CN105700475A - Data processing method for realizing machine tool robustness thermal error compensation of wide-range environment temperature - Google Patents

Abstract

The invention discloses a data processing method for realizing machine tool robustness thermal error compensation of wide-range environment temperature. The data processing method comprises the steps that 1. a modeling temperature independent variable Xk is extracted; 2. standardized processing is performed on the Xk, and an expression of a main component Zk is obtained through standardized temperature independent variable Xk* conversion; 3. previous p main components are extracted for participating in modeling; 4. standardized processing is performed on spindle thermal deformation quantity Sj, and standardized thermal deformation quantity Sj* and a multivariate linear regression equation between the previous p main components are established; 5. the Sj* and the regression equation between the previous p main components are converted into an equation of the Sj* and the Xk*; and 6. the regression equation of the Sj* and the Xk* is converted into an equation of the Sj and the Xk, and a thermal error compensation model is established; and the prediction performance of the thermal error model is further analyzed. Influence of coupling performance between the temperature independent variables on distortion of a model independent variable estimation value can be suppressed so that the data processing method has high prediction precision and prediction robustness.

Description

Data processing method for realizing machine tool robustness thermal error compensation of large-range environment temperature

Technical Field

The invention relates to a data processing method for machine tool robust thermal error compensation, in particular to a data processing method for compensating and predicting a machine tool thermal error in a large-range environment temperature.

Background

The numerical control machine tool is important equipment in the manufacturing field, and the processing performance of the numerical control machine tool is one of main marks of the development level of the national manufacturing industry. In the machining process of the numerical control machine tool, due to thermal errors caused by unbalanced temperature rise of all parts of the machine tool, the relative correct position between the cutter and the workpiece is changed, and the machining errors of the workpiece are caused. According to statistics, the thermal error of the numerical control machine tool accounts for about 40-70% of the total error of the machine tool. The processing method of modeling the thermal error data of the numerical control machine tool and compensating in advance through the numerical control system is an effective and economic means for improving the machining precision of the machine tool.

The most commonly adopted method for processing the thermal error data of the machine tool at present is a multiple linear regression based on the least square principle. The method is based on the analysis principle of minimum sum of squared residuals, the established model has high linear fitting accuracy, and the algorithm is easy to realize, so that the method is commonly used for modeling and processing the thermal error data of the machine tool. However, due to the complex non-linear and time-varying characteristics of the machine tool temperature, the machine tool temperature field has different characteristics under different environmental temperature conditions, which mainly show the difference of the coupling among the heat sources of the machine tool, and the larger the difference of the environmental temperatures is, the larger the difference of the coupling of the machine tool temperature field is. Therefore, when the temperature of the environment where the machine tool is located changes greatly, the coupling degree between the temperature sensors changes greatly, which causes distortion of the least square estimation value of each variable in the multiple regression model, which is a disadvantage of the multiple linear regression processing method.

In 2011, mianmao et al proposed a method for processing machine thermal error data by using a time series modeling technique (see the literature, "research on thermal error time series modeling technique for precision data machine tool", from 2011 national precision engineering academy). The method has the advantages that the temperature lag term is incorporated into the thermal error model, so that the linear fitting accuracy of the model is obviously improved, but the method is also limited to the application in a small-range environment temperature range. When the temperature change of the environment where the numerical control machine tool is located is large, the coupling performance among the temperature sensors can have more serious influence on the time series model than the multiple linear regression, and the prediction robustness performance of the thermal error prediction model is seriously reduced.

Disclosure of Invention

The invention aims to overcome the defects in the prior art and provide a data processing method for realizing the robust thermal error compensation of a machine tool in a large range of environmental temperatures. Because the data processing method adopts principal components of the element independent variables to replace the independent variables to establish the thermal error model, and the covariance among the principal components is zero, which means that the principal components are not related to each other, the data processing method provided by the invention well inhibits the influence of the coupling among the temperature independent variables on the distortion of the estimation value of the model independent variables, thereby ensuring the high prediction accuracy and the prediction robustness of the thermal error model established by the data processing method provided by the invention under the condition of a large range of environment and having important practical engineering application.

The invention provides a data processing method for machine tool robustness thermal error compensation in a large range of environment temperature, which comprises the following steps:

a data processing method for realizing robust thermal error compensation of a machine tool in a wide range of environmental temperatures is characterized by comprising the following steps of:

the method comprises the following steps: obtaining a temperature variable DeltaT of a machine tool_iAnd amount of thermal deformation of spindle S_jWherein i is 1, 2, …, m.

The m refers to the number of temperature variables. J is the axial direction of the main shaft of the machine tool, and j is X, Y and/or Z. X, Y, Z respectively represent the X-axis of the machine tool spindle, the Y-axis of the spindle, and the Z-axis of the spindle.

By temperature variation Δ T_iAnd amount of thermal deformation of spindle S_jObtaining the independent temperature variable X_k，k＝1，2，…，n。

The temperature independent variable X_kTo build a thermal error compensation model. N is the number of temperature independent variables, and n is less than or equal to m.

Step two: for the temperature independent variable X obtained in the step one_kCarrying out standardization treatment to obtain a standardized temperature independent variable X_k ^*。

And from a normalized temperature independent variable X_k ^*Converted to obtain a principal component Z_kIs described in (1).

The principal component Z_kRefers to the independent variable X of the temperature by orthogonal transformation_k ^*And a set of variables obtained after conversion is not linearly related.

Step three: from principal component Z obtained in step two_kThe first p principal components Z are extracted₁，Z₂，…，Z_pAnd p is less than or equal to n for the next thermal error modeling.

