Calculation Method for Optimal Fitting Paraboloid of Radio Telescope under Structural Deformation

Abstract

The present invention relates to the field of radio telescope accuracy calculation, and discloses a method for calculating the best-fitting paraboloid of a radio telescope under the action of structural deformation, including: Step 1: Design the original paraboloid equation of the OXYZ coordinate system and the best-fitting paraboloid equation of the #imgabs0# coordinate system; Step 2: Obtain the equation of the best-fitting paraboloid under the original designed paraboloid according to the displacement #imgabs1# of the vertex, the rotation angle #imgabs2# of the focal axis, and the change amount #imgabs3# of the focal length; Step 3: Calculate the influence matrix M of the node displacement of the best-fitting paraboloid x , M y , M z , and calculate the undetermined parameters of the best-fitting paraboloid according to the three-dimensional displacements Δ x , Δ y , Δ z of the node and the influence matrix of the node displacement of the best-fitting paraboloid; Step 4: Substitute the undetermined parameters to obtain the equation of the best-fitting paraboloid. This solution can quickly calculate the undetermined parameters of the best-fitting paraboloid given the node displacement of the reflector surface.

Description

Technical Field

[0001] The present invention relates to the field of radio telescope accuracy calculation, and particularly to a method for calculating the best-fitting paraboloid of a radio telescope under the action of structural deformation. Background Art

[0002] When the designed parabolic reflector deforms under the action of load, as shown in the appendix Figure 24 , a point A (x 0 , y 0 , z 0 ) moves to point B (x 0 + u a , y 0 + v a , z 0 + w a ). And the displacement from A to B can be decomposed into two parts (A - B = A - C + C - B): the former part is due to the rigid body displacement of the paraboloid (A - C represents rotation and translation) and the change of the focal length of the paraboloid; the latter part is due to the elastic deformation of each point on the reflector (C - B). Among them, only the latter affects the geometric shape of the reflector. There are countless paraboloids that meet the fitting standard, but there must be one paraboloid that can make the root mean square value of the semi-optical path difference of each point on the reflector with respect to this paraboloid (abbreviated as RMS, used as a measure of accuracy) reach the minimum. Such a fitting paraboloid is called the best-fitting paraboloid.

[0003] A radio telescope is a basic device for observing and studying radio waves from celestial bodies. In actual work, the parabolic reflector of its antenna cannot completely maintain coincidence with the designed paraboloid, and there are certain errors. These errors will significantly affect the working performance of the radio antenna. The reasons for the difference between the reflector and the designed paraboloid are various, including the deformation of the reflector caused by various factors such as the self-weight 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-fitting paraboloid of a radio telescope under the action of structural deformation, which is used to identify the influence of each error source on the best-fitting paraboloid to solve the problem of large errors in the designed paraboloid.

[0005] The invention provides a method for calculating the best-fitting paraboloid of a radio telescope under the action of structural deformation, and the method includes:

[0006] Step 1: Let OXYZ be the coordinate system of the original designed paraboloid, be the coordinate system of the best-fitting paraboloid, and the origin and are their vertices respectively, and are their respective focal axes, and let be the coordinates of a point in space in coordinate system; be the coordinates of a point in space in coordinate system, and design the original paraboloid equation and the best-fitting paraboloid equation;

[0007] Step 2: Obtain the equation of the best-fitting paraboloid under the original designed paraboloid according to the displacement of the vertex, the rotation angle of the focal axis, and the change in focal length;

[0008] Step 3: Calculate the influence of the displacements in the X, Y, and Z directions of all nodes on the best-fitting paraboloid, and obtain the influence matrix M x of the best-fitting paraboloid nodes, M y of the best-fitting paraboloid nodes, M z of the best-fitting paraboloid nodes. According to the displacements Δ x in the X direction, Δ y in the Y direction, Δ z in the Z direction of the nodes and the influence matrix of the best-fitting paraboloid node displacements, calculate the undetermined parameter vector of the best-fitting paraboloid, ;

[0009] Step 4: Obtain the equation of the best-fitting paraboloid according to the undetermined parameter vector.

