Off-axis aspherical mirror polishing method, device and equipment and storage medium

Abstract

The invention relates to the technical field of machine tool machining, and discloses an off-axis aspherical mirror polishing method, device and equipment and a storage medium, and the method comprises the steps: building a female aspherical geometric model, and dividing an off-axis region based on the female aspherical geometric model and a preset design requirement; performing trajectory planning based on the off-axis region to obtain a polishing path; constructing a removal function model based on the off-axis region; obtaining a surface shape residual error from the off-axis aspherical mirror to be polished, and solving and obtaining residence time based on the removal function model, the surface shape residual error and the polishing path; constructing a five-axis machine tool kinematic model; the residence time and the polishing path are substituted into the five-axis machine tool kinematic model to obtain polishing control parameters; through the process design of off-axis area positioning, path planning, modeling removal, residence solution and coordinate conversion, stable control over the machining precision of the asymmetric structure of the off-axis aspherical mirror is achieved, and the high-precision manufacturing requirement of a high-performance optical system for the off-axis aspherical mirror is stably met.

Description

Technical Field

[0001] This invention relates to the field of machine tool processing technology, and in particular to a method, apparatus, equipment and storage medium for polishing off-axis aspherical mirrors. Background Technology

[0002] Off-axis aspherical mirrors are increasingly used in modern high-performance optical systems (such as space telescopes, extreme ultraviolet lithography, and infrared imaging systems) because they can simultaneously eliminate central occlusion and aberrations in optical systems. However, their manufacturing process faces significant challenges: off-axis aspherical mirrors require nanometer-level precision in lens surface shape and sub-nanometer-level surface roughness. Furthermore, during processing, the designed surface shape in the workpiece coordinate system must be accurately transformed to the machine tool motion coordinate system. The complex coordinate transformation further increases the difficulty of high-precision processing, making it impossible to reliably meet the high-precision manufacturing requirements of off-axis aspherical mirrors. Summary of the Invention

[0003] To overcome the shortcomings of the prior art, the present invention aims to provide a method, apparatus, device and storage medium for polishing off-axis aspherical mirrors. The present invention achieves stable control of the processing accuracy of the asymmetric structure of off-axis aspherical mirrors through the process design of off-axis region positioning, path planning, removal modeling, dwell solution and precise coordinate transformation, so as to stably meet the high-precision manufacturing requirements of high-performance optical systems for off-axis aspherical mirrors.

[0004] The first aspect of this invention provides a method for polishing an off-axis aspherical mirror, the method comprising the steps of: establishing a parent aspherical geometric model, and dividing an off-axis region based on the parent aspherical model and preset design requirements; performing trajectory planning based on the off-axis region to obtain a polishing path; constructing a removal function model based on the off-axis region; obtaining the surface residual from the off-axis aspherical mirror to be polished, and solving for the dwell time based on the removal function model, the surface residual, and the polishing path; constructing a five-axis machine tool kinematic model; and substituting the dwell time and polishing path into the five-axis machine tool kinematic model to obtain polishing control parameters.

[0005] Optionally, in a first implementation of the first aspect of the present invention, the step of establishing a parent aspherical geometric model and dividing the off-axis region based on the parent aspherical model and preset design requirements includes: superimposing the quadratic surface equation with higher-order aspherical terms in a global coordinate system to obtain the parent aspherical geometric model; determining the off-axis amount and the aperture size of the off-axis region of the off-axis aspherical mirror according to preset design requirements; and dividing the off-axis region to be polished on the parent aspherical model based on the off-axis amount and the aperture size of the off-axis region.

[0006] Optionally, in the second implementation form of the first aspect of the present application, the trajectory planning based on the off-axis region to obtain the polishing path comprises: obtaining point cloud data from the off-axis region, and extracting the inner and outer contour polygons of the off-axis region from the point cloud data by using a convex hull algorithm to obtain a boundary contour; determining the tilt angle of the off-axis region according to the polishing processing requirement; performing a rotation transformation on the point set in the boundary contour based on the tilt angle, and combining the rotated point set into a closed polygon to obtain a processing contour; performing a free-form surface fitting transformation on the processing contour to form polishing surface shape data; and generating the polishing path based on the polishing surface shape data and a preset raster path planning method.

[0007] Optionally, in the third implementation form of the first aspect of the present application, the construction of the removal function model based on the off-axis region comprises: establishing a pressure distribution model and a velocity distribution model based on the off-axis region; coupling the pressure distribution model and the velocity distribution model to construct an initial removal function model; performing experimental calibration based on the initial removal function model to obtain actual removal depth data; and fitting and correcting the model parameters of the initial removal function model by using the actual removal depth data to obtain the removal function model.

[0008] Optionally, in the fourth implementation form of the first aspect of the present application, the obtaining of the surface shape residual from the off-axis aspheric mirror to be polished and the solving of the residence time based on the removal function model, the surface shape residual and the polishing path comprise: obtaining the actual three-dimensional coordinates of the off-axis aspheric mirror to be polished, and comparing the actual three-dimensional coordinates with the coordinates of the off-axis region to calculate the surface shape residual of the off-axis aspheric mirror to be polished; for each processing region on the polishing path, constructing a convolution relationship among the removal function model, the surface shape residual and the residence time; and performing iterative updating on the convolution relationship based on a preset iterative deconvolution method to obtain the residence time.

[0009] Optionally, in the fifth implementation form of the first aspect of the present application, the construction of the kinematics model of the five-axis machine tool comprises: establishing a workpiece coordinate system and a machine tool coordinate system of a five-axis double-rotary table type machine tool, and determining the origins of the workpiece coordinate system and the machine tool coordinate system respectively; solving the spatial offset distance between the origin of the machine tool coordinate system and the origin of the workpiece coordinate system to complete the calibration of the origin of the workpiece coordinate system; taking the origin of the workpiece coordinate system as a rotation reference to construct a total transformation matrix; and deriving the coordinate conversion relationship of the tool position points between the workpiece coordinate system and the machine tool coordinate system based on the spatial offset distance and the total transformation matrix to complete the construction of the kinematics model of the five-axis machine tool.

[0010] Optionally, in a sixth implementation form of the first aspect of the application, the substituting the dwell time and the polishing path into the five-axis machine tool kinematics model to obtain the polishing control parameter comprises: extracting coordinate data of each polishing tool position on the polishing path, and establishing a corresponding relationship with a corresponding dwell time to form a correlation data set; and substituting the correlation data set into the five-axis machine tool kinematics model to generate the polishing control parameter.

