# A Review of Thermal Error Modeling Methods for Machine Tools
## 3 Discussion  
In this section, the advantages and disadvantages of the models are summarized in the following table (Table 1). Specifically, LS is an earlier way to compensate the thermal errors of machine tools with mature theory and simple model structure, so it is widely used. However, it has less independent variables and is not quite in line with the actual situation and its predictive ability is limited under complex conditions. The modeling process of the MRA is not only simple in structure, but also reliable in performance. The MRA model for a machine tool with simple structure and working conditions has higher accuracy. However, when there are a lot of variables, the calculation time of MRA is long. However, the correlation between the thermal error and the temperature is only considered by the MRA, which will lead to the temperature variable coupling and reduce the accuracy of the model. Modeling based on the grey system theory is simpler, more convenient, and does not depend on massive and complete data information. However, it is a model that predicts its own development based on its own original data. So taking different data in the original data sequence, the model will be very different. Generally, neural networks have higher prediction accuracy compared to LS and MRA. Among them, the BP neural network had the high accuracy, but it had a slow convergence speed and it was easy to fall into a local minimum. The initial value was also very difficult to determine. Because most of the parameters can be given better \begin{table}
\begin{tabular}{l l l} \hline
**Modeling Method** & **Advantages** & **Disadvantages** & **References** \\ \hline \multirow{3}{*}{LS} & \begin{tabular}{l} Mature theory. \\ Simple model structure. \\ Widely used. \\ \end{tabular} & \begin{tabular}{l} Less independent variables. \\ Low predictive ability under \\ complex conditions. \\ Calculation time is too long with \\ more variables. \\ Temperature variable coupling. \\ When changing the input, the model will be \\ very different. \\ Slow convergence speed. \\ Easy to fall into a local minimum \\ The initial value is very difficult to determine. \\ \end{tabular} & [18; 19; 20; 21] \\ \multirow{3}{*}{MRA} & \begin{tabular}{l} Simple model structure. \\ Reliable in performance. \\ \end{tabular} & \begin{tabular}{l} Calculation time is too long with \\ more variables. \\ Temperature variable coupling. \\ When changing the input, the model will be \\ very different. \\ Slow convergence speed. \\ Easy to fall into a local minimum \\ The initial value is very difficult to determine. \\ \end{tabular} & [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35] \\ \multirow{3}{*}{Grey system} & \begin{tabular}{l} Simple modeling. \\ Do not rely on massive and complete data information. \\ Slow convergence speed. \\ Easy to fall into a local minimum \\ The initial value is very difficult to determine. \\ \end{tabular} & [36; 37; 38; 39; 40; 41; 42] \\ \multirow{3}{*}{NN} & \begin{tabular}{l} BIP \\ Single structure design. \\ Faster training speed. \\ \end{tabular} & \
\begin{tabular}{l} The key feature functions are more difficult \\ to extract. \\ Worse generalization performance. \\ Not easy to select parameters. \\ A lot of computing resources. \\ slow convergence speed. \\ Increase the difficulty of modeling. \\ \end{tabular} & [57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83] \\ \multirow{3}{*}{Wbrid model} &
\begin{tabular}{l} Combine the advantages of the separated two modeling \\ methods.Good versatility. \\ High accuracy. \\ Strong robustness. \\ \end{tabular} & Increase the difficulty of modeling. & [84; 85; 86; 87; 88; 89; 90] \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of different thermal error modeling methods. initial values during initialization according to actual problems, the structure design of RBF neural network is simple and the training speed is faster.