Gear chamfering track calculation method

Abstract

The invention relates to the field of gear machining, and discloses a gear chamfering track calculation method, which comprises the following steps of: 1, defining geometric parameters used in a machine tool chamfering process, and defining a machine tool coordinate system, a workpiece coordinate system and coordinate systems of different forms of a cutter; 2, converting the coordinates of the point of the milling cutter in the workpiece coordinate system into a machine tool coordinate system through a coordinate system; 3, after the milling cutter is inclined, the section of the cutter is an ellipse, and ellipse parameters and ellipse center point coordinates are solved; and 4, solving a linkage track array. The invention provides a calculation method capable of quickly and accurately chamfering the edge of the gear.

Description

Technical Field

[0001] The invention relates to the field of gear processing, and in particular to a method for calculating a gear chamfer trajectory. Background Art

[0002] Gears are commonly used components in industrial production. Gears and other toothed workpieces used in automobiles and other devices have high precision requirements. At the same time, the edges of the gears need to be chamfered, that is, the chamfering process. Now, two methods are usually used to chamfer gears, including rolling and vertical milling. The rolling method uses an efficient and continuous rolling method. Although it can achieve a higher shape and efficiency, it has limitations in small batch processing and complex shape processing. The rolling method cannot process all parts. In addition, the tool cost is too high. Customized tools are required for each type of gear. The vertical milling method uses intermittent tools, and the edge roughness after milling is not good. Therefore, it is necessary to propose a calculation method that can quickly and accurately chamfer the edges of gears. Summary of the invention

[0003] The present invention aims to provide a gear chamfer trajectory calculation method, so as to propose a calculation method capable of quickly and accurately chamfering the gear edge.

[0004] To achieve the above object, the present invention adopts the following technical solution: a method for calculating a gear chamfer trajectory, comprising the following steps:

[0005] Step 1: Define the geometric parameters used in the chamfering process of the machine tool, and define the coordinate system of the machine tool, the workpiece coordinate system, and the coordinate systems of different tool forms;

[0006] Step 2: transform the coordinates of the milling cutter tip in the workpiece coordinate system into the machine tool coordinate system through the coordinate system;

[0007] Step 3: After the milling cutter is tilted, the cross section of the cutter is an ellipse, and the ellipse parameters and the coordinates of the center point of the ellipse are solved;

[0008] Step 4: Solve the linkage trajectory array.

[0009] The beneficial effects of this program are:

[0010] By defining and calculating the description of the machine tool, workpiece and tool shape, solving the ellipse parameters and tangency points of the milling cutter and the workpiece tooth profile surface, the tool starts chamfering when the ellipse section contacts the tooth profile surface, ensuring high precision during the processing. In this technical solution, the characteristic that the cross section of the tool is an ellipse when it is tilted is used to calculate the coordinates of the center point of the ellipse, so that the tool can accurately be tangent to the tooth profile surface when it is in different shapes, thereby achieving a smooth chamfering and deburring process. The solution of the ellipse center point provides an accurate reference for each workpiece rotation and tool feed, ensuring the processing continuity after multiple feeds and rotations. This method is suitable for gears of different sizes and complexities, has strong adaptability, and can handle different types of gear edge chamfering and deburring requirements. The milling method in this technical solution can solve the problem that the rolling method cannot chamfer the gear, such as double teeth, internal teeth or shaft teeth with steps too close to the tooth root, and has good structural adaptability, and can effectively chamfer most gear workpieces; and using a milling cutter to chamfer the gear can improve the surface finish of the gear edge.

[0011] Preferably, as an improvement, the geometric parameter definition in step 1 is specifically as follows:

[0012] Bs is the rotation angle of the machine tool B axis, which is determined by the chamfer size at the bottom of the tooth root;

[0013] As is the rotation angle of the machine tool A axis, which is determined by the helix angle of the workpiece;

[0014] CA is the tool angle, which is the angle between the milling cutter axis and the workpiece end face;

[0015] CB is the ellipse axis deflection angle, which is the angle between the projection line of the milling cutter axis on the end face of the workpiece and the end face of the workpiece;

[0016] C is the workpiece rotation angle, that is, the machine tool table rotation angle. Incremental rotation C refers to multiple small rotations of angles C1, C2...Cn. The total angle is C = C1+C2+...+Cn.