[0011] The second aspect of the application provides a polishing device for off-axis aspheric mirrors, comprising: a region division module, configured to establish a parent aspheric surface geometric model, and divide an off-axis region based on the parent aspheric surface model and a preset design requirement; a trajectory planning module, configured to perform trajectory planning based on the off-axis region to obtain a polishing path; a function construction module, configured to construct a removal function model based on the off-axis region; a time solving module, configured to obtain a surface shape residual from an off-axis aspheric mirror to be polished, and solve a dwell time based on the removal function model, the surface shape residual and the polishing path; a model construction module, configured to construct a five-axis machine tool kinematics model; and a parameter generation module, configured to substitute the dwell time and the polishing path into the five-axis machine tool kinematics model to obtain a polishing control parameter.

[0012] The third aspect of the application provides a polishing device for off-axis aspheric mirrors, comprising: a memory and at least one processor, wherein the memory stores instructions; and the at least one processor invokes the instructions in the memory to enable the computer device to perform each step of the off-axis aspheric mirror polishing method described above.

[0013] The fourth aspect of the application provides a computer readable storage medium, wherein the computer readable storage medium stores instructions, and the instructions are executed by a processor to implement each step of the off-axis aspheric mirror polishing method described above.

[0014] In the technical solution of the application, first, a parent aspheric geometric model is established, and an off-axis region is divided based on preset design requirements to determine the machining range, thereby adapting the asymmetric structure of the off-axis asphere from the source; then, a polishing path is obtained based on the off-axis region to ensure that the path adapts to the asymmetric profile of the off-axis region and reduces invalid machining travel; subsequently, a removal function model is constructed based on the off-axis region to make the removal function match the surface shape characteristics of the region, and then the surface shape residual of the mirror to be polished is obtained, and the removal function model and the polishing path are combined to solve the dwell time, so that the dwell time of each polishing site can be matched with the surface shape residual, and the material removal amount is accurately controllable; finally, a five-axis machine tool kinematics model is constructed, and the dwell time and the polishing path are substituted into the model to obtain polishing control parameters, thereby realizing accurate conversion of the workpiece coordinate system to the machine tool coordinate system and solving the precision deviation problem caused by complex coordinate conversion. Through the process design of off-axis region positioning, path planning, removal modeling, dwell solving, and accurate coordinate conversion, the application realizes stable control of the machining precision of the asymmetric structure of the off-axis aspheric mirror, and stably meets the high-precision manufacturing requirements of the off-axis aspheric mirror for high-performance optical systems. BRIEF DESCRIPTION OF DRAWINGS

[0015] The above and / or additional aspects and advantages of the present application will become apparent and more readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings, in which: Figure 1 The first flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 2 The second flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 3 The third flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 4 The fourth flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 5 The fifth flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 6 The sixth flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 7 The seventh flowchart of the off-axis aspheric mirror polishing method provided for the embodiments of the present application; Figure 8 The structure schematic diagram of the off-axis aspheric mirror polishing device provided for the embodiments of the present application; Figure 9 The structure schematic diagram of the off-axis aspheric mirror polishing device provided for the embodiments of the present application; DETAILED DESCRIPTION

[0016] The application provides a method, device and equipment for polishing an off-axis aspheric mirror and a storage medium.

[0017] The terms "first", "second", "third", "fourth" and the like in the description, claims, and drawings of the application, if any, are used for distinguishing between similar objects and not necessarily for describing a specific sequential or chronological order. It is to be understood that the use of these terms herein is to be construed as interchangeable in order to distinguish between the similar objects. It is also to be understood that the term "comprising" or "having" or any variation thereof, is intended to cover a non-exclusive inclusion, such that a process, method, system, product or apparatus that comprises a list of steps or units is not necessarily limited to those steps or units but can include other not expressly listed steps or units. It is further understood that the use of the terms "including", "comprising", or "having" and variations thereof, does not limit the scope of the application to the specific features so described, and is intended to cover the situation where additional not specified features are present.

[0018] For the purpose of facilitating understanding, the specific flow of the embodiments of the application is described below, and reference is made to Figure 1 One embodiment of the method for polishing an off-axis aspheric mirror in the embodiments of the application comprises the following steps. The method for polishing an off-axis aspheric mirror comprises the following steps. 101. Establishing a parent aspheric surface geometric model, and dividing an off-axis region based on the parent aspheric surface model and preset design requirements. In this embodiment, a complete parent aspheric surface geometric model is constructed by adding high-order aspheric surface related parameters based on a quadric surface in a global coordinate system, so as to describe the overall surface shape of the parent aspheric surface. Then, the off-axis region to be processed is determined and divided on the parent aspheric surface according to the preset design requirements of the optical system, such as the offset degree of the off-axis region and the required aperture size, so as to clearly define the range of the region and determine the geometric basis for subsequent processing.

[0019] 102. Trajectory planning based on the off-axis region to obtain a polishing path. In this embodiment, considering the feature that the off-axis aspheric mirror has no rotational symmetry, a grating path is selected to plan the polishing path. The contour information of the off-axis region is obtained to determine the boundary range of processing, and then the path interval is set according to the requirement of the surface shape accuracy, so that the polishing tool can move according to the planned path to completely cover the entire off-axis region and form a polishing path.

[0020] 103. Constructing a removal function model based on the off-axis region. In this embodiment, the removal function model is constructed by using the air bag polishing method. A spare test piece made of the same material as the off-axis aspheric mirror to be polished is used to perform a static indentation experiment in the idle area of the machine tool. The air bag tool is allowed to stay on the test piece and polish to form a polishing indentation. The three-dimensional data is obtained by scanning the indentation with the equipment. After data processing, the removal law of the indentation is fitted with a suitable model, the model parameters are determined, and finally the removal function model reflecting the material removal characteristics of the air bag tool in the polishing of the off-axis area is obtained.

[0021] 104. Obtain the surface shape residual from the off-axis aspheric mirror to be polished, and solve the residence time based on the removal function model, the surface shape residual, and the polishing path; In this embodiment, the actual surface shape data of the off-axis aspheric mirror to be polished is obtained by scanning it with a detection device, and then compared with the theoretical surface shape data of the off-axis area to calculate the thickness of the material that needs to be removed at each position, i.e. the surface shape residual. Then, by combining the previously constructed removal function model, the obtained surface shape residual, and the planned polishing path, a suitable algorithm is used to establish the relationship between them, and the residence time of the polishing tool at each position on the polishing path is solved, i.e. the residence time.

