A curved surface polishing track uniform covering method, device and readable storage medium

Abstract

The present application belongs to the technical field of polishing trajectory planning, and relates to a curved surface polishing trajectory uniform covering method, device and readable storage medium: the I-ring plane polishing trajectory of a curved surface to be processed is mapped to a grid model to obtain an I-ring curved surface trajectory; a reference point is selected in the first ring curved surface trajectory, each trajectory point in the first ring curved surface trajectory is adjusted until the Euclidean distance between each trajectory point and the reference point is equal to a target distance, and the first polishing trajectory is obtained; i is initialized as 2; for each trajectory point in the i-ring curved surface trajectory, a corresponding reference point is selected in the (i-1)-th polishing trajectory, and each trajectory point is adjusted until the Euclidean distance between each trajectory point in the i-ring curved surface trajectory and the corresponding reference point is equal to the target distance, and the i-th polishing trajectory is obtained; i is updated as i+1, and the trajectory points in the i-ring curved surface trajectory are adjusted until i=I, and the target polishing trajectory is obtained.

Description

Technical Field

[0001] This invention relates to the field of polishing trajectory planning technology, and in particular to a method, apparatus and computer-readable storage medium for uniformly covering a curved surface polishing trajectory. Background Technology

[0002] Complex curved surface parts have wide applications in aerospace, automotive, and other fields. High-precision machining of curved surfaces to obtain high-precision, high-quality parts is crucial. During curved surface machining, polishing trajectory planning is closely related to machining accuracy and efficiency. Good trajectory planning ensures that the polishing tool uniformly covers and adapts to areas with different curvatures and shapes throughout the machining process, thereby improving the accuracy of the curved surface parts. Therefore, curved surface polishing trajectory planning is an important guarantee for high-precision machining of curved surfaces.

[0003] Existing polishing trajectory planning methods mainly include the isoparametric method, the isosection line method, and the projection method. The isoparametric method parameterizes the surface to be processed using parameters u and v, which typically vary within a finite interval. Different values ​​of u and v within this interval generate points on the surface. Given the equation of the surface, the isoparametric method extracts one parameter from u and v in any direction and substitutes it into the surface equation to solve for the other parameter, thus directly generating the polishing trajectory. The isosection line method works by using a set of parallel planes intersecting the surface in Cartesian space. The polishing trajectory is obtained by solving for the intersection line between the corresponding plane and the surface to be processed. In the actual planning process, it is necessary to first obtain the surface equation of the surface to be processed, and then solve a set of plane equations and the surface equation simultaneously to obtain the coordinate points on the intersection line of the plane and the surface as trajectory points. The equal residual height method selects a region in the surface to be processed and extracts the boundary points of the region as initial data. From the selected initial data, a series of equally spaced plane trajectories are generated according to the coordinate values ​​of two directions. Then, the generated plane trajectories are solved simultaneously with the surface equation to determine the corresponding trajectory points on the surface, thus obtaining the polishing trajectory. These polishing trajectory planning methods all require obtaining the surface equation of the surface to be processed first. However, for some complex surfaces, their mathematical properties are difficult to obtain, making it impossible to establish the corresponding surface equation. Therefore, for some complex surfaces for which the surface equation cannot be obtained, the existing methods cannot generate the corresponding polishing trajectory, thus making it impossible to process the surface to be processed.

[0004] In summary, existing surface polishing trajectory planning methods all require the surface equation of the surface to be processed to generate the polishing trajectory. For complex surfaces whose surface equation cannot be obtained, the corresponding polishing trajectory cannot be generated, thus making it impossible to process the surface to be processed. Summary of the Invention

[0005] Therefore, the technical problem to be solved by the present invention is to overcome the problem that the existing surface polishing trajectory planning methods all require the surface equation of the surface to be processed to generate the polishing trajectory. For complex surfaces whose surface equation cannot be obtained, the corresponding polishing trajectory cannot be generated, which leads to the inability to process the surface to be processed.

[0006] To solve the above technical problems, the present invention provides a method for uniformly covering the polishing trajectory of a curved surface, comprising:

[0007] S10: Discretize the surface to be processed to obtain a mesh model of the surface to be processed;

[0008] S20: Obtain the I-circle planar polishing trajectory of the surface to be processed, and map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory;

[0009] S30: Select a reference point in the first circular surface trajectory, and adjust each trajectory point in the first circular surface trajectory until the Euclidean distance between each trajectory point in the first circular surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first circular surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first circular surface trajectory is taken as the reference point; if the first circular surface trajectory is an asymmetrical trajectory, the first trajectory point in the first circular surface trajectory is taken as the reference point;

[0010] S40: Initialize i = 2;

[0011] S50: For each trajectory point in the i-th round of the surface trajectory, select the trajectory point with the closest Euclidean distance in the (i-1)-th polishing trajectory as the reference point, and adjust each trajectory point in the i-th round of the surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory.

[0012] S60: Update i = i + 1, and return to execute step S50 until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories;

[0013] The target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

[0014] Preferably, step S10 further includes: extracting the vertex coordinates of each triangular mesh in the mesh model, and calculating the area and unit normal vector of each triangular mesh.

[0015] Preferably, the calculation steps for the polishing contact circle radius at the trajectory point include:

[0016] Step 1: Use the triangular mesh containing the trajectory point as the target triangular mesh, and initialize m = 1;

[0017] Step 2: Take the m-th vertex in the target triangle mesh as the target vertex, and calculate the average unit normal vector of the target vertex based on the area and unit normal vector of the triangle mesh with the target vertex as the common vertex in the mesh model;

[0018] Step 3: Obtain the tangent plane of the surface to be processed at the target vertex, and based on the average unit normal vector of the target vertex, calculate the unit tangent vector of the curve between any vertex in the target triangle mesh other than the target vertex and the target vertex onto the tangent plane;

[0019] Step 4: Based on the average unit normal vector of the target vertex, the curve between any vertex in the target triangle mesh other than the target vertex and the target vertex, and the coordinates of the target vertex, calculate the normal curvature of the target vertex along the direction of the unit tangent vector;

[0020] Step 5: Construct a curvature tensor based on the average unit normal vector of the target vertex and the normal curvature of the target vertex along the direction of the unit tangent vector; and calculate the Gaussian curvature and average curvature at the target vertex based on the first and second eigenvalues ​​of the curvature tensor.

