## 5 Simulation
### Case 2  
The form of Singer acceleration model [25] is as follows:  
\[\mathbf{x}_{k}=\mathbf{F}\mathbf{x}_{k-1}+\mathbf{G}\mathbf{\omega}_{k-1} \tag{84}\]  
\[\mathbf{z}_{k}=\left[\begin{array}{c}r_{k}\\ \theta_{k}\end{array}\right]=\left[\begin{array}{c}\sqrt{\varsigma_{k}^{2}+ \zeta_{k}^{2}}\\ \tan^{-1}\!\left(\frac{\varsigma_{k}}{\varsigma_{k}}\right)\end{array}\right]+ \mathbf{\xi}_{k} \tag{85}\]  
where \(\mathbf{x}=\left[\begin{array}{cccccc}\zeta&\zeta&\zeta&\dot{\zeta}&a_{x}&a_{y} \end{array}\right]^{\mathrm{T}},\zeta,\dot{\zeta},\zeta\) and \(\dot{\zeta}\) denote the same definition as in case 1. \(a_{x}\) and \(a_{y}\) are the acceleration values; The transition matrices \(\mathbf{F}\) and \(\mathbf{G}\) are given as follows:  
\[\mathbf{F}=\left[\begin{array}{cccccc}1&0&T&0&\left(\alpha T-1+e^{-\alpha T} \right)/\alpha^{2}&0\\ 0&1&0&T&0&\left(\alpha T-1+e^{-\alpha T}\right)/\alpha^{2}\\ 0&0&1&0&\left(1-e^{-\alpha T}\right)/\alpha&0\\ 0&0&0&1&0&\left(1-e^{-\alpha T}\right)/\alpha\\ 0&0&0&0&e^{-\alpha T}&0\\ \end{array}\right] \tag{86}\]  
\[\mathbf{G}=\left[\begin{array}{cccccc}0.5T^{2}&0&T&0&1&0\\ 0&0.5T^{2}&0&T&0&1\end{array}\right]^{\mathrm{T}} \tag{87}\]  
where \(T=0.1\)s and the maneuver frequency \(\alpha=0.0045\). The initialization is selected as:  
\[\begin{array}{l}\mathbf{x}_{0}=\left[\begin{array}{cccccc}10000\mathrm{m}&790 0\mathrm{m}&-6\mathrm{m/s}&-10\mathrm{m/s}&0\mathrm{m/s}^{2}&0\mathrm{m/s}^{2} \end{array}\right]^{\mathrm{T}}\\ \mathbf{P}_{0}=\mathrm{diag}\!\left(\!\left[\begin{array}{cccccc}1\mathrm{m}^{2 }&1\mathrm{m}^{2}&0.1\mathrm{m}^{2}/\mathrm{s}^{2}&0.1\mathrm{m}^{2}/\mathrm{ s}^{2}&0.01\mathrm{m}^{2}/\mathrm{s}^{4}&0.01\mathrm{m}^{2}/\mathrm{s}^{4} \end{array}\right]\!\right)\end{array} \tag{88}\]  
The variance matrix \(\mathbf{Q}=\mathbf{G}\left[\begin{array}{cccccc}0.04&0.09&\right]\mathbf{G}^{\mathrm{T}}\), and \(\mathbf{R}=diag\!\left(\!\left[\begin{array}{cccccc}(300\mathrm{m})^{2}&(10 \mathrm{mrad})^{2}&\right]\!\right)\!\right)\).  
In case 2, the probability of \(\gamma_{k}=1\) was set as \(p_{k}=0.4\), i.e., the probability of the time-delayed measurement increased. Moreover, the noises have been enlarged. The comparison between the true trajectory and estimated trajectories is shown in Figure 5, and it is clear that the estimated trajectory of ISTUF was closest to the real trajectory. The trajectory of IUKF has been divergent, because of the effect of heavy-tailed noise.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Filters** & **ARMSE of** & **ARMSE of** & **ARMSE of Turn** & **Time Consuming** \\  & **Position** & **Velocity** & **Rate** & **Time Consuming** \\ \hline IUKF & 14.779 m & 2.866 m/s & 0.607\({}^{\circ}\)/s & 0.0481 s \\ RSTUF & 2.342 m & 0.578 m/s & 0.051\({}^{\circ}\)/s & 0.0494 s \\ ISTUF & 1.841 m & 0.386 m/s & 0.027\({}^{\circ}\)/s & 0.0519 s \\ \hline \hline \end{tabular}
\end{table}
Table 2: The ARMSE and time consuming of the three filters.  
The RMSEs of the estimating results are shown in Figures 6-8. The figures indicate that the RMSE of IUKF had been divergent, i.e., the enlarged heavy-tailed noise seriously affected the performance of filters under GF framework. The RMSE of RSTUF was also larger than that of ISTUF obviously. As the probability of time-delayed measurement had increased, the effect of the phenomenon was aggravated.  
Figure 5: Comparison of the trajectories.  
Figure 6: RMSE of estimating position.  
Figure 7: RMSE of estimating velocity.  
The ARMSE of the three filters is listed in Table 3. It is clear that the ARMSE of ISTUF was less than IUKF and RSTUF, and the ARMSE of IUKF was too large. As a result, ISTUF was more suitable for a target tracking system with time-delayed measurement and time-correlated heavy-tailed noises.