CN103268082B - Thermal error modeling method based on gray linear regression - Google Patents
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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
Technical Field
The invention belongs to the field of application of error compensation of numerical control machines, and particularly relates to a modeling method of a gray linear regression combination model of thermal errors of a precise horizontal machining center.
Background
The thermal error refers to a machining error generated by the thermal deformation of machine tool parts caused by the temperature rise of a machine tool and the change of the relative position between a workpiece and a cutter, and the main research contents of the thermal error are thermal deformation theoretical analysis, measurement in thermal error compensation, key point optimization, thermal error modeling and thermal error compensation implementation. The thermal error is the largest error source of the numerical control machine tool and is also an important influence factor of the machining precision, and accounts for about 40-70% of the total error of the machine tool. Therefore, the thermal error must be reasonably and effectively controlled, a thermal error model with higher precision is established as far as possible, and the method is a key technology for realizing thermal error compensation and improving the machining precision of the machine tool.
With the continuous development of numerical control machines, higher and higher requirements are put forward on machining precision, and in recent years, researchers at home and abroad make a great deal of research on how to reduce the thermal error of the numerical control machine. The thermal error modeling method includes an artificial neural network model based on a genetic algorithm, a fuzzy logic model, a gray prediction model, a linear regression model and the like. However, these thermal error models are single and cannot fully express the overall appearance of thermal error data, the adopted linear regression model has the disadvantages of no exponential increase and difficulty in describing linear variation trend, the gray prediction model has no linear factors, and the thermal error raw data has both linear and non-linear factors, so that the established thermal error model must have the capability of handling linear and non-linear problems.
Disclosure of Invention
The invention aims to overcome the defects of modeling, and provides a modeling method with higher precision than a gray thermal error model, namely a thermal error modeling method based on gray linear regression, and a BP neural network is introduced to correct the residual error so as to obtain a more accurate thermal error predicted value.
The invention adopts the following technical means for solving the problems:
a thermal error modeling method based on gray linear regression comprises the following steps:
1) introducing a linear equation on the basis of a gray thermal error model to construct a gray linear regression combination model:
the thermal error trend of the numerical control machine tool can be changed by constructionAnalyzing a state differential equation, adopting a gray thermal error model to perform algebraic sum calculation on thermal error original data due to uncertainty of thermal error, and processing gray variables of the thermal error original data to weaken randomness in the thermal error original data so as to generate a thermal error predicted value with strong regularity; from the time response sequence equation of the gray thermal error model, let And introducing a linear equation, wherein the equation of the gray linear regression combination thermal error model is as follows:
wherein, X(0)=(x(0)(1),x(0)(2),…,x(0)(n)) is the thermal error raw data sequence, andhaving X(0)Of the accumulation sequence X(1)=(x(1)(1),x(1)(2),…，x（1）(n)). At the same timeI.e. representing the mean of adjacent data, the sequence z is generated immediately adjacent to the mean(1)=(z(1)(2),z(1)(3),…,z(1)(k))。Is x of the equation(0)(k)+az(1)(k) = b response sequence; a. b is a parameter calculated by a least square method, wherein, -a is a development coefficient, and b is a gray effect amount; v, l1，l2Is a simplified parameter obtained,. l3For the parameters introduced by the linear equation,is an estimated value which is an average value of the respective values of v; x(0)(1) And x(0)(1) Have the same meaning and all represent corresponding data elements in the thermal error data sequence;is calculated thermal error data, t can be 1,2, …, n is a natural number larger than 1;
2) solving gray linear regression combined model parameters by using a least square method:
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</math> And is also provided with Ym(t) = Z (t + m) -Z (t), m is a natural number of not less than 1, that is
Obtained by the above formula, v = ln [ Y ]m(t+1)/Ym(t)]Then there is an estimated value
Order to Then, L = (A) is obtained by the least square methodTA)-1ATX(1)Thus, the parameters of the gray linear regression combination model are obtained;
3) thermal error prediction using a gray linear regression model:
substituting the obtained gray linear regression combination model parameters into the equationObtaining a predicted value of the thermal error data through accumulation and subtraction calculation;
4) and correcting the residual error of the combined model by using a BP neural network, so that the prediction precision is improved:
and (3) predicting and correcting the residual error of the gray linear regression combination thermal error model by adopting a BP neural network, wherein the residual error value is the difference value between a predicted value and an actually measured value, namely Matlab software is used for operation to obtain the residual error predicted value of the gray linear regression combination thermal error model, so that the predicted value of an actual value is obtained, the prediction accuracy is improved, and the method has important significance for the thermal error compensation of the numerical control machine tool.
