## 3 Experimental Tests and Results
### ZERO(tm) Machine Tests
#### 3.2.1 First-Order Model Validation of ZERO(tm) Machine Tests
The model's input was the difference between the ambient temperature and the initial temperature, while the output represented the measurement error in the Y-axis position relative to the initial measurement  
Table 8 presents the results of the identification process, including the system gains and time constants derived from the experimental data recorded at three different selected points. Further details can be found in Figure 14.  
The thermal time constants at the three Y-measured positions are found to be quite similar. When \(U\times A\) is calculated for all cases, values ranging from 48 to 53 W/K are obtained. Assuming a specific heat capacity \(C_{p}\) of 460 J/kg\(\cdot\)K, as specified in EN 1561:2024 for grey cast iron [38], a ram mass (m) of 1580 kg, and an effective surface area of the ram in contact with the environment of approximately 5 m\({}^{2}\) (\(A\)), the convection coefficient (\(U\)) is estimated to be around 10 W/m\({}^{2}\cdot\)K. This appears to be a reasonable value for natural convection at moderate temperatures  
Once the model was developed, an initial analysis was conducted by comparing the model's output with the measured data to assess its performance. Figure 19 presents the training data used to identify the model parameters as expressed in Equation (5). This includes the measured E\({}_{\text{Y}}\) error evolution at Point P_11 alongside the simulated values representing the output of the adjusted first-order model at the same point. This point corresponds to the maximum extension of the ram. Figure 20 presents a second analysis conducted to validate the first-order model. This involved comparing the model's output, generated using ambient temperature data from the initial test period from 12 to 15 February as the input, against the MIM output recorded during the same period. Figure 19: Training data to identify the first-order model parameters and the corresponding model results. From 23 to 26 February. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Position** & **Maximum Ram Extension** & **Middle Ram Extension** & **Minimum Ram Extension** \\ \hline Measured Point & & & \\ Travel Y (mm) & \(-\)100 & \(-\)700 & \(-\)1300 \\ k (mm/\({}^{\circ}\)K) & 0.02 & 0.014 & 0.011 \\ \(\tau\) (s) & 13,773 & 14,653 & 15,079 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Identified system parameters. Several metrics were employed to evaluate the quality of the model, demonstrating that the output, namely the machine error variation, is directly related to the model's input, which is the recorded ambient temperature. The following metrics were used:  
\[MAE=\frac{1}{N}\times\sum_{i=1}^{N}\lvert y_{i}-\hat{y}_{i}\rvert \tag{6}\]  
\[RMSE=\sqrt{\sum_{i=1}^{N}\frac{\left(y_{i}-\hat{y}_{i}\right)^{2}}{N}} \tag{7}\]  
herein, the following definitions apply:  
* \(y_{i}\): Measured value at time instant i. In this case, it represents the ETVE in the Y direction obtained from the MIIM measurement;
* \(\hat{y}_{i}\): Estimated or predicted value at time instant i. This is the estimated E\({}_{\text{TVE}}\) in the Y direction, as calculated by the adjusted first-order model;
* \(N\): Number of data points; in this case, 100 MIIM results;
* Mean absolute error (_MAE_): Calculated as the sum of the absolute differences between the predicted values (calculated by the model) and the actual measured values divided by the sample size;
* Root mean square error (_RMSE_): The quadratic mean of the differences between the measured values and the predicted values. Table 9 presents the numerical values of the established metrics, comparing the measured data with the predictions generated by the first-order model for the two analyses. The comparison between the model's output and the measured data revealed a strong correlation, demonstrating that the model accurately predicts the machine's error variation, which is primarily influenced by the ambient temperature.