Cutting system error compensation method for cylindrical fresnel structure ultra-precision machining

Abstract

The application discloses a cutting system error compensation method for cylindrical Fresnel structure ultra-precision machining and belongs to the technical field of ultra-precision machining. The method comprises the following steps: S1, analyzing a cylindrical Fresnel five-axis machine motion chain, establishing a five-axis machine tool kinematics model and a machine tool error model; S2, using a Fourier amplitude sensitivity analysis method, analyzing the sensitivity of 15 position-independent geometric errors of the five-axis machine tool to the cylindrical Fresnel shape precision, finding a plurality of geometric errors that have the greatest influence on machining, S3, using a micrometer and a test cutting method to identify the geometric errors, and measuring the geometric error values; S4, proposing a five-axis machine tool error compensation iteration algorithm to compensate for the error of the motion amount of each axis. The method realizes error compensation of the cutting system for cylindrical Fresnel structure ultra-precision machining, so as to ensure the precision and rigidity of cylindrical Fresnel machining.

Description

Technical Field

[0001] This invention belongs to the field of ultra-precision machining technology, and specifically relates to a cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures. Background Technology

[0002] Fresnel structures have a wide range of applications. Their advantages, such as light weight and small size, make them widely used in many fields, including solar energy focusing and macroscopic and microscopic imaging. Examples include focusing lenses in concentrated photovoltaic systems and anti-glare screens for laser televisions. With the rapid development of technology, the demand for large-diameter Fresnel optical components is increasing. Cylindrical Fresnel machining methods can reduce machine tool size and mold costs. Combined with roll-to-roll replication technology, Fresnel components can be produced efficiently and in large quantities. Currently, in the process of machining cylindrical Fresnel using the contour method, the two manual displacement stages used to adjust the tool position lead to low overall machine tool rigidity and are prone to problems such as cutting chatter, thus affecting the accuracy and rigidity of cylindrical Fresnel machining. Summary of the Invention

[0003] The purpose of this invention is to overcome the problems of low overall machine tool rigidity and cutting chatter caused by the two manual displacement stages used to adjust the tool position in the existing contour machining of cylindrical Fresnel structures. Therefore, this invention provides a cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures. The cutting system is a five-axis machine tool.

[0004] The method of the present invention compensates for machining errors caused by tool tip eccentricity without adding any tool adjustment mechanism, so as to ensure the accuracy and rigidity of the cylindrical Fresnel machining.

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

[0006] An error compensation iterative method for ultra-precision machining cutting systems of cylindrical Fresnel structures includes the following steps:

[0007] S1. Analyze the kinematic chain of the cylindrical Fresnel five-axis machine tool, and establish the kinematic model and error model of the five-axis machine tool.

[0008] S2. Using the Fourier amplitude sensitivity analysis method, analyze the sensitivity of 15 position-independent geometric errors of a five-axis machine tool to the accuracy of the cylindrical Fresnel shape, and find the geometric errors that have the greatest impact on machining.

[0009] S3. Use a dial indicator and trial cutting method to identify geometric errors and measure the geometric error values;

[0010] S4. A five-axis machine tool error compensation iterative algorithm is proposed to compensate for errors in the motion quantities of the five axes: A-axis, B-axis, C-axis, Y-axis, and Z-axis.

[0011] Furthermore, in step S1, the method for analyzing the kinematic chain of the five-axis machine tool and establishing the kinematic model and error model of the five-axis machine tool is as follows:

[0012] Based on screw theory, a forward kinematics model of the machine tool is established. Given the motion quantities of the five axes (A, B, C, Y, and Z), the coordinates of the tool in the workpiece coordinate system are solved, yielding the homogeneous transformation matrices of the workpiece kinematic chain and the tool kinematic chain.

[0013]

[0014]

[0015]

[0016]

[0017]

[0018] in, and The exponential forms of the homogeneous motion matrices for the C-axis, Z-axis, X-axis, A-axis, and B-axis, respectively; g bw (0) and g bt (0) are the homogeneous matrices of the workpiece and the tool in the initial state, respectively; θ a ,θ b ,θ c ,θ x ,θ z These represent the motion quantities along the A-axis, B-axis, C-axis, X-axis, and Z-axis, respectively; w0 = (w 0x ,w 0y ,w 0z ) T and t0=(t 0x ,t 0y ,t 0z ) T These are the coordinates of the workpiece coordinate system and the tool coordinate system in the absolute coordinate system under the initial state; g bw and g bt These are the homogeneous transformation matrices of the workpiece and tool in their normal states, respectively. The motion of the five axes of the machine tool is derived from the homogeneous transformation matrices of the workpiece and tool in their initial states. 0x ,w 0y ,w 0z These represent the x, y, and z coordinates of the workpiece coordinate system origin in the absolute coordinate system under the initial state; t 0x ,t 0y ,t 0z These are the x, y, and z coordinates of the tool coordinate system origin in the absolute coordinate system under the initial state;

[0019] A geometric error and tool pose error analysis was performed on the five axes of a five-axis machine tool, establishing 15 position-independent geometric errors. These 15 position-independent geometric errors are: perpendicularity errors SX and SZ of the X and Z axes. xz The angular error ε of the A-axis around the Y-axis ba and the angular error ε of the A-axis around the Z-axis ca ; Angular error ε of B-axis around X-axis ab Angular error ε of B-axis around Z-axis cb And the offset error δ of the B-axis relative to the ideal A-axis zb ; Angle error ε of the C-axis around the X-axis ac Angular error ε of C-axis around Y-axis bc And the offset error δyc of the C-axis relative to the ideal A-axis; the position error and attitude error of the tool in the initial state, where the position error of the tool tip coordinate is the offset along the three coordinate axes δxt, δyt and δzt, and the attitude error of the tool is the rotation angle εat, εbt and εct of the tool around the three coordinate axes;

[0020] Multiplying the error transformation matrices sequentially yields the total error transformation matrix for all five axes:

[0021]

[0022] in, and These are the total error transformation matrices for the translational axes X and Z, respectively; and This is the total error transformation matrix for the rotation axes A, B, and C;

[0023] It is S xz The transformation matrix, S xz This indicates the perpendicularity error between the X and Z axes. It is the transformation matrix of δyc. It is ε ac The transformation matrix, It is ε bc The transformation matrix, It is ε ba The transformation matrix, It is ε ca The transformation matrix, It is δ zb The transformation matrix, It is ε ab The transformation matrix, It is ε cb The transformation matrix, It is the transformation matrix of δxt. It is the transformation matrix of δyt. It is the transformation matrix of δzt. It is the transformation matrix of εat. It is the transformation matrix of εbt. It is the transformation matrix of εct; I 4×4 It is a 4×4 diagonal matrix of 0 and 1, with 1s on the diagonal and 0s on the rest.

