An adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-shaped end mill.

Abstract

This invention discloses an adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-head end mill. Specifically, the method involves: First, constructing the cutting edge equations of the peripheral and end edges of the arc-head end mill based on structural feature decomposition, and establishing a parameterized model of the second flank face through uniform offset densification of the cutting edges. Then, based on the established grinding wheel posture model, determining the grinding interference by calculating the grinding contact area between the grinding wheel and the first flank face of the end mill. Finally, an adaptive adjustment algorithm for the grinding wheel grinding trajectory is used to iteratively correct the grinding wheel lift angle and swing angle, avoiding abrupt changes in grinding posture and grinding interference, thus achieving precise grinding of the flank face. This invention effectively avoids abrupt changes in grinding posture and grinding interference, resulting in a smooth and continuous flank face surface with good quality, achieving precise grinding of the flank face.

Description

Technical Field

[0001] This invention belongs to the technical field of end mill structure design, and particularly relates to an adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-head end mill. Background Technology

[0002] The circular end mill is a type of solid end mill. Compared to ball end mills, it offers greater rigidity and higher machining efficiency when machining parts with rounded corners. Compared to flat end mills, it exhibits higher resistance to chipping when machining hard materials and performing high-speed, high-feed machining. Due to its low manufacturing cost and high material removal rate, it is increasingly used in the aerospace, automotive, and mold industries for efficient machining of free-form surfaces. The flank face of the circular end mill is a critical part of the cutting process; its structural design plays a decisive role in the tool's lifespan, machining efficiency, and quality.

[0003] The grinding of the flank face of a circular arc head end mill includes two aspects: the construction and optimization of the mathematical model of the end mill and the calculation method of the grinding trajectory and posture. Regarding the construction and optimization of the mathematical model of the end mill, Chen Tao [1] defined the geometric features such as the flank face of the circular arc head end mill with rounded corners, established the mathematical model of the end mill profile, and derived the parametric equation of the spiral cutting edge. Han [2] proposed a method of generalized modeling of the end edge to address the structural characteristics of the tooth center and end inclination of the circular arc head end mill, so that the cutting lines of different parts can maintain continuity, and derived the tool path for flank face grinding based on the parametric model. Chen [3] proposed a design and manufacturing method of a circular arc end mill with double circular arc features to address the limitations of ordinary circular arc end mills in machining, and established a mathematical model of the end mill cutting edge. Regarding the calculation method of the grinding trajectory and posture of the flank face, Tang Jun et al. [4] studied a flank face trajectory algorithm of the arc head end mill using parallel grinding wheel for the grinding process of the flank face of the arc head end mill. Liu Jianjun et al. [5] established a mathematical model of the end cutting edge line of the arc head end mill and studied the grinding process of the end cutting edge flank face. They proposed a calculation method for the motion trajectory of the grinding wheel and the tool axis vector, which can improve the manufacturing accuracy of the tool and reduce the design cycle. Ma [6] analyzed the error of the flank face based on the contour characteristics of the grinding wheel wear and proposed a flank face grinding trajectory compensation algorithm based on the grinding wheel wear, which can reduce the influence of grinding wheel wear on the flank face grinding process. In summary, the current research on the flank face grinding process of the arc head end mill is mainly based on the segmented design of the flank face of the end mill, and no unified optimization iterative design method has been formed. Due to the independence of the grinding process of each part, problems such as grinding interference and sudden change of grinding wheel grinding mode may occur, resulting in discontinuity of the flank face and poor surface quality.

[0004] References

[0005] [1] Chen Tao, Li Xianchuang, Wang Guangyue, Liu Xianli. Key technologies for the design of rounded end mills with rounded corners [J]. Journal of Harbin University of Science and Technology, 2017, 22(01):75-79.

[0006] [2]Lei Han et al. Research on parametric modeling and grindingmethods of bottom edge of toroid-shaped end-milling cutter[J]. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of EngineeringManufacture, 2019, 233(1): 31-43

[0007] [3]Tao Chen et al. Design and fabrication of double-circular-arctorus milling cutter[J]. The International Journal of Advanced Manufacturing Technology, 2015, 80(1-4): 567-579.

[0008] [4] Tang Jun, Ma Zhongbao, Luo Bin, Jiang Lei. Algorithm for trajectory of the back face of a circular arc head end mill ground by parallel grinding wheel [J]. Manufacturing Technology & Machine Tool, 2021(09):86-91.

[0009] [5] Liu Jianjun, Li Rong, Cheng Xuefeng, Jin Xiaobo, Ding Guofu. Research on CNC grinding simulation technology of end mill edge of circular arc head end mill [J]. Modern Manufacturing Engineering, 2012(10):84-89.

[0010] [6]Ma Yuhao et al. A compensation algorithm of tool path for grindingwheel wear in solid cutting tool flank grinding[J]. Proceedings of the Institution of Mechanical Engineers, 2022, 236(3): 245-254. Summary of the Invention

[0011] To address the problems existing in the prior art, this invention provides an adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-head end mill.

[0012] An adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-shaped end mill includes the following steps:

[0013] Step 1: Define the geometric parameters of the flank face of the arc-head end mill.

[0014] End mill starting radius R W The tool radius at the starting position of the helical cutting edge of the end mill.

[0015] Circumferential cutting length L W The length of the peripheral cutting edge of the end mill along the axial direction.

[0016] Taper angle k: The angle between the generatrix of the tooth rotation profile and the axis of the end mill.

[0017] Helix angle β: The angle between the helix cutting edge tangent vector and the tool rotation profile generatrix.

[0018] End blade inclination angle The angle between the straight cutting edge line of a circular arc end mill and the perpendicular plane to the tool axis.

[0019] Tooth offset center h: The shortest distance between the straight cutting edge line of the end mill and the axis of the end mill.

[0020] Tooth over center amount The projection of the line connecting the end point of the straight cutting edge of the end blade and the axis of the tool onto the straight cutting edge direction of the end blade.

[0021] End-edge radius r: the radius of the generatrix of the end-edge arc surface.

[0022] Back face width : The profile length of the flank face on the normal section of the cutting edge, i.e., the length of the intersection line between the flank face and the normal section of the cutting edge. If multiple flank faces exist, the widths of the flank faces are respectively... , where n represents the number of back faces.