[0022] 105. Construct a five-axis machine tool kinematics model; In this embodiment, for the type of five-axis machine tool used, the machine tool coordinate system and the workpiece coordinate system are established respectively, and the origins and axis directions of the two coordinate systems are determined. By deriving the transformation matrix of the machine tool rotating shaft and considering the positional offset between the two coordinate systems, the total transformation matrix is constructed to form the conversion relationship between the tool position point coordinates in the workpiece coordinate system and the movement instructions of each axis of the machine tool, thereby completing the construction of the five-axis machine tool kinematics model.

[0023] 106. Substitute the residence time and the polishing path into the five-axis machine tool kinematics model to obtain the polishing control parameters; In this embodiment, the coordinates of each position on the polishing path are first associated with the corresponding residence time to form a complete data set. Then, this data set is substituted into the constructed five-axis machine tool kinematics model to calculate the positions and corresponding movement angles that each movement axis of the machine tool needs to reach. At the same time, the movement speed of the machine tool is determined in combination with the residence time and the polishing path, and finally the polishing control parameters that can be used to control the five-axis machine tool are formed.

[0024] In the embodiment of the present application, first, a parent aspheric geometric model is established, and the off-axis region is divided based on the preset design requirements to determine the to-be-processed range, so as to adapt the asymmetric structure of the off-axis asphere from the source; then, the polishing path is obtained based on the off-axis region to ensure that the path adapts to the asymmetric profile of the off-axis region and reduces the invalid processing travel; subsequently, the removal function model is constructed based on the off-axis region to make the removal function match the surface shape characteristics of the region, and then the residence time is solved by combining the removal function model and the polishing path based on the surface shape residual of the to-be-polished mirror, so that the residence time of each polishing site can be matched with the surface shape residual to ensure that the material removal amount is accurately controllable; finally, the kinematic model of the five-axis machine tool is constructed, and the residence time and the polishing path are substituted into the kinematic model to obtain the polishing control parameters, so as to realize the accurate conversion of the workpiece coordinate system to the machine tool coordinate system and solve the precision deviation problem caused by the complex coordinate conversion; through the process design of the off-axis region positioning, path planning, removal modeling, residence solving and accurate coordinate conversion, the present application realizes the stable control of the machining precision of the asymmetric structure of the off-axis aspheric mirror and stably meets the high-precision manufacturing requirements of the off-axis aspheric mirror of the high-performance optical system.

[0025] Please refer to Figure 2 In the embodiment of the present application, the two embodiments of the off-axis aspheric mirror polishing method include: 201. In a global coordinate system, a quadratic surface equation and a high-order aspheric term are superimposed to obtain a parent aspheric geometric model. In the embodiment, the global coordinate system (O-XYZ) is set as a right-handed rectangular coordinate system, the coordinate origin O is located at the vertex of the parent aspheric surface, the Z-axis is the optical axis of the parent aspheric surface, and the X-axis and the Y-axis are in the vertex tangent plane of the parent aspheric surface. The quadratic surface equation adopts a standard circular quadratic surface form, and the expression is: ; ; Wherein, R is the curvature radius of the vertex of the parent aspheric surface, r is the radial distance of a certain point in the global coordinate system to the Z-axis, and k is the conic coefficient of the quadratic surface (k=0 for a spherical surface, k<0 for a hyperboloid, and k>0 for an ellipsoid). The high-order aspheric term adopts an even polynomial form, and the expression is: ; Wherein, is a high-order aspheric coefficient, which is used to correct the surface shape deviation of the quadratic surface and adapt to the high-precision design requirements of the off-axis aspheric surface. The quadratic surface equation and the high-order aspheric term are superimposed to obtain the complete expression of the parent aspheric geometric model: ; The principle of using the superposition mode is that the quadric surface provides a basic profile of the parent aspheric surface, and the high-order aspheric surface term compensates for the aberration of the optical system to ensure that the parent aspheric surface has excellent optical performance; the advantage is that the model has high precision and strong parameterization, and is convenient for subsequent division of the off-axis region and calculation of the surface shape error.

[0026] 202. Determine the off-axis amount and the off-axis region aperture size of the off-axis aspheric mirror according to the preset design requirements; In this embodiment, the preset design requirements are determined based on the performance indicators of the target optical system, including the field of view, the clear aperture, the aberration correction requirement and the assembly space limitation of the optical system; The off-axis amount and the off-axis region aperture size are determined according to the light transmission requirement of the optical system to ensure that the distribution of the off-axis region on the parent aspheric surface is continuous and unobstructed; the off-axis amount is defined as the perpendicular distance between the optical axis of the off-axis aspheric mirror and the optical axis of the parent aspheric surface (global coordinate system Z axis); the off-axis region aperture size includes the radial aperture and the circumferential angle range; This step reversely deduces the parameters based on the design requirements of the off-axis aspheric mirror, so that the off-axis amount and the aperture size not only meet the optical performance requirements, but also adapt to the feasibility of subsequent processing, avoiding the problems of processing rework or substandard optical performance caused by unreasonable parameter design.

[0027] 203. Divide the off-axis region to be polished on the parent aspheric surface model based on the off-axis amount and the off-axis region aperture size; In this embodiment, the optical axis of the parent aspheric surface model is taken as the reference, and the center position of the off-axis region is located on the parent aspheric surface according to the set off-axis amount; then, the continuous sub-aperture range is determined based on the center position according to the determined off-axis region aperture size, and the range is the off-axis region to be polished.

[0028] Please refer to Figure 3 The three embodiments of the off-axis aspheric mirror polishing method in the embodiment of the application include: 301. Obtain point cloud data from the off-axis region, and extract the inner and outer contour polygons of the off-axis region from the point cloud data using a convex hull algorithm to obtain a boundary contour; In this embodiment, the off-axis region defined on the parent aspheric surface is fully scanned by a three-dimensional scanning device first, and point cloud data uniformly distributed in the region is collected to completely restore the surface shape features of the off-axis region; the point cloud density is set according to the aperture size of the off-axis region and the polishing accuracy requirement; Subsequently, a convex hull algorithm is used to extract the contour of the collected point cloud data. The algorithm automatically filters out the outermost boundary points of the off-axis region by finding the smallest convex polygon that can contain all the data points in the point cloud. Meanwhile, to deal with the internal avoidance structure or invalid region that may exist in the off-axis region, a distance threshold is set to remove the internal redundant point cloud and extract the internal boundary points. Finally, the closed inner and outer contour polygons are formed, and the combined boundary contour of the off-axis region is obtained.

