A super-precision grinding tool path planning method based on phase shift regulation

Abstract

The application discloses a kind of based on phase shift regulation and control's ultra-precision grinding tool path planning method.The method includes the following steps: S1, according to the rotational speed of workpiece and the active oscillation frequency of grinding wheel, calculate phase shift, solve phase according to phase shift, solve helical groove interval S according to tool feed speed and workpiece rotational speed;S2, according to the phase shift effect generated by grinding wheel oscillation, solve position parameter;S3, carry out coordinate transformation, change grinding wheel coordinate system X'-Y'-Z' and workpiece coordinate system X''-Y''-Z'' into fixed global coordinate system X-Y-Z;S4, change phase shift, calculate corresponding workpiece surface microtexture under different phase shift, propose tool path planning method.A kind of based on phase shift regulation and control's ultra-precision grinding tool path planning method provided by the application can form deterministic microtexture on the machining surface by controlling phase shift under grinding wheel oscillation, to realize the controllable manufacturing of microtexture on the machining surface in ultra-precision grinding.

Description

Technical Field

[0001] This invention belongs to the field of ultra-precision grinding, and relates to tool oscillation, and particularly to an ultra-precision grinding tool path planning method based on phase shift control. Background Technology

[0002] With the development of science and technology and technological innovation, high-precision optical components with complex structures are playing an increasingly important role in optical applications. Advanced optical materials are mostly hard and brittle, and ultra-precision grinding technology, due to its excellent processing capabilities for various hard and brittle materials, is widely used in the processing of optical components to improve their forming accuracy and surface finish. However, during the grinding process, the vibration of the grinding wheel has a significant impact on the formation of the machined surface. The imbalance of the grinding wheel is the main cause of grinding wheel vibration. Even with meticulous dressing and precise dynamic balancing, it is difficult to completely eliminate grinding wheel vibration. Furthermore, because the vibration of the grinding wheel is always present, the machined surface of the workpiece will always produce a series of vibrational textures, affecting surface quality. Therefore, analyzing the influence of the tool path on the micro-texture of the machined surface under relative grinding wheel vibration, in order to find the generation principle and control method of micro-textures, is an urgent need in the field of ultra-precision grinding.

[0003] Currently, research on surface quality in ultra-precision grinding is mainly divided into two categories: passive vibration error and vibration-assisted grinding. The focus is on the formation of surface ripples and the prediction of surface roughness. Research on the micro-textures generated by grinding wheel oscillation on the machined surface is still very lacking, and there is no universal toolpath planning method to control the micro-textures generated by grinding wheel oscillation on the machined surface. Summary of the Invention

[0004] To address the aforementioned problems, the present invention aims to provide a method for planning the path of ultra-precision grinding tools based on phase shift control. Under the oscillation of the grinding wheel, the phase shift is controlled to form a deterministic micro-texture on the machined surface, thereby achieving controllable manufacturing of the micro-texture of the machined surface in ultra-precision grinding.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for planning ultra-precision grinding toolpaths based on phase-shift control, wherein the toolpath of the grinding wheel on the workpiece surface exhibits a helical structure, and the planning method includes the following steps:

[0007] S1. Calculate the phase shift Ψ and phase based on the workpiece rotation speed, tool feed rate, and grinding wheel oscillation frequency. And the spiral groove spacing S;

[0008] S2. Solve for the position parameters of the grinding wheel in the workpiece coordinate system X″-Y″-Z″ based on the phase shift effect caused by the grinding wheel oscillation.

[0009] S3. Perform coordinate transformation to transform the grinding wheel coordinate system X′-Y′-Z′ and the workpiece coordinate system X″-Y″-Z″ into a fixed global coordinate system XYZ, and solve for the focus of the cutting profile of adjacent grinding wheels.

[0010] S4. Change the phase shift, calculate the corresponding micro-texture of the workpiece surface under different phase shifts, and perform tool path planning. The micro-texture is the ripple trajectory generated by the oscillation of the grinding wheel on the workpiece surface. The geometric parameters of the ripple trajectory are determined by the position parameters in S2.

[0011] Further, in S1, the grinding wheel oscillation frequency is equal to the grinding wheel rotation speed, the phase shift is the fractional part of the ratio of the grinding wheel rotation speed to the workpiece rotation speed, and the phase... It is represented by the phase shift angle.

