Polishing track planning method for sub-aperture polishing

Abstract

The invention relates to the technical field of polishing trajectory planning, in particular to a polishing trajectory planning method for sub-aperture polishing, which comprises the following steps of: constructing a polishing material removal function based on a polishing load, a polishing tool rotating speed, a feeding speed, a polishing attitude angle, a polishing tool and a workpiece material attribute; optimizing the initial polishing tracks one by one according to a polishing material removal function; and the residence time is solved. According to the method, the controllability of distribution of the material removal amount in the time-space double domains in the polishing process is achieved, and the curved surface polishing quality and stability are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of polishing trajectory planning, and in particular to a polishing trajectory planning method for sub-aperture polishing. Background Art

[0002] The solution of the material removal function during the polishing process is the basis for achieving deterministic polishing. The accuracy of the material removal function calculation determines the consistency of material removal during the polishing process, and has a crucial impact on the correction of surface shape errors.

[0003] The essence of polishing trajectory planning is to achieve precise spatial distribution of material removal by controlling the relative movement between the polishing tool and the workpiece. The planning of the polishing trajectory is an intersection and integration of related theories such as path topological structures (such as spiral lines, raster lines, random paths, etc.), kinematic parameters (speed, feed rate, etc.), and contact mechanics parameters (pressure distribution, attitude adjustment, etc.). The research on polishing trajectory planning plays an important role in enabling high-precision complex surface manufacturing and promoting the upgrading of intelligent manufacturing. First of all, polishing trajectory planning is the basis for precise distribution of removal amount. The correction of surface shape errors requires dynamically adjusting the tool dwell time according to the local error amplitude. If the trajectory spacing is fixed, insufficient overlap of the removal functions of adjacent paths will cause "peak-valley" residual errors. Moreover, if the material removal amount cannot match the surface shape error distribution, the shape correction process will become blind. Secondly, polishing trajectory planning is the guarantee of contact mechanics consistency. Polishing of complex surfaces requires the tool path to strictly follow the local curvature characteristics. Otherwise, the spatial heterogeneity of contact pressure and removal function will lead to surface shape distortion. Trajectory planning that ignores geometric adaptation will result in local over-polishing or under-polishing, and the surface accuracy will also get out of control due to mechanical instability. Thirdly, trajectory planning is the core carrier of process robustness. Without dynamic coverage ability, the polishing trajectory will lead to the collapse of process robustness and directly cause surface shape accuracy drift. Finally, as the hub of the polishing process system integration, trajectory planning can effectively decompose the influence of multiple parameters such as speed, pressure, and rotation speed on polishing quality, and achieve the coordination of process parameters.

[0004] The accurate calculation of the dwell time during the polishing process is the core link for controlling the material removal distribution in precision polishing, and directly determines the key quality indicators such as the surface shape accuracy, surface roughness, and subsurface damage of the workpiece after polishing. The dwell time distribution determines the spatial distribution of the material removal amount. If the calculation error causes the dwell time to deviate from the theoretical value, it will directly amplify the surface shape residuals. At the same time, the high-frequency oscillation of the dwell time (caused by deconvolution noise) is transmitted as periodic fluctuations of the removal amount through convolution. Therefore, on the premise that both the material removal function and the polishing trajectory reach the optimal solution, effectively improving the calculation accuracy of the dwell time can effectively guarantee the lower limit of the final polishing quality.

[0005] However, in the existing polishing technologies, the derivation of the material removal function is mostly based on the Preston empirical equation. The Preston coefficient is determined by on-site sampling, and then the material removal function is solved. This method is limited by equipment, sensor accuracy, and operator experience in practical engineering applications. The sampling results often have large errors, resulting in unstable calculation results of the removal function, which seriously limits the accuracy of the polished sample. Although the existing polishing trajectory generation methods can achieve complete coverage of the trajectory on the workpiece surface during the polishing process, they ignore the linkage relationship between the trajectory and the material removal function during the polishing process and cannot be used as the optimal solution for the trajectory during the polishing process. As a result, the obtained trajectory does not have the forward-looking attribute of material removal, seriously affecting the polishing quality of the curved surface. Finally, the existing dwell time is mostly calculated based on the least squares method. Due to the instability of its solution results, high computational complexity, and the lack of association with the kinematic characteristics of processing equipment such as machine tools, the practical application value of the solved dwell time is too low.

[0006] In summary, the existing polishing trajectory planning methods have problems inconsistent with expectations in the solution of the material removal function, the generation of the polishing trajectory, and the solution of the dwell time, which limit the accuracy and stability of curved surface polishing and reduce the quality of curved surface polishing. Summary of the Invention

[0007] Therefore, the technical problem to be solved by the present invention is to overcome the problems in the existing technology that are inconsistent with expectations in the solution of the material removal function, the generation of the polishing trajectory, and the solution of the dwell time, which limit the accuracy and stability of curved surface polishing and reduce the quality of curved surface polishing.

[0008] To solve the above technical problems, the present invention provides a polishing trajectory planning method for sub-aperture polishing, including:

[0009] Calculate the relative linear velocity of any point in the polishing contact area based on the curvature radius, elastic modulus, Poisson's ratio, feed rate, attitude angle, angular velocity, and polishing load of the polishing tool; use the ratio of the Archard wear coefficient of the workpiece to be polished to the hardness of the workpiece material as the Preston coefficient in the Preston empirical equation, and substitute the relative linear velocity of a point in the polishing contact area into the Preston empirical equation to obtain the initial polishing material removal function;

[0010] Establish a spiral initial polishing trajectory with a total number of turns of I'. The number of trajectory points on each turn is the same, and the polishing contact areas of each turn of the trajectory do not overlap; optimize the second turn of the trajectory with the first turn of the trajectory, and optimize the (i'+1)-th turn of the trajectory with the optimized i'-th turn of the trajectory, where i' = 2, 3,..., I' - 1, including:

[0011] Select a trajectory point in the optimized i'-th circle trajectory that has a preset angular difference from the phase angle of the currently to-be-adjusted trajectory point as the reference point for the currently to-be-adjusted trajectory point in the (i'+1)-th circle trajectory;

[0012] Bisect the line connecting the reference point and the currently to-be-adjusted trajectory point, and extract the bisection points; calculate the material removal amount functions of the reference point and each bisection point using the initial polishing material removal function; superimpose the material removal amount functions of the bisection points with the material removal amount function of the reference point, and record the distance value between the minimum inflection point value and the extreme point value of the superimposed result as the PV value;

[0013] Calculate the PV value of each bisection point, and select the bisection point corresponding to the minimum PV value as the adjusted trajectory point;

[0014] Select dwell points according to the trajectory points of the optimized polishing trajectory; decompose the initial polishing material removal function into the tool influence function and the dwell time relationship of the dwell points, and establish a dwell time equation based on the dwell point coordinates and the control point coordinates within the polishing contact area corresponding to the dwell points; use the Lasso regression method to solve the dwell time of the dwell points.

