Analytical method for solving the cutting area of end mill based on point cloud in five-axis nc machine tool

Abstract

The present application belongs to the field of five-axis NC machine tool machining, and proposes an analytical method for solving the cutting area of end mill based on point cloud. Firstly, the workpiece coordinate system, the normal coordinate system of inclined plane and the tool coordinate system are established, and the coordinate transformation relationship is determined through the rotation matrix. Then the geometric equations of the cutting area involving cylindrical surface, the swept surface of the previous tool path and the unprocessed surface are determined. The intersection of the above surfaces with the cylindrical surface is obtained, and two space intersection lines are obtained. Combined with the third line segment obtained by cutting the bottom circle of the cylindrical surface, the complete cutting area is formed. In order to overcome the shortcomings of traditional analytical method such as complex intersection, difficult cutting and large amount of calculation, the present application introduces the point cloud method, which disperses the continuous geometric body into a regular distributed space point set, and solves the problem by algebraic operation of point coordinates instead of complex analytical equation. The nearest neighbor search is used to realize the intersection positioning. This method significantly improves the operation speed of solving the cutting area, while maintaining high accuracy, and is suitable for the situation of variable tool posture in five-axis machining, providing an efficient and reliable means for cutting force modeling and machining optimization.

Description

Technical Field

[0001] This invention belongs to the field of five-axis CNC machine tool processing in mechanical manufacturing, and relates to an analytical method for solving the cutting contact area of ​​an end mill based on the point cloud method in five-axis CNC machine tools. Background Technology

[0002] When machining complex curved surfaces on a five-axis machine tool, the time-varying nature of the tool posture makes the contact area between the tool and the workpiece during the machining process complex and varied. Extracting the milling contact area is crucial for studying the milling force, machining error and vibration system stability in five-axis machining.

[0003] There are three main methods for extracting the tool-workpiece contact area in five-axis milling: solid modeling, infinitesimal discretization modeling, and numerical analytical methods. Solid modeling, based on CAM software, uses Boolean operations to determine the overlap between the tool and workpiece, thus extracting the contact area. While solid modeling offers higher accuracy than the other two methods, its simulation efficiency is low because it requires continuous updates to the workpiece model. Z-map discretization uses a series of uniformly distributed parallel vectors to discretize the workpiece, similar to a lawnmower process. These discretized vectors are shortened after being swept by the tool. By replacing logical operations in three-dimensional space with comparisons of Z-coordinate values, the computational efficiency of the discretization method is significantly improved. However, high discretization resolution consumes a large amount of memory, significantly increasing the demands on the computer. Analytical methods extract the contact area by numerically calculating the intersection curves of the tool and workpiece boundaries. However, the analytical intersection method is complex when geometry is superimposed, and the geometric equations need frequent rewriting after tool rotation. Furthermore, the local contact area is only a small part of the overall geometry, making analytical trimming difficult. These issues will increase computation time when generating the contact area.

[0004] To address the aforementioned shortcomings, this invention proposes an analytical method for solving the end mill cutting contact area based on point cloud analysis. This method combines discretization and analytical methods, discretizing the continuous geometry into a finite number of spatial points and handling complex geometric relationships through algebraic operations on the point coordinates. This overcomes the drawbacks of traditional analytical methods, which directly perform calculations on geometric equations, resulting in high computational load and slow program execution speed, thus significantly improving the computational speed. Summary of the Invention

[0005] The specific embodiments of the present invention are described in detail below with reference to the technical solutions and accompanying drawings.

[0006] An analytical method for solving the cutting contact area of ​​an end mill in a five-axis CNC machine tool based on point cloud method is specifically divided into the following steps: Step 1: Define the workpiece coordinate system, the oblique plane normal coordinate system, and the tool coordinate system, and calculate the transformation relationship between the three coordinate systems using the rotation matrix.

[0007] To analyze the condition of the tool contact area on the inclined plane, it is necessary to know the transformation relationship between the three coordinate systems, namely the workpiece coordinate system. Normal coordinate system of oblique plane and tool coordinate system The geometric transformation relationships between the three coordinate systems are as follows: Figure 1 As shown.

