Blade surface numerical control polishing method for controlling material removal depth
By constructing discrete convolution and optimization models during the polishing process of blade surfaces, and precisely planning the feed rate, the problem of uneven material removal depth in blade surface polishing was solved, thus improving processing accuracy and consistency.
Patent Information
- Application Number
- CN202610372150.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-03-25
- Publication Date
- 2026-06-30
AI Technical Summary
Existing technologies cannot meet the need for localized and customized removal of original errors in the polishing process of blade surfaces, resulting in uneven material removal depth and affecting contour accuracy and consistency.
By constructing a discrete convolution model of the residence time of discrete points of the polishing trajectory in the parameter domain of the blade surface and the tool influence function, and introducing residence time boundary constraints and difference penalty terms, a constrained least squares optimization model is established to finely plan the feed rate to achieve control of the material removal depth.
It effectively compensates for the differences in local contact state of the blade surface, improves the controllable accuracy of material removal, ensures the consistency and accuracy of the processed surface quality, and can flexibly customize the feed speed of the polishing trajectory according to the initial error distribution.
Smart Images
Figure CN122299465A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of CNC polishing of blade surfaces, and particularly relates to a CNC polishing method for controlling the material removal depth of blade surfaces. Background Technology
[0002] Aero-engine blade components (such as integral bladed disks and blade rings) are core parts of aero-engines, and their surface precision and quality directly affect the engine's aerodynamic characteristics and structural fatigue life. Due to the complex free-form surface characteristics of blades, current manufacturing processes typically employ precision milling to ensure initial geometric accuracy, while subsequent polishing primarily improves surface roughness. Traditional manual polishing methods heavily rely on worker experience, resulting in inconsistent processing quality that fails to meet the consistency requirements of high-precision parts. Currently, automated polishing methods, utilizing industrial robots and CNC machine tools combined with various flexible grinding mechanisms, are widely proposed and applied. In these methods, the equipment typically processes according to a preset trajectory, but process parameters (such as pressure and speed) are often kept constant throughout the polishing process. This "uniform parameter throughout" method relies heavily on extensive trial-and-error experiments in the early stages to control the material removal depth; as long as the overall error is within acceptable limits, active control during the process is lacking.
[0003] Existing technologies primarily focus on achieving constant-force contact between the polishing tool and the blade profile. However, due to significant differences in local curvature and contact posture across the blade surface, the material removal rate varies at different locations even with a constant overall contact force. This not only leads to uneven material removal depth across the superimposed surface, affecting the original blade's contour accuracy, but also fails to meet the need for customized localized removal of original part errors. Summary of the Invention
[0004] The purpose of this invention is to provide a CNC polishing method for blade curved surfaces that controls the material removal depth, so as to solve the problem that the existing technology cannot meet the needs of localized customized removal of original errors in parts.
[0005] This invention employs the following technical solution: a CNC polishing method for controlling the material removal depth of blade curved surfaces, comprising: Step 1: Obtain the target removal distribution covering the entire blade surface parameter space; Obtain a mapping model for predicting the influence function of surface tools; Step 2: Within the blade surface parameter domain, a discrete convolution model is established for the dwell time of discrete points on the polishing trajectory, the target removal distribution, and the predicted tool influence function. The upper and lower bounds of the dwell time constraint and the differential penalty term to ensure speed stability are introduced. A constrained least squares model for dwell time optimization is constructed and solved to obtain the optimal dwell time. Finally, the dwell time is mapped to the feed speed of the CNC system based on the length of the trajectory segment.