[0010] In an optional manner, step 3 specifically includes:

[0011] Step (1): Define the influence of the displacements of each node of the reflector surface on the undetermined parameters of the best-fitting paraboloid as the influence coefficient of the best-fitting paraboloid node displacements, and calculate the influence of the displacements in the X, Y, and Z directions of all nodes on the best-fitting paraboloid;

[0012] Step (2): Obtain the influence matrix M x of the best-fitting paraboloid nodes, M y of the best-fitting paraboloid nodes, M z of the best-fitting paraboloid nodes according to the influence of the displacements in the X, Y, and Z directions of all nodes on the best-fitting paraboloid;

[0013] Step (3): Obtain the given displacements Δ x in the X direction, Δ y in the Y direction, Δ z in the Z direction, and calculate the undetermined parameters of the best-fitting paraboloid according to the given displacements and the influence matrix of the best-fitting paraboloid nodes.

[0014] In an optional manner, the influence matrix M x of the best-fitting paraboloid nodes, M y of the best-fitting paraboloid nodes, Mz is:

[0015]

[0016]

[0017]

[0018] wherein, U a_nx , V a_nx , W a_nx , , , h nx respectively represent the influence of the displacement of the nth node in the x direction on each undetermined parameter; U a_ny , V a_ny , W a_ny , , , h ny respectively represent the influence of the displacement of the nth node in the y direction on each undetermined parameter; U a_nz , V a_nz , W a_n , , , h nz respectively represent the influence of the displacement of the nth node in the z direction on each undetermined parameter.

[0019] In an alternative embodiment, the original designed paraboloid equation is: ; the equation of the best-fitting paraboloid in the coordinate system of the original designed paraboloid is: .

[0020] In an alternative embodiment, in step 3, the deformed paraboloid equation is compared with the best-fitting paraboloid equation to obtain the normal deviation equation, the normal deviation equation is linearized, and the least squares method is used to fit the best paraboloid to obtain the basis functions: , , , , , ; the coefficient matrix is: ; the undetermined parameter vector: ; the offset vector: ; then the residual term: ; the least squares method is used to solve the undetermined parameter vector: .

[0021] In an alternative embodiment, the normal deviation of the deformed paraboloid with respect to the best-fitting paraboloid is expressed as: , let , , , , , , , , , , and then let , , then: .

[0022] In an alternative manner, the specific steps of the equation of the best-fitting paraboloid in the original designed paraboloid coordinate system are as follows: The equation of the best-fitting paraboloid is: , perform coordinate transformation on the points on the best-fitting paraboloid: , obtain the equation of the best-fitting paraboloid in the original designed paraboloid coordinate system according to the equation of the best-fitting paraboloid and the coordinates of the points on the best-fitting paraboloid: .

[0023] According to the present invention, each parameter in the undetermined parameter vector is independently and linearly superposable under the influence of the displacements of each point. It is possible to calculate the influence of the displacements on the undetermined parameters of the best-fitting paraboloid for all nodes on the reflecting surface; and it is possible to separate the influences of error sources such as gravity, environmental factors (wind, temperature), and manufacturing errors on the best-fitting paraboloid; using this method, when a set of displacement deformations of the reflecting surface nodes is given, the undetermined parameters of the best-fitting paraboloid can be quickly calculated.

[0024] The characteristics of the present invention can further produce the following advantageous effects:

[0025] The influence of the reflecting surface node displacements can be quantified: (1) Improvement of surface shape accuracy: Through quantitative analysis, the relationship between the node displacements and the parameters of the best-fitting paraboloid is established, laying a foundation for the precise adjustment of the best-fitting paraboloid and helping to improve the shape accuracy of the reflecting surface. (2) Fault diagnosis and repair, the quantified displacement influence can help quickly locate the problem nodes.

[0026] The influences of each load action on the best-fitting paraboloid can be considered separately: (1) Error sources can be separated. By separately analyzing each load factor, it is possible to more accurately determine their influences on the shape of the reflecting surface. (2) Provide a basis for an efficient adjustment strategy. When the system performance deteriorates, it is possible to quickly identify which load factor causes the problem, facilitating fault diagnosis and maintenance. (3) Improve the design efficiency. By separately considering each load factor, simulation and calculation can be carried out more quickly.

