloid according to the X-direction displacement delta x, the Y-direction displacement delta y, the Z-direction displacement delta z of the nodes and the node displacement influence matrixes of the optimal matching paraboloid; and 4, substituting undetermined parameters to obtain an optimal matching paraboloid equation. According to the scheme, the undetermined parameters of the optimal matching paraboloid can be quickly calculated by giving the node displacement of the reflecting surface.

Description

Technical Field

[0001] The invention relates to the field of radio telescope accuracy calculation, and in particular to a method for calculating the best matching parabola of a radio telescope under the action of structural deformation. Background Art

[0002] The designed parabolic reflector deforms under the action of load, such as the attached Fig.24 , and point A (x0, y0, z0) moves to point B (x0+u a ,y0+v a ,z0+w a ), and the displacement AB can be decomposed into two parts (AB=A-C+CB): the first part is caused by the rigid body displacement of the parabola (AC represents rotation and movement) and the change of the focal length of the parabola; the second part is caused by the elastic deformation of each point on the reflecting surface (CB). Among them, only the latter affects the geometric shape of the reflecting surface. There are countless paraboloids with fitting standards, but there must be a parabola that can minimize the root mean square value (RMS for short, which is used as a measure of accuracy) of the half-optical path difference of each point on the reflecting surface to the parabola. Such a fitting parabola is called the best matching parabola.

[0003] Radio telescopes are basic equipment for observing and studying radio waves from celestial bodies. In actual work, the parabolic reflector of its antenna cannot completely keep consistent with the designed parabola, and there are certain errors. These errors will significantly affect the working performance of the radio antenna. There are many reasons for the difference between the reflector and the designed parabola, including deformation of the reflector caused by various factors such as the deadweight of the structure, non-uniform temperature effect of sunlight, and wind load during the use of the antenna. Summary of the invention

[0004] In view of the above problems, the present invention provides a method for calculating the best matching parabola of a radio telescope under the action of structural deformation, which is used to identify the influence of various error sources on the best matching parabola to solve the problem of large errors in designing the parabola.

[0005] The invention provides a method for calculating the best matching parabola of a radio telescope under structural deformation, the method comprising: Step 1: Let OXYZ be the coordinate system of the original designed parabola. is the coordinate system that best fits the parabola, with the origin and are their vertices, and are their focal axes respectively, and For a space point Coordinates in the coordinate system; For a space point Coordinates in the coordinate system, and design the original parabola equation and the best matching parabola equation; Step 2: Based on the displacement of the vertex , focal axis rotation angle and the change in focal length Obtain the equation of the best fitting parabola under the original design parabola; Step 3: Calculate the influence of the X, Y, and Z displacements of all nodes on the best fitting parabola, and obtain the best fitting parabola node displacement influence matrix M x 、M y 、M z , according to the node X, Y, Z displacement Δ x , Δ y , Δ z The best matching parabola node displacement influence matrix is ​​used to calculate the best matching parabola parameter vector , ; Step 4: Obtain the best fitting parabola equation based on the unknown parameter vector.

[0006] In an optional manner, the step 3 specifically includes: Step (1): define the influence of each node displacement of the reflection surface on the undetermined parameters of the best fitting parabola as the node displacement influence coefficient of the best fitting parabola, and calculate the influence of the displacement of all nodes in the three directions of X, Y, and Z on the best fitting parabola; Step (2): Based on the influence of the X, Y, and Z displacements of all nodes on the best fitting parabola, the best parabola node displacement influence matrix M is obtained. x 、M y 、M z ; Step (3): Calculate the given displacement Δ x , Δ y , Δ z , and calculate the best matching parabola parameters to be determined based on the given displacement and the best parabola node influence matrix . In an optional manner, the optimal parabola node displacement influence matrix M x 、M y 、M z for: Among them, U a_nx 、V a_nx , W a_nx , , 、hnx They represent the influence of the displacement in the x direction of node n on each undetermined parameter; U a_ny 、V a_ny , W a_ny , , 、h ny They represent the influence of the displacement of node n in the y direction on each undetermined parameter; U a_nz 、V a_nz , W a_n , , 、h nz They respectively represent the influence of the z-direction displacement of node n on each unknown parameter.

[0007] In an optional manner, the design original parabola equation is: ; The equation of the best matching parabola in the original design parabola coordinate system is: .