[0023] Back face angle : is the angle between the profile of the flank face on the cutting edge normal section and the normal section of the end mill axis, where the flank angles are respectively... .

[0024] Step 2: Establish a coordinate system.

[0025] Workpiece Coordinate System (WCS)

[0026] Workpiece coordinate system O W -X W Y W Z W With the tool rotation axis as ZW The axis is defined by the end face where the starting position of the circumferential cutting edge line is located. W O W Y W Plane, with the center of the end face circle as the origin O of the coordinate system. W .

[0027] End-edge coordinate system EECS

[0028] The end-edge coordinate system is based on the tool rotation axis as Z. d The axis is defined by the interface between the end cutting edge and the peripheral cutting edge. d O d Y d The origin O of the coordinate system is the center of the intersection surface. d .

[0029] End-edge principal section coordinate system MSCS

[0030] With the cutting edge point P0 as the origin O of the coordinate system mt Let the tangent to the longitude line at the current location be Z. mt The axis is defined by the tangent to the latitude line at the current location. mt axis.

[0031] End-edge section coordinate system NSCS

[0032] Let the principal section coordinate system revolve around X mt Rotate the axis to establish the coordinate system O of the end-edge normal section. mt1 -X mt1 Y mt1 Z mt1 The end-edge section coordinate system has the cutting edge point P0 as the origin O. mt1 Z is the tangent of the cutting edge. mt1 axis, Y mt1 The shaft is tangent to the end-edge arc surface and perpendicular to the cutting line.

[0033] Step 3: Coordinate system transformation matrix.

[0034] Transform from the end-edge section coordinate system NSCS to the end-edge principal section coordinate system MSCS

[0035] First, set the tangent vector F of the current position of the cutting edge. p The coordinate system Z is connected to the cutting edge, with the line passing through point P0 and tangent to the arc surface, pointing towards the end edge coordinate system. d The vector F of the axis d They are defined as follows:

[0036] (1)

[0037] In the formula This represents the coordinates of point P0 on the cutting edge in the end-edge coordinate system EECS.

[0038] So, Fp With F d The included angle is the angle that needs to be rotated around X when transforming from the end-edge section coordinate system to the principal section coordinate system. mt1 The rotation angle of the axis; therefore, the transformation matrix R mt1-mt As shown below:

[0039] (2)

[0040] Transform the end-edge principal section coordinate system MSCS to the end-edge coordinate system EECS.

[0041] Define R mt-d T mt-d Let be the rotation and translation matrices used to transform the end-edge principal section coordinate system to the end-edge coordinate system, respectively. Then:

[0042] (3)

[0043] Transform the end-edge coordinate system EECS to the workpiece coordinate system WCS.

[0044] Define R d-w T d-w Let be the rotation matrix and displacement matrix for transforming the end-edge coordinate system to the workpiece coordinate system, respectively. Then:

[0045] (4)

[0046] In the formula, The angle through which the cutting edge of the end mill rotates around the axis of the end mill, from the starting point of the circumferential cutting edge to the end point of the circumferential cutting edge.

[0047] Step 4: Establish a mathematical model of the back face of the end mill with a circular arc head.

[0048] Equation of the end-edge circular arc cutting edge line:

[0049] (5)

[0050] In the formula, θ represents the latitude angle of P0 on the end-edge cutting line, and R... d This is the distance between the center of the end-edge arc and the tool axis. R represents the rotation angle at point P0 on the arc-shaped cutting edge. d and The expression is as follows:

[0051] (6)

[0052] (7)

[0053] Equation of the cutting edge line of a plane curve:

[0054] (8)

[0055] In the formula This indicates the straight cutting edge of the end edge in the X-axis of the end edge coordinate system. d O d Y d Projection in the plane and the end-edge coordinate system X d The included angle of the axis.

[0056] Equation of the straight cutting edge line:

[0057] (9)

[0058] In the formula, P P2_d With F P2_d These represent the final point P of the curved section of the end edge. 2d The coordinate vectors at point P 2d Tangent vector at point P 2d The coordinates are obtained using equation (8). This indicates the amount of tooth passing through the center.

[0059] Step 5: Establish the first flank width control line and its offset curve family.

[0060] The width and angle of the first flank face are determined by the relative positions of point P1 and point P0. Point P1 is the control point of the first flank face on the cross section. The curve formed by the control points on each cross section is the control line C1 of the first flank face. The coordinate vector of point P1 is represented as follows:

[0061] (10)

[0062] In the formula, and These represent the widths of the first and second flank faces, respectively. and These are the first rear angle and the second rear angle.

[0063] The width of the flank face is calculated by establishing the offset curve C2 of the control line on the second flank face. Point P2 on C2 is represented in the end-edge normal section coordinate system as follows:

[0064] (11)

[0065] Based on the coordinate system transformation relationship proposed above, namely equation (8), point P2 can be represented in the end-edge coordinate system as:

[0066] (12)

[0067] Combining equations (4) and (12), point P2 can be represented in the workpiece coordinate system as follows:

[0068] (13)

[0069] According to equation (11), when the first flank cutting edge line remains unchanged, the position of the offset curve of the flank control line is changed from... Decision; therefore, by changing the step size in equal increments This yields multiple offset curves on the second flank face, also known as the offset curve family. The intersection of the grinding wheel with any one of these curves indicates that the grinding wheel is in contact with the second flank face, thereby allowing the width of the flank face to be detected.

[0070] Step 6: Define the grinding wheel posture.

[0071] Grinding wheel tilt angle

[0072] Connect the cutting edge point P0 with the grinding wheel axis vector F. g0 The parallel vector is defined as the X-axis of the flank face in the end-edge normal section coordinate system. mt1 O mt1 Y mt1 Normal vector F under the plane g1 normal vector F g1 With the grinding wheel coordinate system X s aligning the axes, the grinding wheel is rotated around F. g1 The angle rotated clockwise is defined as the swing angle. .

[0073] Grinding wheel lifting angle

[0074] The vector pointing from P0 to P1 after the swing angle transformation is defined as the tangent vector F of the grinding wheel at the cutting edge point. t1 , rotate the grinding wheel around F t1 That is, the grinding wheel coordinate system Y s The angle through which the axis rotates counterclockwise is defined as the lift angle. .

[0075] Step 7: Model the contour of the large end face of the grinding wheel.

[0076] The center O of the large end face of the grinding wheel under the end-edge section. g With axis vector F g coordinate.

