A contour error pre-compensation method based on global analytical reconstruction of NC machining paths

Abstract

The present invention belongs to the technical field of numerical control machining, and proposes a contour error pre-compensation method based on global analytical reconstruction of numerical control machining paths. The nominal path is interpolated, and the transient tracking error at each interpolation point is predicted according to the transient error response of the servo system; the contour error of each interpolation point is used as the initial condition for spline path reconstruction, and the reconstructed spline path is interpolated with parameters such as the nominal path; the actual tool position linear expression of the reconstructed spline path control point is obtained according to the analytical relationship between the transient tracking error and the interpolation instruction; the directed distance between the actual tool position and the corresponding root point on the nominal path is defined as the contour error vector, and a contour error pre-compensation model is established according to the analytical relationship between the contour error vector and the reconstructed spline path control point; the minimum contour error is taken as the optimization target, and the reconstructed spline path control point is taken as the optimization variable, and the least squares algorithm is used to reconstruct the entire spline path, thereby realizing accurate pre-compensation of the contour error.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical control machining, and in particular to a contour error pre-compensation method based on global analytical reconstruction of numerical control machining paths. Background Art

[0002] At present, most of the complex curved surface structural parts in my country's aerospace, energy and power fields are made by CNC machining. Due to the different open-loop gains of each drive axis, the coordinated movement of each axis cannot follow the reference command. Especially in high-speed machining, due to bandwidth limitations, the closed-loop system is difficult to accurately track the rapidly changing position instructions, which inevitably produces contour errors. Therefore, contour error control is crucial to ensure the machining accuracy of the final product. The contour error pre-compensation method can effectively improve the machining accuracy of parts without changing the internal structure of the machine tool and reducing the machining efficiency, and has therefore attracted widespread attention from scholars and the industry.

[0003] A literature search of the prior art found that the patent "Contour error pre-compensation method based on CNC machining path minimization correction, CN 110032142 A" invented a contour error pre-compensation method suitable for five axes, which obtains the optimized compensation amount by establishing a quadratic programming model with the minimum tool position compensation as the optimization goal and the contour error as the boundary constraint condition, and then uses the optimized compensation amount to correct the nominal tool position sequence to achieve contour error pre-compensation. The patent "A multi-axis machining contour error pre-compensation method based on interpolation data, CN 112731865 A" invented an iterative CNC machining contour error pre-compensation method, which uses the contour error vector to correct the command interpolation points in the G code segment that do not meet the accuracy requirements, thereby improving the contour accuracy. In addition, the paper "Generalized Taylor series expansion for free-form two-dimensional contour error compensation, International Journal of Machine Tools & Manufacture, 2012" proposed a generalized Taylor series expansion method for free-form two-dimensional contour error compensation. This method uses the Taylor expansion method to establish an analytical relationship between the compensation amount and the contour error, and then corrects the interpolation points to achieve pre-compensation of the contour error. The various existing contour error pre-compensation methods are mostly based on the contour error vector or the optimized compensation amount to perform point-by-point mirror correction on the discrete tool position points on the tool path, which can effectively reduce the CNC contour error, but lacks overall control of the machining path, and the improvement of contour accuracy is limited. At present, the global analytical reconstruction method of the machining path for reducing contour errors for given CNC machining equipment and nominal machining paths has not yet been involved. Summary of the invention

[0004] In view of the shortcomings of existing contour error pre-compensation technology, the present invention invents a contour error pre-compensation method based on global analytical reconstruction of CNC machining paths. Different from existing discrete methods such as mirror compensation, the proposed method aims to analytically reconstruct the overall tool path under zero contour error conditions, thereby effectively improving the contour accuracy of CNC machining. This method analytically establishes a contour error pre-compensation model for the global machining path, thereby converting the complex contour error pre-compensation problem into a reconstruction and solution problem of the actual spline path control points, and realizing global optimization adjustment of the contour error vector.

