## 5 Performance Evaluations
### Accuracy
#### 5.2.1 Average Localization Error
Therefore, the localization error of the proposed model is also low. The HILBERT model does not reach the borders of the ROI. Thus, a higher error rate is obtained. In Table 3, the same test is performed using the APT localization method. Generally, it is observed that the performance of path models improved when this method is used and this is the expected result. Since accuracy is the priority in APT, three UNNs closest to the beacon are considered. Therefore, it is possible to achieve higher accuracy in results. In this scenario, although the path planning models' error performances are similar, the H-curve gives the best result, and the Z curve contains slightly more error than the others. The reason why the Z curve performs the worst in this scenario is that the resolution is \(R=1\), and the localization method is APT. As mentioned, APT uses the three closest beacons for localization. However, there are places with long distances between two beacons in the Z-curve trajectory. Therefore, the resolution should be low for the Z-curve to give good results in APT. This situation can be seen in Figure 12b. In this study, the average localization error variation with two variables given in Table 1 is also investigated separately: resolution (\(R\)) and std (\(\sigma\)). Firstly, 250 UNNs simulations with different resolutions ranging from 0.5 to 2.5 are performed. Figure 12 shows the average localization error according to the resolution values for path models and two different localization algorithms. As can be seen from Figure 12a, the proposed NHexCurves model showed good error performance at low-resolution values for the WCL method. The proposed model stands out compared to other models, especially with a value of \(R=0.5\). As is known, WCL takes the weighted average of the beacons in the coverage area to determine the locations of UNNs. If paying attention, there are many turns in the proposed model. Beacons are also published on these turns. This increases the probability of finding more beacons than the uniformly distributed ROI for the WCL method, especially when the resolution decreases. Therefore, the performance of the proposed model can be higher at low resolutions. After \(R=1\), the H-curve and LMAT models showed the best error performance. NHexCurves comes after these models. The performance of the H-curve and LMAT models are similar. In general, error values are obtained slightly higher due to the WCL method. The worst result is again achieved with the HILBERT model. Figure 12b shows the results of the same scenario for the APT method. Here the error values are generally similar, and far fewer errors are obtained than in the WCL. In the proposed model, the distance between the beacons is less than the other models. This is because NHexCurves contains more \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Stat. Char.** & **Hilbert** & **LMAT** & **H** & **M** & **Z** & **NHex** \\ \hline mean & 1.4268 & 1.4954 & * 1.3347 & 1.4909 & 1.6815 & 1.4925 \\ median & 1.4077 & 1.5029 & * 1.3214 & 1.4644 & 1.6637 & 1.4644 \\ std & 0.1784 & 0.1611 & * 0.1493 & 0.1889 & 0.1767 & 0.1738 \\ var & 0.0318 & 0.0260 & * 0.0223 & 0.0357 & 0.0312 & 0.0302 \\ min error & * 0.9783 & 1.0489 & 1.0091 & 1.0956 & 1.2005 & 1.1397 \\ max error & 1.9102 & 1.8845 & * 1.6325 & 1.8920 & 2.0494 & 1.8749 \\ \hline \hline \end{tabular} \({}^{*}\) is the best value.  
\end{table} 
Table 3: Statistical characterization of all mobile path planning models in terms of localization error for APT. triangles in different sizes than other models. For this reason, NHexCurves show high performance at low resolutions. Additionally, interesting results have been encountered in this scenario. It has been observed that the LMAT model, which shows stable results after \(R=1\), gives higher localization error at resolution values lower than this resolution value compared to other models.