## Appendix B Special Remarks on the Moving Head
### Remarks on the Performance of [PERSON]'s Method  
This basic performance analysis of the proposed calibration algorithm in conjunction with the usage of [PERSON]'s method as a non-linear solving algorithm is conducted as an example of the first iteration step, outlined in Section 4.1. The computational effort of the second step is one-third smaller due to one less iteration variable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Index** & **X in mm** & **Y in mm** & **Z in mm** & \(\sigma_{LT}\) **in mm** \\ \hline
1 & \(-\)306.4154 & \(-\)2032.1464 & \(-\)972.1941 & 0.00188634 \\ \hline
2 & \(-\)306.4316 & \(-\)2032.1124 & \(-\)972.2038 & 0.00220966 \\ \hline
3 & \(-\)306.2175 & \(-\)2032.1691 & \(-\)971.9397 & 0.00263319 \\ \hline
4 & \(-\)306.2847 & \(-\)2032.1653 & \(-\)971.9680 & 0.00199971 \\ \hline
5 & \(-\)306.2746 & \(-\)2032.1472 & \(-\)971.9647 & 0.00221153 \\ \hline
6 & \(-\)306.2349 & \(-\)2032.1175 & \(-\)971.9473 & 0.00211342 \\ \hline
7 & \(-\)306.2486 & \(-\)2032.1570 & \(-\)971.9434 & 0.00228444 \\ \hline
8 & \(-\)306.2201 & \(-\)2032.1406 & \(-\)971.9422 & 0.00236737 \\ \hline \(\mu\) & \(-\)306.290925 & \(-\)2032.144438 & \(-\)972.0129 & 0.002213208 \\ \hline \(\sigma\) & 0.079785928 & 0.019298765 & 0.107880628 & 0.000214366 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measurements of the moving head pose repeatability. The coordinates exist in the laser tracker coordinate system. \(\sigma_{LT}\) is the measurement uncertainty reported by the laser tracker. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Index** & _pan_ & _finepan_ & _tilt_ & _finetilt_ & **X in m** & **Y in m** & **Z in m** & \(e_{t}\) **in mm** & \(e_{u}\) **in mm** \\ \hline
1 & 192 & 202 & 45 & 84 & \(-\)1.7443 & 0.2477 & \(-\)0.3072 & 5 & 5 \\ \hline
2 & 185 & 246 & 55 & 235 & \(-\)1.6257 & 0.5788 & \(-\)0.2365 & 3 & 6 \\ \hline
3 & 173 & 29 & 62 & 0 & \(-\)1.5049 & 1.0239 & \(-\)0.1801 & 3 & 4 \\ \hline
4 & 160 & 209 & 65 & 190 & \(-\)1.3521 & 1.4272 & \(-\)0.0178 & 3 & 3 \\ \hline
5 & 150 & 226 & 58 & 10 & \(-\)1.2126 & 1.9556 & 0.1109 & 4 & 6 \\ \hline
6 & 153 & 73 & 50 & 73 & \(-\)1.3299 & 1.9238 & \(-\)0.0744 & 3 & 6 \\ \hline
7 & 160 & 36 & 52 & 166 & \(-\)1.4477 & 1.5623 & \(-\)0.1810 & 6 & 6 \\ \hline
8 & 172 & 22 & 50 & 241 & \(-\)1.6119 & 1.1009 & \(-\)0.3073 & 5 & 6 \\ \hline
9 & 180 & 236 & 44 & 190 & \(-\)1.7302 & 0.7689 & \(-\)0.3858 & 4 & 4 \\ \hline
10 & 189 & 52 & 35 & 0 & \(-\)1.8702 & 0.3832 & \(-\)0.4739 & 5 & 5 \\ \hline
11 & 160 & 170 & 45 & 225 & \(-\)1.5453 & 1.6221 & \(-\)0.2528 & 5 & 5 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline
**\(\mu\)** & & & & & & & 4.18 & 5.09 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline
**\(\sigma\)** & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reference points in moving head DMX Channel values, measured coordinates, and lower and upper position error between jogged and calculated reference points. The calibration process in Section 4 starts with \(N\) reference points. From these \(N\) reference points, \(N_{pairs}\) duplicate free pairs are created. \[N_{pairs}=\frac{N(N-1)}{2} \tag{10}\]  
From each pair, two Equations (11) and (12) are created, which form the function vector Equations (16) and (19). Equation (10) shows the dimension of the function vector \(\vec{f}\). \[dim(\vec{f})=2\cdot N_{pairs}=N(N-1) \tag{11}\]  
Since the Jacobian matrix is created from the function vector, the dimension of the function vector and the major dimension of the Jacobian are identical, and the minor dimension is equal to the number of iteration variables (three for the first iteration step and two for the second iteration step). Equation (11) shows the number of elements in the Jacobian matrix. \[N_{I}=3\cdot N_{pairs}=3\cdot N(N-1) \tag{12}\]  
The number of elements in the Jacobian matrix is relevant to the substitution of the symbolic values in the \(sympy\) implementation. The substitution is performed in every iteration step.