# Relational Positioning Method for 2D and 3D Ad Hoc Sensor Networks in Industry 4.0
## 4 Simulations and Results
### The 3D Evaluation
This analysis was carried out for \(n=43\) randomly deployed nodes in the tetrahedron illustrated in Figure 5b, with \(AN_{1}\) to \(AN_{4}\) deployed as stated before and \(AN_{5}\), \(AN_{6}\), \(AN_{7}\) deployed at the centroid of planes confirmed by the corresponding sets (\(AN_{1}\), \(AN_{2}\), \(AN_{3}\)), (\(AN_{2}\), \(AN_{3}\), \(AN_{4}\)), and (\(AN_{1}\), \(AN_{2}\), \(AN_{4}\)), with \(d=10\). As anticipated, when we increase from 4 ANs to 5 ANs, we can see improvements from approximately 5% to 12% for all the cases reported. Additionally, when we increase from 5 ANs to 6 ANs, improvements from 2% to 6.6% are achieved. Finally, when we increase from 6 ANs to 7 ANs, improvements from 1.6% to 2.84% are observed. Simulations were performed by \begin{table}
\begin{tabular}{c c c c c} \hline \(r_{0}\)**(m)** & **2** & **2.5** & **3.333** & **5** \\ \hline \(h=10\) m & 0.216 & 0.1869 & 0.204213 & 0.2275 \\ \(h=11\) m & 0.224 & 0.196075 & 0.218312 & 0.262 \\ \(h=12\) m & 0.23 & 0.2002 & 0.220778 & 0.2623 \\ \(h=13\) m & 0.22356 & 0.193516 & 0.233377 & 0.2672 \\ \hline \end{tabular}
\end{table} 
Table 4: Error (\(\varepsilon/d\)) analysis for different tetrahedron (4 ANs) heights \(h\) and transmission ranges \(r_{0}\). \begin{table}
\begin{tabular}{c c c c c} \hline \(r_{0}\)**(m)** & **2** & **2.5** & **3.333** & **5** \\ \hline \(x_{3}=d\) sin\(\frac{\pi}{3}\) & 0.2160 & 0.1869 & 0.2042 & 0.2275 \\ \(x_{3}=d\) sin\(\frac{\pi}{3}\) + 1 & 0.2129 & 0.1899 & 0.2080 & 0.2364 \\ \(x_{3}=d\) sin\(\frac{\pi}{3}\) + 2 & 0.2232 & 0.2067 & 0.2157 & 0.2269 \\ \(x_{3}=d\) sin\(\frac{\pi}{3}\) + 3 & 0.2160 & 0.1821 & 0.1997 & 0.2370 \\ \(x_{3}=d\) sin\(\frac{\pi}{3}\) + 4 & 0.2106 & 0.1879 & 0.1927 & 0.2273 \\ \(x_moving \(AN_{5}\) on the plane conformed by the set \((AN_{1}\), \(AN_{2}\), \(AN_{3})\), and it was observed that the achieved improvement decreases as the \(AN_{5}\) approaches \(AN_{1}\), \(AN_{2}\), or \(AN_{3}\). A similar experiment was conducted regarding \(AN_{6}\) and \(AN_{7}\), and similar behavior was observed. For these scenarios, the entropy was also calculated, and results were very similar to those reported in Table 4 for \(L=1\). Note that when the number of ANs increases, there is an improvement in the location. However, for a given reachability ratio \(r_{0}\), the further increase in the number of ANs provides marginal improvement. Thus far, we have focused on evaluations for indoor scenarios, which are frequently used on Industry 4.0 applications. However, there are some applications in smart industries and sensor networks in which outdoor environments have to be considered. Since the  
Figure 10: Error \((\varepsilon/d)\) analysis for different transmission ranges \(r_{0}\) for four to seven ANs with \(L=1\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(r_{0}\)**(m)** & **2** & **2.5** & **3.333** & **5** \\ \hline \(H_{L=1}\) & 2.8 & 2.66 & 2.86 & 3.04 \\ \(H_{L=1/2}\) & 4.94 & 4.7112 & 4.78 & 5.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Entropy analysis for 4 ANs with different transmission ranges, \(r_{0}\), and discretization unit \(L\). Figure 9: Error analysis for different values of the discretization unit \(L\) and different transmission ranges \(r_{0}\), using four ANs in a 3D scenario, for \(d=h=10\) m.  
proposed relational positioning method does not rely on range estimation, there is no need to consider signal impairments associated with outdoor scenarios. However, the scenario size \(d\), as well as the transmission range r0, have to be increased in order to assess outdoor networks. As shown in Figure 11, we evaluated the performance of the relational positioning method for \(d\) = 20 m, 30 m, and for transmission ranges \(r_{0}\) from 6 m to 12 m for the basic set up, \(n\) = 43, \(L\) = 1, and regular deployment of ANs (\(AN_{1}\)(0,0,0), \(AN_{2}\)(\(d\),0,0), \(AN_{3}\)(\(d\) cos\(\frac{\pi}{3}\), \(d\) sin\(\frac{\pi}{3}\), 0) and \(AN_{4}\)(\(\frac{d}{2}\), \(\frac{d}{2}\) tan\(\frac{\pi}{6}\), \(d\))).