Reflector surface shape Zernike coefficient solving and residual analysis method based on MPC unit

Abstract

The invention relates to the field of space camera reflector analysis, in particular to a reflector surface shape Zernike coefficient solving and residual analysis method based on an MPC unit, and the method comprises the steps: creating a node set in a reflector finite element model; node coordinates on the optical surface of the reflector are extracted; establishing a reference coordinate system, and constructing an optical surface node reference coordinate conversion function; performing normalization processing on the node reference coordinates and the node reference displacement of the optical surface; an optical surface Zernike coefficient solving function is constructed; creating an MPC unit of the Zernike coefficient of the optical surface; a Zernike fitting residual error RMS value solving function is constructed; and the MPC unit and the Zernike fitting residual error RMS value solving function are imported into the finite element model of the reflector to calculate and obtain a Zernike coefficient of the surface shape of the reflector and an RMS value of the Zernike fitting residual error. The optimization design efficiency of the space camera reflector is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of space camera reflector analysis, and in particular relates to a reflector surface Zernike coefficient solution and residual analysis method based on an MPC unit. Background Art

[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 that characterize the deformation of the mirror under various working conditions are important indicators for the design of space optical cameras. During the design process, simulation analysis is usually performed using the finite element method. Existing commercial finite element analysis software does not yet have the function of solving and analyzing the Zernike coefficients of the optical surface of the reflector. It is usually necessary to export and calculate the coordinates of the mirror node and the displacement data of the reflector surface to complete the Zernike coefficient solution and residual analysis, which seriously affects the iterative optimization efficiency of the reflector design. Summary of the invention

[0003] In view of this, the present invention aims to provide a method for solving the Zernike coefficients of the reflector surface shape and residual analysis based on the MPC unit, so as to solve the technical problem that the existing commercial finite element analysis software does not have the function of solving and analyzing the Zernike coefficients of the optical surface of the reflector, which seriously affects the iterative optimization efficiency of the reflector design.

[0004] To achieve the above object, the technical solution created by the present invention is implemented as follows: A method for solving Zernike coefficients of a reflector surface and analyzing residual errors based on an MPC unit comprises the following steps: S1: Create a node set in the finite element model of the reflector; S2: extracting the node numbers of all nodes located on the optical surface of the reflector, and extracting the spatial coordinates of the nodes corresponding to the node numbers in the global coordinate system; S3: With the optical axis as s Axis, taking the intersection of the optical surface of the reflector and the optical axis as the origin, establishing a reference coordinate system for characterizing the optical surface of the reflector, constructing a reference coordinate conversion function for the nodes on the optical surface of the reflector, converting the spatial coordinates of the nodes on the optical surface of the reflector in the global coordinate system to the reference coordinate system, and obtaining the reference coordinates and node reference displacements of the nodes on the optical surface of the reflector; S4: normalizing the reference coordinates and reference displacements of the nodes on the optical surface of the reflector to obtain normalized cylindrical coordinates and normalized vector height displacements of the nodes on the optical surface of the reflector; S5: Use the least square method to perform Zernike fitting on the optical surface of the reflector to obtain the Zernike polynomial, substitute the normalized cylindrical coordinates and normalized vector height displacement of the nodes on the optical surface of the reflector into the Zernike polynomial to construct the Zernike coefficient solution function of the optical surface of the reflector; S6: creating an MPC unit for characterizing the Zernike coefficient of the optical surface of the reflector on the node set based on the Zernike coefficient solution function of the optical surface of the reflector; S7: Based on the Zernike coefficient solution function of the optical surface of the reflector, a Zernike fitting residual RMS value solution function for characterizing the optical surface shape of the reflector is constructed; S8: The MPC unit and the Zernike fitting residual RMS value solving function are imported into the finite element model of the reflector, and the Zernike coefficient of the optical surface shape of the reflector and the RMS value of the Zernike fitting residual are calculated by the finite element solver.

[0005] Furthermore, the reflector optical surface node reference coordinate conversion function established in step S3 is: ; in, u i , v i , s i Represents the optical surface nodes of the reflector i exist u Axis direction, v Axis direction, s Reference coordinates in the axis direction; Trans、Rotu、Rotv、Rots They are the translation and rotation of the global coordinate system relative to the reference coordinate system. u Axis direction, v Axis direction, s Homogeneous coordinate transformation matrix for axis rotation.

