# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview  
[PERSON]  
1 Department of Electronics and Communications Engineering, PES College of Engineering, Mandya 571401, India; [EMAIL_ADDRESS]  
[PERSON]  
2 Independent Researcher, Anacortes, WA 98221, USA  
times gravitational acceleration [3; 4; 5]. The maneuverability of a target increases with the \(g\)-value. The maximum \(g\)-value of a fighter aircraft is about \(9g\). In this study, we take into account aircraft capable of withstanding accelerations ranging from three times to six times gravitational acceleration. A maneuvering target usually has multiple motions during its travel. This poses a challenge for a filtering algorithm to estimate the states of such targets. In real-world problems, the times at which the motion of a maneuvering target changes are not known. Therefore, the trajectory estimation of a target undergoing various motion dynamics during its maneuvers is significantly harder than that of a target with a single motion. The Kalman filter (KF) [6; 7; 8; 9; 10; 11], extended KF (EKF) [7; 8; 9; 10; 11; 12; 13], unscented KF (UKF) [10; 11; 14; 15], Gauss-Hermite filter (GHF) [16], central difference filter (CDF) [16], divided difference filter (DDF) [17], cubature Kalman filter (CKF) [18], and particle filter (PF) [19; 20; 21; 22] are single-model filters based on the Bayesian framework. When a target moves with a single motion, these filters are applicable to the filtering problem. The KF operates on the premise of linear dynamics and observation models, incorporating both process noise and measurement noise modeled as additive Gaussian distributions. Given these assumptions and a prior Gaussian distribution, the KF is considered optimal when evaluated under the criterion of minimizing the mean square error (MSE) [7; 8; 20]. In scenarios where the system dynamics and/or observation models exhibit nonlinear characteristics, the original KF equations are not applicable to the filtering problem, and the EKF, UKF, GHF, CDF, DDF, CKF, and PF are used instead. These filters are approximate or sub-optimal in the MMSE sense. The EKF, UKF, GHF, CDF, DDF, and CKF are based on Gaussian approximation [16; 23], and the PF is based on Monte Carlo techniques [19; 20; 21; 22]. The EKF, UKF, and CKF are widely used in nonlinear filtering. Therefore, we have chosen these three filtering techniques for tracking an aircraft that engages in complex maneuvers. The computational cost to attain an equivalent level of precision to that of these three filters is a few hundred times more for the PF [24]. Therefore, we have excluded the PF from this study. Detailed explanations are given later. The EKF [9; 10; 11; 12; 13] garnered significant attention following its successful application in the Apollo lunar program. The EKF uses the linearization of the time-evolution function and measurement function by using the Taylor series expansion [7; 8]. Previous researchers have found that when the dynamic and measurement models are highly nonlinear and the measurement accuracy is low, the EKF could potentially exhibit sub-optimal performance, and there is a risk that the filter might experience divergence [14; 15]. [PERSON] and [PERSON] introduced the UKF, which incorporates unscented transformation (UT) [14; 15], to obtain an improved filtering algorithm. The UKF approximates the posterior probability density function (pdf) by utilizing a collection of \(2n+1\) sigma points, which are deterministically selected, alongside their respective weights, where \(n\) is the dimension of the state vector. By using a third-degree spherical-radial cubature rule, [PERSON] and [PERSON] [19] introduced an innovative approach to the formulation of the CKF. The CKF has \(2n\) deterministic cubature points and associated weights, where each weight is \(1/2n\). The CKF is regarded as a simple, stable, and effective algorithm for a nonlinear filtering problem based on sound theoretical foundations.