[0024] Multiplying the geometric error matrix of each axis by the motion matrix on the left, we obtain the actual motion matrix of each axis. Substituting this into equation (3), we obtain the geometric error model of the cylindrical Fresnel five-axis machine tool:

[0025]

[0026] Combined with the ideal homogeneous coordinates of the blade tip point P = (0,0,0,1) T The direction vector of the cutting edge is L = (0,0,0,1). T And by combining the motion quantities of each axis, the actual tool tip position P′ and the cutting edge direction L′ are calculated, as shown in equations (8) and (9):

[0027] P'=g wt P (8)

[0028] L'=g wt L (9)

[0029] Among them, g wt This represents the coordinate matrix of the origin of the tool coordinate system in the workpiece coordinate system.

[0030] Furthermore, in step S2, the Fourier amplitude sensitivity analysis method is used to sequentially analyze the sensitivity of 15 position-independent geometric errors (PIGEs) of the five-axis machine tool to the accuracy of the cylindrical Fresnel shape. The method for identifying the geometric errors that have the greatest impact on machining is as follows:

[0031] The three shape parameters of the cylindrical Fresnel structure—depth Δh, radius r, and working angle β—are used as the objective functions for sensitivity analysis.

[0032]

[0033] Among them, P 0ρ P represents the polar radius coordinates of the tool tip. 0θ Let P be the polar coordinate of the tool tip. 0z Let r0 be the Fresnel radius of the tool tip, r1 be the Fresnel radius of a point on the cutting edge, and R be the Z-coordinate of the tool tip. roll Let P be the radius of the roller die. 1ρ The polar radius coordinates of point P1;

[0034] For each objective function, establish its variable space search curve:

[0035]

[0036] Where: θ is a vector composed of motion quantities along each axis; δ is a vector composed of 15 geometric errors, sampled using an oscillatory sampling function, and is a function of the search curve parameter s; Δh(s), r(s), and β(s) represent the depth of the Fresnel ring; the radius of the Fresnel ring; the working angle at a certain point on the Fresnel ring; f Δh f r f β These functions represent the search curve functions for the depth of a Fresnel ring, the radius of a Fresnel ring, and the working angle at a specific point on the Fresnel ring.

[0037] The final analysis revealed that the main geometric error affecting the Fresnel shape accuracy of the five-axis machine tool is the deviation δ of the tool along the X and Z directions in the initial state. xt and δ zt The deviation of the B-axis along the Z-direction δ zb The deviation δ of the C-axis along the Y direction yc .

[0038] Furthermore, in step S3, the geometric error is identified using a dial indicator and a trial-cut method. The method for measuring the geometric error value is as follows:

[0039] A standard cylinder is mounted on the A-axis and adjusted to be concentric with it. A cylindrical surface with the same diameter as the standard cylinder is machined in-situ on a five-axis machine tool. This cylindrical surface is at the same height as the C-axis. The A-axis is mounted on the worktable of the five-axis machine tool. A dial indicator is placed on both the standard cylindrical surface and the machined cylindrical surface. The dial indicator holder is placed on a high-precision plate. The difference in the dial indicator readings on both sides is the height difference δ between the A-axis and the C-axis. yc ;

[0040] The initial eccentricity error of the B-axis is caused by δ xt and δ zt Composed of; when the B-axis rotates, the eccentricity error manifests as changes in the X and Z coordinates of the tool tip. The change in the X coordinate of the tool tip is calculated using the tool setting method. For each angle θ rotated by the B-axis... b Move the X-axis of the machine tool until the tool tip just touches the workpiece surface, record the position measured by the X-axis grating, rotate the B-axis several times, and perform multiple tool adjustments to obtain the relationship between the X-axis coordinate of the tool tip and the B-axis angle.

[0041] The tool tip eccentricity error of the A-axis is caused by δ yt δ zt δ zbThe Z-coordinate of the tool tip in the A-axis coordinate system varies depending on the B-axis angle. Therefore, the eccentricity error of the A-axis needs to be identified each time the B-axis rotates by a certain angle. The identification method is as follows: when the A-axis is at 0°, with the rake face upward, the Z-axis with the tool tip draws a horizontal line along the axial direction of the roller mold. Then, returning to the starting point, the C-axis rotates the workpiece in the positive direction, and the tool tip draws a vertical line circumferentially on the roller surface. Then, the tool is retracted, the A-axis rotates 180°, the rake face downward, and the tool tip draws another horizontal line axially on the roller. Then, the C-axis is reversed, and another vertical line is drawn circumferentially on the roller. The four lines form a grid. This structure is replicated on a PET polymer film, and the width Δy and length Δz of the grid are measured under a microscope to calculate δ. yt δ zt δ zb error.

[0042] Furthermore, δ is calculated. yt δ zt δ zb The error formula is:

[0043] In the XZ plane, the position of the tool tip moves around the B-axis as the B-axis angle changes. Initially, the A-axis coincides with the X-axis, and the tool tip is on the Z-axis with its X-coordinate being zero. A coordinate system O along the A-axis is established. A X A Z A The position of the tool tip in the A-axis coordinate system is (x t ,z t ), x t The position is obtained by reading the X-axis position of the tool, while the z-axis position is obtained by reading the X-axis position of the tool. t The length of the test-cut mesh, Δz, is obtained by measuring the length of the test-cut mesh. t The relationship is:

[0044] Δz=2z t

[0045] In the formula: Δz is the length of the test mesh (mm); z t The Z position of the tool tip in the A-axis coordinate system, in mm;

[0046] According to all (X) t Z t Coordinates, in O A X A Z A Plot the trajectory of the tool tip as it rotates along the B-axis within the coordinate system. Fit this circular trajectory using the least squares method. The center point of the circle is the centerline of the B-axis, thus establishing the B-axis coordinate system O. B X B Z B Then the initial position of the tool tip in the B-axis coordinate system is δ. xtand δ zt The distance between the centerline of axis B and the centerline of axis A is δ. zb Based on the centers of the B-axis and A-axis, an absolute coordinate system OXZ is obtained. The distance between the two horizontal lines of the test cut structure is the position error δ of the tool tip in the Y direction. yt Twice the value of δ, calculate δ. yt for:

[0047]

[0048] In the formula: Δy is the width (mm) of the test mesh.

[0049] Furthermore, in step S4, a five-axis machine tool error compensation iterative algorithm is proposed, and the method for error compensation of the motion of each axis is as follows:

[0050] Perform inverse kinematics on a five-axis machine tool to solve for the motion quantities of the five axes;

[0051]

[0052] Where x0, y0, and z0 are the XYZ coordinates of the tool tip in the workpiece coordinate system. ψ and θ are two Euler angles of the cutting edge in the workpiece coordinate system; b θ c θ a These represent the rotation angles of the B-axis, C-axis, and A-axis, respectively.