[0006] Furthermore, the translational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system is: ; in, T u , T v , T s They represent the origin of the global coordinate system in the reference coordinate system. u axis, v axis, s Coordinate values ​​in three directions of the axis; The global coordinate system is relative to the reference coordinate system. uThe homogeneous coordinate transformation matrix for axis rotation is: ; The global coordinate system is relative to the reference coordinate system. v The homogeneous coordinate transformation matrix for axis rotation is: ; The global coordinate system is relative to the reference coordinate system. s The homogeneous coordinate transformation matrix for axis rotation is: ; in, R u , R v , R s They represent the relative position of the global coordinate system to the reference coordinate system. u Axis direction, v Axis direction, s The rotation angle about the axis.

[0007] Furthermore, the node reference displacement of the node on the optical surface of the reflector in step S3 in the reference coordinate system is expressed as: ; in, d ui , d vi , d si They represent the nodes of the optical surface of the reflector in the reference coordinate system. u Axis direction, v Axis direction, s The reference displacement value in the axial direction, d xi , d yi , d zi They respectively represent the global displacement values ​​of the nodes on the optical surface of the reflector in the x-axis direction, y-axis direction, and z-axis direction in the global coordinate system.

[0008] Furthermore, in step S4, the reflector optical surface node i The normalized cylindrical coordinates of ρ i , θ i ,s i ) is expressed as:

[0009] Where D is the diameter of the reflector. i Indicatesi nodes; Normalized vector height displacement of the nodes on the optical surface of the mirror sag i It is expressed as: ; in, λ Indicates the operating wavelength of the reflector.

[0010] Furthermore, the Zernike polynomial in step S5 is expressed as: ; in, represents the displacement of the nodes on the optical surface of the reflector in the direction of the sagittal height, n Expressed as the radial coefficient, m represents the angular coefficient, n and m are integers, and n ≥ m , n - m is an even number, A nm and B nm All represent Zernike polynomial coefficients, represents the radial function, which is expressed as: ; The normalized cylindrical coordinates ( ρ i , θ i ,s i ) and normalized vector height displacement sag i Substituting into the above formula, we get: ; in, sag Represents the displacement column vector of all nodes on the optical surface of the reflector, , T represents the transpose symbol; A are the Zernike polynomial coefficients, , H is the coefficient matrix, which is expressed as: .

[0011] Furthermore, the constructed Zernike coefficient solution function of the reflector optical surface is: .

[0012] Furthermore, in step S1, the node set is set q Elements in q j of x The displacement value corresponds to the reflector j Term Zernike coefficient, extract node set q The x-direction displacement of all nodes in q x , then the node set q All nodes in x The mapping relationship between the axial displacement and the node reference displacement of all nodes of the optical mirror surface of the reflector is expressed as: .

[0013] Furthermore, the Zernike fitting residual RMS value solution function constructed in step S7 is expressed as: ; in, N Represents the number of nodes on the optical surface of the mirror.

[0014] Compared with the prior art, the invention can achieve the following beneficial effects: The main purpose of the present invention is to provide a method for solving the Zernike coefficients of the reflector surface shape and residual analysis based on an MPC unit. By creating an MPC unit for characterizing the Zernike coefficients of the optical surface of the reflector, and constructing a reflector surface shape Zernike coefficient solution function and a surface shape residual RMS value solution function in a finite element analysis software, the Zernike coefficient solution of the reflector optical surface and the RMS value solution of the surface shape residual can be directly achieved, thereby improving the optimization design efficiency of the space camera reflector. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The drawings constituting part of the present invention are used to provide a further understanding of the present invention. The exemplary embodiments and descriptions of the present invention are used to explain the present invention and do not constitute an improper limitation on the present invention. In the drawings: Figure 1 A schematic diagram of a process for solving the Zernike coefficients of a reflector surface and analyzing residual errors based on an MPC unit according to an embodiment of the present invention; Figure 2 A schematic diagram of selected MPC unit parameters according to an embodiment of the present invention; Figure 3 A schematic diagram of a function for solving the residual RMS value of a Zernike fitting of a selected surface shape according to an embodiment of the present invention; Figure 4A schematic diagram of the solution result of the Zernike coefficient of the optical surface profile of the reflector described in the embodiment of the present invention; Figure 5 This is a schematic diagram of the solution results of the Zernike coefficients of the optical surface shape of the reflector obtained according to the traditional analysis method. DETAILED DESCRIPTION

[0016] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation of the present invention.