[0008] In an optional manner, in step 3, the deformed parabola equation is compared with the best fitting parabola equation to obtain the normal deviation equation, the normal deviation equation is linearized, and the best parabola is fitted using the least squares method to obtain the basis function: , , , , , ; The coefficient matrix is: ; Undetermined parameter vector: ; Offset vector: ; then the residual term: ; Use the least squares method to solve the unknown parameter vector: .

[0009] In an alternative embodiment, the normal deviation of the deformed parabola from the best fitting parabola is The expression is: ,make , , , , , , , , , , and then , ,but: .

[0010] In an optional manner, the specific steps of the equation of the best fitting parabola in the original design parabola coordinate system are as follows: The equation of the best fitting parabola is: , transform the coordinates of the points on the best fitting parabola: , according to the best fitting parabola equation and the coordinates of the points on the best fitting parabola, the equation of the best fitting parabola in the original design parabola coordinate system is obtained: .

[0011] The present invention is based on the displacement of each point of each parameter in the undetermined parameter vector. The influence of the displacement of all nodes on the reflecting surface can be calculated for the influence of their displacement on the parameters of the best matching parabola to be determined; and the influence of error sources such as gravity, environmental factors (wind, temperature) and manufacturing errors on the best matching parabola can be separated; using this method, when a set of reflecting surface node displacements and deformations is given, the parameters of the best matching parabola to be determined can be quickly calculated.

[0012] The characteristics of the present invention can further produce the following advantages: The influence of node displacement on the reflector surface can be quantified: (1) Improvement of surface shape accuracy: Through quantitative analysis, the relationship between node displacement and the parameters of the best matching parabola is established, which lays the foundation for the precise adjustment of the best matching parabola and helps to improve the shape accuracy of the reflector surface. (2) Fault diagnosis and maintenance, quantified displacement influence can help quickly locate the problem node.

[0013] The effects of each load on the best-fit parabola can be considered separately: (1) The sources of error can be separated. By analyzing each load factor separately, their effects on the shape of the reflector can be determined more accurately. (2) This provides a basis for efficient adjustment strategies. When system performance degrades, it is possible to quickly identify which load factor is causing the problem, facilitating fault diagnosis and maintenance. (3) Design efficiency can be improved. By considering each load factor separately, simulation and calculation can be performed faster.

[0014] Fast calculation of the best-fit parabola: (1) Easy to adjust in real time. The displacement of the reflector node has independent and additive effects on the best-fit parabola. Given the displacement of the reflector node, the parameters of the best-fit parabola can be quickly calculated. This can be used in real-time systems to maintain optimal performance. (2) Enhanced system adaptability. The fast calculation method can quickly adapt to different working conditions and requirements, improving the flexibility and reliability of the antenna system.

[0015] The above description is only an overview of the technical solution of the present invention. In order to more clearly understand the technical means of the present invention, it can be implemented according to the contents of the specification. In order to make the above and other purposes, features and advantages of the present invention more obvious and easy to understand, the specific implementation methods of the present invention are listed below. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] The accompanying drawings are only used to illustrate the embodiments and are not to be considered as limiting the present invention. In addition, the same reference symbols are used to represent the same components throughout the accompanying drawings. In the accompanying drawings: Figure 1 A schematic flow chart of a method for calculating the best matching parabola of a radio telescope under structural deformation provided by the present invention is shown.

[0017] Figure 2 The schematic diagram of the structure of the 65m radio telescope of the present invention is shown.

[0018] Figure 3 The figure shows the parameters U to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the X direction. a The impact diagram of .

[0019] Figure 4 The figure shows the parameters V to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the X direction. a The impact diagram of .

[0020] Figure 5 The figure shows the parameters W to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the X direction. a The impact diagram of .

[0021] Figure 6 The diagram shows the influence of the node of the reflection surface of the present invention on the parameter h when the displacement of 1 m occurs in the X direction.

[0022] Figure 7 The following diagram shows the parameters to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the X direction. The impact diagram of .

[0023] Figure 8 The following diagram shows the parameters to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the X direction. The impact diagram of .

[0024] Fig. 9 The figure shows the parameters U to be determined when the nodes of the reflection surface of the present invention are displaced by 1 m in the Y direction. a The impact diagram of .

[0025] Fig.10 The figure shows the parameters V to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the Y direction.a The impact diagram of .

[0026] Fig.11 The figure shows the parameters W to be determined when the nodes of the reflection surface of the present invention are displaced by 1 m in the Y direction. a The impact diagram of .

[0027] Fig.12 The diagram shows the influence of the node of the reflection surface of the present invention on the determined parameter h when the node is displaced by 1 m in the Y direction.