[0008] Workpiece coordinate system and the oblique plane normal coordinate system The conversion relationship between them is as follows: Figure 2 As shown, the calculation is performed by multiplying the row vectors by the rotation matrix on the right: [Calculation based on the workpiece coordinate system] Transform to the oblique plane normal coordinate system It needs to be rotated around the X-axis first. Angle, then rotate around the Z-axis Angle (here) Angle and Angles are also defined as slope angles and relative feed angles. The calculation in row vector form is as follows: (1) in (2) Inclined plane normal coordinate system and tool coordinate system The transformation relationship between them is calculated according to the convention of multiplying the row vectors by the rotation matrix on the right: From the oblique plane normal coordinate system Transform to tool coordinate system It needs to be rotated around the X-axis first. Angle, then rotate around the Y-axis Angle (here) Angle and The angle is also defined as the tool's rake angle and side tilt angle. The calculation in row vector form is as follows: (3) in (4) Step 2: Determine the equations of the geometric surfaces of the cutting area, including the cylindrical surface, the swept surface of the previous tool path, and the unmachined surface.

[0009] The derivation of the U-plane (the intersection with the previous tool sweep surface) is divided into four steps based on the tool coordinate system. The first step is to write out the transformation matrix T. (5) Put the second column (corresponding to) What is extracted is the component of the feed direction in the tool coordinate system (denoted as a vector). ): (6) This means that when along The displacement components within the tool coordinate system during directional movement.

[0010] The second step is to model the center path of the previous toolpath trajectory, treating the previous trajectory as a straight line, with the center moving with parameter t in the tool coordinate system. This indicates the lateral offset of the previous trajectory relative to the current trajectory in tool coordinates.

[0011] (7) Step distance s is the lateral offset in the workpiece coordinate system (along...) Therefore, in the tool coordinate system: (8) The third step is to represent the equation of the family circle in the tool coordinate system. The tool bottom circle lies in the "tool bottom plane" (plane) in the tool system. (Center height)), for parameters The equations for the family of circles are: (9) And because the plane containing the circle is perpendicular to the tool axis, the points on the circle The component is equal to the center of the circle. Quantity: (10) To obtain the envelope of the swept surface, the implicit equation can be obtained at a point where there exists a parameter t such that both equations hold simultaneously (and t is eliminated).

[0012] The fourth step is to eliminate the parameter t, from: (11) have to: (12) Bring this in have to: (13) Substituting into the above equation, we get: (14) Regarding the derivation of the V-plane (intersection with the unmachined surface), the V-plane is the part where the tool's rotating surface intersects with the unmachined surface of the workpiece. Since the tool's rotating surface is related to its rake angle and tilt angle in the workpiece coordinate system, α and β need to be considered in the intersection line modeling process. Therefore, the intersection line is described as a function of the tool's rake angle α and tilt angle β.

[0013] Given a vector (0, 0, 1) rotated α° around the x-axis and then β° around the y-axis, the resulting vector is rotated by the following angle around the z-axis: (15) In the oblique plane normal coordinate system In the equation, the expression for surface V is: (16) Writing the plane implicitly: (17) Right now:

[0014] Active rotation transforms the plane normal vector as follows: (18) Finally, the coordinates are transformed to the tool coordinate system. The middle part is: (19) Step 3: Take the intersection line of the previous tool path sweep surface and the unmachined surface with the cylindrical surface, then limit the range to cut out the intersection line part that makes up the cutting contact area, and finally find the intersection point of the two intersection lines with the bottom circle of the cylindrical surface. The line segment formed by connecting the two intersection points is the third line segment that makes up the cutting contact area. The above three line segments make up the tool cutting contact area.

[0015] Then, the solved surface U and the cylindrical surface Intersecting lines yield the intersection line, which can be generated in MATLAB as follows: Figure 3 As shown.

[0016] Then, the solved surface V and the cylindrical surface Intersecting lines yield the intersection line, which can be generated in MATLAB as follows: Figure 4 As shown.