[0006] The beneficial effects of this invention are: This invention constructs a discrete convolution model of the dwell time of discrete points on the polishing trajectory within the parameter domain and the tool influence function. For the target material removal depth, it introduces dwell time boundary constraints and difference penalty terms and establishes a constrained least squares optimization model to achieve fine planning of the feed rate. This invention can customize the removal depth according to the surface features and initial error of the blade, and significantly improve the controllable accuracy of the material removal amount on complex curved surfaces while ensuring the stability of CNC machining. This invention achieves effective compensation for local contact state differences during free-form surface polishing by precisely planning and solving the feed rate. Unlike traditional constant-speed polishing methods, this invention can flexibly customize the feed rate at each point on the polishing trajectory according to the initial error distribution of the part, thereby accurately achieving the target uniform or non-uniform material removal depth. When there are large local contour errors on the part surface, this invention can adjust the feed rate to increase or decrease the amount of local material removal, achieving on-demand error correction and accuracy compensation. At the same time, the introduction of differential penalty terms ensures the smoothness of speed switching, improving the surface accuracy of the part while ensuring the consistency of the processed surface quality. Attached Figure Description
[0007] Figure 1 This is a schematic diagram of the polishing blade trajectory of the integral bladed disk in this invention; Figure 2 This is a schematic diagram of the target material removal map generation process in this invention, wherein (a) is the cross-sectional profile error in the uv parameter domain of the surface, (b) is the error interpolation cloud map in the uv parameter domain of the surface, and (c) is the target removal cloud map of the blade surface. Figure 3 This is a flowchart of the influence function of the surface polishing tool in this invention; wherein, TIF p TIF represents the influence function of the planar polishing tool. c This represents the influence function of the surface polishing tool; Figure 4 This is a schematic diagram of the tool influence function and sampling point distribution at the dwell point within the parameter domain in this invention; Figure 5 This is a diagram showing the feed rate planning results in this invention; Figure 6This is the post-processed NC code of the present invention; Figure 7 Analysis of blade cross-sectional profile error and material removal depth before and after polishing in this invention. Detailed Implementation
[0008] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.
[0009] This invention discloses a CNC polishing method for controlling the material removal depth of blade curved surfaces, including step 1 and step 2.
[0010] Step 1 includes: Step 101: Obtain the target removal distribution covering the entire blade surface parameter space; Step 102: Obtain the mapping relationship model for predicting the influence function of the surface tool.
[0011] In step 101, when obtaining the target removal distribution covering the entire blade surface parameter space, there are two options: a uniform material removal mode and a mode that considers discrete contour errors.
[0012] The discrete profile error model is considered as follows: First, the directed distance from the registered measurement point to the corresponding theoretical cross-sectional line of the blade to be polished is defined as the initial profile error. The initial profile errors on the cross-sectional curves of the blade base and blade back are then mapped to the corresponding surface parameter coordinates of the blade base and blade back, thereby obtaining a discrete profile error dataset. Interpolation calculations are performed based on the discrete profile error dataset to generate a target removal distribution covering the entire blade surface parameter space. The specific steps are as follows: (1) Obtain initial contour error data. Use a coordinate measuring machine or non-contact optical measuring equipment to perform precise measurement on the milled blade to be polished; perform probe radius compensation on the original measurement point and data registration between the measurement coordinate system and the theoretical model coordinate system; calculate the directed distance from the registered measurement point to the corresponding theoretical cross-section line of the blade to be polished, and define it as the initial contour error value of the point.
[0013] (2) Error Space Mapping. Based on the boundary points of the blade cross-section regions, the initial contour errors on the cross-sectional curves of the blade base and blade back are mapped to the corresponding blade base and blade back surface parameter coordinates (u, v), thereby obtaining a discrete contour error dataset. Among them, (u i , v i Let be the coordinates of the i-th measurement point in the parameter domain, Er i This represents the initial profile error at that point.
[0014] As attached Figure 2As shown, to achieve proactive error compensation, interpolation calculations are performed based on the discrete contour error dataset to generate a target removal distribution Hr(u,v) covering the entire surface parameter space. During interpolation: within the parameter coordinate measurement range, the natural neighborhood interpolation method is used to ensure the smoothness of error transition; outside the measurement coverage range, the nearest neighbor interpolation method is used for extension. The final generated target material removal distribution map serves as the input target for subsequent residence time calculations, guiding the refined planning of the feed rate.
[0015] Uniform material removal mode: Set the target removal amount across the entire region to a constant value Hr(u,v) = Hr set Hr set The settings should take into account the minimum removal thickness required to improve surface roughness, as well as the initial error range and process allowance, to ensure that the residual contour error after final removal meets the tolerance limits specified in the drawings.