[0029] 302. Determine the inclination angle of the off-axis region according to the polishing processing requirements; In this embodiment, the polishing processing requirements mainly include the effective contact range of the polishing tool, the optimal fitting angle of the tool and the mirror surface, the avoidance requirement to avoid processing interference, and the uniform correction requirement of the surface shape error; The inclination angle needs to ensure that the contact area between the tool and the surface of the off-axis region remains stable, while avoiding excessive local polishing caused by too small contact area or affecting the surface shape correction accuracy caused by too large contact area. If there are positions with sudden curvature changes in the off-axis region, the inclination angle needs to be dynamically adjusted according to the local curvature to ensure that the tool is always tangent to the surface or maintains a preset fitting angle, avoiding processing interference. The specific value of the inclination angle can be determined by simulating the motion trajectory of the polishing tool in the off-axis region, ensuring that the tool can cover the entire off-axis region without interference, and the overlap rate of the processing path meets the requirements of uniform correction of the surface shape.

[0030] 303. Rotate the point set in the boundary contour based on the inclination angle, and combine the rotated point set into a closed polygon to obtain a processing contour; In this embodiment, the rotation transformation takes the center of the off-axis region as the rotation reference, and uniformly rotates all the point cloud data points in the boundary contour and its internal points according to the determined inclination angle. During the rotation process, the relative positional relationship between the point cloud data is maintained unchanged, and only the overall attitude is adjusted to avoid distortion of the surface shape features. After the rotation is completed, the boundary points of the transformed point cloud data are re-extracted, and they are connected in sequence in clockwise or counterclockwise order to form a new closed polygon. This polygon is the processing contour that adapts to the polishing processing attitude.

[0031] 304. Perform free-form surface fitting transformation on the processing contour to form a polishing surface shape data; In this embodiment, the least squares method is used for surface reconstruction of all point cloud data in the processing contour in the free-form surface fitting, and the smooth surface equation is constructed to convert the discrete point cloud data into continuous surface shape data. During the fitting process, the processing contour is used as the boundary constraint condition to ensure that the fitted surface is strictly limited within the processing contour range. The free-form surface fitting can convert the discrete point cloud data into continuous and calculable surface shape model, and completely restore the actual surface shape of the off-axis region.

[0032] 305、generating the polishing path based on the polishing surface shape data and a preset grating path planning method; In the embodiment, first, the line distance of the grating path is determined according to the diameter of the polishing tool and the machining precision requirement, the machining overlap rate between adjacent paths is ensured, and the polishing blind area is avoided; then, the extension direction of the grating path is set based on the longest side of the machining contour, so that the path direction is consistent with the main curvature direction of the off-axis region, and the curvature adaptation difficulty of the polishing tool in the movement process is reduced; finally, the continuous path is generated on the fitted polishing surface shape data according to the set line distance and direction.

[0033] Please refer to Figure 4 , the four embodiments of the off-axis aspheric mirror polishing method in the embodiment of the application include: 401、establishing a pressure distribution model and a velocity distribution model based on the off-axis region; In the embodiment, the core of establishing the pressure distribution model is to first determine the radius of the contact area (a circle), and then describe the pressure gradient in the contact area; There are two methods for calculating the contact circle radius; The first method is a solving method based on an empirical formula, and the empirical formula is as follows: ; In the formula, is the contact circle radius; C is a comprehensive constant (related to the tool flexibility and the air bag material, which needs to be calibrated by experiment); is the air bag pressure; k is the local curvature of the off-axis region; Then, a standard spherical test piece with a known curvature is preset, single-station polishing is performed on it by applying air bag pressure, and the actual radius of the polished spot is measured by a microscope; the radius is substituted into the empirical formula, and the comprehensive constant can be inversely deduced; then, for the curvature of any position in the off-axis region, the empirical formula is substituted, and the contact circle radius can be calculated by the formula; The second method is to solve by using the geometric relationship between the amount of depression and the removal diameter, as follows: The tool radius , the local curvature radius of the off-axis region , and the tool depression amount form a triangle, and the three sides of the triangle are , , , the contact circle radius is the right angle side of the triangle; then, based on the geometric relationship of the triangle, the specific value of the contact circle radius can be deduced; The pressure distribution description in the contact area is based on the Hertz contact distribution law: ; wherein, is the normal pressure at the point with distance from the center of the contact circle; is the peak pressure at the center of the contact circle; is the radial distance from the center of the contact circle to the point; The pressure in the contact circle is not uniformly distributed, but follows the Hertz contact pressure distribution rule, that is, gradually decreases from the center to the edge of the contact circle; the core feature of the pressure distribution is that the center peak pressure is the largest and the edge pressure is zero, and the Hertz contact pressure distribution rule can truly reflect the mechanical properties when the flexible air bag contacts the off-axis aspheric surface, and avoids the calculation deviation of the removal amount caused by the assumption of uniform pressure; The velocity distribution model is used to calculate the relative velocity of any point in the contact area; during polishing, the air bag tool has two movements: rotational movement (angular velocity ) around its own axis and translational movement (linear velocity ) along the polishing path; For air bag polishing, the tool axis is usually aligned with the normal line of the workpiece (perpendicular incidence), and the relative velocity of the contact area can be approximated as: ; is the radial distance from the center of the contact circle to the point; If the workpiece is also rotating, the angular velocity and the linear velocity vectors need to be superimposed; In the contact area, the farther the distance from the point to the tool rotation axis, the greater the rotational linear velocity, combined with the direction and size of the translational linear velocity, finally forming a velocity gradient distribution from the contact center to the edge; the velocity distribution model completely covers the core parameters of the polishing movement, and provides accurate velocity basis for subsequent removal rate calculation.

