## 1 Introduction  
The positioning accuracy of a modern machine tool (MT) is one of the most important requirements to guarantee the machining precision and quality [1]. The accuracy of a MT related to the geometrical deviations may depend up to 60% on the stability of the structure and for the rest on workpiece fixturing and tools [2; 3; 4]. In the last few years, researches have addressed their studies to mechatronic systems with the intention of compensating these undesired effects [5]. The principal causes of errors may be classified as geometrical errors and thermo-mechanical strains [6; 7; 8; 9; 10]. The first group is represented as the shape deviation that disturbs the moveable slides of the MT and may be assumed to be independent to the time. The second class is the result of external factors acting on the frame which are fundamentally time-correlated. This work focuses on temperature oscillations that can origin three-dimensional thermal grades in MT structures and can produce non-linear distortions over time, impacting significantly on the machine accuracy [11; 12]. The thermal distortions of the device components are generated by temperature gradients existing inside the structure. In order to model the temperature spreading and to compensate the consecutive distortion, it is important to get the temperature mapping in real-time, in particular when ultra-high precision machining is required. Although thermal errors might be reduced by using structure materials with a low coefficient of thermal expansion, it is useful to consider a more reasonable method of decreasing thermal error such as an error compensation system. Recent studies have been carried out to introduce Polymeric Matrix Composite (PMC) fiber reinforced materials. Significant results related to mass reduction, stiffness increment, and damping increase have been achieved. Nevertheless, the growing of customer requirements in terms of cutting machine performances has pushed the designers to investigate novel solutions. Calibration is a method where the existing thermal deviations are compensated by correcting the position of the MT points based on the magnitude of the thermal-mechanical error. The state of art shows many works in the field of thermal-structural error compensation [13]. Researchers have developed various procedures such as a simulation method [14; 15; 16] in modelling the thermal characteristics. Obtaining a mathematical representation can be a complex process because it is problematic to produce the edge limitations and get the characteristic of heat transmission. The test of the machine tool is still required to calibrate the model for a successful application of the technique. Different model structures have been used to calculate thermal errors such as Multiple Regression Analysis (MRA) [17], artificial neural networks (ANN) [18], fuzzy logic [19], an adaptive neuro-fuzzy implication [20; 21], and a grouping of diverse modelling procedures [22; 23]. Early work by [PERSON] et al. [15] used both a MRA model and an ANN model for thermal error compensation of a horizontal machining center. Thanks to their experimental results, it was possible to decrease the thermal error from 200 \(\upmu\)m to 30 \(\upmu\)m.  
[PERSON] [22] applied a Hierarchy-Genetic-Algorithm (HGA) trained neural network with the aim of mapping the temperature change against the thermal response of the MT. Experimental results pointed out that the temperature inaccuracy model could moderate the deviation to smaller than 10 \(\upmu\)m under real conditions. Nowadays, there is a number of different approaches to solve this problem [22], which can be grouped in two main categories: design-based techniques and compensation-based methods. The first category collects all those good practice design techniques that permit the development of a MT based on a structure that is stable, stiff, and able to damp vibrations, thereby limiting excessive thermal gradients.