[0016] Step 102: Obtain the mapping model for predicting the influence function of the surface tool, as shown in the appendix. Figure 3 As shown, the specific steps include: Step 1021: Calculate the principal curvature parameter pairs at uniformly discrete points on the freeform surface of the blade to be polished. Perform K-means clustering analysis and convex hull calculation on the principal curvature parameter pairs. Generate M cluster center points as central curvature points based on their distribution characteristics, and extract N dispersed convex hull vertices as boundary curvature points. Then, design a parabolic test piece that approximates the freeform surface of the blade to be polished. Specifically, this step involves: First, calculating the principal curvature parameter pairs at uniformly discrete points on the freeform surface of the blade to be polished. .in and Let be the minimum principal curvature and the maximum principal curvature at discrete points, respectively, satisfying... Second, K-means clustering analysis and convex hull calculation were performed on the principal curvature parameter CP. Based on its distribution characteristics, eight cluster center curvature points were selected as representative experimental groups; six dispersed convex hull vertices were selected as boundary curvature points. Third, for the cluster center curvature points and boundary curvature points, a parabolic test piece approximating the freeform surface of the blade to be polished was designed based on the principles of differential geometry. Its surface equation is: Where a and b are the shape coefficients of the parabolic surface, and their relationship with the principal curvature parameters of the freeform surface is as follows: Fourth, surface modeling was performed using UG / NX software. TC17 titanium alloy, consistent with the blade material, was selected as the experimental material, and a test block with the target curvature characteristics was prepared through precision milling. In addition, a plane polishing benchmark experimental group was simultaneously set up to extract standard material removal characteristics of the abrasive blade wheel under zero-curvature plane polishing conditions.
[0017] Based on the parabolic test piece, an abrasive cloth flap wheel with a diameter of 15mm and a width of 12mm was used. The polishing process parameters were set as follows: spindle speed 7000r / min, contact radius 8.3mm, and abrasive grit size P400. The typical contact directions were set as the directions of the maximum and minimum principal curvature of the elliptical point, and the directions of the maximum and minimum principal curvature and asymptotic direction of the hyperbolic point, to ensure that the experimental group covered the typical geometric feature directions of the flattest and most curved surfaces.
[0018] Step 1022: Before polishing, measure the surface topography point cloud data of the parabolic test piece, then perform fixed-point polishing, and measure the surface topography point cloud data of the parabolic test piece after polishing. Specifically, install the abrasive cloth flap wheel on the machine tool spindle, and perform a fixed-point polishing experiment according to preset process parameters and contact posture. The dwell time setting needs to be pre-calibrated to ensure that, under the current process parameters, each experimental point can produce material removal marks with sufficient signal-to-noise ratio and clear boundary features, thereby ensuring the accuracy of subsequent data extraction and fitting. Before and after polishing, a high-precision white light interferometer is used to measure the surface topography point cloud data of the parabolic test piece.
[0019] Step 1023: Based on the surface topography point cloud data of the parabolic test piece before and after polishing, calculate and obtain the surface deviation distribution of the parabolic test piece relative to the theoretical parabolic surface before and after polishing. Combine coordinate mapping, plane projection and gradient-based region extraction to obtain the material removal distribution data of the contact area. Fit the material removal distribution data of the contact area and then characterize the tool influence function in the form of a curved surface.
[0020] Step 1023 is as follows: First, the error between the measurement coordinate system and the machining coordinate system is eliminated by point cloud registration. That is, a positioning groove is formed by drilling and scribing the center line on the boundary of the parabolic test piece. The groove positioning of the point cloud data and the surface fine registration are used to achieve accurate matching between the scanned point cloud data and the theoretical surface.
[0021] Second: Calculate the surface deviation distribution before and after polishing; first, calculate the directed distance from the measurement point to the parabolic surface of the parabolic test piece. ; Where sign is the sign function. For the registered measurement points, This is the closest point on the theoretical surface. Where f(x,y) is the equation of the parabola. ; in, The sign of the curvature is determined by the curvature, and a and b are the shape coefficients of the parabolic surface.
[0022] Third: Project the surface deviation distribution onto the plane through coordinate mapping, that is: project the three-dimensional deviation value of each point. According to its planar projection position (x i , y i Mapping The shape of the peeled surface is obtained, with x, y as position coordinates and deviation as... The data is point cloud data with height z.
[0023] Fourth: Perform region extraction based on the gradient change of the removed features. The stripped data is the material removal distribution data of the contact area caused by polishing. That is: according to the material removal depth threshold (e.g., a directional deviation of less than -0.01mm is considered to have produced substantial material removal, otherwise it is considered to be noise error), the point cloud is binarized in the plane, the boundary of the effective removal area is determined by the image edge detection algorithm, and the three-dimensional point cloud data that produced material removal is extracted.