[0015] Preferably, calculate the relative linear velocity of any point within the polishing contact area based on the curvature radius, elastic modulus, Poisson's ratio, feed rate, attitude angle, angular velocity, and polishing load of the polishing tool, including:

[0016] Calculate the deformation of the polishing tool based on the curvature radius, elastic modulus, Poisson's ratio, and polishing load of the polishing tool. The formula is:

[0017]

[0018] where ρ is the curvature radius of the polishing tool, E1 is the elastic modulus of the polishing tool, μ1 is the Poisson's ratio of the polishing tool, and Q is the polishing load;

[0019] Calculate the actual downward pressure of the polishing tool based on the curvature radius and deformation of the polishing tool. The formula is:

[0020]

[0021] where OC t represents the actual downward pressure of the polishing tool;

[0022] Taking the center of the polishing contact area as the origin and the direction of the feed rate of the polishing tool as the x-axis direction to establish a coordinate system, calculate the relative linear velocity of any point within the polishing contact area based on the actual downward pressure of the polishing tool. The formula is:

[0023]

[0024] where v sRepresents the relative linear velocity of the coordinate point (x k , y k ) within the polished contact area, where v a is the feed rate of the polishing tool, v ax and v ay are the velocity components of v a along the x-axis and y-axis respectively; ω is the angular velocity of the polishing tool; σ and λ are the attitude angles of the polishing tool, where σ is the polishing inclination angle and λ is the polishing declination angle.

[0025] Preferably, the formula for the initial polishing material removal function is:

[0026]

[0027] where h is the initial polishing material removal function, k abr is the Archard wear coefficient of the workpiece to be polished, H v is the hardness of the workpiece material, v a is the feed rate of the polishing tool, Q is the polishing load, σ is the polishing inclination angle, ω is the angular velocity of the polishing tool; E(·) is the integral symbol of the second kind of ellipse; a and b are the major axis and minor axis of the polished contact area respectively.

[0028] Preferably, the line connecting the reference point and the current trajectory point to be adjusted is equally divided to extract the equally divided points, including:

[0029] The number of equally divided points is D1 = c + (i' - 1)Δ, where i' is the loop number of the trajectory to which the reference point belongs, c is the initial number of equally divided points; Δ is the fixed increment;

[0030] A constraint condition is added to the spacing value between the reference point and the equally divided points: a1 < Dis < a1 + a, where a1 is the major axis of the polished contact area of the reference point, Dis is the spatial distance between the reference point and the equally divided points; a is the major axis of the polished contact area of the equally divided points.

[0031] Preferably, the material removal amount functions of the reference point and each equally divided point are calculated using the initial polishing material removal function, including:

[0032] Solve the normal vectors of the triangular meshes where the reference point and each equally divided point are located, rotate the initial polishing material removal function according to the normal vectors of the reference point and each equally divided point respectively, and then translate according to the coordinates of the reference point and each equally divided point to obtain the material removal amount functions of the normal vectors of the reference point and each equally divided point correspondingly.

[0033] Preferably, the material removal amount function of the equally divided point is superimposed with the material removal amount function of the reference point, and the superimposed result is denoted as H(x, y); the first partial derivative H x(x, y), the partial derivative of the superposition result H(x, y) with respect to y gives the first-order partial derivative H y (x, y); H x (x, y) the partial derivative of H with respect to x gives the second-order partial derivative H xx (x, y), H x (x, y) the partial derivative of H with respect to y gives the second-order partial derivative H xy (x, y), H y (x, y) the partial derivative of H with respect to y gives the second-order partial derivative H yy (x, y);

[0034] When there exists a point (x C , y C ) that satisfies H xx (x C , y C )H yy (x C , y C ) - [H xy (x C , y C )] 2 > 0 and H xx (x C , y C ) > 0, the point (x C , y C , H(x C , y C )) is an extreme point of the superposition result H(x, y), where x C and y C are the x - coordinate and y - coordinate of the extreme point respectively, and H(x C , y C ) is the material removal amount at the extreme point;

[0035] When there exists a point (x I , y I ) that satisfies:

[0036]

[0037] The point (x I , y I , H(x I , y I )) is an inflection point of the superposition result H(x, y), where H u (x, y) is the material removal amount function of the equal - division point u, x I and y I are the x - coordinate and y - coordinate of the inflection point respectively, and H(x I , y I ) is the material removal amount at the inflection point.

[0038] Preferably, the initial polishing material removal function is decomposed into a tool influence function and a dwell time relationship at the dwell points, including:

[0039]

[0040] where, TIF i represents the tool influence function at the i-th dwell point, t i represents the dwell time at the i-th dwell point, v ai represents the feed rate of the polishing tool at the i-th dwell point, l i represents the trajectory step size;

[0041] A dwell time equation is established based on the dwell point coordinates and the control point coordinates within the polishing contact area corresponding to the dwell points, including:

[0042]

[0043] where, N represents the number of dwell points, M represents the number of control points within the polishing contact area corresponding to the dwell points; TIF MN represents the tool influence function value of the M-th control point within the polishing contact area corresponding to the N-th dwell point, t N represents the dwell time at the N-th dwell point; T represents the matrix form of the tool influence function, t represents the matrix form of the dwell time, and e represents the residual error.

[0044] Preferably, the Lasso regression method is used to solve the dwell time at the dwell points, including:

[0045] The Lasso regression algorithm is used to construct an objective function, and the objective function is split into the sum of squared residuals and a regularization term, expressed as:

[0046]

[0047] where, is the sum of squared residuals, g(t) = η||t||1 represents the regularization term, η represents the regularization parameter, and ||t||1 represents the L1 norm;

[0048] The gradient of the sum of squared residuals is solved, and the gradient and Lipschitz constant of the sum of squared residuals are calculated; the gradient descent step size of the sum of squared residuals is calculated according to the gradient and Lipschitz constant of the sum of squared residuals;

[0049] The proximal operator of the regularization term is solved and decomposed into a differentiable iterative form;

[0050] The gradient descent step size of the sum of squared residuals is substituted into the differentiable iterative form of the proximal operator of the regularization term, and the dwell time is iteratively solved until the dwell time converges.

