# Mid-State Kalman Filter for Nonlinear Problems
## 5 Simulation
### Simulations Results  
The following two very long-range target tracking scenarios are designed for different radar performances:  
Figure 6: Measurement Distribution and the filter estimation error ellipse. (1) \(r=2700\) km, \(\delta_{r}=10\) m, \(\delta_{\theta}=0.3^{\circ}\), \(\beta\)= 3.7; (2) \(r=2700\) km, \(\delta_{r}=0.1\) m, \(\delta_{\theta}=0.3^{\circ}\), \(\beta\)= 371  
Figure 7 shows the measurement distribution of two simulation scenarios. Simulation scenario 1 uses narrowband radar to track very long-range targets. This is set according to the situation of missile target tracking by narrowband radar in engineering. The measurement distribution is Gaussian distribution. Simulation scenario 2 uses wide-band radar to track very long-range targets. In scenario 2, the typical CL problem occurs. The experiment uses 100 Monte Carlo simulations, the initial target position is \(x_{0}=1050\) km, \(y_{0}=2500\) km, and the initial velocity is \(v_{x}=200\)\(m/s\),\(v_{y}=300\)\(m/s\). The traditional nonlinear filters EKF, UKF, and CKF are used for performance comparison simulation. The sampling time interval of the four filters is T = 0.1 \(s\) and the tracking time is 80 \(s\). RMSE, CRLB theory, and consistency metric ANEES are used to test the filtering performance. The convergence speed of the filter is greatly related to the initial state, so the two-point starting method is adopted. The initial state and covariance are obtained by differentiating the two measurements at the first time and the second time. Table 1 lists the range and velocity estimation RMSE of different filters in the tracking scenario (1) when the measurement uncertainty region is Gaussian distribution. In order to compare the computational complexity, the table also lists the Average State Estimation Time Consumption (ASETC) of different filters. It can be seen from Table 1 that the range measurement accuracy of the MSKF is similar to that of the UKF and CKF, which shows that the MSKF is also suitable for nonlinear target tracking in general. The angle measurement accuracy of the CKF and MSKF is higher than the EKF and UKF. The time cost of the MSKF is slightly higher than that of the other filters, but it increases the accuracy of radar angle measurement. In Table 2, the MSKF improves the angle measurement accuracy and achieves accurate tracking of a very long-range target compared with other filters. In summary, the MSKF can be used in various situations and, for special nonlinear scenarios, it can also effectively improve radar tracking accuracy. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter** & **Range RMSE/m** & **Angle RMSE/\({}^{c}\)** & **ASETC/s** \\ \hline EKF & 2.487 & 0.2224 & 0.0295 \\ \hline UKF & 2.506 & 0.0986 & 0.0450 \\ \hline CKF & 2.502 & 0.0266 & 0.0426 \\ \hline MSKF & 2.598 & 0.0208 & 0.0691 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different filtering accuracy and time consumption in scenario (1). Figures 8-11 shows the simulation results of tracking very long-range targets with filters in scenario (2). At the time \(\beta=371\), the uncertainty region of measurement is CL distribution. It can be seen from Figures 8 and 9 that for very long-range targets, the measurement uncertainty region presents a severe non-Gaussian shape and the three filters except EKF can accurately track the target. In order to achieve the linear approximation of the system model, the EKF ignores the higher-order terms of [PERSON]'s expansion. Therefore, the EKF causes a significant error in the estimated state posterior distribution due to the truncation error. The UKF and CKF design a small number of deterministic sampling points and approximate the posterior probability of the state by calculating the statistical characteristics of these sampling points after passing through the nonlinear system so the target state is estimated accurately. For the tracking scenario (2), the angle estimation accuracy of the CKF is higher than that of the UKF.