Thermal Displacement Prediction Model Based on Logistic Map Bifurcation, Newton’s Method, and Particle Swarm Optimization for Computer Numerical Control Three-Axis Milling Machine Tools

In order to meet the increasing demand for elevated cutting precision in machining, this study developed an accurate thermal displacement (error) prediction model that identifies temperature-sensitive points (TSPs). The performance levels of different models and optimization algorithms in thermal error prediction and TSP selection, respectively, were compared to identify a suitable model and optimization algorithm. The best error prediction results were obtained with a long short-term memory (LSTM) model, and the best optimization results were obtained with a novel optimization algorithm produced through the combination of particle swarm optimization (PSO), Newton’s method, and logistic map bifurcation (LMB). Newton’s method is a numerical method that is faster than PSO. This method can, however, produce accurate results only when suitable initial values are selected; furthermore, it can exhibit divergence. In order to overcome these limitations, Newton’s method was combined with PSO, thereby achieving rapid convergence and minimizing the effect of the initial values on the final results. An LSTM model with the aforementioned novel algorithm exhibited excellent results in thermal error prediction, with the coefficient of determination ( $R^{2}$ ), mean absolute error (MAE), and root-mean-square error (RMSE) being 0.99, 1.23, and $1.57~\mu $ m, respectively. These results indicate that the proposed model can be applied to thermal compensation processes in industrial manufacturing.