[0053] Rearranging terms in equation (3), we get:

[0054]

[0055] Let p = (0,0,0,1) T Let p be the intersection of axis A and axis B in the initial state. Rotation along axes A and B will not change the position of p. Then:

[0056]

[0057] Where p is the intersection of the A-axis and B-axis in the initial state, and is also the homogeneous coordinate of the tool tip in the initial state;

[0058] Therefore, equation (13) can be written as:

[0059]

[0060] Will and Substituting the matrix into equation (15) yields:

[0061]

[0062] Solving equation (16), we get:

[0063]

[0064] Substitute equation (17) into equation (12), and... and Move it to the left and you get:

[0065]

[0066] Take a point q = (0,0,1,1) on the cutting edge. T Multiplying this by equation (18) gives:

[0067]

[0068] Subtracting p from both sides of equation (18) yields: q is a point on the cutting edge;

[0069]

[0070] make:

[0071]

[0072] In equation (21), u and v have no practical meaning;

[0073] Equation (20) can be expressed as:

[0074]

[0075] Solve for the motion quantities θ along the A-axis and B-axis. a and θ b The rotation angles are as follows:

[0076]

[0077] In equation (23), ω a Let ω be the homogeneous coordinate of the A-axis; b The coordinates of the B-axis are homogeneous.

[0078] The parameters are expressed as follows:

[0079]

[0080] In equation (24), z is just an algebraic number. After simplifying the equation, z can be expressed as:

[0081] z=k1ω b +k2ω a +k3(ω b ×ω a (25)

[0082] In equation (25), k1, k2, and k3 are just algebras to simplify the equation;

[0083]

[0084] Based on the sensitivity analysis results, only the five geometric errors in the geometric error model are considered: δ xt δ yt δ zt δ zb and δ yc Then the geometric error model (7) of the five-axis machine tool becomes:

[0085]

[0086] The homogeneous transformation matrices for the tool and the workpiece are as follows:

[0087]

[0088] Furthermore, an iterative algorithm is proposed for error compensation of this configuration of five-axis machine tool.

[0089] The iterative solution approach is as follows: given the target position of the tool tip and the target orientation of the cutting edge f = (x0, y0, z0, ... ψ0) T ,in, ψ0 and ψ0 are the two Euler angles of the cutting edge in the workpiece coordinate system in the initial state;

[0090] The iteration ends when the coordinate error of the three positions of the tool tip is less than 1 nm and the angular error of the two Euler angles of the cutting edge is less than 1 arcsecond. The motion quantities of each axis are solved using an ideal, error-free inverse kinematic model. As the initial value for iteration, the initial value is substituted into the geometric error model (7) to obtain the actual position of the tool tip and the actual posture of the cutting edge. ψ 0 These represent the three coordinate positions of the blade tip and the two Euler angles, respectively.

[0091] Determine if the iteration termination condition is met. If not, begin the iteration. Let i be the iteration number, and the motion quantities of each axis obtained after the i-th iteration are... With x i Given an initial value, solve for x. i+1 Then put x i+1 Substitute the actual tool pose into the geometric error mode (7). It then checks whether the iteration termination condition is met. If it is met, the final result is obtained; otherwise, the iteration continues.

[0092] The calculation process of iterative solution is as follows: Suppose that i iterations have been performed, use the result of the i-th iteration as the initial value, and solve for the result of the (i+1)-th iteration;

[0093] Multiply both sides of equation (27) by the initial state point p = (0,0,0,1). T have:

[0094]

[0095] make:

[0096]

[0097] in and These represent the symbols for the three coordinate values;

[0098] Equation (29) can be simplified to:

[0099]

[0100] There are two conditions:

[0101]

[0102] In equation (32), ω c The coordinates of the C-axis are homogeneous coordinates. Expanding equation (32) will provide the solution. and for:

[0103]

[0104] and These represent the Z-axis and X-axis motion amounts after the (i+1)th iteration, respectively.

[0105] make:

[0106]

[0107] Then equation (31) becomes:

[0108]

[0109] make:

[0110]

[0111] Seek for:

[0112]

[0113] The solution and Substituting into equation (27), we multiply both sides by the direction vector l = (0,0,1,0). T ,get:

[0114]

[0115] make:

[0116]

[0117] in r i and l i It's just an algebra, a simplified method

[0118] The process simplifies equation (38) to:

[0119]

[0120] Therefore, the solution can be obtained. and

[0121]

[0122] and Let A and B represent the motion amounts of the (i+1)th iteration after the first iteration, respectively.

[0123] This completes one iteration and yields a new solution vector. Then use the motion quantities of each axis obtained from the new iteration to solve for the tool pose. x i+1 Substitute into equation (27) and multiply both sides by the initial state p = (0,0,0,1). T and l = (0,0,1,0) T The new tool tip position and cutting edge posture are obtained as follows:

[0124]

[0125] in These are the newly obtained homogeneous coordinates of the blade tip. The newly obtained cutting edge posture:

[0126]

[0127] Therefore, the coordinates of the tool tip and the Euler angles of the cutting edge are obtained as follows:

[0128]

[0129] The difference between the solved tool tip coordinates and cutting edge Euler angle coordinates and the given coordinates after iteration is the condition for terminating the iteration, as shown in equation (45):

[0130]

[0131] If equation (45) is satisfied, the iteration terminates, and the result obtained in the (i+1)th iteration is the final result; otherwise, the iteration terminates with x. i+1 The result is used as the initial value to iterate again until the iteration termination condition is met;

[0132] The error compensation for each motion axis is calculated as follows:

[0133]

[0134] In equation (46), Δθ x ,Δθ z ,Δθ a ,Δθ b ,Δθ c These represent the differences in motion amounts across the five axes before and after the iteration; These represent the motion amounts of the five axes after the iteration minus the motion amounts of the five axes before the iteration.