[0017] The ellipse is the cross-sectional shape of the intersection line between the rod-shaped milling cutter and the end face of the workpiece. The minor semi-axis of the ellipse is e. b The size is equal to the radius of the stick milling cutter, the short axis is b, and the long semi-axis is e a is the distance from the end face of the workpiece to the center of the stick milling cutter, the major axis is a, the minor semi-axis is e b With the major axis e a The included angle is CA.

[0018] The beneficial effects are: through the precise definition and control of key geometric parameters, it is convenient to accurately calculate and control the feed and rotation angles, which not only improves the intelligence and automation level of the processing process, but also improves the accuracy and efficiency of gear edge chamfering and deburring, significantly improves the consistency of processing, and reduces the errors and complexity in the processing process.

[0019] Preferably, as an improvement, the coordinate system definition in step 1 is specifically as follows:

[0020] (O b -x b ,y b ,z b ) is the coordinate system when the tool has no deflection angle on the A-axis and B-axis. It is used to mathematically describe the position of the tool tip. The origin of the coordinate system is located at the center of the machine tool origin table. The coordinate axis x b Coincident with the machine tool B1 axis, same direction, y b There is an angle Bs with the y-axis of the machine tool, and the direction is opposite;

[0021] (O a -x a ,y a ,z a ) is the coordinate system when the tool has a deflection angle on the A axis and no deflection angle on the B axis. The specific meaning is (O b -x b ,y b ,z b ) coordinate system is O b The point is the origin, y b The axis is the coordinate system obtained by rotating the axis by an angle Bs, which is used to mathematically describe the position of the tool tip. a -x a ,y a ,z a ) Coordinate system origin O a and (O b -x b ,y b ,z b ) The origin of the coordinate system is O b Coincident, coordinate axis x a Coincident with the machine tool B1 axis, in the same direction, coordinate axis y a Parallel to the machine tool x-axis and in the opposite direction;

[0022] (O g -x g ,y g ,z g ): Workpiece coordinate system, used to mathematically describe the profile of the gear. The origin of the workpiece coordinate system is O g Located at the center of the workpiece, coordinate axis y g Coincident with the machine tool y-axis, in the same direction, x g Coincident with the machine tool x-axis, in the opposite direction, coordinate axis z g Coincident with the machine tool z-axis;

[0023] (O m -xm ,y m ,z m ) is the machine tool coordinate system, and the A axis is perpendicular to the y axis of the machine tool coordinate system. m -z m Plane, B axis is perpendicular to the x axis of the machine tool coordinate system m -z m Plane is used to mathematically describe the machining motion trajectory.

[0024] The beneficial effects are as follows: in the chamfering process, the motion trajectory of the tool needs to be precisely controlled. By defining the coordinate system under different states, it is convenient to mathematically express and calculate the tool position, angle change and processing trajectory, especially the mathematical description of the tool tip position, which ensures the precise control of the position during the processing. By combining the conversion of different coordinate systems with the tool motion trajectory, the consistency of the processing can be ensured, especially when multiple rotations and feeds are required, the precise calculation of the tangent point can be performed.

[0025] Preferably, as an improvement, the specific steps of step 2 are:

[0026] Coordinate system (O b -x b ,y b ,z b ) to (Oa-x a ,y a ,z a ) is:

[0027]

[0028] Coordinate system (O a -x a ,y a ,z a ) to (O h -x h ,y h ,z h ) is:

[0029]

[0030] Coordinate system (O h -x h ,y h ,z h ) to (O g -x g ,y g ,z g ) is:

[0031]

[0032] Coordinate system (Og -x g ,y g ,z g ) to (O m -x m ,y m ,z m ) is:

[0033]

[0034] The milling cutter tip is set in the coordinate system (O b -x b ,y b ,z b ) is: (L1,L2,0);

[0035] Transform this point to the machine tool coordinate system (O m -x m ,y m ,z m ), through the coordinate system transformation, the position coordinate point of the milling cutter tip in the machine tool coordinate system is obtained, which is expressed as (X m ,Y m ,Z m );

[0036] [X m , Y m , Z m , 1]=M mg M gh M ha M ab [L1,L2,0,1]

[0037] The beneficial effect is that the initial position of the milling cutter is accurately converted into the machine tool coordinate system through a continuous coordinate transformation matrix, so that an accurate processing path can be calculated.