[0005] The technical scheme of the present invention is as follows: a contour error pre-compensation method based on global analytical reconstruction of CNC machining paths, which performs motion planning on the nominal path according to the motion limits of each drive axis of the CNC machine tool and the machining bow height difference; uses an interpolation algorithm to interpolate the nominal path, and predicts the transient tracking error at each interpolation point according to the transient error response of the servo system to obtain the contour error estimation of each interpolation point; uses the obtained contour errors of each interpolation point as the initial conditions for reconstructing a new spline path, and interpolates the reconstructed spline path in a manner of sampling parameters such as the nominal path after motion planning; based on The analytical relationship between the predicted transient tracking error and the interpolation instruction of the nominal path is used to obtain the linear expression of the actual tool position of the reconstructed spline path control point; by defining the directed distance between the actual tool position and the corresponding root point on the nominal path as the contour error vector, a contour error pre-compensation model is established according to the analytical relationship between the contour error vector and the reconstructed spline path control point; with the minimum contour error on the entire machining path as the optimization goal and the reconstructed spline path control point as the optimization variable, the least squares algorithm is used to reconstruct the entire spline path, thereby realizing accurate pre-compensation of the contour error.

[0006] The contour error pre-compensation method based on global analytical reconstruction of the NC machining path comprises the following steps:

[0007] Step 1: The total number of segments of the nominal path is N, and the nominal path is segmented; H The parameter length corresponding to the number of backtracking points h of the nominal path segment is N H >1, then the starting point and end point of the segment interval corresponding to the parameter are obtained by formula (1):

[0008]

[0009]

[0010] in, For N H Parameters corresponding to the starting position of the segment nominal path, For NH The end position of the segment corresponds to the parameter;

[0011] Step 2: Use NURBS curve to describe the Nth H The nominal path of the segment, the expression of the NURBS curve is

[0012]

[0013] Where u is the normalized nominal path parameter, and u∈[0,1]; ω i (i=0,1,...,n) is the weight factor of the corresponding control vertex; Θ i are the control vertices of the nominal path curve control polygon, i = 0, 1, ..., n, n is the total number of control points; k is the order of the NURBS spline curve, N i,k (u) is the NURBS spline basis function;

[0014]

[0015] Step 3: According to the motion limits of each axis of the machine tool and the tolerance of the machining bow height, the motion planning is performed for the nominal path of the NH segment, and then the nominal path is interpolated based on the interpolation algorithm to generate an interpolation sequence CL = {CL t |CL t =(P x,t ,P y,t ,P z,t ),t=1,...,l},P x,t ,P y,t ,P z,t Represent the x, y, and z coordinates of the interpolation point t respectively, and the interpolation period is T s , l represents the total number of interpolation points;

[0016] Step 4: Tracking error prediction;

[0017] The position closed-loop servo drive system controlled by the proportional controller simplifies the single-axis feed drive system into a first-order inertial system when the damping ratio ξ∈[0.707,1], and its transfer function G(s) is as follows:

[0018]

[0019] Among them, T τ It is expressed as a time constant, τ represents the x, y, and z directions; s is a complex number;

[0020] Known interpolation point CL t =(P x,t ,P y,t ,Pz,t) and interpolation period T s , then the speed of each axis of the machine tool is expressed as

[0021]

[0022] Substituting equation (2) into equation (4), we can obtain

[0023]

[0024] The interpolation instruction CL of the nominal path t As a ramp signal, with a time interval T s For a single-axis feed drive system that continuously delivers to three directions, the transient tracking error is expressed as;

[0025]

[0026] in, Θ τi It is the spline trajectory control point of the single-axis feed drive system;

[0027] Thus, the actual tool position CL is obtained rt :

[0028]

[0029] P rx,t , P ry,t , P rz,t They represent the x, y, and z coordinates of the actual tool position respectively;

[0030] Step 5: Contour error prediction;

[0031] According to formula (7), the actual tool position CL of the nominal path is obtained rt =[P rx,t ,P ry,t ,P rz,t ], the closest point between the actual tool position point and the nominal path is P f , according to the contour error definition, the vector CL rt -P f With P f The tangent vector at is perpendicular, that is, (CL rt -P f )·dP(u f ) / du=0, where u f P f The corresponding parameters are determined by iterative solution method. f , then the nearest point

[0032] Step 6: Establish a contour error pre-compensation model;

[0033] According to step 2, we get the Nth H The mathematical expression of the actual spline path reconstructed by the segment is

[0034]

[0035] in, The control vertices of the control polygon for the actual spline path curve reconstructed. is the total number of control points; is the weight factor of the corresponding control vertex;