[0034] 402, coupling the pressure distribution model and the velocity distribution model to construct an initial removal function model; In this embodiment, the expression of the Preston equation is as follows: ; wherein, is the removal depth in time; is the Preston coefficient; is the pressure; is the relative velocity; Substituting the pressure distribution model (pressure) and velocity distribution model (relative velocity) obtained in step 401 into the Preston equation, and combining the process characteristics of airbag polishing, an initial removal function model suitable for the off-axis region is constructed. For the typical working condition where the airbag polishing tool is in perpendicular contact (90°) with the off-axis aspherical mirror, the following specific form can be used for characterization: ; in, Let (x, y) be the pressure characterization term at any point (x, y) within the contact area. The peak pressure at the contact center between the airbag and the off-axis region can be calculated using the pressure distribution model in step 401. The rotational angular velocity of the airbag workpiece itself; The composite linear velocity characterization term for any point (x, y) within the contact area is derived from the velocity distribution model in step 401. The radius of the airbag tool; This represents the actual contact area between the tool and the workpiece. The Gaussian width parameter is used to characterize the spatial distribution range of the removal function and is adapted to the local surface curvature of the off-axis region. This is the pressure decay index, used to correct the influence of asymmetric surface shape in off-axis regions on pressure distribution; Let K be the infinitesimal element of the polishing dwell time; K is an engineering correction factor, which can be regarded as the Preston coefficient. It is used to correct the deviation between theoretical models and engineering practice; In the initial removal function model, The Preston coefficient is a key parameter to be determined. Its value is related to the material properties of the off-axis aspherical mirror to be polished, the concentration of abrasive particles in the polishing fluid, the temperature, and the material of the airbag. In this embodiment, an initial value is preset first, and subsequent precise optimization will be carried out through experimental calibration. The initial removal function model simultaneously considers the mechanical properties (pressure distribution) and motion properties (velocity distribution) of the off-axis region, and fully takes into account the influence of the local curvature of the off-axis region on the contact pressure. This avoids the shortcomings of traditional removal function models that ignore the characteristics of asymmetric surfaces, and can accurately describe the removal patterns at different locations in the off-axis region.

[0035] 403. Conduct experimental calibration based on the initial removal function model to obtain actual removal depth data; In this embodiment, the core of the experimental calibration is to obtain removal data through real polishing experiments, so as to provide a basis for subsequent adjustment of the Preston coefficient; During calibration, a standard specimen made of the same material as the off-axis aspherical mirror to be polished is selected. The surface of the specimen contains multiple representative areas with curvature consistent with the key positions of the off-axis region. This ensures that the calibration results can cover the curvature variation range of the off-axis region and avoids insufficient model adaptability caused by single curvature calibration. The experimental conditions were configured according to the parameters of the initial removal function model, including airbag pressure, tool rotation angular velocity, translation speed, polishing time, etc., to ensure that the experimental conditions were completely consistent with the subsequent actual polishing conditions. The polishing time setting must meet the removal depth to achieve the detection accuracy requirements, so as to avoid data errors caused by insufficient removal. A dwell polishing experiment was conducted in each representative curvature region of the standard specimen. The tool was held in a fixed posture (perpendicular to the surface of the specimen) and stayed at the target position for a set time to form an independent circular polishing spot. The experiment was repeated at least 3 times for each curvature region to reduce random error. After each dwell experiment, a circular polishing spot will be formed on the surface of the specimen. A white light interferometer is used to perform a three-dimensional morphology scan on each polishing spot and collect the three-dimensional coordinate data of each discrete polishing spot. The acquired 3D data then undergoes three steps of processing: The first step is to determine the center of the polishing spot. From the removal depth of all data points, find the point with the largest removal depth and record the horizontal and vertical coordinates of this deepest point as the center coordinates of the imprint. The second step is to calculate the radial distance from each point to the center of the polishing spot. Subtract the horizontal coordinate of the center from the horizontal coordinate of each data point to obtain the horizontal deviation of the point relative to the center. Similarly, calculate the longitudinal deviation relative to the center, and then use the Pythagorean theorem to obtain the straight-line distance from each point to the center of the imprint, i.e., the radial distance.

[0036] The third step is to convert the removal rate per unit time by dividing the actual removal depth of each point by the total polishing time to obtain the amount of material removed per unit time at that point (i.e., the removal rate per unit time). Finally, the measured dataset of "radial distance - removal rate per unit time" (i.e., the actual removal depth data) is compiled.

[0037] 404. Use actual removal depth data to fit and correct the model parameters of the initial removal function model to obtain the removal function model; In this embodiment, based on the initial removal function model, the experimental parameters (pressure, speed, polishing time) set in step 403 are substituted to calculate the theoretical removal rate per unit time for each measured data point (corresponding to different radial distances), forming a theoretical dataset.

[0038] Subsequently, by comparing the theoretical dataset and the experimental dataset, if the theoretical removal rate is generally higher than the experimental value, it indicates that the initial settings are correct. Too large, need to be adjusted properly; if the overall theoretical removal rate is lower than the measured value, it needs to be adjusted upwards ; at the same time, for the representative area of different curvature in the off-axis area, the deviation is analyzed respectively, if the deviation of a certain curvature area is obviously larger than that of other areas, further consideration is needed whether it changes with the curvature, providing direction for subsequent parameter optimization.

[0039] Then, the least square method is used to construct the objective function, and the square sum of the deviation between the theoretical and measured removal rates is minimized as the goal, and the is continuously adjusted through iterative calculation until the square sum of the deviation converges to a preset threshold, at which time the obtained is the optimal parameter; the optimal is substituted into the initial removal function model to obtain the corrected removal function model.

[0040] Please refer to Figure 5 , the five embodiments of the off-axis aspheric mirror polishing method in the embodiment of the application include: 501. Obtain the actual three-dimensional coordinates of the off-axis aspheric mirror to be polished, and compare the actual three-dimensional coordinates with the coordinates of the off-axis region to calculate the surface shape residual error of the off-axis aspheric mirror to be polished; In this embodiment, first, the laser interferometer is used to perform full-area three-dimensional scanning on the unpolished off-axis aspheric mirror, and the actual three-dimensional coordinates of the discrete points uniformly distributed on the mirror surface are collected to restore the actual surface shape of the mirror; then, the theoretical coordinates of the off-axis region defined in step 203 are retrieved, the actual three-dimensional coordinates measured are aligned with the theoretical coordinates at the corresponding positions according to the same space reference, and the coordinate difference of each corresponding point in the optical axis direction (Z-axis direction) is calculated, which is the surface shape residual error of the point; for the points not directly collected in the scanning area, the residual error value can be calculated by interpolation to form the surface shape residual error distribution data covering the entire off-axis region.

[0041] 502. For each processing area on the polishing path, a convolution relationship between the removal function model, the surface shape residual error and the residence time is constructed; In this embodiment, the variables corresponding to the removal function model, the surface shape residual error and the residence time are first defined as follows: Surface shape residual error (target removal amount) , representing the material thickness to be removed in any processing area on the polishing path, is the known input data; Residence time , representing the residence time of the polishing tool in the processing area, is the key parameter to be solved, which directly determines the actual removal amount of the area; Removal function , which represents the material removal rate of the polishing tool on the to-be-processed region per unit time, is a known process parameter obtained by experiment calibration and parameter fitting in step 404; The removal effect of the tool in the polishing process has a spatial diffusion characteristic, that is, the removal of the tool in a region not only affects the current position but also has a certain removal effect on the surrounding region. This diffusion effect needs to be described by convolution operation. Therefore, the target removal amount of any to-be-processed region is the residence time and the convolution result of the removal function in space, and the formula is as follows: ; However, in practice, because the residence time needs to be solved, when the target removal amount and the removal function are known, solving the residence time is a deconvolution problem .

