## 5 Performance Evaluations
### Accuracy
#### 5.2.1 Average Localization Error  
The average localization error is used to assess how accurate the estimated position obtained is. The ratio of the sum of the localization errors of all the localized UNNs in the ROI to the number of nodes determines the average error rate. The localization error for node \(i\) is calculated as follows:  
\[\text{error}_{\text{(i)}}=\sqrt{\left(\text{x}_{\text{i}}-\text{u}_{\text{i}} \right)^{2}+\left(\text{y}_{\text{i}}-\text{v}_{\text{i}}\right)^{2}} \tag{10}\]  
(x\({}_{\text{i}}\), y\({}_{\text{i}}\)) are the actual coordinates of the node discussed and (u\({}_{\text{i}}\), v\({}_{\text{i}}\)) are the estimated coordinates of the same node. Therefore, \(\text{error}_{\text{avg}}\) is given by the average localization error equation for the total N sensor nodes. \[\text{error}_{\text{avg}}=(\text{ }\sum_{\text{i=1}}^{\text{N}}\text{error}_{ \text{(i)}})/\text{N} \tag{11}\]  
However, in order to compare the average localization error in each path planning model discussed, performance testing is first performed in an environment with 250 UNNs with a CR of 12.5 m. The resolution value is taken as \(R\) = 1 and the std of the noise as \(\sigma\) = 7. Figure 11 shows the performance of path planning models for 50 simulations in terms of the localization error. As shown in the figures, the models' error performances obtained are close to each other. For this reason, to establish more accurate error analysis, statistical characterization of localization errors of path planning models has been made. Tables 2 and 3 show the statistical characterization of different static path planning models for WCL and APT, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Value** & **Unit** \\ \hline Network size & \(S\) & \(100\times 100\) & m\({}^{2}\) \\ Number of UNNs & \(N\) & 250 & - \\ Number of MAs & \(M\) & 1 & - \\ Resolutions & \(R\) & 0.5, 0.75, 1, 1.5, 2, 2.5 & - \\ Path loss exponent & \(\gamma\) & 3.3 & - \\ Std. dev. of noise & \(\sigma\) & \(3,5,7,9\) & - \\ Power loss at \(d_{0}\) & \(P_{L}(d_{0})\) & 55 & dB \\ Reference point & \(d_{0}\) & 1 & m \\ Transmission power & \(P_{trans}\) & \(-20<P_{trans}<10\) & dBm \\ Simulation run & - & 50 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 11: Localization errors of all mobile path planning models in (**a**) WCL, and (**b**) APT. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Stat. Char.** & **Hilbert** & **LMAT** & **H** & **M** & **Z** & **NHex** \\ \hline mean & 3.9088 & 3.6584 & * 3.6089 & 3.9644 & 3.8419 & 3.6247 \\ median & 3.9287 & 3.6673 & * 3.6010 & 3.9704 & 3.8347 & 3.6406 \\ std & 0.1358 & 0.1217 & 0.1205 & * 0.1144 & 0.1249 & 0.1224 \\ var & 0.0184 & 0.0148 & 0.0145 & * 0.0131 & 0.0156 & 0.0150 \\ min error & 3.5992 & * 3.3378 & 3.3948 & 3.6854 & 3.6056 & 3.3737 \\ max error & 4.2907 & 3.8806 & * 3.8559 & 4.2104 & 4.2000 & 3.8576 \\ \hline \hline \end{tabular}  
* is the best value.  
\end{table} 
Table 2: Statistical characterization of all mobile path planning models in terms of localization error for WCL. As can be seen in Table 2, according to most statistical evaluation criteria, the H-curve path planning model showed the best error performance compared to other models. After this model, the best error results are obtained with the proposed NHexCurves and LMAT models, respectively. While the other models' error performances are similar, HILBERT model is noticeably worse than the other models. If we pay attention to the path trajectories of the H-curve and LMAT models in Figure 4, it is seen that the MA travels the boundaries of the ROI. Hence, in these models, MAs share the beacon signal at the borders. This is why these models localize UNNs with less error. In the proposed model NHexCurves, beacons are shared on both sides of the ROI. At the same time, since there are too many turns in this model, the number of beacons increases with the shared beacons when they hit the obstacle.