[0021] Step 6: Calculate the principal curvature and secondary curvature of the surface at the target vertex based on the Gaussian curvature and mean curvature at the target vertex;

[0022] Step 7: Update m = m + 1, and return to execute step 2 until m = 3, to obtain the principal curvature and secondary curvature of the surface at each vertex in the target triangular mesh;

[0023] Step 8: Based on the area of ​​the sub-triangle formed by the trajectory point and each vertex in the target triangle mesh, the area of ​​the target triangle mesh, and the principal curvature and secondary curvature of the surface at each vertex in the target triangle mesh, calculate the principal curvature and secondary curvature of the surface at the trajectory point.

[0024] Step 9: Calculate the radius of the polishing contact circle at the trajectory point based on the principal curvature and secondary curvature of the surface at the trajectory point, and the principal curvature and secondary curvature of the polishing tool.

[0025] Preferably, the formula for calculating the average unit normal vector of the target vertex is:

[0026]

[0027] Where n represents the average unit normal vector of the target vertex; A gN represents the area of ​​the g-th triangular mesh that shares a common vertex with the target vertex; g This represents the unit normal vector of the g-th triangular mesh that shares a common vertex with the target vertex;

[0028] The formula for calculating the unit tangent vector on the tangent plane by projecting the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex is as follows:

[0029]

[0030] Where t represents the unit tangent vector projected onto the tangent plane from the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex; OV represents the vector formed by the target vertex O and the vertex V in the target triangular mesh;

[0031] The formula for calculating the normal curvature of the target vertex along the direction of the unit tangent vector is:

[0032]

[0033] Where, k n (t) represents the normal curvature of the target vertex along the direction of the unit tangent vector t; r(c) represents the curve between the target vertex O and the vertex V in the target triangular mesh;

[0034] The curvature tensor is represented as:

[0035] L = k n (t)(I-nn T ),

[0036] Where L represents the curvature tensor; I represents the identity matrix; and T represents the transpose of the matrix.

[0037] The formulas for calculating the Gaussian curvature and mean curvature at the target vertex are as follows:

[0038]

[0039] Among them, K G Let λ0 represent the Gaussian curvature at the target vertex; H represents the average curvature at the target vertex; λ1 represents the first eigenvalue of the curvature tensor; λ2 represents the second eigenvalue of the curvature tensor.

[0040] The formulas for calculating the principal curvature and secondary curvature of the surface at the target vertex are as follows:

[0041]

[0042] Where k1 represents the principal curvature of the surface at the target vertex; k2 represents the secondary curvature of the surface at the target vertex;

[0043] The formulas for calculating the principal curvature and secondary curvature of the surface at the trajectory point are as follows:

[0044]

[0045] Where K1 represents the principal curvature of the surface at the trajectory point; K2 represents the secondary curvature of the surface at the trajectory point; τ A This represents the ratio of the area of ​​the sub-triangle formed by vertices A and B in the target triangle mesh to the area of ​​the target triangle mesh; k 1A τ represents the principal curvature of the surface at vertex A in the target triangular mesh; B This represents the ratio of the area of ​​the sub-triangle formed by vertices B and C in the target triangle mesh to the area of ​​the target triangle mesh; k 1B τ represents the principal curvature of the surface at vertex B in the target triangular mesh; C This represents the ratio of the area of ​​the sub-triangle formed by vertices C and A in the target triangle mesh to the area of ​​the target triangle mesh; k 1C k represents the principal curvature of the surface at vertex C in the target triangular mesh. 2A k represents the surface sub-curvature at vertex A in the target triangular mesh. 2B k represents the surface sub-curvature at vertex B in the target triangular mesh. 2C This represents the surface sub-curvature at vertex C in the target triangular mesh;

[0046] The formula for calculating the radius of the polishing contact circle at the trajectory point is:

[0047] a=(3h 2 εQR / πE) 1 / 3 ,

[0048] Where a represents the radius of the polishing contact circle at the trajectory point; h represents the first empirical coefficient; ε represents the second empirical coefficient; Q represents the polishing force applied to the surface to be processed; R represents the curvature coefficient, which is equal to the reciprocal of the sum of the principal curvature, secondary curvature of the polishing tool, the principal curvature of the surface at the trajectory point, and the negative curvature of the surface; and E represents the equivalent elastic modulus.

[0049] Preferably, the formula for calculating the target distance is:

[0050] D = |a m -a n -J|,

[0051] Where D represents the target distance; a m Indicates the radius of the polishing contact circle at trajectory point m; a n J represents the radius of the polishing contact circle at the reference point n corresponding to trajectory point m; J represents the preset polishing trajectory overlap amount.

[0052] Preferably, adjusting the trajectory points includes:

[0053] Calculate the Euclidean distance between the trajectory point and the reference point, and determine whether the Euclidean distance is equal to the target distance between the trajectory point and the reference point;

[0054] If the Euclidean distance is not equal to the target distance, then a plane is spanned based on the line connecting the trajectory point and the reference point, and the unit normal vector of the triangle mesh containing the trajectory point;

[0055] Select a target point inside or on the extension of the line connecting the trajectory point and the reference point. Generate a circle in the plane with the reference point as the center and the distance between the target point and the reference point as the radius.

[0056] Obtain all intersection points between the generated circle and the surface to be processed, and take the intersection point with the smallest Euclidean distance to the trajectory point as the target trajectory point.

[0057] Preferably, after selecting the intersection point with the smallest Euclidean distance to the trajectory point as the target trajectory point, the method further includes:

[0058] Calculate the polishing contact radius of the target trajectory point, the Euclidean distance between the target trajectory point and the reference point, and the target distance between the target trajectory point and the reference point;

[0059] Determine whether the Euclidean distance between the target trajectory point and the reference point is equal to the target distance between the target trajectory point and the reference point. If not, readjust the target trajectory point.

[0060] Preferably, selecting a target point within or on the extension of the line connecting the trajectory point and the reference point includes:

[0061] If the Euclidean distance between the trajectory point and the reference point is greater than the target distance between the trajectory point and the reference point, then a target point is selected on the extension line connecting the trajectory point and the reference point;

[0062] If the Euclidean distance between the trajectory point and the reference point is less than the target distance between the trajectory point and the reference point, then a target point is selected within the line connecting the trajectory point and the reference point.

[0063] The present invention also provides a device for uniformly covering the polishing trajectory of a curved surface, comprising:

[0064] The surface discretization module is used to discretize the surface to be processed, and obtain the mesh model of the surface to be processed;

[0065] The trajectory mapping module is used to obtain the I-circle planar polishing trajectory of the surface to be processed, and to map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory.