[0038] Preferably, as an improvement, the specific steps of step three are:

[0039] Solve for the ellipse parameters;

[0040] e b =d / 2

[0041]

[0042] The calculation formula for the tool cross-section ellipse is:

[0043]

[0044] where e b is the minor semi-axis of the ellipse, e aa is the semi-major axis of the ellipse, b is the minor axis of the ellipse, and a is the major axis of the ellipse.

[0045] The beneficial effect is: by calculating in detail the elliptical cross-section at the intersection of the rod milling cutter and the end face of the workpiece, the major and minor axes and angles of the ellipse are determined. Through these parameters, the tool can be ensured to be in precise contact with the workpiece at different rotational positions, so that the processing path can accurately fit the contour edge of the workpiece, ensuring the uniformity of deburring and chamfering.

[0046] Preferably, as an improvement, step 4 is specifically as follows:

[0047] 4.1 Calculation of the tangent point of the ellipse. Solve in the workpiece coordinate system. The workpiece contour is tangent to the ellipse, and the center of the ellipse is the center point of the tool tip motion trajectory. When the workpiece rotates by an angle C1, C2...Cn, the corresponding position of the center point of the ellipse tangent to the rotating tooth profile will also change. The corresponding position of the center point of the ellipse tangent to the rotating tooth profile is (X g1 ,Y g1 ,Z g1 )(X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn );

[0048] 4.2 The coordinate point (X g1 ,Y g1 ,Z g1 ), (X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn ) are transformed into the machine tool coordinate system one by one to obtain a series of coordinate points (X m1 ,Y m1 ,Z m1 ), (X m2 ,Y m2 ,Z m2 )……(X mn ,Y mn ,Z mn );

[0049] 4.3 Add up the point positions found in the above two steps to obtain the linkage trajectory array of the four coordinate axes X, Y, Z, C, expressed as {C1, X m1 +X m , Y m1 +Y m ,Z m1 +Z m},{C2,X m2+X m , Y m2 +Y m ,Z m2 +Z m}...{Cn,X mn +X m , Y mn +Y m ,Z mn +Z m}.

[0050] The beneficial effect is that during the chamfering and deburring process, the position of the center of the ellipse will change as the workpiece rotates. Mathematical calculations can accurately determine the center point of the ellipse where the tool is tangent to the workpiece at each rotation angle. Since the contour of the workpiece is precisely tangent to the elliptical cross-section of the tool, it is ensured that the burrs can be completely removed, thereby improving the processing quality of the product. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 This is a flow chart of a method according to an embodiment of the present invention;

[0052] Figure 2 A schematic diagram of the inclination of the milling cutter in an embodiment of the present invention;

[0053] Figure 3 Schematic diagram of various coordinate systems in embodiments of the present invention. DETAILED DESCRIPTION

[0054] The following is further described in detail through specific implementation methods:

[0055] Example

[0056] The embodiment is basically as follows Figure 1 As shown, Figure 1 A method for calculating a gear chamfer trajectory is shown, comprising the following steps:

[0057] Step 1: Define the machine tool coordinate system, workpiece coordinate system and coordinate systems of different tool forms;

[0058] The geometric parameters are defined as follows:

[0059] Bs is the rotation angle of the machine tool B axis, that is, the deflection angle of the milling cutter, which is determined by the chamfer size at the bottom of the tooth root;

[0060] As is the rotation angle of the machine tool A axis, that is, the inclination angle of the milling cutter, which is determined by the helix angle of the workpiece;

[0061] CA is the tool angle, which is the angle between the axis of the milling cutter and the end face of the workpiece;

[0062] CB is the ellipse axis deflection angle, which is the angle between the projection line of the milling cutter axis on the end face of the workpiece and the end face of the workpiece;

[0063] C is the workpiece rotation angle, that is, the machine tool table rotation angle. Incremental rotation C refers to multiple small rotations of angles C1, C2...Cn. The total angle is C = C1+C2+...+Cn.

[0064] The ellipse is the cross-sectional shape of the intersection line between the rod-shaped milling cutter and the end face of the workpiece. The minor semi-axis of the ellipse is e. b The size is equal to the radius of the stick milling cutter, the short axis is b, and the long semi-axis is e a is the distance from the end face of the workpiece to the center of the stick milling cutter, the major axis is a, the minor semi-axis is e b With the major axis e a The included angle is CA. Through the precise definition and control of the above key geometric parameters, it is convenient to accurately calculate and control the feed and rotation angles, which not only improves the intelligence and automation level of the processing process, but also improves the accuracy and efficiency of gear edge chamfering and deburring, significantly improves the consistency of processing, and reduces the error and complexity of the processing process.

