A high-steepness aspheric surface partition polishing method based on curvature characteristics

Abstract

The application provides a high-steepness aspheric surface partition polishing method based on curvature characteristics, comprising the following steps: calculating the relative principal curvature radius of two main directions of a workpiece surface contact area to determine an elliptical contact size, and constructing a pressure model based on Hertz theory; calculating a relative speed of each point by subtracting a linear vector of a workpiece and tool rotation speed; combining pressure, speed and environment parameters, constructing a removal function based on Preston theory; establishing a discrete linear equation of removal depth, removal function and residence time, and inversely solving the residence time by using a preconditioned projection conjugate gradient method under constraints; extracting the principal curvature, change rate and radial distance by using a central difference method to construct a feature vector and standardize it, determining an optimal clustering number K by using an elbow rule, and performing K-means clustering partition and merging optimization; configuring process parameters for each partition to generate a removal function, combining surface shape error to calculate discrete residence time, and converting the discrete residence time into numerical control feeding code provided with a speed upper limit to perform processing.

Description

Technical Field

[0001] This invention relates to a polishing technique, and more particularly to a high-steep aspherical partition polishing method based on curvature characteristics. Background Technology

[0002] High-steepness aspherical optical elements offer advantages such as drag reduction and range extension, improved stealth performance, enhanced optical detection capabilities, and increased target field of view. They can effectively balance the contradictions between aerodynamic requirements, stealth effects, and optical performance of aircraft. Conventional aspherical polishing methods use a single removal function to calculate dwell time and surface shape accuracy. However, the curvature effect of steep, deep-cavity structures directly affects the actual polishing removal function, leading to deviations between the actual removal amount and the theoretical model. This results in significant errors in surface shape prediction, requiring multiple polishing cycles, resulting in numerous polishing steps, long cycles, low efficiency, and difficulty in guaranteeing processing accuracy.

[0003] In recent years, research on the problems of removal function distortion, large surface shape prediction errors, and low efficiency caused by curvature effects in polishing has mainly focused on removal function modeling and correction, and residence time optimization. Dynamic removal function models considering local curvature have been established to correct theoretical biases and construct efficient and high-precision residence time calculation models. However, existing methods rarely simultaneously consider the removal function and polishing residence time models under varying curvature in deterministic polishing research on high-steep aspherical surfaces, and there is a lack of verification on high-steep aspherical components, thus failing to form a complete deterministic polishing method. Summary of the Invention

[0004] The purpose of this invention is to provide a high-steep aspherical surface partitioning polishing method based on curvature characteristics, comprising:

[0005] Step S100: By calculating the relative principal radii of curvature in the two principal directions of the contact area on the workpiece surface, the characteristic dimensions of the elliptical contact area are obtained, and then a polishing pressure distribution model is constructed based on the Hertz contact model.

[0006] Step S200: Combining kinematics and geometric dimensions, subtract the linear vector of the workpiece rotation speed from the polishing tool rotation speed to calculate the relative velocity at each point in the contact area;

[0007] Step S300: Introduce polishing environment parameters, combine the pressure distribution model of Step 1 and the relative rate distribution model of Step 2, and construct a removal function for the amount of material removed per unit time based on Preston theory;

[0008] Step S400: Establish the discrete linear superposition equation of removal depth, removal function and dwell time. Use the preprocessing projection conjugate gradient method. By introducing a symmetric positive definite preprocessor to reduce the spectral condition number of the coefficient matrix, the dwell time solution is transformed into a quadratic functional minimization problem under the constraints.

[0009] Step S500: The central difference method is used to construct a feature vector by combining the principal curvature, curvature change rate and radial distance of the sampling points and then standardize it. The elbow rule is used to combine the sum of squares within the cluster to determine the optimal number of clusters K. The Euclidean distance is used to iteratively update the cluster center until convergence, thereby realizing the optimization of the component surface region division and merging.

[0010] Step S600: Set the polishing process parameters for each partition to generate the corresponding removal function model, and calculate the discrete dwell time of each partition by combining the surface error data.

[0011] In step S700, the discrete dwell time is converted into CNC feed code, and a constraint upper limit is set on the feed speed to avoid excessive shaping.