[0066] The first trajectory adjustment module is used to select a reference point in the first-round curved surface trajectory and adjust each trajectory point in the first-round curved surface trajectory until the Euclidean distance between each trajectory point in the first-round curved surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first-round curved surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first-round curved surface trajectory is used as the reference point; if the first-round curved surface trajectory is an asymmetrical trajectory, the first trajectory point in the first-round curved surface trajectory is used as the reference point;

[0067] The data initialization module is used to initialize i = 2;

[0068] The second trajectory adjustment module is used to select the trajectory point with the closest Euclidean distance to each trajectory point in the i-th round of the surface trajectory as the reference point in the (i-1)th polishing trajectory, and adjust each trajectory point in the i-th round of the surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory.

[0069] The target polishing trajectory acquisition module is used to update i = i + 1 and return to the step of executing the second trajectory adjustment module until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories; wherein, the target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

[0070] The present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the above-described method for uniformly covering curved surface polishing trajectories.

[0071] The method for uniformly covering the polishing trajectory of a curved surface provided in this application first discretizes the surface to be processed to obtain a mesh model of the surface. Then, the trajectory points on the pre-generated planar trajectory are mapped to the mesh model to obtain the surface trajectory. Since the surface to be processed has complex curvature and shape, the planar trajectory points cannot be fully adapted to the concavity and curvature of the surface after being mapped to the surface model. Therefore, after obtaining the surface trajectory, this application first selects a reference point in the first circle of the surface trajectory and adjusts each trajectory point in the first circle of the trajectory so that the Euclidean distance between each trajectory point and the reference point is equal to the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point, and the pre-set overlap amount of the polishing trajectory. The absolute value of the value is used to adapt the trajectory points to the concavity and curvature of the surface. After obtaining the first polishing trajectory, for any trajectory point on the second surface trajectory, the trajectory point with the closest Euclidean distance on the first polishing trajectory is used as the reference point. The trajectory points on the second surface trajectory are then adjusted to obtain the second polishing trajectory. This process is repeated to complete the adjustment of all surface trajectories, so that the final target polishing trajectory uniformly covers the surface to be processed, enabling high-precision polishing of the surface. This application directly maps the planar trajectory onto the surface model and adjusts the position of the trajectory points, without needing to obtain the surface equation of the surface to be processed, thus realizing the generation of surface polishing trajectories without surface equations. Attached Figure Description

[0072] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings, wherein:

[0073] Figure 1 Flowchart of the method for uniform coverage of curved surface polishing trajectory provided in this application;

[0074] Figure 2 A schematic diagram illustrating the mapping of the planar polishing trajectory provided in this application to a mesh model; wherein, Figure 2 (a) in the diagram illustrates the mapping principle of mapping trajectory points on the planar polishing trajectory to the mesh model. Figure 2 (b) in the diagram is a schematic diagram of the mapping result of the trajectory points on the planar polishing trajectory to the mesh model;

[0075] Figure 3 A schematic diagram illustrating the calculation principle of the average normal vector and normal curvature at each triangular mesh vertex on the mesh model provided in this application; wherein... Figure 3 (a) in the diagram is a schematic diagram illustrating the principle of calculating the average normal vector at the vertices of a triangular mesh. Figure 3 (b) in the diagram is a schematic diagram illustrating the principle of calculating the normal curvature of the vertices of the triangular mesh along the unit tangent vector direction;

[0076] Figure 4This is a schematic diagram illustrating the principle of determining the adjustment range and direction of trajectory points on a curved surface trajectory, as provided in this application; wherein... Figure 4 (a) in the diagram illustrates the principle of trajectory point adjustment amplitude and direction determination when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is greater than 0. Figure 4 (b) in the figure is a schematic diagram of the trajectory point adjustment range and direction determination principle when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is less than 0;

[0077] Figure 5 This application provides a schematic diagram illustrating the principle of target trajectory point location retrieval; wherein, Figure 5 (a) in the diagram illustrates the principle of target trajectory point location retrieval when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is greater than 0. Figure 5 (b) in the figure is a schematic diagram of the target trajectory point position retrieval principle when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is less than 0;

[0078] Figure 6 This is a schematic diagram showing the uniform coverage polishing trajectory results for different trajectory overlap amounts after adjusting the trajectory points on the curved surface trajectory provided in this application; wherein, Figure 6 (a) in the figure is a schematic diagram of the uniform coverage polishing trajectory result obtained for an asymmetric trajectory with zero trajectory overlap. Figure 6 (b) in the diagram is a schematic diagram of the uniform coverage polishing trajectory result obtained for an asymmetric trajectory with a trajectory overlap of 1. Figure 6 (c) in the diagram is a schematic diagram of the uniform coverage polishing trajectory result obtained for a symmetrical trajectory with zero trajectory overlap. Figure 6 (d) in the figure is a schematic diagram of the uniform coverage polishing trajectory result obtained for a symmetrical trajectory with a trajectory overlap of 1;

[0079] Figure 7 A schematic diagram illustrating the process of uniformly covering the curved surface polishing trajectory provided in this application;

[0080] Figure 8 A schematic diagram of the device for uniformly covering the curved surface polishing trajectory provided in this application. Detailed Implementation

[0081] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0082] Please refer to Figure 1 , Figure 1 The flowchart of the method for uniformly covering the polishing trajectory of a curved surface provided in this application is shown. The method specifically includes:

[0083] S10: Discretize the surface to be processed to obtain a mesh model of the surface to be processed;

[0084] S20: Obtain the I-circle planar polishing trajectory of the surface to be processed, and map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory;

[0085] S30: Select a reference point in the first circular surface trajectory, and adjust each trajectory point in the first circular surface trajectory until the Euclidean distance between each trajectory point in the first circular surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first circular surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first circular surface trajectory is taken as the reference point; if the first circular surface trajectory is an asymmetrical trajectory, the first trajectory point in the first circular surface trajectory is taken as the reference point;

[0086] Preferably, the first-round surface trajectory is the surface trajectory closest to the center of the surface to be processed; in some embodiments of this application, when adjusting the first-round surface trajectory, if the surface trajectory is a symmetrical trajectory, the trajectory point located at the center of symmetry in the first-round surface trajectory is taken as the reference point; if the surface trajectory is an asymmetrical trajectory, the first generated trajectory point is taken as the reference point, wherein the first generated trajectory point is the first trajectory point in the first-round surface trajectory;

[0087] S40: Initialize i = 2;

[0088] S50: For each trajectory point in the i-th round of the surface trajectory, select the trajectory point with the closest Euclidean distance in the (i-1)-th polishing trajectory as the reference point, and adjust each trajectory point in the i-th round of the surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory.