[0135] Compared with the prior art, the beneficial effects of the present invention are: the present invention compensates for the machining error caused by tool tip eccentricity by using an error compensation method without adding any tool adjustment mechanism, so as to ensure the accuracy and rigidity of the cylindrical Fresnel machining. Attached Figure Description

[0136] Figure 1 This is a schematic diagram of a cutting system for ultra-precision machining of cylindrical Fresnel structures according to the present invention in the absolute coordinate system OXY plane;

[0137] Figure 2 This is a schematic diagram of a cutting system for ultra-precision machining of cylindrical Fresnel structures according to the present invention in the absolute coordinate system OXZ plane;

[0138] Figure 3 A schematic diagram illustrating the measurement of height error between the A-axis and C-axis using a dial indicator;

[0139] Figure 4 For the purpose of identifying δ xt δ yt δ zt δ zb Trial cutting scheme diagram with four error parameters;

[0140] Figure 5 A diagram showing the coordinates of points p and q;

[0141] Figure 6This is a flowchart of the five-axis machine tool error compensation iterative algorithm of the present invention. Detailed Implementation

[0142] Specific implementation method one: as follows Figure 1 , Figure 2 As shown, this embodiment discloses an error compensation iterative method for an ultra-precision machining cutting system for cylindrical Fresnel structures, which includes the following steps:

[0143] S1. Analyze the kinematic chain of the cylindrical Fresnel five-axis machine tool (existing equipment), and establish the kinematic model and error model of the five-axis machine tool;

[0144] S2. Using the Fourier amplitude sensitivity analysis method, analyze the sensitivity of 15 position-independent geometric errors (PIGEs) of a five-axis machine tool to the accuracy of the cylindrical Fresnel shape, and find the geometric errors that have the greatest impact on machining.

[0145] S3. Use a dial indicator and trial cutting method to identify geometric errors and measure the geometric error values;

[0146] S4. A five-axis machine tool error compensation iterative algorithm is proposed to compensate for errors in the motion quantities of the five axes: A-axis, B-axis, C-axis, Y-axis, and Z-axis.

[0147] Furthermore, in step S1, the method for analyzing the kinematic chain of the five-axis machine tool and establishing the kinematic model and error model of the five-axis machine tool is as follows:

[0148] Based on screw theory, a forward kinematics model of the machine tool is established. Given the motion quantities of the five axes (A, B, C, Y, and Z), the coordinates of the tool in the workpiece coordinate system are solved, yielding the homogeneous transformation matrices of the workpiece kinematic chain and the tool kinematic chain.

[0149]

[0150]

[0151]

[0152]

[0153]

[0154] in, and The exponential forms of the homogeneous motion matrices for the C-axis, Z-axis, X-axis, A-axis, and B-axis, respectively; g bw (0) and g bt (0) are the homogeneous matrices of the workpiece and the tool in the initial state, respectively; θ a ,θ b,θ c ,θ x ,θ z These represent the motion quantities along the A-axis, B-axis, C-axis, X-axis, and Z-axis, respectively; w0 = (w 0x ,w 0y ,w 0z ) T and t0=(t 0x ,t 0y ,t 0z ) T These are the coordinates of the workpiece coordinate system and the tool coordinate system in the absolute coordinate system under the initial state; g bw and g bt These are the homogeneous transformation matrices of the workpiece and tool in their normal states, respectively. The motion of the five axes of the machine tool is derived from the homogeneous transformation matrices of the workpiece and tool in their initial states. 0x ,w 0y ,w 0z These represent the x, y, and z coordinates of the workpiece coordinate system origin in the absolute coordinate system under the initial state; t 0x ,t 0y ,t 0z These are the x, y, and z coordinates of the tool coordinate system origin in the absolute coordinate system under the initial state;

[0155] A geometric error and tool pose error analysis was performed on the five axes of a five-axis machine tool, establishing 15 position-independent geometric errors. These 15 position-independent geometric errors are: perpendicularity errors SX and SZ of the X and Z axes. xz The angular error ε of the A-axis around the Y-axis ba and the angular error ε of the A-axis around the Z-axis ca ; Angular error ε of B-axis around X-axis ab Angular error ε of B-axis around Z-axis cb And the offset error ε of the B-axis relative to the ideal A-axis zb ; Angle error ε of the C-axis around the X-axis ac Angular error ε of C-axis around Y-axis bc And the offset error δyc of the C-axis relative to the ideal A-axis; the position error and attitude error of the tool in the initial state, where the position error of the tool tip coordinate is the offset along the three coordinate axes δxt, δyt and δzt, and the attitude error of the tool is the rotation angle εat, εbt and εct of the tool around the three coordinate axes;

[0156] Multiplying the error transformation matrices sequentially yields the total error transformation matrix for all five axes:

[0157]

[0158] in, and These are the total error transformation matrices for the translational axes X and Z, respectively; and This is the total error transformation matrix for the rotation axes A, B, and C;

[0159] It is S xz The transformation matrix, S xz This indicates the perpendicularity error between the X and Z axes. It is the transformation matrix of δyc. It is ε ac The transformation matrix, It is ε bc The transformation matrix, It is ε ba The transformation matrix, It is ε ca The transformation matrix, It is δ zb The transformation matrix, It is ε ab The transformation matrix, It is ε cb The transformation matrix, It is the transformation matrix of δxt. It is the transformation matrix of δyt. It is the transformation matrix of δzt. It is the transformation matrix of εat. It is the transformation matrix of εbt. It is the transformation matrix of εct; I 4×4 It is a 4×4 diagonal matrix of 0 and 1, with 1s on the diagonal and 0s on the rest.

[0160] Multiplying the geometric error matrix of each axis by the motion matrix on the left, we obtain the actual motion matrix of each axis. Substituting this into equation (3), we obtain the geometric error model of the cylindrical Fresnel five-axis machine tool:

[0161]

[0162] Combined with the ideal homogeneous coordinates of the blade tip point P = (0,0,0,1) T The direction vector of the cutting edge is L = (0,0,0,1). T And by combining the motion of each axis, the actual tool tip position P′ and the cutting edge direction L′ are calculated, as shown in equations (8) and (9):

[0163] P'=g wt P (8)

[0164] L'=g wt L (9)

[0165] Among them, gwt This represents the coordinate matrix of the origin of the tool coordinate system in the workpiece coordinate system.

[0166] Furthermore, in step S2, the Fourier amplitude sensitivity analysis method is used to sequentially analyze the sensitivity of 15 position-independent geometric errors (PIGEs) of the five-axis machine tool to the accuracy of the cylindrical Fresnel shape. The method for identifying the geometric errors that have the greatest impact on machining is as follows:

[0167] Using the three shape parameters of the cylindrical Fresnel shape—depth Δh, radius r, and working angle β—as the objective function for sensitivity analysis:

[0168]

[0169] Among them, P 0ρ P represents the polar radius coordinates of the tool tip. 0θ Let P be the polar coordinate of the tool tip. 0z Let r be the Z-coordinate of the tool tip, r0 be the Fresnel radius of the tool tip, and r1 be the Fresnel radius of a point on the cutting edge (let this point be P1, which differs from the tool tip only in its Z-coordinate; the tool tip's Z-coordinate is 0, while this point's Z-coordinate is -0.2). roll Let P be the radius of the roller die. 1ρ The polar radius coordinates of point P1;

[0170] For each objective function, establish its variable space search curve:

[0171]

[0172] Where: θ is a vector composed of motion quantities along each axis; δ is a vector composed of 15 geometric errors, sampled using an oscillatory sampling function, and is a function of the search curve parameter s; Δh(s), r(s), and β(s) represent the depth of the Fresnel ring; the radius of the Fresnel ring; the working angle at a certain point on the Fresnel ring; f Δh f r f β These functions represent the search curves for the depth of a Fresnel ring, the radius of a Fresnel ring, and the working angle at a specific point on the Fresnel ring.