[0042] 503. The convolution relationship is iteratively updated based on a preset iterative deconvolution method to obtain the residence time; In this embodiment, the Lucy-Richardson algorithm is used as the iterative deconvolution method, and the deviation between the theoretical and actual removal amounts is gradually reduced through multiple iterations to finally obtain the residence time , and the iterative formula is as follows: ; wherein is the residence time estimation value in the kth iteration, is the residence time estimation value in the (k+1)th iteration; The specific process of iteration is as follows: Firstly, the surface shape residual, the removal function and the residence time are converted into a two-dimensional array in a unified format; Subsequently, the residence time is initialized, and the initial values of all elements are uniformly set to 1 (or the initial values are set according to experience); Then, in each iteration, firstly, the current theoretical removal amount is calculated, that is, the residence time in the kth iteration is convoluted with the removal function to obtain the theoretical removal amount in the kth iteration; and then the target removal amount is convoluted with the theoretical removal amount Divide the corresponding elements to obtain the ratio result; if any element in the ratio result is greater than 1, it means that the current theoretical removal amount in that area is less than the target value, and the dwell time needs to be increased; if it is less than 1, the dwell time needs to be decreased; the ratio result provides the direction and magnitude basis for adjusting the dwell time; flip the removal function matrix P, and then perform a convolution operation between the ratio result and the flipped removal function to obtain the dwell time correction coefficient; finally, the dwell time of the kth iteration is... Multiply by the dwell time correction factor to obtain the dwell time for the (k+1)th time. This completes one iteration update; In each convolution operation, zero-filling or mirror-filling is performed on the boundary regions of the off-axis region. Zero-filling means filling the extended regions of the matrix beyond the off-axis region with zero values. This method can simulate the actual situation that "the tool has no removal effect after going beyond the off-axis region", avoiding the influence of meaningless removal calculations outside the boundary on the results. If the residual surface shape of the off-axis region edge is large, mirror-filling can be used to avoid insufficient boundary correction caused by zero-filling. That is, the values ​​of the extended regions outside the boundary are mirrored according to the values ​​of the boundary elements of the off-axis region, ensuring that the removal calculation of the boundary region is more in line with the actual processing conditions. To ensure the accuracy and efficiency of the iteration, a stopping condition is set. The iteration stops when any condition is met, and the final dwell time is obtained when the iteration stops. The stopping conditions are as follows: The change threshold condition is used to calculate the root mean square (RMS) difference between the dwell time matrix of the (k+1)th iteration and the kth iteration. If this value is less than the preset threshold, it means that the change in dwell time has become extremely small, and further iteration will have limited effect on improving accuracy. Maximum iteration count condition: Set the maximum number of iterations. If the number of iterations reaches the maximum value but the change threshold condition is not met, the iteration should also be terminated.

[0043] Please see Figure 6 The six embodiments of the off-axis aspherical mirror polishing method in this invention include: 601. Establish the workpiece coordinate system and the machine tool coordinate system for a five-axis dual rotary table machine tool, and determine the origin of the workpiece coordinate system and the origin of the machine tool coordinate system respectively; In this embodiment, the machine tool coordinate system ( - The origin is established with reference to the machine tool's own mechanical reference. It is set at the center of the fixed base of the dual rotary table (i.e., the geometric center of the machine tool). The axis moves along the longitudinal translation direction of the machine tool table. The axis moves along the lateral direction of the worktable. The axis moves vertically along the spindle; the machine tool coordinate system is the machine tool's own inherent coordinate system, whose origin and axis direction are calibrated and fixed at the machine tool factory and can be directly read through the machine tool control system. Workpiece coordinate system ( - The coordinate system is established with reference to the machining datum of the off-axis aspherical mirror to be polished; O1 is the actual rotation center when the workpiece is rotated by the dual rotary tables, and its position is determined by the mechanical structure of the dual rotary tables; the workpiece coordinate system... axis, The axes are respectively related to the machine tool coordinate system. axis, The axes are parallel.

[0044] 602. Solve for the spatial offset distance between the origin of the machine tool coordinate system and the origin of the workpiece coordinate system to complete the calibration of the origin of the workpiece coordinate system; In this embodiment, the spatial offset distance between the origin of the machine tool coordinate system and the origin of the workpiece coordinate system is... The essence of value calibration is to determine the spatial relationship between the "physical platform reference" and the "rotation center"; if not calibrated... Value, directly with Calculating the tool position coordinates for a rotational reference will result in: Rotational transformation reference error, tool axis around Rotation rather than This does not match the actual mechanical motion trajectory of the machine tool, causing deviation in the tool axis posture. Inaccurate coordinate transformation leads to a mismatch between the linear and rotary axis motion commands when the machine tool executes G-code, ultimately resulting in excessive machining errors for off-axis aspherical mirror surfaces. Therefore, calibration (Solve) The core purpose of this value is to establish the relationship between the physical platform reference and the rotation reference, providing a precise basis for reference offset compensation for subsequent kinematic models; only by clearly defining this offset distance can we ensure that subsequent tool position rotation transformations revolve around the true rotation center. This process is carried out to avoid machining posture errors caused by misalignment of the reference point; Calibration The specific steps are as follows: First, a standard sphere with radius r (simulating the reference feature of the workpiece to be processed) is selected and fixed on the dual rotary table physical platform, ensuring that the center of the standard sphere is aligned with the reference feature of the workpiece. On the same Z-axis; for relatively of The axis offset is... value; For a point on a standard sphere Rotation Then the point in the machine tool coordinate system Axis coordinates; For a point on a standard sphere Rotation Then the point in the machine tool coordinate system Axis coordinates; y represents the orbit around a point on a standard sphere. Before rotation, the point in the machine tool coordinate system Axis coordinates; z represents a point on a standard sphere orbiting around... Before rotation, the point in the machine tool coordinate system Axis coordinates; The relationship between the above variables can be expressed by the following formula: ; ; ; ; Furthermore, due to the relationship between the arc length and chord length of the standard ball's rotation, It can be expressed by the following expression: ; in, The initial state of the standard ball ( The coordinates of the center of the sphere at 0°; By combining the above expressions, we can obtain the answer. .