[0065] like Figure 3 As shown in the figure, the coordinate system is defined as follows:

[0066] (O b -x b ,y b ,z b ) is the coordinate system when the tool has no deflection angle on the A-axis and B-axis. It is used to mathematically describe the position of the tool tip. The origin of the coordinate system is located at the center of the machine tool origin table. The coordinate axis x b Coincident with the machine tool B1 axis, same direction, y b There is an angle Bs with the y-axis of the machine tool, and the direction is opposite;

[0067] (O a -x a ,y a ,z a ) is the coordinate system when the tool has a deflection angle on the A axis and no deflection angle on the B axis. The specific meaning is (O b -x b ,y b ,z b ) coordinate system is O b The point is the origin, y b The axis is the coordinate system obtained by rotating the axis by an angle Bs, which is used to mathematically describe the position of the tool tip. a -x a ,y a ,z a ) Coordinate system origin O a and (O b -x b ,y b ,z b ) The origin of the coordinate system is Ob Coincident, coordinate axis x a Coincident with the machine tool B1 axis, in the same direction, coordinate axis y a Parallel to the machine tool x-axis and in the opposite direction;

[0068] (O g -x g ,y g ,z g ): Workpiece coordinate system, used to mathematically describe the profile of the gear. The origin of the workpiece coordinate system is O g Located at the center of the workpiece, coordinate axis y g Coincident with the machine tool y-axis, in the same direction, x g Coincident with the machine tool x-axis, in the opposite direction, coordinate axis z g Coincident with the machine tool z-axis;

[0069] (O m -x m ,y m ,z m ) is the machine tool coordinate system, and the A axis is perpendicular to the y axis of the machine tool coordinate system. m -z m Plane, B axis is perpendicular to the x axis of the machine tool coordinate system m -z m Plane is used to mathematically describe the machining motion trajectory.

[0070] In chamfering processing, the motion trajectory of the tool needs to be precisely controlled. By defining coordinate systems under different states, it is convenient to mathematically express and calculate the tool position, angle change and processing trajectory, especially the mathematical description of the tool tip position, which ensures precise control of the position during the processing. By combining the conversion of different coordinate systems with the tool motion trajectory, the consistency of processing can be ensured, especially when multiple rotations and feeds are required, the precise calculation of the tangent point can be performed.

[0071] Step 2: transform the coordinates of the milling cutter tip in the workpiece coordinate system into the machine tool coordinate system through the coordinate system;

[0072] Coordinate system (O b -x b ,y b ,z b ) to (Oa-x a ,y a ,z a ) is:

[0073]

[0074] Coordinate system (O a -x a ,y a ,za ) to (O h -x h ,y h ,z h ) is:

[0075]

[0076] Coordinate system (O h -x h ,y h ,z h ) to (O g -x g ,y g ,z g ) is:

[0077]

[0078] Coordinate system (O g -x g ,y g ,z g ) to (O m -x m ,y m ,z m ) is:

[0079]

[0080] The milling cutter tip is set in the coordinate system (O b -x b ,y b ,z b ) is: (L1,L2,0);

[0081] Transform this point to the machine tool coordinate system (O m -x m ,y m ,z m ), through the coordinate system transformation, the position coordinate point of the milling cutter tip in the machine tool coordinate system is obtained, which is expressed as (X m ,Y m ,Z m ):

[0082] [X m , Y m , Z m , 1]=M mg M gh M ha M ab [L1,L2,0,1]

[0083] Step 3: After the milling cutter is tilted, the cross section of the cutter is an ellipse, and the ellipse parameters and the coordinates of the center point of the ellipse are solved;

[0084] Solve for the ellipse parameters;

[0085] e b =d / 2

[0086]

[0087] The calculation formula for the tool cross-section ellipse is:

[0088]

[0089] where e b is the minor semi-axis of the ellipse, e a a is the semi-major axis of the ellipse, b is the minor axis of the ellipse, and a is the major axis of the ellipse.

[0090] like Figure 2 As shown in the figure, by calculating in detail the elliptical cross-section at the intersection of the stick milling cutter and the end face of the workpiece, the major and minor axes and angles of the ellipse are determined. Through these parameters, the tool can be guaranteed to accurately contact the workpiece at different rotational positions, so that the machining path can accurately fit the contour edge of the workpiece, ensuring the uniformity of deburring and chamfering.