[0012] Furthermore, step S100 specifically includes the following steps:

[0013] Step S110: The polishing area is an elliptical area. Calculate its principal curvature radii in two directions. The calculation formula is shown in (1).

[0014] (1),

[0015] Among them, R x R is the principal radius of curvature in the x-direction within the elliptical contact region. y R is the relative principal curvature radius in the y-direction within the elliptical contact region. p R is the radius of the polishing tool. tx and R ty These are the relative principal radii of curvature of the aspherical element at the polishing point in the x and y directions, respectively.

[0016] Step S120: Based on the contact mechanics theory, the characteristic dimensions with semi-axis lengths a and b in the elliptical contact area are obtained. a and b have the relationship shown in formula (2). According to this set of equations, a and b are obtained by solving formulas (3) and (4) simultaneously.

[0017] (2),

[0018] (3),

[0019] (4),

[0020] Where F is the polishing pressure, R e E is the equivalent radius of curvature of the elliptical contact region. * E represents the equivalent elastic modulus between the polishing tool and the aspherical component. t E p The Young's modulus of the polishing tool and the aspherical component, respectively.t ν p Poisson's ratio for the polishing tool and the aspherical component;

[0021] Step S130: Calculate the pressure P(x,y) distribution model in the above elliptical contact area based on the Hertzian theory model, as shown in formula (5).

[0022] (5).

[0023] Further, step S200 specifically includes: during polishing, the relative velocity within the elliptical contact area is determined by subtracting the linear vectors of the workpiece rotation speed and the polishing tool rotation speed; combining kinematics and geometric dimensions, the relative linear velocity v at any point in the contact area is calculated. r (x,y),

[0024] (6),

[0025] Among them, v r Let v be the relative linear velocity at any point in the contact area. p v is the linear velocity of the workpiece. t ω is the linear velocity of the polishing tool. p ω t r p and r t These are the workpiece rotation speed, tool rotation speed, workpiece contact point gyration radius, and tool contact point gyration radius, respectively.

[0026] Furthermore, step S300 specifically includes:

[0027] Based on Preston's theory, the amount of material removed per unit time at a polishing contact point (x,y) ΔZ(x,y) is proportional to the relative pressure P(x,y) and relative velocity V(x,y) of the tool and workpiece, and is subject to the polishing environment parameter K. p To assess the impact of material removal, a removal function R(x,y) for the amount of material removed per unit time is constructed.

[0028] (7),

[0029] Where T is the unit time and t is the independent variable in the integration process.

[0030] Furthermore, step S400 specifically includes the following steps:

[0031] Step S410: Establish a discretized relationship model between the expected removal depth E, removal function R and residence time T on the component surface, and transform the continuous superposition process of material removal into a discrete linear superposition. The linear equation is shown in formula (8).

[0032] (8),

[0033] Wherein, the total number of points in the surface data of the component is s, and the error value of the i-th point is E. i The removal function at point j and the material removal function at point i are denoted as r. ij Remove function r ij The dwell time at point j is denoted as t. j ;

[0034] Step S420: The preprocessing projection conjugate gradient method is used to solve for the constrained polishing dwell time vector U. Solving the linear equation system RT=E is equivalent to minimizing a quadratic functional, the minimum point of which is the solution to the equation system. The minimized quadratic functional is:

[0035] (9),

[0036] Step S430: Substitute the symmetric positive definite preprocessor M into the standard conjugate gradient method, and let... ,in Let M be the positive definite square root, satisfying Substituting it into formula (9) yields:

[0037] (10)

[0038] The preprocessed coefficient matrix is The right-hand side after preprocessing is The constraint set is The optimization problem after preprocessing is shown in equation (10).

[0039] (11).

[0040] Furthermore, step S500 specifically includes:

[0041] Step S510: Combining the rate of curvature change and radial distance, the principal curvature feature vector of each sampling point is calculated using the central difference method. ,

[0042] (12),

[0043] Where Δr is the sampling step size, and κ1 is the radial principal curvature. The radial principal curvature change rate is given by r, where r is the radial distance from the sampling point to the center of the element.

