MPC unit-based mirror surface shape zernike coefficient solving and residual analysis method

Abstract

The present application relates to the field of space camera mirror analysis, and particularly relates to a mirror surface shape Zernike coefficient solving and residual error analysis method based on MPC unit, comprising: creating a node set in a mirror finite element model; extracting node coordinates on a mirror optical surface; establishing a reference coordinate system, and constructing an optical surface node reference coordinate conversion function; performing normalization processing on optical surface node reference coordinates and node reference displacements; constructing an optical surface Zernike coefficient solving function; creating an MPC unit of the optical surface Zernike coefficient; constructing a Zernike fitting residual error RMS value solving function; and importing the MPC unit and the Zernike fitting residual error RMS value solving function into the mirror finite element model to calculate the Zernike coefficient of the mirror surface shape and the RMS value of the Zernike fitting residual error. The present application improves the space camera mirror optimization design efficiency.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of spatial camera mirror analysis, and particularly relates to a mirror surface shape Zernike coefficient solving and residual analysis method based on MPC units. BACKGROUND

[0002] Zernike polynomials are consistent with the phase difference polynomials observed in optical detection, and are often used to describe the wavefront characteristics of optical systems. Therefore, the Zernike coefficients of the mirror surface deformation of the mirror under various working conditions are important indicators for the design of spatial optical cameras. In the design process, simulation analysis is usually performed by using the finite element method. The existing finite element analysis commercial software does not have the function of solving and analyzing the Zernike coefficients of the mirror optical surface, and the mirror surface node coordinates and displacement data need to be exported for calculation to solve and analyze the Zernike coefficients, which seriously affects the iterative optimization efficiency of the mirror design. SUMMARY

[0003] Therefore, the application aims to provide a mirror surface shape Zernike coefficient solving and residual analysis method based on MPC units to solve the technical problem that the existing finite element analysis commercial software does not have the function of solving and analyzing the Zernike coefficients of the mirror optical surface, which seriously affects the iterative optimization efficiency of the mirror design.

[0004] To achieve the above-mentioned purpose, the technical scheme of the application is as follows:

[0005] A mirror surface shape Zernike coefficient solving and residual analysis method based on MPC units, comprising the following steps:

[0006] S1: creating a node set in the finite element model of the mirror;

[0007] S2: extracting the node numbers of all nodes located on the optical surface of the mirror, and extracting the spatial coordinates of the nodes corresponding to the node numbers in the global coordinate system;

[0008] S3: establishing a reference coordinate system for characterizing the optical surface of the mirror with the optical axis as the s axis and the intersection of the optical surface of the mirror and the optical axis as the origin, constructing a node reference coordinate conversion function of the optical surface of the mirror, converting the spatial coordinates of the nodes of the optical surface of the mirror in the global coordinate system to the reference coordinate system, and obtaining the reference coordinates and node reference displacements of the nodes of the optical surface of the mirror;

[0009] S4: performing normalization processing on the reference coordinates and node reference displacements of the nodes of the optical surface of the mirror to obtain the normalized cylindrical coordinates and normalized sag displacements of the nodes of the optical surface of the mirror;

[0010] S5: The Zernike polynomial is obtained by fitting the optical surface of the mirror using the least squares method. The normalized cylindrical coordinates and normalized vector displacements of the nodes of the optical surface of the mirror are substituted into the Zernike polynomial to construct the solution function of the Zernike coefficients of the optical surface of the mirror.

[0011] S6: Create MPC cells for characterizing the Zernike coefficients of the mirror optical surface based on the solution function of the Zernike coefficients of the mirror optical surface;

[0012] S7: Construct a function to solve the RMS value of the Zernike fitting residual for characterizing the surface shape of the optical surface of the mirror based on the Zernike coefficient solution function of the mirror optical surface;

[0013] S8: Import the MPC unit and the Zernike fitting residual RMS value solution function into the finite element model of the mirror, and use the finite element solver to calculate the Zernike coefficients and the RMS value of the Zernike fitting residual of the optical surface shape of the mirror.

[0014] Furthermore, the transformation function for the reference coordinates of the optical surface nodes of the mirror established in step S3 is:

[0015] ;

[0016] in, u i , v i , s i These represent the optical surface nodes of the mirror. i exist u Axial direction, v Axial direction, s Reference coordinates in the axial direction; Trans, Rotu, Rotv, Rots These represent the translation and revolution of the global coordinate system relative to the reference coordinate system, respectively. u Axial direction, v Axial direction, s Homogeneous coordinate transformation matrix for rotation along the axis.

