## 5 Simulation and Discussion
### Scenario 2  
To prove the tracking performance of VB-TPHD in nonlinear scenarios and the effect of update window length \(L\) on filtering, we design a two-dimensional extended trajectory space scenario with unknown and time-varying noise covariance. Suppose there are two targets that can form trajectories in the scenario and each target makes a constant turning motion. The dynamic and measurement model are shown in the Equations (57) and (58):  
\[F_{k}= \begin{bmatrix}1&\frac{\sin(\omega)}{\omega}&0&\frac{-(1-\cos(\omega))}{ \omega}&0\\ 0&\frac{-\cos(\omega)}{\omega}&1&\frac{\sin(\omega)}{\omega}&0\\ 0&\cos(\omega)&0&-\sin(\omega)&0\\ 0&\sin(\omega)&0&\cos(\omega)&0\\ 0&0&0&0&1\end{bmatrix} ,\,B_{k}= \begin{bmatrix}\sigma_{v}\frac{\Delta^{2}}{ \gamma}&0&0\\ \sigma_{v}\Delta&0&0\\ 0&\sigma_{v}\frac{\Delta^{2}}{\gamma}&0\\ 0&\sigma_{v}\Delta&0\\ 0&0&\sigma_{t}\Delta\end{bmatrix} , \tag{60}\]  
Figure 4: Comparison of adaptive and false estimation of AGM-VB-TPHD as well as AGM-tag-VB-PHD under GOSPA metric. (**a**) RMS GOSPA error of filters. (**b**) RMS GOSPA localization cost of filters. (**c**) RMS GOSPA missed target cost of filters. (**d**) RMS GOSPA false target cost of filters. \[Q_{k}=B_{k}\times{B_{k}}^{T},\quad h_{k}(x_{k},0)=\left[ \begin{matrix}\arctan( \frac{p_{x_{k}}}{p_{x_{k}}})\\ \sqrt{p_{x_{k}}^{2}+p_{y,k}^{2}}\end{matrix} \right],\ R_{k}=\text{diag}(\sigma_{ v_{k}}^{2},\sigma_{v_{k}}^{2})\]  
where, \(\Delta~{}=~{}1\) s, \(\sigma_{v}~{}=~{}1.8\) m/s represents the velocity standard deviation of the process noise and \(\sigma_{t}=(\pi/180)^{2}\) rad/s represents the angular velocity standard deviation of the process noise. \(\sigma_{v}\), and \(\sigma_{v_{k}}\) represent the unknown parameters of the standard deviation of the measurement noise. The former four dimensions of the state quantity of the target in the extended space are the same as the linear model and the fifth dimension is the angular velocity of the target. The first and second dimensions are the azimuth and distance of the relevant observation point (set as the origin of coordinates). Next are some settings for the scenario parameters. The scenario size is set to [0, \(\pi/2\)] rad \(\times\) [0, 1200] m and the total tracking duration is set to 100 s. The trajectory survival probability is set as \(p_{s}~{}=~{}0.99\) and the detection probability is set as \(p_{D}~{}=~{}0.99\). The clutter generated in each scan obeys the uniform distribution and its quantity obeys the Poisson distribution with parameter \(\lambda\) = 3. The initial state, time of birth and time of death and measurement noise covariance of the trajectory in the extended state space are shown in Table 2. Note that target 2 is divided by the bound of time 50 and the standard deviation of measurement noise in the first period is diag(\(\pi\)/180,2) and in the second period is diag(1.5\(\pi\)/180,3). The threshold of pruning is set as \(\varepsilon_{p}~{}=~{}10^{-4}\), the threshold of absorption is set as \(\varepsilon_{\mu}~{}=~{}4\) and the maximum number of Gaussian components is set as \(J_{\text{max}}~{}=~{}70\). The covariance matrix for quantitative measurement of noise obeys the inverse Gamma distribution. The initial parameters are \(\alpha_{1}~{}=~{}1\), \(\beta_{1}~{}=~{}\pi/180~{}\times~{}0.01\), \(\alpha_{2}~{}=~{}\beta_{2}~{}=~{}1\), and the given attenuation factor is \(\rho\) = 0.95. Similar to the linear scenario simulation, we also conducted 1000 Monte Carlo runs on the trajectory set with MATLAB 2021a on 2.50 GHz Intel I5 laptop. In scenario 2, the trajectory metric [40] is used to evaluate the performance of the extended trajectory estimation, and the parameters are set as \(p\) = 2, \(c\) = 10, \(\gamma~{}=~{}1\). The target RMS trajectory metric error (TM) consists of localization, missed targets, false targets and track switching. Figures 5-9 show the length \(L\) of the update window is set to 5.