[0091] Step 4: Solve the linkage trajectory array.

[0092] 4.1 Calculation of the tangent point of the ellipse. Solve in the workpiece coordinate system. The workpiece contour is tangent to the ellipse, and the center of the ellipse is the center point of the tool tip motion trajectory. When the workpiece rotates by an angle C1, C2...Cn, the corresponding position of the center point of the ellipse tangent to the rotating tooth profile will also change. The corresponding position of the center point of the ellipse tangent to the rotating tooth profile is (X g1 ,Y g1 ,Z g1 )(X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn );

[0093] 4.2 The coordinate point (X g1 ,Y g1 ,Z g1 ), (X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn ) are transformed into the machine tool coordinate system one by one to obtain a series of coordinate points (X m1 ,Ym1 ,Z m1 ), (X m2 ,Y m2 ,Z m2 )……(X mn ,Y mn ,Z mn );

[0094] 4.3 Add up the point positions found in the above two steps to obtain the linkage trajectory array of the four coordinate axes X, Y, Z, C, expressed as {C1, X m1 +X m , Y m1 +Y m ,Z m1 +Z m},{C2,X m2 +X m , Y m2 +Y m ,Z m2 +Z m}...{Cn,X mn +X m , Y mn +Y m ,Z mn +Z m}.

[0095] During the chamfering and deburring process, the position of the center of the ellipse will change as the workpiece rotates. Mathematical calculations can be used to determine the center point of the ellipse where the tool is tangent to the workpiece at each rotation angle. Since the contour of the workpiece is precisely tangent to the elliptical cross-section of the tool, the burrs can be completely removed, thereby improving the processing quality of the product.

[0096] By defining and calculating the description of the machine tool, workpiece and tool shape, solving the ellipse parameters and tangency points of the milling cutter and the workpiece tooth profile surface, the tool starts chamfering when the ellipse section contacts the tooth profile surface, ensuring high precision during the processing. In this technical solution, the characteristic that the cross section of the tool is an ellipse when it is tilted is used to calculate the coordinates of the center point of the ellipse, so that the tool can accurately be tangent to the tooth profile surface when it is in different shapes, thereby achieving a smooth chamfering and deburring process. The solution of the ellipse center point provides an accurate reference for each workpiece rotation and tool feed, ensuring the processing continuity after multiple feeds and rotations. This method is suitable for gears of different sizes and complexities, has strong adaptability, and can handle different types of gear edge chamfering and deburring requirements. The milling method in this technical solution can solve the problem that the rolling method cannot chamfer the gear, such as double teeth, internal teeth or shaft teeth with steps too close to the tooth root, and has good structural adaptability, and can effectively chamfer most gear workpieces; and using a milling cutter to chamfer the gear can improve the surface finish of the gear edge.

[0097] The above is only an embodiment of the present invention, and the common knowledge such as the known specific technical solutions and / or characteristics in the solution is not described in detail here. It should be pointed out that for those skilled in the art, without departing from the technical solution of the present invention, several modifications and improvements can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.

Claims

1. A method for calculating a gear chamfer trajectory, characterized in that: The following steps are involved: Step 1: Define the geometric parameters used in the chamfering process of the machine tool, and define the coordinate system of the machine tool, the workpiece coordinate system, and the coordinate systems of different tool forms; Step 2: transform the coordinates of the milling cutter tip in the workpiece coordinate system into the machine tool coordinate system through the coordinate system; Step 3: After the milling cutter is tilted, the cross section of the cutter is an ellipse, and the ellipse parameters and the coordinates of the center point of the ellipse are solved; Step 4: Solve the linkage trajectory array.

2. A gear chamfer trajectory calculation method according to claim 1, characterized in that: The geometric parameters in step 1 are defined as follows: Bs is the rotation angle of the machine tool B axis, which is determined by the chamfer size at the bottom of the tooth root; As is the rotation angle of the machine tool A axis, which is determined by the helix angle of the workpiece; CA is the tool angle, which is the angle between the milling cutter axis and the workpiece end face; CB is the ellipse axis deflection angle, which is the angle between the projection line of the milling cutter axis on the end face of the workpiece and the end face of the workpiece; C is the rotation angle of the workpiece, that is, the rotation angle of the machine tool worktable. Incremental rotation C refers to multiple small rotations of angles C1, C2...Cn. The total angle is C=C1+C2+...+Cn.

