In this section, we illustrate the wide applicability of the proposed robust method using synthetic (inter)arrival time data from three different localization techniques: Toa, Tdoa and Tdst. The performance is evaluated using the localization error

$$\begin{aligned} \Delta ({\widehat{\varvec{x}}})~=~||\varvec{x}-{\widehat{\varvec{x}}}|| \end{aligned}$$

where \(\varvec{x}\) is the node location of interest.

We observe \(n~=~100\) measurements from (7). The Los distribution is

$$\begin{aligned} p_{\textsc {los}}(s,\varvec{z})=\underbrace{{\mathcal {N}}\left( \varvec{\mu }(s,\varvec{\theta }_\circ ),\sigma _{\textsc {Los}}^2 {\mathbf {Q}} \right) }_{p_{\textsc {los}}(\varvec{z}|s)}~\underbrace{{\mathcal {U}}(s)}_{p_{\textsc {los}}(s)}, \end{aligned}$$

(17)

where adjacent timing measurement errors have a correlation structure given by \({\mathbf {Q}}\). The uniform distribution \({\mathcal {U}}(s)\) draws \(s\) from a set \({\mathcal {S}}\). For the corrupting Nlos distribution, we use

$$\begin{aligned} p_{\textsc {nlos}}(s,\varvec{z})=\underbrace{{\mathcal {E}}(\varvec{\mu }(s,\varvec{\theta }_\circ )+\mu _{\textsc {Nlos}})}_{p_{\textsc {nlos}}(\varvec{z}|s)}~\underbrace{{\mathcal {U}}(s)}_{p_{\textsc {nlos}}(s)}, \end{aligned}$$

(18)

so that measurements are drawn from an exponential distribution with bias *μ*_{Nlos}. We set *σ*_{Los} = 3 ns and *μ*_{Nlos} = 75 ns. Unless otherwise specified, the unknown corruption fraction is set to \(\epsilon = 15\%\).

*Remark:* The code for the experiments is available at [23].

### 4.1
Toa: two-way ranging anchor nodes

Consider a wireless network consisting of \(N~=~8\) nodes as shown in Fig. 1a. Since the Toa measurements are uncorrelated, we have that \({\mathbf {Q}}= {\mathbf {I}}\). The unknown location of interest is

$$\begin{aligned} \varvec{\theta }_\circ ~=~\{\varvec{x}_{0} \} = \left\{ [5,~5]^\top \right\} . \end{aligned}$$

The set \({\mathcal {S}}\) consists of two sequences \(s_{0}~=~\{6,5,7,8\}\) and \(s_{1}~=~\{4,3,2,1\}\) , where the node numbers are given in Fig. 1a. The sequences have been selected so that the anchor nodes are not coaligned in either \(s_0\) or \(s_1\).

In the case of Toa, the measurement model (1) admits an (overparameterized) linear form which is ideal for classical methods in robust statistics, such as the Huber method [9]. This method is tailored for Toa measurements with distributional form (1) and therefore provides a useful benchmark. We compare it to the standard nonlinear least-squares method (8) and the proposed method (12).

Figure 3a shows the cumulative distribution functions (cdf) of the localization error \(\Delta ({\widehat{\varvec{x}}})\) estimated from 100 Monte Carlo simulations, setting \({\widetilde{\epsilon }}~=~20\%\) in Algorithm 1. As expected there is a severe performance degradation of the standard method. To investigate the sensitivity of the results to the unknown corruption fraction, we also plot the root-mean-square error (Rmse), i.e., \(\sqrt{{{\,\mathrm{{\mathbb {E}}}\,}}[\Delta ^{2}({\widehat{\varvec{x}}})]}\), versus \(\epsilon\) for all three methods. For the proposed method, we set a very conservative upper bound \({\widetilde{\epsilon }}~=~50\%\) in Algorithm 1. Figure 3b shows the Rmse for the different methods, where we see that the robust method is insensitive to \(\epsilon\), with a graceful rise in Rmse when \(\epsilon\) exceeds \({\widetilde{\epsilon }}\). In sum, the proposed method outperforms the standard nonlinear least-squares method and is close to the benchmark provided by the Toa-tailored Huber method.

The results are corroborated also in Fig. 4 which shows Rmse as a function of \(\varvec{x}_0\). It can be seen that the robust method is more sensitive than the benchmark only near the edges of the vertical boundaries, where the resolution of time-differences decreases.

### 4.2
Tdoa: synchronous anchor nodes

We consider again the network in Fig. 1a, but now the self-localizing node is a passive receiver. Since Tdoa measurements are correlated, we set \({\mathbf {Q}}\) with 1s along the diagonal and 1/3 along the off-diagonals. The unknown location of interest is

$$\begin{aligned} \varvec{\theta }_\circ ~=~\{\varvec{x}_{0} \} = \left\{ [5,~5]^\top \right\} . \end{aligned}$$

As in the case of Toa above, we use the sequence set \({\mathcal {S}}\) with \(s_{0}~=~\{6,5,7,8\}\) and \(s_{1}~=~\{4,3,2,1\}\).

Note that the measurement model (2) is nonlinear in \(\varvec{\theta }\) and does not admit a linear re-parameterization. Thus, the Huber method is not readily applicable for Tdoa as it requires tuning an alternative numerical search techniques. We instead compare against a robust localization method tailored for Tdoa in [4], which is based on assuming a bound on the measurement error and then estimating the location which minimizes the worst-case error by solving a semi-definite programming (SDP) problem. In addition, we compare our proposed method (12) with the standard nonlinear least-squares method (8) as well.