[0089] S60: Update i = i + 1, and return to execute step S50 until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories;

[0090] The target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

[0091] The method for uniformly covering the polishing trajectory of a curved surface provided in this application first discretizes the surface to be processed to obtain a mesh model of the surface. Then, the trajectory points on the pre-generated planar trajectory are mapped to the mesh model to obtain the surface trajectory. Since the surface to be processed has complex curvature and shape, the planar trajectory points cannot be fully adapted to the concavity and curvature of the surface after being mapped to the surface model. Therefore, after obtaining the surface trajectory, this application selects a reference point in the first round of the surface trajectory and adjusts each trajectory point in the first round of the trajectory so that the Euclidean distance between each trajectory point and the reference point is equal to the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point, and the pre-set overlap amount of the polishing trajectory. The absolute value of the value is used to adapt the trajectory points to the concavity and curvature of the surface. After obtaining the first polishing trajectory, for any trajectory point on the second surface trajectory, the trajectory point with the closest Euclidean distance on the first polishing trajectory is used as the reference point. The trajectory points on the second surface trajectory are then adjusted to obtain the second polishing trajectory. This process is repeated to complete the adjustment of all surface trajectories, so that the final target polishing trajectory uniformly covers the surface to be processed, enabling high-precision polishing of the surface. This application directly maps the planar trajectory onto the surface model and adjusts the position of the trajectory points, without needing to obtain the surface equation of the surface to be processed, thus realizing the generation of surface polishing trajectories without surface equations.

[0092] Specifically, the mesh model obtained after discretizing the surface to be processed is a discrete set of several triangular meshes, represented as: S={T1,T1,...,T w}, w∈N + In some embodiments of this application, after obtaining the mesh model of the surface to be processed in step S10, the method further includes: extracting the vertex coordinates of each triangular mesh in the mesh model, and calculating the area and unit normal vector of each triangular mesh.

[0093] For example, the vertex coordinates of each triangular mesh are represented as: P w1 (x w1 ,y w1 ,z w1 ),P w2 (x w2 ,y w2 ,z w2 ),P w3 (x w3 ,y w3 ,z w3 );

[0094] Specifically, the formula for calculating the unit normal vector of a triangular mesh is:

[0095]

[0096] Optionally, in some embodiments, the unit normal vector of each triangular mesh is decomposed along the spatial coordinate direction to obtain the normal vector components r on each coordinate axis. x r y r z This allows us to obtain the mathematical model of the surface to be processed:

[0097] S=∑ w r wx x+r wy y+r wz z+d w .

[0098] Specifically, in step S20, the I-circle planar polishing trajectory of the surface to be processed can be generated as needed by any planar surface equation. Each circle of planar polishing trajectory consists of multiple trajectory points, and the plane where the trajectory points are located is usually selected as a plane perpendicular to the processing direction.

[0099] For example, such as Figure 2 The diagram shown is a schematic representation of the planar polishing trajectory mapped to the mesh model provided in this application. Figure 2 (a) in the diagram illustrates the mapping principle of mapping trajectory points on the planar polishing trajectory to the mesh model. Figure 2 (b) in the diagram is a schematic diagram of the mapping result of the trajectory points on the planar polishing trajectory to the mesh model;

[0100] Optionally, during the mapping process, the triangular mesh can be projected onto the plane containing the trajectory points in the opposite direction of the mapping direction to check whether the triangular mesh is parallel to the mapping direction, causing the trajectory points on the plane polishing trajectory to fail to be mapped into the mesh model; specifically, the vertices P of the triangular mesh are... w1 P w2 P w3 The vertex P′ is obtained by projecting the mapping vector m in the opposite direction onto the plane containing the planar polishing trajectory. w1 、P′ w2 、P′ w3 :

[0101]

[0102] Define vectors U and V, where U = P′ w1 -P′ w2 V = P' w1 -P′ w3 If U×V≠0, it indicates that the triangular mesh is a valid triangular mesh in the direction of the mapping vector m;

[0103] After determining that the triangular mesh is valid in the direction of the mapping vector m, for any trajectory point P on the planar polishing trajectory, if the vector satisfy: This demonstrates that the trajectory point P can be mapped to the triangular mesh.

[0104] Furthermore, mapping the trajectory point P to a triangular mesh yields the trajectory point P on the surface trajectory. M P M The coordinates are represented as:

[0105]

[0106] Where, r x r y r z The m represents the normal vector components of the unit normal vector r of the triangular mesh along each coordinate axis; x m y m z The coordinates represent the normal vector components of the mapping vector m on each coordinate axis; x0, y0, and z0 represent the coordinates of the trajectory point P; x w1 y w1 y w1 This represents the coordinates of a vertex of a triangular mesh.

[0107] Specifically, in some embodiments of this application, the calculation process for the radius of the polishing contact circle at the trajectory point includes:

[0108] Step 1: Use the triangular mesh containing the trajectory point as the target triangular mesh, and initialize m = 1;

[0109] Step 2: Take the m-th vertex in the target triangle mesh as the target vertex, and calculate the average unit normal vector of the target vertex based on the area and unit normal vector of the triangle mesh with the target vertex as the common vertex in the mesh model;

[0110] Specifically, the formula for calculating the average unit normal vector of the target vertex is:

[0111]

[0112] Where n represents the average unit normal vector of the target vertex; A g N represents the area of ​​the g-th triangular mesh that shares a common vertex with the target vertex; g This represents the unit normal vector of the g-th triangular mesh that shares a common vertex with the target vertex;

[0113] Step 3: Obtain the tangent plane of the surface to be processed at the target vertex, and based on the average unit normal vector of the target vertex, calculate the unit tangent vector of the curve projection between any vertex in the target triangular mesh other than the target vertex and the target vertex onto the tangent plane;

[0114] Specifically, the formula for calculating the unit tangent vector projected onto the tangent plane from the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex is as follows:

[0115]

[0116] Where t represents the unit tangent vector projected onto the tangent plane from the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex; OV represents the vector formed by the target vertex O and the vertex V in the target triangular mesh;

[0117] Step 4: Based on the average unit normal vector of the target vertex, the curve between any vertex in the target triangle mesh other than the target vertex and the target vertex, and the coordinates of the target vertex, calculate the normal curvature of the target vertex along the unit tangent vector direction;