Step four: for the heat deformation quantity S of the main shaft in the step one_jPerforming standardization process to obtain standard thermal deformation S_j ^*。

And to the normalized heat distortion amount S_j ^*And the principal component Z for thermal error modeling obtained in step 3₁，Z₂，…，Z_pPerforming multiple linear regression analysis to obtain S_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)

Step five: s obtained in the fourth step_j ^*With respect to Z₁，Z₂，…，Z_pIs converted into S_j ^*With respect to X_k ^*The expression equation of (1).

Step six: s obtained in the fifth step_j ^*With respect to X_k ^*Is converted into S_jWith respect to X_kIs the expression equation of_jWith respect to X_kThe expression equation of (a) is the thermal error model.

Further, the thermal error model established in step six, namely S_jWith respect to X_kThe expression equation of (a) analyzes the predicted performance of the thermal error model.

Wherein, the first step specifically comprises the following steps:

1.1 increment Δ T by temperature value of temperature sensor_iFor temperature variation, Δ T is calculated according to the correlation coefficient equation (1)_iAmount of thermal deformation S from the main shaft_jThe value of the correlation coefficient therebetween.

In the formula (1), r_ijIs DeltaT_iAnd S_jValue of the correlation coefficient between, T (Δ T)_i) Is DeltaT_iLength of (1), Δ T_iqIs DeltaT_iThe value of the q-th value in the sequence,is DeltaT_iAverage of the sequences, S_iqIs S_jThe value of the q-th value in the sequence,is S_jAverage of the sequences. Temperature variable Δ T_iIs collected from the machine tool in operation by a temperature sensor. Temperature variable Δ T_iLength of sequence and amount of thermal deformation S_jThe sequences are of the same length.

1.2 extracting n temperature variables with the maximum relational numerical values in the step 1.1 as temperature independent variables X_k. The temperature independent variable X_kUsed to model thermal errors. The value of n is not more than m.

Wherein, the second step specifically comprises the following steps:

2.1 for the extracted temperature independent variable X, using equation (2)_kPerforming standardization treatment to obtain a standardized temperature independent variable X_k ^*。

${X_{kq}}^{*}=\frac{X_{kq}-{\stackrel{\‾}{X}}_k}{\sqrt{\frac{1}{t\left({X_k}^{*}\right)}{\Σ}_{q=1}^{t\left({X_k}^{*}\right)}{(X_{kq}-{\stackrel{\‾}{X}}_k)}^2}},(k=1,2,\mathrm{...},n,q=1,2,\mathrm{...},t({X_k}^{*}\left)\right)---\left(2\right)$

In the formula (2), X_kq ^*Is X_k ^*Value qth in the sequence, X_kqIs X_kThe value of the q-th value in the sequence,is X_kAverage of the sequences, t (X)_k ^*) Is X_k ^*The length of the sequence.

2.2 calculating the correlation coefficient value between any two normalized temperature independent variables by using the correlation coefficient formula (1) in the step 1.1 to obtain a correlation matrix R of the normalized temperature independent variables_n×n(3) Specifically, the following is made.

In the formula (3), r_abDenotes the normalized temperature independent variable X_a ^*And X_b ^*The correlation coefficient value of (a).

2.3 from the formula R_n×nu and λ, where u ═ λ u holds, are eigenvectors and eigenvalues of the correlation matrix R, respectively, and the correlation matrix R is obtained_n×nCharacteristic vector u of_kAnd a characteristic value lambda_kWherein k is 1, 2, …, n.

2.4 normalized temperature independent variable X obtained according to step 2_k ^*And a feature vector u_kThe normalized temperature independent variable X is obtained from the formula (4)_k ^*And a feature vector u_kPrincipal component Z of the composition_kThe expression of (c) is specifically:

$\left\{\begin{array}{c}{Z}_1=u_{11}{X_1}^{*}+u_{21}{X_2}^{*}+\mathrm{...}+u_{k1}{X_k}^{*}+\mathrm{...}+u_{n1}{X_n}^{*}\\ Z_2=u_{12}{X_1}^{*}+u_{22}{X_2}^{*}+\mathrm{...}+u_{k2}{X_k}^{*}+\mathrm{...}+u_{n2}{X_n}^{*}\\ .\\ .\\ .\\ Z_k=u_{1k}{X_1}^{*}+u_{2k}{X_2}^{*}+\mathrm{...}+u_{kk}{X_k}^{*}+\mathrm{...}+u_{nk}{X_n}^{*}\\ .\\ .\\ .\\ Z_n=u_{1n}{X_1}^{*}+u_{2n}{X_2}^{*}+\mathrm{...}+u_{kn}{X_k}^{*}+\mathrm{...}+u_{nn}{X_n}^{*}\end{array}\right.---\left(4\right)$

in the formula (4), Var (Z)_k)＝λ_k，Cov(Z_i，Z_j) 0(i ≠ j), i.e. there is no correlation between the principal components.