[0017] Furthermore, the translational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system is:

[0018] ;

[0019] in, T u , T v , T srespectively represent the coordinate values of the three directions of the global coordinate system origin in the reference coordinate system under the base u axis, v axis, s axis;

[0020] The homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system rotating around the u axis direction is:

[0021] ;

[0022] The homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system rotating around the v axis direction is:

[0023] ;

[0024] The homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system rotating around the s axis direction is:

[0025] ;

[0026] wherein, R u , R v , R s respectively represent the rotation angles of the global coordinate system relative to the reference coordinate system in the u axis direction, v axis direction, s axis direction.

[0027] Further, the node reference displacement of the mirror optical surface node in the reference coordinate system in step S3 is represented as:

[0028] ;

[0029] wherein, d ui , d vi , d si respectively represent the reference displacement values of the mirror optical surface node in the reference coordinate system in the u axis direction, v axis direction, s axis direction, d xi , d yi , d zi respectively represent the global displacement values of the mirror optical surface node in the global coordinate system in the x-axis direction, y-axis direction and z-axis direction.

[0030] Further, the normalized cylindrical coordinates of the mirror optical surface node in step S4 i ρ i , θ i ,s i are expressed as:

[0031]

[0032] where D represents the aperture of the mirror, i represents the first node; i

[0033] The normalized sag displacement of the mirror optical surface node in step S4 sag i is expressed as:

[0034] ;

[0035] where λ represents the working wavelength of the mirror.

[0036] Further, the Zernike polynomial in step S5 is expressed as:

[0037] ;

[0038] where represents the sag direction displacement of the mirror optical surface node, n is expressed as a radial coefficient, m is expressed as an angular coefficient, n and m are integers, and n ≥ m , n - m is even, A nm and B nm represent Zernike polynomial coefficients, represents a radial function, which is expressed as:

[0039] ;

[0040] Substituting the normalized cylindrical coordinates ρ i , θ i ,s i and the normalized sag displacement sag i into the above formula, we obtain: ​​

[0041] ;

[0042] wherein, sag represents the displacement column vector of all node heights of the mirror optical surface,

[0043] , T represents a transpose symbol; A is a Zernike polynomial coefficient, , H is a coefficient matrix, which is expressed as:

[0044] .

[0045] Further, the constructed mirror optical surface Zernike coefficient solving function is:

[0046] .

[0047] Further, in step S1, the node set q is set as the node set q j , the x displacement value corresponding to the first j Zernike coefficient of the mirror is extracted, the x-direction displacement of all nodes in the node set q is extracted, and is recorded as q x , and the mapping relationship between the q displacement of all nodes in the node set x and the node reference displacement of all nodes of the mirror optical surface is expressed as:

[0048] .

[0049] Further, the Zernike fitting residual RMS value solving function constructed in step S7 is expressed as:

[0050] ;

[0051] wherein, N represents the number of nodes of the mirror optical surface.

[0052] Compared with the prior art, the application can achieve the following beneficial effects:

[0053] The main objective of this invention is to provide a method for solving the Zernike coefficients and performing residual analysis on the surface of a mirror based on MPC elements. By creating MPC elements to characterize the Zernike coefficients of the optical surface of the mirror, and constructing a function for solving the Zernike coefficients and the RMS value of the surface residual in finite element analysis software, the method enables direct solving of the Zernike coefficients and the RMS value of the surface residual of the mirror's optical surface, thereby improving the efficiency of the optimization design of space camera mirrors. Attached Figure Description

[0054] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0055] Figure 1 A flowchart illustrating the method for solving the Zernike coefficients and performing residual analysis on the surface of a reflective mirror based on MPC units, as described in an embodiment of the present invention.