[0036] The reconstructed actual spline path is interpolated by sampling parameters such as the nominal path. According to the transient error response of the servo system, the transient tracking error is predicted, and the transient tracking error at each interpolation point is obtained as follows;

[0037]

[0038] According to equations (8) and (9), the mathematical expression of the actual tool position after pre-compensation is:

[0039]

[0040] in,

[0041] Define the contour error vector as the actual tool position after pre-compensation To the nearest point P corresponding to the nominal path f The directed distance between them, then the contour error vector is;

[0042]

[0043] In order to make the actual tool position after pre-compensation fall on the nominal path, so as to achieve the purpose of zero contour error after pre-compensation, the condition satisfied by formula (11) is

[0044]

[0045] Step 7: Solve the control points;

[0046] According to step 6, the condition for satisfying the contour error of 0 at the interpolation point is:

[0047] ε T ε=0 (13)

[0048] To suppress the N H The contour error of the nominal path segment is used to improve the contour performance of the part and minimize the sum of the squares of the contour errors of all interpolation points Ω, that is,

[0049]

[0050] In order to obtain the control point that meets the conditions, the g-th control point of the reconstructed spline path is derived to obtain the following formula:

[0051]

[0052] in, u m 、u j Represent the parameter values ​​corresponding to the mth and jth interpolation points respectively;

[0053] Formula (15) can be rewritten as:

[0054] ((N * ) T N * )Θ * =γ (16)

[0055] in,

[0056]

[0057]

[0058] The control points are solved by:

[0059] Θ * =((N * ) T N * ) -1 Y (17)

[0060] Step 8: local control point correction and spline path segment splicing;

[0061] In order to ensure the continuity of the previous spline path and the current spline path at the splicing point, the control points of the reconstructed spline path are corrected again; according to C 2 Continuity Conditions and DeBoer-Cox Algorithm, N H The control points of the segment tool nominal path are obtained by the following formula;

[0062]

[0063] in, For N H The k+1th parameter after re-parameterization of the segment reconstruction path, the Nth H-1 and N H The B-spline expression of the segment displacement curve is:

[0064]

[0065] Repeat the above steps until all segmented spline path control points are optimized.

[0066] In particular, when the weight factor ω=1, the nominal path can be represented by a k-order B-spline curve.

[0067] Beneficial effects of the present invention: The contour error pre-compensation method based on global analytical reconstruction of the CNC machining path proposed in the present invention can effectively reduce the contour error of the CNC machining path. Compared with the prior art, the compensation effect is obvious. The method proposed in the present invention analyzes and reconstructs the overall tool path under the condition of zero contour error, thereby effectively improving the CNC machining contour accuracy. The method analytically establishes a contour error pre-compensation model for the global machining path, thereby converting the complex contour error pre-compensation problem into a reconstruction and solution problem of the actual spline path control points, thereby realizing global optimization adjustment of the contour error vector. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a schematic diagram of a contour error pre-compensation method based on global analytical reconstruction of a numerical control machining path according to the present invention.

[0069] Figure 2 It is a processing path diagram represented by NURBS curve. Among them, the X-axis represents the x-axis coordinate of the processing path, the Y-axis represents the y-axis coordinate of the processing path, and the Z-axis represents the z-axis coordinate of the processing path, and the unit is mm.

[0070] Figure 3 It is the contour error diagram before CNC machining path compensation. The X-axis represents the machining time t, in seconds; the Y-axis represents the corresponding absolute value of the contour error, in mm.

[0071] Figure 4 This is the contour error diagram after compensation based on the traditional contour error pre-compensation method. The X-axis represents the processing time t in seconds, and the Y-axis represents the corresponding absolute value of the contour error in mm.

[0072] Figure 5 It is the contour error diagram after compensation by the method proposed in the present invention, wherein the X-axis represents the processing time t, in seconds; the Y-axis represents the corresponding absolute value of the contour error, in mm. DETAILED DESCRIPTION

[0073] The present invention is further described below in conjunction with the accompanying drawings and embodiments, but they are not intended to limit the present invention.