[0027] Fast calculation of the best - fitting paraboloid: (1) Facilitating real - time adjustment. The influence of the reflector node displacement on the best - fitting paraboloid is independent and superimposable. Given the reflector node displacement, the undetermined parameters of the best - fitting paraboloid can be quickly calculated, which can be used in real - time systems to maintain the best performance. (2) Enhancing system adaptability. The fast calculation method can quickly adapt to different working conditions and requirements, improving the flexibility and reliability of the antenna system.

[0028] The above description is only an overview of the technical solution of the present invention. In order to understand the technical means of the present invention more clearly, it can be implemented according to the content of the specification. And in order to make the above and other objects, features, and advantages of the present invention more obvious and understandable, the specific embodiments of the present invention are specifically given below. Brief Description of the Drawings

[0029] The drawings are only used to illustrate the embodiments and are not considered as a limitation to the present invention. Moreover, throughout the drawings, the same reference numerals are used to represent the same components. In the drawings:

[0030] Figure 1 A schematic flow chart of a method for calculating the best - fitting paraboloid of a radio telescope under the action of a structural deformation provided by the present invention is shown.

[0031] Figure 2 A schematic structural diagram of the 65m radio telescope of the present invention is shown.

[0032] Figure 3 The influence diagram of the undetermined parameter U when the reflector nodes of the present invention have a 1m displacement in the X - direction respectively is shown. a is shown.

[0033] Figure 4 The influence diagram of the undetermined parameter V when the reflector nodes of the present invention have a 1m displacement in the X - direction respectively is shown. a is shown.

[0034] Figure 5 The influence diagram of the undetermined parameter W when the reflector nodes of the present invention have a 1m displacement in the X - direction respectively is shown. a is shown.

[0035] Figure 6 The influence diagram of the undetermined parameter h when the reflector nodes of the present invention have a 1m displacement in the X - direction respectively is shown.

[0036] Figure 7 The influence diagram of the undetermined parameter when the reflector nodes of the present invention have a 1m displacement in the X - direction respectively is shown.

[0037] Figure 8 The influence diagram of the undetermined parameter The impact diagram of .

[0038] Figure 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 .

[0039] Figure 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 .

[0040] Figure 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 .

[0041] Figure 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.

[0042] Figure 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 .

[0043] Figure 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 .

[0044] Figure 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 .

[0045] Figure 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 .

[0046] Figure 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 .

[0047] Figure 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.

[0048] Figure 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 .

[0049] Figure 20The 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 .

[0050] Figure 21 The X-direction displacement of the node of the reflection surface of the present invention is shown.

[0051] Figure 22 The Y-direction displacement of the node of the reflection surface of the present invention is shown.

[0052] Figure 23 The Z-direction displacement of the node of the reflection surface of the present invention is shown.

[0053] Figure 24 A schematic diagram showing the deviation of the deformed parabola from the best matching parabola of the present invention is shown. DETAILED DESCRIPTION

[0054] 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.

[0055] 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:

[0056] 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:

[0057] The design parabola equation is: (1)

[0058] The best fitting parabola equation is: (2).

[0059] 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.

[0060] This step specifically includes: First, perform coordinate transformation on the points on the best-fitting paraboloid. Since are all infinitesimals, their higher-order infinitesimals can be ignored, so the coordinate transformation equation is: (3), and then substitute formula (3) into formula (2) to obtain the equation of the best-fitting paraboloid in the original designed paraboloid coordinate system:

[0061] (5).

[0062] Step 31: Define the influence of the displacements of each node of the reflector on the undetermined parameters of the best-fitting paraboloid as the influence coefficient of the best-fitting paraboloid node displacement, and calculate the influence of the X, Y, and Z displacements of all nodes on the best-fitting paraboloid;

[0063] In this scheme, a 65m radio telescope is taken as an example, and its structure is as shown in the appendix Figure 2 . Let the 1104 nodes of its reflector be displaced by 1m in the X, Y, and Z directions respectively, and calculate their influence on the 6 undetermined parameters of the best-fitting paraboloid. As shown in the appendix Figure 3 - appendix Figure 8 , the influence of the reflector node displaced by 1m in the X direction on the 6 undetermined parameters of the best-fitting paraboloid can be obtained; as shown in the appendix Figure 9 - appendix Figure 14 , the influence of the reflector node displaced by 1m in the Y direction on the 6 undetermined parameters of the best-fitting paraboloid can be obtained; as shown in the appendix Figure 15 - appendix Figure 20 , the influence of the reflector node displaced by 1m in the Z direction on the 6 undetermined parameters of the best-fitting paraboloid can be obtained.