[0012] The tool feed rate and the workpiece rotation speed are given values. The tool path of the grinding wheel on the workpiece surface is a spiral structure. The spiral groove interval S is the ratio of the tool feed rate to the workpiece rotation speed.

[0013] Furthermore, in S2, the position parameters are solved based on the phase shift effect generated by the grinding wheel oscillation. The position parameters specifically include: the grinding wheel cutting point, the instantaneous distance ρ(i,j) between the grinding wheel cutting point and the workpiece rotation center, the position coordinates of the grinding wheel cutting arc center relative to the workpiece surface, the grinding wheel cutting profile coordinates, and the geometric parameters of adjacent grinding wheel cutting arcs.

[0014] The grinding wheel cutting point is the relative cutting point between the grinding wheel and the workpiece in the XY plane without considering grinding wheel oscillation.

[0015] The instantaneous distance ρ(i,j) between the grinding wheel cutting point and the workpiece rotation center is determined based on the grinding wheel cutting point.

[0016] Considering the grinding wheel oscillation, the position coordinates of the center of the grinding wheel cutting arc relative to the workpiece surface are determined according to the phase. Since the grinding wheel oscillation causes the cutting depth of the grinding wheel on the Z-axis to change periodically, the position trajectory of the center of the grinding wheel cutting arc relative to the workpiece surface can be approximated as a sinusoidal motion.

[0017] The coordinates of the grinding wheel cutting profile are determined by combining the position coordinates of the center of the grinding wheel cutting arc relative to the workpiece surface with the radius of the grinding wheel cutting arc and the phase.

[0018] The geometric parameters of adjacent grinding wheel cutting arcs are determined based on the position coordinates of the center of the grinding wheel cutting arc relative to the workpiece surface, including: the center distance between adjacent grinding wheel cutting arcs, the angle parameters, and the coordinates of the intersection point of adjacent cutting arcs.

[0019] Specifically, the adjacent grinding wheel cutting arcs refer to two grinding wheel cutting arcs with a distance of S in the X direction equal to the helical groove interval S.

[0020] Furthermore, the coordinate transformation described in S3 specifically includes: performing coordinate transformation on the cutting point of the grinding wheel and performing coordinate transformation on the coordinates of the intersection points of the cutting arcs of adjacent grinding wheels.

[0021] Furthermore, in S4, the phase shift is changed, and the corresponding micro-texture of the workpiece surface under different phase shifts is analyzed. A tool path planning method is proposed, which specifically includes:

[0022] The microtexture is a wavy trajectory generated by the oscillation of the grinding wheel on the surface of the workpiece, and the geometric parameters of the wavy trajectory are determined by the position parameters in S2.

[0023] When the ratio of the grinding wheel speed to the workpiece speed is an integer, that is, when the phase shift is 0, the ripple trajectory on the workpiece surface is linearly distributed.

[0024] When the ratio of the grinding wheel speed to the workpiece speed is not an integer, that is, the phase shift exists and is equal to the fractional part of the ratio, the ripple patterns on the workpiece surface are arranged in an alternating pattern.

[0025] The phase shift greater than 0 and less than 0.5 is defined as the first phase shift. When the first phase shift exists, the number of ripple tracks on the workpiece surface is equal to the integer part of the ratio of the grinding wheel speed to the workpiece speed.

[0026] A phase shift greater than or equal to 0.5 is defined as a second phase shift. When a second phase shift exists, the number of ripple tracks on the workpiece surface is equal to twice the integer part of the ratio of the grinding wheel speed to the workpiece speed.

[0027] Compared with the prior art, the beneficial effects of the present invention are:

[0028] This invention discloses a tool path planning method for ultra-precision grinding based on phase shift control. This method establishes an ultra-precision grinding model under given grinding wheel speed, workpiece speed, and tool feed rate based on S1, S2, and S3. Then, by changing the phase shift, it analyzes the micro-texture of the workpiece surface under different phase shifts, and proposes a tool path planning method by combining the grinding wheel cutting arc position parameters in the global coordinate system. The modeling method of this invention is simple; by controlling the phase shift, it changes the geometric morphology of the workpiece surface micro-texture, achieving controllable manufacturing of the workpiece surface micro-texture in ultra-precision grinding. Attached Figure Description

[0029] To more clearly and intuitively illustrate the embodiments and technical solutions of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1 This is a schematic diagram of the ultra-precision grinding model provided by the present invention.

