## 1 Introduction
LaserTRACER adopts multilateration for spatial coordinate measuring, as shown in Figure 3. The measurement equation is as below [12]: \[l_{\mathrm{ij}}+l_{\mathrm{oj}}+w_{\mathrm{ij}}=s_{\mathrm{j}}\sqrt{\left(x_{ \mathrm{i}}-x_{0\mathrm{j}}\right)^{2}+\left(y_{\mathrm{i}}-y_{0\mathrm{j}} \right)^{2}+\left(z_{\mathrm{i}}-z_{0\mathrm{j}}\right)^{2}}\] (2) where \(\left(x_{i},\,y_{i},\,z_{i}\right)\) is the coordinate of the \(i\)-th measurement point; \(\left(x_{0\mathrm{j}},\,y_{0\mathrm{j}},\,z_{0\mathrm{j}}\right)\) is the position of the \(j\)-th LaserTRACER; \(l_{ij}\) is the measured length when the target stops at the \(i\)-th measurement point; \(l_{0\mathrm{j}}\) is the initial distance between the target and the \(j\)-th LaserTRACER; \(s_{j}\) is the scale factor for \(j\)-th LaserTRACER, which can be determined through calibration; and \(w_{ij}\) is the residual error. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{**Working Range (m)**} & \multirow{2}{*}{**Angular Accuracy**} & \multicolumn{2}{c}{**Distance Accuracy**} \\ \cline{3-4}  & & & **ADM** & **IFM** \\ \hline Leica AT960 & 40/160 & \(\pm\)15 μm + 6 μm/m & \(\pm\)0.5 μm/m (AIFM) \\ Leica AT930 & 160 & \(\pm\)15 μm + 6 μm/m & \(\pm\)10 μm & \(\pm\)0.4 μm + 0.3 μm/m \\ Leica AT901 & 50/160 & \(\pm\)15 μm + 6 μm/m & \(\pm\)10 μm & - \\ Leica AT402 & 320 & \(\pm\)15 μm + 6 μm/m & \(\pm\)10 μm & - \\ Leica AT401 & 320 & \(\pm\)15 μm + 6 μm/m & \(\pm\)10 μm & - \\ Leica LTD600 & 40 & \(\pm\)25 μm & \(\pm\)25 μm & \(\pm\)10 μm \(\pm\)0.5 μm/m \\ API Tracker3 & 30/80/\(\sim\)120 & 3.5 μm/m & \(\pm\)15 μm & whthin \(\pm\)0.5 μm/m \\ API Radian & 40/100/\(\sim\)160 & 3.5 μm/m & \(\pm\)10 μm & whthin \(\pm\)0.5 μm/m \\ API Omnitrac2 & 160/200 & 3.5 μm/m & \(\pm\)15 μm & - \\ FARO ION & 30/40/55 & 20 μm + 5 μm/m & 16 μm + 0.8 μm/m & 4 μm + 0.8 μm/m \\ FARO Vantage & 30/60/80 & 20 μm + 5 μm/m & 16 μm + 0.8 μm/m & - \\ \hline Etalon & \multirow{2}{*}{0.2–20} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & 0.2 μm + 0.3 μm/m \\ LaserTRACER & & & (Measuring Uncertainty, \(k\) = 2) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of measurement uncertainty for different auto-tracking laser interferometers (ATLIs) (the value is from manufacturer catalogs). (Note: ADM = Absolute Distance Measurement; IFM = Interferometer; AIFM = Absolute Distance Measurement + Interferometer). The error terms of machine tool can be measured by multilateration combined with error model (or error mapping) of the machine tool [20; 21]. There is a condition that should be stratified for using multilateration and LaserTRACER to measure the spatial coordinate of the target [13; 22]:  
\[5\times m+3\times n\leq m\times n \tag{3}\]  
where \(m\) represented the number of used LaserTRACER and \(n\) represented the number of the measurement points. From Equation (3), we can know that the equation can be solved if and only if \(m\geq 4\) and \(n\geq 20\). Please note that the measurement points should be independent of each other. ISO 230-2 states that the test result should be corrected according to the measurement uncertainty. ISO 230-9 also shows a detailed explanation and ISO 230-2 gives two examples. The measurement uncertainties come from the measuring instrument, the compensation of the machine tool temperature, the environmental temperature variation error, and the misalignment of the measuring instrument.