[0173] The final analysis revealed that the main geometrical error affecting the Fresnel shape accuracy of the five-axis machine tool is the deviation δ of the tool along the X and Z directions in the initial state. xt and δ zt The deviation of the B-axis along the Z-direction δ zb The deviation δ of the C-axis along the Y direction yc .

[0174] Furthermore, in step S3, the geometric error is identified using a dial indicator and a trial-cut method. The method for measuring the geometric error value is as follows:

[0175] like Figure 3 As shown, a standard cylinder is mounted on the A-axis and adjusted to be concentric with it. A cylindrical surface with the same diameter as the standard cylinder is machined in-situ on a five-axis machine tool (i.e., the cylinder is machined in-situ). This cylindrical surface is at the same height as the C-axis. The A-axis is mounted on the worktable of the five-axis machine tool. A dial indicator is placed on both the standard cylindrical surface and the machined cylindrical surface. The dial indicator holder is placed on a high-precision plate. The difference in readings on both sides of the dial indicator is the height difference δ between the A-axis and the C-axis. yc ;

[0176] Used to identify δ xt δ yt δ zt δ zb The trial cutting scheme with four error factors is as follows: Figure 4 As shown, the eccentricity error of the B-axis in the initial state is caused by δ xt and δ zt Composed of;

[0177] When the B-axis rotates, the eccentricity error manifests as changes in the X and Z coordinates of the tool tip. The change in the X coordinate of the tool tip is calculated using the tool setting method. This change occurs for each angle θ that the B-axis rotates. b Move the X-axis of the machine tool until the tool tip just touches the workpiece surface, record the position measured by the X-axis grating, rotate the B-axis several times, and perform multiple tool adjustments to obtain the relationship between the X-axis coordinate of the tool tip and the B-axis angle.

[0178] The tool tip eccentricity error of the A-axis is caused by δ yt δ zt δ zb The Z-coordinate of the tool tip in the A-axis coordinate system varies with the B-axis angle. Therefore, the eccentricity error of the A-axis must be identified each time the B-axis rotates by a certain angle. The identification method is as follows: when the A-axis is at 0°, with the rake face upward, the Z-axis with the tool tip draws a horizontal line along the axial direction of the roller mold. Then, returning to the starting point, the C-axis rotates the workpiece in the positive direction (clockwise), and the tool tip draws a vertical line circumferentially on the roller surface. Then, the tool is retracted, the A-axis rotates 180°, the rake face downward, and the tool tip draws another horizontal line axially on the roller. Then, the C-axis reverses, and another vertical line is drawn circumferentially on the roller. The four lines form a grid. This structure is replicated on a PET polymer film, and the width Δy and length Δz of the grid are measured under a microscope to calculate δ. yt δ zt δ zb error.

[0179] Furthermore, δ is calculated. yt δ zt δ zb The error formula is:

[0180] In the XZ plane, the position of the tool tip moves around the B-axis as the B-axis angle changes. Initially, the A-axis coincides with the X-axis, and the tool tip is on the Z-axis with its X-coordinate being zero. A coordinate system O along the A-axis is established. A X A Z A ,like Figure 4 As shown; the position of the tool tip in the A-axis coordinate system is (x t ,z t ), x t The position is obtained by reading the X-axis position of the tool, while the z-axis position is obtained by reading the X-axis position of the tool. t The length of the test-cut mesh, Δz, is obtained by measuring the length of the test-cut mesh. t The relationship is:

[0181] Δz=2z t

[0182] In the formula: Δz is the length of the test mesh (mm); z t The Z position of the tool tip in the A-axis coordinate system, in mm;

[0183] According to all (X) t Z t Coordinates, in O A X A Z A Plot the trajectory of the tool tip as it rotates along the B-axis within the coordinate system. Fit this circular trajectory using the least squares method. The center point of the circle is the centerline of the B-axis, thus establishing the B-axis coordinate system O. B X B Z B Then the initial position of the tool tip in the B-axis coordinate system is δ. xt and δ zt The distance between the centerline of axis B and the centerline of axis A is δ. zb Based on the centers of the B-axis and A-axis, an absolute coordinate system OXZ is obtained. The distance between the two horizontal lines of the test cut structure is the position error δ of the tool tip in the Y direction. yt Twice the value of δ, calculate δ. yt for:

[0184]

[0185] In the formula: Δy is the width (mm) of the test mesh.

[0186] Furthermore, in step S4, a five-axis machine tool error compensation iterative algorithm is proposed, and the method for error compensation of the motion of each axis is as follows:

[0187] Perform inverse kinematics on a five-axis machine tool to solve for the motion quantities of the five axes;

[0188]

[0189] Where x0, y0, and z0 are the XYZ coordinates of the tool tip in the workpiece coordinate system. ψ and θ are two Euler angles of the cutting edge in the workpiece coordinate system; b θ c θ a These represent the rotation angles of the B-axis, C-axis, and A-axis, respectively.

[0190] Rearranging terms in equation (3), we get:

[0191]

[0192] Let p = (0,0,0,1) T The intersection of the A-axis and B-axis in the initial state, such as... Figure 5 As shown, rotation along axes A and B will not change the position of p, therefore:

[0193]

[0194] Where p is the intersection of the A-axis and B-axis in the initial state, and is also the homogeneous coordinate of the tool tip in the initial state;

[0195] Therefore, equation (13) can be written as:

[0196]

[0197] Will and Substituting the matrix into equation (15) yields:

[0198]

[0199] Solving equation (16), we get:

[0200]

[0201] Substitute equation (17) into equation (12), and... and Move it to the left and you get:

[0202]

[0203] Take a point q = (0,0,1,1) on the cutting edge. T ,like Figure 5 As shown, multiplying by equation (18) simultaneously, we get:

[0204]

[0205] Subtracting p from both sides of equation (18) yields: q is a point on the cutting edge;

[0206]

[0207] make:

[0208]

[0209] In equation (21), u and v have no practical meaning;

[0210] Equation (20) can be expressed as:

[0211]

[0212] Solve for the motion quantities θ along the A-axis and B-axis. a and θ b The rotation angles are as follows:

[0213]

[0214] In equation (23), ω a Let ω be the homogeneous coordinate of the A-axis; b The coordinates of the B-axis are homogeneous.