[0017] The present invention will be described in detail below with reference to the accompanying drawings and in combination with embodiments.

[0018] like Figure 1 As shown, the method for solving the Zernike coefficients of the reflector surface shape and residual analysis based on the MPC unit provided by the embodiment of the present invention includes the following steps: S1: Create a node set in the mirror finite element model.

[0019] Set the node set to q , and the node set q Elements in q j of x The displacement value corresponds to the reflector j Term Zernike coefficient, extract node set q The x-direction displacement of all nodes in q x .

[0020] For example: Create a node set at the coordinate origin (0,0,0) q , node set q The nodes in are numbered 9001001~9001028.

[0021] S2: Extract the node numbers of some nodes located on the optical surface of the reflector, and extract the spatial coordinates of the nodes corresponding to the node numbers in the global coordinate system.

[0022] For example: Extract the node number range of 312 nodes on the optical surface of the reflector: 101~412, and the node coordinates of each node are ( x i , y i , z i ), i Indicates the node number.

[0023] S3: With the optical axis ass Axis, with the intersection of the reflector optical surface and the optical axis as the origin, establishes a reference coordinate system for characterizing the reflector optical surface, constructs a reference coordinate conversion function for the reflector optical surface nodes, converts the spatial coordinates of the reflector optical surface nodes located in the global coordinate system to the reference coordinate system, and obtains the reference coordinates and node reference displacements of the reflector optical surface nodes.

[0024] The established reflector optical surface node reference coordinate conversion function is: ; in, u i , v i , s i Represents the optical surface nodes of the reflector i exist u Axis direction, v Axis direction, s Reference coordinates in the axis direction; Trans、Rotu、Rotv、Rots They are the translation and rotation of the global coordinate system relative to the reference coordinate system. u Axis direction, v Axis direction, s Homogeneous coordinate transformation matrix for axis rotation.

[0025] The translational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system is: ; in, T u , T v , T s They represent the origin of the global coordinate system in the reference coordinate system. u axis, v axis, s Coordinate values ​​in three directions of the axis.

[0026] The global coordinate system is relative to the reference coordinate system. u The homogeneous coordinate transformation matrix for axis rotation is: .

[0027] The global coordinate system is relative to the reference coordinate system. v The homogeneous coordinate transformation matrix for axis rotation is: .

[0028] The global coordinate system is relative to the reference coordinate system. s The homogeneous coordinate transformation matrix for axis rotation is: ; in, R u , R v , R s They represent the relative position of the global coordinate system to the reference coordinate system. u Axis direction, v Axis direction, s The rotation angle about the axis.

[0029] The node reference displacement of the reflector optical surface node in the reference coordinate system is expressed as: ; in, d ui , d vi , d si They represent the nodes of the optical surface of the reflector in the reference coordinate system. u Axis direction, v Axis direction, s The reference displacement value in the axial direction, d xi , d yi , d zi They respectively represent the global displacement values ​​of the nodes on the optical surface of the reflector in the x-axis direction, y-axis direction, and z-axis direction in the global coordinate system.

[0030] S4: normalizing the reference coordinates and reference displacements of the nodes on the optical surface of the reflector to obtain normalized cylindrical coordinates and normalized vector height displacements of the nodes on the optical surface of the reflector.

[0031] Optical Surface Nodes for Reflectors i 、Node reference coordinates( u i , v i , s i ), node base displacement( d ui , d vi , d si ) is normalized to obtain the optical surface nodes of the reflector i The normalized cylindrical coordinates of ρ i , θ i ,s i ) and normalized vector height displacement sag i .

[0032] Mirror Optical Surface Node i The normalized cylindrical coordinates of ρ i , θ i ,s i ) is expressed as:

[0033] Where D is the diameter of the reflector. i Indicates i nodes; Mirror Optical Surface Node i Normalized vector height displacement sag i It is expressed as: ; in, λ Indicates the operating wavelength of the reflector.

[0034] S5: Use the least squares method to perform Zernike fitting on the optical surface of the reflector to obtain the Zernike polynomial, substitute the normalized cylindrical coordinates and normalized vector height displacement of the nodes on the optical surface of the reflector into the Zernike polynomial to construct the Zernike coefficient solution function of the optical surface of the reflector.