Wherein, the third step specifically comprises:

let the first p principal components Z₁，Z₂，…，Z_pHas a cumulative variance contribution rate of Vcc_pP is less than or equal to n, let the cumulative variance contribution rate Vcc_pNot less than 85%, calculating the value of p, and determining the specific principal component Z according to the value of p₁，Z₂，…，Z_pExtracted for further thermal error modeling. Wherein the cumulative variance contribution rate Vcc of the first p principal components_pThe formula of (1) is:

${\mathrm{Vcc}}_p={\Σ}_{i=1}^p{\mathrm{\λ}}_i/{\Σ}_{i=1}^n{\mathrm{\λ}}_i---\left(5\right)$

wherein, the fourth step specifically comprises:

4.1 thermal deformation amount S of spindle by using equation (6)_jPerforming standardization process to obtain standardized heat distortion S_j ^*。

${S_{jq}}^{*}=\frac{S_{jq}-{\stackrel{\‾}{S}}_j}{\sqrt{\frac{1}{t\left({S_j}^{*}\right)}{\Σ}_{q=1}^{t\left({S_j}^{*}\right)}{(S_{jq}-{\stackrel{\‾}{S}}_j)}^2}}---\left(6\right)$

In the formula (6), S_jq ^*Is S_j ^*The qth value in the sequence, S_jqIs S_jThe value of the q-th value in the sequence,is S_jAverage of the sequences, t (S)_j ^*) Is S_j ^*The length of the sequence.

4.2 pairs of normalized Heat distortion amount S_j ^*And a principal component Z participating in thermal error modeling₁，Z₂，…，Z_pPerforming least squares-based multiple linear regression analysis to obtain S_j ^*With respect to Z₁，Z₂，…，Z_pIs expressed as equation (7). Obtaining regression coefficient estimated value of each element in the model by utilizing multiple linear regression analysis based on least square principleS_j ^*With respect to Z₁，Z₂，…，Z_pEquation of expression (2)And regression coefficient estimation valueRespectively as follows:

${{\hat{S}}_j}^{*}={\widehat{\mathrm{\α}}}_1{Z}_1+{\widehat{\mathrm{\α}}}_2{Z}_2+\mathrm{...}+{\widehat{\mathrm{\α}}}_p{Z}_p---\left(7\right)$

${\widehat{\mathrm{\α}}}_p={\left({Z_p}^T{Z}_p\right)}^{-1}{Z_p}^T{S_j}^{*}---\left(8\right)$

wherein, the fifth step specifically comprises:

using the principal component Z obtained in step two_kThe expression of (A) is to obtain the step fourTo S_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)Conversion to S_j ^*With respect to X_k ^*(k is 1, 2, …, n) expression equationS_j ^*With respect to Xk^*Equation of (2)The method comprises the following specific steps:

$\begin{array}{c}{{\hat{S}}_j}^{*}={\widehat{\mathrm{\α}}}_1{Z}_1+{\widehat{\mathrm{\α}}}_2{Z}_2+\mathrm{...}+{\widehat{\mathrm{\α}}}_p{Z}_p\\ ={\widehat{\mathrm{\α}}}_1(u_{11}{X_1}^{*}+u_{21}{X_2}^{*}+\mathrm{...}+u_{k1}{X_k}^{*}+\mathrm{...}+u_{n1}{X_n}^{*})+{\widehat{\mathrm{\α}}}_2(u_{12}{X_1}^{*}+u_{22}{X_2}^{*}+\mathrm{...}+u_{k2}{X_k}^{*}+\mathrm{...}+u_{n2}{X_n}^{*})\\ +\mathrm{...}+{\widehat{\mathrm{\α}}}_p(u_{1n}{X_1}^{*}+u_{2n}{X_2}^{*}+\mathrm{...}+u_{kp}{X_k}^{*}+\mathrm{...}+u_{np}{X_n}^{*})\\ ={\widehat{\mathrm{\β}}}_1{X_1}^{*}+{\widehat{\mathrm{\β}}}_2{X_2}^{*}+\mathrm{...}+{\widehat{\mathrm{\β}}}_k{X_k}^{*}+\mathrm{...}+{\widehat{\mathrm{\β}}}_n{X_n}^{*}\end{array}---\left(9\right)$

in the formula (9), the reaction mixture is,

wherein, the sixth step specifically includes:

to temperature independent variable X_kFormula X for normalization_kq ^*And the amount of thermal deformation S to the spindle_jNormalized formula S_jq ^*S from step five_j ^*With respect to X_k ^*Equation of expression (c) transforms S_jWith respect to X_kAnd (4) completing the establishment of a thermal error prediction model according to the expression equation (10). Wherein S is_jWith respect to X_kEquation (10) is:

${\hat{S}}_j={\widehat{\mathrm{\θ}}}_0+{\widehat{\mathrm{\θ}}}_1{X}_1+{\widehat{\mathrm{\θ}}}_2{X}_2+\mathrm{...}+{\widehat{\mathrm{\θ}}}_k{X}_k+\mathrm{...}+{\widehat{\mathrm{\θ}}}_n{X}_n---\left(10\right)$

${\widehat{\mathrm{\θ}}}_k=\frac{S_{s_j}}{S_{x_k}}{\widehat{\mathrm{\β}}}_k---\left(11\right)$

${\mathrm{\θ}}_0=\stackrel{\‾}{S_j}-{\Σ}_{k=1}^n{\widehat{\mathrm{\θ}}}_k{\stackrel{\‾}{X}}_k---\left(12\right)$

in the formula (11), the reaction mixture is,is S_jThe discrete standard deviation of the sequence is determined,is X_kDiscrete standard deviation of sequences.

After completing step six:

inputting the temperature independent variable value measured in the next experiment into a thermal error prediction model (10) to obtain a predicted value of the thermal deformationAccording toAnd a measured value S of the amount of thermal deformation_jAnd calculating to obtain a residual value and a residual standard difference value, obtaining the prediction performance of the prediction model (10), and finally finishing the data processing of the thermal error compensation of the machine tool. Wherein the residual meansAnd S_jThe calculation formula of the difference value of (2) and the residual standard deviation is shown in formula (13).