[0051] Preferably, substituting the gradient descent step size of the sum of squared residuals into the differentiable iterative form of the proximal operator of the regularization term, the (k + 1)-th iteration is expressed as:

[0052] t (k+1) =sign(z (k) )·max(|z (k) -ηα,0|)

[0053] where t (k+1) is the residence time of the (k + 1)-th iteration, z (k) represents the gradient descent step size of the sum of squared residuals of the k-th iteration, sign(·) is the soft threshold function, α = 1 / l, and L is the Lipschitz constant of the sum of squared residuals;

[0054] If |f[t (k+1) -f[t (k) |<ξ, where t (k) is the residence time of the k-th iteration and ξ is a preset threshold, then the objective function converges, and t (k+1) is the optimal solution; otherwise, continue iterative solution.

[0055] Preferably, a regularization method is used to smooth the obtained residence time, including:

[0056] Construct the objective function for smoothing:

[0057]

[0058] where t i and respectively represent the residence times before and after smoothing of the i-th residence point, and respectively represent the residence times after smoothing of the (i - 1)-th and (i + 1)-th residence points, β represents the smoothing intensity coefficient, and N represents the number of residence points;

[0059] Introduce the second-order difference matrix to convert the objective function for smoothing into a matrix operation form, and use the Cholesky decomposition method to solve it to obtain the smoothed residence time.

[0060] The above technical solutions of the present invention have the following beneficial effects compared with the prior art:

[0061] A polishing trajectory planning method for sub-aperture polishing according to the present invention first re-solves the polishing material removal function, and establishes a polishing material removal function model with polishing load, polishing tool rotation speed, feed speed, polishing attitude angle, and polishing tool and workpiece material properties as inputs. This not only improves the prediction accuracy of material removal during polishing, but also can adapt to the polishing processes of different materials, realizing the controllability of material removal during polishing. Further, the present invention combines the derived polishing material removal function to adjust and optimize each generated initial polishing trajectory one by one, and can obtain a polishing trajectory that dynamically covers the surface according to the contact area and material removal depth, so that a higher surface accuracy and lower PV value can be obtained within a unit processing cycle, meeting the requirements of rapid convergence of surface accuracy and surface quality simultaneously during fine polishing. Moreover, the present invention constructs a dwell time equation according to the derived polishing material removal function and solves it using the Lasso regression method, which not only realizes the controllability of the distribution of material removal amount in the time-space double domain during polishing, effectively reducing the surface shape residual after polishing, but also avoids the periodic fluctuation problem of the removal amount caused by convolution transfer in the conventional dwell time solving algorithm, improving the accuracy and stability of surface polishing.

[0062] Further, the present invention introduces a regularization optimization method to smooth the obtained dwell time, making the finally obtained dwell time more in line with the kinematic characteristics of the equipment during the actual polishing process. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to make the content of the present invention easier to be clearly understood, the following further elaborates on the present invention in detail according to specific embodiments of the present invention and in conjunction with the drawings, where:

[0064] Figure 1 is a flowchart of a polishing trajectory planning method for sub-aperture polishing of the present invention;

[0065] Figure 2 is a schematic diagram of the velocity distribution of any point within the polishing contact area;

[0066] Figure 3 is a flowchart of the polishing trajectory optimization algorithm of the present invention;

[0067] Figure 4 is a schematic diagram for determining the PV value of surface polishing based on the overlap amount between polishing trajectories;

[0068] Figure 5 is a comparison diagram before and after polishing trajectory optimization, where Figure 5 in (a) is the polishing trajectory diagram before optimization, Figure 5 in (b) is the polishing trajectory diagram after optimization;

[0069] Figure 6 It is a flow chart for solving the dwell time;

[0070] Figure 7 It is a schematic diagram of control points and trajectory points;

[0071] Figure 8 It is an example diagram of the statistical result of the initial residual error of the sample to be polished;

[0072] Figure 9 It is a schematic diagram of the surface quality of the sample after dwell time polishing obtained by using the Lasso regression method;

[0073] Figure 10 It is a schematic diagram of the surface quality of the sample after dwell time polishing smoothed by using the regularization optimization method. Specific implementation manners

[0074] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited are not intended to limit the present invention.

[0075] Referring to Figure 1 as shown, the present invention provides a polishing trajectory planning method for sub-aperture polishing, including:

[0076] S1: Calculate the relative linear velocity of any point in the polishing contact area based on the curvature radius, elastic modulus, Poisson's ratio, feed speed, attitude angle, angular velocity, and polishing load of the polishing tool; the ratio of the Archard wear coefficient of the workpiece to be polished to the hardness of the workpiece material is used as the Preston coefficient in the Preston empirical equation, and the relative linear velocity of a point in the polishing contact area is substituted into the Preston empirical equation to obtain the initial polishing material removal function.

[0077] The currently widely used polishing material removal function based on the Preston empirical formula is: R(x, y) = kP(x, y)V(x, y). In this formula, R(x, y) is the material removal rate, k is the Preston coefficient (an empirical constant related to the material, abrasive, and polishing fluid), P(x, y) is the contact pressure distribution between the polishing tool and the workpiece, and V(x, y) is the relative sliding velocity distribution between the tool and the workpiece. During the solution process, it is first necessary to calibrate the Preston coefficient k through experiments. By performing fixed-point polishing experiments under a fixed pressure P0 and speed V0, the removal depth ΔZ and time t are measured. As shown in this formula: k = ΔZ / (P0V0t).

[0078] After that, a pressure distribution model P(x, y) is established according to the Hertz contact theory, and the removal function R(x, y) is solved in combination with the machine tool motion parameters (rotation speed, feed speed) and the velocity distribution in the polishing contact area.

[0079] However, this material removal function calculation method has obvious calculation defects: First, this method is a linear solution method, but the material removal amount commonly seen in the actual polishing process exhibits non-linear characteristics; Second, due to the multi-physical field coupling effect, the Preston coefficient is inconsistent for the same workpiece under the same process parameters but different test conditions.

[0080] Therefore, the present invention constructs a polishing material removal model based on the workpiece material characteristics, polishing pressure distribution, polishing tool material characteristics, rotational speed, and polishing feed rate, specifically referring to Figure 2 . Assume C t is the center of the spherical polishing tool, O is the center of the polishing contact area, and the direction of the feed rate v a is taken as the x-axis direction to establish a coordinate system O-xyz (the z-axis direction is the same as the Z-axis direction in the global coordinate system). The angle between the axis Z t of the polishing tool and the z-axis is defined as the polishing inclination angle σ. ω' is the component of the angular velocity ω of the polishing tool in the xOy plane, and the angle between ω' and the x-axis is defined as the polishing declination angle λ. The distance between C t and point O is OC t , representing the actual downward pressure of the polishing tool. Assume that the feed rate and relative linear velocity at a point k(x k , y k ) within the polishing contact area are v a and v r respectively, and the distance between point k and C t is r.