[0074] The schematic diagram of a contour error pre-compensation method based on global analytical reconstruction of NC machining paths proposed by the present invention is shown in FIG. Figure 1As shown in the figure. First, the nominal path is motion planned according to the motion limits of each drive axis of the CNC machine tool and the machining bow height difference, and then the nominal path is interpolated by the interpolation algorithm. According to the transient error response of the servo system, the transient tracking error at each interpolation point is predicted, thereby realizing the contour error estimation of each interpolation point. Further, the obtained contour errors of each interpolation point are used as the initial conditions for the reconstruction of the new spline path, and the reconstructed spline path is interpolated in a way of sampling parameters such as the nominal path. According to the analytical relationship between the transient tracking error and the interpolation instruction of the input nominal path, the linear expression of the actual tool position of the spline path control point is obtained. By defining the directed distance between the actual tool position and the corresponding root point on the nominal path as the contour error vector, the contour error pre-compensation model is established according to the analytical relationship between the contour error vector and the reconstructed spline path control point. Finally, with the minimum contour error on the entire machining path as the optimization goal, the control points of the reconstructed actual spline path are used as the optimization variables, and the least squares algorithm is used to reconstruct the entire spline path, thereby realizing the accurate pre-compensation of the contour error.

[0075] The implementation example is a two-axis spline path, such as Figure 2 The initial feed rate is 20 mm / s, and the sub-axis acceleration constraint range is [-300 mm / s 2 ,300mm / s 2 ], the axis jump constraint range is [-6000mm / s 3 ,6000 / s 3 ], the bow height difference is 0.002mm, and the sampling time is 4ms.

[0076] Step 1: The total number of path segments is N. The nominal path is adaptively segmented. Let the Nth H (N H >1) The parameter length corresponding to the number of backtracking points h of the segment path Then the starting point and end point of the corresponding parameters of the segment interval can be obtained by formula (1):

[0077]

[0078] in, For N H Parameters corresponding to the starting position of the segment nominal path, For N H The end position of the segment corresponds to the parameter;

[0079] Step 2: Use NURBS curve to H In particular, when the weight factor ω = 1, the nominal path can be represented by a k-order B-spline curve. The expression of the k-order B-spline curve is

[0080]

[0081] Where u is the normalized nominal path parameter, and u∈[0,1]; Θ i are the control vertices of the nominal path curve control polygon, i = 0, 1, ..., n, where n is the total number of control points;

[0082]

[0083] Step 3: According to the motion limit of each axis of the machine tool and the tolerance of the machining bow height, H The nominal path is used for motion planning, and then the nominal path is interpolated based on the interpolation algorithm to generate an interpolation sequence CL = {CL t |CL t =(Q x,t ,Q y,t ,Q z,t ),t=1,...,l}, the interpolation period is T s , l represents the total number of interpolation points;

[0084] Step 4: Tracking error prediction;

[0085] When the damping ratio ξ∈[0.707,1] is used for the position closed-loop servo drive system controlled by the proportional controller, the drive system can be simplified to a first-order inertial system, and its transfer function G(s) is as follows:

[0086]

[0087] Among them, T τ Expressed as a time constant;

[0088] Known interpolation point CL t =(P x,t ,P y,t ,P z,t ) and interpolation period T s , then the speed of each axis of the machine tool is expressed as

[0089]

[0090] Substituting equation (2) into equation (4), we can obtain

[0091]

[0092] The interpolation instruction CL of the nominal path t As a ramp signal, with a time interval T s The transient tracking error of a single-axis feed drive system that continuously delivers in three directions is expressed as

[0093]

[0094] in, Θτ i It is the spline trajectory control point of the single-axis feed drive system;

[0095] Thus, the actual tool position CL is obtained rt :

[0096]

[0097] Step 5: Contour error prediction;

[0098] According to formula (7), the actual tool position CL of the nominal path can be obtained: rt =[P rx,t ,P ry,t ,P rz,t ], let the closest point between the actual tool position point and the nominal path be P f , according to the contour error definition, the vector CL rt -P f With P f The tangent vector at is perpendicular, that is, (CL rt -P f )·dP(u f ) / du=0, where u f P f The corresponding parameters are determined by iterative solution method. f , then the nearest point

[0099] Step 6: Establishment of contour error pre-compensation model;

[0100] According to step 2, we can get the Nth H The mathematical expression of the actual spline path reconstructed by the segment is

[0101]

