CN101446994A - Modeling method of thermal error least squares support vector machine of numerically-controlled machine tool - Google Patents

Abstract

The invention discloses a modeling method of a thermal error least squares support vector machine of a numerically-controlled machine tool. The method comprises the following steps: (1) selecting a kernel function and determining parameters; and (2) establishing a thermal error model of the machine tool according to least squares support vector machine theory. A compensating system of the invention has the advantages of simple structure and reliable application; and the modeling method of the least squares support vector machine improves the model precision and generalization capability and overcomes defects of the existing prediction methods, such as low precision, low generalization capability and the like. The modeling method has higher prediction precision when the sample size is small and even the sample data are small, thus reducing dependence on experience. Meanwhile, the method improves the self-learning ability of the system, the thermal error model obtained by training can reflect the manufacturing procedure change of the machine tool, and the method has the advantages of adaptability, low hardware requirements for the thermal error compensating system, simple structure and good reliability.

Description

The modeling method of thermal error least squares support vector machine of numerically-controlled machine

Technical field

The present invention relates to a kind of modeling method of thermal error least squares support vector machine of numerically-controlled machine.

Background technology

One of basic technology that numerical control machining tool heat error control is accurate and ultraprecise is processed.Machine tool thermal error compensation key step is: the execution and the error compensation Evaluation on effect of the foundation of the detection of error source and analysis, error motion comprehensive mathematical model, the identification of error element, error compensation.

In heat error compensation, the foundation of hot error model is committed step.The experiment modeling is a hot error modeling method the most commonly used, promptly utilizes hot error information that experiment records and lathe temperature value and carries out the match modeling with the principle of least square.Yet, machine tool thermal error depends on to a great extent such as multiple factors such as the use of processing conditions, process-cycle, liquid coolant and surrounding environment, there is reciprocation, from the statistics angle, machine tool thermal error presents nonlinear relationship with the variation of temperature and working time, its distribution then is abnormal, and is jiggly.Therefore adopt the match modeling method accurately to set up hot error mathematic model and have suitable limitation.

In recent years, particularly expert system, neural network theory and fuzzy system theory etc. have also applied in the hot error modeling.Hot error model commonly used has multivariate regression analysis model, neural network model, comprehensive least square modeling, Orthogonal Experiment and Design modeling, recursion modeling or the like.Because hot error became when having usually, multifactor, characteristics such as operating mode uncertainty make the modeling method of development in recent years have certain limitation.And based on the method for analyzing and modeling that the machine tool thermal error of least square method supporting vector machine (LS-SVM) is predicted, the LS-SVM method is applied in the numerical control machining tool heat error Study on Forecast.This new method can overcome the shortcoming of traditional Forecasting Methodology, has very high precision and generalization ability; According to this forecast model, it is more effective that the numerically-controlled machine real-Time Compensation becomes.

Summary of the invention

The object of the present invention is to provide a kind of modeling method of thermal error least squares support vector machine of numerically-controlled machine.

The technical solution used in the present invention comprises the following steps:

(1) parameter is selected:

The LS-SVM algorithm at first will be selected a kernel function, and determines following correlation parameter: for kernel function, select RBF nuclear: K (x _i, x _j)=exp[-(x _i-x _j) ²/ (2 σ ²)], it has only a undetermined parameter σ, its value is big more, speed of convergence is fast more, but the model that obtains thus can make all predicted values trend towards the mean value of span when prediction, the square error of this moment can not reflect real each point data, for the LS-SVM that adopts RBF nuclear, major parameter is regularization parameter γ and kernel function width cs, and these two parameters have determined study and the generalization ability of LS-SVM;

Select RBF nuclear, problem can be simplified to: seek the combination of adjustable parameter γ and σ, make LS-SVM that estimated performance be arranged, intercepting γ and each a bit of interval of σ, on these the two sections interval two dimensional surfaces that constitute, be criterion with the accuracy rate, do search fully, then can determine a unique [σ, γ] combination, corresponding high-accuracy is though this accuracy rate is not necessarily at (∞, + optimum solution on ∞), but a satisfactory solution;

1. determine the span of initial parameter: in span, choose parameter value, make up parameter to (γ i, σ i) two-dimensional grid plane, wherein i=1,2 ... f, j=1,2, ... g, for example two parameters are respectively chosen 20 numerical value, and it is right to constitute 20 * 20 grid plans and 400 parameters; Selection of parameter has two kinds of methods: the 1st kind is to determine two parameter range earlier, again parameter is carried out even value; The 2nd kind is to determine that according to the feature and the experience of training sample parameter is to value;