[0044] Step S520: Construct a set of principal curvature feature samples. Formula (13) is used to convert the mean of each feature dimension in the feature sample set to 0 and the standard deviation to 1, thereby eliminating the influence of dimensions and obtaining a standardized feature vector set.

[0045] (13)

[0046] Where n is the number of sampling points, j is the feature index, and μ j For the standardized mean, σ j The characteristic standard deviation, For the feature sample set, These are the standardized eigenvalues;

[0047] Step S530: Use the sum of squares within clusters (WCSS) metric to measure the clustering effect and reflect the compactness of samples within the same cluster. Let the number of clusters be K, and the k-th cluster contain n... k There are samples, and the cluster sample set is . The cluster center is Then the sum of squares of the k-th cluster is

[0048] (14)

[0049] Total WCSS:

[0050] (15)

[0051] V' represents the average value of all sample points in the k-th cluster after standardization along the j-th feature dimension, where d is the feature space dimension index. j Let K be the standardized eigenvalue of the j-th dimension; the number of clusters K is determined using the elbow rule;

[0052] Step S540: Using Euclidean distance as a similarity measure, calculate the straight-line distance between the standardized feature vector of each sampling point and the center vector of each cluster, and assign each sampling point to the cluster closest to it to form K initial partitions;

[0053] Step S550: Update the cluster center vectors of each cluster with the goal of minimizing the total WCSS, and set the updated cluster center vectors as the arithmetic mean of the features of all samples within the corresponding cluster; iterate through steps S540 and S550 until the update magnitude of the cluster center vectors in two adjacent iterations is less than a preset convergence threshold. Or, if the preset maximum number of iterations is reached, the algorithm converges and locks the boundaries of each partition;

[0054] Step S560: Based on the converged clustering results, the high-steep aspherical element is physically divided into regions, and the edge micro-partitions with a sampling point ratio lower than the preset processing statistical threshold are merged into their adjacent partitions to complete the partition optimization and merging.

[0055] Further, step S600 specifically includes: setting the polishing removal function for each partition, calculating the polishing dwell time for the high-steep aspherical element; setting the polishing tool speed, workpiece speed, indentation, and precession angle to generate a polishing removal function model; and calculating the discrete dwell time using the known surface error data and the polishing removal function model for each partition.

[0056] Further, step S700 specifically includes: setting the current dwell point after discretization as P. i (x i ), which is adjacent to the previous point P i-1 The spacing is Δl i-1 , and the adjacent point P i+1 The spacing is Δl i ;

[0057] At the current point of residence P i (x i The length unit L corresponding to ) i =(Δl i-1 +Δl i Set the polishing feed speed v within the range of ) / 2. i Constant, and the feed rate satisfies v i =L i / t i , where t i For the current station P i (x i Discrete dwell time;

[0058] Introducing a speed constraint upper limit v max When the calculated feed rate v i >v max At that time, the current station P will be... i (x i The feed rate at point ) is forcibly corrected to v. max .

[0059] Compared with the prior art, the present invention has the following significant advantages: (1) The present invention uses the preprocessing projection conjugate gradient method to solve the constrained polishing dwell time, which improves the efficiency of solving the dwell time corresponding to the large sparse surface detection matrix; (2) In response to the high steep curvature effect, the present invention proposes a K-means clustering high steep aspheric surface polishing partition model based on the principal curvature feature; for different principal curvature feature partitions, different polishing process parameters are used to ensure the adaptability of the polishing removal function to the variable curvature, thereby improving the polishing accuracy and surface convergence effect of high steep aspheric surfaces. Attached Figure Description

[0060] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0061] like Figure 1 As shown, a high-steep aspherical partition polishing method based on curvature characteristics includes the following steps.

[0062] Step 1: Based on the principal curvature radius and Hertzian contact theory, construct a contact pressure model in the high-steep aspherical polishing area: First, calculate the characteristic size of the elliptical contact area by calculating the relative principal curvature radii in the two principal directions of the contact area, and then construct a polishing pressure distribution model based on the Hertzian contact model.