[0064] Step 32: Obtain the influence matrix M x 、M y 、M z of the best paraboloid node displacement according to the influence of the X, Y, and Z displacements of all nodes on the best-fitting paraboloid.

[0065] Among them, according to the description of the best paraboloid in the background technology, the fitting standard in the radio telescope back surface accuracy analysis is relative to the best paraboloid, rather than the original designed paraboloid. The root mean square value of the half optical path difference between the actually deformed reflector and each point on the best-fitting paraboloid is the smallest. The root mean square value of the half optical path difference, called RMS, is the most important index for evaluating the accuracy of the radio telescope reflector.

[0066] In order to obtain the inherent law between scattered discrete data or use the current data to predict the expected data, a continuous curve is usually used to approximately depict or analogize the functional relationship between the coordinates represented by a discrete point group on a plane. The corresponding problems in high-dimensional spaces also fall into this category. Taking the data in two-dimensional space as an example, assume and are a pair of observed quantities, and the function is used to reflect the quantity and relationship. is often called the fitting model, where are undetermined parameters. Assuming that the functional relationship has been determined, but the elements within the vector are undetermined parameters. The multiple sets of data obtained are , and through these sets of data, the best estimate of the parameter vector can be obtained, and thus the best fitting function is obtained. It can be seen that curve fitting is to formulate the data of discrete points.

[0067] There are many methods to solve the fitting curve. If the theoretical function is a linear function of each parameter , it is called linear fitting, otherwise it is called a non-linear model. For a linear model, the parameters are generally determined by establishing and solving a system of equations, and thus the fitting curve is obtained.

[0068] There are many criteria for measuring the goodness of fit. The most common approach is to select the parameter such that the weighted sum of squares of the residuals (or deviations) (6) between the fitting model and the actual observed values reaches the minimum. At this time, the obtained curve is called the fitting curve of the data in the sense of weighted least squares.

[0069] It can be seen that the linear fitting method needs to use the least squares method and solve the corresponding system of equations to determine the parameters. Now, for two-dimensional data, this method will be briefly introduced.

[0070] Suppose we use to fit the data set and are not equal. Generally, it is linearly combined by linearly independent functions (basis functions). It can be expressed as: .

[0071] Substitute the fitting data into , and a system of linear equations is obtained:

[0072]

[0073] When solving a system of linear equations, it is usually required that the number of unknowns is equal to the number of equations. If the number of equations is more than the number of unknowns, there will be a situation where the equation has no solution, which is called an overdetermined system of equations. The above system of equations , satisfying the conditions of the overdetermined system of equations, and since are not equal, so this system of equations has no solution. That is, there does not exist a curve such that all points on the data set are on the curve. Therefore, only an approximate solution can be sought. This approximate solution is not an approximation of the exact solution (there is no exact solution here). When using the least squares method to fit a curve, it is to find a set of parameters such that the sum of the squares of the errors between the fitting function and the data point set is minimized. At this time, the curve fitted by this set of parameters is called the optimal approximate solution of the overdetermined system of equations.

[0074] Each row of the above linear system of equations can be written as form.

[0075] Define the coefficient matrix , the vector of undetermined parameters , , then .

[0076] According to the definition of the least squares method, now a set of coefficients needs to be found such that the inner product of the residual vector with itself is minimized, that is .

[0077] can be regarded as a quadratic continuous function of the vector as the independent variable. Therefore, there exists a set of parameters such that reaches the minimum. At this time, only :

[0078] Using matrix operations,

[0079]

[0080] Taking the partial derivative with respect to the coefficient vector a, using the rules of matrix partial derivative, we can obtain;

[0081]

[0082] Furthermore, we can obtain:

[0083]

[0084] Here, define the matrix , it can be seen that the matrix is order square matrix, and the matrix satisfies , further analysis can prove that the matrix is a symmetric positive definite matrix, and this matrix must have a unique solution (properties of symmetric positive definite matrices). Further analysis can obtain that the matrix is a normal matrix, so the above-generated system of linear equations is called a normal system of equations. This system of equations has a unique solution.