3. A gear chamfer trajectory calculation method according to claim 2, characterized in that: The coordinate system definition in step 1 is as follows: (O b -x b ,y b ,z b ) is the coordinate system when the tool has no deflection angle on the A-axis and B-axis. It is used to mathematically describe the position of the tool tip. The origin of the coordinate system is located at the center of the machine tool origin table. The coordinate axis x b Coincident with the machine tool B1 axis, same direction, y b There is an angle Bs with the y-axis of the machine tool, and the direction is opposite; (O a -x a ,y a ,z a ) is the coordinate system when the tool has a deflection angle on the A axis and no deflection angle on the B axis. The specific meaning is (O b -x b ,y b ,z b ) coordinate system is O b The point is the origin, y b The axis is the coordinate system obtained by rotating the axis by an angle Bs, which is used to mathematically describe the position of the tool tip. a -x a ,y a ,z a ) Coordinate system origin O a and (O b -x b ,y b ,z b ) The origin of the coordinate system is O b Coincident, coordinate axis x a Coincident with the machine tool B1 axis, in the same direction, coordinate axis y a Parallel to the machine tool x-axis and in the opposite direction; (O g -x g ,y g ,z g ): Workpiece coordinate system, used to mathematically describe the profile of the gear. The origin of the workpiece coordinate system is O g Located at the center of the workpiece, coordinate axis y g Coincident with the machine tool y-axis, in the same direction, x g Coincident with the machine tool x-axis, in the opposite direction, coordinate axis z g Coincident with the machine tool z-axis; (O m -x m ,y m ,z m ) is the machine tool coordinate system, and the A axis is perpendicular to the y axis of the machine tool coordinate system. m -z m Plane, B axis is perpendicular to the x axis of the machine coordinate system m -z m Plane is used to mathematically describe the machining motion trajectory.

4. A gear chamfer trajectory calculation method according to claim 3, characterized in that: The specific steps of step 2 are: Coordinate system (O b -x b ,y b ,z b ) to (Oa-x a ,y a ,z a ) is: Coordinate system (O a -x a ,y a ,z a ) to (O h -x h ,y h ,z h ) is: Coordinate system (O h -x h ,y h ,z h ) to (O g -x g ,y g ,z g ) is: Coordinate system (O g -x g ,y g ,z g ) to (O m -x m ,y m ,z m ) is: The milling cutter tip is set in the coordinate system (O b -x b ,y b ,z b ) is: (L1,L2,0); Transform this point to the machine tool coordinate system (O m -x m ,y m ,z m ), through the coordinate system transformation, the position coordinate point of the milling cutter tip in the machine tool coordinate system is obtained, which is expressed as (X m ,Y m ,Z m ): [X m ,AND m ,Z m ,1]=M mg M gh M ha M ab [L1,L2,0,1]。 5. A gear chamfer trajectory calculation method according to claim 4, characterized in that: The specific steps of step three are: Solve for the ellipse parameters; e b =d / 2 The calculation formula for the tool cross-section ellipse is:

6. A gear chamfer trajectory calculation method according to claim 5, characterized in that: Step 4 is as follows: 4.1 Calculation of the tangent point of the ellipse. Solve in the workpiece coordinate system. The workpiece contour is tangent to the ellipse, and the center of the ellipse is the center point of the tool tip motion trajectory. When the workpiece rotates by an angle C1, C2...Cn, the corresponding position of the center point of the ellipse tangent to the rotating tooth profile will also change. The corresponding position of the center point of the ellipse tangent to the rotating tooth profile is (X g1 ,Y g1 ,Z g1 )(X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn ); 4.2 The coordinate point (X g1 ,Y g1 ,Z g1 ), (X g2 ,Y g2 ,Z g2 )……(X gn ,Y gn ,Z gn ) are transformed into the machine tool coordinate system one by one to obtain a series of coordinate points (X m1 ,Y m1 ,Z m1 ), (X m2 ,Y m2 ,Z m2 )……(X mn ,Y mn ,Z mn ); 4.3 Add up the point positions found in the above two steps to obtain the linkage trajectory array of the four coordinate axes X, Y, Z, C, expressed as {C1, X m1 +X m , Y m1 +Y m ,Z m1 +Z m },{C2,X m2 +X m , Y m2 +Y m ,Z m2 +Z m }...{Cn, X mn +X m , Y mn +Y m ,Z mn +Z m }.

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