## 4 Results
### System Parameters Computation  
To perform the calibration, the receiver was placed in a total of six positions in the coverage area of the two emitters. In this case, in each position the signal impact points of emitters 2 and 4 were obtained. The receiver positions used to obtain the parameters are shown in Figure 6. The points were chosen to try to cover as much of the area as possible (in the corners) and by placing some points near the center. A total of six points was chosen as a trade-off between using minimal points to calibrate (faster calibration process) and obtaining an acceptable solution. The number of parameters to obtain was 13 (six coordinates of the two emitters, plus the three Euler angles, plus the three intrinsic parameters and the distortion parameter). Using six receiver positions, we got 12 values of \((x_{i})\) and 12 \((y_{i})\) (using two emitters there are two times more impact points than receiver positions), so we used 24 equations to obtain 13 parameters. To know the receiver positions used in the calibration, the ground truth data obtained by the total stations were used. To emulate a real measurement in a real environment \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Index of Emitter** & 1 & 2 & 3 & 4 \\
**Modulation Frequency (kHz)** & 6 & 8 & 10 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Index and frequency modulation of the emitters. Figure 5: Positions of the emitters, receivers (green), and total stations in the test environment. Figure 6: Points used in the calibration process. where total stations are not available, a Gaussian random error with a standard deviation of 5 mm (value obtained experimentally after several measurements by usual measuring instruments compared to the total station) was added to these positions. It was also mandated that the receiver always moved in the same plane. To calibrate, it is necessary to know the rotation of the receiver at each point, and when measuring in a real environment there can always be small errors when positioning the receiver. For this reason, in the calibration process we fixed the rotation, with a value of \(-90^{\circ}\), when, in fact, the angles measured by the TS vary from \(-88^{\circ}\) to \(-92^{\circ}\) (which means that we had errors of \(\pm 2^{\circ}\) in the measurements that the calibration method must tolerate). To consider that possibly the PSD is not coplanar with the plane of movement (Figure 1), the calibration process had to obtain the Euler angles \(\alpha\), \(\beta\), and \(\gamma\) to correct those possible orientation errors, which were fixed in all positions. The range of values that these angles can take was limited to \(\pm 2^{\circ}\), since we estimated that this would be the maximum error that we would have between the two planes. The calibration procedure provides the coordinates of the emitters. A range of possible values has been set for the emitter coordinates with respect to those measured with the total station of \(\pm 2\) cm. This value was chosen because after performing some manual measurements and comparing them with the real values, the maximum error obtained when measuring the emitter coordinates manually was \(\pm 2\) cm. This value is higher than the one established in the receiver coordinates, as manual measurement of the ceiling increases the errors. The calibration process was set up so that the \(z\) coordinates of the emitters were the same. Finally, the calibration returned the parameters of the system PSD + lens, such as focal length \(f\), optical center \(C_{x}\) and \(C_{y}\), and radial distortion parameter \(k\). In summary, the input and output values of the calibration method for this specific case were:  
* The inputs:
* The coordinates of the six different positions of the receiver (Figure 6) with a deviation error of 5 mm in the receiver position (to emulate manual positioning). * For each receiver position, the impact points obtained from the signals of two emitters (number 2 and number 4) have been used.