Method for predicting three-dimensional surface topography of a workpiece during turning

Abstract

The application discloses a kind of turning processing workpiece surface three-dimensional topography prediction method, comprising the following steps: step one: the Marching Cubes contour surface of target surface is constructed: according to the range of target surface, constructs a cubic grid;The value of implicit function is operated on each grid point in cubic grid;According to the sign of implicit function value, based on Marching Cubes algorithm, generate triangular patches and the corresponding vertex value of triangle;Step two: the target surface is uniformly randomly sampled: according to the established triangular contour surface grid, uniformly randomly in each triangular grid inside sampling point is obtained, to realize the uniform random sampling of target surface;Step three: three-dimensional topography prediction is carried out to target surface: based on the tool position data of complex free surface path planning, tool path is modeled, and residual height on each random sampling point is calculated, realizes three-dimensional topography prediction to target surface.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of turning machining, and particularly relates to a turning machining workpiece surface three-dimensional morphology prediction method. BACKGROUND

[0002] Ultra-precision turning machining has become the main machining method in the fields of military industry and optics due to its high machining precision and good surface quality. With the progress of science and technology, multi-axis ultra-precision machining and other machining methods have gradually emerged and have made a name for themselves in the machining of some complex free-form surfaces.

[0003] However, the machining time of ultra-precision turning machining is long, and if the process parameters are selected incorrectly during the machining process, the time for adjustment and modification will be extremely long, greatly exceeding the predetermined production cycle. Therefore, before ultra-precision turning machining, if the three-dimensional morphology after machining can be predicted, the preparation work and the process system of the entire ultra-precision turning machining will be greatly improved. In the prior art, the ultra-precision turning surface three-dimensional morphology prediction method mainly focuses on regular curved surfaces such as planes, and the method used is mainly based on the projection of Archimedes spirals in a conventional coordinate system. This method will cause the loss of local morphology details for complex free-form surfaces during machining, especially for some high-slope regions in the curved surface, and cannot accurately predict the three-dimensional surface quality of the machined workpiece. SUMMARY

[0004] Therefore, the purpose of the present application is to provide a turning machining workpiece surface three-dimensional morphology prediction method which can be effectively applied to complex free-form surfaces containing high slopes and accurately predict the three-dimensional morphology of the turning machining workpiece surface.

[0005] To achieve the above purpose, the present application provides the following technical solution:

[0006] A turning machining workpiece surface three-dimensional morphology prediction method comprises the following steps:

[0007] Step 1: Constructing the Marching Cubes isosurface of the target curved surface

[0008] According to the range of the target curved surface, a cubic grid with x, y and z axes as grid values is constructed; according to the target curved surface equation, the value of the implicit function at each grid point in the cubic grid is calculated; for each small square in the cubic grid, based on the Marching Cubes algorithm, the triangular patches and the corresponding vertex values of the triangles are generated according to the sign of the implicit function value, and the Marching Cubes isosurface of the target curved surface is constructed;

[0009] Step 2: Uniformly and randomly sampling the target curved surface

[0010] According to the established triangular equivalent surface mesh, a sampling point is randomly obtained in each triangular mesh to realize uniform random sampling of the target surface;

[0011] Step three: three-dimensional topography prediction of the target surface

[0012] Based on the tool position data of the complex free-form surface path planning, the tool path is modeled, and the residual height on each random sampling point is calculated to realize three-dimensional topography prediction of the target surface.

[0013] Further, in the step two, the method for randomly obtaining a sampling point in the triangular mesh is as follows:

[0014] 21) Randomly select a triangular equivalent surface by area

[0015] The area of each triangular patch is calculated by the Heron formula, and a random number u between 0 and the total area N0 of all triangular patches is uniformly and randomly generated; there must exist a k∈[1,n] such that:

[0016] s1+s2+…+s k-1 ≤u<s1+s2+…+s k

[0017] Wherein, S i represents the area of the i-th triangular patch, and i∈[1,n]; n represents the number of triangular patches in the triangular mesh;

[0018] Then, the sampling point is taken from the k-th triangular patch;

[0019] 22) Uniformly and randomly obtain a sampling point in the triangular patch

[0020] Two uniformly distributed random numbers a1,a2 between 0 and 1 are generated, and in the triangular patch ΔABC, the point P is taken as:

[0021]

[0022] Determine whether the point P is located in ΔABC: if yes, the point P is a uniformly random sampling point in the parallelogram with AB and AC as adjacent edges; if not, rotate the point P by 180° to obtain the point P′, and take P′ as a uniformly random sampling point in the parallelogram with AB and AC as adjacent edges; point O represents the origin of the workpiece coordinate system;

[0023] For each sampling point, a Newton step is taken along the current implicit function gradient direction each time until convergence is obtained, and a uniformly random sampling point falling on the target surface is obtained.