[0031] Figure 2 The flowchart of the ultra-precision grinding tool path planning method based on phase shift control provided by the present invention is shown.

[0032] Figure 3 This is a schematic diagram of the geometric shape of the micro-texture on the surface of the workpiece provided by the present invention.

[0033] Figure 4 This is a schematic diagram of the coordinate definition and cutting contour intersection of the workpiece surface microtexture formation model provided by the present invention.

[0034] Figure 5 This is a schematic diagram illustrating the principle of micro-texture formation on the workpiece surface provided by the present invention.

[0035] Figure 6 This is a schematic diagram of the coordinate transformation provided by the present invention.

[0036] Figure 7 This is a schematic diagram illustrating the phase shift principle provided by the present invention.

[0037] Figure 8 This is a schematic diagram of the dual phase shift mechanism provided by the present invention. Detailed Implementation

[0038] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings and examples. Obviously, the embodiments described are merely some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0039] In the description of this invention, it should be noted that the terms "first" and "second" mentioned in the embodiments of this invention are for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Therefore, a feature defined with "first" and "second" may explicitly or implicitly include one or more of that feature.

[0040] like Figure 1 As shown, the formation of micro-textures on the workpiece surface during ultra-precision grinding is caused by the following: (Reference) Figure 1 In (a), since the tool feed rate and workpiece rotation speed are given values, the tool path of the grinding wheel on the workpiece surface is helical; (Refer to...) Figure 1 In (b), the cutting depth of the grinding wheel changes due to its oscillation, resulting in a wavy structure on the workpiece surface. Therefore, the micro-texture of the workpiece surface can be determined by knowing the grinding wheel speed, tool feed rate, workpiece speed, and grinding wheel oscillation frequency.

[0041] like Figure 2 As shown, this invention provides a method for ultra-precision grinding tool path planning based on phase shift control, comprising the following steps:

[0042] S1. The tool path of the grinding wheel on the workpiece surface exhibits a helical structure. Based on the workpiece rotation speed and the grinding wheel oscillation frequency, the phase shift Ψ is calculated, and the phase is determined from the phase shift Ψ. Meanwhile, the helical groove spacing S is calculated based on the tool feed rate and workpiece rotation speed.

[0043] Wherein, the grinding wheel oscillation frequency is equal to the frequency of the actively applied periodic motion in the Z direction (i.e., the Z-axis direction of the machine tool), the phase shift Ψ is the fractional part of the ratio of the grinding wheel oscillation frequency to the workpiece rotational speed frequency, and the phase... It is represented by the angle of phase shift Ψ, that is

[0044] refer to Figure 3 Since the tool feed rate and workpiece rotation speed are given values, and the tool path of the grinding wheel on the workpiece surface is a helical structure, the helical groove interval S is the ratio of the tool feed rate to the workpiece rotation speed.

[0045] S2. Solve for the position parameters of the grinding wheel in the workpiece coordinate system X″-Y″-Z″ based on the phase shift effect generated by the grinding wheel oscillation.

[0046] The positional parameters solved in this step specifically include: the position coordinates of the grinding wheel cutting point, the distance between the grinding wheel cutting point and the workpiece rotation center, the position coordinates of the grinding wheel cutting arc center relative to the workpiece surface, the grinding wheel cutting profile coordinates, and the geometric parameters of adjacent grinding wheel cutting arcs. The geometric parameters of adjacent grinding wheel cutting arcs include: the center-to-center distance between adjacent grinding wheel cutting arcs, the angle parameter, and the coordinates of the intersection point of adjacent cutting arcs. Adjacent grinding wheel cutting arcs are two grinding wheel cutting arcs spaced S apart in the X direction (i.e., the X-axis direction of the machine tool).