2. import each mesh node parameter to (γ _i, σ _i) in LS-SVM, adopt learning sample to train, and output study error, get the nodal value (γ of least error correspondence _i, σ _i) _EminFor optimized parameter right;

3. if the precision of training does not reach needed requirement, then with (γ _i, σ _i) _EminBe the center, make up new two-dimensional network plane, choose the close parameter value of numerical value and further train, thereby obtain more high-precision training result; By that analogy, can construct multilayer parameter optimization grid plan, continue to optimize the least square method supporting vector machine parameter, up to reaching the precision that needs;

(2) based on the hot error model of least square method supporting vector machine, prediction error value:

LS-SVM returns and estimates to be expressed as form:

$f\left(x\right)=\underset{i=1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\α}}_{i}K(x,x_i)+b.$

Kernel function K (x in the formula _i, x _j) be the RBF nuclear in the step (1), α, b is then solved by following formula:

$\left[\begin{array}{cccc}0& 1& \&CenterDot;\&CenterDot;\&CenterDot;& 1\\ 1& K(x_1,x_1)+1/\mathrm{\&gamma;}& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_1,x_l)\\ \&CenterDot;& \text{\&CenterDot;}& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ 1& K(x_l,x_l)& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_l,x_l)+1/\mathrm{\&gamma;}\end{array}\right]\left[\begin{array}{c}b\\ {\mathrm{\&alpha;}}_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{\&alpha;}}_l\end{array}\right]=\left[\begin{array}{c}0\\ {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="A200810163141E00091.png"\; state="\{\{state\}\}">y</figure-callout>}}_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y_l\end{array}\right].$

The beneficial effect that the present invention has is:

Modeling method based on least square method supporting vector machine is different from traditional match modeling, support vector machine regression modeling theory is applied to machine tool thermal error modeling field, model accuracy and generalization ability have been improved, part shortcoming such as it is low to have overcome existing Forecasting Methodology precision, and generalization ability is low.

Based on the modeling method of least square method supporting vector machine hour, have higher forecast precision equally,, also can carry out the prediction of match fast and accurately, reduced dependence experience to it even on small sample data basis in sample size.

Support vector machine method has improved the self-learning capability of system, and the hot error model that training obtains can reflect that the machine tooling process changes, and has adaptivity.

Heat error compensation system hardware demand is lower, and is simple in structure, has good reliability.

Description of drawings

Fig. 1 workflow diagram of the present invention.

Fig. 2 sample data is gathered and the modeling method of least squares support schematic diagram.

Fig. 3 is that the embodiment of the invention adopts the hot sum of errors of least square support model prediction to survey hot error comparison diagram.

Embodiment

The present invention is further illustrated below in conjunction with accompanying drawing and implementation process.

It is a kind of inference method based on Statistical Learning Theory for a hot error modeling method of the present invention, realizes according to following steps, as shown in Figure 1:

At first consider to produce the correlative factor of hot error, need to determine the measuring point of image data, lathe is carried out the data sample collection.The related data of each measuring point in the harvester bed operating process under the condition of approximate actual condition.The sample data acquisition system as shown in Figure 2, and is general, and temperature data is obtained by temperature sensor, and thermal deformation is by the laser displacement sensor collection.Repeatedly repeat this process, each time monitoring gained data are carried out modeling on PC.

Modeling process at first carries out parameter and selects.The LS-SVM algorithm at first will be selected a kernel function, and definite correlation parameter.For kernel function, generally select RBF nuclear: K (x _i, x _j)=exp[-(x _i-x _j) ²/ (2 σ ²)].It has only a undetermined parameter σ, its value is big more, speed of convergence is fast more, but the model that obtains thus, when prediction, can make all predicted values trend towards some values, this is worth the mean value of span often, though the square error of this moment and little but can not reflect real each point data.For adopting the radially LS-SVM of base nuclear, major parameter is regularization parameter γ and kernel function width cs, and these two parameters have determined study and the generalization ability of LS-SVM to a great extent.Select RBF nuclear, problem can be simplified to: seek suitable adjustable parameter γ and the combination of σ, make LS-SVM that best estimated performance be arranged.If can intercept each a bit of interval of γ and σ, on these the two sections interval two dimensional surfaces that constitute, be criterion with the accuracy rate, do search fully, just can determine only [σ, a γ] combination, correspondence high-accuracy.Though this accuracy rate is not necessarily in (∞ ,+optimum solution on ∞), but an acceptable satisfactory solution.Here propose a kind of dynamic self-adapting optimized Algorithm, model parameter is selected to be optimized.Concrete steps are as follows:

1. determine the span of initial parameter: in span, choose parameter value, make up parameter to (γ _i, σ _i) the two-dimensional grid plane, i=1 wherein, 2 ... f, j=1,2 ... g.For example two parameters are respectively chosen 20 numerical value, and it is right to constitute 20 * 20 grid plans and 400 parameters.Selection of parameter has two kinds of methods: the 1st kind is to determine two parameter range earlier, carries out even value according to the desired parameters logarithm again; The 2nd kind is to determine that according to the feature and the experience of training sample parameter is to value.

2. import each mesh node parameter to (γ _i, σ _i) in LS-SVM, adopt learning sample to train, and output study error.Get the nodal value (γ of least error correspondence _i, σ _i) _EminFor optimized parameter right.

3. if the precision of training does not reach needed requirement, then with (γ _i, σ _i) _EminBe the center, make up new two-dimensional network plane, choose the close parameter value of numerical value and further train, thereby obtain more high-precision training result.By that analogy, can construct multilayer parameter optimization grid plan, continue to optimize the least square method supporting vector machine parameter, up to reaching the precision that needs.

After parameter is selected, set up hot error model based on least square method supporting vector machine.

LS-SVM returns and estimates to be expressed as form:

$f\left(x\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i}K(x,x_i)+b.$

Kernel function K (x in the formula _i, x _j) be the RBF nuclear in the step (1), α, b is then solved by following formula:

$\left[\begin{array}{cccc}0& 1& \&CenterDot;\&CenterDot;\&CenterDot;& 1\\ 1& K(x_1,x_1)+1/\mathrm{\&gamma;}& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_1,x_l)\\ \&CenterDot;& \text{\&CenterDot;}& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ 1& K(x_l,x_1)& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_l,x_l)+1/\mathrm{\&gamma;}\end{array}\right]\left[\begin{array}{c}b\\ {\mathrm{\&alpha;}}_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{\&alpha;}}_l\end{array}\right]=\left[\begin{array}{c}0\\ {\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="A200810163141E00091.png"\; state="\{\{state\}\}">y</figure-callout>}}_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y_l\end{array}\right].$

For improving the robustness of model, estimate it is to be weighted processing to returning, utilize the error variance ξ of the preceding LS-SVM of weighting of following formula acquisition _iDetermine weighting coefficient v _i:

$v_i=\left[\begin{array}{cc}1& \mathrm{if}|{\mathrm{\&xi;}}_i/\hat{s}|\&le;c_1\\ \frac{c_2-|{\mathrm{\&xi;}}_i/\hat{s}|}{c_2-{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="A200810163141E00091.png"\; state="\{\{state\}\}">c</figure-callout>}}_1}& \mathrm{if}{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="A200810163141E00091.png"\; state="\{\{state\}\}">c</figure-callout>}}_1\&le;|{\mathrm{\&xi;}}_i/\hat{s}|\&le;{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="A200810163141E00101.png"\; state="\{\{state\}\}">c</figure-callout>}}_2\\ {10}^{-4}& \mathrm{otherwise}\end{array}\right]$

In the formula

Be LS-SVM error variance ξ _iThe Robust Estimation value of standard variance, general value is:

$\hat{s}=1.483\mathrm{MAD}\left(x_i\right)$

MAD (x in the formula _i) be data x _iMedian absolute deviation.Constant C ₁, C ₂Usually value is: C ₁=2.5 and C ₂=3.

At last according to weight v _iBe weighted the least square method supporting vector machine training, obtain the regression modeling model:

$f\left(x\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i^{*}K(x,x_i)+b^{*}$

Can calculate the machine tool thermal error predicted value by this formula.According to this predicted value, export it to digital control system, realize error compensation.

Embodiments of the invention are below described.

Embodiment:

An XHK-714F numerical control machining center is carried out hot error modeling analysis.Machine tool chief axis thermal deformation data are gathered by laser displacement sensor (LK-150H).Temperature field measuring system is made up of 14 intelligent temperature sensors, ARM7 embedded system platform (FS44B0XLII) and liquid crystal display.Repeatedly the repeated test machining center moves 6 hours continuously under simulated condition, shuts down 1 hour temperature rise and axial hot error condition of main shaft in the process, obtains 70 groups of data altogether.

