## 5 Methods and Models
### Miner Tracking Algorithm  
In underground mines, tracking is still an unsolved problem as most of the applied techniques rely on RSS. The harsh environment and uneven surfaces of mine walls make RSS measurements highly vulnerable to errors. In this study, this effect was minimized using an RSS range-based weighted centroid approach by creating a tradeoff between accuracy and computational complexity. This algorithm utilizes RSS values of mobile nodes according to their logarithmic distances from SNs. Our model follows a linear relationship between RSSI and logarithmic distance given by  
\[RSSI(dBm)=A-10n\log_{10}(d) \tag{9}\]  
To enhance accuracy, the present system measures power from at least three reference SNs, resulting in a rough estimation of MN. Refinement is done using a triangulation approach of centroid localization. SNs with known coordinates \(N_{k}=N_{l}(x,y)\) should be defined over the entire system to help estimate the position of MN located within the coverage range of SNs, given by  
\[Pk(x_{k},y_{k})=(1/n)\sum_{l=1}^{n}N_{l}(x_{k},y_{k}) \tag{10}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pt ID** & _SeqF_ & **Pt _Num_** & **Node ID** & **Next** & \(\ldots\) & **Node ID** & **Next** \\ \hline
1 & 1 & 1 & 2 & 2 & \(\ldots\) & 2 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 6: _LocalizInfo_ packet, field name, and size for each column.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**1 Byte** & **1 Byte** & **1 Byte** & **Variable** \\ \hline RFD (0\(\times\)0a) & _Limit_ & _Type_ & _PaRm_ \\ \hline \hline \end{tabular}
\end{table}
Table 4: Frame format for SNs.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**1 Byte** & **1 Byte** & **2 Byte** & **1 Byte** & **Variable** & \(\ldots\) & **2 Byte** & **1 Byte** & **Variable** \\ \hline RN (0\(\times\)0b) & _Num_ & _Addr1_ & _Type 1_ & _PaRm 1_ & \(\ldots\) & _Addr n_ & _Type n_ & _PaRm n_ \\ \hline \hline \end{tabular}
\end{table}
Table 5: Frame format on RNs.  
This localization only considers the coverage range of SN, while ignoring RSS values of MN. These results are calculated based on equal distances for all reference nodes. Therefore, we considered advanced algorithms based on weighted centroids, in which each SN contributes by assigning a weight factor to the RSS. Position estimation of MN within the coverage range of each SNs is given by  
\[P_{i}(x,y)=\frac{\sum_{j=1}^{n}{{{A}}{P}\left({{w}_{ij.AP_{j}}(x,y)}\right)}}{ \sum_{j=1}^{n}{{{A}}{P}{w}_{ij}}} \tag{11}\]  
The object weight is calculated using  
\[w_{i}=1/{{d}_{ij}}^{g} \tag{12}\]  
Here, \({{d}_{ij}}\) is the distance between MN and SN as determined from RSSI, and \(g\) is the degree of contribution of each anchor node, usually set to 1. The normalized weight of \(w_{i}\) can be given by  
\[W_{i}=\frac{W_{i}}{\sum_{j=i}^{m}{{{W}}_{j}}}=\frac{\sqrt{{{\left({10^{RSSI_{i }/10}}\right)}^{g}}}}{\sum_{j=1}^{m}{\sqrt{{{\left({10^{RSSI_{j}/10}}\right)}^ {g}}}}} \tag{13}\]  
The estimated positions of the target node are  
\[X_{est}=\sum_{i=1}^{m}(W_{i}\times x_{i})Y_{est}=\sum_{i=1}^{m}(W_{i}\times y_ {i}) \tag{14}\]  
Equation (14) represents the estimated position of a target node. The benefit of using this technique is that it does not require any calculation of a path loss component or other parameters. Thus, it provides the advantages of low-complexity and high accuracy.