1. Introduction
The trajectory estimation of a highly maneuvering fighter aircraft is a challenging problem. By utilizing data obtained from an air moving target indicator (AMTI) radar system, we delve into estimating the trajectory of such a highly maneuvering fighter aircraft. The AMTI radar is on another aircraft, and it measures the range, azimuth, and radial velocity of a target. The AMTI measurement model is nonlinear since the AMTI measurements are nonlinear functions of the target state. The fighter aircraft is capable of flying with motions of nearly constant velocity (NCV) and nearly constant turn (NCT). The NCV model is linear, while the NCT model is nonlinear. The filtering problem is nonlinear since the measurement and dynamic models are nonlinear. An accelerating target is called a maneuvering target [
1]. Aircraft that have an acceleration of 3
g or higher are considered to be highly maneuvering, where
g is gravitational acceleration and is quantified as 9.8 m/
. Typical commercial aircraft are restricted to velocities that do not exceed the speed of one Mach, where 1 Mach = 343 m/s [
2]. For example, a Boeing 747-8i, Airbus A380, Boeing 787, and a Boeing 777 have maximum speeds of 0.86 Mach, 0.85 Mach, 0.85 Mach, and 0.84 Mach, respectively [
2]. Commercial aircraft are typically capable of achieving a maximum rotational acceleration equivalent to approximately three 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 9
g. 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]. Jullier and Uhlmann 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
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, Arasaratnam and Haykin [
19] introduced an innovative approach to the formulation of the CKF. The CKF has 2
n deterministic cubature points and associated weights, where each weight is 1/2
n. The CKF is regarded as a simple, stable, and effective algorithm for a nonlinear filtering problem based on sound theoretical foundations.
The NCV is a nonmaneuvering motion, whereas nearly constant acceleration (NCA), nearly constant jerk (NCJ), and NCT with an unknown angular velocity are maneuvering motions [
1,
8]. The models for NCA and NCJ present linear characteristics when considering the Cartesian co-ordinates of the target state. Conversely, the NCT model introduces nonlinearity into the target state representation if the angular velocity remains unspecified. In general, a target can move with a sequence of nonmaneuvering and maneuvering motions. In scenarios where a target exhibits a variety of movement patterns, the times of change in motion are usually unknown. For such a target, the continuous-valued kinematic state (known as the base state) and the discrete-valued mode (known as the modal state) or dynamic model change with time [
8]. This class of problems is known as the hybrid state estimation or the jump Markov state estimation problem [
8,
20,
25]. A dynamic model describes the time evolution of the base state by using a stochastic difference equation. The modal state can assume a finite number of possible values and can switch from one mode to another over time. A homogeneous Markov chain with a known transition probability matrix (TPM) describes changes in the modal state [
8,
20,
25]. In a hybrid state estimation algorithm, a bank of mode-matched filters is run in parallel, and an algorithm to manage the co-operation among the filters is used. The generalized pseudo-Bayesian (GPB) [
8,
25] algorithm and the interacting multiple model (IMM) algorithm were proposed by Blom et al. [
26,
27] and are examples of hybrid state estimation algorithms. The
kth-order GPB algorithm is denoted by GPBk and requires
filters, where
is the number of models, and
k is the number of sampling periods over which mode switching is considered. In contrast, the IMM algorithm always uses
filters, and the performance of the IMM algorithm is comparable to that of the GPB2 algorithm (which uses
filters) at a much reduced computational cost. The computational complexity of the IMM algorithm is nearly linear in the number of models with a small quadratic term. An attractive feature of the IMM algorithm is its modularity [
25], which allows it to be used with any filter, such as the KF, EKF, UKF, CKF, etc. The IMM approach has been effectively utilized in a broad spectrum of applications [
28,
29,
30,
31,
32,
33,
34,
35,
36,
37]. Comprehensive insights into the algorithm, along with an extensive review, can be found in [
8] and [
25], respectively.