[0077] (14)

[0078] In the formula, O Og0_mt1 and F g0_mt1 Representing O g With F g The coordinate vector in the initial attitude.

[0079] The lifting angle during the grinding process With swing angle After substituting into the calculation, we get O g With F g coordinate vector and :

[0080] (15)

[0081] In the formula , , ;

[0082] Substituting the calculated coordinates of the grinding wheel's large end face center and axis vector into the coordinate system transformation equations (2)-(4) will transform it to the workpiece coordinate system; then, the axis vector F obtained will be used to transform it to the workpiece coordinate system. g The center of the large end face of the grinding wheel, O g Coordinates and grinding wheel radius R g This yields the equation for the profile of the large end face of the grinding wheel:

[0083] (16)

[0084] In the formula, x, y, and z represent the solutions to the equation, that is, the coordinates of points on the large end face profile of the grinding wheel. and F g With O g The coordinate vector in the workpiece coordinate system X W Y W Z W Components in direction.

[0085] Step 8: Adaptive adjustment of grinding wheel posture based on grinding zone control.

[0086] The interference of the grinding wheel on the second flank face is detected by using the cutting edge equation and the outer circle contour equation of the large end face of the grinding wheel:

[0087] Equation (17) is a detection formula for determining whether the offset point of the cutting edge on the curved surface of the second back cutter is in contact with the large end face of the grinding wheel. In the formula, d1 is the distance between the offset point of the cutting edge and the center of the large end face of the grinding wheel and the radius R of the grinding wheel. g The difference is d2, which is the distance between the offset point P2 of the cutting edge and the plane where the large end face of the grinding wheel is located. If d1=d2=0, it means that the offset curve C2 of the back face control line composed of P2 intersects the outline circle of the large end face of the grinding wheel when machining the first back face. That is, the grinding wheel touches the second back face and causes interference. At this time, the attitude of the grinding wheel is adaptively adjusted.

[0088] (17)

[0089] Determine whether the large end face of the grinding wheel is in contact with the control line C1 on the back face under the current posture:

[0090] Equation (18) is the detection formula for the contact state of the back face control line. The distance between the back face control point and the center of the large end face of the grinding wheel is detected by d3, and the distance between the back face control point and the large end face of the grinding wheel is detected by d4. If d4=0 and d3≤0, it means that the large end face of the grinding wheel is still in contact with the back face control line C1 at the current position, and the posture adjustment continues; if the above conditions are not met, it means that the normal processing at the current position will be affected, and the posture adjustment is stopped.

[0091] (18)

[0092] During the machining of the back face, the grinding wheel posture needs to be adaptively detected and adjusted multiple times at each position until the large end face contour circle of the grinding wheel no longer intersects with the offset curve C2 on the second back face before proceeding to the next position.

[0093] Furthermore, the specific methods for adjusting the grinding wheel posture are as follows:

[0094] Step 1: Set the initial values ​​of the grinding wheel attitude and calculate the grinding wheel coordinates.

[0095] Before performing adaptive adjustments, the attitude of the grinding wheel, i.e., the lifting angle, must first be set. With swing angle The initial values ​​of the swing angle and the lifting angle are set to -5° and 10° respectively, and the coordinates O of the center of the large end face of the grinding wheel at the current position are calculated based on the grinding wheel posture. g With axis vector coordinates F g .

[0096] Step 2: Determine the current grinding state of the grinding wheel.

[0097] By O g With F g Substituting into equations (16) and (17) yields the circular profile equation of the large end face of the grinding wheel and determines whether the large end face of the grinding wheel is in contact with the offset curve C2 of the control line of the back face. If it is in contact, it indicates that interference has occurred, and the process proceeds to Step 3 and subsequent adjustments. If it is not in contact, it indicates that no interference has occurred, and the process proceeds directly to the calculation and adjustment of the next machining position.

[0098] Step 3: Angle Adjustment Adjustment

[0099] When adjusting the grinding wheel posture, the first consideration should be the grinding wheel tilt angle. Impact on the processing; by increasing the swing angle The value moves the grinding wheel away from the offset curve C2, and the adjustment range is [value missing]. The angle is 0.1°. After each adjustment, it must be determined whether the grinding wheel affects the normal machining at the current position, i.e., whether it contacts the control line C1 of the back face at the current position. If it does not affect normal machining, jump back to Step 2 and check for interference again. If it affects normal machining, it means that it is no longer possible to adjust the angle by only adjusting the swing angle. Adjust the grinding wheel posture, then proceed to Step 4, lifting the angle. Adjustment process: When the result of Step 2 is that no interference occurs, skip to Step 5 to end the adjustment.

[0100] Step 4: Raise the corner Adjustment

[0101] Due to the lifting angle Changes occur, swing angle The attitude change process changes, at which point the swing angle should be adjusted. The value is set to the initial adjustment value, i.e., -5°, for the tilt angle. Adjustment range Also 0.1°, lift angle Each adjustment must be evaluated to determine if it affects the processing. If it does not, proceed back to Step 2 to determine if interference has occurred and make subsequent adjustments to the swing angle until no interference or tilting occurs. The adjustment continues until it affects the normal processing at the current position; if the angle is raised... Adjustments to the first flank face will also affect normal machining, indicating that the swing angle has been modified. With the angle Neither can completely prevent the grinding wheel from interfering with the second flank face. Therefore, the angle of the grinding position should be adjusted. With the angle Set all values ​​to the values ​​from the last adjustment, then proceed to Step 5 to finish the adjustment.

[0102] Step 5: End the adaptive adjustment process and proceed to the calculation of the next position.

[0103] The beneficial technical effects of this invention are as follows:

[0104] The method of this invention can effectively avoid problems such as abrupt changes in grinding posture and grinding interference. The machined flank face has a smooth and continuous surface with good quality, and achieves precise grinding of the flank face. Attached Figure Description

[0105] Figure 1 This is a schematic diagram of the back face of a circular arc head end mill.

[0106] Figure 2 This is a schematic diagram of the workpiece coordinate system. In the figure, (a) is the isometric view of the end mill, and (b) is the parameter description of the end mill.

[0107] Figure 3 This is a schematic diagram of the end-edge coordinate system, the end-edge principal section, and the normal section coordinate system.

[0108] Figure 4 This refers to the cutting edge line of an arc-head end mill.