According to carrying out data acquisition, get main shaft temperature and change T ₀, spindle motor measuring point temperature rise T ₁, ball-screw measuring point temperature rise T ₂, column measuring point temperature rise T ₃, bed piece measuring point temperature rise T ₄, environment temperature T ₅, with main shaft axial error D ₁, main shaft radial error D ₂Together, constitute the sample data set, utilize the dynamic self-adapting optimized Algorithm, model parameter is optimized selection.According to the cause and effect dependence between the variable, sample data is carried out initial training, calculate the corresponding respectively ξ of 70 groups of data _i=α _i/ γ.Then according to ξ _iThe Distribution calculation standard deviation that goes out to have Robust Estimation

Basis again And ξ _iDetermine weight v _iAt last according to weight v _iSample is carried out LS-SVM training, the hot error model of this machining center main shaft.Shown in Fig. 3 a, provided main machine spindle axially and shown in Fig. 3 b radially the modeling value and the comparable situation of measured value, the mean absolute percentage error that axial deformation predicts the outcome is 1.33%, the mean absolute percentage error that radial deformation predicts the outcome is 1.62%, proves that this method has good modeling accuracy.

Claims (1)

1, a kind of modeling method of thermal error least squares support vector machine of numerically-controlled machine is characterized in that, comprises the following steps:

(1) parameter is selected:

The LS-SVM algorithm at first will be selected a kernel function, and determines following correlation parameter: for kernel function, select RBF nuclear: K (x _i, x _j)=exp[-(x _i-x _j) ²/ (2 σ ²)], it has only a undetermined parameter σ, its value is big more, speed of convergence is fast more, but the model that obtains thus can make all predicted values trend towards the mean value of span when prediction, the square error of this moment can not reflect real each point data, for the LS-SVM that adopts RBF nuclear, major parameter is regularization parameter γ and kernel function width cs, and these two parameters have determined study and the generalization ability of LS-SVM;

Select RBF nuclear, problem can be simplified to: seek the combination of adjustable parameter γ and σ, make LS-SVM that estimated performance be arranged, intercepting γ and each a bit of interval of σ, on these the two sections interval two dimensional surfaces that constitute, be criterion with the accuracy rate, do search fully, then can determine a unique [σ, γ] combination, corresponding high-accuracy is though this accuracy rate is not necessarily at (∞, + optimum solution on ∞), but a satisfactory solution;

1. determine the span of initial parameter: in span, choose parameter value, make up parameter to (γ _i, σ _i) the two-dimensional grid plane, i=1 wherein, 2 ... f, j=1,2 ... g, for example two parameters are respectively chosen 20 numerical value, and it is right to constitute 20 * 20 grid plans and 400 parameters; Selection of parameter has two kinds of methods: the 1st kind is to determine two parameter range earlier, again parameter is carried out even value; The 2nd kind is to determine that according to the feature and the experience of training sample parameter is to value;

2. import each mesh node parameter to (γ _i, σ _i) in LS-SVM, adopt learning sample to train, and output study error, get the nodal value (γ of least error correspondence _i, σ _i) _EminFor optimized parameter right;

3. if the precision of training does not reach needed requirement, then with (γ _i, σ _i) _EminBe the center, make up new two-dimensional network plane, choose the close parameter value of numerical value and further train, thereby obtain more high-precision training result; By that analogy, can construct multilayer parameter optimization grid plan, continue to optimize the least square method supporting vector machine parameter, up to reaching the precision that needs;

(2) based on the hot error model of least square method supporting vector machine, prediction error value:

LS-SVM returns and estimates to be expressed as form:

$f\left(x\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i}K(x,x_i)+b.$

Kernel function K (x in the formula _i, x _j) be the RBF nuclear in the step (1), α, b is then solved by following formula:

$\left[\begin{array}{cccc}0& 1& \&CenterDot;\&CenterDot;\&CenterDot;& 1\\ 1& K(x_1,x_1)+1/\mathrm{\&gamma;}& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_1,x_l)\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& \&CenterDot;& \&CenterDot;\\ 1& K(x_l,x_1)& \&CenterDot;\&CenterDot;\&CenterDot;& K(x_l,x_l)+1/\mathrm{\&gamma;}\end{array}\right]\left[\begin{array}{c}b\\ {\mathrm{\&alpha;}}_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {\mathrm{\&alpha;}}_l\end{array}\right]=\left[\begin{array}{c}0\\ y_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ y_l\end{array}\right].$

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