[0028] Fig.13 The following diagram shows the parameters to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the Y direction. The impact diagram of .

[0029] Fig.14 The following diagram shows the parameters to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the Y direction. The impact diagram of .

[0030] Fig.15 The figure shows the parameters U to be determined when the nodes of the reflection surface of the present invention are displaced by 1 m in the Z direction. a The impact diagram of .

[0031] Fig.16 The figure shows the parameters V to be determined when the nodes of the reflection surface of the present invention are displaced by 1 m in the Z direction. a The impact diagram of .

[0032] Fig.17 The figure shows the parameters W to be determined when the nodes of the reflector surface of the present invention are displaced by 1 m in the Z direction. a The impact diagram of .

[0033] Fig.18 The diagram shows the influence of the node of the reflection surface of the present invention on the determined parameter h when the displacement of 1 m occurs in the Z direction.

[0034] Fig.19 The following diagram shows the parameters to be determined when the nodes of the reflective surface of the present invention are displaced by 1 m in the Z direction. The impact diagram of .

[0035] Fig. 20 The following diagram shows the parameters to be determined when the nodes of the reflective surface of the present invention are displaced by 1 m in the Z direction. The impact diagram of .

[0036] Fig.21 The X-direction displacement of the node of the reflection surface of the present invention is shown.

[0037] Fig. 22 The Y-direction displacement of the node of the reflection surface of the present invention is shown.

[0038] Fig.23 The Z-direction displacement of the node of the reflection surface of the present invention is shown.

[0039] Fig.24 A schematic diagram showing the deviation of the deformed parabola from the best matching parabola of the present invention is shown. DETAILED DESCRIPTION

[0040] Exemplary embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited to the embodiments set forth herein.

[0041] Figure 1 A flow chart of a method for calculating the best matching parabola of a radio telescope under structural deformation of the present invention is shown, and the method comprises the following steps: Step 1: Let OXYZ be the coordinate system of the original designed parabola. is the coordinate system that best fits the parabola, with the origin and are their vertices, and are their focal axes respectively, and For a space point Coordinates in the coordinate system; For a space point The coordinates in the coordinate system, and design the original parabola equation and the best matching parabola equation. The specific formula is as follows: The design parabola equation is: (1) The best fitting parabola equation is: (2).

[0042] Step 2: Based on the displacement of the vertex , focal axis rotation angle and the change in focal length Get the equation of the best fitting parabola under the original design parabola.

[0043] This step specifically includes: firstly, coordinate transformation of the points on the best matching parabola. are all traces, and their higher-order traces can be ignored, so the coordinate transformation equation is: (3), then substitute formula (3) into formula (2) to obtain the equation of the best fitting parabola in the original design parabola coordinate system: (5).

[0044] Step 31: define the influence of each node displacement of the reflection surface on the undetermined parameters of the best matching parabola as the node displacement influence coefficient of the best matching parabola, and calculate the influence of the displacement of all nodes in the three directions of X, Y, and Z on the best matching parabola; In this scheme, a 65m radio telescope is used as an example. Its structure is shown in the attached figure. Figure 2 As shown in the figure, the 1104 nodes of the reflective surface are displaced by 1m in the X, Y, and Z directions respectively, and the influence on the six undetermined parameters of the best matching parabola is calculated. Figure 3 -Attached Figure 8 As shown in Figure 2, the influence of a 1m displacement of the reflector node in the X direction on the six undetermined parameters of the best matching parabola can be obtained; Fig. 9 -Attached Fig.14 , we can obtain the influence of the displacement of 1m of the reflection surface node in the Y direction on the six unknown parameters of the best matching parabola; as shown in the attached Fig.15 -Attached Fig. 20 , we can obtain the influence of the 1m displacement of the reflection surface node in the Z direction on the six unknown parameters of the best matching parabola.

[0045] Step 32: According to the influence of the displacements of all nodes in the three directions of X, Y, and Z on the best matching parabola, the best parabola node displacement influence matrix M is obtained. x 、M y 、M z .

[0046] Among them, according to the description of the best parabola in the background technology, the fitting standard in the accuracy analysis of the back surface of the radio telescope is relative to the best parabola, not the original design parabola. The root mean square value of the half-optical path difference of each point on the actual deformed reflector relative to the best matching parabola is the smallest. The root mean square value of the half-optical path difference is called RMS, which is the most important indicator for evaluating the accuracy of the radio telescope reflector.

