Step 3.
Location Estimation: After obtaining the points\(p_{1}^{l},\ p_{1}^{m},\ ...,\ p_{1}^{o},\ and\ CN\) (as shown in Fig. 6) still the unknown node \(U(x,y)\)may exist somewhere beyond the points \(p_{1}^{l},\ p_{1}^{m},\ ...,\ p_{1}^{o},\ and\ CN\). After getting the base distance value, now we calculate comprehensive distance value by using base distance value to localize the unknown node \({}^{\prime}U^{\prime}\). So, for location estimation of unknown node \({}^{\prime}U^{\prime}\), consider\(D_{i}\ (where\ D_{i}=D_{1},\ D_{2},\ \ldots,\ and\ so\ on)\) as a distance value in-between the anchor nodes \((A_{n}-K)\) and the respective intersecting points \(p_{i}\). Since the unknown node \({}^{\prime}U^{\prime}\)may be beyond the intersecting points, therefore we add the respective comprehensive distance \(e_{i}\) in \(D_{i}\), where\(e_{i}=e_{1},e_{2},\ \ldots,\ and\ so\ on\).\(\ \)Here \(e_{i}\) is a necessary comprehensive value to reach on a consensus from the anchor nodes with the calculated distance \(D_{i}\). The centroid CNis also considered as a reference point to participate in the location estimation. It implies that the unknown node \({}^{\prime}U^{\prime}\) may exist at a nominal distance, say \(e_{n}\), from \(CN(C_{x},\ C_{y})\). Now\(e_{i}\) updates with \(e_{n}\), therefore\(e_{i}=\ e_{1},e_{2},\ \ldots,and\ e_{n}\). Hence \(e_{i}\) is a necessary error required to calculate; such that\(\sum{e_{i}\approx 0}\). Consequently by considering error\(e_{i}\) in \(D_{i}\) a system of Euclidean distance between anchor nodes and the unknown node \(U(x,y)\) can be written by equation (13)-
\begin{equation} \begin{matrix}\left.\ \begin{matrix}\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=\left(D_{1}+e_{1}\right)^{2}\\ {(x-x_{2})}^{2}+{(y-y_{2})}^{2}={(D_{2}+e_{2})}^{2}\\ \begin{matrix}\vdots\\ {(x-x_{n-1})}^{2}+{(y-y_{n-1})}^{2}={(D_{n-1}+e_{n-1})}^{2}\\ \end{matrix}\\ \end{matrix}\right\}&(13)\\ \end{matrix}\nonumber \\ \end{equation}
where \(\left(n-1\right)=N(A_{n}-K)\)
Similarly, the distance equation between unknown node \({}^{\prime}U^{\prime}\) andCN can be written by equation (14)-
\({(x-C_{x})}^{2}+{(y-C_{y})}^{2}={(e_{n})}^{2}\) (14)
Now, subtract the equation (14) from every equation of (13) one by one, we get equation (15)-
\begin{equation} \begin{matrix}\left.\ \begin{matrix}2\left(C_{x}-x_{1}\right)x+2\left(C_{y}-y_{1}\right)y-2D_{1}e_{1}+\left(e_{n}^{2}-e_{1}^{2}\right)=\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{1}^{2}+y_{1}^{2}\right)+D_{1}^{2}\\ 2\left(C_{x}-x_{2}\right)x+2\left(C_{y}-y_{2}\right)y-2D_{2}e_{2}+\left(e_{n}^{2}-e_{2}^{2}\right)=\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{2}^{2}+y_{2}^{2}\right)+D_{2}^{2}\\ \begin{matrix}\vdots\\ 2\left(C_{x}-x_{n-1}\right)x+2\left(C_{y}-y_{n-1}\right)y-2D_{n-1}e_{n-1}+\left(e_{n}^{2}-e_{n-1}^{2}\right)=\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{n-1}^{2}+y_{n-1}^{2}\right)+D_{n-1}^{2}\\ \end{matrix}\\ \end{matrix}\right\}&(15)\\ \end{matrix}\nonumber \\ \end{equation}
Further divide system of equation (15) by \(\text{BL}^{2}\) to minimize a term \((e_{n}^{2}-e_{i}^{2})\) where BL is a border length of the region under considerations.
After dividing the equation (15) by \(\text{BL}^{2}\), we get the equation (16).
\begin{equation} \begin{matrix}\left.\ \begin{matrix}\frac{2\left(C_{x}-x_{1}\right)x}{\text{BL}^{2}}+\ \frac{2\left(C_{y}-y_{1}\right)y}{\text{BL}^{2}}-\ \frac{2D_{1}e_{1}}{\text{BL}^{2}}+\frac{e_{n}^{2}-e_{1}^{2}}{\text{BL}^{2}}=\ \frac{\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{1}^{2}+y_{1}^{2}\right)+D_{1}^{2}}{\text{BL}^{2}}\\ \frac{2\left(C_{x}-x_{2}\right)x}{\text{BL}^{2}}+\ \frac{2\left(C_{y}-y_{2}\right)y}{\text{BL}^{2}}-\ \frac{2D_{2}e_{2}}{\text{BL}^{2}}+\frac{e_{n}^{2}-e_{2}^{2}}{\text{BL}^{2}}=\ \frac{\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{2}^{2}+y_{2}^{2}\right)+D_{2}^{2}}{\text{BL}^{2}}\\ \begin{matrix}\vdots\\ \frac{2\left(C_{x}-x_{n-1}\right)x}{\text{BL}^{2}}+\ \frac{2\left(C_{y}-y_{n-1}\right)y}{\text{BL}^{2}}-\ \frac{2D_{n-1}e_{n-1}}{\text{BL}^{2}}+\frac{e_{n}^{2}-e_{n-1}^{2}}{\text{BL}^{2}}=\ \frac{\left(C_{x}^{2}+C_{y}^{2}\right)-\left(x_{n-1}^{2}+y_{n-1}^{2}\right)+D_{n-1}^{2}}{\text{BL}^{2}}\\ \end{matrix}\\ \end{matrix}\right\}&(16)\\ \end{matrix}\nonumber \\ \end{equation}
Since \(\text{BL}^{2}\)>> \(e_{1,2,\ldots,\ n}^{2}\) , the term\(\frac{e_{n}^{2}-e_{1,\ 2,\ \ldots,\ n-1}^{2}}{\text{BL}^{2}}\)