[0085] Since , where is the coefficient matrix. It can be seen that since the elements of the coefficient matrix are obtained from the measured data points , and the of the data points are known, so as long as the function form is given, each element of can be calculated. Then each element of the parameter vector to be solved can be expressed as:

[0086] , where can be obtained according to the coefficient matrix . Therefore, it can be shown that the parameter to be solved is a linear combination of. Therefore, it can be seen that the of each measured data point for the undetermined parameter

[0087] When using the linear fitting method to fit a function, the commonly used basis functions can be taken as: ; ; and so on. Here, taking the power function as an example, the basis function is taken as: .

[0088] The fitting polynomial is:

[0089] According to the definition of the least squares method, that is, through the data of the given points, to determine the coefficient such that the sum of the squares of the errors at each point is minimized. Substituting n data points into the polynomial P(x), an overdetermined system of n equations with m + 1 unknowns is obtained.

[0090]

[0091]

[0092] According to the above derivation, .

[0093] , it can be seen that is a symmetric positive definite matrix, and this matrix must have a unique solution (property of symmetric positive definite matrix). Therefore, the undetermined parameters can also be written as form.

[0094] For the linear fitting problem in high dimensions, this rule also applies. Taking the fitting of a surface by a three-dimensional space function as an example, it is briefly proved below. The principle for other higher dimensions is the same.

[0095]

[0096] The measured data points are in total pieces. Suppose the coordinates of known points are passed, and the least squares method is used to fit the best-fitting surface. The undetermined parameter vector ,

[0097] Then the residual term can be obtained.

[0098]

[0099] Take the partial derivative of the undetermined parameter vector Take the partial derivative, and we can get

[0100]

[0101]

[0102]

[0103] It can be seen that when using the least squares method for linear fitting in high-dimensional space, this rule is also satisfied.

[0104] This scheme uses the least squares method to fit the best-fitting paraboloid. The deviation of the deformed paraboloid from the best-fitting paraboloid is as Figure 24 shown. Ignoring the higher-order infinitesimals, the expression of the normal deviation is:[[]]

[0105] ,

[0106] Let , , , , , , , , , , and then let , , then: .

[0107] It is known that the best-fitting paraboloid of the antenna is fitted by points on the antenna reflector surface, then each point has , where is the independent variable, is the dependent variable. When using the least squares method to fit the best-fitting paraboloid, the basis functions can be obtained as: , , , , , ; The coefficient matrix is: ; The vector of parameters to be determined: ; The offset vector: ; Then the residual term: ; Using the least squares method to solve the vector of parameters to be determined: .

[0108] As can be seen from the above, each term in the coefficient matrix is calculated according to , and is the independent variable and its value is given. Therefore, the coefficient matrix can be considered a constant matrix under the premise of given , and all elements in the matrix are constants. And these dependent variables only appear in the offset vector, and the other parameters in the offset vector can also be considered known constants. And are all linear terms in the offset vector, then the elements in the vector of parameters to be determined can all be expressed as linear combinations of each variable. It can also be said that each term in the vector of parameters to be determined is only linearly related to the displacements of each point . Therefore, each parameter in the vector of parameters to be determined is independently affected by the displacements of each point and can be linearly superposed.

[0109] According to the above conclusion, it is calculated that the influence matrix M of the best-fitting paraboloid node displacement x , M y , M z is:

[0110]

[0111]

[0112]

[0113] where, where U a_nx , V a_nx , W a_nx , , , h nx respectively represent the influence of the x-direction displacement of the nth node on each undetermined parameter; U a_ny , V a_ny , W a_ny , , , h ny respectively represent the influence of the y-direction displacement of the nth node on each undetermined parameter; U a_nz , V a_nz , W a_nz , , , h nz respectively represent the influence of the z-direction displacement of the nth node on each undetermined parameter. After calculation in step 31, after calculating the influence of the X, Y, and Z three-direction displacements of 1104 nodes of the 65m radio telescope on the pointing accuracy by 1m, through assembly, the displacement influence matrix M x , M y , M z of the best-fitting paraboloid of the radio telescope can be obtained. The specific values can be seen in Appendix Figure 3 - Appendix Figure 20 . Figures 3 - 20 The units of the abscissa and ordinate of the left coordinate axis are both m, Figures 3 - 6 , Figures 9 - 12 and Figures 15 - 18 the unit of the rightmost axis of Figure 7 , Figure 8 , Figure 13 , Figure 14 , Figure 19 , Figure 20 the rightmost axis is degree (°).