[0215] The parameters are expressed as follows:

[0216]

[0217] In equation (24), z is just an algebraic number. After simplifying the equation, z can be expressed as:

[0218] z=k1ω b +k2ω a +k3(ω b ×ω a (25)

[0219] In equation (25), k1, k2, and k3 are just algebras to simplify the equation;

[0220]

[0221] Based on the sensitivity analysis results, only the five PIGEs geometric errors in the geometric error model are considered: δ xt δ yt δ zt δ zb and δ yc Then the geometric error model (7) of the five-axis machine tool becomes:

[0222]

[0223] The homogeneous transformation matrices for the tool and the workpiece are as follows:

[0224]

[0225] Furthermore, an iterative algorithm is proposed for error compensation of the five-axis machine tool with this configuration. The flowchart of the iterative solution process is shown in Figure (6).

[0226] The iterative solution approach is to find the target position of the tool tip and the target orientation of the cutting edge. in, ψ0 and ψ0 are the two Euler angles of the cutting edge in the workpiece coordinate system in the initial state;

[0227] The iteration ends when the error of the three coordinates of the tool tip (x, y, z, the difference between the two calculated values, which must be small) is less than 1 nm, and the angle error of the two Euler angles of the cutting edge is less than 1 arcsecond. The motion quantities of each axis are then solved using an ideal, error-free inverse kinematic model. As the initial value for iteration, the initial value is substituted into the geometric error model (7) to obtain the actual position of the tool tip and the actual posture of the cutting edge. ψ 0 These represent the three coordinate positions of the blade tip and the two Euler angles, respectively.

[0228] Determine if the iteration termination condition is met. If not, begin the iteration. Let i be the iteration number, and the motion quantities of each axis obtained after the i-th iteration are... With x i Given an initial value, solve for x. i+1 Then put x i+1 Substitute the actual tool pose into the geometric error mode (7). It then checks whether the iteration termination condition is met. If it is met, the final result is obtained; otherwise, the iteration continues.

[0229] The calculation process of iterative solution is as follows: Suppose that i iterations have been performed, use the result of the i-th iteration as the initial value, and solve for the result of the (i+1)-th iteration;

[0230] Multiply both sides of equation (27) by the initial state point p = (0,0,0,1). T have:

[0231]

[0232] make:

[0233]

[0234] in and These represent the symbols for the three coordinate values;

[0235] Equation (29) can be simplified to:

[0236]

[0237] There are two conditions:

[0238]

[0239] In equation (32), ω c The coordinates of the C-axis are homogeneous coordinates. Expanding equation (32) will provide the solution. and for:

[0240]

[0241] and These represent the Z-axis and X-axis motion amounts after the (i+1)th iteration, respectively.

[0242] make:

[0243]

[0244] Then equation (31) becomes:

[0245]

[0246] make:

[0247]

[0248] Seek for:

[0249]

[0250] The solution and Substituting into equation (27), we multiply both sides by the direction vector l = (0,0,1,0). T ,get:

[0251]

[0252] make:

[0253]

[0254] in r i and l i It's just an algebra, a simplified method

[0255] The process simplifies equation (38) to:

[0256]

[0257] Therefore, the solution can be obtained. and

[0258]

[0259] and These represent the motion amounts along the A-axis and B-axis respectively after the (i+1)th iteration;

[0260] This completes one iteration and yields a new solution vector. Then use the motion quantities of each axis obtained from the new iteration to solve for the tool pose. x i+1 Substitute into equation (27) and multiply both sides by the initial state p = (0,0,0,1). T and l = (0,0,1,0) T The new tool tip position and cutting edge orientation are obtained as follows:

[0261]

[0262] in These are the newly obtained homogeneous coordinates of the blade tip. The newly obtained cutting edge posture:

[0263]

[0264] Therefore, the coordinates of the tool tip and the Euler angles of the cutting edge are obtained as follows:

[0265]

[0266] The difference between the solved tool tip coordinates and cutting edge Euler angle coordinates and the given coordinates after iteration is the condition for terminating the iteration, as shown in equation (45):

[0267]

[0268] If equation (45) is satisfied, the iteration terminates, and the result obtained in the (i+1)th iteration is the final result; otherwise, the iteration terminates with x. i+1 The result is used as the initial value to iterate again until the iteration termination condition is met;

[0269] The error compensation for each motion axis is calculated as follows:

[0270]

[0271] In equation (46), Δθ x ,Δθ z ,Δθa ,Δθ b ,Δθ c These represent the differences in motion amounts across the five axes before and after the iteration; These represent the motion amounts of the five axes after the iteration minus the motion amounts of the five axes before the iteration.

[0272] Figure 6 In the given target position of the tool tip and target orientation of the cutting edge... in, Let ψ0 be the two Euler angles of the cutting edge in the workpiece coordinate system under the initial state. The iteration ends when the coordinate error of the three positions of the tool tip is less than 1 nm (the three coordinate positions of the tool tip, x, y, z, are the difference between the values ​​obtained in the previous two iterations, and the difference must be small), and the angle error of the two Euler angles of the cutting edge is less than 1 arcsecond. The motion quantities of each axis are solved using an ideal, error-free inverse kinematic model. As the initial value for iteration, the initial value is substituted into the geometric error model (7) to obtain the actual position of the tool tip and the actual posture of the cutting edge.

[0273] Determine if the iteration termination condition is met. If not, begin the iteration. Let i be the iteration number, and the motion quantities of each axis obtained after the i-th iteration are... With x i Given an initial value, solve for x. i+1 Then put x i+1 Substitute the geometric error model (7) into the equation to find the actual tool pose. Then determine whether the iteration termination condition is met. If it is met, the final result is obtained; otherwise, the iteration continues.