[0081] According to the conclusion of elasticity mechanics, the deformation of the polishing tool is calculated based on the curvature radius, elastic modulus, Poisson's ratio, and polishing load of the polishing tool, and the formula is:

[0082]

[0083] where ρ is the curvature radius of the polishing tool, E1 is the elastic modulus of the polishing tool, μ1 is the Poisson's ratio of the polishing tool, and Q is the polishing load.

[0084] The Euclidean distance value between point O and point C t is calculated based on the curvature radius and deformation of the polishing tool, that is, the actual downward pressure of the polishing tool, and the formula is

[0085]

[0086] where OC t represents the actual downward pressure of the polishing tool.

[0087] Then the calculation formula for the relative linear velocity of point k is:

[0088]

[0089] wherein, v s represents the relative linear velocity of the coordinate point (x k , y k ) in the polishing contact area, v a is the feed rate of the polishing tool, v ax and v ay are the velocity components of v a on the x-axis and y-axis respectively; ω is the angular velocity of the polishing tool; σ and λ are the attitude angles of the polishing tool, where σ is the polishing inclination angle and λ is the polishing declination angle.

[0090] When the polishing tool moves on the surface to be polished at a feed rate v a , the unit time dt can be expressed by the unit displacement dl and the velocity v a : dt = dl / v a .

[0091] Since the polishing contact area is an elliptical area, define a as the major axis of the polishing contact area and b as the minor axis of the polishing contact area; the major axis a is perpendicular to the advancing direction of the tool. Using the polar coordinate transformation strategy, let x = arcosθ and y = brcosθ, and substitute the relative linear velocity of a point in the polishing contact area into the Preston empirical equation dh = k p VPdt, where V is the relative linear velocity distribution in the polishing contact area, P is the polishing pressure distribution, k p = k abr / H v is the Preston coefficient, k abr is the Archard wear coefficient of the workpiece to be polished, and H v is the hardness of the workpiece material.

[0092] Integrate and solve the Preston empirical equation to calculate the initial polishing material removal function in the local coordinate system O-xyz:

[0093]

[0094] wherein, h is the initial polishing material removal function, k abr is the Archard wear coefficient of the workpiece to be polished, H v is the hardness of the workpiece material, v a is the feed rate of the polishing tool, Q is the polishing load, σ is the polishing inclination angle, ω is the angular velocity of the polishing tool; E(·) is the integral symbol of the second kind of elliptic; a and b are the major axis and minor axis of the polishing contact area respectively; A, B, C are parameter combinations related to OC t , σ, λ, a, b.

[0095] The values of the major axis and minor axis of the polished contact area are affected by the curvature radius ρ of the polishing tool, the elastic modulus e1 of the polishing tool, and the Poisson's ratio μ1 of the polishing tool.

[0096] This step optimizes the conventional Preston empirical equation, derives a more accurate polishing material removal function, improves the estimation accuracy of material removal during the polishing process, and realizes the controllability of material removal during the polishing process.

[0097] S2: Optimize the polishing trajectory according to the initial polishing material removal function.

[0098] Existing polishing trajectory planning methods include isoparametric trajectory planning, constant cross-section trajectory planning, and equal residual height trajectory planning.

[0099] Isoparametric trajectory planning generates a tool path directly along the other parameter of the parametric surface S(u, v) by keeping one parameter of the surface S(u, v) unchanged (that is, one of the parameters u or v remains unchanged), and has a wide range of applications under the condition that the u and v parameters of the machined surface are determined. Constant cross-section trajectory planning uses a set of parallel planes to intersect the machined surface in the Cartesian space, and solves the intersection line of the corresponding plane and the machined surface as the tool trajectory during the machining process. Different from the isoparametric method, the constant cross-section method can be used for the generation of machining trajectories of relatively complex surfaces and triangular mesh models. Equal residual height trajectory planning is for the tool trajectory planning during the surface machining process. Taking any surface boundary as the initial machining path, each tool contact in the next trajectory is calculated based on the tool contact on the current trajectory, and the residual height at each point on the surface is the same.

[0100] Existing trajectory planning methods can achieve complete coverage of the trajectory on the workpiece surface during the polishing process, but ignore the linkage relationship between the trajectory and the polishing material removal function during the polishing process, and cannot be used as the optimal solution for the trajectory during the polishing process.

[0101] Therefore, the present invention optimizes the polishing trajectory according to the initial polishing material removal function, referring to Figure 3 As shown, the S2 specifically includes the following steps:

[0102] S21: Establish a spiral initial polishing trajectory with a total number of turns I', the number of trajectory points on each turn is the same, and the polished contact areas of each turn do not overlap, ensuring complete coverage of the workpiece by the polishing trajectory and no over-polishing / under-polishing problems. The non-overlap of the polished contact areas of each turn means that the overlap amount in the major axis direction of the polished contact area is 0.

[0103] S22: Optimize the second loop of trajectories using the first loop of trajectories, and optimize the (i'+1)-th loop of trajectories using the optimized i'-th loop of trajectories, where i' = 2, 3, …, I'-1.

[0104] Specifically, for the first loop of trajectories, the trajectory points with a phase angle ≤ 180° are not adjusted, and the trajectory points with a phase angle > 180° are adjusted based on the first trajectory point; any trajectory point on the first loop of trajectories uses the starting point of the helix as the adjustment reference. For the remaining trajectories, the trajectory points of the (i'+1)-th polished trajectory are adjusted using the trajectory points of the already optimized i'-th trajectory as a reference, that is, a trajectory point with a phase angle difference of a preset angle value from the currently to-be-adjusted trajectory point in the optimized i'-th loop of trajectories is selected as the reference point for the currently to-be-adjusted trajectory point in the (i'+1)-th loop of trajectories. During each optimization process, the adjacent next trajectory is optimized based on the trajectory completed in the previous optimization as a reference, that is, each optimization operation only targets one unoptimized trajectory.

[0105] Preferably, the preset angle value is 360°.

[0106] Refer to Figure 4 As shown, the trajectory points P1(x1, y1, z1) and P2(x2, y2, z2) are respectively the trajectory points on adjacent trajectories with a phase angle difference of 360°, P2 is the currently to-be-adjusted trajectory point, and P1 is the reference point for P2.

[0107] Within the connection interval between the reference point and the to-be-adjusted trajectory point, there must exist a point P such that the PV value between the reference point and this point is less than the PV value between the reference point and any other point within the connection interval.

[0108] Specifically, taking the trajectory points P1(x1, y1, z1) and P2(x2, y2, z2) as an example, S22 includes the following steps:

[0109] S221: Divide the connection line between the reference point and the to-be-adjusted trajectory point into equal parts, and extract multiple equal division points.