[0063] Step 1.1: Approximate the polished area as an elliptical area and calculate its principal radii of curvature in two directions. The calculation formula is shown in (1).

[0064] (1),

[0065] Among them, R x R is the principal radius of curvature in the x-direction within the elliptical contact region. y R is the relative principal curvature radius in the y-direction within the elliptical contact region. p R is the radius of the polishing tool. tx and R ty These represent the relative principal radii of curvature of the aspherical element at the polishing point in the x and y directions, respectively.

[0066] Step 1.2, the lengths of the elliptical semi-axis a and b are the characteristic dimensions of the polishing area. Based on the contact mechanics theory, the characteristic dimensions a and b have the relationship shown in formula (2). According to this set of equations, a and b are obtained by solving formulas (3) and (4) simultaneously.

[0067] (2),

[0068] (3),

[0069] (4),

[0070] Where F is the polishing pressure, R e E is the equivalent radius of curvature of the elliptical contact region. * E represents the equivalent elastic modulus between the polishing tool and the aspherical component. t E p The Young's modulus of the polishing tool and the aspherical component, respectively; ν t ν t Let F be the Poisson's ratio between the polishing tool and the aspherical element. The polishing pressure F is related to the indentation δ and the precession angle α of the polishing tool, i.e., F = f(δ, α).

[0071] Step 1.3: Based on the Hertzian theory model, the pressure P(x,y) distribution model in the above elliptical contact region is calculated, as shown in formula (5).

[0072] (5).

[0073] Step 2: Construct a relative velocity distribution model within the polishing area. During polishing, the relative velocity within the contact area is determined by the linear subtraction of the workpiece rotation speed and the polishing tool rotation speed. Combining kinematics and geometric dimensions, calculate the relative velocity v at each point. r (x,y),

[0074] (6),

[0075] Among them, v r Let v be the relative linear velocity at any point in the contact area. p v is the linear velocity of the workpiece. t ω is the linear velocity of the polishing tool. p ω p r p and r t These are the workpiece rotation speed, tool rotation speed, workpiece contact point gyration radius, and tool contact point gyration radius, respectively.

[0076] Step 3: Construct a removal function model within the polishing area. Based on Preston's theory, the material removal amount ΔZ(x,y) at a polishing contact point (x,y) per unit time is proportional to the relative pressure P(x,y) and relative velocity V(x,y) of the tool and workpiece, and is subject to the polishing environment parameter K. p The effect is shown in the following equation:

[0077] (7).

[0078] Step 4: Construct the residence time model of the projection conjugate gradient method. First, a discrete linear relationship model between material removal depth, removal function, and residence time is established. Then, the preprocessed projection conjugate gradient method is used to transform the residence time solution into a quadratic functional minimization problem. By introducing a symmetric positive definite preprocessor to reduce the spectral condition number of the coefficient matrix and optimize the iterative convergence, the polishing residence time is finally solved efficiently under constraints.

[0079] Step 4.1: Establish a discretized relationship model between the expected removal depth E, the removal function R and the residence time T on the component surface, and transform the continuous superposition process of material removal into a discrete linear superposition. The linear equation is shown in formula (8).

[0080] (8),

[0081] The total number of points in the surface data of the component is s; the error value of the i-th point is E. i The removal function at point j and the material removal function at point i are denoted as r. ij Remove function r ij The dwell time at point j is denoted as t. j .

[0082] Step 4.2: Solve for the constrained polishing dwell time vector U using the preprocessing projection conjugate gradient method. Solving the linear equation system RT=E is equivalent to minimizing a quadratic functional; its minimum point is the solution to the equation system. Therefore, set the gradient to zero.

[0083] (9),

[0084] Step 4.3: Substitute the symmetric positive definite preprocessor M into the standard conjugate gradient method; its convergence rate is determined by the spectral condition number of R. It is determined that the larger the condition number, the slower the convergence. The introduced preprocessing matrix M can reduce the condition number of the equivalent matrix, thus accelerating convergence. Let... ,in Let M be the positive definite square root, satisfying Substituting it into formula (9) yields:

[0085] (10);

[0086] The preprocessed coefficient matrix is The right-hand side after preprocessing is The constraint set is The optimization problem after preprocessing is as follows:

[0087] (11);

[0088] R M The spectral condition number is much smaller than R, resulting in a significant improvement in the iteration convergence speed. Correspondingly, the original residual is transformed... Replace with preprocessed residuals To adapt to the preprocessing framework.