[0024] Fifth: The point cloud data of the contact area material removal is transformed into an analytical expression, thereby obtaining the tool influence function, which is represented as a feature vector containing surface coefficients and range boundary parameters. In this embodiment, a cubic polynomial function is used to perform surface fitting on the contact area material removal point cloud data. During fitting, the effective boundary of the surface is truncated according to the actual range of the point cloud distribution. Finally, the tool influence function of each experimental sample is represented as a feature vector containing surface coefficients and range boundary parameters. The tool influence function is: in, These are the polynomial coefficients.
[0025] When fitting the material removal distribution data in the contact area, a Gaussian or polynomial function is used.
[0026] Step 1024: Establish a mapping model between the theoretical surface height vector of the parabolic specimen and the principal components of the tool influence function coefficients using principal component analysis and radial basis function interpolation.
[0027] Step 1024 specifically involves: recombining the feature vectors obtained from each group of fixed-point polishing experiments into a high-dimensional sample data matrix. (n groups of samples, p variables in each group). First, the data are normalized to eliminate differences in dimensions, and then principal component analysis is used to identify the main direction of variable changes.
[0028] Principal Component Analysis (PCA) is performed using the singular value decomposition (SVD) of the data matrix X: ; in, It is a left singular vector matrix; The main diagonal elements are singular values. ; V is a right singular vector matrix, and the first k principal components are the first k columns of V, corresponding to the largest k singular values.
[0029] Eigenvalues With singular values The relationship is: The number of principal components retained, i.e., the number of variables after dimensionality reduction, is determined by the cumulative variance explained ratio. The cumulative variance explained ratio of the first k principal components is then calculated. for: By setting the cumulative variance explanation rate threshold to 95%, the first k columns of V are extracted, and the original high-dimensional matrix is compressed into a low-dimensional principal component matrix that can represent the core features of morphological distortion removal.
[0030] A local coordinate system is established with the theoretical contact center of the polishing tool on the curved surface as the origin, where y T Positive axis direction and tool axis direction Conversely, z T The axis is normal to the center of the surface (n) (from the point of contact on the surface to the center of the tool). Tool axis direction. The counterclockwise angle between the surface coordinate system O-xyz and the positive x-axis (the direction of minimum principal curvature dmin) is denoted as This characterizes the contact direction between the grinding wheel and the curved surface. Within the contact area of the planar polishing, 40 feature sampling points are discretely extracted according to a preset regular grid. The height value sequence of the sampling points is then... As an input feature vector characterizing the local shape of a workpiece.
[0031] A radial basis function interpolation method is employed, using the discrete height vector z as the independent variable and the principal components as the dependent variable. An interpolation function is established between the theoretical surface height vector and the principal components of the tool influence function, constructing a nonlinear mapping from the geometric feature space to a space devoid of topographic features. The interpolation function is expressed as follows: Among them, w iThese are the weight coefficients corresponding to the i-th basis function; r = ||xx i || represents the relationship between variable x and sample point x. i The Euclidean distance; N is the total number of sample points; The radial basis functions are represented using the following Gaussian kernel function: in, It is a shape parameter that controls the width of the radial basis function, and is estimated by the following formula: ; Where, x i For the i-th sample point, x j For the j-th sample point, mean is the mean value; The interpolation function can be written in matrix form as follows: in, ; It is an N×N interpolation matrix, calculated as follows: The solution is: Step 1025: Input the theoretical surface height value vector of the local contact area of the freeform surface to be polished, and obtain the predicted tool influence function using the mapping relationship model, such as... Figure 4 As shown.
[0032] Step 2: Within the blade surface parameter domain, establish a discrete convolution model of the dwell time of discrete points on the polishing trajectory, the target removal distribution, and the predicted tool influence function. Introduce upper and lower bounds for dwell time constraints and a differential penalty term to ensure speed stability. Construct and solve a constrained least squares model for dwell time optimization to obtain the optimal dwell time. Finally, map the dwell time to the feed speed of the CNC system based on the length of the trajectory segment.
[0033] like Figure 1 As shown, for open integral bladed disk parts, the isoparametric method is used to generate a smooth, continuous tool path that accurately follows the surface shape and has a quality comparable to the surface, ensuring machining stability and surface quality. A v-axis reciprocating polishing path is used to polish the blades, avoiding prolonged tool dwell and frequent tool advances and retractions near the leading and trailing edge boundaries. Circular paths are used for the initial and final tool advances and retractions in the blade tip region to achieve smooth tool entry and exit.