[0110] To simplify the calculation difficulty, extract the X coordinates and Y coordinates of the reference point and the adjustment point, and establish a two-dimensional connection line. Divide this two-dimensional connection line into equal parts, and the number of equal division points D1 = c + (i'-1)Δ, where i' is the loop number of the trajectory to which the reference point belongs, c is the initial number of equal division points; Δ is a fixed increment. Since the distance between the (i + 1)-th loop of trajectories and the i-th loop of trajectories will increase after each adjustment, adding the fixed increment Δ ensures the retrieval accuracy.

[0111] Preferably, during the retrieval process, in order to avoid the unknown influence of the (i + 2)-th loop trajectory with too small trajectory spacing on the existing conclusions, a constraint condition is added to the spacing value between the reference point and the equally divided points: a1 < Dis < a1 + a, where a1 is the major axis of the polishing contact area of the reference point, Dis is the spatial distance between the reference point and the equally divided points; a is the major axis of the polishing contact area of the equally divided points.

[0112] Since the coordinates of the reference point P1(x1, y1, z1) and the trajectory point P2(x2, y2, z2) to be adjusted are known, in the XOY plane, the coordinates of the u-th equally divided point on the line connecting the two points can be directly solved by the following formula:

[0113]

[0114] where, x u and y u are the abscissa and ordinate of the u-th equally divided point respectively, x1 and y1 are the abscissa and ordinate of the reference point respectively, and x2 and y2 are the abscissa and ordinate of the adjustment point respectively.

[0115] S222: Calculate the material removal amount functions of the reference point and each equally divided point with the initial polishing material removal function, including:

[0116] Solve the normal vectors of the triangular meshes where the reference point and each equally divided point are located, rotate the initial polishing material removal function according to the normal vectors of the reference point and each equally divided point respectively, and then translate it according to the coordinates of the reference point and each equally divided point to obtain the material removal amount functions of the normal vectors of the reference point and each equally divided point correspondingly.

[0117] Taking the u-th equally divided point as an example, map the u-th equally divided point in the XOY plane back to the mesh surface to obtain point P u (x u , y u , z u ), and solve the normal vector n u of the triangular mesh where the equally divided point P u is located. Rotate the initial removal function according to the normal vector values at the reference point P1 and the equally divided point P u respectively to obtain the function Then rotate the function according to the coordinate values on the respective X, Y, and Z coordinate axes of the reference point P1 and the equally divided point P u respectively to obtain the actual polishing removal function model u at the reference point P1 and the equally divided point P as shown in the following formula:

[0118]

[0119] where the matrix R is the rotation matrix.

[0120] S223: Superimpose the material removal amount function of the equally divided points and the material removal amount function of the reference point, and record the distance value between the minimum inflection point value and the extreme point value of the superimposed result as the PV value.

[0121] Superimpose the material removal amount function of the reference point P1 and the equally divided point P u . The superimposed result is:

[0122]

[0123] where is the material removal amount function of the u-th equally divided point, is the material removal amount function of the reference point.

[0124] Take the partial derivative of the superimposed result H(x, y) with respect to x to obtain the first-order partial derivative H x (x, y), and take the partial derivative of the superimposed result H(x, y) with respect to y to obtain the first-order partial derivative H y (x, y); Take the partial derivative of H x (x, y) with respect to x to obtain the second-order partial derivative H xx (x, y), take the partial derivative of H x (x, y) with respect to y to obtain the second-order partial derivative H xy (x, y), take the partial derivative of H y (x, y) with respect to y to obtain the second-order partial derivative H yy (x, y). Construct the Hessian matrix H e , as shown in the following formula:

[0125]

[0126] Calculate the determinant D’ = det(H e ) = H e H xx H yy -(H xy ) 2 . When there exists a point (x C , y C ) that satisfies H xx (x C , y C )H yy (x C , y C ) - [H xy (x C , y C )] 2 > 0 and H xx (x C , y C ) > 0, the point (x C , yC , H(x C , y C )) is the extreme point of the superposition result H(x, y), where x C and y C are the x - coordinate and y - coordinate of the extreme point respectively, and H(x C , y C ) is the material removal amount of the extreme point.

[0127] When there exists a point (x I , y I ) that satisfies:

[0128]

[0129] The point (x I , y I , H(x I , y I )) is the inflection point of the superposition result H(x, y), where H u (x, y) is the material removal amount function of the equal - division point u, x I and y I are the x - coordinate and y - coordinate of the inflection point respectively, and H(x I , y I ) is the material removal amount of the inflection point.

[0130] Let C1 = |H(x C , y C )|, C2 = min|H(x I , y I )|, then the PV value after polishing between the reference point P1 and the equal - division point P u is PV = C1 - C2.

[0131] S224: Calculate the PV value of each equal - division point, and select the equal - division point corresponding to the minimum PV value as the adjusted trajectory point.

[0132] The calculation of the optimal adjustment point is the problem of solving min(C1 - C2). By comparing the PV value calculation results at the positions of each equal - division point that meet the constraint conditions, the optimal adjustment point can be retrieved.

[0133] S225: Repeat S221 to S224 until the optimization of each trajectory point to be adjusted for the last trajectory is completed. The optimal polishing trajectory obtained based on the optimization of the polishing trajectory with a constant overlap of 0 is shown in Figure 5 , where Figure 5 (a) in it is the polishing trajectory diagram before optimization, Figure 5 (b) in it is the polishing trajectory diagram after optimization.

[0134] This step combines the influence of the material removal function, trajectory, and process parameters on the PV value after polishing. The proposed optimal polishing trajectory optimization algorithm applicable to polishing operations can achieve higher surface form accuracy and lower PV value in a unit processing cycle. Moreover, during the operation of the trajectory optimization algorithm in this step, the optimization direction of the trajectory point positions is consistent, and the optimization result is unique. The trajectory generated after optimization by this algorithm is a polishing trajectory that dynamically covers the surface according to the contact area and material removal depth, which can meet the requirements of rapid convergence of both the surface form accuracy and surface quality during fine polishing.

[0135] S3: Select the dwell points according to the trajectory points of the optimized polishing trajectory; decompose the initial polishing material removal function into the tool influence function of the dwell points and the dwell time relationship formula, and establish a dwell time equation based on the dwell point coordinates and the control point coordinates within the polishing contact area corresponding to the dwell points; use the Lasso regression method to solve the dwell time of the dwell points.

