# 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
It can be remarked that the proposed scheme shows a good performance through iterations. It reaches an average RMSE of 23.15 m in 3 iterations for R = 35 m and an average RMSE of 14.29 m also in 3 iterations for R = 45 m.

### Range-Based Multi-Hop Localization over Irregular Topologies of Sensor Networks

This section analyzes the accuracy and iterations performance of the proposed RWNM scheme under anisotropic networks. A benchmark of 10 independent multi-hop networks

Figure 6: (**a**) \(RMSEavg\) and (**b**) \(Iter\_avg\) after running five iterative algorithms over 10 multi-hop isotropic networks with \(R=45\) m and \(nfe=0.3\). Outliers levels are increased by steps of 10% from 0% to 50%. Figure 7: \(RMSEavg\) vs. Iterations of algorithms using \(\Theta=20\%\) and \(nfe=0.1\) considering the average of 10 independent isotropic networks. (**a**) Corresponds to R = 35 m and (**b**) R = 45 m.

like that presented in Figure 1a is considered for testing. In all networks, it is known that there is at least one path between every pair of nodes around the circle shape. The analysis and tests for this section run also in the same way as that analyzed in Section 6.1. The first test consists of evaluating the error on position estimates and the number of iterations spent by the distributed algorithms. The stopping criterion is assumed equal in all algorithms (i.e., \(\tau=10^{-2}\)). Initial estimates average of the 10 networks is around 102.1 m. Using the combination \(nfe\) and R, 10 independent networks are created, and each one is run in the distributed algorithms to finally averaging the results (i.e., \(RMSEavg\) and \(Iter\_avg\)). Figure 8 shows the results for \(nfe\) = 0.1 and R = 35 m.

Despite having a large initial error on position estimates (i.e., \(\sim\)102.1 m), most algorithms (except the SOCP scheme) end up with a \(RMSEavg\) below to 22 m of error on position estimates without presence of outliers (i.e., \(\Theta\) = 0%). The combinations RELM+LM and DV-Hop+LM are linearly affected with increasing outliers. The combination RELM+SOCP is unaffected by outliers, but its \(RMSEavg\) is too high compared with the remaining four combinations. For example, when there are not outliers (i.e., \(\Theta\) = 0%) in the estimated distances between sensors, the best result of the estimated positions is obtained by the combination DV-Hop+LM reaching a \(RMSEavg\) = 15.24 m followed by the DV-Hop+RWNM scheme with 16.25 m. However, the accuracy on position estimates affects the former method linearly with the increase of outliers, providing a \(RMSEavg\) = 31.06 m at \(\Theta\) = 50%. On the other hand, the combination DV-Hop+RWNM is less affected by the presence of outliers. It ends up with a \(RMSEavg\) = 16.94 m for a \(\Theta\) = 50%. It should also be recognized that the best \(RMSEavg\)-\(Iter\_avg\) performance is obtained by this combination, as shown the Table 6 with only 4.15 iterations and a RMSE = 16.49 m, on average. Figure 8: (**a**) \(RMSEavg\) and (**b**) \(Iter\_avg\) after running five iterative algorithms over 10 multi-hop anisotropic networks with R = 35 m and \(nfe\) = 0.1. Outliers levels are increased by steps of 10% from 0% to 50%. The next step is to increase the noise level on distance estimates from \(nfe=0.1\) to \(nfe=0.3\) in each one of the 10 benchmark networks and carry out the test like in the previous step. Figure 9 shows that all combinations almost have the same behavior. The \(RMSEavg\) of the combination DV-Hop+RWNM slightly increases as the quantity of outliers also does. It starts with a \(RMSEavg=16.47\) m at \(\Theta=0\%\) and finishes with \(RMSEavg=17.19\) m at \(\Theta=50\%\). Furthermore, this combination gets the best trade-off between \(RMSEavg\) and \(Iter\_avg\) as summarized in Table 7.