CN103268082A - Thermal error modeling method based on gray linear regression - Google Patents

Abstract

The invention relates to a thermal error modeling method based on gray linear regression. The method comprises the following steps that first, on the basis of a gray thermal error model, a linear equation is introduced, a gray linear regression combination model is constructed; second, a least square method is used for solving a gray linear regression combination model parameter; third, the gray linear regression model is used for thermal error prediction; fourth, a BP nerve network is used for amending combination model residual errors, and prediction accuracy is improved. According to the method, the shortcoming that a linear regression model does not have exponential growth and cannot describe linear changing trend easily, and a gray thermal error model does not have a linear factor can be overcome, good capacity for solving linear and nonlinear problems is achieved, good effect is achieved for thermal error prediction on an accurate horizontal type machining center is achieved, linear factors and nonlinear factors of thermal error data are considered, the shortcoming of an original single gray model is overcome, and an accurate thermal error prediction value and high fitting degree are acquired.

Description

A kind of hot error modeling method based on the grey linear regression

Technical field

The invention belongs to the application of numerically-controlled machine error compensation, be specifically related to the modeling method of the linear regression combination model of grey of the hot error of a kind of accurate horizontal Machining centers.

Background technology

Hot error refers to cause the machine tool element thermal deformation by the rising of lathe temperature, and causing relative position variation between workpiece and cutter and the mismachining tolerance of generation, its main research contents has measurement, key point optimization, hot error modeling and the heat error compensation in thermal deformation theoretical analysis, the heat error compensation to implement five major parts.Hot error is the error source of numerically-controlled machine maximum, also is machining precision significant effects factor, accounts for 40%～70% of lathe total error.Therefore, must control hot error rationally and effectively, set up the higher hot error model of precision as far as possible, be to realize heat error compensation, improves the gordian technique of machine finish.

Along with the continuous development of numerically-controlled machine, machining precision has been proposed more and more higher requirement, Chinese scholars has been carried out a large amount of research for how reducing numerical control machining tool heat error in recent years.Wherein, employed hot error modeling method comprises the artificial nerve network model based on genetic algorithm, fuzzy logic model, grey forecasting model, linear regression model (LRM) etc.But these hot error models are more single, can not express the overall picture of hot error information fully, the linear regression model (LRM) that adopts has the shortcoming that does not have exponential increase and be difficult to describe linear trends of change, grey forecasting model does not have linear factor, and hot error raw data exists linearity and non-linear factor simultaneously, thereby the hot error model of setting up must have the ability of handling linear and nonlinear problem.

Summary of the invention

The objective of the invention is to overcome the weak point of above-mentioned modeling, provide a kind of and have more high-precision modeling method than the hot error model of grey, it is a kind of hot error modeling method based on the grey linear regression, and introduce the BP neural network simultaneously its residual error is revised, and then obtain hot error prediction value more accurately.

The present invention adopts following technological means for dealing with problems:

A kind of hot error modeling method based on the grey linear regression comprises the following steps:

1) on the basis of the hot error model of grey, introduce linear equation, make up the linear regression combination model of grey:

Numerical control machining tool heat error trend can be analyzed by making up the dynamic differential equation, because hot error has uncertainty, adopt the hot error model of grey, hot error raw data is made algebraic sum to be calculated, and handle its grey variable, to weaken the randomness in the hot error raw data, has strong regular hot error prediction value thereby generate; By the hot error model of grey time response the sequence equation as can be known, the order ${\hat{X}}^{\left(1\right)}(k+1)=(X^{\left(0\right)}\left(1\right)-\frac{b}{a})e^{-\mathrm{ak}}+\frac{b}{a}=l_1{e}^{\mathrm{vk}}+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">l</figure-callout>}}_2$ , to introduce linear equation and know, the equation of the hot error model of the linear regression combination of grey is:

Wherein, X ⁽⁰⁾=(x ⁽⁰⁾(1), x ⁽⁰⁾(2) ..., x ⁽⁰⁾(n)) be hot error original data sequence, and , X is arranged ⁽⁰⁾The sequence X that adds up ⁽¹⁾=(x ⁽¹⁾(1), x ⁽¹⁾(2) ..., x ⁽¹⁾(n)).Simultaneously

, namely represent the mean value of adjacent data then to be close to average formation sequence z ⁽¹⁾=(z ⁽¹⁾(2), z ⁽¹⁾(3) ..., z ⁽¹⁾(k)).