Claims

1. A method for compensating cutting system errors in ultra-precision machining of cylindrical Fresnel structures, characterized in that, The method includes the following steps: S1. Analyze the kinematic chain of the cylindrical Fresnel five-axis machine tool, and establish the kinematic model and error model of the five-axis machine tool. S2. Using the Fourier amplitude sensitivity analysis method, analyze the sensitivity of 15 position-independent geometric errors of a five-axis machine tool to the accuracy of the cylindrical Fresnel shape, and find the geometric errors that have the greatest impact on machining. S3. Use a dial indicator and trial cutting method to identify geometric errors and measure the geometric error values; S4. A five-axis machine tool error compensation iterative algorithm is proposed to compensate for errors in the motion quantities of the five axes: A-axis, B-axis, C-axis, Y-axis, and Z-axis; specifically: Perform inverse kinematics on a five-axis machine tool to solve for the motion quantities of the five axes; The iterative algorithm is used for error compensation in five-axis machine tools. The iterative solution approach is to find the target position of the tool tip and the target orientation of the cutting edge. ,in, and These are the two Euler angles of the cutting edge in the workpiece coordinate system under the initial state; The iteration ends when the coordinate error of the three positions of the tool tip is less than 1 nm and the angular error of the two Euler angles of the cutting edge is less than 1 arcsecond. The motion quantities of each axis are solved using an ideal, error-free inverse kinematic model. As the initial value for iteration, the initial value is substituted into the geometric error model (7) to obtain the actual position of the tool tip and the actual posture of the cutting edge. , These represent the three coordinate positions of the knife tip and the two Euler angles, respectively. Determine if the iteration termination condition is met. If not, begin the iteration. Let i be the iteration number, and the motion quantities of each axis obtained after the i-th iteration are... ;by Given the initial value, solve. , and then Substitute the actual tool pose into the geometric error mode (7). The iteration is then checked to determine if the termination condition is met. If it is met, the final result is obtained; otherwise, the iteration continues. The calculation process of iterative solution is as follows: Suppose that i iterations have been performed, use the result of the i-th iteration as the initial value, and solve for the result of the (i+1)-th iteration; Multiply both sides of equation (27) by the initial state point have: （29） make: （30） in , , These represent the symbols for the three coordinate values; Equation (29) can be simplified to: （31） There are two conditions: （32） In equation (32) The homogeneous coordinates of the C-axis can be solved by expanding equation (32). for: （33） and These represent the Z-axis and X-axis motion amounts after the (i+1)th iteration, respectively. make: （34） Then equation (31) becomes: （35） make: （36） Seek for: （37） The solution Substituting into equation (27), both sides are multiplied by the direction vector. ,get: （38） make: （39） in , ; and It's just an algebra, simplifying the equation; simplifying equation (38) to: （40） Therefore, the solution can be obtained. : （41） and Let A and B represent the motion amounts of the (i+1)th iteration after the first iteration, respectively. This completes one iteration and yields a new solution vector. Then, the tool pose is solved using the motion quantities of each axis obtained from the new iteration. ,Will Substitute into equation (27) and multiply by the initial state on both sides simultaneously. and The new tool tip position and cutting edge orientation are obtained as follows: （42） in These are the newly obtained homogeneous coordinates of the blade tip. The newly obtained cutting edge posture: （43） Therefore, the coordinates of the tool tip and the Euler angles of the cutting edge are obtained as follows: （44） The difference between the solved tool tip coordinates and cutting edge Euler angle coordinates and the given coordinates after iteration is the condition for terminating the iteration, as shown in equation (45): （45） If equation (45) is satisfied, the iteration terminates, and the result obtained in the (i+1)th iteration is the final result; otherwise, the iteration terminates. The result is used as the initial value to iterate again until the iteration termination condition is met; The error compensation for each motion axis is calculated as follows: （46） In equation (46), These represent the differences in motion amounts across the five axes before and after the iteration; These represent the motion amounts of the five axes after the iteration minus the motion amounts of the five axes before the iteration.

2. The method for compensating cutting system errors for ultra-precision machining of cylindrical Fresnel structures according to claim 1, characterized in that: In step S1, the method for analyzing the kinematic chain of the five-axis machine tool and establishing the kinematic model and error model of the five-axis machine tool is as follows: Based on screw theory, a forward kinematics model of the machine tool is established. Given the motion quantities of the five axes (A, B, C, Y, and Z), the coordinates of the tool in the workpiece coordinate system are solved, yielding the homogeneous transformation matrices of the workpiece kinematic chain and the tool kinematic chain. （1） （2） （3） （4） （5） in, , , , and These are the exponential forms of the homogeneous motion matrices for the C-axis, Z-axis, X-axis, A-axis, and B-axis, respectively. and These are the homogeneous matrices of the workpiece and the tool in their initial states, respectively. These represent the motion quantities along the A-axis, B-axis, C-axis, X-axis, and Z-axis, respectively. and These are the coordinates of the workpiece coordinate system and the tool coordinate system in the absolute coordinate system under the initial state; These are the homogeneous transformation matrices of the workpiece and the tool in their normal state, respectively. The motion of the five axes of the machine tool is derived from the homogeneous transformation matrices of the workpiece and the tool in their initial state. These are the x, y, and z coordinates of the origin of the workpiece coordinate system in the absolute coordinate system under the initial state; These are the x, y, and z coordinates of the tool coordinate system origin in the absolute coordinate system under the initial state; A geometric error and tool pose error analysis was performed on the five axes of a five-axis machine tool, establishing 15 position-independent geometric errors. These 15 position-independent geometric errors are: perpendicularity errors SX and SZ of the X and Z axes. xz Angular error of A-axis around Y-axis and the angular error of the A-axis around the Z-axis ; Angle error of B-axis around X-axis Angular error of B-axis around Z-axis And the offset error of the B-axis relative to the ideal A-axis ; Angle error ε of the C-axis around the X-axis ac Angular error ε of C-axis around Y-axis bc And the offset error δyc of the C-axis relative to the ideal A-axis; the position error and attitude error of the tool in the initial state, where the position error of the tool tip coordinate is the offset along the three coordinate axes δxt, δyt and δzt, and the attitude error of the tool is the rotation angle εat, εbt and εct of the tool around the three coordinate axes; Multiplying the error transformation matrices sequentially yields the total error transformation matrix for all five axes: （6） in, and These are the total error transformation matrices for the translational axes X and Z, respectively; , and The total error transformation matrix for rotation axes A, B, and C; yes The transformation matrix, This indicates the perpendicularity error between the X and Z axes. It is the transformation matrix of δyc. It is ε ac The transformation matrix, It is ε bc The transformation matrix, yes The transformation matrix, yes The transformation matrix, yes The transformation matrix, yes The transformation matrix, yes The transformation matrix, It is the transformation matrix of δxt. It is the transformation matrix of δyt. It is the transformation matrix of δzt. It is the transformation matrix of εat. It is the transformation matrix of εbt. It is the transformation matrix of εct; I 4×4 It is a 4×4 diagonal matrix of 0 and 1, with 1s on the diagonal and 0s on the rest. Multiplying the geometric error matrix of each axis by the motion matrix on the left, we obtain the actual motion matrix of each axis. Substituting this into equation (3), we obtain the geometric error model of the cylindrical Fresnel five-axis machine tool: （7） Combining the homogeneous coordinates of the ideal blade tip and the direction vector of the cutting edge And by combining the motion of each axis, the actual tool tip position P′ and the cutting edge direction L′ are calculated, as shown in equations (8) and (9): （8） （9） in, This represents the coordinate matrix of the origin of the tool coordinate system in the workpiece coordinate system.

