## 5 Positioning Results
### Position Estimation  
By performing detection at all points in the plane and utilizing the channel characterization measurements, the average deviation along the \(x\) and \(y\) axes is determined. Additionally, the distance between the measured and the real point in the plane is calculated, resulting in an average error smaller than 3 cm. The obtained measurements are  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Straight Line** & \(\mathbf{V_{RMS}}\) & **Equation** \\ \hline
1 & \(V_{RMS}>14\) & \(V_{RMS}=-1.12\cdot d+59.41\) \\ \hline
2 & \(14\geq V_{RMS}>12\) & \(V_{RMS}=-0.85\cdot d+47.85\) \\ \hline
3 & \(12\geq V_{RMS}>10\) & \(V_{RMS}=-0.54\cdot d+33.99\) \\ \hline
4 & \(10\geq V_{RMS}>8\) & \(V_{RMS}=-0.57\cdot d+35.40\) \\ \hline
5 & \(8\geq V_{RMS}\) & \(V_{RMS}=-0.44\cdot d+28.71\) \\ \hline \end{tabular}
\end{table}
Table 4: Distance estimation from received signal strength on top photodiode.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Frame 1 (\%)** & **Frame 2 (\%)** & **Correlation (\%)** \\ \hline
**Measure 1** & 89.25 & 90.50 & 99.25 \\ \hline
**Measure 2** & 91.00 & 90.75 & 98.75 \\ \hline
**Mean** & 90.13 & 90.63 & 99.00 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results obtained to determine decoding code efficiency  
Figure 9: \(V_{rms}\) vs. distance curve derived from plane characterization, with fitted mathematical model lines.  
presented in the first row of Table 6. Furthermore, Figure 10 provides a clearer depiction of the positioning error for the correlated measurements across various points in the plane.  
Evaluating the effectiveness of position estimation using the side photodiode method is crucial. Channel characterization shows that information from at least three transmitters is consistently available at all points. By removing one transmitter, we can assess the accuracy of this method with only three transmitters and compare with the results obtained when calculating one transmitter's distance via side photodiodes.  
Table 6 presents the results from various iterations, showing that the average error is below 3 cm considering four transmitters. When removing one transmitter, the error increases to 3.30 cm. In scenarios with only two transmitters, from the top photodiode and estimating a third's distance (L\({}^{\prime}\)), the average error rises to 6.38 cm as shown on Table 7, nearly double the error with four transmitters but still acceptable given the challenging conditions. These results demonstrate the system's effectiveness and adaptability.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Luminaries** & **X Axis (cm)** & **Y Axis (cm)** & **Position (cm)** \\ \hline
4 & 2.48 & 1.28 & 2.95 \\ \hline
3 (L1 = 0) & 2.66 & 1.44 & 3.29 \\ \hline
3 (L2 = 0) & 2.56 & 1.25 & 3.07 \\ \hline
3 (L3 = 0) & 2.89 & 1.56 & 3.65 \\ \hline
3 (L4 = 0) & 2.55 & 1.50 & 3.20 \\ \hline Mean & 2.66 & 1.43 & 3.30 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Mean deviation of the position on the \(x\) and \(y\) axes.  
Figure 10: Error in the positioning of correlated measurements at different points on the plane.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Luminaries** & **X Axis (cm)** & **Y Axis (cm)** & **Position (cm)** \\ \hline L1\({}^{\prime}\)-L2-L3 & 4.55 cm & 4.44 cm & 6.71 cm \\ \hline L1\({}^{\prime}\)-L2-L4 & 5.09 cm & 0.93 cm & 5.41 cm \\ \hline L1\({}^{\prime}\)-L3-L4 & 3.67 cm & 5.06 cm & 6.69 cm \\ \hline L2\({}^{\prime}\)-L1-L3 & 8.08 cm & 1.11 cm & 8.25 cm \\ \hline L2\({}^{\prime}\)-L1-L4 & 7.13 cm & 5.81 cm & 9.66 cm \\ \hline L2\({}^{\prime}\)-L3-L4 & 2.74 cm & 5.69 cm & 7.58 cm \\ \hline L3\({}^{\prime}\)-L1-L2 & 4.66 cm & 3.58 cm & 6.55 cm \\ \hline L3\({}^{\prime}\)-L1-L4 & 7.56 cm & 3.45 cm & 8.50 cm \\ \hline \hline \end{tabular}
\end{table}
Table 7: Mean deviation of the position on the \(x\) and \(y\) axes for extreme conditions.