[0024] Further, the step three includes the following steps:

[0025] 31) solving the sampling point tool pose

[0026] The tool nose of the tool is modeled as a circular sheet, and the effective sampling point tool pose of the tool when cutting the cutting segment is determined by the fixed point iteration method according to the tool axis vector at the two end points of the cutting segment, and the tool center point position is determined;

[0027] 32) calculating the residual height of the sampling point;

[0028] 33) predicting the three-dimensional morphology of the target surface turning process:

[0029] For each sampling point, the lowest value of the residual height calculated by all line segments is taken as the final residual height, and the three-dimensional morphology of the target surface under the turning process parameters is obtained.

[0030] Further, in step 31), the starting point M of the cutting segment is 0, the end point N is 1, and the center point H of the tool when the effective sampling point p on the cutting segment is cut is at position t, t∈[0,1]; then the tool center point H satisfies:

[0031]

[0032] Wherein, point O represents the origin of the workpiece coordinate system;

[0033] Suppose each rotation axis is rotating at a constant speed, then the tool direction when point p is cut is spherical interpolation:

[0034]

[0035]

[0036] Wherein, is the tool axis vector at the starting point M of the cutting segment; is the tool axis vector at the end point N of the cutting segment;

[0037] The rake face of the tool is parallel or coplanar with the workpiece rotation axis, the tool axis vector is (0, 0, 1), and the tool face normal vector is:

[0038]

[0039] Let be the direction of the tool axis movement, then its unit vector is:

[0040]

[0041] When the tool cuts the sampling point p, since the sampling point p is in the plane with the normal , the equation can be obtained:

[0042]

[0043] Where l represents the distance traveled by the cutting tool on the cutting segment when it cuts to the sampling point p;

[0044] but:

[0045]

[0046]

[0047] The fixed-point method is used for iterative solution: given t, new t is continuously obtained according to the above process until the value of t converges and no longer changes. Then, the value of t satisfies that the tool head direction and the cutting direction are consistent when cutting the sampling point p. Finally, the tool pose and the center point H of the tool rake face are obtained at the cutting sampling point p.

[0048] Furthermore, in step 32), the rake face of the lathe tool is constructed as a circle with radius r, and a normal to the target machining surface is drawn through the sampling point p. The two intersection points of the normal to the target surface and the circle are P1 and P2, respectively. P1 is located between sampling points p and P2. Then the residual height pP1 is:

[0049] t1=t c -t 1c

[0050] Where t1 represents the residual height; t c express exist Projection on; t 1c Indicates P1H in The projection on; and:

[0051]

[0052]

[0053]

[0054] Where d0 represents the distance between the center point H of the rake face and the straight line P1P2.

[0055] Furthermore, in step 33), the three-dimensional roughness of the entire target surface is:

[0056]

[0057]

[0058] in, δ represents the average residual height across the entire workpiece surface. irepresents the residual height at the i th sampling point; S a represents the three-dimensional roughness of the machined workpiece surface; m represents the number of randomly sampled points of the workpiece surface;

[0059] The target surface profile is:

[0060] P = maxδ i -minδ i

[0061] Wherein, P represents the target surface profile.

[0062] The beneficial effects of the present application are:

[0063] The turning machining workpiece surface three-dimensional topography prediction method of the present application uniformly and randomly samples the target complex free surface based on the Marching Cubes algorithm, avoiding the problem of local topography detail loss in the local high slope area under the projection of the traditional Cartesian coordinate system. The Marching Cubes and fixed point iteration methods used in the present application are universal, considering the actual motion of three-axis and multi-axis turning machining, and finally calculating the residual height of the target surface through geometric relationship derivation, so that the present application can be applied to three-axis, four-axis or even five-axis machining of any complex free surface, greatly simplifying the determination of process parameters before turning machining and trial cutting, and having important significance for the improvement of the overall process of ultra-precision turning. BRIEF DESCRIPTION OF DRAWINGS

[0064] In order to make the purpose, technical scheme and beneficial effects of the present application clearer, the present application provides the following drawings for illustration:

[0065] Figure 1 It is a flow chart of the turning machining workpiece surface three-dimensional topography prediction method of the present application;

[0066] Figure 2 It is a schematic diagram of adjacent tool path cutting section MN;

[0067] Figure 3 It is a schematic diagram of five-axis turning machining;

[0068] Figure 4 It is a geometric relationship diagram for solving residual height. DETAILED DESCRIPTION

[0069] The present application will be further described below in combination with the drawings and specific embodiments, so that those skilled in the art can better understand the present application and implement it, but the embodiments are not limiting the present application.