[0034] Within the blade surface parameter domain, a discrete convolution model is established for the dwell time of discrete points on the polishing trajectory, the target removal distribution, and the tool influence function.
[0035] As attached Figure 5As shown, the polishing abrasive moves along a defined polishing trajectory P(u,v) on the target surface. The discrete points on the trajectory are the dwell points, and the sampling points on the surface are the points where the polishing abrasive moves. Total material removal depth generated at the site It is a linear superposition of the local material removal depths produced by the tool influence functions at all dwell points at the current sampling point, specifically expressed as: (1) in, Indicates the mold at point Surface sampling points during residence time Material removal depth; Indicates the mold at point Duration of stay; n s n represents the total number of surface sampling points; d This indicates the total number of outposts.
[0036] Will , and Let them be denoted as r ki , t i and h k (k=1, 2, ... n) s i=1, 2, ... n d The discrete convolution process for material removal can be expressed as a system of linear equations in matrix form: (2) Where R is a convolution coefficient matrix with the number of rows equal to the total number of sampling points and the number of columns equal to the total number of dwell points.
[0037] in, (3) Where, r ki (k=1, 2, ... n s i=1, 2, ... n d ) represents the material removal depth at the k-th sampling point on the surface when the abrasive tool remains at the i-th dwell point for a unit of time; t i (i=1, 2, ... n d Let h be the dwell time at the i-th stop. k (k=1, 2, ... n s ) represents the removal depth of the k-th sampling point.
[0038] In the surface parameter domain, the steps for calculating R are as follows: Parametric domain of the target polished surface Uniform discretization is performed to obtain the set of sampling grid points. Determine the trajectory of the discrete polishing tool contact. and the corresponding tool axis direction Using discrete tool contacts as dwell points Calculate the principal curvature of the surface at that point. and normal n k Using the surface tool's influence function, based on the principal curvature... and contact direction angle The tool influence function at the current trajectory dwell point is calculated, and each sampling grid point is used. Along the normal n k Projection transformation to the coordinate system of the tool's influence function ,like If the value falls within the support region of the tool's influence function, then the depth calculation is removed. ,like If r falls outside the support region of the tool's influence function, then ki =0. Completing the calculation for all sampling points completes the calculation of column R, obtaining the contribution of that stationary point to the removal of all sampling points. For all stationary points... Repeat this process to complete the calculation of the entire convolution coefficient matrix R.
[0039] in, ; in, These are the polynomial coefficients.
[0040] Step 202: Introduce upper and lower bounds for dwell time constraints and a differential penalty term to ensure velocity stability, construct and solve the constrained least squares model for dwell time optimization, and then obtain the optimal dwell time.
[0041] To ensure a smooth transition of dwell time at discrete points on the trajectory within a reasonable range, the solution to the dwell time series is transformed into a constrained least squares optimization problem by introducing upper and lower bounds on the dwell time and a difference penalty term: (4) In the optimization objective, R, t, and h are the material removal convolution coefficient matrix, the dwell time series vector, and the target removal depth vector, respectively; D1 and D2 are the first-order difference matrix and the second-order difference matrix, as shown in equations (5) and (6). and The corresponding weighting coefficients are used to limit the rate of change and acceleration of adjacent residence times through penalty terms, thereby achieving first-order and second-order control of the smoothness of the residence time series.
[0042] (5) (6) In the constraints, the lower bound t of the dwell time min Determined by the maximum feed rate, i.e., for all dwell points, tmin =1 / V f,max The upper bound of the dwell time t max The depth of removal is determined by the maximum removal depth of the tool influence function at the dwell point and the target removal depth at the corresponding point, i.e.: (7) Among them, t i,max Let represent the maximum dwell time at the i-th dwell point, and k represent the index of the strongest position corresponding to the i-th dwell point; This represents the target removal depth of the discrete point at the k-th sampling point; This indicates the material removal depth at the surface sampling point per unit residence time; This represents the material removal depth at the j-th sampling point on the surface when the abrasive tool resides at the i-th dwell point for a unit of time. This is because the removal depth at each discrete point is generated by the superposition of the tool influence functions at multiple dwell points. To avoid excessive removal at a dwell point where its influence is strongest, the maximum dwell time must be limited to the upper limit of the time calculated based on the target removal depth at that location.
