# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
## 5 Simulation
### Simulation Scenarios
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(\rho\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \(\sigma_{\theta}(\times 10^{-5\circ})\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \end{tabular}
\end{table} 
Table 6: Standard deviation \(\sigma_{\theta}\) against \(\rho\) at \(V=10^{-4}\) m/s. Simulation results of the time-averaged RMSEs and BNorms of the target position and velocity estimates and the bearing noise standard deviations is shown in Figure 8 where PL-MMSE has lower errors in the RMSE performance with \(\text{RMSE}_{avg}^{pos}=2.526\times 10^{-4}\) m, \(\text{RMSE}_{avg}^{vel}=1.783\times 10^{-5}\) m/s. The evolution of RMSEs, BNorms of the target position and velocity estimates in time \(kT\) for \(\sigma_{\theta}=7^{\circ}\) at \(V=10^{-4}\) m/s is presented in Figure 9.  
It is noticeable that the curve trends in Figures 8 and 9 are similar to Figures 4 and 5 because of the weak model nonlinearity caused by small velocity. The performance metric values of PL-MMSE gradually approach \(\text{RMSE}_{k}^{pos}=3.099\times 10^{-4}\) m, \(\text{RMSE}_{k}^{vel}=2.053\times 10^{-5}\) m/s, \(\text{BNorm}_{k}^{pos}=5.774\times 10^{-5}\) m and \(\text{BNorm}_{k}^{vel}=3.497\times 10^{-6}\) m/s, which are superior to other unconstrained algorithms in Figure 9. It is also demonstrated that the constrained PL-MMSE is better than the unconstrained PL-MMSE and other constrained filters at all bearing noise levels under the arc section in Figures 8 and 9. Performance comparisons of four algorithms combined with the linear approximation method and the second-order  
Figure 8: Time-averaged RMSEs, BNorms and bearing noise standard deviation at \(V=10^{-4}\) m/s for the PL-MMSE, PLKF, BC-PLKF and IVKF algorithms, as well as, the PL-MMSE algorithm with nonlinear constraints proposed in the paper. Figure 9: RMSEs, BNorms and time \(kT\) for \(\sigma_{\theta}=7^{\circ}\) at \(V=10^{-4}\) m/s for the PL-MMSE, PLKF, BC-PLKF and IVKF algorithms, as well as, the proposed PL-MMSE algorithm with nonlinear constraints in the paper. approximation method for \(\sigma_{\theta}=7^{\circ}\) at \(V=10^{-4}\) m/s are shown in Figures 10 and 11, respectively, which present that PL-MMSE-L and PL-MMSE-S have fewer errors than other constrained algorithms at a large bearing noise level. Table 7 presents the results of the time-averaged RMSEs and BNorms of different filters for \(\sigma_{\theta}=7^{\circ}\) at \(V=10^{-4}\) m/s. The RMSE and BNorm performance of PL-MMSE-L are \(\text{RMSE}^{pos}_{avg}=1.579\times 10^{-4}\) m, \(\text{RMSE}^{vel}_{avg}=1.415\times 10^{-5}\) m/s, \(\text{BNorm}^{pos}_{avg}=1.518\times 10^{-5}\) m and \(\text{BNorm}^{vel}_{avg}=1.590\times 10^{-6}\) m/s. The PL-MMSE algorithm combined with the linear approximation method is better than the second-order approximation method as \(\text{BNorm}^{vel}_{avg}=1.708\times 10^{-6}\) m/s of the PL-MMSE-S is greater than \(\text{BNorm}^{vel}_{avg}=1.590\times 10^{-6}\) m/s of the PL-MMSE-L in Table 7 due to the weak nonlinearity. For \(V=0.2\) m/s, we have the true initial state \(x_{1}=[200\ 0\ 0.2]^{T}\) and the initial covariance matrix \(P_{1|1}=\text{diag}([10^{-3}\ 10^{-  
Simulation results of the time-averaged RMSEs and BNorms of the target position and velocity estimates against \(\sigma_{\theta}\) are presented in Figure 12.  
The performance of PL-MMSE and PL-MMSE with constraints is better than other filters, where \(\text{RMSE}_{avg}^{pos}=0.3346\) m of PL-MMSE and \(\text{RMSE}_{avg}^{pos}=0.2954\) m of PL-MMSE-S are less than other algorithms for \(\sigma_{\theta}=3^{\circ}\). The evolution of RMSEs, BNorms of the target position and velocity estimates in time \(kT\) for \(\sigma_{\theta}=7^{\circ}\) at \(V=0.2\) m/s is shown in Figure 13.  
Figure 12: Time-averaged RMSEs, BNorms and bearing noise standard deviation at \(V=0.2\) m/s for the PL-MMSE, PLKF, BC-PLKF and IVKF algorithms, as well as, the PL-MMSE algorithm with nonlinear constraints proposed in the paper.