[0102] The reconstructed actual spline path is interpolated in a way that the parameters are equal to the nominal path. According to the transient error response of the servo system, the transient tracking error is predicted, and the transient tracking error at each interpolation point can be obtained as;

[0103]

[0104] According to equations (8) and (9), the mathematical expression of the actual tool position after pre-compensation is:

[0105]

[0106] The contour error vector is defined as the closest point P corresponding to the actual tool position after compensation to the nominal path. fThe pointed distance between them, then the contour error vector is

[0107]

[0108] In order to make the actual tool position after pre-compensation fall on the nominal path, so as to achieve the purpose of zero contour error after pre-compensation, the condition that equation (11) needs to satisfy is:

[0109]

[0110] Step 7: Solve the control points;

[0111] According to step 6, the condition for satisfying the contour error of 0 at the interpolation point can be obtained as follows:

[0112] ε T ε=0 (13)

[0113] To suppress the N H The contour error of the segment path is used to improve the contour performance of the part, and the sum of the squares of the contour errors of all interpolation points Ω is minimized, then

[0114]

[0115] In order to obtain the control points that meet the conditions, the g-th control point is derived to obtain the following formula:

[0116]

[0117] in, u m 、u j Represent the parameter values ​​corresponding to the mth and jth interpolation points respectively;

[0118] Then formula (15) can be rewritten as:

[0119] ((N * ) T N * )Θ*=Υ (16)

[0120] in,

[0121]

[0122]

[0123] The control points can be solved by the following formula:

[0124] Θ * =((N * ) T N * ) -1 Y (17)

[0125] Step 8: local control point correction and spline path segment splicing;

[0126] In order to ensure the continuity of the previous spline path and the current spline path at the splicing point, the control points need to be corrected again. 2 Continuity Conditions and DeBoer-Cox Algorithm, N H The control points of the segment tool path can be obtained by the following formula

[0127]

[0128] Among them, the B-spline expression of the displacement curves of the NH-1 and NH segments is:

[0129]

[0130] Step 9: Repeat the above steps until all segmented spline path control points are optimized.

[0131] like Figure 3 As shown in Figure 1, it is the contour error diagram before CNC machining path compensation, and the maximum contour error is 0.0727mm. Figure 4 This is the contour error diagram after compensation based on the traditional contour error pre-compensation method. After compensation using the existing technology, the maximum contour error is 0.0194mm; Figure 5 This is the contour error diagram after compensation by the method proposed in the present invention. The maximum contour error is 0.0117 mm. Figure 4 Compared with the previous example, it can be improved by about 10.5%.

Claims

1. A contour error pre-compensation method based on global analytical reconstruction of CNC machining paths, characterized in that: The motion planning of the nominal path is carried out according to the motion limit of each driving axis of the CNC machine tool and the machining bow height difference; the interpolation algorithm is used to interpolate the nominal path, and the transient tracking error at each interpolation point is predicted according to the transient error response of the servo system, and the contour error estimation of each interpolation point is obtained; the contour error of each interpolation point is used as the initial condition for reconstructing the new spline path, and the reconstructed spline path is interpolated in a manner of sampling parameters such as the nominal path after motion planning; based on the analytical relationship between the predicted transient tracking error and the interpolation instruction of the nominal path, the linear expression of the actual tool position of the reconstructed spline path control point is obtained; by defining the directed distance between the actual tool position and the corresponding root point on the nominal path as the contour error vector, a contour error pre-compensation model is established according to the analytical relationship between the contour error vector and the reconstructed spline path control point; with the minimum contour error on the entire machining path as the optimization goal, the reconstructed spline path control point is used as the optimization variable, and the least squares algorithm is used to reconstruct the entire spline path, thereby realizing accurate pre-compensation of the contour error.