[0109] Figure 5 This is a schematic diagram of the cross-section of the back face.

[0110] Figure 6 This refers to the control lines of the back face and their offset curves.

[0111] Figure 7 This is a schematic diagram of the grinding wheel's tilt angle.

[0112] Figure 8 This is a schematic diagram of the grinding wheel lifting angle.

[0113] Figure 9 This is a schematic diagram of the grinding wheel attitude adjustment process. In the diagram, (a) shows the swing angle adjustment process, and (b) shows the swing angle adjustment process.

[0114] Figure 10 This is a flowchart for adjusting the attitude of the grinding wheel.

[0115] Figure 11 The figures show a comparison of simulation results for machining the flank face. In the figures, (a) and (c) show the simulation results of machining the flank face with the algorithm adjusted for the front cutting edge, and (b) and (d) show the simulation results of machining the flank face with the algorithm adjusted for the rear cutting edge.

[0116] Figure 12 The figures show a comparison of actual machining results. In the figures, (a) and (c) show the actual machining results of the algorithm adjusting the front cutting edge and the flank face, and (b) and (d) show the actual machining results of the algorithm adjusting the rear cutting edge and the flank face. Detailed Implementation

[0117] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0118] The present invention provides an adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-head end mill, comprising the following steps:

[0119] Step 1: Define the geometric parameters of the flank face of the arc-head end mill.

[0120] To more completely and accurately analyze and study the structure of the flank face of a circular arc head end mill, combined with Figure 1 A schematic diagram of the flank face of an arc-head end mill, defining the following parameters:

[0121] End mill starting radius R W The tool radius at the starting position of the helical cutting edge of the end mill.

[0122] Circumferential cutting length L WThe length of the peripheral cutting edge of the end mill along the axial direction.

[0123] Taper angle k: The angle between the generatrix of the tooth rotation profile and the axis of the end mill.

[0124] Helix angle β: The angle between the helix cutting edge tangent vector and the tool rotation profile generatrix.

[0125] End blade inclination angle The angle between the straight cutting edge line of a circular arc end mill and the perpendicular plane to the tool axis.

[0126] Tooth offset center h: The shortest distance between the straight cutting edge line of the end mill and the axis of the end mill.

[0127] Tooth over center amount The projection of the line connecting the end point of the straight cutting edge of the end blade and the axis of the tool onto the straight cutting edge direction of the end blade.

[0128] End-edge radius r: the radius of the generatrix of the end-edge arc surface.

[0129] Back face width : The profile length of the flank face on the normal section of the cutting edge, i.e., the length of the intersection line between the flank face and the normal section of the cutting edge. If multiple flank faces exist, the widths of the flank faces are respectively... , where n represents the number of back faces.

[0130] Back face angle : is the angle between the profile of the flank face on the cutting edge normal section and the normal section of the end mill axis, where the flank angles are respectively... .

[0131] Step 2: Establish a coordinate system.

[0132] Workpiece coordinate system (WCS) (e.g.) Figure 2 (As shown)

[0133] Workpiece coordinate system O W -X W Y W Z W With the tool rotation axis as Z W The axis is defined by the end face where the starting position of the circumferential cutting edge line is located. W O W Y W Plane, with the center of the end face circle as the origin O of the coordinate system. W The expression of the cutting edges of each part of the arc-head end mill and the definition of the grinding wheel posture must ultimately be transformed into the workpiece coordinate system for description.

[0134] End-edge coordinate system EECS (e.g.) Figure 3 (As shown)

[0135] Since it is difficult to directly model the cutting edge of the end cutting edge in the workpiece coordinate system, we first model it in the end cutting edge coordinate system O. d -X d Y d Z d The cutting edge of the end-edge is modeled and then transformed into the workpiece coordinate system through coordinate system transformation. The end-edge coordinate system is based on the tool rotation axis as Z. d The axis is defined by the interface between the end cutting edge and the peripheral cutting edge. d O d Y d The origin O of the coordinate system is the center of the intersection surface. d .

[0136] End-edge principal section coordinate system MSCS (e.g.) Figure 3 (As shown)

[0137] Define the coordinate system O of the end-edge principal section mt -X mt Y mt Z mt This describes the shape of the cutting edge's rear face and the grinding wheel's posture. The origin O of the coordinate system is set at the cutting edge point P0. mt Let the tangent to the longitude line at the current location be Z. mt The axis is defined by the tangent to the latitude line at the current location. mt axis.

[0138] End-edge section coordinate system NSCS (e.g.) Figure 3 (As shown)

[0139] In the end-cutting section, due to the significant variation in the curvature of the cutting edge, while the principal section coordinate system can describe the position of each cutting edge point, it cannot accurately express the shape and width of the flank face. Therefore, the principal section coordinate system is revolved around the X-axis. mt Rotate the axis to establish the coordinate system O of the end-edge normal section. mt1 -X mt1 Y mt1 Z mt1 The end-edge section coordinate system has the cutting edge point P0 as the origin O. mt1 Z is the tangent of the cutting edge. mt1 axis, Y mt1 The shaft is tangent to the end-edge arc surface and perpendicular to the cutting line.

[0140] Step 3: Coordinate system transformation matrix.

[0141] To facilitate tool setting on the grinding wheel and post-processing on CNC grinding machines, the grinding trajectory in the linear cutting edge coordinate system of the drill tip needs to be transformed to the workpiece coordinate system.

[0142] Transform from the end-edge section coordinate system NSCS to the end-edge principal section coordinate system MSCS

[0143] First, set the tangent vector F of the current position of the cutting edge. p The coordinate system Z is connected to the cutting edge, with the line passing through point P0 and tangent to the arc surface, pointing towards the end edge coordinate system. d The vector F of the axis d They are defined as follows:

[0144] (1)

[0145] In the formula This represents the coordinates of point P0 on the cutting edge in the end-edge coordinate system EECS.

[0146] So, F p With F d The included angle is the angle that needs to be rotated around X when transforming from the end-edge section coordinate system to the principal section coordinate system. mt1 The rotation angle of the axis; therefore, the transformation matrix R mt1-mt As shown below:

[0147] (2)

[0148] Transform the end-edge principal section coordinate system MSCS to the end-edge coordinate system EECS.