If a single-model filter (e.g., KF, EKF, UKF, or CKF) is used for a maneuvering target, then a high level of process noise will be chosen with a nonmaneuvering motion (e.g., NCV) used to handle maneuvering motion. For this choice, the filter will perform well in the maneuvering segments of the trajectory, but its performance will be poor in the nonmaneuvering segments. If a low level of process noise is chosen, then the filter will perform well in the nonmaneuvering segments, but it will have large errors in the maneuvering segments. On the other hand, the IMM filter will perform well in the nonmaneuvering and maneuvering segments since it probabilistically chooses the motion models. The IMM algorithm also finds application in multi-target tracking (MTT), which refers to the problem of jointly estimating the number of targets and their states from noisy sensor measurements in the presence of a false alarm and a sensor probability of detection that is less than unity [
38,
39,
40,
41,
42]. MTT problems occur in both civilian and military domains and arise in air traffic control (ATC), ground surveillance, maritime surveillance, airborne surveillance, space–object tracking, autonomous vehicles, ballistic missile defense, [
8,
25,
38,
40,
43,
44,
45], robotics, remote sensing, computer vision, biomedical research, etc. In general, a tracking scenario includes nonmaneuvering and maneuvering targets. Well-known MTT algorithms are the joint probabilistic data association filter (JPDAF) [
38,
43], multiple hypothesis tracker (MHT) [
38,
39,
40,
41,
44], and random finite set (RFS)-based algorithms [
41,
42,
45]. The IMM algorithm has been successfully used in the multiscan JPDAF [
43], MHT [
46] and RFS-based algorithms [
47]. If a single-model filtering algorithm is used in a multi-target tracker for maneuvering targets, then the measurements will not be associated with predicted tracks due to model mismatch, and tracks will be terminated after a few scans. This will create many tracks corresponding to a true target trajectory. This phenomenon is known as track fragmentation. The fraction of correct measurements in a track, known as the track purity, will also be affected. In general, an IMM-based multi-target tracker will have a significant advantage over a single-model filter-based tracker. Since computer hardware and software systems have improved significantly over the last few decades, IMM-based multi-target trackers can solve complex tracking problems involving nonmaneuvering and maneuvering targets.
The organization of the study is as follows.
Section 2 presents the single-model nonlinear filtering algorithms. The hybrid state estimation needed for a maneuvering target is explained in
Section 3.
Section 4 and
Section 5 describe the dynamic and measurement models used in the current problem. The EKF, UKF, and CKF are explained in
Section 6. The IMM algorithm based on the CKF and PCRLB are discussed in
Section 7 and
Section 8, respectively. The numerical simulation and results for the 3
g, 4
g, 5
g, and 6
g cases for various operating conditions are presented in
Section 10. Finally,
Section 11 provides a summary of the work in the study and discusses the future directions of this research.
5. Measurement Model
The AMTI radar measures the range, bearing (or azimuth), and radial velocity of the target [
52].
Figure 1 [
51] depicts the true range,
, azimuth,
, and radial velocity,
of the target at time
. Assume that the radar moves with CV. At time
, the state vector of the AMTI radar is given by
The position and velocity vectors of the target denoted by
and
respectively, at time
are specified by
Let
denote the position vector of the AMTI radar at time
:
The range vector of the target from the AMTI radar at time
is
Then, the range of the target from the AMTI radar at time
is given by
Let
denote the unit vector along the radar line-of-sight (RLOS). Then,
The radial velocity of the target
at time
is given by [
24,
52]
Hence, the radial velocity represents the component of the target’s velocity that is projected onto the RLOS (
Figure 1).
At time
, given the target speed
and the angle between the target velocity and the RLOS
, the radial velocity of the target
can be calculated as
Figure 1 and Equation (
44) demonstrate that the radial velocity exhibits its peak values of
and
when the target moves in the same or opposite direction as the RLOS. It vanishes to zero when the target’s velocity is perpendicular to the RLOS. The true trajectories of the AMTI radar and aircraft moving at 3
g, 4
g, 5
g, and 6
g are shown in
Figure 2, where the red and blue lines indicate the trajectories of the aircraft and AMTI radar, respectively. An open circle and a diamond indicate the start and end points of a trajectory, respectively. Each aircraft trajectory encompasses a succession of motion phases, including segments of NCV, NCT+, another NCV, NCT-, and concluding with another NCV. In
Figure 3, different segments of the trajectory of an aircraft moving at 3
g are presented, and the durations of various segments with start and end times are also given in
Table 1.
At time
, the radar measurement vector
is defined as
The measurement model for the AMTI radar is defined by [
52]
where
is the nonlinear measurement function
In (
46),
is a zero-mean white Gaussian measurement noise with the covariance matrix
. The corresponding definitions of
are given as follows:
where
are the measurement errors’ standard deviations (SDs) for range, azimuth, and radial velocity, respectively. The dynamic model of the radar moving with CV is formulated by
where
is given by (
27).