[0118] Specifically, for the curve r(c) between any vertex in the target triangular mesh and the target vertex (excluding the target vertex), r(0) = 0, r′(0) = t, r″(0) = k n (t), where k n (t) represents the normal curvature of the target vertex along the direction of the unit tangent vector t. When the curve r(c) has a fixed length, r′(c) is always perpendicular to r(c). Therefore, the following formula is used to perform a Taylor expansion on the surface expression r = r(c):

[0119]

[0120] o(C 3 ),

[0121] Further transformation of the above equation yields: Furthermore, we can obtain:

[0122]

[0123] According to the curvature k of the same curve n The inner product of (t)n and the tangent t is always 0, therefore:

[0124] [r(c)-O][r(c)-O]=(tC)(tC)+o(C 3 ),

[0125] Further simplification of the above equation yields: ||r(c)-P|| 2 =C 2 +o(C 3 ),

[0126] Based on the above derivation, the formula for calculating the normal curvature of the target vertex along the direction of the unit tangent vector is:

[0127]

[0128] Where, k n (t) represents the normal curvature of the target vertex along the direction of the unit tangent vector t; r(c) represents the curve between the target vertex O and the vertex V in the target triangular mesh;

[0129] Step 5: Construct a curvature tensor based on the average unit normal vector of the target vertex and the normal curvature of the target vertex along the direction of the unit tangent vector; and calculate the Gaussian curvature and average curvature at the target vertex based on the first and second eigenvalues ​​of the curvature tensor.

[0130] Specifically, the curvature tensor is represented as:

[0131] L = k n (t)(I-nn T ),

[0132] Where L represents the curvature tensor; I represents the identity matrix; and T represents the transpose of the matrix.

[0133] The formulas for calculating the Gaussian curvature and mean curvature at the target vertex are as follows:

[0134]

[0135] Among them, K G Let λ0 represent the Gaussian curvature at the target vertex; H represents the average curvature at the target vertex; λ1 represents the first eigenvalue of the curvature tensor; λ2 represents the second eigenvalue of the curvature tensor.

[0136] Step 6: Calculate the principal curvature and secondary curvature of the surface at the target vertex based on the Gaussian curvature and mean curvature at the target vertex;

[0137] Specifically, the formulas for calculating the principal curvature and secondary curvature of the surface at the target vertex are as follows:

[0138]

[0139] Where k1 represents the principal curvature of the surface at the target vertex; k2 represents the secondary curvature of the surface at the target vertex;

[0140] Step 7: Update m = m + 1, and return to execute step 2 until m = 3, to obtain the principal curvature and secondary curvature of the surface at each vertex in the target triangular mesh;

[0141] Step 8: Based on the area of ​​the sub-triangle formed by the trajectory point and each vertex in the target triangle mesh, the area of ​​the target triangle mesh, and the principal curvature and secondary curvature of the surface at each vertex in the target triangle mesh, calculate the principal curvature and secondary curvature of the surface at the trajectory point.

[0142] Specifically, the formulas for calculating the principal curvature and secondary curvature of the surface at the trajectory point are as follows:

[0143]

[0144] Where K1 represents the principal curvature of the surface at the trajectory point; K2 represents the secondary curvature of the surface at the trajectory point; τ A This represents the ratio of the area of ​​the sub-triangle formed by vertices A and B in the target triangle mesh to the area of ​​the target triangle mesh; k 1A τ represents the principal curvature of the surface at vertex A in the target triangular mesh; B This represents the ratio of the area of ​​the sub-triangle formed by vertices B and C in the target triangle mesh to the area of ​​the target triangle mesh; k 1B τ represents the principal curvature of the surface at vertex B in the target triangular mesh; C This represents the ratio of the area of ​​the sub-triangle formed by vertices C and A in the target triangle mesh to the area of ​​the target triangle mesh; k 1C k represents the principal curvature of the surface at vertex C in the target triangular mesh. 2A k represents the surface sub-curvature at vertex A in the target triangular mesh. 2B k represents the surface sub-curvature at vertex B in the target triangular mesh. 2C This represents the surface sub-curvature at vertex C in the target triangular mesh;

[0145] Step 9: Based on the principal and secondary curvatures of the surface at the trajectory point, and the principal and secondary curvatures of the polishing tool, calculate the radius of the polishing contact circle at the trajectory point.

[0146] Specifically, the formula for calculating the radius of the polishing contact circle at the trajectory point is:

[0147] a=(3h 2 εQR / πE) 1 / 3 ,

[0148] Where a represents the radius of the polishing contact circle at the trajectory point; h represents the first empirical coefficient; ε represents the second empirical coefficient; Q represents the polishing force applied to the surface to be processed; R represents the curvature coefficient, which is equal to the reciprocal of the sum of the principal curvature, secondary curvature of the polishing tool, the principal curvature of the surface at the trajectory point, and the negative curvature of the surface; and E represents the equivalent elastic modulus.

[0149] Specifically, the formula for calculating the equivalent elastic modulus E is:

[0150] 1 / E=(1-μ1 2 ) / E1-(1-μ2 2 ) / E2,

[0151] Where E1 and μ1 represent the elastic modulus and Poisson's ratio of the polishing tool head; E2 and μ2 represent the elastic modulus and Poisson's ratio of the surface to be processed.

[0152] For example, such as Figure 3 The diagram shown illustrates the principle of calculating the average normal vector and normal curvature at each triangular mesh vertex in the mesh model provided in this application. Figure 3 (a) in the figure is a schematic diagram illustrating the principle of calculating the average normal vector at the vertices of the triangular mesh. These represent the unit normal vectors of the triangular mesh that share vertex O. Figure 3 Figure (b) is a schematic diagram illustrating the principle of calculating the normal curvature of a triangular mesh vertex along the unit tangent vector direction. In the figure, t represents the curve OV. i The unit tangent vector projected onto the tangent plane, where T represents the tangent plane.

[0153] Furthermore, based on the polishing contact circle radius at the trajectory point calculated in the above embodiment, the formula for calculating the target distance between the trajectory point and the reference point is as follows:

[0154] D = |a m -a n -J|,

[0155] Where D represents the target distance; a m Indicates the radius of the polishing contact circle at trajectory point m; a n J represents the radius of the polishing contact circle at the reference point n corresponding to trajectory point m; J represents the preset polishing trajectory overlap amount.

[0156] Optionally, in some embodiments of this application, each trajectory point in the curved surface trajectory can be randomly adjusted until the Euclidean distance between the trajectory point and the corresponding reference point is equal to the target distance; however, since the number of trajectory points is large, the random adjustment method is inefficient.