[0047] Specifically, refer to Figure 3 The grinding wheel cutting point is the relative cutting point between the grinding wheel and the workpiece on the XY plane of the global coordinate system XYZ, without considering grinding wheel oscillation. The equation for calculating the position coordinates of the cutting point is:

[0048]

[0049] In the formula, θ(i,j) is the angular position of the grinding wheel, ρ(i,j) is the position of the grinding wheel relative to the workpiece's rotation center, i.e., the distance between the grinding wheel's cutting point and the workpiece's rotation center, Δθ is the workpiece's rotation angle, and R... w V2 is the workpiece radius, V2 is the workpiece rotation speed, ω2 is the workpiece angular velocity, and i,j are the discrete point numbers from the workpiece center at equal angular and equal radial distances, respectively.

[0050] Furthermore, the distance between the grinding wheel cutting point and the workpiece rotation center is determined based on the position coordinates of the grinding wheel cutting point. The calculation formula is as follows:

[0051]

[0052] In the formula, V f This represents the feed speed of the grinding wheel.

[0053] Further, refer to Figure 4 Because the grinding wheel oscillation causes the cutting depth of the grinding wheel on the Z-axis (global coordinate system XYZ) to change periodically, the position trajectory of the center of the grinding wheel cutting arc relative to the workpiece surface can be approximated as a sinusoidal motion. Considering the grinding wheel oscillation, the center O of the grinding wheel cutting arc is determined based on the phase. s The system of equations for the position coordinates {x(i,j), y(i,j), z(i,j)} relative to the workpiece surface is as follows:

[0054]

[0055] In the formula, S′ is the grinding wheel feed rate, S′=V f / V2, A is the amplitude of grinding wheel oscillation, f w ω1 is the grinding wheel rotation frequency, V1 is the grinding wheel angular velocity, and N1 is the grinding wheel spindle speed; N1 is the total number of points discrete at equal angles, and N2 is the total number of points discrete at equal radial intervals.

[0056] Furthermore, based on the position coordinates {x(i,j),y(i,j),z(i,j)} of the center of the grinding wheel cutting arc relative to the workpiece surface, combined with the radius r of the grinding wheel cutting arc... s and the phase The coordinates of the grinding wheel cutting profile are determined, and the set of profile coordinate equations is as follows:

[0057]

[0058] In the formula, z(x) i ,y j ) is a discrete point (x) i ,y j The amplitude of grinding wheel oscillation at point (r) s The radius of the arc cut by the grinding wheel;

[0059] Furthermore, the oscillation of the grinding wheel causes a periodic change in the cutting depth of the grinding wheel's cutting arc, resulting in a regular change in the geometric parameters between adjacent grinding wheel cutting arcs. (Refer to...) Figure 5 The geometric parameters of adjacent grinding wheel cutting arcs are determined based on the position coordinates {x(i,j),y(i,j),z(i,j)} of the center of the grinding wheel cutting arc relative to the workpiece surface. These parameters include the center distance between adjacent grinding wheel cutting arcs, the angle parameters, and the coordinates of the intersection point of adjacent cutting arcs.

[0060] Specifically, the formula for calculating the center distance between the cutting arcs of adjacent grinding wheels is:

[0061]

[0062] In the formula, O i,j+1 and O i,j Let O be the center of the cutting arc of the adjacent grinding wheels. In the coordinate system X′-Y′-Z′, O i,j The coordinates are (x′) i,j ,z′ i,j ), O i,j+1 The coordinates are (x′) i,j+1 ,z′ i,j+1 ).

[0063] The angular parameters between the cutting arcs of adjacent grinding wheels include α, β, α', and β', and their calculation formulas are as follows:

[0064]

[0065]

[0066] Furthermore, since the formation of the micro-texture on the workpiece surface is affected by the interference effect of adjacent grinding wheel cutting arcs, it is necessary to determine the coordinates of the intersection point of adjacent grinding wheel cutting arcs. Adjacent cutting arcs have two intersection points, and the coordinate equation of intersection point C(i,j+1) is:

[0067] x i,j =x′ i,j -r s cos(α+β)

[0068]

[0069] The coordinate equation of the intersection point C(i,j-1) is:

[0070] x i,j-1 =x′ i,j +r s cos(α′+β′)

[0071]

[0072] Specifically, adjacent grinding wheel cutting arcs refer to two grinding wheel cutting arcs with a spacing of helical groove interval S in the X direction.