Be the x of equation ⁽⁰⁾(k)+az ⁽¹⁾(k)=the b response sequence; A, b are by the least square method parameters calculated, and wherein ,-a is the development coefficient, and b is the grey action; V, l ₁, l ₂Be the parameter of simplifying gained, l ₃The parameter of introducing for linear equation,

It is the estimated value of mean value conduct of each value of v; X ⁽⁰⁾(1) and x ⁽⁰⁾(1) has identical meanings, all represent data element corresponding in the hot error information sequence;

Be through calculating the hot error information of gained, t is desirable 1,2 ..., n, n are the natural number greater than 1;

2) utilize least square method, find the solution the linear regression combination model parameter of grey:

If $Z\left(t\right)={\hat{X}}^{\left(1\right)}(t+1)-{\hat{X}}^{\left(1\right)}\left(t\right)=l_1{e}^{\mathrm{vt}}(e^v-1)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">l</figure-callout>}}_2,t=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n-1.$ Establish Y again _m(t)=and Z (t+m)-Z (t), m is not less than 1 natural number, namely

$\left\{\begin{array}{c}{Y}_m\left(t\right)=l_1{e}^{\mathrm{vt}}(e^{\mathrm{vm}}-1)(e^v-1)\\ Y_m(t+1)=l_1{e}^{v(t+1)}(e^{\mathrm{vm}}-1)(e^v-1)\end{array}\right.$

Got v=ln[Y by following formula _m(t+1)/Y _m(t)], estimated value is then arranged

Order $f\left(t\right)=e^{\hat{v}t},$ $X^{\left(1\right)}=\left[\begin{array}{c}{x}^{\left(1\right)}\left(1\right)\\ x^{\left(1\right)}\left(2\right)\\ .\\ .\\ .\\ x^{\left(1\right)}\left(n\right)\end{array}\right],$ $L=\left[\begin{array}{c}{l}_1\\ l_2\\ {\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\end{array}\right],$ $A=\left[\begin{array}{ccc}f\left(1\right)& 1& 1\\ f\left(2\right)& 2& 1\\ .& .& .\\ .& .& .\\ .& .& .\\ f\left(n\right)& \mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">n</figure-callout>}& 1\end{array}\right],$ Then utilize least square method to try to achieve, L=(A ^TA) ^-1A ^TX ⁽¹⁾, so try to achieve the linear regression combination model parameter of grey;

3) utilize the grey linear regression model (LRM) to carry out hot error prediction:

The linear regression combination model parameter of the grey that utilization is tried to achieve, the substitution equation

, through the tired predicted value that calculates hot error information that subtracts;

4) utilize the BP neural network that the built-up pattern residual error is revised, improve precision of prediction:

Adopt the BP neural network that the hot error model residual error of the linear regression combination of grey is predicted correction, wherein residual values is the difference of predicted value and measured value, namely use Matlab software to carry out computing, obtain the residual prediction value of the hot error model of the linear regression combination of grey, thereby obtain the predicted value to actual value, improve the degree of accuracy of prediction, this is significant to numerical control machine heat error compensation.

Beneficial effect

Method of the present invention can be improved not to be had exponential increase and is difficult to describe the shortcoming of linear trends of change and the deficiency that the hot error model of grey does not have linear factor in the linear regression model (LRM), ability with good treatment linearity and nonlinear problem, hot error prediction to accurate horizontal Machining centers has been obtained good effect, not only consider the linear factor of hot error information but also considered its non-linear factor, improved the shortcoming of original single gray model, obtained hot error prediction value and higher degree of fitting more accurately, significant for numerical control machine heat error compensation.

Description of drawings

Fig. 1 is based on the hot error modeling method flow diagram of grey linear regression;

Fig. 2 (a)-(b) detects figure for the hot error experiments of the present invention;

Fig. 3 embodiment of the invention BP neural network residual prediction realization flow figure;

The hot error model predicted value of Fig. 4 embodiment of the invention comparison diagram;

The hot error model residual values of Fig. 5 embodiment of the invention comparison diagram;

Embodiment

A kind of hot error modeling method flow diagram based on the grey linear regression of the embodiment of the invention is described further step of the present invention below in conjunction with process flow diagram as shown in Figure 1.Concrete implementation step is as follows:

The first step: on the basis of the hot error model of grey, introduce linear equation, make up the linear regression combination model of grey;

Numerical control machining tool heat error trend can be analyzed by making up the dynamic differential equation, because hot error has uncertainty, adopt the hot error model of grey, hot error raw data is made algebraic sum to be calculated, and handle its grey variable, to weaken the randomness in the hot error raw data, has strong regular hot error prediction value thereby generate.