[0149] Define R mt-d T mt-d Let be the rotation and translation matrices used to transform the end-edge principal section coordinate system to the end-edge coordinate system, respectively. Then:

[0150] (3)

[0151] Transform the end-edge coordinate system EECS to the workpiece coordinate system WCS.

[0152] Define R d-w T d-w Let be the rotation matrix and displacement matrix for transforming the end-edge coordinate system to the workpiece coordinate system, respectively. Then:

[0153] (4)

[0154] In the formula, The angle through which the cutting edge of the end mill rotates around the axis of the end mill, from the starting point of the circumferential cutting edge to the end point of the circumferential cutting edge.

[0155] Step 4: Establish a mathematical model of the back face of the end mill with a circular arc head.

[0156] Referring to the modeling method of the end mill cutting edge proposed by Tang Jun et al. [4], the cutting edge equations of each part of the end mill can be established as shown below. The cutting edge of the end mill cutting edge is as follows: Figure 4 As shown.

[0157] Equation of the end-edge circular arc cutting edge line:

[0158] (5)

[0159] In the formula, θ represents the latitude angle of P0 on the end-edge cutting line, and R... d This is the distance between the center of the end-edge arc and the tool axis. R represents the rotation angle at point P0 on the arc-shaped cutting edge. d and The expression is as follows:

[0160] (6)

[0161] (7)

[0162] Equation of the cutting edge line of a plane curve:

[0163] (8)

[0164] In the formula This indicates the straight cutting edge of the end edge in the X-axis of the end edge coordinate system. d O d Y d Projection in the plane and the end-edge coordinate system X d The included angle of the axis.

[0165] Equation of the straight cutting edge line:

[0166] (9)

[0167] In the formula, P P2_d With F P2_d These represent the final point P of the curved section of the end edge. 2d The coordinate vectors at point P 2d Tangent vector at point P 2d The coordinates are obtained using equation (8). This indicates the amount of tooth passing through the center.

[0168] Step 5: Establish the first flank width control line and its offset curve family.

[0169] To detect whether the large end face profile of the cup-shaped grinding wheel interferes with the second flank face, this invention analyzes the interference state using the control line of the first flank face and its family of offset curves. The control line of the first flank face is the theoretical intersection line of the first flank face and the second flank face, and the family of offset curves consists of multiple equidistant offset curves of the control line on the second flank face.

[0170] like Figure 5As shown, the width and angle of the first flank face are determined by the relative positions of points P1 and P0. Point P1 is the control point of the first flank face on the cross section, and the curve formed by the control points on each cross section is the control line C1 of the first flank face. The coordinate vector of point P1 is represented as follows:

[0171] (10)

[0172] In the formula, and These represent the widths of the first and second flank faces, respectively. and These are the first rear angle and the second rear angle.

[0173] The width of the flank face is calculated by establishing the offset curve C2 of the control line on the second flank face. Point P2 on C2 is represented in the end-edge normal section coordinate system as follows:

[0174] (11)

[0175] Based on the coordinate system transformation relationship proposed above, namely equation (8), point P2 can be represented in the end-edge coordinate system as:

[0176] (12)

[0177] Combining equations (4) and (12), point P2 can be represented in the workpiece coordinate system as follows:

[0178] (13)

[0179] According to equation (11), when the first flank cutting edge line remains unchanged, the position of the offset curve of the flank control line is changed from... Decision; therefore, by changing the step size in equal increments This yields multiple offset curves on the second flank face, i.e., a family of offset curves (such as...). Figure 6 (As shown in the figure); the intersection of the grinding wheel with any of the curves indicates that the grinding wheel is in contact with the second flank face, thereby detecting the width of the flank face.

[0180] Step 6: Define the grinding wheel posture.

[0181] During the grinding process of the end-edge relief face, due to the dense arrangement of the end-edge cutting lines and the complex curvature changes, grinding interference is inevitable if the grinding wheel maintains its initial posture. This invention addresses this by defining and calculating the grinding wheel tilt angle. With the angle This enables dynamic adjustment of the grinding wheel's posture.

[0182] Grinding wheel tilt angle

[0183] like Figure 7 As shown, the point P0 on the cutting edge line is aligned with the grinding wheel axis vector F. g0 The parallel vector is defined as the X-axis of the flank face in the end-edge normal section coordinate system. mt1 O mt1 Y mt1 Normal vector F under the plane g1 normal vector F g1 With the grinding wheel coordinate system X s aligning the axes, the grinding wheel is rotated around F. g1 The angle rotated clockwise is defined as the swing angle. .

[0184] Grinding wheel lifting angle

[0185] like Figure 8 As shown, the vector pointing from P0 to P1 after the swing angle transformation is defined as the tangent vector F of the grinding wheel at the cutting edge point. t1 , rotate the grinding wheel around F t1 That is, the grinding wheel coordinate system Y s The angle through which the axis rotates counterclockwise is defined as the lift angle. .

[0186] Step 7: Model the contour of the large end face of the grinding wheel.

[0187] The derivation of the equation for the profile circle of the large end face of the grinding wheel requires the large end face radius R of the grinding wheel. g In addition to determining the dimensions, it is also necessary to use the coordinates O of the grinding wheel center. g Determine the grinding wheel position and use the grinding wheel axis vector F g Determine the grinding wheel posture. According to the definition and derivation process of grinding wheel posture in reference [4], the center O of the large end face of the grinding wheel under the end-edge normal section can be obtained. g With axis vector F g coordinate.

[0188] (14)

[0189] In the formula, O Og0_mt1 and F g0_mt1 Representing O g With F g The coordinate vector in the initial attitude.

[0190] The lifting angle during the grinding process With swing angle After substituting into the calculation, we get O g With F g coordinate vector and :

[0191] (15)

[0192] In the formula , , ;

[0193] Substituting the calculated coordinates of the grinding wheel's large end face center and axis vector into the coordinate system transformation equations (2)-(4) will transform it to the workpiece coordinate system; then, the axis vector F obtained will be used to transform it to the workpiece coordinate system. g The center of the large end face of the grinding wheel, O g Coordinates and grinding wheel radius R g This yields the equation for the profile of the large end face of the grinding wheel:

[0194] (16)

[0195] In the formula, x, y, and z represent the solutions to the equation, that is, the coordinates of points on the large end face profile of the grinding wheel. and F g With O g The coordinate vector in the workpiece coordinate system X W Y W Z W Components in direction.