$SD=\sqrt[2]{{\Σ}_{q=1}^{t\left(S_j\right)}\frac{{(S_{jq}-{\hat{S}}_{jq})}^2}{t\left(S_j\right)-n-1}}---\left(13\right)$

In formula (13), SD is the residual standard deviation, S_jqIs S_jThe value of the q-th value in the sequence,is composed ofThe qth value in the sequence, t (S)_j) Is S_jOrThe length of the sequence. S_jLength of sequence andthe sequences are of the same length.

Compared with the prior art, the invention has the following benefits: the method for establishing the regression model by using the principal components of the original variables to replace the original variables effectively avoids the coupling effect among the temperature independent variables, has better prediction precision and robustness on the condition of large-range environmental temperature, and provides a good data processing idea for the thermal error compensation application of the numerical control machine.

Drawings

FIG. 1 is a flow chart of a data processing method for robust thermal error compensation of a machine tool for a wide range of ambient temperatures according to the present invention.

FIG. 2 shows 10 temperature variation data obtained from a Leaderway-V450 numerical control machine experiment.

FIG. 3 shows Z-axis thermal deformation data obtained from a Leaderway-V450 numerical control machine tool experiment.

FIG. 4 shows the measured thermal error, predicted thermal error and residual data at an initial ambient temperature of 9.19 deg.C, 19.94 deg.C and 29.19 deg.C, respectively, and a spindle speed of 4000 rpm.

Detailed Description

The invention is described in further detail below with reference to the figures and the specific examples.

Referring to fig. 1, a data processing method for realizing robust thermal error compensation of a machine tool in a wide range of environmental temperatures comprises the following steps:

the method comprises the following steps: obtaining a temperature variable DeltaT of a machine tool_iAnd amount of thermal deformation of spindle S_jWherein i is 1, 2, …, m.

The m refers to the number of temperature variables. J is the axial direction of the main shaft of the machine tool, and j is X, Y and/or Z. X, Y, Z respectively represent the X-axis of the machine tool spindle, the Y-axis of the spindle, and the Z-axis of the spindle.

By temperature variation Δ T_iAnd amount of thermal deformation of spindle S_jObtaining the independent temperature variable X_k，k＝1，2，…，n。

The temperature independent variable X_kTo build a thermal error compensation model. N is the number of temperature independent variables, and n is less than or equal to m.

Step two: for the temperature independent variable X obtained in the step one_kCarrying out standardization treatment to obtain a standardized temperature independent variable X_k ^*。

And from a normalized temperature independent variable X_k ^*Converted to obtain a principal component Z_kIs described in (1).

The principal component Z_kRefers to the independent variable X of the temperature by orthogonal transformation_k ^*And a set of variables obtained after conversion is not linearly related.

Step three: from principal component Z obtained in step two_kThe first p principal components Z are extracted₁，Z₂，…，Z_pAnd p is less than or equal to n for the next thermal error modeling.

Step four: for the heat deformation quantity S of the main shaft in the step one_jGo on markStandardized processing to obtain standardized heat distortion S_j ^*。

And to the normalized heat distortion amount S_j ^*And the principal component Z for thermal error modeling obtained in step 3₁，Z₂，…，Z_pPerforming multiple linear regression analysis to obtain S_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)

Step five: s obtained in the fourth step_j ^*With respect to Z₁，Z₂，…，Z_pIs converted into S_j ^*With respect to X_k ^*The expression equation of (1).

Step six: s obtained in the fifth step_j ^*With respect to X_k ^*Is converted into S_jWith respect to X_kIs the expression equation of_jWith respect to X_kThe expression equation of (a) is the thermal error model.

Further, the step one specifically includes:

1.1 obtaining the temperature variable DeltaT of the machine tool_iAnd amount of thermal deformation of spindle S_jCalculating the temperature variable DeltaT_iAnd amount of thermal deformation of spindle S_jThe value of the correlation coefficient therebetween. The correlation coefficient formula (1) is as follows:

in the formula (1), r_ijIs DeltaT_iAnd S_jValue of the correlation coefficient between, T (Δ T)_i) Is DeltaT_iLength of (1), Δ T_iqIs DeltaT_iThe value of the q-th value in the sequence,is DeltaT_iAverage of the sequences, S_iqIs S_jThe value of the q-th value in the sequence,is S_jAverage of the sequences. Temperature variable Δ T_iIs collected from the machine tool in operation by a temperature sensor. Temperature variable Δ T_iLength of sequence and amount of thermal deformation S_jThe sequences are of the same length.

1.2 extracting n temperature variables with the maximum relational numerical values in the step 1.1 as temperature independent variables X_k. The temperature independent variable X_kUsed to model thermal errors. N is not more than m, and the preferable scheme is that the value of n is 2-4.

3. The data processing method for robust thermal error compensation of a machine tool for achieving wide range of ambient temperatures according to claim 1, wherein the second step is specifically:

2.1 pairs of the temperature independent variable X obtained in step one_kCarrying out standardization treatment to obtain a standardized temperature independent variable X_k ^*. To temperature independent variable X_kThe equation (2) for the normalization is as follows:

${X_{kq}}^{*}=\frac{X_{kq}-{\stackrel{\‾}{X}}_k}{\sqrt{\frac{1}{t\left({X_k}^{*}\right)}{\Σ}_{q=1}^{t\left({X_k}^{*}\right)}{(X_{kq}-{\stackrel{\‾}{X}}_k)}^2}},(k=1,2,\mathrm{...},n,q=1,2,\mathrm{...},t({X_k}^{*}\left)\right)---\left(2\right)$

in the formula (2), X_kq ^*Is X_k ^*Value qth in the sequence, X_kqIs X_kThe value of the q-th value in the sequence,is X_kAverage of the sequences, t (X)_k ^*) Is X_k ^*The length of the sequence.

