# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
## 5 Simulation
### Simulation Scenarios
It is evident that the constrained PL-MMSE has more stable and accurate tracking performance at large bearing noise levels under the arc section. Comparisons of four algorithms combined with the linear approximation method and the second-order approximation method for \(\sigma_{\theta}=7^{\circ}\) at \(V=0.2\) m/s are provided in Figures 14 and 15, respectively. The time-averaged RMSEs and BNorms of different filters for \(\sigma_{\theta}=7^{\circ}\) are presented in Table 9. It is remarkable that the PL-MMSE with constraints has less errors than other filters, where \(\text{RMSE}_{avg}^{pos}\) of PL-MMSE-L is 0.176 m and \(\text{RMSE}_{avg}^{pos}\) of PL-MMSE-S is 0.176 m. On contrary to the first experiment with \(V=10^{-4}\) m/s, the algorithms combined with the second-order approximation method perform better than the algorithms combined with the linear approximation method in Table 9, where \(\text{RMSE}_{avg}^{pos}\) of PL-MMSE-S is less than \(\text{RMSE}_{avg}^{pos}\) of PL-MMSE-L for \(\sigma_{\theta}=7^{\circ}\) at the scale of \(10^{-6}\) because of strong nonlinearity. Figure 14: Time-averaged RMSEs, BNorms and bearing noise standard deviation at \(V=0.2\) m/s for the four algorithms combined with the linear approximation method. Figure 9.0 RMSEs, BNorms and time \(kT\) for \(\sigma_{\theta}=7^{\circ}\) at \(V=0.2\) 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. It is worthwhile to point out that the errors of the filters are basically unchanged and slightly decrease as the noise level rises in Figure 12 when the velocity is relatively large. The cause of this phenomenon is the error in the state update equation introduced by the linearization of the arc movement. When \(V=0.2\) m/s, the error from the state update equation becomes the main source of the filter estimate error, which relatively reduces the effect of the bearing noise and breaks the average algorithm performance trend. **Remark 2**. : _It is noticeable that the magnitudes of the velocity in the second scenario are both small. There are two reasons for such a setting. Firstly, the PL-MMSE is still a linear Kalman filter under linear and nonlinear constraints. There is discrepancy between the TMA result from the linear motion model (9) and the actual arc trajectory in the circular section. The faster the target moves in an interval, the greater the discrepancy will be. Secondly, the sampling frequency of the sensor in reality is much higher than that in our experiment. For example, the sampling frequency of the radar is generally between 1 and 15 GHz. When the sampling frequency increases, the relative speed of the target also rises up proportionally to maintain the same traveling distance. Hence, if we set the sampling time \(T=10^{-5}\) s rather than \(0.1\) s as in the simulation, the relative speed of the target becomes \(V=1\) m/s and \(2000\) m/s, respectively, which are quite common in practice. The transformation of the sampling frequency and target velocity corresponding to a real radar demonstrates that the setting is meaningful._  
_Since the distance the target moves in an interval is small, the corresponding RMSEs are low. Nevertheless, simulation results show that the RMSEs of our constrained algorithms are much smaller than those of other filters._  
Figure 9.1 Time-averaged RMSEs, BNorms and bearing noise standard deviation at \(V=0.2\) m/s for the four algorithms combined with the second-order approximation method.