[0196] Step 8: Adaptive adjustment of grinding wheel posture based on grinding zone control (e.g.) Figure 9 (As shown).

[0197] While keeping the second flank angle constant, the width l1 of the first flank can be significantly changed by adjusting the grinding wheel's machining posture. To maintain a consistent width of the first flank, the wheel's tilt angle needs to be adjusted according to the actual machining conditions. With the angle Adjust the angle.

[0198] The interference of the grinding wheel on the second flank face is detected by using the cutting edge equation and the outer circle contour equation of the large end face of the grinding wheel:

[0199] Equation (17) is a detection formula for determining whether the offset point of the cutting edge on the curved surface of the second back cutter is in contact with the large end face of the grinding wheel. In the formula, d1 is the distance between the offset point of the cutting edge and the center of the large end face of the grinding wheel and the radius R of the grinding wheel. g The difference is d2, which is the distance between the offset point P2 of the cutting edge and the plane where the large end face of the grinding wheel is located. If d1=d2=0, it means that the offset curve C2 of the back face control line composed of P2 intersects the outline circle of the large end face of the grinding wheel when machining the first back face. That is, the grinding wheel touches the second back face and causes interference. At this time, the attitude of the grinding wheel is adaptively adjusted.

[0200] (17)

[0201] Determine whether the large end face of the grinding wheel is in contact with the control line C1 on the back face under the current posture:

[0202] Equation (18) is the detection formula for the contact state of the back face control line. The distance between the back face control point and the center of the large end face of the grinding wheel is detected by d3, and the distance between the back face control point and the large end face of the grinding wheel is detected by d4. If d4=0 and d3≤0, it means that the large end face of the grinding wheel is still in contact with the back face control line C1 at the current position, and the posture adjustment continues; if the above conditions are not met, it means that the normal processing at the current position will be affected, and the posture adjustment is stopped.

[0203] (18)

[0204] During the machining of the back face, the grinding wheel posture needs to be adaptively detected and adjusted multiple times at each position until the large end face contour circle of the grinding wheel no longer intersects with the offset curve C2 on the second back face before proceeding to the next position.

[0205] Adjusting the grinding wheel posture, as follows Figure 10 As shown, the details are as follows:

[0206] Step 1: Set the initial values ​​of the grinding wheel attitude and calculate the grinding wheel coordinates.

[0207] Before performing adaptive adjustments, the attitude of the grinding wheel, i.e., the lifting angle, must first be set. With swing angle The initial values ​​of the swing angle and the lifting angle are set to -5° and 10° respectively, and the coordinates O of the center of the large end face of the grinding wheel at the current position are calculated based on the grinding wheel posture. g With axis vector coordinates F g .

[0208] Step 2: Determine the current grinding state of the grinding wheel.

[0209] By O g With F g Substituting into equations (16) and (17) yields the circular profile equation of the large end face of the grinding wheel and determines whether the large end face of the grinding wheel is in contact with the offset curve C2 of the control line of the back face. If it is in contact, it indicates that interference has occurred, and the process proceeds to Step 3 and subsequent adjustments. If it is not in contact, it indicates that no interference has occurred, and the process proceeds directly to the calculation and adjustment of the next machining position.

[0210] Step 3: Angle Adjustment Adjustment

[0211] When adjusting the grinding wheel posture, the first consideration should be the grinding wheel tilt angle. Impact on the processing; by increasing the swing angle The value moves the grinding wheel away from the offset curve C2, and the adjustment range is [value missing]. The angle is 0.1°. After each adjustment, it must be determined whether the grinding wheel affects the normal machining at the current position, i.e., whether it contacts the control line C1 of the back face at the current position. If it does not affect normal machining, jump back to Step 2 and check for interference again. If it affects normal machining, it means that it is no longer possible to adjust the angle by only adjusting the swing angle. Adjust the grinding wheel posture, then proceed to Step 4, lifting the angle. Adjustment process: When the result of Step 2 is that no interference occurs, skip to Step 5 to end the adjustment.

[0212] Step 4: Raise the corner Adjustment

[0213] Due to the lifting angle Changes occur, swing angle The attitude change process changes, at which point the swing angle should be adjusted. The value is set to the initial adjustment value, i.e., -5°, for the tilt angle. Adjustment range Also 0.1°, lift angle Each adjustment must be evaluated to determine if it affects the processing. If it does not, proceed back to Step 2 to determine if interference has occurred and make subsequent adjustments to the swing angle until no interference or tilting occurs. The adjustment continues until it affects the normal processing at the current position; if the angle is raised... Adjustments to the first flank face will also affect normal machining, indicating that the swing angle has been modified. With the angle Neither can completely prevent the grinding wheel from interfering with the second flank face. Therefore, the angle of the grinding position should be adjusted. With the angle Set all values ​​to the values ​​from the last adjustment, then proceed to Step 5 to finish the adjustment.

[0214] Step 5: End the adaptive adjustment process and proceed to the calculation of the next position.

[0215] Simulation verification:

[0216] The simulation verification of the adaptive adjustment method for the grinding trajectory of the flank face of the integral arc head end mill was carried out in Vericut 8.0 simulation software. The blank and process parameters set are shown in Table 1 below.

[0217] Table 1. Geometric parameters and grinding process parameters of the flank face of a double-edged circular arc end mill.

[0218]

[0219] The adaptive adjustment algorithm proposed in this invention is used to adjust the grinding wheel pose, outputting the corresponding toolpath file. Machining simulation is then performed in the Vericut 8.0 simulation environment, and the results are compared with those without the algorithm adjustment. The simulation comparison results before and after algorithm adjustment are attached. Figure 11 As shown in the attached figure, the toolpath file used in the simulation machining was post-processed and compared with the actual machining. The comparison results of the actual machining are as follows. Figure 12 As shown in Table 2, the measurement results of the key geometric parameters of the end mill are obtained by measuring the machined end mill on the PG-1000 tool measuring instrument. This is compared with the measured values ​​of the flank face width of each part of the end mill end cutting edge shown in Table 3.

[0220] Table 2 Key geometric parameters of end mills

[0221]

[0222] Table 3 Comparison of machining errors for the width of the flank face

[0223]

[0224] The comparison images and measurement results of the simulation and actual machining show that the difference in the flank face of the end mill is small before and after using the adaptive algorithm. However, without the adaptive algorithm, the width of the flank face of the end mill cannot be maintained well, and the transition at the junction of different cutting lines is not smooth. After using the adaptive algorithm, the consistency of the flank face width of the arc portion of the end mill end mill and the continuity of the flank face at the junction of different cutting lines are significantly improved.