If X ⁽⁰⁾=(x ⁽⁰⁾(1), x ⁽⁰⁾(2) ..., x ⁽⁰⁾(n)) be a hot error original data sequence, X ⁽¹⁾Be X ⁽⁰⁾The sequence that adds up, X is then arranged ⁽¹⁾=(x ⁽¹⁾(1), x ⁽¹⁾(2) ..., x ⁽¹⁾(n)).Wherein,

With season

, namely represent the mean value of adjacent data then to be close to average formation sequence z ⁽¹⁾=(z ⁽¹⁾(2), z ⁽¹⁾(3) ..., z ⁽¹⁾(k)).If $\hat{a}={[a,b]}^T$ For Argument List and $Y=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ .\\ .\\ .\\ x^{\left(0\right)}\left(n\right)\end{array}\right],$ $B=\left[\begin{array}{cc}-z^{\left(1\right)}\left(2\right)& 1\\ {-z}^{\left(1\right)}\left(3\right)& 1\\ .& .\\ .& .\\ .& .\\ {-z}^{\left(1\right)}\left(n\right)& 1\end{array}\right],$ X then ⁽⁰⁾(k)+az ⁽¹⁾(k)=the least-squares estimation Argument List of b satisfies Wherein, a, b are by the least square method parameters calculated, and-a is the development coefficient, and b is the grey action.And differential equation sequence time response of the hot error model of grey is

Thereby by the hot error model of grey time response the sequence equation as can be known, the order

, to introduce linear equation and know, the equation of the hot error model of the linear regression combination of grey is:

Wherein, v, l ₁, l ₂Be the parameter of simplifying gained, l ₃The parameter of introducing for linear equation, It is the estimated value of mean value conduct of each value of v.

Second step: utilize and descend square law most, find the solution the linear regression combination model parameter of grey;

If $Z\left(t\right)={\hat{X}}^{\left(1\right)}(t+1)-{\hat{X}}^{\left(1\right)}\left(t\right)=l_1{e}^{\mathrm{vt}}(e^v-1)+{\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">l</figure-callout>}}_2,t=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n-1.$ Establish Y again _m(t)=Z (t+m)-Z (t), namely

$\left\{\begin{array}{c}{Y}_m\left(t\right)=l_1{e}^{\mathrm{vt}}(e^{\mathrm{vm}}-1)(e^v-1)\\ Y_m(t+1)=l_1{e}^{v(t+1)}(e^{\mathrm{vm}}-1)(e^v-1)\end{array}\right.$

Got v=ln[Y by following formula _m(t+1)/Y _m(t)], estimated value is then arranged

Order $f\left(t\right)=e^{\hat{v}t},$ $X^{\left(1\right)}=\left[\begin{array}{c}{x}^{\left(1\right)}\left(1\right)\\ x^{\left(1\right)}\left(2\right)\\ .\\ .\\ .\\ x^{\left(1\right)}\left(n\right)\end{array}\right],$ $L=\left[\begin{array}{c}{l}_1\\ l_2\\ {\mathrm{<figure-callout\; id="3"\; label="l"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">l</figure-callout>}}_3\end{array}\right],$ $A=\left[\begin{array}{ccc}f\left(1\right)& 1& 1\\ f\left(2\right)& 2& 1\\ .& .& .\\ .& .& .\\ .& .& .\\ f\left(n\right)& \mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA00003197078100031.png,HDA00003197078100032.png"\; state="\{\{state\}\}">n</figure-callout>}& 1\end{array}\right],$ Then utilize least square method to try to achieve, L=(A ^TA) ^-1A ^TX ⁽¹⁾To be detected more than the hot error information substitution that obtains by hot error experiments various, so try to achieve the linear regression combination model parameter of concrete grey.Hot error experiments detects figure as shown in Figure 2, and wherein figure (a) is the location drawing of institute's target temperature sensor on the accurate horizontal Machining centers, and figure (b) is the hot error-detecting figure of main shaft displacement, and wherein m is not less than 1 natural number, and n is the natural number greater than 1.

The 3rd step: utilize the grey linear regression model (LRM) to carry out hot error prediction;

The linear regression combination model parameter of the grey that utilization is tried to achieve, the substitution equation

, through the tired predicted value that calculates hot error information that subtracts.

The 4th step: utilize the BP neural network that the built-up pattern residual error is revised, improve precision of prediction.

Adopt the BP neural network that the hot error model residual error of the linear regression combination of grey is predicted correction, wherein residual values is the difference of predicted value and measured value.According to principle and the learning algorithm thereof of BP neural network, the process flow diagram of its realization residual prediction as shown in Figure 3.Use Matlab software to carry out computing, obtain the more accurate residual prediction value of the linear hot error model of regression combination of grey, thereby obtain the prediction to actual value.

By to the prediction of hot error measured value with to the correction of residual values, draw hot error model predicted value comparison diagram as shown in Figure 4, and hot error model residual values comparison diagram as shown in Figure 5.By Fig. 4 and Fig. 5 as can be known, the predicted value of the hot error model of the linear regression combination of grey is than the precision of prediction height of the hot error model of grey, and after the residual error in the hot error model of the linear regression combination of grey was revised with the BP neural network, the precision of prediction of its hot error amount and the fitting degree of original hot error amount were higher.

