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Self learning-empowered thermal error control method of precision machine tools based on digital twin

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Abstract

To improve machining accuracy of complex parts, a self learning-empowered thermal error control method of precision machine tools is presented based on digital twin. The memory of thermal error is theoretically and numerically revealed by error mechanism analysis, and then the applicability of long-short-term memory (LSTM) neural network (NN) in the training of the self-learning error model is proved. To improve the predictive accuracy, the Bayesian optimization algorithm is used to optimize such hyper-parameters as the epoch size, batch size, and the number of hidden nodes of the LSTM NN model. Then the self-learning prediction model of thermal error is proposed based on Bayesian-LSTM NN. The fitting and prediction performance of the proposed Bayesian-LSTM NN is better than that of such models as the LSTM NN with random hyperparameters, back propagation NN, multiple linear regression analysis (MLRA), and least square support vector machine (LSSVM). Finally, the self learning-empowered error control method is proposed based on digital twin, and the Bayesian-LSTM NN error control model is embedded into the self learning-empowered error control framework to realize the real-time thermal error prediction and control. When the predicted thermal error is greater than the preset machining error, the control components are recalculated automatically, and inserted into the machining instructions. It is shown that the machining error can be reduced effectively by the self learning-empowered error control method, which is vital for precision machining of complex parts and improvement of the intelligence level.

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Abbreviations

T 0 :

Ambient temperature

\(c\) :

Specific heat capacity

h :

Convective coefficient

\(\rho\) :

Density

\(d_{0}\) :

Shaft diameter

\(\lambda\) :

Thermal conductivity

\(T(0)\) :

Temperature of heat source

T(x,t):

Temperature of shaft

\(\Delta T(x,t)\) :

Temperature difference

\(L\) :

Shaft length

\( \gamma _{t} = f\left( {x_{t} } \right) + \varepsilon _{t} \) :

Observations

\(x_{t + 1}\) :

The next evaluation points

γ t :

The total loss

α t(x:D 1:t):

The acquisition function

\(y^{*}\) :

The current optimal solution

\(f_{t}\) :

Gate

\(x\) :

The input vector

\(W\) :

Weight matrix

\(b\) :

Deviation matrix

\(h\) :

Output vector

\(y\) :

Dependent variable

\(x = (x_{1} ,x_{2} , \cdots ,x_{m} )\) :

Independent variables

\(b_{0} ,b_{1} , \cdots ,b_{m}\) :

Pending regression parameters

\(J\) :

Minimization function of structure risk;

\(\omega\) :

Weight vector

\(\xi_{i}\) :

Error variable

\(\gamma\) :

Adjustable parameter

\(x_{i}\) :

Input

\(\alpha\) :

Thermal expansion coefficient

\(\tau\) :

Time constant

\(f\) :

Unknown target function

\(D_{1:t} = \{ (x_{1} ,y_{1} ),(x_{2} ,y_{2} ), \cdots ,(x_{t} ,y_{t} )\}\) :

Collection of observations

\(p(D_{1:t} \left| f \right.)\) :

Likelihood distribution of \(y\)

\(p(f)\) :

Priori probability distribution of \(f\)

\(p(D_{1:t} )^{\prime}\) :

Marginal likelihood distribution of \(f\)

\(p(f\left| {D_{1:t} } \right.)\) :

Posteriori probability distribution of \(f\)

\(x_{t}\) :

Decision vector;

\(\varepsilon_{t}\) :

Error of observations

\(x_{t + 1}\) :

Next evaluation points

\(\gamma_{t}\) :

Total loss

\(\alpha_{t} (x:D_{1:t} )\) :

Acquisition function

\(y^{*}\) :

Current optimal solution

\(i_{t}\) :

Input gate

\(\tanh\) :

Activation function

\(C_{t - 1}\) :

The cell status at the time \(t - 1\)

\(\tilde{c}_{t}\) :

Storage status of unit at the time of \(t\)

\(x_{t}\) :

The input at the time \(t\)

O t :

Unit status at the time \(t\)

\(\sigma\) :

Activation function

\(n\) :

Sets of error data

\(m\) :

Number of typical temperature variables

\(\varepsilon\) :

A random error

\(\varphi \left( \cdot \right)\) :

Map** function

\(b\) :

Deviation

\(l\) :

Input length

\(y_{i}\) :

Target quantity

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (51905057), the Natural Science Foundation Project of Chongqing, Chongqing Science and Technology Commission (cstc2019jcyj-msxmX0050), the Fundamental Research Funds for the Central Universities (2020CDJQY-A036), and the Venture & Innovation Support Program for Chongqing Overseas Returnees (cx2019054), and State Key Laboratory for Manufacturing Systems Engineering of **'an Jiaotong University (sklms2020016).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, 400044, China

    Chi Ma, Hongquan Gui & Jialan Liu

  2. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China

    Chi Ma, Hongquan Gui & Jialan Liu

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  1. Chi Ma
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Correspondence to Chi Ma.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work; there is no professional or other personal interest of any nature or kind in any product or company that could be construed as influencing the position presented in, or the review of, the manuscript. Jialan Liu and Hongquan Gui contributed equally to this manuscript.

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Ma, C., Gui, H. & Liu, J. Self learning-empowered thermal error control method of precision machine tools based on digital twin. J Intell Manuf 34, 695–717 (2023). https://doi.org/10.1007/s10845-021-01821-z

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