[0047] In order to obtain the inherent rules between scattered data or use current data to predict the expected data, continuous curves are usually used to approximate or compare the functional relationship between the coordinates represented by the discrete point groups on the plane. The corresponding problems in high-dimensional space also belong to this category. Taking the data in two-dimensional space as an example, suppose and is a pair of observed quantities, the function To reflect the amount and The relationship between. Often called the fitted model, is a parameter to be determined. Assume that the function relationship has been determined, but The elements in the vector are the parameters to be determined. The multiple sets of data obtained are , through this The group data can be used to find the parameter vector The best estimate of , thus obtaining the best fitting function It can be seen that curve fitting is to formulate the data of discrete points.

[0048] There are many ways to solve the fitting curve. If the theoretical function Each parameter If it is a linear function, it is called a linear fit, otherwise it is called a nonlinear model. For a linear model, the parameters are generally determined by establishing and solving a set of equations to obtain a fitting curve.

[0049] There are many measures of goodness of fit, the most common approach is to choose the parameter The residual (or deviation) between the fitted model and the actual observed value at each point The weighted sum of squares of (6) reaches a minimum. At this time, the required curve is called the fitting curve of the data in the sense of weighted least squares.

[0050] It can be seen that the linear fitting method needs to use the least squares method and solve the corresponding set of equations to determine the parameters. Now this method is briefly introduced for two-dimensional data.

[0051] Assume that To fit the data set and Not equal. Generally by Linearly independent function (basis functions) linear combination. It can be expressed as: .

[0052] Fitting the data Substitution Among them, we get the linear equation system: When solving a linear system of equations, it is usually required that the number of unknowns is equal to the number of equations. If the number of equations is greater than the number of unknowns, there is a situation where the equations have no solution, which is called an overdetermined system of equations. , satisfies the conditions of the overdetermined system of equations, and since are not equal, so the system of equations has no solution. In other words, there is no curve such that all points in the data set are on the curve. Therefore, we can only seek an approximate solution, which is not an approximation of the exact solution (there is no exact solution here). When the least squares method is used to fit the curve, we are looking for a set of parameters that minimizes the sum of the squares of the errors between the fitting function and the data point set. At this time, the curve fitted by this set of parameters is called the optimal approximate solution to the overdetermined system of equations.

[0053] Each row of the above linear equations can be written as form.

[0054] Define the coefficient matrix , parameter vector to be determined , ,but .

[0055] According to the definition of the least squares method, we now need to find a set of coefficients , so that the inner product of the residual vector and itself is minimized, that is, .

[0056] Can be considered as a vector is a quadratic continuous function of the independent variable, so there exists a set of parameters , so that To reach the minimum, only : Using matrix operations, To find the partial derivative of the coefficient vector a, we can use the matrix partial derivative rule to get; It can be concluded that: Here we define the matrix , we can see that the matrix for order square matrix, and the matrix satisfy , further analysis can prove that the matrix is a symmetric positive definite matrix, which must have a unique solution (symmetric positive definite matrix property). Further analysis can yield the matrix is a normal matrix, so the linear equations generated above are called normal equations. This equation system has a unique solution.

[0057] because ,in is the coefficient matrix. It can be seen that since the coefficient matrix The elements are based on the measured data points Find the data point is known, so as long as Functional form, Each element of can be calculated, then the parameter vector to be determined is Each element of can be expressed as: ,in According to the coefficient matrix So we can explain the parameters to be found. for Therefore, it can be seen that each measured data point For the undetermined parameters The effects are independent of each other and can be added linearly.

[0058] When using the linear fitting method to fit a function, the commonly used basis functions are Desirable: ; ; Here we take the power function as an example and take the basis function as: .

[0059] The fitting polynomial is: It is defined by the least squares method, that is, through the data of the given points , to determine the coefficient , so that the sum of square errors of each point is minimized. Substitute n data points into the polynomial P (x) to obtain a polynomial with m+1 unknowns An overdetermined system of n equations.

[0060] According to the above derivation, .

[0061] , it can be seen is a symmetric positive definite matrix, which has a unique solution (symmetric positive definite matrix property). Therefore, the unknown parameters can also be written as form.

[0062] This law also applies to linear fitting problems in high dimensions. The following uses the three-dimensional space function fitting surface as an example to briefly prove that the same principle applies to other higher dimensions.

[0063] The measured data points are total , assuming that through The coordinates of the known points are used to fit the best fitting surface using the least squares method. , Then we can get the residual term .

[0064] Find the partial derivative of the unknown parameter vector Taking partial derivative, we get It can be seen that this law is also satisfied when the least squares linear fitting is used in high-dimensional space.