[0017] To solve for the bottom circle segment of the tool, the intersection line of the two planes mentioned above should be used to define the range of the intersection line that forms the cutting contact area. Then, the intersection point with the bottom surface of the cylinder should be found. By combining the two obtained intersection points, the third intersection line segment of the tangent region can be obtained. In MATLAB, the program can generate the following... Figure 5 As shown.

[0018] Step 4: Introduce the point cloud method to solve the point coordinate problem. This involves discretizing the continuous geometry into a finite number of spatial points and handling complex geometric relationships through algebraic operations on the point coordinates. The cylindrical surface generated by the analytical method, the swept surface of the previous tool path, and the unmachined surface are represented as a set of regularly distributed surface points.

[0019] Directly using analytical methods presents several challenges: the superposition of geometric shapes complicates analytical intersections; geometric equations need frequent rewriting after tool rotation; and local contact areas are often only a small part of the overall geometry, making analytical trimming difficult. These issues prolong computation time when generating the cutting contact area in the program. To address these problems, we employ the point cloud method. The core idea of ​​the point cloud method is to discretize the continuous geometry into a finite number of spatial points and handle complex geometric relationships through algebraic operations on the point coordinates.

[0020] When representing the surface of a cylinder using point clouds, the cylinder is not treated as a continuous curved surface, but rather as a set of regularly distributed surface points: (20) Discrete points on a cylinder using angle parameters And the height parameter z represents that each For each point on the surface of a cylinder, all points together constitute the point cloud of the cylinder's surface. The density of the point cloud is controlled by a resolution parameter; the higher the density, the more accurate the approximation of the real geometry.

[0021] Representation of unprocessed surfaces in point clouds: (twenty one) Complex surfaces employ implicit forms: (twenty two) F is a nonlinear function. At the point cloud level, the surface is not solved directly, but is treated as a nonlinear scalar field constraint.

[0022] Step 5: Perform geometric intersection analysis on the surface generated by the analytical method at the point cloud level, that is, select points that simultaneously satisfy the geometric constraints from one or more point cloud sets.

[0023] The essence of the geometric intersection problem is to find points in one or more point cloud sets that simultaneously satisfy geometric constraints. It is not about solving a system of equations, but rather a problem of selecting, solving, and matching points.

[0024] The point cloud concept for intersection point localization: Both intersecting lines exist in the form of point clouds. The intersection point is not obtained directly analytically, but is determined through nearest neighbor search. (twenty three) The principle of rigid body transformation of point clouds: The attitude changes of all geometric bodies are achieved through point cloud coordinate transformation, rather than re-deriving geometric equations. Rotation of the geometric body no longer depends on analytical formula changes, but only on linear algebraic operations. For any set of point clouds... After rotating by an angle α along the Z-axis, we obtain .

[0025] I. Implementation approach for point cloud intersection of cylinder and plane: (1) The point cloud on the surface of a cylinder can be parameterized as follows: (twenty four) Representation of a plane in a point cloud: (25) (2) Fix the point cloud position on the cylindrical surface and discretize the angle parameters. Generate a cylindrical circumferential point cloud: (26) (3) Substitute into the plane equation and calculate the corresponding : (27) This step is equivalent to finding points in the point cloud on the cylindrical surface that satisfy the planar constraints.

[0026] (4) Perform constraint filtering on the obtained point cloud:

[0027] The final point set is obtained as follows: (28) This is the discrete intersection point cloud of the plane and the cylinder.

[0028] II. Point cloud implementation approach for finding the intersection of a cylinder and a curved surface: (1) Complex surfaces adopt implicit forms: (29) F is a nonlinear function. At the point cloud level, the surface is not solved directly, but is treated as a nonlinear scalar field constraint.

[0029] (2) Fix the point cloud position on the cylindrical surface and discretize the angle parameters. Generate a cylindrical circumferential point cloud: (30) (3) Fixed Search for the intersection point along the z-direction, which is the solution: (31) (4) Perform constraint filtering on the obtained point cloud:

[0030] The final point set is obtained as follows: (32) Comparison between traditional analytical methods and analytical methods combined with point cloud methods: When using MATLAB to calculate the tangent region using traditional analytical methods and analytical methods combined with point cloud methods, the traditional analytical method directly performs calculations on the geometric analytical equations, resulting in a large amount of computation and slow program execution speed. In contrast, the analytical method combined with point cloud methods handles complex geometric calculations through algebraic operations on point coordinates, resulting in faster calculation speed.