[0157] For the reasons mentioned above, as a preferred embodiment, adjusting the trajectory points in this application includes:

[0158] Calculate the Euclidean distance between the trajectory point and the reference point, and determine whether the Euclidean distance is equal to the target distance between the trajectory point and the reference point;

[0159] If the Euclidean distance is not equal to the target distance, then a plane is spanned by the line connecting the trajectory point and the reference point and the unit normal vector of the triangle mesh containing the trajectory point;

[0160] Select a target point inside or on the extension of the line connecting the trajectory point and the reference point. With the reference point as the center and the distance between the target point and the reference point as the radius, generate a circle in the stretched plane.

[0161] Obtain all intersection points between the generated circle and the surface to be processed, and take the intersection point with the smallest Euclidean distance to the trajectory point as the target trajectory point.

[0162] Preferably, in some embodiments of this application, if the Euclidean distance between the trajectory point and the reference point is greater than the target distance between the trajectory point and the reference point, a target point is selected on the extension line of the line connecting the trajectory point and the reference point; if the Euclidean distance between the trajectory point and the reference point is less than the target distance between the trajectory point and the reference point, a target point is selected inside the line connecting the trajectory point and the reference point.

[0163] Furthermore, after selecting the intersection point with the smallest Euclidean distance from the trajectory point as the target trajectory point, the following is also included:

[0164] Calculate the polishing contact radius of the target trajectory point, the Euclidean distance between the target trajectory point and the reference point, and the target distance between the target trajectory point and the reference point;

[0165] Determine whether the Euclidean distance between the target trajectory point and the reference point is equal to the target distance between the target trajectory point and the reference point. If not, readjust the target trajectory point.

[0166] like Figure 4 The diagram shown is a schematic diagram illustrating the principle of adjusting the trajectory points on the curved surface trajectory and determining the direction provided in this application. Figure 4 (a) in the diagram illustrates the principle of trajectory point adjustment amplitude and direction determination when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is greater than 0. Figure 4 (b) in the figure is a schematic diagram of the trajectory point adjustment range and direction determination principle when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is less than 0.

[0167] like Figure 5 The diagram shown is a schematic diagram illustrating the target trajectory point location retrieval principle provided in this application. Figure 5 (a) in the diagram illustrates the principle of target trajectory point location retrieval when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is greater than 0. Figure 5 (b) in the diagram is a schematic diagram of the target trajectory point location retrieval principle when the difference between the Euclidean distance between the trajectory point and the reference point and the target distance is less than 0.

[0168] like Figure 6 The diagram shows the result of uniformly covering the polishing trajectory with different amounts of trajectory overlap after adjusting the trajectory points on the curved surface trajectory using the method provided in this application. Figure 6 (a) in the figure is a schematic diagram of the uniform coverage polishing trajectory result obtained for an asymmetric trajectory with zero trajectory overlap. Figure 6 (b) in the diagram is a schematic diagram of the uniform coverage polishing trajectory result obtained for an asymmetric trajectory with a trajectory overlap of 1. Figure 6 (c) in the diagram is a schematic diagram of the uniform coverage polishing trajectory result obtained for a symmetrical trajectory with zero trajectory overlap. Figure 6 (d) in the figure is a schematic diagram of the uniform coverage polishing trajectory obtained for a symmetrical trajectory with a trajectory overlap of 1. As can be seen from the figure, for different polishing trajectory overlaps, the method provided in this application can obtain the corresponding uniform coverage polishing trajectory, which confirms the effectiveness of this application.

[0169] like Figure 7 The diagram illustrates the process of uniformly covering the polishing trajectory of a curved surface according to an embodiment of this application. Specifically, a mesh model of the surface to be processed is obtained using computer graphics software, and the vertex coordinates of all triangular meshes in the mesh surface are further read. The triangular meshes are projected onto the plane where the trajectory points are located in the opposite direction of the mapping direction to detect whether the triangular meshes are parallel to the mapping direction and cannot be mapped to the plane trajectory. Based on the original triangular mesh vertices and the unit mapping vector, the vertex coordinates of the triangular meshes projected onto the plane where the trajectory points are located in the opposite direction of the mapping direction are obtained. The vertices obtained by projecting the triangular meshes onto the plane are used to construct the corresponding triangle edge vectors, and the validity of the triangular meshes in the mapping direction is determined by the vector relationship. If the triangle is a valid triangle, it can be mapped to the corresponding plane trajectory, and the surface trajectory is obtained by the proposed formula for calculating the coordinates of the mapped trajectory points.

[0170] The mapped trajectories are then stored and optimized according to their generation order. Given the principal and secondary curvatures of the surface at each trajectory point, the polishing contact circle radius of each trajectory point is calculated. A reference point is set for trajectory point position adjustment. The relationships between the polishing contact circle radius at each trajectory point, the polishing circle radius at the reference point, the target Euclidean distance between the two points, the Euclidean distance between the two points, and the set polishing trajectory overlap are checked to ensure they meet requirements. For trajectory points whose positions do not meet requirements, position adjustments are made. The adjustment range and direction are determined based on the deviation between the given constraint value and the actual value. A plane is spanned using the line connecting the two points and the normal vector of the triangular mesh. A target point is found inside or on the extension of the line connecting the two points, such that the distance relationship between this point and the reference point satisfies the constraint conditions. A circle is generated in the spanned plane with the reference point as the center and the Euclidean distance between the reference point and the newly retrieved point as the radius. The relationship between this circle and the reference point is then extracted. The point closest to the reference point among the intersections of the mesh surfaces is taken as the target trajectory point. The polishing contact circle at the location of the target trajectory point is calculated, and the relationship between the point and the reference point is checked again to see if it meets the requirements. If it does, the target trajectory point is the required optimized trajectory point. If it does not, the search continues according to the search strategy until a trajectory point that meets the requirements is found. The optimized trajectory of the first circle serves as the seed path for subsequent trajectory adjustments. After obtaining the seed path, the positions of the trajectory points on the second circle trajectory are adjusted: for any trajectory point on the second circle trajectory, the point with the closest Euclidean distance to that point can be retrieved on the seed path as the reference point for position adjustment. The size of the polishing contact area at the locations of the two points is calculated, and it is determined whether the corresponding calculation results meet the given constraints. For trajectory points that do not meet the constraints, position optimization is performed according to the above steps. The adjusted second circle trajectory is used as the new seed path to adjust the third trajectory, and so on, to complete all trajectory adjustments.

