# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## 10 Numerical Simulations and Results
### Numerical Results for \(T=0.5\) s  
The RMS position and velocity errors for the IMM-EKF, IMM-UKF, IMM-CKF, and PCRLB in 3\(g\), 4\(g\), 5\(g\), and 6\(g\) cases are presented in Figures 5 and 6, respectively. In Figures 5 and 6, we observe that the RMS position and velocity errors from various filters are nearly the same. In order to understand the results in Figures 5 and 6, we have presented the time indices of dynamic models in Table 4. From Figures 5 and 6, it is noted that the RMS errors in position and velocity during the NCV phases approximate the respective PCRLB values closely. Nonetheless, minor discrepancies exist between the RMS errors and their corresponding PCRLB values within the NCT+ and NCT- phases. \begin{table}
\begin{tabular}{c c} \hline
**Variable** & **Value** \\ \hline Prior mean position & (1000, 20,000) m \\ Prior mean velocity & (374.332, 0) m/s \\ Variance of prior position & (40\({}^{2}\), 40\({}^{2}\)) m\({}^{2}\) \\ Variance of prior velocity & (4\({}^{2}\), 4\({}^{2}\)) (m/s)\({}^{2}\) \\ NCV: PSD of process noise (\(q_{1}\)) & 0.1 m\({}^{2}\)/s\({}^{3}\) \\ NCT: PSD of process noise (\(q_{2}\)) & 0.01 m\({}^{2}\)/s\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Aircraft parameters. Figure 5: RMS position error in (**a**) 3\(g\), (**b**) 4\(g\), (**c**) 5\(g\), and (**d**) 6\(g\) cases for \(T=0.5\) s.  
In order to see the differences in the RMS errors of various filters, we have plotted the RMS position and velocity differences (EKF-UKF) and (EKF-CKF) in Figures 7a and 7b, respectively. Figure 7 illustrates that the IMM-UKF and IMM-CKF filters demonstrate a comparable level of precision, and they both marginally outperform the IMM-EKF in terms of accuracy. The spikes in differences occur during the NCT+ and NCT- segments. Figure 8 displays the ANEES for the three IMM filters under consideration across three distinct acceleration scenarios, all bounded by the 99% confidence intervals. It is evident from the figure that the ANEES remains within the 99% confidence limits for segments \begin{table}
\begin{tabular}{c c} \hline
**Time Indices** & **Dynamic Model** \\ \hline
1–121 & NCV \\
121–181 & NCT+ \\
181–301 & NCV \\
301–361 & NCT- \\
361–481 & NCV \\ \hline \end{tabular}
\end{table} 
Table 4: Time indices of dynamic models for \(T=0.5\) s.  
Figure 6: RMS position error in (a) 3\(g\), (b) 4\(g\), (c) 5\(g\), and (d) 6\(g\) cases for \(T=0.5\) s.  
Figure 7: (a) RMS position difference; (b) RMS velocity difference for 4\(g\), azimuth SD = 1 mrad, and \(T=0.5\) s.  
where the aircraft's motion is characterized by NCV. Conversely, for the NCT segments, the ANEES is observed to exceed these limits. This discrepancy suggests that while the filters' computed covariance matrices align well with the estimation errors during the NCV phases, they fail to maintain the same consistency during the NCT+ and NCT- phases. Further research is warranted to address this issue and enhance the covariance calculations within the NCT segments for improved filtering performance. A magnified view of ANEES for the 4\(g\) case is shown in Figure 9a, and the covariance consistency metric \(\zeta_{k}\) is plotted in Figure 9b. The plot of \(\zeta_{k}\) in Figure 9b indicates that the IMM-UKF and IMM-CKF-calculated covariances have better covariance consistency than that of the IMM-EKF. In the first Monte Carlo run, Figure 10 depicts the mode probabilities for the 3\(g\), 4\(g\), and 5\(g\) cases for the IMM-CKF. When comparing the mode probabilities in Figure 10 with the results in Table 3, we find that the mode probabilities for the first NCV, NCT+, second NCV, NCT-, and third NCV segments are calculated accurately. Moreover, the mode probabilities approach a value of one in most cases. A similar trend is observed for the IMM-EKF and IMM-UKF. Table 5 presents the RTAMS position and velocity errors and corresponding PCRLBs for the NCT- and last NCV segment.