Sum up by above instance analysis: the inventive method can be improved not to be had exponential increase and is difficult to describe the shortcoming of linear trends of change and the deficiency that the hot error model of grey does not have linear factor in the linear regression model (LRM), ability with good treatment linearity and nonlinear problem, hot error prediction to accurate horizontal Machining centers has been obtained good effect, not only consider the linear factor of hot error information but also considered its non-linear factor, improved the shortcoming of original single gray model, obtained hot error prediction value and higher degree of fitting more accurately, significant for numerical control machine heat error compensation.

Claims (1)

1. the hot error modeling method based on the grey linear regression is characterized in that, comprises the following steps:

1) on the basis of the hot error model of grey, introduce linear equation, make up the linear regression combination model of grey:

Numerical control machining tool heat error trend can be analyzed by making up the dynamic differential equation, because hot error has uncertainty, adopt the hot error model of grey, hot error raw data is made algebraic sum to be calculated, and handle its grey variable, to weaken the randomness in the hot error raw data, has strong regular hot error prediction value thereby generate; By the hot error model of grey time response the sequence equation as can be known, the order ${\hat{X}}^{\left(1\right)}(k+1)=(x^{\left(0\right)}\left(1\right)-\frac{b}{a})e^{-\mathrm{ak}}+\frac{b}{a}=l_1{e}^{\mathrm{vk}}+l_2$ , to introduce linear equation and know, the equation of the hot error model of the linear regression combination of grey is:

Wherein, X ⁽⁰⁾=(x ⁽⁰⁾(1), x ⁽⁰⁾(2) ..., x ⁽⁰⁾(n)) be hot error original data sequence, and

, X is arranged ⁽⁰⁾The sequence X that adds up ⁽¹⁾=(x ⁽¹⁾(1), x ⁽¹⁾(2) ..., x ⁽¹⁾(n)), simultaneously

, namely represent the mean value of adjacent data then to be close to average formation sequence z ⁽¹⁾=(z ⁽¹⁾(2), z ⁽¹⁾(3) ..., z ⁽¹⁾(k)),

It is EQUATION x ⁽⁰⁾(k)+az ⁽¹⁾(k)=response sequence of b; A, b are by the least square method parameters calculated, and wherein ,-a is the development coefficient, and b is the grey action; V, l ₁, l ₂Be the parameter of simplifying gained, l ₃The parameter of introducing for linear equation,

It is the estimated value of mean value conduct of each value of v; X ⁽⁰⁾(1) and x ⁽⁰⁾(1) has identical meanings, all represent data element corresponding in the hot error information sequence;

Be that t gets 1,2 through the hot error information of calculating gained ..., n, n are the natural number greater than 1;

2) utilize least square method, find the solution the linear regression combination model parameter of grey:

If

, establish Y again _m(t)=and Z (t+m)-Z (t), m is not less than 1 natural number, namely

$\left\{\begin{array}{c}{Y}_m\left(t\right)=l_1{e}^{\mathrm{vt}}(e^{\mathrm{vm}}-1)(e^v-1)\\ Y_m(t+1)=l_1{e}^{v(t+1)}(e^{\mathrm{vm}}-1)(e^v-1)\end{array}\right.$

Got v=ln[Y by following formula _m(t+1)/Y _m(t)], estimated value is then arranged

Order $f\left(t\right)=e^{\hat{v}t},$ $X^{\left(1\right)}=\left[\begin{array}{c}{x}^{\left(1\right)}\left(1\right)\\ x^{\left(1\right)}\left(2\right)\\ .\\ .\\ .\\ x^{\left(1\right)}\left(n\right)\end{array}\right],$ $L=\left[\begin{array}{c}{l}_1\\ l_2\\ l_3\end{array}\right],$ $A=\left[\begin{array}{ccc}f\left(1\right)& 1& 1\\ f\left(2\right)& 2& 1\\ .& .& .\\ .& .& .\\ .& .& .\\ f\left(n\right)& n& 1\end{array}\right],$ Then utilize least square method to try to achieve, L=(A ^TA) ^-1A ^TX ⁽¹⁾, so try to achieve the linear regression combination model parameter of grey;

3) utilize the grey linear regression model (LRM) to carry out hot error prediction:

The linear regression combination model parameter of the grey that utilization is tried to achieve, the substitution equation

, through the tired predicted value that calculates hot error information that subtracts;

4) utilize the BP neural network that the built-up pattern residual error is revised, improve precision of prediction:

Adopt the BP neural network that the hot error model residual error of the linear regression combination of grey is predicted correction, wherein residual values is the difference of predicted value and measured value, namely use Matlab software to carry out computing, obtain the residual prediction value of the hot error model of the linear regression combination of grey, thereby obtain the predicted value to actual value, improve the degree of accuracy of prediction.

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