[0031] The time taken by the traditional analytical method and the analytical method combined with the point cloud method were compared ten times, and the data is recorded in Table 1. The average value is then plotted as a bar chart. Figure 6 As shown in the figure. Experimental verification shows that the analytical method combining point cloud analysis takes significantly less time than the traditional analytical method.

[0032] Table 1 method Traditional analytical method Traditional analytical method combined with point cloud method Traditional analytical method Traditional analytical method combined with point cloud method Traditional analytical method Traditional analytical method combined with point cloud method 1st time 4.424595 0.264627 4.531288 0.261917 4.417421 0.266307 2nd time 4.401749 0.268089 4.404881 0.263051 4.423432 0.37604 3rd 4.528288 0.281976 4.513024 0.287761 4.503111 0.281222 4th 4.556205 0.26666 4.429926 0.293498 4.42556 0.257194 5th 4.405067 0.239038 4.376171 0.262058 4.534889 0.249076 6th 4.311883 0.256212 4.521792 0.286553 4.600422 0.285792 7th 4.315377 0.29475 4.502145 0.251599 4.517801 0.256822 8th 4.370978 0.248618 4.469781 0.282628 4.504668 0.276573 9th 4.638864 0.247337 4.481704 0.276978 4.511406 0.293509 10th 4.304023 0.269235 4.514093 0.241105 4.449744 0.249288 10 averages 4.425703 0.2636542 4.474481 0.270715 4.488845 0.279182 Attached Figure Description

[0033] Figure 1 It represents the transformation relationship between the workpiece coordinate system, the oblique plane normal coordinate system, and the tool coordinate system.

[0034] Figure 2 This is a schematic diagram showing the spatial relationship between the workpiece coordinate system and the oblique plane normal coordinate system.

[0035] Figure 3 It is the intersection line generated in MATLAB when a cylindrical surface intersects with the sweep surface of the previous tool path.

[0036] Figure 4 It refers to generating the intersection line between a cylindrical surface and an unprocessed surface in MATLAB.

[0037] Figure 5 This involves generating a tangential region composed of three line segments in MATLAB.

[0038] Figure 6 It is a bar chart made up of the average of ten data points. Detailed Implementation

[0039] To address the problems existing in the prior art, this invention provides an analytical method for solving the end mill contact area based on the point cloud method during planar machining on a five-axis CNC machine tool, which consists of the following steps: Step 1: Define the workpiece coordinate system, the oblique plane normal coordinate system, and the tool coordinate system, and calculate the transformation relationship between the three coordinate systems using the rotation matrix.

[0040] Step 2: Determine the equations of the geometric surfaces of the cutting area, including the cylindrical surface, the swept surface of the previous tool path, and the unmachined surface.

[0041] Step 3: Take the intersection line of the previous tool path sweep surface and the unmachined surface with the cylindrical surface, then limit the range to cut out the intersection line part that makes up the cutting contact area, and finally find the intersection point of the two intersection lines with the bottom circle of the cylindrical surface. The line segment formed by connecting the two intersection points is the third line segment that makes up the cutting contact area. The above three line segments make up the tool cutting contact area.

[0042] Step 4: Introduce the point cloud method to solve the point coordinate problem. This involves discretizing the continuous geometry into a finite number of spatial points and handling complex geometric relationships through algebraic operations on the point coordinates. The cylindrical surface generated by the analytical method, the swept surface of the previous tool path, and the unmachined surface are represented as a set of regularly distributed surface points.

[0043] Step 5: Perform geometric intersection analysis on the surface generated by the analytical method at the point cloud level, that is, select points that simultaneously satisfy the geometric constraints from one or more point cloud sets.