2. The contour error pre-compensation method based on global analytical reconstruction of NC machining paths according to claim 1 is characterized in that: The following steps are involved: Step 1: The total number of segments of the nominal path is N, and the nominal path is segmented; H The parameter length corresponding to the number of backtracking points h of the nominal path segment is Then the starting point and end point of the segment interval corresponding to the parameter are obtained by formula (1): in, For N H Parameters corresponding to the starting position of the segment nominal path, For N H The end position of the segment corresponds to the parameter; Step 2: Use NURBS curve to describe the Nth H The nominal path of the segment, the expression of the NURBS curve is Where u is the normalized nominal path parameter, and u∈[0,1]; ω i (i=0,1,...,n) is the weight factor of the corresponding control vertex; Θ i are the control vertices of the nominal path curve control polygon, i = 0, 1, ..., n, n is the total number of control points; k is the order of the NURBS spline curve, N i,k (u) is the NURBS spline basis function; Step 3: According to the motion limit of each axis of the machine tool and the tolerance of the machining bow height, H The nominal path is used for motion planning, and then the nominal path is interpolated based on the interpolation algorithm to generate an interpolation sequence CL = {CL t |CL t =(P x,t ,P y,t ,P z,t ),t=1,...,l},P x,t ,P y,t ,P z,t Represent the x, y, and z coordinates of the interpolation point t respectively, and the interpolation period is T s , l represents the total number of interpolation points; Step 4: Tracking error prediction; The position closed-loop servo drive system controlled by the proportional controller simplifies the single-axis feed drive system into a first-order inertial system when the damping ratio ξ∈[0.707,1], and its transfer function G(s) is as follows: Among them, T τ It is expressed as a time constant, τ represents the x, y, and z directions; s is a complex number; Known interpolation point CL t =(P x,t ,P y,t ,P z,t ) and interpolation period T s , then the speed of each axis of the machine tool is expressed as Substituting equation (2) into equation (4), we can obtain The interpolation instruction CL of the nominal path t As a ramp signal, with a time interval T s For a single-axis feed drive system that continuously delivers to three directions, the transient tracking error is expressed as; in, j∈[1,h];Θ τi It is the spline trajectory control point of the single-axis feed drive system; Thus, the actual tool position CL is obtained rt : P rx,t , P ry,t , P rz,t They represent the x, y, and z coordinates of the actual tool position respectively; Step 5: Contour error prediction; According to formula (7), the actual tool position CL of the nominal path is obtained rt =[P rx,t ,P ry,t ,P rz,t ], the closest point between the actual tool position point and the nominal path is P f , according to the contour error definition, the vector CL rt -P f With P f The tangent vector at is perpendicular, that is, (CL rt -P f )·dP(u f ) / du=0, where u f P f The corresponding parameters are determined by iterative solution method. f , then the nearest point Step 6: Establish a contour error pre-compensation model; According to step 2, we get the Nth H The mathematical expression of the actual spline path reconstructed by the segment is in, The control vertices of the control polygon for the actual spline path curve reconstructed. is the total number of control points; is the weight factor of the corresponding control vertex; The reconstructed actual spline path is interpolated in the same way as the nominal path parameter sampling. According to the transient error response of the servo system, the transient tracking error is predicted, and the transient tracking error at each interpolation point is obtained as follows; According to equations (8) and (9), the mathematical expression of the actual tool position after pre-compensation is: in, Define the contour error vector as the actual tool position after pre-compensation To the nearest point P corresponding to the nominal path f The directed distance between them, then the contour error vector is; In order to make the actual tool position after pre-compensation fall on the nominal path, so as to achieve the purpose of zero contour error after pre-compensation, the condition satisfied by formula (11) is Step 7: Solve the control points; According to step 6, the condition for satisfying the contour error of 0 at the interpolation point is: e T ε=0(13) To suppress the N H The contour error of the nominal path segment is used to improve the contour performance of the part and minimize the sum of the squares of the contour errors of all interpolation points Ω, that is, In order to obtain the control point that meets the conditions, the g-th control point of the reconstructed spline path is derived to obtain the following formula: in, u m 、u j Represent the parameter values ​​corresponding to the mth and jth interpolation points respectively; Formula (15) can be rewritten as: ((N * ) T N * )I * =Y(16) among them, The control points are solved by: I * =((N * ) T N * ) -1 Y(17) Step 8: local control point correction and spline path segment splicing; In order to ensure the continuity of the previous spline path and the current spline path at the splicing point, the control points of the reconstructed spline path are corrected again; according to C 2 Continuity Conditions and DeBoer-Cox Algorithm, N H The control points of the segment tool nominal path are obtained by the following formula; in, For N H The k+1th parameter after re-parameterization of the segment reconstruction path, the Nth H-1 and N H The B-spline expression of the segment displacement curve is: Repeat the above steps until all segmented spline path control points are optimized.

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