7. IMM Algorithm Based on CKF
In order to estimate the state of a maneuvering target in a nonlinear filtering problem, the IMM-EKF [
29,
30,
31,
53,
54,
55,
56], IMM-UKF [
32,
54,
55,
57], IMM-CKF [
2,
3,
58], and IMM-cubature quadrature Kalman filter (IMM-CQKF) [
4] are commonly used. The IMM filtering algorithm maintains a common structure across its variations, such as the IMM-EKF, IMM-UKF, and IMM-CKF, with only minor distinctions. In order to conserve space, this section focuses on detailing the IMM-CKF filter. The IMM filter is designed to leverage a collection of kinematic models, encompassing all potential aircraft maneuvers across any trajectory segment [
8,
20,
25,
27]. By incorporating a parallel execution of three distinct CKFs, the IMM filter adeptly manages the NCV, NCT+, and NCT- motion models.
Suppose at time
, the state estimates
, the covariance matrices
, and the model probabilities
are available, where
. We define the TPM as
with elements
and the probability of the
ith model as
. Next, we summarize the various steps of the IMM filter [
8,
20,
25,
27] to compute the model probabilities, state estimates, and covariances of all three filters at time
.
Step 1: Compute the Mixing Probabilities
Given the model probabilities at time
and transition probabilities, compute the mixing probabilities
Step 2: Compute the Mixed Initial States and Covariances
Compute the mixed initial states and covariances,
Step 3: Compute the Predicted State Estimates and Covariances
The predicted state estimate and covariance for the NCV model are given, respectively, by
Since the NCT+ and NCT- models are nonlinear, the predicted state estimates and covariances are calculated by using cubature points. Thus, the predicted state estimates are given by
where the
ith predicted cubature point
is calculated using (
65)–(
71). Then, the predicted covariances are calculated by
Step 4: Compute the Innovation Covariance and Cross-covariance
For all three models, the predicted measurement and innovation covariance are given, respectively, by
where
is calculated using (
72)–(
74). The cross-covariance is calculated by
Step 5: Compute the Model Likelihoods
Calculate the model likelihoods by
where
is the innovation and is given by
Step 6: Compute the Updated States and Covariances
where the Kalman gain is given by
Step 7: Update the Mode Probabilities
Update the model probabilities by
Step 8: Compute the Combined State Estimate and Covariance
The combined state estimate and covariance are given, respectively, by
A schematic diagram of the IMM filter based on the CKF is shown in
Figure 4.
A large number of publications using the IMM estimator are available in the literature. A few selected relevant papers are discussed based on their technical contribution and results. Li and Bar-Shalom [
29] considered a maneuvering aircraft in the air traffic control (ATC) problem with accelerations of 0.6
g, 1
g, and 2
g. The measurements of the ATC radar are range and azimuth, which were converted to unbiased Cartesian position measurements. They used two versions of the IMM-EKF with the NCV and NCT motions. Kastella and Biscuso [
30] performed an extensive study on the ATC problem and showed that the IMM-EKF with the NCV and NCT motions had superior performance over the adaptive single model KF. The maximum acceleration for the high-performance turn was 3.3
g. Generic maneuvering aircraft with accelerations of 1
g, 1.5
g, 2.5
g, and 3
g were studied in [
31] using radar range, azimuth, and range rate measurements. The authors used the EKF-based switching grid and adaptive grid IMM filters. Blackman et al. [
32] analyzed a highly maneuvering aircraft with a maximum acceleration of 3g using the NCT motion.
9. Measures of Performance (MoP)
In order to evaluate the performance of the IMM-EKF, IMM-UKF, and IMM-CKF filters, we calculate the RMS position and velocity errors, ANEES [
8,
24,
62,
63], RTAMS errors [
12,
18,
51,
63], covariance consistency metric, and CPU times using Monte Carlo simulation. Given the true state
and estimated state
of the aircraft, the RMS position and velocity errors at time
are defined, respectively, by
where
M denotes the number of Monte Carlo runs.
The root time-averaged mean square (RTAMS) position and velocity errors [
5,
20,
52] are defined, respectively, by
where
and
are the first and the last indices of the trajectory segment, respectively.
At time
, the ANEES is calculated as [
8,
24,
62,
63]
where
n denotes the dimension of the state, and
is the covariance corresponding to
The criterion for determining that the estimation error is consistent with the calculated covariance of the filter at time
is that
lies within the two-sided 99% confidence interval. For 1000 Monte Carlo runs and a four-dimensional state vector, the 99% confidence interval is [0.943, 1.058]. Since the nonlinear filters are approximate, the filter calculated covariance, in general, is different from the actual mean square error matrix (MSEM). The average of the covariance matrix at time
is
The MSEM at time
is specified by
At time
, to determine the covariance consistency of a filter, we compute the ratio
If
is close to unity, then the filter-calculated covariance is a good measure of the MSEM. Thus,
can be regarded as a second metric for covariance consistency.
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].