[0045] 603. Construct the total transformation matrix with the origin of the workpiece coordinate system as the rotation reference; In this embodiment, constructing the total transformation matrix involves three steps. Transformation matrix Then transform the matrix and transformation matrix The product is the total transformation matrix; First, construct the loop Transformation matrix of rotation angle B ; Around The purpose of rotating the axis by angle B is to change the tool axis direction vector ( ) Go to The formula for calculating angle B in the plane is: ; With as the rotation reference, the corresponding homogeneous transformation matrix of the rotation is: ; Subsequently, a transformation matrix of rotating by an angle A around the axis is constructed ; After completing the rotation by an angle B, the tool axis vector is made consistent with the axis by rotating by an angle A around the axis, and the calculation formula of the angle A is: ; wherein, , and are the differences in the components of the tool axis vector; With as the rotation reference, the corresponding homogeneous transformation matrix of the rotation is: ; Finally, a total transformation matrix is constructed, the rotation sequence is “first rotating B and then rotating A”, and both are based on , the total transformation matrix is the product of and : ; Thus, the total transformation matrix is obtained.

[0046] 604、Based on the spatial offset distance and the total transformation matrix, the coordinate conversion relationship of the tool position points between the workpiece coordinate system and the machine tool coordinate system is derived to complete the construction of the kinematic model of the five-axis machine tool; In this embodiment, the coordinate conversion formula of the tool position points is first constructed, and then the spatial offset distance and the total transformation matrix are combined to complete the construction of the kinematic model of the five-axis machine tool, which is as follows: First, the tool position points are defined as vectors: In the workpiece coordinate system, the homogeneous coordinates of the tool position points are: ; wherein, , and are local coordinates; In the machine tool coordinate system, the homogeneous coordinates of the tool position points are: ; Subsequently, based on the fact that the tool position points need to be first rotated around (the total transformation matrix M), the coordinate conversion formula of the tool position points can be constructed:​ ; That is: ; The formula expansion calculation can obtain: ; Further combined with the spatial offset can be obtained: ; Wherein, is the machine tool spindle reference compensation value; Therefore, combined with the above coordinate conversion formula and the control logic of the double rotary table rotation angle (A, B), a complete five-axis machine tool kinematics model is formed; the five-axis machine tool kinematics model can accurately convert the tool position point coordinates and the tool axis posture in the workpiece coordinate system into linear axis instructions and rotary axis instructions that can be executed by the machine tool, that is, complete machine tool control code is obtained.

[0047] Please refer to Figure 7 , seven embodiments of the polishing method of the off-axis aspherical mirror in the embodiment of the application include: 701, coordinate data of each polishing tool position point on the polishing path is extracted, and a corresponding relationship with the corresponding residence time is established to form an associated data set; In this embodiment, the coordinate data of all polishing tool position points is extracted from the polishing path determined in step 202 according to the preset path interval, and the coordinate data is the coordinate in the workpiece coordinate system; secondly, the residence time obtained by solving in step 503 is called, and the residence time value corresponding to each tool position point is located in the residence time according to the coordinate data of each tool position point; finally, the coordinate data of each tool position point and the residence time value are combined in order to form an associated data set.

[0048] 702, the associated data set is substituted into the five-axis machine tool kinematics model to generate polishing control parameters; In this embodiment, the coordinate data and residence time of each tool position point in the associated data set are read first, and the coordinate data and rotation angle are substituted into the coordinate conversion formula of the five-axis machine tool kinematics model combined with the tool axis posture requirement (rotation angles A and B) corresponding to the tool position point to calculate the machine tool control code of the tool position point in the machine tool coordinate system; Secondly, according to the residence time and the distance between adjacent tool position points on the polishing path, the travel speed parameter of the machine tool at the tool position point is calculated: Finally, the polishing control parameters of all tool position points are generated in the order of tool position points of the polishing path, combined with the machine tool control code and the travel speed parameters. The polishing control parameters can be imported into the five-axis machine tool control system to control the five-axis machine tool to realize the polishing machining of the off-axis aspheric mirror.

[0049] The polishing method for the off-axis aspheric mirror in the embodiment of the application is described above, and the polishing device for the off-axis aspheric mirror in the embodiment of the application is described below. Please refer to Figure 8 The polishing device for the off-axis aspheric mirror in the embodiment of the application includes: The region division module 801 is configured to establish a parent aspheric surface geometric model and divide an off-axis region based on the parent aspheric surface model and a preset design requirement. The trajectory planning module 802 is configured to perform trajectory planning based on the off-axis region to obtain a polishing path. The function construction module 803 is configured to construct a removal function model based on the off-axis region. The time solving module 804 is configured to obtain a surface shape residual from the off-axis aspheric mirror to be polished and solve a dwell time based on the removal function model, the surface shape residual and the polishing path. The model construction module 805 is configured to construct a five-axis machine tool kinematics model. The parameter generation module 806 is configured to substitute the dwell time and the polishing path into the five-axis machine tool kinematics model to obtain polishing control parameters. In the embodiment, first, the region division module 801 establishes a parent aspheric surface geometric model and divides an off-axis region based on a preset design requirement to determine a machining range. Then, the trajectory planning module 802 performs trajectory planning based on the off-axis region to obtain a polishing path. Subsequently, the function construction module 803 constructs a removal function model based on the off-axis region, so that the removal function can match the surface shape characteristics of the region. Then, the time solving module 804 obtains a surface shape residual of the mirror to be polished, and solves a dwell time based on the removal function model and the polishing path. Finally, the model construction module 805 constructs a five-axis machine tool kinematics model, and the parameter generation module 806 substitutes the dwell time and the polishing path into the five-axis machine tool kinematics model to obtain polishing control parameters, so as to realize accurate conversion of the workpiece coordinate system to the machine tool coordinate system and solve the precision deviation problem caused by complex coordinate conversion. Through the process design of off-axis region positioning, path planning, removal modeling, dwell solving and accurate coordinate conversion, the embodiment of the application realizes stable control of the machining precision of the off-axis aspheric mirror with an asymmetric structure, and stably meets the high-precision manufacturing requirement of the off-axis aspheric mirror for high-performance optical systems.

[0050] Figure 9is a structural schematic diagram of a polishing device for off-axis aspheric mirror provided by an embodiment of the present application. The polishing device for off-axis aspheric mirror 900 can have great differences due to different configurations or performances, and can include one or more processors (central processing units, CPU) 910 (for example, one or more processors) and a memory 920, one or more storage media 930 (for example, one or more mass storage devices) storing application programs 933 or data 932. The memory 920 and the storage media 930 can be temporary storage or persistent storage. The programs stored in the storage media 930 can include one or more modules (not shown in the figure), and each module can include a series of instruction operations in the polishing device for off-axis aspheric mirror 900. Further, the processor 910 can be configured to communicate with the storage media 930 and execute the series of instruction operations in the storage media 930 on the polishing device for off-axis aspheric mirror 900, so as to implement the steps of the off-axis aspheric mirror polishing method provided by the above-mentioned method embodiments.