[0056] Figure 2 A schematic diagram of selected MPC unit parameters as described in the embodiments of the present invention;

[0057] Figure 3 A schematic diagram of the function for solving the RMS value of the Zernike fitting residual of the surface shape as described in the embodiment of the present invention;

[0058] Figure 4 A schematic diagram showing the solution results of the Zernike coefficient of the optical surface profile of the mirror described in the embodiment of the present invention;

[0059] Figure 5 This is a schematic diagram showing the solution results of the Zernike coefficients of the optical surface profile of a mirror obtained according to traditional analysis methods. Detailed Implementation

[0060] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0061] The invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0062] like Figure 1 As shown, the method for solving the Zernike coefficients and performing residual analysis on the reflective mirror shape based on MPC units provided by this invention includes the following steps:

[0063] S1: Create a node set in the finite element model of the reflector.

[0064] Set the node set as q and node set q elements in q j of x The displacement value corresponds to the first reflector. j Zernike coefficients are used to extract the node set. q The x-direction displacement of all nodes in the middle is denoted as . q x .

[0065] For example: Create a node set at the origin (0,0,0). q Node set q The node numbers are 9001001 to 9001028.

[0066] S2: Extract the node numbers of some nodes located on the optical surface of the mirror, and extract the spatial coordinates of the nodes corresponding to the node numbers in the global coordinate system.

[0067] For example: Extract the node number range of 312 nodes from the optical surface of the mirror: 101~412, and the node coordinates of each node are ( x i , y i , z i ), i Indicates the node number.

[0068] S3: With the optical axis as s A reference coordinate system is established with the intersection of the optical surface of the mirror and the optical axis as the origin to characterize the optical surface of the mirror. A reference coordinate transformation function for the nodes of the optical surface of the mirror is constructed to transform the spatial coordinates of the nodes of the optical surface of the mirror located in the global coordinate system to the reference coordinate system, thereby obtaining the reference coordinates and reference displacements of the nodes of the optical surface of the mirror.

[0069] The established transformation function for the reference coordinates of the optical surface nodes of the mirror is as follows:

[0070] ;

[0071] in, u i , v i , s i These represent the optical surface nodes of the mirror. i exist u Axial direction, v Axial direction, sReference coordinate in the axial direction Trans, Rotu, Rotv, Rots respectively represent the translational coordinates of the global coordinate system relative to the reference coordinate system in the x, y, z directions respectively. u axial direction, v axial direction, s axial direction.

[0072] The translational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system is:

[0073] ;

[0074] wherein, T u , T v , T s respectively represent the coordinates of the global coordinate system origin in the reference coordinate system in the x, y, z directions respectively. u axial direction, v axial direction, s axial direction.

[0075] The rotational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system around the x-axis direction is: u

[0076] .

[0077] The rotational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system around the y-axis direction is: v

[0078] .

[0079] The rotational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system around the z-axis direction is: s

[0080] ;

[0081] wherein, R u , R v , R s respectively represent the rotation angles of the global coordinate system relative to the reference coordinate system in the x, y, z directions respectively. u axial direction, v axial direction, s axial direction.

[0082] The node reference displacement of the mirror optical surface node in the reference coordinate system is represented as:

[0083] ;

[0084] ​​​in, d ui , d vi , d si These represent the optical surface nodes of the mirror in the reference coordinate system. u Axial direction, v Axial direction, s Reference displacement value in the axial direction, d xi , d yi , d zi These represent the global displacement values ​​of the optical surface nodes of the mirror in the x-axis, y-axis, and z-axis directions, respectively, in the global coordinate system.

[0085] S4: Normalize the reference coordinates and reference displacements of the optical surface nodes of the mirror to obtain the normalized cylindrical coordinates and normalized sag displacements of the optical surface nodes of the mirror.

[0086] Optical surface nodes of the mirror i Node reference coordinates ( u i , v i , s i ), nodal reference displacement ( d ui , d vi , d si Normalization is performed to obtain the optical surface nodes of the mirror. i Normalized cylindrical coordinates ( ρ i , θ i ,s i ) and normalized vector height displacement sag i .

[0087] Mirror optical surface nodes i Normalized cylindrical coordinates ( ρ i , θ i ,s i ) is represented as:

[0088]

[0089] Where D represents the aperture of the reflecting mirror. i Indicates the first i One node;

[0090] Normalised sag displacement of mirror optical surface node i sag i is expressed as:

[0091] ;

[0092] wherein, λ denotes the working wavelength of the mirror.

[0093] S5: Zernike fitting is performed on the mirror optical surface by using the least square method to obtain the Zernike polynomial, and the normalised cylindrical coordinates and the normalised sag displacement of the mirror optical surface node are substituted into the Zernike polynomial to construct a Zernike coefficient solving function of the mirror optical surface.

