# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
## 5 Simulation
### Simulation Scenarios  
Designed algorithms are tested in two scenarios in this subsection. As can be observed in Figure 3a, the target moves on a straight line in the first scenario while the target moves on an arc in the second scenario, as shown in Figure 3b with nearly constant velocity magnitude \(V\). Under the known angle of the vehicle, the constrained matrix \(G\) and the vector \(g\) are governed by  
\[G = \begin{bmatrix}1&-\tan\theta&0&0\\ 0&0&1&-\tan\theta\end{bmatrix} , \tag{72}\] \[g =[0,0]^{T}. \tag{73}\]  
The constrained estimate can be produced by setting \(W=I\) and \(W=P^{-1}\). In the simulation, the sampling interval T is set to 0.1 s. The total time span is the 20 s. The angle \(\theta\) is set to \(\pi/4\). The velocity \(V\) is set to 12 m/s. The target initial position is \([0\ 0]^{T}\) m. The true initial state is set to \(x_{1}= \begin{bmatrix}0\ 0\ 0\ 6\sqrt{2}\ 6\sqrt{2}\end{bmatrix} ^{T}\). The initial covariance matrix is \(P_{1|1}=\text{diag}([1\ 1\ 0.01\ 0.01])\). In addition, the covariances of process noise \(\omega_{k-1}\) and bearing noise \(e_{k}\) are given by  
\[\omega_{k-1} =\lambda\mathcal{N}(\mu_{x1},Q_{1})+(1-\lambda)\mathcal{N}(\mu_{ x2},Q_{2}), \tag{74}\] \[e_{k} =\rho^{2}[\lambda\mathcal{N}(\mu_{z1},R_{1})+(1-\lambda)\mathcal{ N}(\mu_{z2},R_{2})], \tag{75}\]  
respectively, where \(\mu_{x1}^{T}= \begin{bmatrix}0\ 0\ 0\ 0\end{bmatrix} \), \(\mu_{x2}^{T}= \begin{bmatrix}0\ 0\ 0\ 0\end{bmatrix} \), \(\mu_{z1}^{T}=0\), \(\mu_{z2}^{T}=0\), \(Q_{1}=\text{diag}([0\ 0\ 0.15\ 0.15])\), \(Q_{2}=\text{diag}([0\ 0\ 0.23\ 0.23])\), \(R_{1}=0.015\), \(R_{2}=0.019\) and \(\lambda=0.4\). The variable \(\rho\) controls the magnitude of bearing noise \(e_{k}\). By setting \(\rho\) from 1 to 10 with the difference of 1, Table 4 shows the standard deviation \(\sigma_{\theta}\) of bearing noise for the corresponding \(\rho\). Simulation results of the mean RMSEs and BNorms of the target position and velocity estimates against \(\sigma_{\theta}\) are presented in Figure 4.  
It is noticeable that the performance metric values of all algorithms increase and finally tend to be stable with \(\sigma_{\theta}\) in Figure 4, where the performance of PL-MMSE is always better than that of PLKF, BC-PLKF, and IVKF. The RMSE\({}^{pos}_{avg}\) of PL-MMSE is stable at 2.5 m at a large bearing noise level, which significantly outperforms other unconstrained algorithms. The evolution of RMSEs, BNorms of the target position and velocity estimates in time \(kT\) (\(k=1,\ldots 150\)) for \(\sigma_{\theta}=7^{\circ}\) is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\rho\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \(\sigma_{\theta}\) (\({}^{\circ}\)) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Standard deviation \(\sigma_{\theta}\) against \(\rho\) in the first scenario. Figure 3: (**a**) target travels in a straight line along the direction \(\theta\); (**b**) target travels along the circular road with the turning center \((R_{x},R_{y})\) and the radius \(R\). The tracking performance of all algorithms gradually deteriorates with the rising scan under a large bearing noise level. The metric values of PL-MMSE gradually approach RMSE\({}_{k}^{\text{pos}}=3.11\) m, RMSE\({}_{k}^{\text{vel}}=0.147\) m/s, BNorm\({}_{k}^{pos}=0.053\) m, and BNorm\({}_{k}^{vel}=0.011\) m/s, which are remarkably lower than other unconstrained algorithms. It can also be observed that the constrained PL-MMSE is superior to the unconstrained PL-MMSE at all bearing noise levels in Figures 4 and 5, which indicates the constrained algorithm has a better robustness and tracking performance. Comparisons of four algorithms combined with the mean square method and the estimation projection method for \(\sigma_{\theta}=7^{\circ}\) are presented in Figures 6 and 7, respectively, which demonstrate that PL-MMSE-C \((W=I)\) and PL-MMSE-C \((W=P^{-1})\) have less errors than other corresponding constrained algorithms at a large bearing noise level.