# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## 11 Conclusions  
In this study, we offer a comprehensive review of IMM filtering techniques specifically applied to the tracking of a highly maneuvering aircraft that is being monitored by an AMTI radar system situated on a separate aircraft. Single-model filtering and hybrid state-estimation algorithms are first described, and then the dynamic and measurement models associated with the problem are explained. We summarize the EKF, UKF, and CKF used in the IMM estimator and present detailed steps for the IMM-CKF. In order to compare the state estimation accuracies of the IMM-EKF, IMM-UKF, and IMM-CKF, we give a derivation of the PCRLB. Finally, we analyze the performance of various IMM-based filtering algorithms.  
The aircraft has a range of accelerations, specifically \(3g\), \(4g\), \(5g\), and \(6g\). We used the RMS errors in position and velocity, ANEES, a covariance consistency metric, PCRLB, model probabilities, and computational cost as performance metrics to compare the performances of the three IMM-based filters. We also compared the root RTAMS errors of position and velocityand the associated CPU times. Our results show that for the operational measurement accuracy of the AMTI radar (range SD = 10 m; azimuth SD = 1, 2, and 4 mrad; radial velocity SD = 1 m/s), the performances of these three filters are nearly the same for \(T=0.5\ s\). The RMS position and velocity errors are close to the corresponding PCRLB values in the NCV segments, but small differences occur in the NCT segments. The ANEES remains confined within the 99% confidence intervals during the NCV phases, whereas it exceeds these intervals in the NCT phases. The results on mode probabilities show that the three IMM-based filters are able to select the correct dynamic model for various segments of the aircraft trajectory. The RTAMS errors of position and velocity are nearly the same for these three IMM-based filters, and they are slightly higher than the corresponding PCRLB values. The IMM-EKF exhibits the shortest CPU time, with the IMM-UKF and IMM-CKF requiring approximately 1.6 times more CPU time in comparison.  
We also analyze the performance of these three IMM filters at significantly higher measurement SDs and \(T=2\) s for the \(4g\) aircraft. In this case, the performance of the IMM-UKF and IMM-CKF is essentially comparable, whereas the IMM-EKF performs the least effectively among the three. This classical trend of the EKF, UKF, and CKF occurs at much higher SDs, which is outside the operational domain of the AMTI radar.  
We study the performance of the IMM-CKF by varying the azimuth SD with values of 1, 2, and 4 mrad, with a measurement time interval of 0.5 s. We observe that the RMS position and velocity errors and the RTAMS position and velocity errors are close to the corresponding PCRLB values. The performance of the IMM-CKF gracefully degrades as the azimuth SD increases. We also studied the performance of the IMM-CKF by using measurement time intervals of 0.5 and 1.0 s. In both cases, the RMS errors in position and velocity, as well as the RTAMS errors for position and velocity, were found to be in close proximity to their respective PCRLB values.  
Our future work will consider scenarios where the target aircraft and AMTI aircraft move in 3D space. We shall also focus on using improved NCT models and new algorithms for a maneuvering target [66].  
**Author Contributions:** Conceptualization, M.M., M.N.R. and X.T.; methodology, M.M., M.N.R. and X.T.; software, M.N.R. and M.M; validation, M.M., M.N.R. and X.T.; writing--original draft preparation, M.N.R., M.M. and X.T.; writing--review and editing, M.M., M.N.R. and X.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.