3. The cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures according to claim 1, characterized in that: In step S2, the Fourier amplitude sensitivity analysis method is used to analyze the sensitivity of 15 position-independent geometric errors of the five-axis machine tool to the accuracy of the cylindrical Fresnel shape. The method for finding the geometric errors that have the greatest impact on machining is as follows: Using the three shape indicators of a cylindrical Fresnel column: depth radius r and working angle As the objective function of sensitivity analysis: （10） in, The polar radius coordinates of the tool tip are... The polar coordinates of the knife tip point. Let Z be the Z-coordinate of the tool tip. Let Fresnel radius be the point at the tip of the knife. Let be the Fresnel radius at a point on the cutting edge. Where is the radius of the roller mold. The polar radius coordinates of point P1; For each objective function, establish its variable space search curve: （11） Where: θ is a vector composed of the motion quantities of each axis; The vector consists of 15 geometric errors, sampled by an oscillating sampling function, and is a function of the search curve parameter s; , , These represent the depth of the Fresnel ring, the radius of the Fresnel ring, and the working angle at a certain point on the Fresnel ring, respectively. , , These functions represent the search curve functions for the depth of a Fresnel ring, the radius of a Fresnel ring, and the working angle at a specific point on the Fresnel ring. The final analysis revealed that the main geometrical error affecting the Fresnel shape accuracy of the five-axis machine tool is the deviation of the tool along the X and Z directions in the initial state. and B-axis deviation along the Z-direction C-axis deviation along the Y direction .

4. The cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures according to claim 3, characterized in that: In step S3, the geometric error is identified using a dial indicator and a trial-cut method. The method for measuring the geometric error value is as follows: Mount a standard cylinder on axis A and adjust it to be concentric with axis A. On a five-axis machine tool, machine a cylindrical surface with the same diameter as the standard cylinder, with the same height as axis C. Mount axis A on the five-axis machine tool table. Place dial indicators on both the standard cylinder surface and the machined cylinder surface. The dial indicator holder is placed on a high-precision plate. The difference in the dial indicator readings on both sides represents the height difference between axis A and axis C. ; The initial eccentricity error of the B-axis is caused by and Composed of; when the B-axis rotates, the eccentricity error manifests as changes in the X and Z coordinates of the tool tip. The change in the X coordinate of the tool tip is calculated using the tool setting method. For each angle the B-axis rotates... Move the X-axis of the machine tool until the tool tip just touches the workpiece surface, record the position measured by the X-axis grating, rotate the B-axis several times, and perform multiple tool adjustments to obtain the relationship between the X-axis coordinate of the tool tip and the B-axis angle. The tool tip eccentricity error of the A-axis is caused by... The composition is such that the Z coordinate of the tool tip in the A-axis coordinate system is different depending on the B-axis angle. Therefore, the eccentricity error of the A-axis must be identified every time the B-axis rotates by an angle. The identification method involves starting at 0° on the A-axis with the rake face upwards, and using the Z-axis with the cutter tip to draw a horizontal line along the roller mold axis. Then, returning to the starting point, the C-axis rotates the workpiece in the positive direction, and the cutter tip draws a vertical line circumferentially on the roller surface. Next, the cutter retracts, the A-axis rotates 180° with the rake face downwards, and the cutter tip draws another horizontal line axially along the roller. Then, the C-axis reverses, drawing another vertical line circumferentially on the roller. These four lines form a grid. This structure is then replicated on a PET polymer film, and the grid width Δy and length Δz are measured under a microscope to calculate... error.

5. The cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures according to claim 4, characterized in that: calculate The error formula is: In the XZ plane, the position of the tool tip moves around the B-axis as the B-axis angle changes. Initially, the A-axis coincides with the X-axis, and the tool tip is on the Z-axis with its X-coordinate being zero. A coordinate system O along the A-axis is established. A X A Z A The position of the tool tip in the A-axis coordinate system is , The position is obtained by reading the X-axis position of the tool. The length of the test-cut mesh is obtained by measuring its length. and The relationship is: In the formula: Δz is the length of the test mesh (mm); The Z position of the tool tip in the A-axis coordinate system, in mm; According to all (X) t Z t Coordinates, in O A X A Z A Plot the trajectory of the tool tip as it rotates along the B-axis within the coordinate system. Fit the circular arc trajectory using the least squares method. The center point of the circle is the centerline of the B-axis. Thus, establish the B-axis coordinate system O. B X B Z B Then the initial position of the tool tip in the B-axis coordinate system is... and The distance between the center line of axis B and the center line of axis A is... Based on the centers of the B-axis and A-axis, an absolute coordinate system OXZ is obtained. The distance between the two horizontal lines of the test cut structure is the position error of the tool tip in the Y direction. Twice the value, find the answer. for: In the formula: Δy is the width (mm) of the test mesh.

6. The cutting system error compensation method for ultra-precision machining of cylindrical Fresnel structures according to claim 1, characterized in that: In step S4, the inverse kinematics of the five-axis machine tool is performed to solve for the motion quantities of the five axes. （12） Where x0, y0, and z0 are the XYZ coordinates of the tool tip in the workpiece coordinate system, and φ and ψ are the two Euler angles of the cutting edge in the workpiece coordinate system; , , These represent the rotation angles of the B-axis, C-axis, and A-axis, respectively. Rearranging equation (3) yields: （13） set up Let p be the intersection of axis A and axis B in the initial state. Rotation along axes A and B will not change the position of p. Then: （14） Where p is the intersection of the A-axis and B-axis in the initial state, and is also the homogeneous coordinate of the tool tip in the initial state; Therefore, equation (13) can be written as: （15） Will Substituting the matrix into equation (15) yields: （16） Solving equation (16), we get: （17） Substitute equation (17) into equation (12), and... Move it to the left and you get: （18） Take a point on the cutting edge Multiplying this by equation (18) gives: （19） Subtracting p from both sides of equation (18) yields: q is a point on the cutting edge; （20） make: （21） In equation (21), u and v have no practical meaning; Then equation (20) can be expressed as: （22） Solve for the motion quantities along the A-axis and B-axis. The rotation angles are as follows: （23） In equation (23), The coordinates of the A-axis are homogeneous. The coordinates of the B-axis are homogeneous. The parameters are expressed as follows: （24） In equation (24), z is just an algebraic number. After simplifying the equation, z can be expressed as: （25） In equation (25), k1, k2, and k3 are simply algebraic terms, simplifying the equation: （26） Based on the sensitivity analysis results, only the five geometric errors in the geometric error model are considered: Then the geometric error model (7) of the five-axis machine tool becomes: （27） The homogeneous transformation matrices for the tool and the workpiece are as follows: （28）。

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