[0073] S3. Perform coordinate transformation to transform the grinding wheel coordinate system X′-Y′-Z′ and the workpiece coordinate system X″-Y″-Z″ into a fixed global coordinate system XYZ, and solve for the focus of the cutting profile of adjacent grinding wheels.

[0074] Specifically, refer to Figure 6 Since the micro-texture on the workpiece surface is the result of the relative motion between the grinding wheel and the workpiece, it is necessary to transform the grinding wheel coordinate system X′-Y′-Z′ and the workpiece coordinate system X″-Y″-Z″ into a fixed global coordinate system XYZ. This coordinate transformation involves transforming the position coordinates of the grinding wheel cutting point and the coordinates of the intersection points of adjacent grinding wheel cutting arcs.

[0075] The coordinate equation of the grinding wheel cutting point in the grinding wheel coordinate system X′-Y′-Z′ is expressed as:

[0076]

[0077] In the formula, ρ represents the radial position of the grinding wheel relative to the workpiece, and R... s γ represents the radius of the grinding wheel, and γ represents the angle between the line connecting the selected point on the cutting arc of the grinding wheel and the center of the cutting arc and the cutting plane.

[0078] The position coordinate equations of points on the workpiece surface in the workpiece coordinate system X″-Y″-Z″ are expressed as follows:

[0079]

[0080] z″=z′=[R s -r s cos(γ)]cos(ω1t)

[0081] Furthermore, after coordinate transformation, the coordinates of the intersection points of adjacent cutting arcs are represented as follows:

[0082]

[0083] S4. By changing the phase shift, analyze the micro-texture of the workpiece surface under different phase shifts, and propose a tool path planning method.

[0084] Specifically, the oscillation of the grinding wheel causes micro-textures on the workpiece surface. The phase shift determines the change in the relative position between the grinding wheel and the workpiece during ultra-precision grinding. Therefore, by controlling the phase shift, a specific micro-texture can be generated on the workpiece surface. The micro-texture is actually the ripple trajectory generated by the grinding wheel oscillation on the workpiece surface. The geometric parameters of the ripple trajectory are determined by the position parameters obtained in S2. (Refer to...) Figure 7 By changing the phase shift, the micro-texture of the workpiece surface under different phase shifts is calculated, and toolpath planning is performed. Specifically, this includes:

[0085] When the ratio of the grinding wheel speed to the workpiece speed is an integer, that is, when the phase shift is 0, the "crests" and "troughs" of the ripples are on the same angular section, and the ripple trajectory on the workpiece surface is linearly distributed.

[0086] When the ratio of the grinding wheel speed to the workpiece speed is not an integer, that is, when the phase shift exists and is equal to the fractional part of the ratio, the ripples will generate an angular increment, causing the ripple trajectories on the workpiece surface to be arranged in an interlaced pattern.

[0087] Specifically, a phase shift greater than 0 and less than 0.5 is defined as the first phase shift. When the first phase shift exists, the number of ripple tracks on the workpiece surface is equal to the integer part of the ratio of the grinding wheel speed to the workpiece speed. A phase shift greater than or equal to 0.5 is defined as the second phase shift. When the second phase shift exists, the number of ripple tracks on the workpiece surface is equal to twice the integer part of the ratio of the grinding wheel speed to the workpiece speed.

[0088] Furthermore, the variation law of the ripple trajectory under different phase shifts is analyzed.

[0089] Specifically, refer to Figure 8 In (a), when the ratio of the grinding wheel speed to the workpiece speed is an integer, the phase shift is 0, and the "crests" and "troughs" of the ripples are at the same radial position and arranged in a straight line; Reference Figure 8 In diagram (b), when the decimal part of the ratio of the grinding wheel speed to the workpiece speed is 0.1, the phase shift is 0.1, and the ripple trajectory is curved; (Reference) Figure 8 In section (c), when the grinding wheel speed is an integer multiple of the workpiece speed plus half of the workpiece speed (i.e., the fractional part of the ratio of grinding wheel speed to workpiece speed is 0.5), a second phase shift exists with a phase shift of 0.5. The ripples on every two spiral grooves S are arranged in a straight line. Figure 8 In the section marked with the green line (c), there is a significant phase difference between adjacent grinding wheel cutting arcs, and the number of wavy trajectories is twice the integer part of the ratio of grinding wheel speed to workpiece speed; (Reference) Figure 8 In the middle (d), when the phase shift is 0.51, Figure 8 In (c), the green line marks the linearly distributed wavy trajectory. Figure 8 In the middle (d), the change becomes a wavy trajectory with a curved distribution, and this wavy trajectory in Figure 8 In section (d), the green line marks the wavy trajectory, and the geometry of the wavy trajectory is... Figure 8 (b) The wavy trajectories have the same geometry.