[0094] The Zernike polynomial is expressed as:

[0095] ;

[0096] wherein, denotes the sag direction displacement of the mirror optical surface node, n is expressed as a radial coefficient, m is expressed as an angular coefficient, n and m are integers, and n ≥ m , n - m is an even number, , , A nm and B nm denote the Zernike polynomial coefficients, , both denote a radial function, which is expressed as:

[0097] ;

[0098] Substitute the normalised cylindrical coordinates ( ρ i , θ i ,s i ) and the normalised sag displacement sag i into the above formula to obtain:

[0099] ;

[0100] wherein, sag ​represents a column vector of all nodal height displacement of the mirror optical surface,

[0101] , T represents a transpose symbol; A is a Zernike polynomial coefficient, , H is a coefficient matrix, which is represented as:

[0102] .

[0103] The constructed Zernike coefficient solving function of the mirror optical surface is represented as:

[0104] .

[0105] S6: Create MPC elements about the node set for characterizing the Zernike coefficients of the mirror optical surface based on the Zernike coefficient solving function of the mirror optical surface.

[0106] MPC (multi-point constraint) elements define the coupling relationship between the degrees of freedom of nodes, that is, to take some degrees of freedom of a node as a standard value, and then to establish a mathematical relationship between some degrees of freedom of other specified nodes and the standard value. MPC elements have been widely used in finite element analysis due to their ability to simulate complex physical phenomena and improve computational efficiency.

[0107] For example: create MPC elements about the node set q based on the Zernike coefficient solving function of the mirror optical surface, as shown in Figure 2 , then the mapping relationship between the q displacement of all nodes in the node set x and the nodal reference displacement of all nodes of the mirror optical surface is represented as:

[0108] .

[0109] S7: Construct a Zernike fitting residual RMS value solving function for characterizing the surface shape of the mirror optical surface based on the Zernike coefficient solving function of the mirror optical surface.

[0110] The constructed Zernike fitting residual RMS value solving function is shown in Figure 3 , and is specifically represented as:

[0111] ;

[0112] wherein, N represents the number of nodes of the mirror optical surface.

[0113] S8: Import the MPC unit and the Zernike fitting residual RMS value solution function into the finite element model of the mirror, and use the finite element solver to calculate the Zernike coefficients and the RMS value of the Zernike fitting residual of the optical surface shape of the mirror.

[0114] The MPC element and Zernike fitting residual RMS value solution function were submitted to the NASTRAN finite element solver for calculation, and the Zernike coefficients of the optical surface shape of the mirror were obtained as shown in Figure 4. The RMS value of the Zernike fitting residual is 0.1191λ. The Zernike coefficients of the optical surface shape of the mirror obtained using traditional analysis methods are shown in Figure 4. Figure 5 As shown, the RMS value of the residual is 0.1194λ. (This is achieved through...) Figure 4 and Figure 5 The comparison shows that the results of the two analysis methods are consistent, indicating that the method for solving the Zernike coefficients and residual analysis of the reflective mirror shape based on MPC elements is practical and effective.

[0115] Although embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

[0116] The specific embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention. Any other corresponding changes and modifications made in accordance with the technical concept of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for solving the Zernike coefficients and performing residual analysis on the surface shape of a reflective mirror based on MPC units, characterized in that, Includes the following steps: S1: Create a node set in the finite element model of the reflector; S2: Extract the node numbers of all nodes located on the optical surface of the mirror, and extract the spatial coordinates of the node corresponding to the node number in the global coordinate system; S3: With the optical axis as s The origin is the intersection of the optical surface of the mirror and the optical axis. A reference coordinate system is established to characterize the optical surface of the mirror. A reference coordinate transformation function for the nodes of the optical surface of the mirror is constructed to transform the spatial coordinates of the nodes of the optical surface of the mirror in the global coordinate system to the reference coordinate system, so as to obtain the reference coordinates and reference displacements of the nodes of the optical surface of the mirror. S4: Normalize the reference coordinates and reference displacements of the optical surface nodes of the mirror to obtain the normalized cylindrical coordinates and normalized sag displacements of the optical surface nodes of the mirror. S5: The Zernike polynomial is obtained by fitting the optical surface of the mirror using the least squares method. The normalized cylindrical coordinates and normalized vector displacements of the nodes of the optical surface of the mirror are substituted into the Zernike polynomial to construct the solution function of the Zernike coefficients of the optical surface of the mirror. S6: Create MPC cells for characterizing the Zernike coefficients of the mirror optical surface based on the solution function of the Zernike coefficients of the mirror optical surface; S7: Construct a function to solve the RMS value of the Zernike fitting residual for characterizing the surface shape of the optical surface of the mirror based on the Zernike coefficient solution function of the mirror optical surface; S8: Import the MPC unit and the Zernike fitting residual RMS value solution function into the finite element model of the mirror, and use the finite element solver to calculate the Zernike coefficients and the RMS value of the Zernike fitting residual of the optical surface shape of the mirror.

2. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 1, characterized in that, The transformation function for the reference coordinates of the optical surface nodes of the mirror in step S3 is: ； in, u i , v i , s i These represent the optical surface nodes of the mirror. i exist u Axial direction, v Axial direction, s Reference coordinates in the axial direction; Trans, Rotu, Rotv, Rots These represent the translation and revolution of the global coordinate system relative to the reference coordinate system, respectively. u Axial direction, v Axial direction, s Homogeneous coordinate transformation matrix for rotation along the axis; x i , y i , z i () represents the node coordinates of each node. i Indicates the node number.

3. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 2, characterized in that, The homogeneous translational transformation matrix of the global coordinate system relative to the reference coordinate system is: ； in, T u , T v , T s These represent the origin of the global coordinate system in the reference coordinate system. u axis, v axis, s The coordinate values ​​in the three directions of the axis; The global coordinate system revolves around the reference coordinate system u The homogeneous coordinate transformation matrix for rotation along the axis is: ； The global coordinate system revolves around the reference coordinate system v The homogeneous coordinate transformation matrix for rotation along the axis is: ； The global coordinate system revolves around the reference coordinate system s The homogeneous coordinate transformation matrix for rotation along the axis is: ； in, R u , R v , R s These represent the global coordinate system relative to the reference coordinate system. u Axial direction, v Axial direction, s Rotation angle along the axis.

4. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 3, characterized in that, In step S3, the nodal reference displacement of the optical surface node of the mirror in the reference coordinate system is expressed as: ； in, d ui , d vi , d si These represent the optical surface nodes of the mirror in the reference coordinate system. u Axial direction, v Axial direction, s Reference displacement value in the axial direction, d xi , d yi , d zi These represent the global displacement values ​​of the optical surface nodes of the mirror in the x-axis, y-axis, and z-axis directions, respectively, in the global coordinate system.

5. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 4, characterized in that, In step S4, the optical surface node of the mirror i Normalized cylindrical coordinates ( ρ i , θ i ,s i ) is represented as: Where D represents the aperture of the reflecting mirror. i Indicates the first i One node; Normalized height displacement of the optical surface nodes of the mirror sag i Represented as: ； in, λ This indicates the operating wavelength of the reflector.

6. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 1, characterized in that, The Zernike polynomial in step S5 is expressed as: ； in, This represents the displacement in the sag direction of the nodes on the optical surface of the mirror. n Represented as radial coefficient, m Indicates the angular coefficient. n and m All are integers, and n ≥ m , n - m Even number, A nm and B nm All represent the coefficients of the Zernike polynomial. The radial function is expressed as: ； Normalized cylindrical coordinates ( ρ i , θ i ,s i ) and normalized vector height displacement sag i Substituting into the above equation, we get: ； in, sag This represents the column vector of elevation displacements at all nodes on the optical surface of the mirror. , T Indicates the transpose symbol; These are the coefficients of the Zernike polynomial. , H The coefficient matrix is ​​represented as follows: 。 7. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 6, characterized in that, The function for solving the Zernike coefficients of the constructed optical surface of the mirror is: 。 8. The method for solving the Zernike coefficients and performing residual analysis of the reflective mirror surface shape based on MPC units according to claim 7, characterized in that, In step S1, the node set is set. q elements in q j of x The displacement value corresponds to the first reflector. j Zernike coefficients are used to extract the node set. q The x-direction displacement of all nodes in the middle is denoted as . q x Then the node set q All nodes x The mapping relationship between the directional displacement and the nodal reference displacements of all nodes on the optical surface of the reflecting mirror is expressed as: 。

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