# 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
Clearly, the accuracy performance of the combination algorithms RELM [35] +LM [36] and DV-Hop+LM are linearly affected with the increase of outliers on the range measurements among sensor nodes. However, without presence of outliers, the best accuracy on position estimates is obtained by the DV-Hop+LM algorithm with a \(RMSEavg=21.83\) m. The worst \(RMSEavg\)-\(Iter\_avg\) performance is carried out by the combination RELM [35] +SOCP [34; 37] algorithms. This is because the SOCP algorithm is highly dependent on the number and location of the anchors. On average, the DV-Hop+DWDSCL [38] obtains the best accuracy performance with 23.36 m followed by the combination DV-Hop-RWNM with 24.04 m. Moreover, this last combination outperforms others in rate of convergence with only 4.25 iterations, in average, which implies more saving energy for the entire network as shown Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(RMSEavg\) **(m)** & \(Iter\_avg\) \\ \hline RELM+SOCP & 42.15 & 15.63 \\ RELM+LM & 33.98 & 8.7 \\ DV-HOP+DWDSCL & **23.36** & 12.81 \\ DV-HOP+LM & 30.70 & 9.01 \\ DV-HOP+RWNM & 24.04 & **4.25** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(RMSEavg\) and \(Iter\_avg\) at different outliers values using R = 35 m and \(nfe=0.1\), the best performances in RMSE average and iteration average are denoted in bold face. Figure 3: (**a**) \(RMSEavg\) and (**b**) \(Iter\_avg\) after running five iterative algorithms over 10 multi-hop isotropic networks with R = 35 m and \(nfe=0.1\). Outliers levels are increased by steps of 10% from 0% to 50%. Now, keeping both the position set of unknown sensors and anchors positions for each one of the 10 benchmark networks and also taking the same set of range connections (i.e., R = 35 m), the distance errors among neighboring sensors are now affected with \(nfe=0.3\), and the results are depicted in Figure 4.

The results show a \(RMSEavg\)-\(Iter\_avg\) behavior very similar to the previous one. The DV-HOP+RWNM scheme still manages to be lower in iterations with 4.06, in average. The average results are shown in Table 3.

On the other hand, increasing the radio range from R = 35 m to R = 45 m, in each sensor node, produces more gathered information of neighboring sensors, and it also presents a reduction in the number of hops between anchors and unknown sensors. Both last effects tend to reduce error on position estimates. Figure 5 shows results with \(nfe=0.1\) and R = 45 m.

It is clear that algorithms reduce the error on position estimates, on average, as shown in Table 4. Figure 5 demonstrates how errors on position estimates tend to increase when the quantity of outliers also increase, as expected. Without outliers (i.e., \(\Theta=0\%\)) on distance estimates, the DV-Hop+LM has the best accuracy performance with \(RMSEavg=9.72\) m, but it increases its error as outliers do. On the other hand, due to the weighted function in the their minimization process, the DV-HOP+DWDSCL and DV-HOP+RWNM schemes are less affected in the \(RMSEavg\), and the RWNM approach requires around 5.05 iterations, on average, to obtain the best \(RMSEavg\) result of 14.78 m.

Figure 4: (**a**) \(RMSEavg\) and (**b**) \(Iter\_avg\) after running five iterative algorithms over 10 multi-hop isotropic networks with R = 35 m and \(nfe=0.3\). Outliers levels are increased by steps of 10% from 0% to 50%. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(RMSEavg\) **(m)** & \(Iter\_avg\) \\ \hline RELM+SOCP & 42.25 & 16.01 \\ RELM+LM & 36.33 & 9.1 \\ DV-HOP+DWDSCL & **24.04** & 9.7 \\ DV-HOP+LM & 33.17 & 10.16 \\ DV-HOP+RWNM & 24.84 & **4.06** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(RMSEavg\) and \(Iter\_avg\) at different outliers values using R = 35 m and \(nfe=0.3\), the best performances in RMSE average and iteration average are denoted in bold face. Figure 7 illustrates the behavior of the \(RMSEavg\) versus number of iterations considering two radio ranges.