2.2 calculating the correlation coefficient value between any two normalized temperature independent variables by using the correlation coefficient formula (1) in the step one to obtain a correlation matrix R of the normalized temperature independent variables_n×n(3) Specifically, the following is made.

In the correlation matrix R_n×n(3) In, r_abDenotes the normalized temperature independent variable X_a ^*And X_b ^*The correlation coefficient value of (a).

2.3 from the formula R_n×nu and λ, where u is λ u, are each a correlation matrix R_n×nThe eigenvectors and eigenvalues of (2) to obtain a correlation matrix R_n×nCharacteristic vector u of_kAnd a characteristic value lambda_kWherein k is 1, 2, …, n.

2.4 normalized temperature independent variable X obtained according to step 2_k ^*And a feature vector u_kObtaining the independent variable X from the normalized temperature_k ^*And a feature vector u_kPrincipal component Z of the composition_kThe expression of (c) is specifically:

$\left\{\begin{array}{c}{Z}_1=u_{11}{X_1}^{*}+u_{21}{X_2}^{*}+\mathrm{...}+u_{k1}{X_k}^{*}+\mathrm{...}+u_{n1}{X_n}^{*}\\ Z_2=u_{12}{X_1}^{*}+u_{22}{X_2}^{*}+\mathrm{...}+u_{k2}{X_k}^{*}+\mathrm{...}+u_{n2}{X_n}^{*}\\ .\\ .\\ .\\ Z_k=u_{1k}{X_1}^{*}+u_{2k}{X_2}^{*}+\mathrm{...}+u_{kk}{X_k}^{*}+\mathrm{...}+u_{nk}{X_n}^{*}\\ .\\ .\\ .\\ Z_n=u_{1n}{X_1}^{*}+u_{2n}{X_2}^{*}+\mathrm{...}+u_{kn}{X_k}^{*}+\mathrm{...}+u_{nn}{X_n}^{*}\end{array}\right.---\left(4\right)$

in the principal component Z_kIn the expression of (A), Var (Z)_k)＝λ_k，Cov(Z_i，Z_j) 0(i ≠ j), i.e. there is no correlation between the principal components.

Further, the third step specifically includes: let the first p principal components Z₁，Z₂，…，Z_pHas a cumulative variance contribution rate of Vcc_pP is less than or equal to n, let the cumulative variance contribution rate Vcc_pNot less than 85%, calculating the value of p, and determining the specific principal component Z according to the value of p₁，Z₂，…，Z_pExtracted for further thermal error modeling. Wherein,cumulative variance contribution rate Vcc of the first p principal components_pThe formula of (1) is:

${\mathrm{Vcc}}_p={\Σ}_{i=1}^p{\mathrm{\λ}}_i/{\Σ}_{i=1}^n{\mathrm{\λ}}_i---\left(5\right).$

further, the fourth step specifically includes:

4.1 Heat distortion amount S of spindle in the first step_jPerforming standardization process to obtain standardized heat distortion S_j ^*. Amount of thermal deformation S to the main shaft_jEquation (6) for normalization is:

${S_{jq}}^{*}=\frac{S_{jq}-{\stackrel{\‾}{S}}_j}{\sqrt{\frac{1}{t\left({S_j}^{*}\right)}{\Σ}_{q=1}^{t\left({S_j}^{*}\right)}{(S_{jq}-{\stackrel{\‾}{S}}_j)}^2}}---\left(6\right).$

in the formula (6), S_jq ^*Is S_j ^*The qth value in the sequence, S_jqIs S_jThe value of the q-th value in the sequence,is S_jAverage of the sequences, t (S)_j ^*) Is S_j ^*The length of the sequence.

4.2 pairs of normalized HeatAmount of deformation S_j ^*And principal component Z for thermal error modeling₁，Z₂，…，Z_pPerforming least squares-based multiple linear regression analysis to obtain S_j ^*With respect to Z₁，Z₂，…，Z_pEquation (7). Obtaining regression coefficient estimated value of each element in the model by utilizing multiple linear regression analysis based on least square principleS_j ^*With respect to Z₁，Z₂，…，Z_pEquation of expression (2)And regression coefficient estimation valueRespectively as follows:

${{\hat{S}}_j}^{*}={\widehat{\mathrm{\α}}}_1{Z}_1+{\widehat{\mathrm{\α}}}_2{Z}_2+\mathrm{...}+{\widehat{\mathrm{\α}}}_p{Z}_p---\left(7\right)$

${\widehat{\mathrm{\α}}}_p={\left({Z_p}^T{Z}_p\right)}^{-1}{Z_p}^T{S_j}^{*}---\left(8\right).$

further, the step five specifically means: using the principal component Z obtained in step two_kS obtained in step four_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)Conversion to S_j ^*With respect to X_k ^*Equation of (2)S_j ^*With respect to X_k ^*Equation of (2)The method comprises the following specific steps:

$\begin{array}{c}{{\hat{S}}_j}^{*}={\widehat{\mathrm{\α}}}_1{Z}_1+{\widehat{\mathrm{\α}}}_2{Z}_2+\mathrm{...}+{\widehat{\mathrm{\α}}}_p{Z}_p\\ ={\widehat{\mathrm{\α}}}_1(u_{11}{X_1}^{*}+u_{21}{X_2}^{*}+\mathrm{...}+u_{k1}{X_k}^{*}+\mathrm{...}+u_{n1}{X_n}^{*})+{\widehat{\mathrm{\α}}}_2(u_{12}{X_1}^{*}+u_{22}{X_2}^{*}+\mathrm{...}+u_{k2}{X_k}^{*}+\mathrm{...}+u_{n2}{X_n}^{*})\\ +\mathrm{...}+{\widehat{\mathrm{\α}}}_p(u_{1n}{X_1}^{*}+u_{2n}{X_2}^{*}+\mathrm{...}+u_{kp}{X_k}^{*}+\mathrm{...}+u_{np}{X_n}^{*})\\ ={\widehat{\mathrm{\β}}}_1{X_1}^{*}+{\widehat{\mathrm{\β}}}_2{X_2}^{*}+\mathrm{...}+{\widehat{\mathrm{\β}}}_k{X_k}^{*}+\mathrm{...}+{\widehat{\mathrm{\β}}}_n{X_n}^{*}\end{array}---\left(9\right)$

in the formula (9), the reaction mixture is,

further, the sixth step specifically includes: by means of independent variable X of temperature_kFormula X for normalization_kq ^*And the amount of thermal deformation S to the spindle_jNormalized formula S_jq ^*S from step five_j ^*With respect to X_k ^*Equation of expression (c) transforms S_jWith respect to X_kAnd (4) completing the establishment of a thermal error prediction model according to the expression equation (10). Wherein S is_jWith respect to X_kEquation (10) is:

${\hat{S}}_j={\widehat{\mathrm{\θ}}}_0+{\widehat{\mathrm{\θ}}}_1{X}_1+{\widehat{\mathrm{\θ}}}_2{X}_2+\mathrm{...}+{\widehat{\mathrm{\θ}}}_k{X}_k+\mathrm{...}+{\widehat{\mathrm{\θ}}}_n{X}_n---\left(10\right)$

${\widehat{\mathrm{\θ}}}_k=\frac{S_{s_j}}{S_{x_k}}{\widehat{\mathrm{\β}}}_k---\left(11\right)$

${\mathrm{\θ}}_0=\stackrel{\‾}{S_j}-{\Σ}_{k=1}^n{\widehat{\mathrm{\θ}}}_k{\stackrel{\‾}{X}}_k---\left(12\right)$

in the formula (11), the reaction mixture is,is S_jThe discrete standard deviation of the sequence is determined,is X_kDiscrete standard deviation of sequences.

Referring to fig. 1, further, the real-time detected temperature value is substituted into the thermal error model (10) to predict and analyze the thermal error of the spindle of the machine tool. Specifically, during prediction pre-analysis, the temperature independent variable value obtained by real-time measurement is input into the thermal error model (10) to obtain the real-time thermal deformation predicted valueAccording toAnd a measured value S of the amount of thermal deformation_jAnd calculating to obtain a residual value and a residual standard difference value, obtaining the prediction performance of the prediction model (10), and finally finishing the data processing of the thermal error compensation of the machine tool. Wherein the residual meansAnd S_jThe residual standard deviation is calculated as shown in equation (13):

$SD=\sqrt[2]{{\Σ}_{q=1}^{t\left(S_j\right)}\frac{{(S_{jq}-{\hat{S}}_{jq})}^2}{t\left(S_j\right)-n-1}}---\left(13\right)$

in formula (13), SD is the residual standard deviation, S_jqIs S_jThe value of the q-th value in the sequence,is composed ofThe qth value in the sequence, t (S)_j) Is S_jOrThe length of the sequence. S_jLength of sequence andthe sequences are of the same length.

In order to more clearly understand the data processing method of the present invention, the following description is further provided with reference to specific embodiments.

The data processing method provided by the invention is applied to the thermal error experimental data of the leader way-V450 type numerical control machine tool.

Fig. 2 is 10 temperature variable data obtained by a leadersway-V450 numerical control machine tool experiment, and fig. 3 is main shaft Z thermal deformation data obtained by the leadersway-V450 numerical control machine tool experiment. During experimental measurement, the initial environment temperature of the machine tool is 9.25 ℃, and the main shaft idles at a constant rotating speed of 4000 rpm.

The first step is as follows: calculating the temperature variable DeltaT according to equation (1)_iZ-direction thermal error S from main axis_zThe value of the correlation coefficient therebetween.

TABLE 1 Δ T_iAnd S_zValue of correlation coefficient therebetween

	ΔT1	ΔT2	ΔT3	ΔT4	ΔT5	ΔT6	ΔT7	ΔT8	ΔT9	ΔT10
											riz	0.90	0.88	0.85	0.89	0.96	0.81	0.86	0.68	0.81	0.74

Selecting the two temperature variables with the largest correlation coefficient as the temperature independent variables of the thermal error modeling, namely X₁＝ΔT₁，X₂＝ΔT₅I.e., n is 2.

The second step is that: to X₁，X₂Performing standardization treatment to obtain standardized temperature variable X₁ ^*，X₂ ^*Of the correlation matrix R_2×2And eigenvalues and eigenvectors of the matrix.

$R_{2\×2}=\left[\begin{array}{cc}1.0000& 0.9265\\ 0.9265& 1.0000\end{array}\right]$

TABLE 2 eigenvalues and eigenvectors

Characteristic value	Feature vector
		λ1＝1.9265	u1＝[0.7071 0.7071]
λ2＝0.0735	u2＝[0.7071 -0.7071]

Thus, the main component Z was obtained₁，Z₂Is described in (1).

