## 6 Working Principle of Volumetric Verification
### Influence of Improving Data Accuracy through Traditional or Weighted Trilateration  
In MT verification, the spatial LT distribution is strongly affected by the MT configuration and visibility of the reflector.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Laser Tracker** & **A** & **B** & **C** \\ \hline Maximum initial error (\(\upmu\)m) & 66.5 & 102.9 & 80.9 \\ Average initial error (\(\upmu\)m) & 33.5 & 54.1 & 40.1 \\ Minimum initial error (\(\upmu\)m) & 8.0 & 6.9 & 8.9 \\ Maximum final error (\(\upmu\)m) & 54.7 & 86.7 & 62.6 \\ Average final error (\(\upmu\)m) & 22.3 & 28.9 & 25.5 \\ Minimum final error (\(\upmu\)m) & 5.1 & 4.4 & 7.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Verification results in relation to laser tracker (LT) used.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Laser Tracker** & **A** & **B** & **C** \\ \hline Measurement volume & - & - & - \\ Horizontal (\({}^{\circ}\)) & \(\pm\)235\({}^{\circ}\) & \(\pm\)320\({}^{\circ}\) & \(\pm\)360\({}^{\circ}\) \\ Vertical (\({}^{\circ}\)) & \(\pm\)45\({}^{\circ}\) & 77\({}^{\circ}\)/\(-\)60\({}^{\circ}\) & 77.9\({}^{\circ}\)/\(-\)52.1\({}^{\circ}\) \\ Range (m) & 0-40 m & 0-30 m & 0–60 m \\ Measurement & - & - & - \\ IFM & Si & Si & No \\ ADM & No & Si & Si \\ Accuracy & - & - & - \\ Angular & 0.14 arcs & 3.5 \(\upmu\)m/m & 20 \(\upmu\)m + 5 \(\upmu\)m/m \\ Absolute accuracy & \(\pm\)10 \(\upmu\)m/m & \(\pm\)5 ppm & 16 \(\upmu\)m + 0.8 \(\upmu\)m/m \\ \hline \hline \end{tabular}
\end{table}
Table 3: Laser tracker specifications.  
To find out whether the new weighted trilateration is better than traditional trilateration reducing the increase of radial noise influences on final MT positioning errors, three LTs were positioned so as to create a spatial angle of approximately 36\({}^{\circ}\) (Figure 12), which is far from the ideal distribution of 90\({}^{\circ}\)[24].  
In this test, all LTs simultaneously measured the same points, avoiding the influences of thermal variation and SMR error when both techniques were compared, because the input data were the same for both. Once the points had been measured and the verification points of each LT had been created, two meshes of points were created. A weighted trilateration mesh was created using Spatial Analyzer software (SA 2015.02.25, New River Kinematic, Williamsburg, VA, USA, 1994), while traditional trilateration was carried out using Matlab algorithms (2013a, MathWorks, Natick, MA, USA, 1984).  
The results in Table 5 show that multilateration provides an average initial error similar to verification with only one LT when using both multilateration techniques. In the same way, the residual error obtained is similar. Consequently, bundle adjustment and trilateration have similar behavior in the verification of this MT.  
Although multilateration might improve the MT's accuracy, as shown by the synthetic results [14], poor relative positioning of LTs does not improve verification results using weighted trilateration. Therefore, the LT's positioning must be studied carefully to avoid unnecessary measurements, as it is the most influential factor.