## 3 Experiments and Discussions
### On-Machine Measurement  
To determine the volumetric error of the five-axis horizontal machine with wBZXYCt topology as shown in Figure 1, a five-standard sphere artefact is purposely designed as shown in Figure 2 with variation of the height of each fiducial. Each calibrated center position of the fiducial is listed in Table 1.  
In order to reduce the thermal effect on the fixture, the designed artefact is transformed and kept in a thermally controlled environment and measured by the on-machine probing system. According to the on-machine measurement system, the artefact was clamped in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ball** & **X (mm)** & **Y (mm)** & **Z (mm)** \\ \hline
1 & 59,31108 & \(-\)0.25560 & 14.99509 \\
2 & \(-\)58.74790 & 0.864597 & 14.29131 \\
3 & \(-\)59.13960 & 59.79868 & 19.31120 \\
4 & 59.49720 & 59.29094 & 24.40756 \\
5 & 28.48006 & 59.94329 & 24.75141 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Centre positions of the fiducials. Figure 5: Position deviations of the fiducials when changing the artefact from a horizontal to a 10-degree (O-\(H\)) position and into a vertical orientation (_V-H_). The error vectors are multiplied by 10,000 in the bottom plot. The top two graphs are position deviations in three directions with their uncertainties. the vertical position on the B-axis as shown in Figure 1. A number of tests were carried out by gradually increasing the rotary B- and C-axis in the measurement process. All the fiducials were probed to establish their center coordinates and the positions (\(x\), \(y\), \(z\), \(b\) and \(c\)) of the corresponding axes were also recorded. Considering the working volume of the measurement and machining conditions, a total of 165 standard sphere centers were measured at the locations of b = 0\({}^{\circ}\), c = 0\({}^{\circ}\); b = 0\({}^{\circ}\), c = \(\pm\)5\({}^{\circ}\); b = \(\pm\)5\({}^{\circ}\), c = 0\({}^{\circ}\); b = \(\pm\)10\({}^{\circ}\), c = 0\({}^{\circ}\); b = \(\pm\)5, c = \(\pm\)5 which was repeated three times in about two hours. As shown in Figure 6, the fiducial center distances in a five-fiducial artefact were compared with the distances in the fiducial-aided CAD (calibrated fiducials and the CAD of the designed surface [18]) by using the cloud point fitting algorithm. The tested positions of the fiducials are determined and transferred into a common frame using the mathematical model. Table 2 shows the compared results including the maximum value (\(r_{\text{max}}\)) and average values (\(r_{\text{avg}}\)) for a total of 10 distances in the artefact under different angular positions of the rotary axes. Value \(r\) is defined in Figure 6.  
As shown in Table 2, in the measured columns, it is clear that the deviations are very small--0.9 \(\upmu\)m at most when the measuring process is carried out with no rotation of the axes--which means that only three linear axes are used to perform the measurement. With the B-axis set to 0\({}^{\circ}\), the error increased with increasing angular positions of the C-axis and  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Position of the Rotary Axes** & \textbackslash{1} **Measured (\(\upmu\)m)** & \textbackslash{2} **Mathematic Model (\(\upmu\)m)** & \textbackslash{1} **–2 (\(\upmu\)m)** \\ \hline
**b** & **c** & \(\mathbf{r_{\text{max}}}\) & \(\mathbf{r_{\text{avg}}}\) & \(\mathbf{r_{\text{max}}}\) & \(\mathbf{r_  
reached 6.5 \(\upmu\)m. In this case, the volumetric errors resulting from the linear axes and errors from the C-axis are integrated into a fiducial when changing the position of the C-axis. It is interesting to note that the error only reaches 6.8 \(\upmu\)m when both angular positions of the rotary axes are combined. The results in the mathematical model suggest that the errors are slightly smaller than those in the measured datasets. This is due to the existing setup errors of the integrated measuring system.