Claims

1. A method for adaptive adjustment of the grinding trajectory of the flank face of an integral arc-head end mill, characterized in that, Includes the following steps: Step 1: Define the geometric parameters of the flank face of the arc-head end mill; End mill starting radius R W The tool radius at the starting position of the helical cutting edge of the end mill; Circumferential cutting edge length L W : The length of the peripheral cutting edge of the end mill along the axial direction; Taper angle k: The angle between the generatrix of the tooth rotation profile and the axis of the end mill; Helix angle β: The angle between the helix cutting edge tangent vector and the tool rotation profile generatrix; End blade inclination angle The angle between the straight cutting edge line of a circular arc end mill and the perpendicular plane to the tool axis; Tooth offset center h: The shortest distance between the straight cutting edge line of the end mill and the axis of the end mill; Tooth over center amount The projection of the line connecting the end point of the straight cutting edge of the end blade and the tool axis onto the straight cutting edge direction of the end blade; End-edge radius r: the radius of the generatrix of the end-edge arc surface; Back face width : The profile length of the flank face on the normal section of the cutting edge, i.e., the length of the intersection line between the flank face and the normal section of the cutting edge. If multiple flank faces exist, the widths of the flank faces are respectively... , where n represents the number of flank faces; Back face angle : is the angle between the profile of the flank face on the cutting edge normal section and the normal section of the end mill axis, where the flank angles are respectively... ; Step 2: Establish a coordinate system; Workpiece Coordinate System (WCS) Workpiece coordinate system O W -X W Y W Z W With the tool rotation axis as Z W The axis is defined by the end face where the starting position of the circumferential cutting edge line is located. W O W Y W Plane, with the center of the end face circle as the origin O of the coordinate system. W ; End-edge coordinate system EECS The end-edge coordinate system is based on the tool rotation axis as Z. d The axis is defined by the interface between the end cutting edge and the peripheral cutting edge. d O d Y d The origin O of the coordinate system is the center of the intersection surface. d ; End-edge principal section coordinate system MSCS With the cutting edge point P0 as the origin O of the coordinate system mt Let the tangent to the longitude line at the current location be Z. mt The axis is defined by the tangent to the latitude line at the current location. mt axis; End-edge section coordinate system NSCS Let the principal section coordinate system revolve around X mt Rotate the axis to establish the coordinate system O of the end-edge normal section. mt1 -X mt1 Y mt1 Z mt1 The end-edge section coordinate system has the cutting edge point P0 as the origin O. mt1 Z is the tangent of the cutting edge. mt1 axis, Y mt1 The shaft is tangent to the end-edge arc surface and perpendicular to the cutting line; Step 3: Coordinate system transformation matrix; Transform from the end-edge section coordinate system NSCS to the end-edge principal section coordinate system MSCS First, set the tangent vector F of the current position of the cutting edge. p The coordinate system Z is connected to the cutting edge, with the line passing through point P0 and tangent to the arc surface, pointing towards the end edge coordinate system. d The vector F of the axis d They are defined as follows: （1） In the formula This indicates the coordinates of point P0 on the cutting edge in the end-edge coordinate system EECS; So, F p With F d The included angle is the angle that needs to be rotated around X when transforming from the end-edge section coordinate system to the principal section coordinate system. mt1 The rotation angle of the axis; therefore, the transformation matrix R mt1-mt As shown below: （2） Transform the end-edge principal section coordinate system MSCS to the end-edge coordinate system EECS; Define R mt-d T mt-d Let be the rotation and translation matrices used to transform the end-edge principal section coordinate system to the end-edge coordinate system, respectively. Then: （3） Transform the end-cutting coordinate system EECS to the workpiece coordinate system WCS; Define R d-w T d-w Let be the rotation matrix and displacement matrix for transforming the end-edge coordinate system to the workpiece coordinate system, respectively. Then: （4） In the formula, It is the angle that the cutting edge of the end mill rotates around the axis of the end mill from the starting point of the peripheral cutting edge to the end point of the peripheral cutting edge. Step 4: Establish a mathematical model of the flank face of the end mill with a circular arc head; Equation of the end-edge circular arc cutting edge line: （5） In the formula, θ represents the latitude angle of P0 on the end cutting edge line, and R d This is the distance between the center of the end-edge arc and the tool axis. R represents the rotation angle at point P0 on the arc-shaped cutting edge. d and The expression is as follows: （6） （7） Equation of the cutting edge line of a plane curve: （8） In the formula This indicates the straight cutting edge of the end edge in the X-axis of the end edge coordinate system. d O d Y d Projection in the plane and the end-edge coordinate system X d The included angle of the axis; Equation of the straight cutting edge line: （9） In the formula, P P2_d With F P2_d These represent the final point P of the curved section of the end edge. 2d The coordinate vectors at point P 2d Tangent vector at point P 2d The coordinates are obtained using equation (8). Indicates the amount of tooth passing through the center; Step 5: Establish the first flank width control line and its offset curve family; The width and angle of the first flank face are determined by the relative positions of point P1 and point P0. Point P1 is the control point of the first flank face on the cross section. The curve formed by the control points on each cross section is the control line C1 of the first flank face. The coordinate vector of point P1 is represented as follows: （10） In the formula, and These represent the widths of the first and second flank faces, respectively. and These are the first rear angle and the second rear angle; The width of the flank face is calculated by establishing the offset curve C2 of the control line on the second flank face. Point P2 on C2 is represented in the end-edge normal section coordinate system as follows: （11） Based on the coordinate system transformation relationship proposed above, namely equation (8), point P2 can be represented in the end-edge coordinate system as: （12） Combining equations (4) and (12), point P2 can be represented in the workpiece coordinate system as follows: （13） According to equation (11), when the first flank cutting edge line remains unchanged, the position of the offset curve of the flank control line is changed from... Decision; therefore, by changing the step size in equal increments This yields multiple offset curves on the second flank face, also known as the offset curve family. The intersection of the grinding wheel with any one of these curves indicates that the grinding wheel is in contact with the second flank face, thereby allowing the width of the flank face to be detected. Step 6: Define the grinding wheel posture; Grinding wheel tilt angle ; Connect the cutting edge point P0 with the grinding wheel axis vector F. g0 The parallel vector is defined as the X-axis of the flank face in the end-edge normal section coordinate system. mt1 O mt1 Y mt1 Normal vector F under the plane g1 normal vector F g1 With the grinding wheel coordinate system X s aligning the axes, the grinding wheel is rotated around F. g1 The angle rotated clockwise is defined as the swing angle. ; Grinding wheel lifting angle ; The vector pointing from P0 to P1 after the swing angle transformation is defined as the tangent vector F of the grinding wheel at the cutting edge point. t1 , rotate the grinding wheel around F t1 That is, the grinding wheel coordinate system Y s The angle through which the axis rotates counterclockwise is defined as the lift angle. ; Step 7: Model the contour of the large end face of the grinding wheel; The center O of the large end face of the grinding wheel under the end-edge section. g With axis vector F g coordinate; （14） In the formula, O Og0_mt1 and F g0_mt1 Representing O g With F g The coordinate vector in the initial attitude; The lifting angle during the grinding process With swing angle After substituting into the calculation, we get O. g With F g coordinate vector and : （15） In the formula , , ; Substituting the calculated coordinates of the grinding wheel's large end face center and axis vector into the coordinate system transformation equations (2)-(4) will transform it to the workpiece coordinate system; then, the axis vector F obtained will be used to transform it to the workpiece coordinate system. g The center of the large end face of the grinding wheel, O g Coordinates and grinding wheel radius R g This yields the equation for the profile of the large end face of the grinding wheel: （16） In the formula, x, y, and z represent the solutions to the equation, that is, the coordinates of points on the large end face profile of the grinding wheel. and F g With O g The coordinate vector in the workpiece coordinate system X W Y W Z W Components in direction; Step 8: Adaptive adjustment of grinding wheel posture based on grinding zone control; The interference of the grinding wheel on the second flank face is detected by using the cutting edge equation and the outer circle contour equation of the large end face of the grinding wheel: Equation (17) is a detection formula for determining whether the offset point of the cutting edge on the curved surface of the second back cutter is in contact with the large end face of the grinding wheel. In the formula, d1 is the distance between the offset point of the cutting edge and the center of the large end face of the grinding wheel and the radius R of the grinding wheel. g The difference is d2, which is the distance between the offset point P2 of the cutting edge and the plane where the large end face of the grinding wheel is located. If d1=d2=0, it means that the offset curve C2 of the back face control line composed of P2 intersects the outline circle of the large end face of the grinding wheel when machining the first back face. That is, the grinding wheel touches the second back face and causes interference. At this time, the attitude of the grinding wheel is adaptively adjusted. （17） Determine whether the large end face of the grinding wheel is in contact with the control line C1 on the back face under the current posture: Equation (18) is the detection formula for the contact state of the back face control line. The distance between the back face control point and the center of the large end face of the grinding wheel is detected by d3, and the distance between the back face control point and the large end face of the grinding wheel is detected by d4. If d4=0 and d3≤0, it means that the large end face of the grinding wheel is still in contact with the back face control line C1 at the current position, and the posture adjustment continues; if the above conditions are not met, it means that the normal processing at the current position will be affected, and the posture adjustment is stopped. （18） During the machining of the back face, the grinding wheel posture needs to be adaptively detected and adjusted multiple times at each position until the large end face contour circle of the grinding wheel no longer intersects with the offset curve C2 on the second back face before proceeding to the next position.