$\left\{\begin{array}{c}{Z}_1=0.7071{X_1}^{*}+0.7071{X_2}^{*}\\ Z_2=0.7071{X_1}^{*}-0.7071{X_2}^{*}\end{array}\right.$

The third step: calculating Z₁，Z₂The cumulative variance contribution of (2), known as Z₁Reaches 96%, so that the main component participating in the thermal error modeling is Z₁。

The fourth step: the thermal deformation S of the spindle is determined by the formula (6)_zPerforming standardization processing to obtain standardized heat distortion S_z ^*Establishing S_z ^*And Z₁Is S_z ^*＝0.5841Z₁。

The fifth step: will S_z ^*And Z₁Is converted into S_z ^*With respect to X₁ ^*And X₂ ^*Is S of_z ^*＝0.4130X₁ ^*+0.4130X₂ ^*。

And a sixth step: will S_z ^*With respect to X₁ ^*And X₂ ^*Is converted into S_zWith respect to X₁And X₂Is S of_z＝11.8384+1.5252X₁+1.9669X₂And finishing the establishment of the thermal error prediction model.

The seventh step: and predicting the Z thermal error of the machine tool main shaft in different running states. Fig. 4 shows a thermal error measurement value 1, a thermal error prediction value 1, and a residual 1 at an initial ambient temperature of 9.19 ℃ and a spindle rotation speed of 4000rpm, a thermal error measurement value 2, a thermal error prediction value 2, and a residual 2 at an initial ambient temperature of 19.94 ℃ and a spindle rotation speed of 4000rpm, and a thermal error measurement value 3, a thermal error prediction value 3, and a residual 3 at an initial ambient temperature of 29.19 ℃ and a spindle rotation speed of 4000 rpm. Meanwhile, the residual standard differences of the prediction model are calculated to be 4.89 μm, 3.78 μm and 4.95 μm respectively. As can be seen from fig. 4, the residual value and the residual standard deviation of the model are kept within a small range no matter the initial environment temperature of the machine tool, so the thermal error data processing method described in this patent has high accuracy and robustness.

Claims (9)

1. A data processing method for realizing robust thermal error compensation of a machine tool in a wide range of environmental temperatures is characterized by comprising the following steps of:

the method comprises the following steps: obtaining a temperature variable DeltaT of a machine tool_iAnd amount of thermal deformation of spindle S_jWherein i is 1, 2, …, m;

the m refers to the number of temperature variables; j is the axial direction of the machine tool spindle, and j is X, Y and/or Z; x, Y, Z respectively representing the X-axis direction of the main shaft of the machine tool, the Y-axis direction of the main shaft and the Z-axis direction of the main shaft;

from temperatureDegree variable Delta T_iAnd amount of thermal deformation of spindle S_jObtaining the independent temperature variable X_k，k＝1，2，…，n；

The temperature independent variable X_kTo build a thermal error compensation model; n is the number of temperature independent variables, and n is less than or equal to m;

step two: for the temperature independent variable X obtained in the step one_kCarrying out standardization treatment to obtain a standardized temperature independent variable X_k ^*；

And from a normalized temperature independent variable X_k ^*Converted to obtain a principal component Z_kThe expression of (1);

the principal component Z_kRefers to the independent variable X of the temperature by orthogonal transformation_k ^*A group of variables obtained after conversion, wherein the group of variables are not linearly related;

step three: from principal component Z obtained in step two_kThe first p principal components Z are extracted₁，Z₂，…，Z_pP is less than or equal to n and is used for modeling the thermal error of the next step;

step four: for the heat deformation quantity S of the main shaft in the step one_jPerforming standardization process to obtain standard thermal deformation S_j ^*；

And to the normalized heat distortion amount S_j ^*And the principal component Z for thermal error modeling obtained in step 3₁，Z₂，…，Z_pPerforming multiple linear regression analysis to obtain S_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)

Step five: s obtained in the fourth step_j ^*With respect to Z₁，Z₂，…，Z_pIs converted into S_j ^*With respect to X_k ^*The expression equation of (1);

step six: s obtained in the fifth step_j ^*With respect to X_k ^*Equation transformation of expression (c)Is S_jWith respect to X_kIs the expression equation of_jWith respect to X_kThe expression equation of (a) is the thermal error model.

2. The data processing method for robust thermal error compensation of a machine tool for a wide range of ambient temperatures according to claim 1, wherein the first step is specifically:

step 1.1 obtaining the temperature variable Delta T of the machine tool_iAnd amount of thermal deformation of spindle S_jCalculating the temperature variable DeltaT_iAnd amount of thermal deformation of spindle S_jThe value of the correlation coefficient therebetween; the correlation coefficient formula (1) is as follows:

in the formula (1), r_ijIs DeltaT_iAnd S_jValue of the correlation coefficient between, T (Δ T)_i) Is DeltaT_iLength of (1), Δ T_iqIs DeltaT_iThe value of the q-th value in the sequence,is DeltaT_iAverage of the sequences, S_jqIs S_jThe value of the q-th value in the sequence,is S_jThe average of the sequences; temperature variable Δ T_iCollected from the machine tool in running state by the temperature sensor; temperature variable Δ T_iLength of sequence and amount of thermal deformation S_jThe length of the sequences is the same;

step 1.2, extracting n temperature variables with the maximum relational numerical values in step 1.1 as temperature independent variables X_k(ii) a The temperature independent variable X_kUsed for establishing a thermal error model; the n is not more than m.