Advantageous effects
The method can improve the defects that the linear regression model has no exponential growth and is difficult to describe the linear change trend and the gray thermal error model has no linear factors, has good capability of processing linear and nonlinear problems, obtains good effect on the thermal error prediction of a precise horizontal machining center, considers the linear factors and the nonlinear factors of thermal error data, improves the defects of the original single gray model, obtains more accurate thermal error predicted value and higher fitting degree, and has important significance on the thermal error compensation of a numerical control machine tool.
Drawings
FIG. 1 is a flow chart of a thermal error modeling method based on gray linear regression;
FIGS. 2(a) - (b) are thermal error experimental detection diagrams of the present invention;
FIG. 3 is a flowchart illustrating a residual prediction implementation of a BP neural network according to an embodiment of the present invention;
FIG. 4 is a comparison graph of predicted values of thermal error models according to an embodiment of the present invention;
FIG. 5 is a comparison graph of residual values of thermal error models according to an embodiment of the present invention;
Detailed Description
A flow chart of a thermal error modeling method based on gray linear regression according to an embodiment of the present invention is shown in fig. 1, and the steps of the present invention are further described below with reference to the flow chart. The specific implementation steps are as follows:
the first step is as follows: introducing a linear equation on the basis of the gray thermal error model, and constructing a gray linear regression combination model;
the thermal error trend of the numerical control machine tool can be analyzed by constructing a dynamic differential equation, because the thermal error has uncertainty, a gray thermal error model is adopted, the thermal error original data is subjected to algebraic sum calculation, and gray variables of the thermal error original data are processed to weaken randomness in the thermal error original data, so that a thermal error predicted value with strong regularity is generated.
Let X(0)=(x(0)(1),x(0)(2),…,x(0)(n)) is a thermal error raw data sequence, X(1)Is X(0)The accumulated sequence of (1) then has X(1)=(x(1)(1),x(1)(2),…,x(1)(n)). Wherein,. At the same time orderI.e. representing the mean of adjacent data, the sequence z is generated immediately adjacent to the mean(1)=(z(1)(2),z(1)(3),…,z(1)(k) ). If it is Is a parameter row and x is then(0)(k)+az(1)(k) The least square estimation parameter column of = b satisfiesWherein, a and b are parameters calculated by a least square method, a is a development coefficient, and b is a gray acting amount. And the time response sequence of the differential equation of the gray thermal error model is. Thereby from the grey thermal error modelCan know the time response sequence equation ofAnd introducing a linear equation, wherein the equation of the gray linear regression combination thermal error model is as follows:. Wherein v, l1，l2Is a simplified parameter obtained,. l3For the parameters introduced by the linear equation,is an estimated value of the average value of the respective values of v.
The second step is that: solving the parameters of the gray linear regression combination model by using the lowest multiplication;
is provided with <math>
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</math> And is also provided with Ym(t) = Z (t + m) -Z (t), i.e.
Obtained by the above formula, v = ln [ Y ]m(t+1)/Ym(t)]Then there is an estimated value
Order to Then, L = (A) is obtained by the least square methodTA)-1ATX(1). The thermal error data obtained by the thermal error experimental detection is substituted into the above formulas, so as to obtain the specific gray linear regression combination model parameters. Fig. 2 is a view showing a thermal error experimental detection diagram, in which (a) is a position diagram of a temperature sensor marked on a precision horizontal machining center, and (b) is a view showing a spindle displacement thermal error detection diagram, in which m is a natural number not less than 1, and n is a natural number greater than 1.
The third step: performing thermal error prediction by using a gray linear regression model;
substituting the obtained gray linear regression combination model parameters into the equationAnd obtaining a predicted value of the thermal error data through accumulation and subtraction calculation.
The fourth step: and correcting the residual error of the combined model by using the BP neural network, so that the prediction precision is improved.
And predicting and correcting the residual error of the gray linear regression combination thermal error model by adopting a BP neural network, wherein the residual value is the difference value between a predicted value and an actually measured value. A flowchart of implementing residual prediction according to the principle of the BP neural network and its learning algorithm is shown in fig. 3. Matlab software is used for operation, and a more accurate residual prediction value of the gray linear regression combination thermal error model is obtained, so that the prediction of an actual value is obtained.
Through the prediction of the measured thermal error values and the correction of the residual values, a comparison graph of the predicted thermal error model values shown in fig. 4 and a comparison graph of the residual values of the thermal error model shown in fig. 5 are obtained. As can be seen from fig. 4 and 5, the prediction value of the gray linear regression combination thermal error model is higher than the prediction accuracy of the gray thermal error model, and after the residuals in the gray linear regression combination thermal error model are corrected by the BP neural network, the degree of fitting between the prediction accuracy of the thermal error values and the original thermal error values is higher.