[0044] The beneficial effects of this invention are as follows: First, it transforms the problem of solving the contact area into the problem of solving the intersection curve between the cylindrical surface, the swept surface of the previous tool path, and the unmachined surface. Second, it introduces the point cloud method to discretize the continuous geometry into a finite number of spatial points, and uses algebraic operations on the point coordinates to handle complex geometric relationships. This solves the problems of complex intersection situations and difficulties in analytical trimming of local contact areas when directly using analytical methods, greatly improving the calculation speed for solving the contact area.

Claims

1. This invention belongs to the field of five-axis CNC machine tool machining in mechanical manufacturing, and relates to planar machining on five-axis CNC machine tools. The process is based on an analytical method for solving the cutting contact area of ​​the end mill using the point cloud method. The specific steps are as follows: Step 1: Define the workpiece coordinate system, the oblique plane normal coordinate system, and the tool coordinate system, and calculate the transformation relationship between the three coordinate systems using the rotation matrix; Workpiece coordinate system Normal coordinate system of oblique plane and tool coordinate system The geometric transformation relationships between the three coordinate systems are as follows; Workpiece coordinate system and the oblique plane normal coordinate system The transformation relationship between them is calculated according to the convention of multiplying the row vector by the rotation matrix on the right: to transform from the workpiece coordinate system to the oblique plane normal coordinate system, it is necessary to first rotate around the X-axis. Angle, then rotate around the Z-axis Angles, calculated in row vector form, are as follows: (1) in (2) Inclined plane normal coordinate system and tool coordinate system The transformation relationship between them is calculated according to the convention of multiplying the row vectors by the rotation matrix on the right: To transform from the oblique plane normal coordinate system to the tool coordinate system, you need to first rotate around the X-axis. Angle, then rotate around the Y-axis Angles, calculated in row vector form, are as follows: (3) in (4) Step 2: Determine the equations of the geometric surfaces of the cutting area, including the cylindrical surface, the swept surface of the previous tool path, and the unmachined surface; The derivation of the U-plane is carried out in four steps, with the tool coordinate system as the reference. The first step is to write out the transformation matrix T. (5) Put column 2 What you extract is the component of the feed direction in the tool coordinate system. : (6) This means that when along Displacement components within the tool coordinate system during directional movement; The second step is to model the center path of the previous toolpath trajectory, treating the previous trajectory as a straight line, with the center moving with parameter t in the tool coordinate system. This indicates the lateral offset of the previous trajectory relative to the current trajectory in tool coordinates; (7) The step distance s is the lower edge of the workpiece coordinate system. The lateral offset, therefore in the tool coordinate system: (8) The third step is to represent the equation of the family circle in the tool coordinate system. The bottom circle of the tool lies on the bottom plane of the tool in the tool system. (Regarding the parameters...) The equations for the family of circles are: (9) And because the plane containing the circle is perpendicular to the tool axis, the points on the circle The component is equal to the center of the circle. Quantity: (10) To obtain the envelope of the swept surface, the implicit equation can be obtained by finding a point where the parameter t exists such that both equations hold simultaneously and eliminating t. The fourth step is to eliminate the parameter t, from: (11) have to: (12) Bring this in have to: (13) Substituting into the above equation, we get: (14) Regarding the derivation of the V-plane, the V-plane is the part where the tool's rotating surface intersects with the unmachined surface of the workpiece. Since the tool's rotating surface is related to its rake angle and tilt angle in the workpiece coordinate system, the intersection line is described as a function of the tool's rake angle α and tilt angle β. Given a vector (0, 0, 1) rotated α° around the x-axis and then β° around the y-axis, the resulting vector is rotated by the following angle around the z-axis: (15) In the oblique plane normal coordinate system In the equation, the expression for surface V is: (16) Writing the plane implicitly: (17) Right now: Active rotation transforms the plane normal vector as follows: (18) Finally, transform to the tool coordinate system The middle part is: (19) Step 3: Take the intersection line of the previous tool path sweep surface and the unmachined surface with the cylindrical surface, then limit the range to intercept the intersection line portion that constitutes the cutting contact area. Finally, find the intersection point of the two intersection lines with the bottom circle of the cylindrical surface. The