[0043] The optimization objective is equivalent to the following quadratic form: (8) Where H is a positive semi-definite (due to (1, (2 ≥ 0)) Hessian matrix, and c is the coefficient vector of the linear terms.
[0044] The constrained least squares optimization problem can be equivalently transformed into a convex quadratic programming problem, which can be solved efficiently using the interior point method.
[0045] Step 203: Map the trajectory segment length to the feed speed of the CNC system.
[0046] Based on the length of the trajectory segment, it is converted into the feed rate v corresponding to the trajectory position. i Used to guide feed rate control during continuous motion, the conversion relationship is as follows: (9) in, Let i be the dwell time of the grinding wheel at point i; (10) Where, d i It is the distance between the i-th discrete stopping point (tool position) of the trajectory and the previous point.
[0047] The feed rate distribution obtained by solving is as follows Figure 5As shown, the feed rate is adaptively adjusted based on the removal characteristics of the local tool influence function: for a given non-uniform material removal target, the optimization process allocates a corresponding feed rate to ensure that the material removal distribution within the coverage area closely matches the target removal map. Simultaneously, under the combined effect of constraints and differential penalty terms, the feed rate curve maintains high smoothness and remains within a reasonable range.
[0048] Based on the calculated feed rate, a post-processing algorithm is used to generate NC polishing code containing variable feed rate information, such as... Figure 6 The F-value control command is shown. This NC code is imported into the CNC machine tool control system to drive the mold to perform machining according to the planned trajectory. After machining, a second measurement is performed on the blade profile. The cross-sectional contour error and material removal depth distribution before and after polishing are shown in the attached figure. Figure 7 As shown. From Figure 7 As can be seen, the method of the present invention is guided by contour error correction. By actively planning the feed rate, the material removal depth is increased by specifically reducing the feed rate in areas with large milling errors (such as the area near the blade tip). The contour error distribution after polishing converges significantly towards zero (theoretical value), and the contour accuracy of the part is effectively improved.
[0049] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A blade surface NC polishing method for controlling a material removal depth, characterized by, include: Step 1: Obtain the target removal distribution covering the entire blade surface parameter space; Obtain a mapping model for predicting the influence function of surface tools; Step 2: Within the blade surface parameter domain, a discrete convolution model is established for the dwell time of discrete points on the polishing trajectory, the target removal distribution, and the predicted tool influence function. The upper and lower bounds of the dwell time constraint and the differential penalty term to ensure speed stability are introduced. A constrained least squares model for dwell time optimization is constructed and solved to obtain the optimal dwell time. Finally, the dwell time is mapped to the feed speed of the CNC system based on the length of the trajectory segment.
2. The method of claim 1, wherein the control of the material removal depth is performed by a blade surface NC polishing method, characterized in that, The specific steps in step 1 to obtain the target removal distribution covering the entire surface parameter space of the blade are as follows: The directed distance from the registered measurement point to the corresponding theoretical cross-sectional line of the blade to be polished is defined as the initial contour error. The initial contour errors on the cross-sectional curves of the blade base and blade back are mapped to the corresponding surface parameter coordinates of the blade base and blade back to obtain a discrete contour error dataset. Interpolation calculation is performed based on the discrete contour error dataset to generate a target removal distribution covering the entire surface parameter space.
3. The method of claim 1, wherein the method is a method of controlling the material removal depth of a blade surface NC polishing method, characterized by, The constrained least squares model for optimizing the solution's residence time constructed in step 2 is as follows: ； where R, t and h are material removal convolution coefficient matrix, residence time sequence vector and target removal depth vector, respectively; D1 and D2 are first-order difference matrix and second-order difference matrix, and are corresponding weight coefficients; t min denotes the lower bound of residence time; t max denotes the upper bound of residence time; t denotes residence time; ， ， in, , Among them, t i,max Let represent the maximum dwell time at the i-th dwell point, and k represent the index of the strongest active position corresponding to the i-th dwell point. This represents the target removal depth of the discrete point at the k-th sampling point; This indicates the material removal depth at the surface sampling point per unit residence time; n represents the material removal depth at the j-th sampling point on the surface when the abrasive tool remains at the i-th dwell point for a unit time. d This indicates the total number of outposts.