[0136] Currently, the commonly used methods for solving the dwell time are mostly based on the least squares method. However, in this process, the dwell time calculation first requires solving the convolution equation: R * T = ΔZ target . Then, construct the least squares objective function. In this case, when the frequency domain response of the removal function R has a low-pass characteristic (high-frequency attenuation), the condition number of the convolution matrix is extremely large, resulting in extremely unstable solution results and generating negative dwell times (unrealizable solutions) in the actual physical state. At the same time, the dwell time calculation method based on the least squares method lacks the constraints of machine tool dynamics, and the calculation results cannot meet the constraints of the machine tool acceleration a max and jerk j max , leading to process parameters exceeding the dynamic capabilities of the machine tool, and ultimately causing a serious deviation of the actual removal amount from the expected result. In addition, since the complexity of solving the N×N linear system by the traditional least squares method is O(N 3 ), it cannot be applied to the real-time solution of the dwell time during the polishing process of a million-level grid model (such as a 1m aperture mirror).

[0137] Therefore, the present invention proposes a trajectory-based polishing dwell time solving method combined with the Lasso regression method. As shown in Figure 6 , the steps of S3 include:

[0138] S31: Select the dwell points according to the trajectory points of the optimized polishing trajectory.

[0139] The dwell points are all selected from the trajectory points, that is, the dense trajectory points are extracted at equal intervals to obtain the dwell points. As shown in Figure 7 , there are multiple control points within the polishing contact area of the dwell points. The control points are obtained by dividing the surface into a grid, and each grid intersection point is a control point.

[0140] S32: Decompose the initial material removal function of polishing into the tool influence function at the dwell points and the dwell time relationship, and establish the dwell time equation according to the coordinates of the dwell points and the coordinates of the control points within the corresponding polishing contact area of the dwell points.

[0141] Since the initial material removal function of polishing can be decomposed into the result of the convolution calculation of the tool influence function and the dwell time h = TIF ** t, where ** represents the convolution operation; therefore, decompose the initial material removal function of polishing into the tool influence function at the dwell points and the dwell time relationship, and the formula is:

[0142]

[0143] where, TIF i represents the tool influence function of the i-th dwell point, t i represents the dwell time of the i-th dwell point, v ai represents the feed rate of the polishing tool at the i-th dwell point, l i represents the trajectory step size, that is, the control distance between two consecutive trajectory points.

[0144] Establish the dwell time equation for each dwell point and the control points within the polishing contact area, including:

[0145]

[0146] where, N represents the number of dwell points, M represents the number of control points within the polishing contact area corresponding to the dwell points; TIF MN represents the value of the tool influence function of the M-th control point within the polishing contact area corresponding to the N-th dwell point, t N represents the dwell time of the N-th dwell point; T represents the matrix form of the tool influence function, t represents the matrix form of the dwell time, and e represents the residual error.

[0147] Substitute the coordinates of the dwell points or control points into the tool influence function, and the coordinate values of the dwell points or control points affect the integral result of, so the calculated values of the tool influence function are different.

[0148] S33: Use the Lasso regression method to solve the dwell time.

[0149] From the definitions of the dwell points and the control points, it can be seen that the total number of control points is generally M times that of the dwell points, resulting in the dwell time equation being an overdetermined equation. In this case, the equation may not have a solution, and only the optimal solution can infinitely approximate the ideal solution.

[0150] S331: By using the Lasso regression algorithm, construct an objective function using the sum of squared residuals, the L1 norm, and the regularization parameter, further transform the dwell time calculation into an extreme value calculation problem of the objective function, and split the objective function into the sum of squared residuals and the regularization term.

[0151] The formula of the objective function is:

[0152]

[0153] Where, is the sum of squared residuals, representing the gap between the predicted value and the actual value. Minimizing the sum of squared residuals can make the prediction result of the model closer to the actual value; represents the regularization term; ||t||1 represents the L1 norm, which can prompt some coefficients to decrease to zero, thus realizing feature selection; η represents the regularization parameter; when η = 0, the objective function is the equation solved by ordinary least squares method, and when η is relatively large, the objective function tends to be simplified.

[0154] Preferably, when the data set is small, the value range of η is λ ∈ [0.1, 1]; when the data set is large, the value range of η is η ∈ [0.01, 1]; when the data set is extremely large, the value range of η is η ∈ [0.001, 0.1].

[0155] S332: Solve the gradient of the sum of squared residuals The formula is:

[0156]

[0157] S333: According to the calculation result of the gradient , further calculate the Lipschitz constant L of the gradient of the sum of squared residuals, and the formula is:

[0158] L = 2λ max (T T T)

[0159] Where, λ max represents the maximum eigenvalue of the matrix T T T.

[0160] S334: Calculate the gradient descent step size of the sum of squared residuals according to the gradient of the sum of squared residuals and the Lipschitz constant, and the formula is:

[0161]

[0162] Where, z (k) is the gradient descent step size, and α = 1 / L.

[0163] S335: For the L1 regularization term g(t), since it is a non-differentiable term, the proximal operator of the regularization term is solved and decomposed into a differentiable iterative form. The formula is as follows:

[0164] t j =prox ag (z) j =sign(z j )maz(|z j -ηα,0|)

[0165] where t j is the j-th component of the dwell time vector t = [t1, t2,..., t N T ; sign(z j ) is the soft threshold function, and the proximal operator independently adjusts each component t j through the soft threshold operation; if |z j |>ηα, then the parameter t j is shrunk to z j -ηα·sign(z j ); if |z j |≤ηα, then the parameter t j is set to zero.

[0166] S336: Substitute the gradient descent step size of the sum of squared residuals into the differentiable iterative form of the proximal operator of the regularization term, and iteratively solve for the dwell time until the dwell time converges.

[0167] Let t (0) =0 as the initial value, and iteratively solve for the dwell time. The (k + 1)-th iteration is expressed as:

[0168] t (k+1) =sign(z (k) )·max(|z (k) -ηα,0|)

[0169] where t (k+1) is the dwell time of the (k + 1)-th iteration, z (k) represents the gradient descent step size of the sum of squared residuals of the k-th iteration, sign(·) is the soft threshold function, α = 1 / L, and L is the Lipschitz constant of the sum of squared residuals;

[0170] After each iteration, check the change between t (k+1) and t (k) . If |f[t (k+1) -f[t (k) |<ξ, where t (k) is the dwell time of the k-th iteration and ξ is the preset threshold, then the objective function converges, and t (k+1) ​That is the optimal solution; otherwise, continue the iterative solution.

[0171] Preferably, a regularization optimization method is used to smooth the obtained dwell time, including:

[0172] According to the already obtained dwell time, construct a regularization optimization objective function. The known dwell time is t = [t1, t2, …, t N T , assuming the smoothed dwell time Construct the objective function for smoothing:

[0173]

[0174] where, t i and respectively represent the dwell times before and after smoothing at the i-th dwell point, and respectively represent the dwell times after smoothing at the (i - 1)-th and (i + 1)-th dwell points, β represents the smoothing intensity coefficient and β > 0, and N represents the number of dwell points; is the fidelity term, is the smoothing term.