[0090] Furthermore, the fractional part of the ratio of the grinding wheel speed to the workpiece speed is used as the phase shift. When the phase shift is not zero, the phase shift will cause the curvature of the ripple trajectory to evolve. Therefore, the tool path can be controlled by controlling the phase shift, thereby machining a specific micro-texture on the workpiece surface.

[0091] The above description is merely illustrative and explanatory of the present invention and its embodiments. Such description is not restrictive, and the accompanying drawings are only one embodiment of the present invention; the actual structure is not limited thereto. If those skilled in the art, inspired by the present invention, design similar structures and embodiments without departing from the spirit of the invention, such designs should fall within the protection scope of the present invention.

Claims

1. A method for ultra-precision grinding tool path planning based on phase shift control, wherein the tool path of the grinding wheel on the workpiece surface exhibits a helical structure, characterized in that, The planning method includes the following steps: S1. Calculate the phase shift Ψ and phase based on the workpiece rotation speed, tool feed rate, and grinding wheel oscillation frequency. And the spiral groove spacing S; wherein, the grinding wheel oscillation frequency is equal to the frequency of the actively applied periodic motion in the Z direction, the phase shift Ψ is the fractional part of the ratio of the grinding wheel oscillation frequency to the workpiece rotation speed frequency, and the phase... It is represented by the angle of phase shift Ψ, that is The tool feed rate and the workpiece rotation speed are given values, the helical groove interval S is the ratio of the tool feed rate to the workpiece rotation speed, and the Z direction is the Z-axis direction of the machine tool; S2. Based on the phase shift effect generated by the grinding wheel oscillation, solve for the position parameters of the grinding wheel in the workpiece coordinate system X″-Y″-Z″. The position parameters include: the position coordinates of the grinding wheel cutting point, the distance between the grinding wheel cutting point and the workpiece rotation center, the position coordinates of the center of the grinding wheel cutting arc relative to the workpiece surface, the coordinates of the grinding wheel cutting profile, and the geometric parameters of adjacent grinding wheel cutting arcs. The geometric parameters of adjacent grinding wheel cutting arcs include: the center distance between adjacent grinding wheel cutting arcs, the angle parameter, and the coordinates of the intersection point of adjacent cutting arcs. The adjacent grinding wheel cutting arcs are two grinding wheel cutting arcs with a spacing of helical groove interval S in the X direction, where the X direction is the X-axis direction of the machine tool. S3. Perform coordinate transformation to transform the grinding wheel coordinate system X′-Y′-Z′ and the workpiece coordinate system X″-Y″-Z″ into a fixed global coordinate system XYZ, and solve for the intersection of adjacent grinding wheel cutting profiles. S4. By changing the phase shift, calculate the corresponding micro-texture of the workpiece surface under different phase shifts, and perform toolpath planning: When the ratio of the grinding wheel speed to the workpiece speed is an integer, that is, when the phase shift is 0, the ripple trajectory on the workpiece surface is linearly distributed. When the ratio of the grinding wheel speed to the workpiece speed is not an integer, that is, the phase shift exists and is equal to the fractional part of the ratio, the ripple trajectories on the workpiece surface are arranged in an interlaced pattern. The microtexture is a wavy trajectory generated by the oscillation of the grinding wheel on the surface of the workpiece, and the geometric parameters of the wavy trajectory are determined by the position parameters in S2.

