## Concept and Kinematic Model Formulation
### The Embedded Distributed Sensors  
Accurate and reliable measurements of the key variables play an important role in compensating for deviations and in calibrating the MT. The information obtained by these variables are considered for model validation and MT error compensation. The temperature sensors are used as inputs to estimate thermal deformations [29; 30]. The spindle speed, the axis feed-rate, the machining time, and other constants may be the main causes of heat sources. A traditional approach is based on the temperature distribution evaluation using thermocouples or resistance thermometers [31; 32]. This method may show a set of limits, in fact when the number of thermocouples increases on the MT surface, many potential undesired effects may be noted (e.g., electrical disturbs), thereby reducing the effectiveness of the calibration method. On the other hand, a limited number of sensor applications does not permit an accurate mapping of the temperature behavior. Nowadays, infrared technologies may be an effective option to measure the thermal gradient on the surfaces of the MT frames [33; 34]. In fact, these systems are able to provide robust measurements of temperature. Nevertheless, they lose information when the system is switched off, compromising the compensation over time. The FBG strain sensors are able to overpass all these limits; in fact they guarantee robust and permanent measurements and they may be installed in multiple positions embedded in the MT, mapping potential thermal gradients completely. In addition, the FBG sensors have the capacity to detect all potential deviations on the tool tip point (TTP), not only originated by temperature variability, but also when other external factors (e.g., cutting effects) are present. In fact, they may measure directly all point to point deviations of the MT structure surfaces [35]. In particular, FBGs are usually insensitive to magnetic and electric inputs. Certain types of optical fibers have high sensitivity on specific wavelengths. Their photo-sensitivity can lead to a variation of refractivity when the fibers are open to a specific wave-length (UV range). Other advantages of FBG sensor application are represented by their lightweight, small size, electromagnetic protection, distance conduction, and electric inaccessibility [36; 37]. These characteristics are particularly suitable in manufacturing conditions. In addition, this technology shows multiplexing capabilities, providing an absolute measure of strain at multiple points simultaneously [38; 39]. The state of art shows a broad range of FBG applications. They are used to detect damages, to monitor structure health, to control stresses in airplanes and rotor blades of wind turbines, or to measure strain in harsh environments [40]. By using these sensors, the modelling process can become simpler, more robust, and more efficient, since the number of sensors can be limited and the effects of thermal hysteresis minimized. [PERSON] et al. [41] used FBG to investigate the effect of temperature  
Figure 1: The adaptive structure concept and the sensor locations (internal view). variations of a machine tool on the shop floor. The FBGs work as an optical filter by selecting a certain portion of the input spectrum of the light to be reflected. They are integrated as sensors in optical fibers. When the part of the FBG sensor is deformed due to external disturbances (e.g., strain or temperature), a change in the grating periods is noted modifying the Bragg wavelength. By measuring the wavelength change, the physical quantities can be measured and the temperature effects should be compensated using a proper system. In this way, a FBG sensor is based on the alterations in the reflective signal, which depends on the core and the periodicity of the grating. The Bragg wavelength is defined as [41]:  
\[\lambda=2\cdot n_{\mathit{eff}}\cdot\Delta; \tag{1}\]  
where \(n_{\mathit{eff}}\) and \(\Delta\) are the refractive value of the material and the grating structure, respectively.