[0035] Zernike polynomials are expressed as: ; in, represents the displacement of the nodes on the optical surface of the reflector in the direction of the sagittal height, n Expressed as the radial coefficient, m represents the angular coefficient, n and m are integers, and n ≥ m , n - m is an even number, , , A nm and B nm All represent Zernike polynomial coefficients, , Both represent radial functions, which can be expressed as: ; The normalized cylindrical coordinates ( ρ i , θ i ,s i ) and normalized vector height displacement sag i Substituting into the above formula, we get: ; in, sag Represents the displacement column vector of all nodes on the optical surface of the reflector, , T represents the transpose symbol; A are the Zernike polynomial coefficients, , H is the coefficient matrix, which is expressed as: .

[0036] Then the constructed Zernike coefficient solution function of the reflector optical surface is expressed as: .

[0037] S6: creating an MPC unit for characterizing the Zernike coefficient of the optical surface of the reflector on the node set based on the Zernike coefficient solution function of the optical surface of the reflector.

[0038] The MPC (multi-point constraint) unit defines the coupling relationship between the degrees of freedom of the nodes, that is, taking several degrees of freedom of a node as the standard value, and then establishing a mathematical relationship between several degrees of freedom of other specified nodes and the standard value. The MPC unit has been widely used in finite element analysis due to its ability to simulate complex physical phenomena and improve computational efficiency.

[0039] For example: Create a node set based on the Zernike coefficient solution function of the optical surface of the mirror q MPC units, such as Figure 2 As shown, the node set q All nodes in x The mapping relationship between the axial displacement and the node reference displacement of all nodes of the optical mirror surface of the reflector is expressed as: .

[0040] S7: Based on the Zernike coefficient solution function of the optical surface of the reflector, a Zernike fitting residual RMS value solution function for characterizing the surface shape of the optical surface of the reflector is constructed.

[0041] The constructed Zernike fitting residual RMS value solution function is as follows Figure 3 As shown, it is specifically expressed as: ; in, N Represents the number of nodes on the optical surface of the mirror.

[0042] S8: The MPC unit and the Zernike fitting residual RMS value solving function are imported into the finite element model of the reflector, and the Zernike coefficient of the optical surface shape of the reflector and the RMS value of the Zernike fitting residual are calculated by the finite element solver.

[0043] The MPC unit and the Zernike fitting residual RMS value solution function are submitted to the NASTRAN finite element solver for calculation, and the Zernike coefficients of the optical surface shape of the reflector are 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 reflector obtained by the traditional analysis method are as follows Figure 5 As shown, the RMS value of the residual is 0.1194λ. Figure 4 and Figure 5 By comparison, it can be seen that the results of the two analysis methods are consistent, indicating that the Zernike coefficient solution and residual analysis method of the reflector surface based on the MPC unit is practical and effective.

[0044] Although the embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and cannot be understood as limiting the present invention. Those skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

[0045] The above specific implementations of the present invention do not constitute a limitation on the protection scope of the present invention. Any other corresponding changes and modifications made based on the technical concept of the present invention should be included in the protection scope of the claims of the present invention.

Claims

1. A method for solving Zernike coefficients and residual analysis of reflector surface shape based on MPC unit, characterized in that: The following steps are involved: S1: Create a node set in the finite element model of the reflector; S2: extracting the node numbers of all nodes located on the optical surface of the reflector, and extracting the spatial coordinates of the nodes corresponding to the node numbers in the global coordinate system; S3: With the optical axis as s Axis, taking the intersection of the optical surface of the reflector and the optical axis as the origin, establishing a reference coordinate system for characterizing the optical surface of the reflector, constructing a reference coordinate conversion function for the nodes on the optical surface of the reflector, converting the spatial coordinates of the nodes on the optical surface of the reflector in the global coordinate system to the reference coordinate system, and obtaining the reference coordinates and node reference displacements of the nodes on the optical surface of the reflector; S4: normalizing the reference coordinates and reference displacements of the nodes on the optical surface of the reflector to obtain normalized cylindrical coordinates and normalized vector height displacements of the nodes on the optical surface of the reflector; S5: Use the least square method to perform Zernike fitting on the optical surface of the reflector to obtain the Zernike polynomial, substitute the normalized cylindrical coordinates and normalized vector height displacement of the nodes on the optical surface of the reflector into the Zernike polynomial to construct the Zernike coefficient solution function of the optical surface of the reflector; S6: creating an MPC unit for characterizing the Zernike coefficient of the optical surface of the reflector on the node set based on the Zernike coefficient solution function of the optical surface of the reflector; S7: Based on the Zernike coefficient solution function of the optical surface of the reflector, a Zernike fitting residual RMS value solution function for characterizing the optical surface shape of the reflector is constructed; S8: The MPC unit and the Zernike fitting residual RMS value solving function are imported into the finite element model of the reflector, and the Zernike coefficient of the optical surface shape of the reflector and the RMS value of the Zernike fitting residual are calculated by the finite element solver.

2. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 1, characterized in that: The reference coordinate conversion function of the optical surface node of the reflector in step S3 is: ； in, u i , v i , s i Represents the optical surface nodes of the reflector i exist u Axis direction, v Axis direction, s Reference coordinates in the axis direction; Trans、Rotu、Rotv、Rots They are the translation and rotation of the global coordinate system relative to the reference coordinate system. u Axis direction, v Axis direction, s Homogeneous coordinate transformation matrix for axis rotation.

3. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 2 is characterized in that: The translational homogeneous coordinate transformation matrix of the global coordinate system relative to the reference coordinate system is: ； in, T u , T v , T s They represent the origin of the global coordinate system in the reference coordinate system. u axis, v axis, s Coordinate values ​​in three directions of the axis; The global coordinate system is relative to the reference coordinate system. u The homogeneous coordinate transformation matrix for axis rotation is: ； The global coordinate system is relative to the reference coordinate system. v The homogeneous coordinate transformation matrix for axis rotation is: ； The global coordinate system is relative to the reference coordinate system. s The homogeneous coordinate transformation matrix for axis rotation is: ； in, R u , R v , R s They represent the relative position of the global coordinate system to the reference coordinate system. u Axis direction, v Axis direction, s The rotation angle about the axis.

4. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 3 is characterized in that: The node reference displacement of the node on the optical surface of the reflector in the reference coordinate system in step S3 is expressed as: ； in, d ui , d vi , d si They represent the nodes of the optical surface of the reflector in the reference coordinate system. u Axis direction, v Axis direction, s The reference displacement value in the axial direction, d xi , d yi , d zi They respectively represent the global displacement values ​​of the nodes on the optical surface of the reflector in the x-axis direction, y-axis direction, and z-axis direction in the global coordinate system.

5. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 1, characterized in that: Mirror optical surface node in step S4 i The normalized cylindrical coordinates of ρ i , θ i ,s i ) is expressed as: Where D is the diameter of the reflector. i Indicates i nodes; Normalized sag displacement of the nodes on the optical surface of the mirror sag i It is expressed as: ； in, λ Indicates the operating wavelength of the reflector.

6. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 1, characterized in that: The Zernike polynomial in step S5 is expressed as: ； in, represents the displacement of the nodes on the optical surface of the reflector in the direction of the sagittal height, n Expressed as the radial coefficient, m represents the angular coefficient, n and m are integers, and n ≥ m , n - m is an even number, A nm and B nm All represent Zernike polynomial coefficients, represents the radial function, which is expressed as: ； The normalized cylindrical coordinates ( ρ i , θ i ,s i ) and normalized vector height displacement sag i Substituting into the above formula, we get: ； in, sag Represents the displacement column vector of all nodes on the optical surface of the reflector, , T represents the transpose symbol; A are the Zernike polynomial coefficients, , H is the coefficient matrix, which is expressed as: 。 7. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 6 is characterized in that: The constructed Zernike coefficient solution function of the reflector optical surface is: 。 8. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 7 is characterized in that: In step S1, set the node set q Elements in q j of x The displacement value corresponds to the reflector j Term Zernike coefficient, extract node set q The x-direction displacement of all nodes in q x , then the node set q All nodes in x The mapping relationship between the axial displacement and the node reference displacement of all nodes of the optical mirror surface of the reflector is expressed as: 。 9. The method for solving the Zernike coefficients of the reflector surface shape and analyzing the residual error based on the MPC unit according to claim 6, characterized in that: The Zernike fitting residual RMS value solution function constructed in step S7 is expressed as: ； in, N Represents the number of nodes on the optical surface of the mirror.

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