[0070] As Figure 1As shown, the turning workpiece surface three-dimensional topography prediction method of the embodiment is based on the Marching Cubes algorithm of computer graphics, and the target workpiece is constructed with a triangular isosurface, and is uniformly and randomly sampled to obtain sampling points on the entire surface. At each sampling point, the ultra-precise machining tool path is modeled, the residual height at each point is calculated, and the three-dimensional topography on the entire free complex surface is accurately predicted. This method can effectively avoid the loss of local topography details at high slopes in traditional prediction models, and is suitable for any free complex surface. Specifically, the turning workpiece surface three-dimensional topography prediction method of the embodiment includes the following steps:

[0071] Step one: Constructing the Marching Cubes isosurface of the target surface

[0072] Because the traditional Cartesian coordinate system will cause local detail loss when sampling the high slope part of the complex free surface, in the embodiment, the Marching cubes method is used to construct the isosurface of the target surface, divide the target surface into a plurality of triangular meshes, and then uniformly and randomly sample in the triangular isosurface. The specific method is as follows:

[0073] First, according to the range of the target surface, a cubic grid of x, y, and z axes is constructed as the grid point value; second, according to the target surface equation, the value of the implicit function is calculated at each grid point in the cubic grid; finally, for each small grid in the cubic grid, according to the sign of the implicit function value, based on the Marching Cubes algorithm, triangular patches and the corresponding vertex values of the triangular patches are generated, and the Marching Cubes isosurface of the target surface is constructed.

[0074] Step two: Uniformly and randomly sampling the target surface

[0075] According to the established triangular isosurface grid, sampling points are uniformly and randomly obtained in each triangular grid to achieve uniform and random sampling of the target surface. In the embodiment, the method of randomly obtaining sampling points in the triangular grid is as follows:

[0076] 21) Randomly selecting triangular isosurfaces by area

[0077] Assume that the total number of triangular patches is n. Because the coordinates of the vertices of each triangle are known, the area of each triangle is calculated by the Heron formula:

[0078]

[0079] The areas S1, S2,..., S n , of each triangular patch are calculated. a, b and c represent the lengths of the three sides of the triangular facet, respectively.

[0080] After calculating the area of each triangular facet by Heron formula, a random number u between 0 and the total area of all triangular facets N0 is generated uniformly and randomly; there must exist a k∈[1, n] such that:

[0081] s1+s2+…+s k-1 ≤u<s1+s2+…+s k

[0082] where S i represents the area of the ith triangular facet, and i∈[1, n]; n represents the number of triangular facets in the triangular mesh;

[0083] Then, the sampling point is taken from the kth triangular facet.

[0084] 22) Randomly taking a sampling point in a triangular facet

[0085] Two random numbers a1 and a2 between 0 and 1 are generated, and a point P is taken in the triangular facet ΔABC as follows:

[0086]

[0087] It is judged whether the point P is located in ΔABC or not: if yes, the point P is a uniformly random sampling point in the parallelogram with AB and AC as adjacent sides; if not, the point P is rotated by 180° to obtain a point P′, and the point P′ is a uniformly random sampling point in the parallelogram with AB and AC as adjacent sides; point O represents the origin of the workpiece coordinate system.

[0088] The points sampled on the rough triangular mesh are obviously still some distance away from the target surface, but these points are uniformly and randomly distributed, and are close enough to the target surface to enable the Newton method to converge quickly. For each sampling point, the Newton step is taken along the current implicit function gradient direction each time:

[0089]

[0090] where p t and p t+1 represent the coordinate positions of the sampling point at the tth and (t+1)th iterations, respectively; f(p t ) represents the value of the sampling point after the tth iteration is substituted into the surface equation; represents the gradient of the position of the sampling point after the tth iteration.

[0091] Until convergence, a uniformly random sampling point on the target surface is obtained.

[0092] Step three: three-dimensional topography prediction of the target surface

[0093] Based on the tool position data of the complex free-form surface path planning, the tool path of the turning tool is modeled, and the residual height at each random sampling point is calculated to realize the three-dimensional topography prediction of the target surface.

[0094] For a line segment composed of two adjacent tool points as shown in Figure 2 , points M and N are two adjacent tool positions, and the area of the sampling point p that produces preferential cutting does not exceed from the midpoint. Therefore, the sampling points with a distance exceeding from the midpoint can be excluded in advance, and only the sampling points within this range are calculated for the participating height.