[0089] Step 5: Construct a polishing partitioning model based on K-means clustering. Using the principal curvature of the sampling points on the steep aspherical surface as a feature, a feature vector is first constructed by combining the central difference method with the rate of curvature change and radial distance. The features are then standardized to unify the clustering weights. The elbow rule combined with the WCSS index is used to determine the optimal number of clusters K. Euclidean distance is used to calculate the sample clustering, and the cluster centers are iteratively updated until the algorithm converges. Finally, the surface region of the component is divided based on the curvature feature, and the partitions are optimized and merged to ensure that each partition meets the processing and statistical requirements.

[0090] Step 5.1: Combining the rate of change of curvature and the radial distance, calculate the principal curvature eigenvector of each sampling point using the central difference method:

[0091] (12),

[0092] Where Δr is the sampling step size, and κ1 is the radial principal curvature. is the radial principal curvature change rate, and r is the radial distance from the sampling point to the center of the element, used to constrain the partition boundary to coincide with the structural features.

[0093] Step 5.2, let the feature sample set be... , where n is the number of sampling points and j is the feature index, the standardization formula is as shown in (13). After standardization, the feature mean = 0 and the standard deviation = 1, ensuring that the weight of each feature on the clustering result is consistent.

[0094] (13)

[0095] Where, μ j For the standardized mean, σ j The characteristic standard deviation, These are the standardized feature values.

[0096] Step 5.3: Determine the number of clusters K based on the elbow rule. The sum of squares within clusters (WCSS) is used to measure the clustering effect and reflect the compactness of samples within the same cluster. Let the number of clusters be K, and the k-th cluster contain n... k There are samples, and the cluster sample set is . The cluster center is Then the sum of squares of the k-th cluster is:

[0097] (14);

[0098] Total WCSS:

[0099] (15);

[0100] WCSS decreases as K increases: when K=1, WCSS is at its maximum, all samples are grouped into one class, and the compactness is the lowest; when K=n, WCSS=0, each sample is in a cluster. The elbow point corresponds to the K value where the decrease in WCSS is sharp, at which point the clustering gain and complexity reach a balance.

[0101] The specific method for determining the optimal number of clusters K using the elbow rule is as follows: calculate the total sum of squares within each cluster when the number of clusters is successively taken as K-1, K, and K+1, and denoted as WCSS respectively. K-1 WCSS K and WCSS K+1Construct a first-order difference change Δ1(K) = WCSS to characterize the rate of WCSS descent. K-1 -WCSS K And the second-order difference change Δ2(K) = Δ1(K) - Δ1(K+1) used to characterize the rate of change of the decline amplitude; when K starts to increase from 1, the K value when the second-order difference change Δ2(K) reaches a local maximum value and the subsequent first-order difference change Δ1(K+1) tends to be flat and always less than the preset rate of change threshold δ is taken as the inflection point of the sharp decrease in the decline amplitude of WCSS, and the K value corresponding to the inflection point is locked as the optimal number of clusters.

[0102] Step 5.4: Use Euclidean distance to measure the similarity between the sample and the cluster center for the standardized feature vector. With cluster center The Euclidean distance is defined as the straight-line distance between two points in space, in d=4-dimensional space:

[0103] (16)

[0104] The sample allocation rule is to Assign to the cluster with the smallest distance, i.e. .

[0105] Step 5.5, the goal of cluster center update is to minimize WCSS, specifically the WCSS with respect to cluster centers. Taking the partial derivative and setting it to zero, we can calculate the cluster center as the mean of the samples within the cluster. The updated cluster center is the mean of the corresponding features of all samples within the cluster, ensuring that WCSS is minimized. The convergence criterion is as follows: Let the cluster center in the t-th iteration be... The cluster center in the (t+1)th iteration is The convergence condition is defined as:

[0106] (17)

[0107] in Preset threshold (recommended) =10 -4 ), This represents the update magnitude of the cluster centers. When the update magnitudes of all cluster centers are less than... The algorithm converges when the number of iterations reaches its limit.