[0065] This scheme uses the least squares method to fit the best fitting parabola. The deviation of the deformed parabola from the best fitting parabola is as follows: Fig.24 As shown, ignoring the higher-order traces, the normal deviation is obtained The expression is: , make , , , , , , , , , , and then , ,but: .

[0066] The best matching parabola for a known antenna is the one on the antenna reflector. If the fitting is done for each point, then each point has ,in is the independent variable, is the dependent variable. When the least squares method is used to fit the best fitting parabola, the basis function can be obtained as: , , , , , ; The coefficient matrix is: ; Undetermined parameter vector: ; Offset vector: ; then the residual term: ; Use the least squares method to solve the unknown parameter vector: .

[0067] From the above, we can see that each item in the coefficient matrix is ​​based on Requested, and is the independent variable, the value is given, so the coefficient matrix is ​​given Under the premise of , it can be considered as a constant matrix, and the elements in the matrix are all constants. These dependent variables only appear in the offset vector, and the other parameters in the offset vector can also be considered as known constants. are all first-order terms in the offset vector, then the elements in the undetermined parameter vector can be expressed as Linear combination of variables, or undetermined parameter vector Each term in Linearly related, therefore, each parameter in the unknown parameter vector is affected by the displacement of each point The effects are independent and linearly additive.

[0068] According to the above conclusions, it is calculated that the optimal parabola node displacement influence matrix M x 、M y 、M z for: Among them, U a_nx 、V a_nx , W a_nx , , 、h nx They represent the influence of the displacement in the x direction of node n on each undetermined parameter; U a_ny 、V a_ny , W a_ny , , 、h ny They represent the influence of the displacement of node n in the y direction on each undetermined parameter; U a_nz 、V a_nz , W a_nz , , 、h nz Respectively represent the influence of the displacement of node n in the z direction on each undetermined parameter. After calculation in step 31, after calculating the influence of the displacement of 1m in the three directions of X, Y, and Z of 1104 nodes of the 65m radio telescope on the pointing accuracy, after assembly, the displacement influence matrix M of the best matching parabola of the radio telescope can be obtained. x 、M y 、M z Specific values ​​can be found in the attached Figure 3 -Attached Fig. 20 , Figure 3-Figure 20 The units of the horizontal and vertical coordinates on the left axis are both m. Figure 3-Figure 6 , Figure 9-12 as well as Figure 15-18The rightmost axis of is in meters. Figure 7 , Figure 8 , Fig.13 , Fig.14 , Fig.19 , Fig. 20 The rightmost axis is in degrees (°).

[0069] Step 33: Obtain the given displacement Δ x , Δ y , Δ z , and calculate the best matching parabola parameter vector based on the given displacement and the best parabola node influence matrix , . The best matching parabola of the 65m radio telescope under the action of gravity only is calculated when the elevation angle is 45°. Fig.21 -Attached Fig.23 , Figure 21-23 The unit of the lower axis value is m, which represents the X, Y, and Z displacements of each node on the reflection surface. Figure 21-23 It is the output of finite element analysis results, where: NODAL SOLUTION indicates the node; STEP=1 indicates the first step of the analysis; SUB=1 indicates sub-step 1; TIME=24(AVG) indicates the time is 24, indicating the average value; RSYS=0 indicates the coordinate system number; DMX indicates the maximum displacement; SMN indicates the minimum stress value; SMX indicates the maximum stress value; UX indicates the displacement in the X direction; UY indicates the displacement in the Y direction; UZ indicates the displacement in the Z direction. The obtained displacement is calculated with the influence matrix, and the undetermined parameter vector of the best matching parabola is: .

[0070] Step 4: Obtain the best fitting parabola equation based on the unknown parameter vector.

[0071] The present invention is based on the displacement of each point of each parameter in the undetermined parameter vector. The influence of the displacement of all nodes on the reflector surface can be calculated for the influence of their displacement on the parameters of the best fitting parabola to be determined; when a set of displacement deformations of the reflector surface nodes is given, the parameters of the best fitting parabola to be determined can be quickly calculated. The influence of the displacement of the reflector surface nodes can be quantified; the source of error is separated: the influence of each load on the best fitting parabola can be considered separately; the best fitting parabola is quickly calculated: the influence of the displacement of the reflector surface nodes on the best fitting parabola is independent and superimposable, and the parameters of the best fitting parabola to be determined can be quickly calculated given the displacement of the reflector surface nodes.

