Kalman filter-driven state observer for thermal error compensation in machine tool digital twins

Sustainable reduction of thermal errors during production is the key challenge in modern high-precision manufacturing. Numerical compensation models provide an energy-efficient solution, but in the case of data-driven models, high-quality experimental data must be time-consuming and expensive to produce, negatively impacting overall productivity. Furthermore, robustness concerns arise in the case of new operating conditions, which were not contained in the training data. This paper presents a novel use of a Kalman filter together with model order reduced finite element models to observe the entire thermal state, which allows the subsequent solution of the mechanical model and computation of the thermal errors in real-time without requiring any training data but instead purely based on the physical system model. The effectiveness of this approach is evaluated using experiments on a thermal test bench with 16 out of 40 temperature sensors employed for observation and demonstrated on a 5-axis machine tool (MT) with 13 out of 25 temperature sensors used. Due to the combination of the reduced order model and Kalman filter these 13 temperature sensors are sufficient to represent a MT mesh of more than 350’000 elements. The entire temperature profile of the thermal test bench is reconstructed to achieve a root mean square error (RMSE) of the unobserved temperature sensors of only 2.7 °C, which accounts for more than 83% of all temperature variations and 1.3 °C for the 5-axis MT. For the thermal error of the thermal test bench, the RMSE could be reduced from $67.4\mu \text{m}$ to $33.3\mu \text{m}$, corresponding to a reduction of 52.7 %. This could be achieved without the need for experimental data for model calibration, in a real-time capable physics-based model.