2. The ultra-precision grinding tool path planning method based on phase shift control according to claim 1, characterized in that, The grinding wheel cutting point is the relative cutting point between the grinding wheel and the workpiece on the XY plane of the global coordinate system XYZ, without considering grinding wheel oscillation. Its position coordinates are: In the formula, θ(i,j) is the angular position of the grinding wheel, ρ(i,j) is the position of the grinding wheel relative to the workpiece's rotation center, i.e., the distance between the grinding wheel's cutting point and the workpiece's rotation center, Δθ is the workpiece's rotation angle, and R... w V2 is the workpiece radius, V2 is the workpiece rotation speed, ω2 is the workpiece angular velocity, and i and j are the discrete point numbers from the center of the workpiece at equal angular and equal radial distances, respectively. The distance between the grinding wheel cutting point and the workpiece rotation center is determined based on the position coordinates of the grinding wheel cutting point, and the calculation formula is as follows: In the formula, V f This refers to the grinding wheel feed rate; The position trajectory of the center of the grinding wheel cutting arc relative to the workpiece surface is approximated as a sinusoidal motion. Considering grinding wheel oscillation, the center O of the grinding wheel cutting arc is determined based on the phase. s The system of equations for the position coordinates {x(i,j), y(i,j), z(i,j)} relative to the workpiece surface is as follows: In the formula, S′ is the grinding wheel feed rate, S′=V f / V2, A is the amplitude of grinding wheel oscillation, f w ω1 is the grinding wheel rotation frequency, V1 is the grinding wheel angular velocity, and N1 is the grinding wheel spindle speed; N1 is the total number of points discrete at equal angles, and N2 is the total number of points discrete at equal radial intervals. The grinding wheel cutting profile coordinates are based on the position coordinates {x(i,j),y(i,j),z(i,j)} combined with the grinding wheel cutting arc radius r. s and the phase The contour coordinate equations are determined as follows: In the formula, z(x) i ,y j ) is a discrete point (x) i ,y j The amplitude of grinding wheel oscillation at point ( ); The geometric parameters of the adjacent grinding wheel cutting arcs are determined based on the position coordinates {x(i,j),y(i,j),z(i,j)}, wherein the formula for calculating the center distance between adjacent grinding wheel cutting arcs is: In the formula, O i,j+1 and O i,j Let O be the center of the cutting arc of the adjacent grinding wheels. In the coordinate system X′-Y′-Z′, O i,j The coordinates are (x′) i,j ,z′ i,j ), O i,j+1 The coordinates are (x′) i,j+1 ,z′ i,j+1 ); The angular parameters between the cutting arcs of adjacent grinding wheels include α, β, α', and β', and their calculation formulas are as follows: Adjacent cutting arcs have two intersection points. The coordinate equation of intersection point C(i,j+1) is: x i,j =x′ i,j -r s cos(α+β) The coordinate equation of the intersection point C(i,j-1) is: x i,j-1 =x′ i,j +r s cos(α′+β′) 3. The ultra-precision grinding tool path planning method based on phase shift control according to claim 1, characterized in that, The coordinate transformation described in S3 includes: performing coordinate transformation on the position coordinates of the grinding wheel cutting point and performing coordinate transformation on the coordinates of the intersection points of the adjacent grinding wheel cutting arcs.

4. The ultra-precision grinding tool path planning method based on phase shift control according to claim 3, characterized in that, The coordinate equation of the grinding wheel cutting point position in the grinding wheel coordinate system X′-Y′-Z′ is expressed as: ρ represents the radial position of the grinding wheel relative to the workpiece, R s γ represents the radius of the grinding wheel, and γ represents the angle between the line connecting the selected point on the cutting arc of the grinding wheel and the center of the cutting arc and the cutting plane. The position coordinate equations of points on the workpiece surface in the workpiece coordinate system X″-Y″-Z″ are expressed as follows: z″=z′=[R s -r s cos(γ)]cos(ω1t) After coordinate transformation, the coordinates of the intersection points of adjacent cutting arcs are represented as follows:

5. The ultra-precision grinding tool path planning method based on phase shift control according to claim 4, characterized in that, The phase shift greater than 0 and less than 0.5 is defined as the first phase shift. When the first phase shift exists, the number of ripple tracks on the workpiece surface is equal to the integer part of the ratio of the grinding wheel speed to the workpiece speed. The phase shift greater than or equal to 0.5 is defined as the second phase shift. When the second phase shift exists, the number of ripple tracks on the workpiece surface is equal to twice the integer part of the ratio of the grinding wheel speed to the workpiece speed.

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