The analysis of the above examples concluded that: the method can improve the defects that the linear regression model has no exponential growth and is difficult to describe the linear change trend and the gray thermal error model has no linear factors, has good capability of processing linear and nonlinear problems, obtains good effect on the thermal error prediction of a precise horizontal machining center, considers the linear factors and the nonlinear factors of thermal error data, improves the defects of the original single gray model, obtains more accurate thermal error prediction value and higher fitting degree, and has important significance on the thermal error compensation of a numerical control machine.
Claims (1)
1. A thermal error modeling method based on gray linear regression is characterized by comprising the following steps:
1) introducing a linear equation on the basis of a gray thermal error model to construct a gray linear regression combination model:
the thermal error trend of the numerical control machine tool can be analyzed by constructing a dynamic differential equation, because the thermal error has uncertainty, a gray thermal error model is adopted, the thermal error original data is algebraically summed, and gray variables of the thermal error original data are processed to weaken the randomness in the thermal error original data, so that the thermal error original data are generatedThe thermal error prediction value with stronger regularity is obtained; from the time response sequence equation of the gray thermal error model, let And introducing a linear equation, wherein the equation of the gray linear regression combination thermal error model is as follows:
wherein, X(0)=(x(0)(1),x(0)(2),…,x(0)(n)) is the thermal error raw data sequence, andhaving X(0)Of the accumulation sequence X(1)=(x(1)(1),x(1)(2),…,x(1)(n)) simultaneouslyI.e. representing adjacent dataMean value, then the sequence z is generated next to the mean value(1)=(z(1)(2),z(1)(3),…,z(1)(k))，Is the equation x(0)(k)+az(1)(k) Response sequence of = b; a. b is a parameter calculated by a least square method, wherein, -a is a development coefficient, and b is a gray effect amount; v, l1，l2Is a simplified parameter obtained,. l3For the parameters introduced by the linear equation,is an estimated value which is an average value of the respective values of v; x(0)(1) And x(0)(1) Have the same meaning and all represent corresponding data elements in the thermal error data sequence;the thermal error data is obtained through calculation, t is 1,2, …, n is a natural number larger than 1;
2) solving gray linear regression combined model parameters by using a least square method:
is provided withAnd is further provided with Ym(t) = Z (t + m) -Z (t), m is a natural number of not less than 1, that is
Obtained by the above formula, v = ln [ Y ]m(t+1)/Ym(t)]Then there is an estimated value
Order to Then, L = (A) is obtained by the least square methodTA)-1ATX(1)Thus, the parameters of the gray linear regression combination model are obtained;
3) thermal error prediction using a gray linear regression model:
substituting the obtained gray linear regression combination model parameters into the equationObtaining a predicted value of the thermal error data through accumulation and subtraction calculation;
4) and correcting the residual error of the combined model by using a BP neural network, so that the prediction precision is improved:
and (3) performing prediction correction on the residual error of the gray linear regression combination thermal error model by adopting a BP neural network, wherein the residual value is the difference value between a predicted value and an actually measured value, namely, Matlab software is used for operation to obtain the residual prediction value of the gray linear regression combination thermal error model, so that the predicted value of an actual value is obtained, and the prediction accuracy is improved.
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| CN103926874B (en) * | 2013-12-03 | 2016-05-25 | 江苏齐航数控机床有限责任公司 | The selection optimization method of numerical control machine heat error compensation modeling temperature point combination |
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| CN104898562A (en) * | 2015-04-27 | 2015-09-09 | 攀枝花学院 | Modeling method of thermal error compensation of numerically-controlled machine tool |
| CN106312103A (en) * | 2015-06-30 | 2017-01-11 | 遵义林棣科技发展有限公司 | Numerical-control lathe control correction method based on command filtering |
| CN105759718B (en) * | 2016-03-21 | 2018-06-29 | 电子科技大学 | Numerical control machining tool heat error online compensation method and system |
| CN105759719B (en) * | 2016-04-20 | 2018-08-24 | 合肥工业大学 | A kind of numerical control machining tool heat error prediction technique and system splitting model based on unbiased esti-mator |
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| CN112147951B (en) * | 2020-09-28 | 2022-09-30 | 沈机（上海）智能系统研发设计有限公司 | Thermal error compensation method for machining equipment, device, system, medium and terminal thereof |
| CN112690771B (en) * | 2020-12-09 | 2022-05-24 | 华南理工大学 | Human face video heart rate detection method using linear regression model |
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