[0114]

[0115]

[0116]

[0117] Step 33: Obtain the given displacements Δ x , Δ y , Δ z , and calculate the vector of the undetermined parameters of the best-fitting paraboloid according to the given displacements and the node influence matrix of the best paraboloid, .

[0118] The best-fitting paraboloid of the 65m radio telescope at a 45° elevation angle when the structure is only under the action of gravity is calculated. Appendix Figure 21 - Appendix Figure 23 . Figures 21 - 23The numerical unit of the lower end shaft is m, which are the X, Y, and Z-direction displacements of each node of the reflecting surface respectively. Figures 21 - 23 For the output of finite element analysis results, where: NODAL SOLUTION represents the node section; STEP = 1 represents the 1st step of the analysis; SUB = 1 represents the sub-step 1; TIME = 24(AVG) represents the time is 24, indicating the average value taken; RSYS = 0 represents the coordinate system number; DMX represents the maximum displacement; SMN represents the minimum stress value; SMX represents the maximum stress value; UX represents the displacement in the X direction; UY represents the displacement in the Y direction; UZ represents the displacement in the Z direction. By operating the obtained displacements with the influence matrix, the undetermined parameter vector of the best-fitting paraboloid can be obtained as .

[0119] Step 4: Obtain the best-fitting paraboloid equation according to the undetermined parameter vector.

[0120] In the present invention, each parameter in the undetermined parameter vector is affected by the displacements of each point independently and linearly superposable. It is possible to calculate the influence of their displacements on the undetermined parameters of the best-fitting paraboloid for all nodes on the reflecting surface; when a set of displacements and deformations of the reflecting surface nodes are given, the undetermined parameters of the best-fitting paraboloid can be quickly calculated. The influence of the reflecting surface node displacements can be quantified; separation of error sources: the influence of each load action on the best-fitting paraboloid can be considered separately; fast calculation of the best-fitting paraboloid: the influence of the reflecting surface node displacements on the best-fitting paraboloid is independent and superposable, and the undetermined parameters of the best-fitting paraboloid can be quickly calculated given the displacements of the reflecting surface nodes.

[0121] In the specification provided here, a large number of specific details are described. However, it can be understood that the embodiments of the present invention can be practiced without these specific details. Similarly, in order to streamline the present invention and help understand one or more of the various inventive aspects, in the above description of the exemplary embodiments of the present invention, the various features of the embodiments of the present invention are sometimes grouped together into a single embodiment, figure, or description thereof. Among them, the claims following the specific implementation manner are hereby expressly incorporated into the specific implementation manner, where each claim itself is regarded as a separate embodiment of the present invention.

[0122] It should be noted that the above embodiments are illustrative of the present invention and not restrictive thereof, and alternative embodiments can be designed by those skilled in the art without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. The word "comprising" does not exclude the presence of elements or steps not listed in the claim. The word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. The present invention can be implemented by means of hardware including several different elements and by means of a suitably programmed computer. In a unit claim listing several devices, several of these devices can be embodied by the same item of hardware. The use of the words first, second, and third, etc. does not denote any order. These words can be interpreted as names. The steps in the above embodiments, unless otherwise specified, should not be construed as limiting 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 according to the undetermined parameter vector; The step 3 specifically includes: step (1): defining 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 calculating the influence of the displacement of all nodes in the three directions of X, Y, and Z on the best matching parabola; Step (2): Each parameter in the undetermined parameter vector of the best fitting parabola is affected by the displacement of each point The influence of the displacement of all nodes on the reflection surface can be calculated for the unknown parameters of the best matching parabola. According to the influence of the displacement of all nodes in the X, Y, and Z directions on the best matching 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 .

2. The method according to claim 1, 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 of node n in the x direction 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.

3. 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: .

4. The method according to claim 3, 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: ; From the above formula, we can get that 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; the unknown 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; As a vector is a quadratic continuous function of the independent variable.

5. The method according to claim 4, characterized in that Normal deviation of the deformed parabola from the best fitting parabola The expression is: ,make , , , , , , , , , , and then , ,but: .

6. The method according to claim 3, 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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