To validate the performance of the proposed algorithm, a highly maneuvering target example has been considered. The proposed algorithm will be compared with the IMMCKF, IMMUKF, and IMM5thCKF algorithm.
4.1. Tracking Model and Measurement Model
Let the state vector at time
k be
, which includes the position (m) and velocity component (m/s) in the x-axis and y-axis. For tracking of the maneuvering target, three models are employed: the constant velocity (CV) model, left constant turn (LCT) model and right constant turn (RCT) model. For constant velocity model, the equation of state is described as:
where
is the white Gaussian process noise with zero mean and nonsingular covariance
.
where the scalar parameter
is the spectral density and set to 1. The constant turn (CT) model is defined as:
where
is the white Gaussian process noise with zero mean and nonsingular covariance
.
where the scalar parameter
is set to 1,
T is the sampling interval,
w stands for the turn rate which is supposed to be known, the right turn rate is defined as
, and the left turn rate is defined as
.
In the experiment, the radar is located at the origin of the plan and equipped to measure range and bearing. Then, the measurement equation can be written as:
where
is the range value at time
k,
is the bearing value at time
k, tan
−1(·) is the inverse tangent function, and
is the white Gaussian measurement noise with zero mean and covariance
.
and
denote the standard deviation of range measurement noise and bearing angle measurement noise, respectively.
4.2. Simulation of the IMM5thSSRCKF
The simulation scene is designed as follows. The sampling interval is T = 1 s and repeats 100 times. The target moves in different state for five periods. The initial position is (15,000 m, 1000 m) and the target starts at 1 s with the velocity (−180 m/s, 200 m/s). From 1 s to 20 s it has motion at constant velocity; from 21 s to 70 s it turns right with ; from 71 s to 120 s it has motion at a constant velocity; from 121 s to 170 s it maneuvers and turns left with ; and from 171 s to 200 s it has motion at constant velocity.
The initial estimates
are generated from the Gaussian distribution
in which the true initial is
= [15,000, −180, 100, 200]
T. The standard deviation of range measurement noise
is 40 m and the standard deviation of bearing angle measurement noise
is 7 mrad. The initial model probability is
= [0.8 0.1 0.1] and the transition probability is given as:
The root mean square error (RMSE) of the target position at time
k and the accumulative RMSE (ARMSE) of estimated position at all times are defined in Equations (33) and (34):
where
is the number of Monte Carlo runs,
N is the total number of sampling points,
is the actual value of the target position at time
and
is the estimated position at time
in
mth Monte-Carlo. The RMSE and the accumulative RMSE in the velocity and acceleration can be defined in the same way. The performance comparison of the four algorithms are tested 200 times in Monte Carlo simulations.
Figure 2 gives the target trajectory and the state estimation generated from a single run of IMMUKF, IMMCKF, IMM5thCKF and IMM5thSSRCKF. As seen from
Figure 2, these four algorithms can track the trajectory of the target.
The RMSEs in position and velocity of the four algorithms are shown in
Figure 3 and
Figure 4, respectively. It can be seen that the proposed IMM5thSSRCKF performs better than the IMMUKF, IMMCKF and IMM5thCKF algorithms when the target moves with CV. The tracking error of target position of the three IMM algorithms would be almost the same when the target moves at constant velocity. The estimation effectiveness of the IMM5thSSRCKF estimator for tracking a maneuvering target outperform greatly than the other two IMM estimators.
To further evaluate the performance of the four algorithms, the ARMSEs of position and velocity of each algorithm are listed in
Table 1. It can be seen from the
Table 1 that IMM5thSSRCKF does better in tracking precision than IMMUKF, IMMCKF and IMM5thCKF, while all of them exhibit no error divergence.
The comparisons of CV mode probability of IMMUKF, IMMCKF, IMM5thCKF and IMM5thSSRCKF are shown in
Figure 5. The mode transitions occur at
t = 21 s,
t = 71 s,
t = 121 s and
t = 171 s, respectively. This figure shows that the IMMUKF, IMMCKF, IMM5thCKF and IMM5thSSRCKF can capture the kinematics of maneuvering when the motion state changes. It can be seen that the mode probabilities of the IMMUKF algorithm are not good at detecting mode transitions. The proposed algorithm and IMM5thCKF algorithm are equally faster at detecting model changes compared with the IMMUKF algorithm and the IMMCKF algorithm.
All the algorithms are implemented on the Intel Core
TM i5-4430 3.0GHZ CPU with 4.00 G RAM.
Table 2 shows the number of points and computational time of IMMUKF, IMMCKF, IMM5thCKF and IMM5thSSRCKF for each run. The points of IMMCKF as well as IMMUKF differ only by one point. As shown in
Table 2, the computational time of the algorithms is approximately proportional to the number of points. It is obvious that the IMM5thSSRCKF algorithm has a slightly lower computational cost than the IMM5thCKF due to the different cubature rule. Although the computation complexity of IMM5thSSRCKF algorithm is larger than IMMUKF and IMMCKF, it can be remedied by more high-speed computer technology.