[0171] The surface curvature at the location of the trajectory point on the mesh model is calculated based on the average unit normal vector, unit tangent vector, and normal curvature of the triangular mesh vertices to construct a curvature tensor: the average unit normal vector of the triangular mesh vertex of each trajectory is calculated, the unit tangent vector at any vertex of the triangular mesh of each trajectory is calculated, the normal curvature along the direction of the unit tangent vector at any vertex of the triangular mesh of each trajectory is calculated, the curvature tensor at the corresponding vertex is constructed based on the unit tangent vector and normal curvature, the eigenvalues ​​of the curvature tensor are solved, and the Gaussian curvature and average curvature at the corresponding vertex are further calculated: given the Gaussian curvature and average curvature at the corresponding vertex, the principal curvature and secondary curvature at that vertex are further calculated. For any triangular mesh, when the principal and secondary curvatures of the surface at the three vertices of the mesh are known, the principal and secondary curvatures of the surface at the location of any trajectory point within the triangular mesh can be solved by fitting using a weight formula.

[0172] Based on the method for uniformly covering curved surface polishing trajectories provided in the above embodiments, this application also provides a device for uniformly covering curved surface polishing trajectories, such as... Figure 8 As shown, the device specifically includes:

[0173] The surface discretization module 10 is used to discretize the surface to be processed to obtain a mesh model of the surface to be processed.

[0174] The trajectory mapping module 20 is used to obtain the I-circle planar polishing trajectory of the surface to be processed, and to map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory.

[0175] The first trajectory adjustment module 30 is used to select a reference point in the first circle of the curved surface trajectory and adjust each trajectory point in the first circle of the curved surface trajectory until the Euclidean distance between each trajectory point in the first circle of the curved surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first circle of the curved surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first circle of the curved surface trajectory is taken as the reference point; if the first circle of the curved surface trajectory is an asymmetrical trajectory, the first trajectory point in the first circle of the curved surface trajectory is taken as the reference point.

[0176] Data initialization module 40 is used to initialize i = 2;

[0177] The second trajectory adjustment module 50 is used to select the trajectory point with the closest Euclidean distance in the (i-1)th polishing trajectory for each trajectory point in the i-th round of the curved surface trajectory as the reference point, and adjust each trajectory point in the i-th round of the curved surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the curved surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory.

[0178] The target polishing trajectory acquisition module 60 is used to update i = i + 1 and return to the step of executing the second trajectory adjustment module until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories; wherein, the target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

[0179] This application also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the above-described method for uniformly covering curved surface polishing trajectories.

[0180] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0181] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0182] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0183] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0184] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method for uniformly covering a curved surface polishing trajectory, characterized in that, include: S10: Discretize the surface to be processed to obtain a mesh model of the surface to be processed; S20: Obtain the I-circle planar polishing trajectory of the surface to be processed, and map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory; S30: Select a reference point in the first circular surface trajectory, and adjust each trajectory point in the first circular surface trajectory until the Euclidean distance between each trajectory point in the first circular surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first circular surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first circular surface trajectory is taken as the reference point; if the first circular surface trajectory is an asymmetrical trajectory, the first trajectory point in the first circular surface trajectory is taken as the reference point; S40: Initialize i = 2; S50: For each trajectory point in the i-th round of the surface trajectory, select the trajectory point with the closest Euclidean distance in the (i-1)-th polishing trajectory as the reference point, and adjust each trajectory point in the i-th round of the surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory. S60: Update i = i + 1, and return to execute step S50 until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories; The target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

2. The method for uniformly covering the curved surface polishing trajectory according to claim 1, characterized in that, Step S10 further includes: extracting the vertex coordinates of each triangular mesh in the mesh model, and calculating the area and unit normal vector of each triangular mesh.

3. The method for uniformly covering the curved surface polishing trajectory according to claim 2, characterized in that, The steps for calculating the radius of the polishing contact circle at the trajectory point include: Step 1: Use the triangular mesh containing the trajectory point as the target triangular mesh, and initialize m = 1; Step 2: Take the m-th vertex in the target triangle mesh as the target vertex, and calculate the average unit normal vector of the target vertex based on the area and unit normal vector of the triangle mesh with the target vertex as the common vertex in the mesh model; Step 3: Obtain the tangent plane of the surface to be processed at the target vertex, and based on the average unit normal vector of the target vertex, calculate the unit tangent vector of the curve between any vertex in the target triangle mesh other than the target vertex and the target vertex onto the tangent plane; Step 4: Based on the average unit normal vector of the target vertex, the curve between any vertex in the target triangle mesh other than the target vertex and the target vertex, and the coordinates of the target vertex, calculate the normal curvature of the target vertex along the direction of the unit tangent vector; Step 5: Construct a curvature tensor based on the average unit normal vector of the target vertex and the normal curvature of the target vertex along the direction of the unit tangent vector; and calculate the Gaussian curvature and average curvature at the target vertex based on the first and second eigenvalues ​​of the curvature tensor. Step 6: Calculate the principal curvature and secondary curvature of the surface at the target vertex based on the Gaussian curvature and mean curvature at the target vertex; Step 7: Update m = m + 1, and return to execute step 2 until m = 3, to obtain the principal curvature and secondary curvature of the surface at each vertex in the target triangular mesh; Step 8: Based on the area of ​​the sub-triangle formed by the trajectory point and each vertex in the target triangle mesh, the area of ​​the target triangle mesh, and the principal curvature and secondary curvature of the surface at each vertex in the target triangle mesh, calculate the principal curvature and secondary curvature of the surface at the trajectory point. Step 9: Calculate the radius of the polishing contact circle at the trajectory point based on the principal curvature and secondary curvature of the surface at the trajectory point, and the principal curvature and secondary curvature of the polishing tool.