Figure 5a shows cdfs of the localization error \(\Delta ({\widehat{\varvec{x}}})\) estimated using 100 Monte Carlo simulations, setting \({\widetilde{\epsilon }}~=~20\%\) in Algorithm 1. The performance characteristics are similar to those in Fig. 3a, but the absolute error levels are not directly comparable due to different measurement setups. The robust Tdoa method of [4] produces larger errors, in fact so large that for 13% of the Monte Carlo runs the SDP becomes infeasible and provides no solution. These runs were omitted for sake of comparison. The sensitivity to the unknown corruption fraction \(\epsilon\) is also shown in Fig. 5b when Rmse is plotted against \(\epsilon\) for both methods. We use a very conservative upper bound \({\widetilde{\epsilon }}~=~50\%\) in Algorithm 1. The proposed method is consistently robust and insensitive to corrupted data in contrast to the standard method for which the errors rise drastically with \(\epsilon\). Here we could not compare against [4] because, again, the SDP became infeasible as the fraction of corrupted samples \(\epsilon\) increased. Figure 6 shows that the proposed method yields substantial error reduction across space.

Finally, we illustrate the ability of the proposed method to effectively isolate corrupted Nlos samples. Since the data is generated synthetically, we can classify each sample from *p*_{Nlos}\((\varvec{z}|s)\) as \({\mathcal {C}}(\varvec{z})=1\) for ‘corrupted’ or \({\mathcal {C}}(\varvec{z})=0\) for ‘normal’. The method solves (12) and learns the probability weights \({\widehat{\varvec{\pi }}}\). If a weight is below a certain threshold, we may classify the corresponding sample \(\varvec{z}\) as an outlier, i.e., \({\widehat{{\mathcal {C}}}}(\varvec{z}) = 1\). We set the weight threshold to \(10^{-5}\) and show the resulting probability of correct detection \(\Pr \{{\widehat{{\mathcal {C}}}}(\varvec{z})=1|{\mathcal {C}}(\varvec{z}) = 1\}\) as well as the probability of false alarm \(\Pr \{{\widehat{{\mathcal {C}}}}(\varvec{z})=1|{\mathcal {C}}(\varvec{z}) = 0\}\) in Fig. 7. We use \({\widetilde{\epsilon }}= 50\%\) and vary the unknown fraction \(\epsilon\), using 50 Monte Carlo runs for each value of \(\epsilon\). It can be seen that the proposed method can effectively isolate Nlos samples with a low false-alarm rate.

### 4.3
Tdst: asynchronous anchor nodes

We consider the network in Fig. 1a, but now the self-localizing node is a passive receiver *and* the anchor nodes are asynchronous. Since Tdst measurements are correlated, we set \({\mathbf {Q}}\) with 1s along the diagonal and 1/3 along the off-diagonals. The unknown location of interest is

$$\begin{aligned} \varvec{\theta }_\circ ~=~\{\varvec{x}_{0} \} = \left\{ [5,~5]^\top \right\} . \end{aligned}$$

We use a set \({\mathcal {S}}\) that consists of four sequences: \(s_0~=~\{6,4,5,3\}\), \(s_1~=~\{3,6,4,5\}\), \(s_2~=~\{5,3,6,4\}\) and \(s_3~=~\{4,5,3,6\}\), following the scheme described in [17].

Similar to Tdoa, the Huber method is not readily applicable to Tdst. We therefore compare only the standard nonlinear least-squares method (8) and the proposed method (12).

Figure 8a shows the cdfs of the localization error \(\Delta ({\widehat{\varvec{x}}})\). The characteristics are similar to those in Fig. 3a. The sensitivity to the unknown corruption fraction \(\epsilon\) is shown in Fig. 8b where Rmse is plotted versus \(\epsilon\) for a very conservative upper bound \({\widetilde{\epsilon }}~=~50\%\) in Algorithm 1. It can be seen that the proposed method also robustifies self-localization in Tdst.

Finally, we consider a more challenging wireless network configuration, where one anchor node is replaced by an auxiliary node (\(N_{a}~=~1\)) at an unknown location as shown in Fig. 9a. The goal is to passively localize auxiliary nodes using asynchronous anchor nodes [16,17,18] in adverse Nlos conditions. The locations of interest are

$$\begin{aligned} \varvec{\theta }_\circ ~=~\{\varvec{x}_0,~\varvec{x}_1\}~=~\left\{ [5,~5]^\top , [-10,~10]^\top \right\} . \end{aligned}$$

We use a set \({\mathcal {S}}~=~\{s_0,~s_1\}\), where \(s_{0} = \{2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8\}\) and

\(s_{1} = \{2, 1 , 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1\}\). The first sequence set \(s_0\) involves only the anchor nodes and ensures that the passive node location is identifiable. The second sequence \(s_1\) ensures that the auxiliary node location is identifiable, see [16].

Figure 9b summarizes the distribution of localization errors \(\Delta ({\widehat{\varvec{x}}})\) for the passive and auxiliary nodes. Note that this is a harder problem that involves two unknown locations \(\varvec{x}_0\) and \(\varvec{x}_1\). Moreover, when estimating \(\varvec{x}_1\) the difficulty is compounded. In Fig. 9c, we see that the localization performance degrades rapidly under Nlos conditions using the standard method, whereas the proposed method is resilient.