[0175] Introduce a second-order difference matrix to convert the objective function of smoothing into a matrix operation form:

[0176]

[0177] where, D is the second-order difference matrix.

[0178] Take the derivative of the matrix form of the objective function of smoothing and set the derivative to zero, and then a linear equation system (I + αD T D)t s = t can be obtained, where I is the identity matrix.

[0179] Use the Cholesky decomposition method to solve the equation t s = (I + αD T D) -1 t to obtain the smoothed dwell time.

[0180] Figure 8 The figure shows an example diagram of the initial residual error statistical result of the sample to be polished. Figure 9 The figure shows that after polishing is performed using the dwell time obtained by solving with the Lasso regression method, Figure 8 the figure shows the distribution of the PV value on the surface of the sample to be polished as an example. Figure 10 The figure shows that after the dwell time is smoothed using the regularization optimization method, Figure 8 ​Distribution of surface PV values of the sample to be polished after polishing. Although the dwell time after smoothing has significantly changed in terms of numerical mutation compared to the dwell time before smoothing, compared to Figure 9 the results shown, Figure 10 the PV value in the polished sample results shown does not fluctuate significantly. The polishing results under the two dwell times are basically the same, but the dwell time after smoothing is more in line with the kinematic characteristics of the processing equipment.

[0181] In summary, for the polishing trajectory planning method of sub-aperture polishing described in the present invention, the polishing material removal function is first re-solved, and a polishing material removal function model with polishing load, polishing tool rotation speed, feed speed, polishing attitude angle, and polishing tool and workpiece material properties as inputs is established. This not only improves the prediction accuracy of material removal during polishing but also can adapt to the polishing processes of different materials, realizing the controllability of material removal during polishing. Further, the present invention combines the derived polishing material removal function to adjust and optimize each generated initial polishing trajectory one by one, and can obtain a polishing trajectory that dynamically covers the surface according to the contact area and material removal depth, enabling a higher surface accuracy and lower PV value to be obtained within a unit processing cycle, meeting the requirements of rapid convergence of both surface accuracy and surface quality during fine polishing. Moreover, the present invention constructs a dwell time equation based on the derived polishing material removal function and solves it using the Lasso regression method, which not only realizes the controllability of the distribution of material removal amount in the time-space double domain during polishing, effectively reducing the surface shape residual after polishing, but also avoids the periodic fluctuation problem of the removal amount caused by convolution transfer in the conventional dwell time solving algorithm, improving the accuracy and stability of surface polishing.

[0182] Further, the present invention introduces a regularization optimization method to smooth the obtained dwell time, making the finally obtained dwell time more in line with the kinematic characteristics of the equipment during the actual polishing process.

[0183] The present invention has good universality. Whether it is a plane, a surface with a known mathematical equation, or a complex surface that is difficult to describe with a specific mathematical equation, the method provided by the present invention can be used for accurate prediction of material removal, generation of the optimal trajectory, and solution of the dwell time during the polishing process.

[0184] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

[0185] The present application is described with reference to the flowcharts and / or block diagrams of methods, apparatuses (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram can be implemented by computer program instructions, and the combination of the flows and / or blocks in the flowchart and / or block diagram can also be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0186] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory generate a manufactured article including instruction means, and the instruction means implements the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0187] These computer program instructions can also be loaded onto a computer or other programmable data processing device, so that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process. Thus, the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in Figure 1 one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0188] Obviously, the above embodiments are only examples for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or variations can be made based on the above description. It is not necessary and impossible to enumerate all the implementation manners here. And the obvious changes or variations derived therefrom are still within the protection scope of the present invention.

Claims

1. A polishing trajectory planning method for sub-aperture polishing, characterized in that, Including: Calculating the relative linear velocity of any point within the polishing contact area based on the curvature radius, elastic modulus, Poisson's ratio, feed rate, attitude angle, angular velocity, and polishing load of the polishing tool; Taking the ratio of the Archard wear coefficient of the workpiece to be polished to the hardness of the workpiece material as the Preston coefficient in the Preston empirical equation, and substituting the relative linear velocity of a point within the polishing contact area into the Preston empirical equation to obtain the initial polishing material removal function; Establishing a spiral initial polishing trajectory with a total number of turns of I', where the number of trajectory points on each turn is the same, and The polishing contact areas of each turn of the trajectory do not overlap; Optimizing the second turn of the trajectory with the first turn of the trajectory, Optimizing the (i'+1)-th turn of the trajectory with the optimized i'-th turn of the trajectory, i' = 2, 3, …, I' - 1, including: Selecting a trajectory point with a phase angle difference of a preset angle value from the current trajectory point to be adjusted in the optimized i'-th turn of the trajectory as the reference point for the current trajectory point to be adjusted in the (i'+1)-th turn of the trajectory; Dividing the line connecting the reference point and the current trajectory point to be adjusted into equal parts, and extracting multiple equally divided points; calculating the material removal amount functions of the reference point and each equally divided point using the initial polishing material removal function; Superposing the material removal amount functions of the equally divided points with the material removal amount function of the reference point, and recording the distance value between the minimum inflection point value and the extreme point value of the superposition result as the PV value; Calculating the PV value of each equally divided point, and selecting the equally divided point corresponding to the minimum PV value as the adjusted trajectory point; Selecting dwell points according to the trajectory points of the optimized polishing trajectory; decomposing the initial polishing material removal function into the tool influence function and the dwell time relationship of the dwell points, and establishing a dwell time equation based on the dwell point coordinates and the control point coordinates within the polishing contact area corresponding to the dwell points; using the Lasso regression method to solve the dwell time of the dwell points.

2. The method for planning a polishing trajectory for sub-aperture polishing according to claim 1, wherein Calculating the relative linear velocity of any point within the polishing contact area based on the curvature radius, elastic modulus, Poisson's ratio, feed rate, attitude angle, angular velocity, and polishing load of the polishing tool, including: Calculating the deformation amount of the polishing tool based on the curvature radius, elastic modulus, Poisson's ratio, and polishing load of the polishing tool, with the formula: Where ρ is the curvature radius of the polishing tool, E1 is the elastic modulus of the polishing tool, μ1 is the Poisson's ratio of the polishing tool, and Q is the polishing load; Calculating the actual downward pressure amount of the polishing tool based on the curvature radius and deformation amount of the polishing tool, with the formula: Among them, OC t represents the actual downward pressure of the polishing tool; Taking the center of the polishing contact area as the origin and the direction of the feed rate of the polishing tool as the x-axis direction to establish a coordinate system, and calculating the relative linear velocity of any point within the polishing contact area based on the actual downward pressure amount of the polishing tool, with the formula: Among them, v s represents the relative linear velocity of the coordinate point (x k , y k ) within the polishing contact area, v a is the feed rate of the polishing tool, v ax and v ay are the velocity components of v a along the x-axis and y-axis respectively; ω is the angular velocity of the polishing tool; σ and λ are the attitude angles of the polishing tool, where σ is the polishing inclination angle and λ is the polishing declination angle.

