# Correction of Thermal Errors in Machine Tools by a Hybrid Model Approach
## 1 Introduction and State of the Art
### Structure Model-Based Correction  
In contrast to the characteristic diagram approach, the structural model-based correction method includes the structural variability and calculates the temperature and displacement field based on the moving structure and current pose. In consequence, heat inputs can be applied to moving components, such as feed axes, in the specific locations where they occur [38; 39]. The structural model is not necessarily based on FEM approaches but also uses physical-based models on a system or digital block simulation, such as Matlab/Simulink(r) or ESI ITI SimulationX(r). It includes partial models for power loss, thermal coupling coefficients, and thermo-elastic displacement and approaches for calculating the axis correction values. A modular strategy for correcting structure models in the context of machine tools has been developed through fundamental research and is undergoing validation on numerous machine components and structures [36; 40; 41; 42; 43]. The modules can be divided into three areas: data acquisition, modeling, and correction, whereby each area is calculated in its own time domain (control real time and thermal real time) that can be calculated decoupled from each other (control internal and control external). In can be shown for a machine component that the thermally induced position error could be reduced by 80-87% for a machine component \"bar axis\" [41]. In ref. [44], the error components of the rotational axes were corrected with the help of a physical model; a reduction in the error between 35% and 38% was achieved.  
Regarding characteristic diagrams, for each load case, as it depends on the workpiece and each machine tool, a suitable characteristic diagram must be determined experimentally, which is highly inefficient [4; 32]. The high costs are a major obstacle to the introduction of correction methods. Simulation-generated characteristic diagrams, on the other hand, allow a certain load case variability right up to the option to generate several characteristics in parallel for multiple load cases and are more cost-efficient. However, the simulations are very time-consuming (48 h for one load case for the simulated DMU80) and the choice of the correct characteristic as a basis for correction as well as methods for changing the characteristics on the running machine are still open research questions. Finally, structure models can include the current load situation and simulate quickly but require a huge modeling effort to achieve sufficient accuracy for thermal error correction.  
Consequently, the development of a cheap and smart hybrid correction methodology that takes the changing load cases (e.g., different workpieces), the specifics of the machine, and the changing environmental conditions into account, allows for a reduction in costs, scrap rate, and inefficient warm-up phases, and, at the same time, an increase in workpiece quality.  
This paper presents a new hybrid model approach that combines the advantages of simulation-generated characteristic diagrams and structural models with a decision algorithm to achieve a cheap and smart correction technology for machine tools.