[0072] In the description provided herein, numerous specific details are described. However, it is understood that embodiments of the present invention may be practiced without these specific details. Similarly, in order to streamline the present invention and aid in understanding one or more of the various inventive aspects, in the above description of exemplary embodiments of the present invention, various features of embodiments of the present invention are sometimes grouped together into a single embodiment, figure, or description thereof. Wherein, the claims that follow the specific embodiments are hereby expressly incorporated into the specific embodiments, wherein each claim itself serves as a separate embodiment of the present invention.

[0073] It should be noted that the above embodiments illustrate the present invention rather than limit it, and that those skilled in the art may devise alternative embodiments without departing from the scope of the appended claims. In the claims, any reference symbol between brackets shall not be construed as a limitation on the claims. The word "comprising" does not exclude the presence of elements or steps not listed in the claims. The word "one" or "an" preceding an element does not exclude the presence of a plurality of such elements. The present invention may be implemented by means of hardware comprising a number of different elements and by means of a suitably programmed computer. In a unit claim enumerating a number of devices, several of these devices may be embodied by the same hardware item. The use of the words first, second, and third, etc. does not indicate any order. These words may be interpreted as names. The steps in the above embodiments, unless otherwise specified, should not be understood as limitations on the order of execution.

Claims

1. A method for calculating the best matching parabola of a radio telescope under structural deformation, characterized in that: The method comprises: Step 1: Let OXYZ be the coordinate system of the original designed parabola. is the coordinate system that best fits the parabola, with the origin and are their vertices, and are their focal axes respectively, and For a space point Coordinates in the coordinate system; For a space point Coordinates in the coordinate system, and design the original parabola equation and the best matching parabola equation; Step 2: Based on the displacement of the vertex , focal axis rotation angle and the change in focal length Obtain the equation of the best fitting parabola under the original design parabola; Step 3: Calculate the influence of the X, Y, and Z displacements of all nodes on the best fitting parabola, and obtain the best fitting parabola node displacement influence matrix M x 、M y 、M z , according to the node X, Y, Z displacement Δ x , Δ y , Δ z The best matching parabola node displacement influence matrix is ​​used to calculate the best matching parabola parameter vector , ; Step 4: Obtain the best fitting parabola equation based on the unknown parameter vector.

2. The method according to claim 1, characterized in that: The step 3 specifically includes: Step (1): define the influence of each node displacement of the reflection surface on the undetermined parameters of the best fitting parabola as the node displacement influence coefficient of the best fitting parabola, and calculate the influence of the displacement of all nodes in the three directions of X, Y, and Z on the best fitting parabola; Step (2): Based on the influence of the X, Y, and Z displacements of all nodes on the best fitting parabola, the best parabola node displacement influence matrix M is obtained. x 、M y 、M z ; Step (3): Calculate the given displacement Δ x , Δ y , Δ z , and calculate the best matching parabola parameters to be determined based on the given displacement and the best parabola node influence matrix .

3. The method according to claim 2, characterized in that Optimal parabola node displacement influence matrix M x 、M y 、M z for: Among them, U a_nx 、V a_nx , W a_nx , , 、h nx They represent the influence of the displacement in the x direction of node n on each undetermined parameter; U a_ny 、V a_ny , W a_ny , , 、h ny They represent the influence of the displacement of node n in the y direction on each undetermined parameter; U a_nz 、V a_nz , W a_nz , , 、h nz They respectively represent the influence of the z-direction displacement of node n on each unknown parameter.

4. The method according to claim 1, characterized in that: The designed original parabola equation is: ; The equation of the best matching parabola in the original design parabola coordinate system is: 。 5. The method according to claim 4, characterized in that In step 3, the deformed parabola equation is compared with the best fitting parabola equation to obtain the normal deviation equation, linearize the normal deviation equation, and use the least squares method to fit the best parabola to obtain the basis function: , , , , , ; The coefficient matrix is: ; Undetermined parameter vector: ; Offset vector: ; then the residual term: ; Use the least squares method to solve the unknown parameter vector: .

6. The method according to claim 5, characterized in that Normal deviation of the deformed parabola from the best fitting parabola The expression is: ,make , , , , , , , , , , and then , ,but: .

7. The method according to claim 4, characterized in that The specific steps of the equation of the best fitting parabola in the original design parabola coordinate system are as follows: The equation of the best fitting parabola is: , transform the coordinates of the points on the best fitting parabola: , according to the best fitting parabola equation and the coordinates of the points on the best fitting parabola, the equation of the best fitting parabola in the original design parabola coordinate system is obtained: 。

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