## 3 Experimental Tests and Results
### ZERO(tm) Machine Tests
#### 3.2.1 First-Order Model Validation of ZERO(tm) Machine Tests  
To explain the measured error values, an initial attempt was made to derive the thermal error transfer function by fitting a simple physical model to the results. This model utilised the ambient temperature as the input and the evolution of the measured error as the output. A similar methodology was recently proposed by [PERSON] et al. [35]. The hypotheses for establishing the model are as follows:  
* The energy balance in each direction is modelled as a single thermal inertia. The heat loss or gain of the machine is assumed to be proportional to the temperature difference between the machine and its environment, with radiative effects being considered negligible;  
Figure 17: Temporal evolution of error E\({}_{Y}\): (**a**) first test from 12 to 15 February and (**b**) second test from 23 to 25 February. Figure 18: Temporal evolution of error E\({}_{Z}\): (**a**) first test from 12 to 15 February and (**b**) second test from 23 to 25 February. * The heat transfer coefficient is treated as a constant, independent of the temperature and direction, whether heat is transferred from the machine to the environment or vice versa;
* The machine's expansion is assumed to be essentially linear concerning the temperature;
* All variables are considered to deviate from the baseline operating conditions, under which the errors were initially defined as zero. By modelling the energy balance in each direction as a single thermal inertia, the calculation is conducted based on the formula presented in Equation (1). \[m\times C_{p}\ \times\frac{dT}{dt}=U\times A\times(T_{env}-T) \tag{1}\]  
wherein the following definitions apply:  
* \(m\): the considered mass of the structure;
* \(C_{p}\): specific heat capacity of the structural material;
* \(U\): the convective heat transfer coefficient;
* \(A\): the effective convective area;
* \(T_{env}\): the environment temperature difference;
* \(T\): the temperature difference of the structure. The length variation with the temperature can be expressed as follows  
\[\alpha\times L_{initial}\times\frac{dT}{dt}=\frac{dL}{dt} \tag{2}\]  
wherein the following definitions apply  
* \(\alpha\): the linear expansion coefficient of the structural material;
* \(L_{initial}\): specific initial length at the instant where the conditions were set to zero. Applying the Laplace transformation to the above Equations (1) and (2), Equations (3) and (4) are obtained. \[m\times C_{p}\times s\times T(s)=U\times A\times(T_{env}(s)-T(s)) \tag{3}\]  
\[\alpha\times L_{initial}\times T(s)=L(s) \tag{4}\]  
Combining Equations (1) and (2) and rearranging them, the following transfer function between the position error and the ambient temperature is obtained:  
\[m\times\frac{L(s)}{T_{ext}(s)}=\frac{\alpha\times L_{initial}\times U\times A}{ U\times A+m\times C_{p}\times s}=\frac{\alpha\times L_{initial}}{1+\frac{m \times C_{p}}{U\times A}s} \tag{5}\]  
wherein:  
* System gain: \(k=\alpha L_{initial}\);
* Time constant: \(\tau=\frac{m\times C_{p}}{U\times A}\). To identify the system, the test function from the MATLAB(tm) R2022a System Identification Toolbox (Mathworks, Natick, MA, USA) was utilised to analyse a three-day data set collected from 23 to 26 February. Data from the second test were selected for the model calibration, as they were gathered during a weekend, thus avoiding interference from other machinery and providing an environment that was likely to exhibit larger temperature differences. The measurements used to adjust the model were taken at three distinct positions along the Y-axis of the ram. The sampling frequency for the training data was 46 min and 24 s, which corresponds to the time required to complete one full MIM cycle.