4. The method for uniformly covering the curved surface polishing trajectory according to claim 3, characterized in that, The formula for calculating the average unit normal vector of the target vertex is: Where n represents the average unit normal vector of the target vertex; A g N represents the area of ​​the g-th triangular mesh that shares a common vertex with the target vertex; g This represents the unit normal vector of the g-th triangular mesh that shares a common vertex with the target vertex; The formula for calculating the unit tangent vector on the tangent plane by projecting the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex is as follows: Where t represents the unit tangent vector projected onto the tangent plane from the curve between any vertex in the target triangular mesh (excluding the target vertex) and the target vertex; OV represents the vector formed by the target vertex O and the vertex V in the target triangular mesh; The formula for calculating the normal curvature of the target vertex along the direction of the unit tangent vector is: Where, k n (t) represents the normal curvature of the target vertex along the direction of the unit tangent vector t; r(c) represents the curve between the target vertex O and the vertex V in the target triangular mesh; The curvature tensor is represented as: L=k n (t)(I-nn T ) Where L represents the curvature tensor; I represents the identity matrix; and T represents the transpose of the matrix. The formulas for calculating the Gaussian curvature and mean curvature at the target vertex are as follows: Among them, K G Let λ0 represent the Gaussian curvature at the target vertex; H represents the average curvature at the target vertex; λ1 represents the first eigenvalue of the curvature tensor; λ2 represents the second eigenvalue of the curvature tensor. The formulas for calculating the principal curvature and secondary curvature of the surface at the target vertex are as follows: Where k1 represents the principal curvature of the surface at the target vertex; k2 represents the secondary curvature of the surface at the target vertex; The formulas for calculating the principal curvature and secondary curvature of the surface at the trajectory point are as follows: Where K1 represents the principal curvature of the surface at the trajectory point; K2 represents the secondary curvature of the surface at the trajectory point; τ A This represents the ratio of the area of ​​the sub-triangle formed by vertices A and B in the target triangle mesh to the area of ​​the target triangle mesh; k 1A τ represents the principal curvature of the surface at vertex A in the target triangular mesh; B This represents the ratio of the area of ​​the sub-triangle formed by vertices B and C in the target triangle mesh to the area of ​​the target triangle mesh; k 1B τ represents the principal curvature of the surface at vertex B in the target triangular mesh; C This represents the ratio of the area of ​​the sub-triangle formed by vertices C and A in the target triangle mesh to the area of ​​the target triangle mesh; k 1C k represents the principal curvature of the surface at vertex C in the target triangular mesh. 2A k represents the surface sub-curvature at vertex A in the target triangular mesh. 2B k represents the surface sub-curvature at vertex B in the target triangular mesh. 2C This represents the surface sub-curvature at vertex C in the target triangular mesh; The formula for calculating the radius of the polishing contact circle at the trajectory point is: a=(3h 2 (eQR / πE) 1 / 3 , Where a represents the radius of the polishing contact circle at the trajectory point; h represents the first empirical coefficient; ε represents the second empirical coefficient; Q represents the polishing force applied to the surface to be processed; R represents the curvature coefficient, which is equal to the reciprocal of the sum of the principal curvature, secondary curvature of the polishing tool, the principal curvature of the surface at the trajectory point, and the negative curvature of the surface; and E represents the equivalent elastic modulus.

5. The method for uniformly covering the curved surface polishing trajectory according to claim 1, characterized in that, The formula for calculating the target distance is: D=|a m -a n -J|, Where D represents the target distance; a m Indicates the radius of the polishing contact circle at trajectory point m; a n J represents the radius of the polishing contact circle at the reference point n corresponding to trajectory point m; J represents the preset polishing trajectory overlap amount.

6. The method for uniformly covering the curved surface polishing trajectory according to claim 5, characterized in that, Adjusting the trajectory points includes: Calculate the Euclidean distance between the trajectory point and the reference point, and determine whether the Euclidean distance is equal to the target distance between the trajectory point and the reference point; If the Euclidean distance is not equal to the target distance, then a plane is spanned based on the line connecting the trajectory point and the reference point, and the unit normal vector of the triangle mesh containing the trajectory point; Select a target point inside or on the extension of the line connecting the trajectory point and the reference point. Generate a circle in the plane with the reference point as the center and the distance between the target point and the reference point as the radius. Obtain all intersection points between the generated circle and the surface to be processed, and take the intersection point with the smallest Euclidean distance to the trajectory point as the target trajectory point.

7. The method for uniformly covering the curved surface polishing trajectory according to claim 6, characterized in that, After taking the intersection point with the smallest Euclidean distance to the aforementioned trajectory point as the target trajectory point, the following is also included: Calculate the polishing contact radius of the target trajectory point, the Euclidean distance between the target trajectory point and the reference point, and the target distance between the target trajectory point and the reference point; Determine whether the Euclidean distance between the target trajectory point and the reference point is equal to the target distance between the target trajectory point and the reference point. If not, readjust the target trajectory point.

8. The method for uniformly covering the curved surface polishing trajectory according to claim 6, characterized in that, Selecting a target point within or on the extension of the line connecting the trajectory point and the reference point includes: If the Euclidean distance between the trajectory point and the reference point is greater than the target distance between the trajectory point and the reference point, then a target point is selected on the extension line connecting the trajectory point and the reference point. If the Euclidean distance between the trajectory point and the reference point is less than the target distance between the trajectory point and the reference point, then a target point is selected within the line connecting the trajectory point and the reference point.

9. A device for uniformly covering curved surface polishing trajectory, characterized in that, include: The surface discretization module is used to discretize the surface to be processed, and obtain the mesh model of the surface to be processed; The trajectory mapping module is used to obtain the I-circle planar polishing trajectory of the surface to be processed, and to map each planar trajectory point on the I-circle planar polishing trajectory to the mesh model to obtain the I-circle surface trajectory. The first trajectory adjustment module is used to select a reference point in the first-round curved surface trajectory and adjust each trajectory point in the first-round curved surface trajectory until the Euclidean distance between each trajectory point in the first-round curved surface trajectory and the reference point is equal to the target distance between the trajectory point and the reference point, thus obtaining the first polishing trajectory; wherein, if the first-round curved surface trajectory is a symmetrical trajectory, the trajectory point located at the symmetrical point in the first-round curved surface trajectory is used as the reference point; if the first-round curved surface trajectory is an asymmetrical trajectory, the first trajectory point in the first-round curved surface trajectory is used as the reference point; The data initialization module is used to initialize i = 2; The second trajectory adjustment module is used to select the trajectory point with the closest Euclidean distance to each trajectory point in the i-th round of the surface trajectory as the reference point in the (i-1)th polishing trajectory, and adjust each trajectory point in the i-th round of the surface trajectory until the Euclidean distance between each trajectory point in the i-th round of the surface trajectory and its corresponding reference point is equal to the target distance between the trajectory point and its corresponding reference point, thus obtaining the i-th polishing trajectory. The target polishing trajectory acquisition module is used to update i = i + 1 and return to the step of executing the second trajectory adjustment module until i = I, and obtain the target polishing trajectory of the surface to be processed based on I polishing trajectories; wherein, the target distance between the trajectory point and the reference point is the absolute value of the difference between the polishing contact circle radius at the trajectory point, the polishing contact circle radius at the reference point corresponding to the trajectory point, and the preset polishing trajectory overlap amount.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the steps of the method for uniformly covering curved surface polishing trajectories as described in any one of claims 1-8.

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