line segment formed by connecting the two intersection points is the third line segment that constitutes the cutting contact area. The above three line segments constitute the tool cutting contact area; connect the solved surface U with the cylindrical surface. The surfaces intersect to obtain the line of intersection; then the solved surface V is intersected with the cylindrical surface. When they intersect, they form an intersection line; To solve for the bottom circle segment of the tool, the intersection line of the two planes mentioned above should be used to define the range of the intersection line that forms the cutting contact area. Then, the intersection point with the bottom surface of the cylinder should be found. By combining the two obtained intersection points, we can obtain the third intersection line segment of the tangent region; Step 4: Introduce the point cloud method to solve the point coordinate problem. That is, the continuous geometry is discretized into a finite number of spatial points. Complex geometric relationships are handled by algebraic operations on the point coordinates. The cylindrical surface generated by the analytical method, the swept surface of the previous tool path, and the unmachined surface are represented as a set of regularly distributed surface points. Directly using analytical methods presents several problems: the superposition of geometric bodies makes analytical intersection calculations complex; geometric equations need to be frequently rewritten after tool rotation; local contact areas are often only a small part of the overall geometry, making analytical trimming difficult; these problems will increase computation time when the program generates the cutting contact area. To address these issues, we use the point cloud method. The core idea of ​​the point cloud method is to discretize the continuous geometry into a finite number of spatial points and handle complex geometric relationships through algebraic operations on the point coordinates. When representing the surface of a cylinder using point clouds, the cylinder is not treated as a continuous curved surface, but rather as a set of regularly distributed surface points: (20) Discrete points on a cylinder using angle parameters And the height parameter z represents that each For each point on the surface of a cylinder, all points together constitute the point cloud of the cylinder's surface. The density of the point cloud is controlled by a resolution parameter; the higher the density, the more accurate the approximation of the real geometry. The representation of an unprocessed surface in a point cloud is as follows: (21) Complex surfaces employ implicit forms: (22) F is a nonlinear function. At the point cloud level, the surface is not solved directly, but is treated as a nonlinear scalar field constraint. Step 5: Perform geometric intersection on the point cloud level on the surface generated by the analytical method, that is, select points that simultaneously satisfy the geometric constraints from one or more point cloud sets; The essence of the geometric intersection problem is to find points in one or more point cloud sets that simultaneously satisfy geometric constraints. It is not about solving a system of equations, but rather a problem of selecting, solving, and matching points. The point cloud concept for intersection point localization: Both intersecting lines exist in the form of point clouds. The intersection point is not obtained directly analytically, but is determined through nearest neighbor search. (23) The principle of rigid body transformation of point clouds: The attitude changes of all geometric objects are achieved through point cloud coordinate transformation, rather than re-deriving geometric equations. The rotation of geometric objects no longer depends on changes in analytical formulas, but only on linear algebraic operations. For any set of point clouds... After rotating by an angle α along the Z-axis, we obtain ; I. Implementation approach for point cloud intersection of cylinder and plane: (1) The point cloud on the surface of a cylinder can be parameterized as follows: (24) Representation of a plane in a point cloud: (25) (2) Fix the point cloud position on the cylindrical surface and discretize the angle parameters. Generate a circumferential point cloud of a cylinder: (26) (3) Substitute into the plane equation and calculate the corresponding : (27) This step is equivalent to finding points that satisfy the planar constraints in the point cloud on the cylindrical surface; (4) Perform constraint filtering on the obtained point cloud: The final point set is obtained as follows: (28) This refers to the discrete intersection point cloud of the plane and the cylinder; II. Point cloud implementation approach for finding the intersection of a cylinder and a curved surface: (1) Complex surfaces adopt implicit forms: (29) F is a nonlinear function. At the point cloud level, the surface is not solved directly, but is treated as a nonlinear scalar field constraint. (2) Fix the point cloud position on the cylindrical surface and discretize the angle parameters. Generate a circumferential point cloud of a cylinder: (30) (3) Fixed Search for the intersection point along the z-direction, which is the solution: (31) (4) Perform constraint filtering on the obtained point cloud: The final point set is obtained. (32)。

Images