3. A method for planning a polishing trajectory of sub-aperture polishing according to claim 2, characterized in that The formula for the initial polishing material removal function is: where h is the initial polishing material removal function, k abr is the Archard wear coefficient of the workpiece to be polished, H v is the hardness of the workpiece material, v a is the feed rate of the polishing tool, Q is the polishing load, σ is the polishing inclination angle, ω is the angular velocity of the polishing tool; E(·) is the integral symbol of the second kind of ellipse; a and b are the major axis and minor axis of the polishing contact area respectively.

4. A polishing trajectory planning method for sub-aperture polishing according to claim 1, characterized in that, Dividing the line connecting the reference point and the current trajectory point to be adjusted into equal parts and extracting the equally divided points, including: The number of equally divided points is D1 = c + (i' - 1)Δ, where i' is the number of turns of the trajectory to which the reference point belongs, c is the number of initial equally divided points; Δ is a fixed increment; Add a constraint to the spacing value between the reference point and the equally divided points: a1 < Dis < a1 + a, where a1 is the major axis of the polishing contact area of the reference point, Dis is the spatial distance between the reference point and the equally divided points; a is the major axis of the polishing contact area of the equally divided points.

5. A polishing trajectory planning method for sub-aperture polishing according to claim 1, characterized in that Calculate the material removal amount functions of the reference point and each equally divided point using the initial polishing material removal function, including: Solve the normal vectors of the triangular meshes where the reference point and each equally divided point are located, rotate the initial polishing material removal function according to the normal vectors of the reference point and each equally divided point respectively, and then translate it according to the coordinates of the reference point and each equally divided point to obtain the material removal amount functions of the normal vectors of the reference point and each equally divided point correspondingly.

6. A polishing trajectory planning method for sub-aperture polishing according to claim 1, characterized in that Overlay the material removal amount function of the equally divided points with the material removal amount function of the reference points, and denote the overlay result as H(x, y); take the partial derivative of the overlay result H(x, y) with respect to x to obtain the first-order partial derivative H x (x, y), and take the partial derivative of the overlay result H(x, y) with respect to y to obtain the first-order partial derivative H y (x, y); H x (x, y) take the partial derivative with respect to x to obtain the second-order partial derivative H xx (x, y), H x (x, y) take the partial derivative with respect to y to obtain the second-order partial derivative H xy (x, y), H y (x, y) take the partial derivative with respect to y to obtain the second-order partial derivative H yy (x, y); When there exists a point (x C , y C ) that satisfies H xx (x C , y C )H yy (x C , y C ) - [H xy (x C , y C )] 2 > 0 and H xx (x C , y C ) > 0, the point (x C , y C , H(x C , y C )) is an extreme point of the superposition result H(x, y), where x C and y C are the x - coordinate and y - coordinate of the extreme point respectively, and H(x C , y C ) is the material removal amount of the extreme point; When there exists a point (x I , y I ) that satisfies: The point (x I , y I , H(x I , y I )) is the inflection point of the superposition result H(x, y), where is the material removal amount function of the u-th equally divided point, x I and y I are the x-coordinate and y-coordinate of the inflection point respectively, and H(x I , y I ) is the material removal amount of the inflection point.

7. A method for planning a polishing trajectory of sub-aperture polishing according to claim 3, characterized in that Decompose the initial polishing material removal function into the tool influence function at the dwell points and the dwell time relationship, including: where, TIF i represents the tool influence function at the i-th dwell point, t i represents the dwell time at the i-th dwell point, v ai represents the feed rate of the polishing tool at the i-th dwell point, l i represents the trajectory step size; Establish the dwell time equation based on the dwell point coordinates and the control point coordinates within the corresponding polishing contact area of the dwell points, including: Where N represents the number of dwell points, and M represents the number of control points within the polishing contact area corresponding to the dwell points; TIF MN represents the tool influence function value of the Mth control point within the polishing contact area corresponding to the Nth dwell point, and t N represents the dwell time of the Nth dwell point; T represents the matrix form of the tool influence function, t represents the matrix form of the dwell time, and e represents the residual error.

8. A polishing trajectory planning method for sub-aperture polishing according to claim 7, characterized in that Solve the dwell time of the dwell points using the Lasso regression method, including: Construct the objective function using the Lasso regression algorithm and split the objective function into the sum of squared residuals and the regularization term, expressed as: Among them, is the sum of squared residuals, \(g(t)=\eta||t||_1\) represents the regularization term, \(\eta\) represents the regularization parameter, and \(||t||_1\) represents the L1 norm; Solve the gradient of the sum of squared residuals and calculate the gradient of the sum of squared residuals and the Lipschitz constant; calculate the gradient descent step size of the sum of squared residuals based on the gradient of the sum of squared residuals and the Lipschitz constant; Solve the proximal operator of the regularization term and decompose it into a differentiable iterative form; Substitute the gradient descent step size of the sum of squared residuals into the differentiable iterative form of the proximal operator of the regularization term, Iteratively solve the dwell time until the dwell time converges.

9. A method for planning a polishing trajectory of sub-aperture polishing according to claim 8, characterized in that, Substitute the gradient descent step size of the sum of squared residuals into the differentiable iterative form of the proximal operator of the regularization term, and the (k + 1)-th iteration is expressed as: t (k+1) = sign(z (k) )·max(|z (k) - ηα, 0|) where t (k+1) is the residence time of the (k + 1)-th iteration, z (k) represents the gradient descent step size of the sum of squared residuals of the k-th iteration, sign(·) is the soft threshold function, α = 1 / L, and L is the Lipschitz constant of the sum of squared residuals; If |f[t (k+1) -f[t (k) | < ξ, where t (k) is the dwell time of the k-th iteration and ξ is a preset threshold, then the objective function converges and t (k+1) is the optimal solution; otherwise, continue the iterative solution.

10. A method for planning a polishing trajectory of sub-aperture polishing according to claim 9, characterized in that Smooth the obtained dwell time using the regularization method, including: Construct the objective function for smoothing: where t i and represent the residence times before and after smoothing of the i-th residence point respectively, and represent the residence times after smoothing of the (i - 1)-th and (i + 1)-th residence points respectively, β represents the smoothing intensity coefficient, and N represents the number of residence points; Introduce the second-order difference matrix to convert the objective function for smoothing into a matrix operation form and solve it using the Cholesky decomposition method to obtain the smoothed dwell time.

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