# A Weighted and Distributed Algorithm for Range-Based Multi-Hop Localization Using a Newton Method
## 4 A Robust and Distributed Localization Algorithm Based on the Newton Method
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(RMSEavg\)** **(m)** & \(Iter\_avg\) \\ \hline RELM+SOCP & 37.89 & 15.35 \\ RELM+LM & 28.33 & 10.06 \\ DV-HOP+DWDSCL & 19.26 & 11 \\ DV-HOP+LM & 23.65 & 8.08 \\ DV-HOP+RWNM & **16.49** & **4.15** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(RMSEavg\) and \(Iter\_avg\) at different levels of outliers \(\Theta\) from 0% to 50%, R = 35 m and \(nfe=0.1\) under anisotropic networks, the best performances in RMSE average and iteration average are denoted in bold face. Figure 9: **(a)**\(RMSEavg\) and **(b)**\(Iter\_avg\) after running five iterative algorithms over 10 multi-hop anisotropic networks with R = 35 m and \(nfe=0.3\). Outliers levels are increased by steps of 10% from 0% to 50%. Another interesting test to analyze consists of increasing the radio range in sensor nodes; so the radio range is set to R = 45 m with \(nfe=0.1\) in all sensor nodes in the 10 independent benchmark networks. Figure 10a shows the accuracy of the estimated positions for this case. As expected, increasing the radio range improves the accuracy of most algorithms at \(\Theta=0\%\). However, as an outlier could be present in a range measurement, it will affect more neighboring sensors in the estimation process, and most algorithms tend to increase their errors as outliers do. The proposed scheme RWNM starts with a \(RMSEavg=16.29\) m at \(\Theta=0\%\) and finishes with a \(RMSEavg=16.39\) m at \(\Theta=50\%\). Table 8 shows that the combination DV-Hop+RWNM continues with the best results for accuracy and iterations with a \(RMSEavg=16.24\) m and 5.33 iterations on average. The final step uses the 10 benchmark networks with R = 45 m and \(nfe=0.3\). Even though the noise level has increased in distance measurements, the final results in both analyzed parameters (i.e., \(RMSEavg\) and \(Iter\_avg\)) remain close similar to the last step as depicted Figure 11. The combinations DV-Hop+RWNM continue with the best accuracy performance with the minimum number of iterations as depicted in Table 9. \begin{table}
\begin{tabular}{c c c} \hline
**Algorithms** & \(RMSEavg\) **(m)** & \(Iter\_avg\) \\ \hline RELM+SOCP & 45.41 & 13.75 \\ RELM+LM & 31.87 & 11.03 \\ DV-HOP+DWDSCL & 19.37 & 10.56 \\ DV-HOP+LM & 26.85 & 9.40 \\ DV-HOP+RWNM & **16.24** & **5.33** \\ \hline \end{tabular}
\end{table} 
Table 8: \(RMSEavg\) and \(Iter\_avg\) at different levels of outliers \(\Theta\) from 0% to 50%, R = 45 m and \(nfe=0.1\) under anisotropic networks, the best performances in RMSE average and iteration average are denoted in bold face. Figure 10: **(a) \(RMSEavg\)** and **(b) \(Iter\_avg\)** after running five iterative algorithms over 10 multi-hop anisotropic networks with R = 45 m and \(nfe=0.1\). Outliers levels are increased by steps of 10% from 0% to 50%. Figure 12 depicts the relation between the \(RMSEavg\) and number of iterations considering two radio ranges for anisotropic networks. It can be observed that the proposed scheme provides the best accuracy and iterations performance. It can be appreciate that the DV-Hop+RWNM algorithms spends only three iterations to reach a \(RMSEavg\) of 17.01 m and 16.2 m for R = 35 m and R = 45 m, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(RMSEavg\) **(m)** & \(Iter\_avg\) \\ \hline RELM+SOCP & 46.22 & 14.01 \\ RELM+LM & 34.68 & 11.30 \\ DV-HOP+DWDSCL & 19.90 & 13.53 \\ DV-HOP+LM & 29.65 & 9.96 \\ DV-HOP+RWNM & **16.75** & **5.2** \\ \hline \hline \end{tabular}
\end{table} 
Table 9: \(RMSEavg\) and \(Iter\_avg\) at different levels of outliers \(\Theta\) from 0% to 50%, R = 45 m and \(nfe=0.3\) under anisotropic networks, the best performances in RMSE average and iteration average are denoted in bold face. Figure 11: (**a**) \(RMSEavg\) and (**b**) \(Iter\_avg\) after running five iterative algorithms over 10 multi-hop anisotropic networks with \(R=45\) m and \(nfe=0.3\). Outliers levels are increased by steps of 10% from 0% to 50%.