[0095]

[0096] 31) Solve the tool position of the sampling point

[0097] Since this embodiment is based on multi-axis or even five-axis linkage ultra-precision turning, the tool is always moving in two rotation directions during the machining process, so the tool position at each sampling point needs to be determined before the participating height is calculated.

[0098] The tool head of the turning tool is modeled as a circular sheet, and point p is an effective sampling point on the cutting segment MN. According to the tool axis vectors at the two endpoints M and N of the cutting segment, the tool position of the turning tool when cutting the effective sampling point p on the cutting segment MN is determined by the fixed point iteration method, and the tool center point position is determined.

[0099] Let the starting point M of the cutting segment be 0 and the ending point N be 1, and the tool center point H of the turning tool when cutting the effective sampling point p on the cutting segment is at position t, t ∈ [0, 1]; then the tool center point H satisfies:

[0100]

[0101] Where point O represents the origin of the workpiece coordinate system;

[0102] Assuming that each rotation axis (tool rotation axis, workpiece rotation axis) is uniformly rotating, the tool direction when the point p is cut is a spherical interpolation:

[0103]

[0104]

[0105] where is the tool axis vector at the starting point M of the cutting segment; to cut the tool axis vector at the end point N of the cutting segment;

[0106] According to the kinematic analysis of ultra-precision turning, for three-axis and four-axis ultra-precision machining, the target workpiece rotation axis is coplanar with the tool rake face, and for five-axis ultra-precision machining, the tool rake face is parallel to the workpiece rotation axis due to the addition of a horizontal axis movement, and the rake face can be rotated through multi-axis linkage, but the rake face is always parallel to the workpiece rotation axis, as shown in FIG. 1. Figure 3 Since the tool rake face is parallel or coplanar to the workpiece rotation axis, the tool axis vector is (0, 0, 1), and the tool face normal vector is:

[0107]

[0108] Let be the direction of the tool axis movement, and its unit vector is:

[0109]

[0110] When the tool cuts to the sampling point p, since the sampling point p is in the plane with the normal , the equation can be obtained:

[0111]

[0112] wherein l represents the distance walked on the cutting segment when the tool cuts to the sampling point p;

[0113] Then:

[0114]

[0115]

[0116] The evaluation of t depends on The tool head vector is determined by the t interpolation, which constitutes a circular dependence. Through the fixed point method iteration: that is, given t, the new t is obtained according to the above process, until the t value converges and no longer changes, then the t value is obtained, which satisfies the cutting sampling point p when the tool head direction and the cutting direction are consistent, and finally the pose of the tool at the cutting sampling point p and the center point H of the tool rake face are obtained.

[0117] 32) Calculate the residual height of the sampling point

[0118] Point H is the center of the rake face, and the rake face is modeled as a circle with a radius r, and its geometric relationship with point p is as follows, the normal line of the target machining surface through p is The intersection point P1 of the ray and the tool head circle is the lowest point of effective cutting, and pP1 is the residual height, which is denoted as t1 as shown in the figure:

[0119] The tool rake face is constructed as a circle with a radius r, and the geometric relationship of the circle with point p is as followsFigure 4 As shown. The normal to the target machining surface is drawn from the sampling point p. The two intersection points of the normal to the target surface and the circle are P1 and P2, respectively. P1 is located between sampling points p and P2. Then the residual height pP1 is:

[0120] t1=t c -t 1c

[0121] Where t1 represents the residual height; t c express exist Projection on; t 1c Indicates P1H in The projection on; and:

[0122]

[0123]

[0124]

[0125] Where d0 represents the distance between the center point H of the rake face and the straight line P1P2.

[0126] 33) Predict the three-dimensional morphology of the target surface during turning:

[0127] For each sampling point, the lowest value among the residual heights calculated from all line segments is taken as the final residual height, thus obtaining the three-dimensional shape of the target surface under the turning process parameters.

[0128] Based on the obtained residual height, the three-dimensional roughness of the entire target surface is calculated as follows:

[0129]

[0130]

[0131] in, δ represents the average residual height across the entire workpiece surface. i S represents the residual height at the i-th sampling point; a This represents the three-dimensional roughness of the workpiece surface; m represents the number of random sampling points on the workpiece surface.

[0132] The target surface profile is:

[0133] P = maxδ i -minδ i

[0134] Where P represents the target surface profile.

[0135] The above-described embodiments are merely preferred embodiments of the present application, and the protection scope of the present application is not limited thereto. Any equivalent substitutions or transformations made by those skilled in the art based on the present application are within the protection scope of the present application. The protection scope of the present application is subject to the claims.

