## 5 Numerical Results
Compared to the GN's results, the RIS's results highlight the superior robustness of the RIS and explain the reason why the RMSEs in Figures 5 and 9 of the RIS are the best. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]2 The RMSE of the UN velocity estimation as the position error of the AN increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]3 The bias of the UN velocity estimation as the position error of the AN increases. The RIS was also found to be resilient to an initial guess. The next simulation fixed the noise and AN position error levels at 10 dB and 0.25 m\({}^{2}\) and examined the influence of the initialization error by varying \(e\) from 10 m to \(10^{4}\) m. The other parameters remained the same as in the previous settings. The convergence probability is shown in \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]4 Figure 14 indicates that the RIS exhibits a higher convergence probability, even when the initial point is distant from the true value. For instance, when \(e\) equals \(10^{3.5}\) m, the RIS method still achieves a convergence probability of 1, whereas the GN method fails to do so. This simulation experiment validates that the RIS is robust to an initial guess. Therefore, although the CFPS cannot provide an accurate solution to start the RIS, it can converge to the optimal and perform better than the GN. We also evaluated the performance as the number of ANs varies. The AN positions were selected from Table 3, with the first \(M\) ANs used for testing. The noise power and position error of ANs are set to \(\sigma^{2}=0.1\) m\({}^{2}\) and \(\sigma_{s}^{2}=1\) m\({}^{2}\), respectively. Notably, the number of unknowns in the first stage is \(2N+4\), and the number of equations is \(M-1\). Therefore, the minimum number of ANs must satisfy \(M-1\geq 2N+4\). For the simulation scenario with \(N=2\), we start with \(M=9\) ANs. Figures 15-18 show the RMSE and bias performances of all compared solutions, which improve as the number of ANs increases. The accuracy of theh CFPS declines significantly when \(M\) is less than 10, yet it still provides a satisfactory initial guess, ensuring that the RIS converges to an optimum. Processing  
\[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]5 Influence of initialization error on convergence. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]6 Convergence probability as the noise power increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]7 Processing times as the number of ANs increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]8 RMSE of the UN velocity estimation as the number of ANs increases. \[RMSE(\mathbf{\mathrm{p}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{p}}^{(t)}-\mathbf{ \mathrm{p}}^{o}\|^{2}} \tag{84}\]9 Bias of the UN velocity estimation as the number of ANs increases. The computational times of generating the results in Figures 5-8 were recorded by Matlab 2022 on a typical personal computer with an AMD 5800H CPU @3.2 GHz, with 16 G of RAM and an AMD Radeon Vega8 GPU. The processing times and relative processing times are depicted in \[RMSE(\mathbf{\mathrm{v}}) =\sqrt{\frac{1}{T}\sum_{t=1}^{T}\|\mathbf{\mathrm{v}}^{(t)}-\mathbf{ \mathrm{v}}^{o}\|^{2}}\]0 Although the RIS is more robust than the GN in measurement noise, AN position error, and initial guess, the processing times of the RIS and GN are comparable. The RIS requires 226% more time than the GN, but the computational load is on the same order of magnitude.