Thermal Error Modeling of CNC Machine Tool Spindle Based on Multiple Regression and Features Selection

The positioning precisions of X, Y, and Z machine directions are susceptible to temperature variations around machine tools to shift the cutter positioning when the CNC machine tool spindles during high-speed rotation. In this context, the study proposes a modeling method of thermal error compensation for the displacement of the cutter position. In the X-direction, the mechanical structure is closer to symmetrical form, which evenly distributes the thermal energy, so the thermal error is always small. Therefore, this study only deals with the thermal error in the Y and Z directions. The explanatory power improvement of the multiple regression model largely depends on the feature selection. The paper proposes the backward elimination (BE) algorithm base on mean squares of K-fold errors minimization as feature selection of multiple regression model to establish thermal error compensation modeling. Firstly, BE fits the complete model with all features, and then deletes the feature one by one using the selected test criterion until deleting any feature cannot improve the model explanatory power. The K-fold Cross Validation (KCV) evaluates model performance in limited training data and be used as a criterion for model selection. KCV cut the data into K subsets to keep k-1 subsets as model training, and the remaining subsets as model validation. The procedure is repeated k-times until the last subset is set as the validation set, then the average error across all k trials is computed. To evaluate each feature to be eliminated through KCV, the smallest mean squares error is selected from the N results to determine the variable for elimination each time, where N is the number of features. The multiple regression model was established by using the features selected for the Y and Z axes. Test results show that the method can reduce the peak-to-peak value of thermal error from about 55 μm to below 14 μm in the Y direction, and in Z direction is from about 74 μm to below 19 μm.