[0108] Step 5.6: Based on curvature characteristics, the high-steep aspherical elements are divided into regions and optimized and merged to ensure that each region has sufficient statistical significance and practical processing value.

[0109] Step 6: Set the polishing removal function for each zone and calculate the polishing dwell time for the high-steep aspherical component. Set process parameters such as polishing tool speed, workpiece speed, indentation, and precession angle to generate the polishing removal function model; calculate the discrete dwell time using the known surface error data and the polishing removal function model for each zone.

[0110] Step 7: Convert the discrete dwell time into CNC feed code recognizable by the machine tool. Assume the discretized dwell point P... i With point P i-1 and point P i+1 The intervals are Δl i-1 and Δl i Then take [P] i (x i )-Δl i-1 / 2, P i (x i )+Δl i / 2] The feed rate v of this length unit i The time of stay remains unchanged, t. i The reciprocal of the value. To prevent excessively high speeds due to excessively short dwell times, an upper limit constraint needs to be added to the feed rate. According to the theoretical model of the polishing removal function, when the speed is greater than 100 mmpmin, the dwell time is too fast and can be considered as having no polishing effect. The program should adjust the polishing trajectory to avoid excessive shaping.

[0111] Example 1: The ideal surface shape of a high-steep aspherical surface is shown in Table 1.

[0112] Table 1 Ideal surface parameters for high-steep aspherical surfaces

[0113]

[0114] The high-steep aspherical element was divided into regions based on curvature characteristics. Referring to Table 2, the radial positions of each region from center to edge are as follows: Region 2 (center) > Region 4 > Region 5 > Region 1 > Region 7 > Region 3 > Region 6 (edge). This region division scheme effectively captures the curvature gradient change of the high-steep aspherical surface from center to edge. The radius of curvature smoothly increases from 26.18 mm at the center to 79.37 mm at the edge, consistent with its geometric characteristics. Significant differences in region quality were observed: the profile coefficients of the central region (Region 2) and the edge region (Region 3) were both higher than 0.77, indicating good internal curvature consistency and serving as high-quality benchmark regions. However, the profile coefficients of the intermediate transition regions (Regions 4, 5, and 7) were all lower than 0.67, revealing significant internal curvature variations and insufficient consistency within these regions, which may affect the stable application of processing parameters.

[0115] Table 2 Results of high-steep aspherical zoning

[0116]

[0117] To further improve the partitioning quality, the original seven partitions were optimized into six. The merged partitioning system significantly improved processing feasibility while maintaining the core geometric feature recognition capability. The main operation involved merging the six smaller edge partitions into the adjacent larger partition three, eliminating invalid processing units and ensuring that each partition had sufficient statistical significance and practical processing value. The merged six partitions exhibited a more reasonable curvature gradient distribution, with the contour coefficient of each partition remaining above 0.66, indicating good consistency within the partitions. In particular, the contour coefficients of the central and edge regions reached 0.7772 and 0.7863, respectively, verifying the high quality of the key region division. The optimized partition boundaries were clearer and more continuous, eliminating the original small gaps.

[0118] The surface shape accuracy (PV) of the surface to be polished was 93.78 μm, and the RMS value was 13.22 μm. Based on K-means and partitioning, with 6 inner regions set and polishing process parameters set separately, the calculated polishing time after rough polishing was 261.37 min, reducing the surface shape error (PV) to 25.86 μm and the RMS to 5.16 μm. The calculated and predicted PV value for fine polishing was 5.862 μm, and the RMS value was 1.556 μm. After exporting the CNC code, the high-steep aspherical surface was machined on a polishing machine. The final polishing accuracy of the high-steep aspherical optical element after fine polishing was 6.471 μm, and the RMS value was 1.887 μm.

