## 4 Results
### Comparison with Formal Calibration  
Once the parameters were obtained from the genetic algorithm, with the incidence points from the two emitters, the positions of the sensors were calculated. We proceeded in a similar way to [47] in order to compare our method with formal calibration. When receiving signals from two emitters, it is possible to know the angles and heights along the plane of the emitters and the plane of movement of the receiver. If these heights are obtained externally with high accuracy, or as in this case, we know them because we have performed the calibration, the positioning results can be improved with respect to cases where the height is calculated in the positioning process. Therefore, both cases will be analyzed: when the height is calculated and when the height is known. Besides, it should be noted that in this section we focus on the accuracy of the position measurement, i.e., only taking into account systematic errors. For this purpose, a high integration time has been used in order to reduce random errors as much as possible. The integration time was configured at 15 s, as it was experimentally demonstrated to provide sufficient absorption of random variations. The study of the accuracy due to random errors was performed in [47]. Since errors due to miscalibration affect only to the measurement accuracy, there is no point in analyzing the precision in this work. Figure 7 shows the calculated positions and the true positions along with the cumulative distribution function (CDF) of the positioning error, measured as the Euclidean distance between the actual point and the calculated point. Figure 8 shows graphically the positioning error in different colors as a function of the error on the receiver coordinates. It can be seen how the errors increase slightly as the receiver moves away from the emitter positions. This effect may be due to the calibration process, which only considers one distortion parameter: the further the receiver moves away from the emitters, the less distortion is well-corrected. As we used four emitters (Figure 5), there were different alternatives to choosing two pairs of emitters. In this work, besides this first case considering one emitter in the center and one in the corner, the same procedure was performed with a different configuration--two emitters placed in the corners. With these two cases we cover practically all the possibilities, since due to the symmetry of the setup, the results of other possible configurations would be similar. Figure 7: Positioning results using emitters 2 and 4. (**a**) Estimated and true receiver positions using calculated height; (**b**) CDF of the positioning error. Table 2 shows the mean, standard deviation, and maximum values of the positioning errors of both case studies. Table 3 shows the same values for a formal calibration obtained by following the procedure given in [47]. It can be seen that the position errors in both cases are low and not very different from those obtained by formal calibration procedure. Mean errors are about 6-13 mm higher when using the genetic algorithms calibration procedure. The accuracy measurement results for the rotation angle are shown similarly to in [47]. To obtain the values, the receiver was located under emitter 2 and was rotated by a total of 34 degrees. The mean, standard deviation, and maximum error in the determination of the rotation angles are detailed in Table 4. Figure 9 shows the calculated rotation angles and the true rotation angles along with the cumulative distribution function (CDF) of the rotation angle error. As can be seen, the error in the measurement of the rotation was below 0.5\({}^{\circ}\) in 80% of cases, with an average value of 0.241\({}^{\circ}\). Errors are also shown for formal calibration was performed. It can be seen that when using the weak calibration, the errors increased slightly; even so the average error was under 0.25\({}^{\circ}\).