3. The data processing method for robust thermal error compensation of a machine tool for achieving wide range of ambient temperatures according to claim 1, wherein the second step is specifically:

step 2.1 for the temperature independent variable X obtained in step one_kCarrying out standardization treatment to obtain a standardized temperature independent variable X_k ^*(ii) a To temperature independent variable X_kThe equation (2) for the normalization is as follows:

in the formula (2), X_kq ^*Is X_k ^*Value qth in the sequence, X_kqIs X_kThe value of the q-th value in the sequence,is X_kAverage of the sequences, t (X)_k ^*) Is X_k ^*The length of the sequence;

step 2.2 calculating the correlation coefficient value between any two normalized temperature independent variables by using the correlation coefficient formula (1) in the step one to obtain a correlation matrix R of the normalized temperature independent variables_n×n(3) Concretely, the following steps are carried out;

in the correlation matrix R_n×n(3) In, r_abDenotes the normalized temperature independent variable X_a ^*And X_b ^*The correlation coefficient value of (a);

step 2.3 by letting the relation R_n×nu and λ, where u is λ u, are each a correlation matrix R_n×nThe eigenvectors and eigenvalues of (2) to obtain a correlation matrix R_n×nCharacteristic vector u of_kAnd a characteristic value lambda_kWherein k is 1, 2, …, n;

step 2.4 normalized temperature independent variable X obtained from step 2_k ^*And a feature vector u_kTo obtainFrom a normalized temperature independent variable X_k ^*And a feature vector u_kPrincipal component Z of the composition_kThe expression of (c) is specifically:

in the principal component Z_kIn the expression of (A), Var (Z)_k)＝λ_k，Cov(Z_i，Z_j) 0(i ≠ j), i.e. there is no correlation between the principal components.

4. The data processing method for robust thermal error compensation of a machine tool for achieving a wide range of ambient temperatures according to claim 1, wherein the third step is specifically: let the first p principal components Z₁，Z₂，…，Z_pHas a cumulative variance contribution rate of Vcc_pP is less than or equal to n, let the cumulative variance contribution rate Vcc_pNot less than 85%, calculating the value of p, and determining the specific principal component Z according to the value of p₁，Z₂，…，Z_pExtracting for next thermal error modeling; wherein the cumulative variance contribution rate Vcc of the first p principal components_pThe formula of (1) is:

5. the data processing method for robust thermal error compensation of a machine tool for achieving a wide range of ambient temperatures according to claim 1, wherein said step four specifically comprises:

step 4.1 Heat distortion amount S of spindle in step one_jPerforming standardization process to obtain standardized heat distortion S_j ^*(ii) a Amount of thermal deformation S to the main shaft_jEquation (6) for normalization is:

in the formula (6), S_jq ^*Is S_j ^*The qth value in the sequence, S_jqIs S_jThe value of the q-th value in the sequence,is S_jAverage of the sequences, t (S)_j ^*) Is S_j ^*The length of the sequence.

Step 4.2 for the normalized Heat distortion S_j ^*And principal component Z for thermal error modeling₁,Z₂,…,Z_pPerforming least squares-based multiple linear regression analysis to obtain S_j ^*With respect to Z₁,Z₂,…,Z_pExpression of (4) equation (7); obtaining regression coefficient estimated value of each element in the model by utilizing multiple linear regression analysis based on least square principleS_j ^*With respect to Z₁，Z₂，…，Z_pEquation of expression (2)And regression coefficient estimation valueRespectively as follows:

6. machine tool for achieving a wide range of ambient temperatures according to claim 1The data processing method for robust thermal error compensation is characterized in that the fifth step specifically includes: using the principal component Z obtained in step two_kS obtained in step four_j ^*With respect to Z₁，Z₂，…，Z_pEquation of (2)Conversion to S_j ^*With respect to X_k ^*Equation of (2)S_j ^*With respect to X_k ^*Equation of (2)(9) The method comprises the following specific steps:

in the formula (9), the reaction mixture is,

7. the data processing method for robust thermal error compensation of a machine tool for achieving a wide range of ambient temperatures according to claim 1, wherein the sixth step is specifically as follows: by means of independent variable X of temperature_kFormula X for normalization_kq ^*And the amount of thermal deformation S to the spindle_jNormalized formula S_jq ^*S from step five_j ^*With respect to X_k ^*Equation of expression (c) transforms S_jWith respect to X_kThe expression equation (10) of (a), completing the establishment of a thermal error prediction model; wherein S is_jWith respect to X_kEquation (10) is:

in the formula (11), the reaction mixture is,is S_jThe discrete standard deviation of the sequence is determined,is X_kDiscrete standard deviation of sequences.

8. The model application method of the data processing method for realizing the robust thermal error compensation of the machine tool in the wide range of environmental temperatures according to claim 1 is characterized in that the thermal error of the spindle of the machine tool is predicted and analyzed by substituting the real-time detected temperature value into the thermal error model.

9. The method of claim 8, wherein the predictive pre-analysis is performed by inputting the temperature independent variable measured in real time into the thermal error model to obtain the predicted value of thermal deformation in real timeAccording toAnd a measured value S of the amount of thermal deformation_jCalculating to obtain a residual value and a residual standard difference value, obtaining the prediction performance of the prediction model, and finally finishing the data processing of the thermal error compensation of the machine tool; wherein the residual meansAnd S_jThe residual standard deviation is calculated as shown in equation (13):

in formula (13), SD is the residual standard deviation, S_jqIs S_jThe value of the q-th value in the sequence,is composed ofThe qth value in the sequence, t (S)_j) Is S_jOrThe length of the sequence; s_jLength of sequence andthe sequences are of the same length.