## 6 Working Principle of Volumetric Verification
It is approximately 2 h. It is because LT should be located on the correct place in order to maximize the workspace of the MT to verify. Moreover, the LT requires as little as 30 min of heating time before measure. Additionally, a numerical control program (NC) taking into consideration characteristic of the MT to verify should be created to automatize data capture. Acquisition or data captured time is presented in Figure 8, 1 h, it is time required to measure all verification points of the NC created previously. This information is used to obtain approximation functions of each MT geometric error during identification process. Identification was carried out approximately in 20 min using optimization process (Section 5). Once it is finished, MT compensation can be carried out. Therefore, the MT verification required approximately 3 h 30 min. On analyzing the initial errors after thermal variations, the results displayed in Figure 9, which represent an error histogram of verification points, show that its current compensation reduces the maximum error from 246.3 to 197.7 \(\upmu\)m, the average volumetric error from 112.9 to 83.1 \(\upmu\)m, and the minimum error from 15.1 to 10.1 \(\upmu\)m. Therefore, it reduces the average positioning error of the MT by 26.5%. The current MT compensation method is based on compensation tables provided by manufacturers, which are not easy to access or use. Therefore, compensation was carried out using an NC post-processor. In the same way, as volumetric verification is a mathematical correction that cannot provide a complete physical meaning to the approximation function obtained, verification can be carried out when MT compensation software is activated (verification with compensation). Table 2 shows the results of the adequacy of approximation with and without previous compensation using simple polynomials of degree three as regression functions. When compensation prior to verification is studied, it can be observed that the maximum error is reduced by 40 \(\upmu\)m, while the average volumetric error is reduced by 30 \(\upmu\)m. In the same way, if \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Compensation Table**} & \multicolumn{3}{c}{**Error (\(\upmu\)m)**} \\ \cline{2-4}  & **Maximum** & **Average** & **Minimum** \\ \hline Compensation inactivated & 246.3 & 112.9 & 15.1 \\ Compensation activated & 197.7 & 83.1 & 10.1 \\ Verification without compensation & 221.5 & 57.4 & 15.1 \\ Verification with compensation & 105.5 & 47.1 & 4.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Verification results using Active Target. Figure 9: Error histograms before verification with MT compensation inactive (**a**) and active (**b**). compensation is inactive in data acquisition (verification without compensation), the average error reduction is reduced to 57.4 \(\upmu\)m. Therefore, volumetric verification improves the accuracy of the MT by 49%, whereas an improvement of 26% was obtained with the previous method. In the same way, if data with prior verification is used, the approximation functions obtained reduce the error to 58%. This is because the residual errors of previous compensation are the result of uncompensated geometric errors. Therefore, the MT has been compensated after volumetric verification, providing an increase of MT positioning accuracy. The histograms in Figure 10 show that the global error of the machine has been reduced. However, to find out how errors are distributed in the MT workspace, color maps such as that in Figure 11 are studied. It shows that volumetric verification provides a homogeneous correction on the whole of the MT workspace, avoiding areas of failure at the ends of the axes of movement. Among the non-geometric errors of the MT such as repeatability, deformations, and so on, one of the most relevant is the backlash error of the axis of movement. It is not introduced in the kinematic model of the machine.