2. The adaptive adjustment method for the grinding trajectory of the flank face of an integral arc-head end mill according to claim 1, characterized in that, The specific method for adjusting the grinding wheel's posture is as follows: Step 1: Set the initial values ​​of the grinding wheel attitude and calculate the grinding wheel coordinates. Before performing adaptive adjustments, the attitude of the grinding wheel, i.e., the lifting angle, must first be set. With swing angle The initial values ​​of the swing angle and the lifting angle are set to -5° and 10° respectively, and the coordinates O of the center of the large end face of the grinding wheel at the current position are calculated based on the grinding wheel posture. g With axis vector coordinates F g ; Step 2: Determine the current grinding state of the grinding wheel. By O g With F g Substituting into equations (16) and (17) yields the circular profile equation of the large end face of the grinding wheel and determines whether the large end face of the grinding wheel is in contact with the offset curve C2 of the control line of the back face. If it is in contact, it indicates that interference has occurred, and the process proceeds to Step 3 and subsequent adjustments. If it is not in contact, it indicates that no interference has occurred, and the process proceeds directly to the calculation and adjustment of the next machining position. Step 3: Angle Adjustment Adjustment When adjusting the grinding wheel posture, the first consideration should be the grinding wheel tilt angle. Impact on the processing; by increasing the swing angle The value moves the grinding wheel away from the offset curve C2, and the adjustment range is [value missing]. The angle is 0.1°. After each adjustment, it is necessary to determine whether the grinding wheel affects the normal machining at the current position, i.e., whether it is in contact with the back face control line C1 at the current position. If it does not affect the normal machining, jump back to Step 2 and check again whether interference occurs. If it affects normal processing, it indicates that it is no longer possible to resolve the issue by simply adjusting the swing angle. Adjust the grinding wheel posture, then proceed to Step 4, lifting the angle. Adjustment process; when Step 2 determines that no interference will occur, proceed to Step 5 to end the adjustment. Step 4: Raise the corner Adjustment Due to the lifting angle Changes occur, swing angle The attitude change process changes, at which point the swing angle should be adjusted. The value is set to the initial adjustment value, i.e., -5°, for the tilt angle. Adjustment range Also 0.1°, lift angle Each adjustment must be evaluated to determine if it affects the processing. If it does not, proceed back to Step 2 to determine if interference has occurred and make subsequent adjustments to the swing angle until no interference or tilting occurs. The adjustment continues until it affects the normal processing at the current position; if the angle is raised... Adjustments to the first flank face will also affect normal machining, indicating that the swing angle has been modified. With the angle Neither can completely prevent the grinding wheel from interfering with the second flank face. Therefore, the angle of the grinding position should be adjusted. With the angle Set all values ​​to the values ​​from the last adjustment, then proceed to Step 5 to finish the adjustment; Step 5: End the adaptive adjustment process and proceed to the calculation of the next position.

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