## 5 Conclusions  
A weak calibration method has been developed to improve the position and orientation accuracy of an indoor positioning system based on a PSD performing AoA measurements from modulated infrastructure lighting. The proposed calibration relies on genetic algorithms to estimate the calibration constants without a priori knowledge of the system parameters and requires only six measurement points within the positioning environment.  
The positioning errors using the proposed calibration process are not significantly degraded (below 2 cm) when compared to those obtained from a far more complex and costly formal calibration. The main specific contributions and features of the proposed weak calibration are:  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Error Using Calculated Height**} & \multicolumn{4}{c}{**Error Using Known Height**} \\ \cline{2-9}
**Indexes of the Emitters** & \multicolumn{2}{c}{**Mean Error**} & \multicolumn{2}{c}{**Std Error**} & \multicolumn{2}{c}{**Mean Error**} & \multicolumn{2}{c}{**Std Error**} \\ \cline{2-9}  & **Mean** & **std** & **Mean** & **std** & **Mean** & **std** & **Mean** & **std** \\ \hline
2–4 & 27.7 mm & 7.1 mm & 16.8 mm & 6.5 mm & 11.1 mm & 2.7 mm & 6.6 mm & 1.2 mm \\ \hline \hline \end{tabular}
\end{table}
Table 7: Mean and error of the 44 receiver positions using calculated and known height and nine calibration points in the case \(1\).  
* The parameters required for correcting the lens distortion with sufficient accuracy can be obtained using only observations on six different positions. This prevents the need for a long and complex formal calibration process that is normally carried out in a measurement bench, thereby increasing the risk of misalignments during deployment. Conversely, the simplified model is unable to account for large lens distortions when the emitters are significantly far away from the sensor.
* Small misalignments in the orientation of the PSD with respect to the plane of motion are automatically accounted for.
* The proposed procedure is robust against inaccuracies in the ground-truth data, yielding relatively low global positioning errors despite small but non-negligible errors in the reference positions.
* The calibration procedure enables positioning with only approximated information about the position of the emitters.  
The evaluation of the positioning system's performance using the proposed weak calibration took place by comparison with the results obtained for the same system following an established calibration procedure. The degradation of accuracy was reasonably minor and largely compensated for by the reduction of calibration complexity and cost. The performance is sufficient for a wide range of indoor positioning applications whose overall cost and scalability should be enhanced by the simplified calibration method proposed herein.  
Conceptualization, A.D.-L.-L.-C. and J.-L.-G.; data curation, A.D.-L.-L.-C., D.S.-M., A.I. and R.G.-V.; formal analysis, A.D.-L.-L.-C. and I.B.-M.; funding acquisition, [PERSON]-L.L.-G. and A.G.-V.; investigation, A.D.-L.-L.-C. and [PERSON]-L.-G.; methodology, A.D.-L.-L.-C. and [PERSON]-L.L.-G.; project administration, J.-L.L.-G.; resources, D.S.-M., [PERSON]-L.-G., A.G.-V. and I.B.-M.; software, A.D.-L.-L.-C., A.I. and [PERSON]-V.; supervision, [PERSON]-L.L.-[PERSON]; validation, A.D.-L.-L.-C. and D.S.-M.; visualization, A.D.-L.-L.-C.; writing, original draft, A.D.-L.-L.-C.; and writing, review and editing, A.D.-L.-L.-C., J.-L.-G. and [PERSON]-V. All authors read and agreed to the published version of the manuscript. This research received no external funding. This research was supported by the Spanish Research Program through the Network REPNIN+ (Ref.: TEC2017-90808-REDT) and the Indoor Location and Positioning System based on the Building Lighting project GUIA (Rej. SBPLY/19/180501/000049). [PERSON] thanks the Posdoctoral program of University of Alcala and GUIA project. The authors declare no conflict of interest.