Claims

1. A method for predicting the three-dimensional topography of a surface of a workpiece being turned, characterized by: It comprises the following steps: Step one: constructing the Marching Cubes isosurface of the target surface According to the range of the target surface, a cubic grid of x, y, z axes as the grid value is constructed; according to the target surface equation, the value of the implicit function at each grid point in the cubic grid is calculated; for each small grid in the cubic grid, based on the Marching Cubes algorithm, triangular patches and the corresponding vertex values of the triangles are generated according to the sign of the implicit function value, and the Marching Cubes isosurface of the target surface is constructed; Step two: uniform random sampling of the target surface According to the established triangular isosurface grid, sampling points are obtained inside each triangular grid uniformly and randomly to achieve uniform random sampling of the target surface; Step three: three-dimensional topography prediction of the target surface Based on the tool position data of complex free-form surface path planning, the tool path is modeled, and the residual height at each random sampling point is calculated to achieve three-dimensional topography prediction of the target surface; The step three comprises the following steps: 31) solving the sampling point tool position The tool bit of the turning tool is modeled as a circular sheet, and the tool position of the turning tool at the effective sampling point on the cutting section is determined by the fixed point iteration method according to the tool axis vector at the two endpoints of the cutting section, and the tool center point position is determined; 32) calculating the residual height of the sampling point; 33) predicting the three-dimensional topography of the target surface after turning: For each sampling point, the lowest value of the residual height calculated by all line segments is taken as the final residual height to obtain the three-dimensional topography of the target surface under the turning process parameters.

2. The method of claim 1, wherein: In the step two, the method of randomly obtaining sampling points in the triangular grid is: 21) randomly selecting triangular isosurfaces by area The area of each triangle piece is calculated by Heron's formula, and a random number between 0 and the sum of all triangle piece areas is uniformly distributed ; there exists a such that:​ wherein, represents the area of the th triangular patch, and ; represents the number of triangular patches within the triangular mesh; then a sampling point is taken from the nth triangular facet; 22) uniformly and randomly obtaining sampling points in the triangular patches Generate two Uniformly distributed random numbers between In the triangular facet middle, take point for: determination point whether located in : if yes, the point is a uniform random sampling point in the parallelogram with , adjacent sides; if not, the point is rotated 180° from the center of the parallelogram with , adjacent sides to obtain the point ; the point is a uniform random sampling point in the parallelogram with , adjacent sides; the point represents the origin of the workpiece coordinate system For each sampling point, a Newton step is taken along the current implicit function gradient direction each time until convergence to obtain a uniformly random sampling point on the target surface.

3. The method of claim 1, wherein: In the step 31), the start point M of the cutting section is 0, the end point N of the cutting section is 1, and the effective sampling points on the cutting section are The center point H of the tool when being cut is at Position, The tool center point H satisfies: wherein the point represents the origin of the workpiece coordinate system; Let each rotation axis rotate at a constant speed, then the point cut by the tool direction is a spherical interpolation: wherein, is the tool axis vector at the start M of the cutting segment; is the tool axis vector at the end N of the cutting segment; The rake face of the turning tool is parallel or coplanar with the workpiece rotation axis, the tool axis vector is (0, 0, 1), and the tool face normal vector is: ； Let is the direction of the movement of the tool shaft, then its unit vector is: When the turning tool cuts to the sampling point , since the sampling point is in the plane with the normal , the equation is obtained: wherein, represents the distance travelled on the cutting segment by the turning tool when cutting to the sampling point represents the distance travelled on the cutting segment by the turning tool when cutting to the sampling point Then: Solve by fixed point method iteration: that is, given , constantly get new according to the above process, until value converges no longer change, then get t value meet cutting sampling point when the tool direction and cutting direction are consistent, ultimately get the pose of the tool and the tool rake face center point H when cutting sampling point .

4. The method of claim 1, wherein: In the step 32), the rake face of the turning tool is constructed as a circle with a radius of and passing through the sampling point and the target machining surface normal , the two intersection points of the circle are and , which are located between the sampling point and , and the residual height is: wherein represents a residual height; represents a projection onto a projection onto represents a projection onto a projection onto wherein, represents the distance between the center point H of the rake face and the straight line .

5. The method of claim 1, wherein: In the step 33), the three-dimensional roughness of the entire target surface is: wherein, represents the average residual height of the entire workpiece surface; represents the residual height at the i-th sampling point; represents the residual height at the i-th sampling point; represents the three-dimensional roughness of the machined workpiece surface; represents the number of randomly sampled points of the workpiece surface; The profile tolerance of the target surface is: wherein, represents the target surface profile.

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