Claims

1. A method for polishing high-steep aspherical surfaces based on curvature characteristics, characterized in that, include: Step S100: By calculating the relative principal radii of curvature in the two principal directions of the contact area on the workpiece surface, the characteristic dimensions of the elliptical contact area are obtained, and then a polishing pressure distribution model is constructed based on the Hertz contact model. Step S200: Combining kinematics and geometric dimensions, subtract the linear vector of the workpiece rotation speed from the polishing tool rotation speed to calculate the relative velocity at each point in the contact area; Step S300: Introduce polishing environment parameters, combine the pressure distribution model of Step 1 and the relative rate distribution model of Step 2, and construct a removal function for the amount of material removed per unit time based on Preston theory; Step S400: Establish the discrete linear superposition equation of removal depth, removal function and dwell time. Use the preprocessing projection conjugate gradient method. By introducing a symmetric positive definite preprocessor to reduce the spectral condition number of the coefficient matrix, the dwell time solution is transformed into a quadratic functional minimization problem under the constraints. Step S500: The central difference method is used to construct a feature vector by combining the principal curvature, curvature change rate and radial distance of the sampling points and then standardize it. The elbow rule is used to combine the sum of squares within the cluster to determine the optimal number of clusters K. The Euclidean distance is used to iteratively update the cluster center until convergence, thereby realizing the optimization of the component surface region division and merging. Step S600: Set the polishing process parameters for each partition to generate the corresponding removal function model, and calculate the discrete dwell time of each partition by combining the surface error data. In step S700, the discrete dwell time is converted into CNC feed code, and a constraint upper limit is set on the feed speed to avoid excessive shaping.

2. The method according to claim 1, characterized in that, Step S100 specifically includes the following steps: Step S110: The polishing area is an elliptical area. Calculate its principal curvature radii in two directions. The calculation formula is shown in (1). （1）， Among them, R x R is the principal radius of curvature in the x-direction within the elliptical contact region. y R is the relative principal curvature radius in the y-direction within the elliptical contact region. p R is the radius of the polishing tool. tx and R ty These are the relative principal radii of curvature of the aspherical element at the polishing point in the x and y directions, respectively. Step S120: Based on the contact mechanics theory, the characteristic dimensions with semi-axis lengths a and b in the elliptical contact area are obtained. a and b have the relationship shown in formula (2). According to this set of equations, a and b are obtained by solving formulas (3) and (4) simultaneously. （2）， （3）， （4）， Where F is the polishing pressure, R e E is the equivalent radius of curvature of the elliptical contact region. * E represents the equivalent elastic modulus between the polishing tool and the aspherical component. t E p The Young's modulus of the polishing tool and the aspherical component, respectively. t ν p Poisson's ratio for the polishing tool and the aspherical component; Step S130: Based on the Hertzian theory model, the pressure P(x,y) distribution model in the above-mentioned elliptical contact area is calculated, as shown in formula (5). （5）。 3. The method according to claim 2, characterized in that, Step S200 specifically includes: during polishing, the relative velocity within the elliptical contact area is determined by subtracting the linear vectors of the workpiece rotation speed and the polishing tool rotation speed. Combining kinematics and geometric dimensions, the relative linear velocity v at any point in the contact area is calculated. r (x,y), （6）， Among them, v r Let v be the relative linear velocity at any point in the contact area. p v is the linear velocity of the workpiece. t ω is the linear velocity of the polishing tool. p ω t r p and r t These are the workpiece rotation speed, tool rotation speed, workpiece contact point gyration radius, and tool contact point gyration radius, respectively.

4. The method according to claim 3, characterized in that, Step S300 specifically includes: Based on Preston's theory, the amount of material removed per unit time at a polishing contact point (x,y) ΔZ(x,y) is proportional to the relative pressure P(x,y) and relative velocity V(x,y) of the tool and workpiece, and is subject to the polishing environment parameter K. p To assess the impact of material removal, a removal function R(x,y) for the amount of material removed per unit time is constructed. （7）， Where T is the unit time and t is the independent variable in the integration process.