[0051] The polishing device for off-axis aspheric mirror 900 can further include one or more power supplies 940, one or more wired or wireless network interfaces 950, one or more input / output interfaces 960, and / or one or more operating systems 931, such as Windows Serve, Mac OS X, Unix, Linux, FreeBSD, and the like. Those skilled in the art can understand that the polishing device for off-axis aspheric mirror 900 can further include other components that are not shown in the figure, and the components shown in the figure can be combined or replaced by other components. Figure 9 The structure of the polishing device for off-axis aspheric mirror shown in the figure does not constitute a limitation on the polishing device for off-axis aspheric mirror, and can include more or fewer components than those shown in the figure, or combine certain components, or different arrangement of components.

[0052] The present application also provides a computer readable storage medium, which can be a non-volatile computer readable storage medium, or a volatile computer readable storage medium. The computer readable storage medium stores instructions, and when the instructions are run on a computer, the computer executes the steps of the off-axis aspheric mirror polishing method.

[0053] Those skilled in the art can clearly understand that, for the convenience and brevity of description, the specific working process of the above-mentioned system or device, unit can refer to the corresponding process in the foregoing method embodiments, which will not be described here.

[0054] The integrated unit, if implemented in the form of a software function unit and sold or used as an independent product, can be stored in a computer readable storage medium. Based on such understanding, the technical solutions of the present application essentially or say the part that contributes to the prior art or the whole or part of the technical solutions can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes a plurality of instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the method described in the various embodiments of the present application. The aforementioned storage medium includes: a U disk, a mobile hard disk, a read-only memory (ROM), a random access memory (RAM), a magnetic disk or an optical disk, and various media that can store program codes.

[0055] Finally, it should be noted that: the above only for the preferred examples of the present application, and not for limiting the present application, although the present application is described in detail with reference to the foregoing examples, for those skilled in the art, it still can be modified to the technical solutions recorded in the foregoing embodiments, or equivalent replacement of some technical features. Any modification, equivalent replacement, improvement, etc. within the spirit and principles of the present application shall be included in the protection scope of the present application.

Claims

1. A method of polishing an off-axis aspheric mirror, comprising: The off-axis aspheric mirror polishing method comprises the steps of: establishing a parent aspheric surface geometric model, and dividing an off-axis region based on the parent aspheric surface model and preset design requirements; planning a trajectory based on the off-axis region to obtain a polishing path; constructing a removal function model based on the off-axis region; obtaining a surface residual from the off-axis aspheric mirror to be polished, and solving a dwell time based on the removal function model, the surface residual and the polishing path; constructing a five-axis machine tool kinematics model; substituting the dwell time and the polishing path into the five-axis machine tool kinematics model to obtain polishing control parameters.

2. The off-axis aspheric mirror polishing method of claim 1, wherein, The method of establishing a parent aspheric surface geometric model and dividing an off-axis region based on the parent aspheric surface model and preset design requirements comprises: stacking a quadratic surface equation and a high-order aspheric surface term in a global coordinate system to obtain the parent aspheric surface geometric model; determining an off-axis amount and an off-axis region aperture size of the off-axis aspheric mirror according to preset design requirements; dividing the off-axis region to be polished on the parent aspheric surface model based on the off-axis amount and the off-axis region aperture size.

3. The off-axis aspheric mirror polishing method of claim 1, wherein, The method of planning a trajectory based on the off-axis region to obtain a polishing path comprises: obtaining point cloud data from the off-axis region, and extracting an inner and outer contour polygon of the off-axis region from the point cloud data by using a convex hull algorithm to obtain a boundary contour; determining an inclination angle of the off-axis region according to polishing processing requirements; rotating the point set in the boundary contour based on the inclination angle, and combining the rotated point set into a closed polygon to obtain a processing contour; performing free-form surface fitting transformation on the processing contour to form polishing surface data; generating the polishing path based on the polishing surface data and a preset raster path planning method.

4. The off-axis aspheric mirror polishing method of claim 1, wherein, The method of constructing a removal function model based on the off-axis region comprises: establishing a pressure distribution model and a velocity distribution model based on the off-axis region; coupling the pressure distribution model and the velocity distribution model to construct an initial removal function model; performing experimental calibration based on the initial removal function model to obtain actual removal depth data; fitting and correcting the model parameters of the initial removal function model by using the actual removal depth data to obtain the removal function model.

5. The off-axis aspheric mirror polishing method of claim 1, wherein, The method of obtaining a surface residual from the off-axis aspheric mirror to be polished, and solving a dwell time based on the removal function model, the surface residual and the polishing path comprises: obtaining actual three-dimensional coordinates of the off-axis aspheric mirror to be polished, and comparing the actual three-dimensional coordinates with the coordinates of the off-axis region to calculate the surface residual of the off-axis aspheric mirror to be polished; constructing a convolution relationship among the removal function model, the surface residual and the dwell time for each processing region on the polishing path; iteratively updating the convolution relationship based on a preset iterative deconvolution method to obtain the dwell time.

6. The off-axis aspheric mirror polishing method of claim 1, wherein, The method of constructing a five-axis machine tool kinematics model comprises: establishing a workpiece coordinate system and a machine tool coordinate system of a five-axis double-turntable machine tool, and determining the workpiece coordinate system origin and the machine tool coordinate system origin; solving a spatial offset distance between the machine tool coordinate system origin and the workpiece coordinate system origin to complete calibration of the workpiece coordinate system origin; constructing a total transformation matrix with the workpiece coordinate system origin as a rotation reference. Based on the spatial offset distance and the total transformation matrix, a coordinate conversion relationship of the tool position points between the workpiece coordinate system and the machine tool coordinate system is derived to complete the construction of the kinematics model of the five-axis machine tool.

7. The off-axis aspheric mirror polishing method of claim 1, wherein, The method comprises: The method comprises: The method comprises:

8. An off-axis aspheric mirror polishing apparatus, characterized by, The method comprises: The method comprises: The method comprises: The method comprises: The method comprises: The method comprises: The method comprises:

9. An off-axis aspheric mirror polishing apparatus, characterized by, The method comprises: The method comprises:

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