5. The method according to claim 4, characterized in that, Step S400 specifically includes the following steps: Step S410: Establish a discretized relationship model between the expected removal depth E, removal function R and residence time T on the component surface, and transform the continuous superposition process of material removal into a discrete linear superposition. The linear equation is shown in formula (8). （8）， Wherein, the total number of points in the surface data of the component is s, and the error value of the i-th point is E. i The removal function at point j and the material removal function at point i are denoted as r. ij Remove function r ij The dwell time at point j is denoted as t. j ; Step S420: The preprocessing projection conjugate gradient method is used to solve for the constrained polishing dwell time vector U. Solving the linear equation system RT=E is equivalent to minimizing a quadratic functional, the minimum point of which is the solution to the equation system. The minimized quadratic functional is: （9）， Step S430: Substitute the symmetric positive definite preprocessor M into the standard conjugate gradient method, and let... ,in Let M be the positive definite square root, satisfying Substituting it into formula (9) yields: （10）， The preprocessed coefficient matrix is The right-hand side after preprocessing is The constraint set is The optimization problem after preprocessing is shown in equation (10). （11）。 6. The method according to claim 5, characterized in that, Step S500 specifically includes: Step S510: Combining the rate of curvature change and radial distance, the principal curvature feature vector of each sampling point is calculated using the central difference method. , （12）， Where Δr is the sampling step size, and κ1 is the radial principal curvature. The radial principal curvature change rate is given by r, where r is the radial distance from the sampling point to the center of the element. Step S520: Construct a set of principal curvature feature samples. Formula (13) is used to convert the mean of each feature dimension in the feature sample set to 0 and the standard deviation to 1, thereby eliminating the influence of dimensions and obtaining a standardized feature vector set. （13）， Where n is the number of sampling points, j is the feature index, and μ j For the standardized mean, σ j The characteristic standard deviation, For the feature sample set, These are the standardized eigenvalues; Step S530: Use the sum of squares within clusters (WCSS) metric to measure the clustering effect and reflect the compactness of samples within the same cluster. Let the number of clusters be K, and the k-th cluster contain n... k There are samples, and the cluster sample set is . The cluster center is Then the sum of squares of the k-th cluster is （14）， Total WCSS: （15）， V' represents the average value of all sample points in the k-th cluster after standardization along the j-th feature dimension, where d is the feature space dimension index. j Let K be the standardized eigenvalue of the j-th dimension; the number of clusters K is determined using the elbow rule; Step S540: Using Euclidean distance as a similarity measure, calculate the straight-line distance between the standardized feature vector of each sampling point and the center vector of each cluster, and assign each sampling point to the cluster closest to it to form K initial partitions; Step S550: Update the cluster center vectors of each cluster with the goal of minimizing the total WCSS, and set the updated cluster center vectors as the arithmetic mean of the features of all samples within the corresponding cluster; iterate through steps S540 and S550 until the update magnitude of the cluster center vectors in two adjacent iterations is less than a preset convergence threshold. Or, if the preset maximum number of iterations is reached, the algorithm converges and locks the boundaries of each partition; Step S560: Based on the converged clustering results, the high-steep aspherical element is physically divided into regions, and the edge micro-partitions with a sampling point ratio lower than the preset processing statistical threshold are merged into their adjacent partitions to complete the partition optimization and merging.

7. The method according to claim 6, characterized in that, Step S600 specifically includes: setting the polishing removal function for each partition, calculating the polishing dwell time for high-steep aspherical components; setting the polishing tool speed, workpiece speed, indentation, and precession angle to generate a polishing removal function model; and calculating the discrete dwell time using known surface error data and the polishing removal function model for each partition.

8. The method according to claim 7, characterized in that, Step S700 specifically includes: setting the current dwell point after discretization as P. i (x i ), which is adjacent to the previous point P i-1 The spacing is Δl i-1 , and the adjacent point P i+1 The spacing is Δl i ; At the current point of residence P i (x i The length unit L corresponding to ) i =(Δl i-1 +Δl i Set the polishing feed speed v within the range of ) / 2. i Constant, and the feed rate satisfies v i =L i / t i , where t i For the current station P i (x i Discrete dwell time; Introducing a speed constraint upper limit v max When the calculated feed rate v i >v max At that time, the current station P will be... i (x i The feed rate at point ) is forcibly corrected to v. max .

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