This section demonstrates the performances of the proposed algorithm by two illustrative examples. In the paper, we mainly compare the estimation performance based on the following benchmarks:
where
K is the entire time steps in every Monte Carlo run and
M represents the total Monte Carlo runs.
4.1. Example 1
Now, a univariate nonstationary growth model (UNGM) that is often used as a benchmark example for nonlinear filtering is considered, whose state and measurement equations are given by
The parameters are set to
.
First, we consider the case in which all the noises are Gaussian
In this example, the parameters are chosen as
.
Table 1 illustrates the
s of
x, defined in Equation (
41), in Gaussian noise. Since all the noises are Gaussian, the UKF gives the smallest
in all filters. Moreover, it is noted that when the kernel bandwidth is small, the MCUKF may result in a worse estimation; in contrast, when the kernel bandwidth becomes larger, its performance will approach that of the UKF. Actually, it has been proved easily that when
, the MCUKF will reduce to the traditional UKF (see Theorem 1). Therefore, one should choose a larger kernel bandwidth in Gaussian noises.
Second, we change the observation noise into a heavy-tailed non-Gaussian noise, with a mixed-Gaussian distribution
Table 2 shows the
s of
x in non-Gaussian measurement noise. As one can see, in this case, when kernel bandwidth is too small or too large, the performance of MCUKF will be not good. However, with a proper kernel bandwidth (say
), the MCUKF can outperform the UKF, achieving the smallest
. Again, when
σ is very large, MCUKF achieves almost the same performance as the UKF.
4.2. Example 2
Finally, we consider a practical example with respect to the relative motion of two spacecrafts, which is illustrated in
Figure 1. One of the spacecrafts is called the chief spacecraft, which is moving on the reference orbit, and the other is the deputy spacecraft. They all revolve around the earth and thus the inertial orbital equations of two spacecraft are given as
where
and
are the position vectors of the chief spacecraft and the deputy spacecraft in ECI coordinate frame,
and
are the norms of
and
, respectively, and
μ is the gravitational parameter of the earth.
The position vector of the deputy spacecraft relative to the chief spacecraft is
Here, we use the Hill coordinate frame, which is a rectangular, Cartesian, dextral rotating frame centered on the chief spacecraft and refer to it as the local-vertical-local-horizontal (LVLH) frame. Then, the motion of the deputy spacecraft relative to the chief spacecraft in the Hill coordinate frame can be described in [
35] as
with
where
x,
y and
z are the radial, in-track and cross-track coordinates of the Hill frame, and
ω denotes the orbital angular velocity of the chief spacecraft.
The radar is set on the chief spacecraft to obtain the measurements, and the measurement coordinate system is showed in
Figure 2. The measurement equations are given as
where
ρ is the relative range between the chief spacecraft and deputy spacecraft,
θ is the azimuth angle, and
is the elevation angle.
The two spacecrafts are on the elliptic low Earth orbits. The initial six orbital elements of the chief spacecraft are showed in
Table 3, and the trajectory of which in ECI coordinate frame is propagated by the fixed-step fourth-order Runge–Kutta algorithm. The prediction of filters is performed every 0.1 s and the measurements record from the radar every 1 s. The state vector
contains the relative position and velocity components in Hill frame, respectively. The initial true state is
and the initial estimates and covariance matrix of the states are chosen as
In this example, 100 independent Monte Carlo runs have been conducted, and, in each case, an elapsed time of 7200 s is considered, that is
. Since correntropy is a local similarity measure, the MCUKF may converge to the optimal solution slowly, especially when the initial values deviate greatly from the true values. Thus, we use the UKF during the first 100 s to make the process settle down and then switch to the MCUKF to continue. The performance of EKF, Huber-EKF (HEKF) [
36], UKF and novel robust UKF (NRUKF) [
16] are shown for comparison with that of the proposed algorithm.
First, we consider the case in which all the noises are Gaussian, that is,
Figure 3 describes the relative motion of the deputy spacecraft in the Hill coordinate frame,
Figure 4 and
Figure 5 demonstrate the
and
, defined in Equations (
42) and (
43), with different filters in Gaussian noises, and
Table 4 summarizes the corresponding
and
, defined in Equations (
44) and (
45) (the parameter is set to
). Those results illustrate that the UKF play the best performance in all filters in this case and the UKF type filters have the better performance than the EKF type counterparts. One can also observe that the robust filters do not perform as well as their non-robust counterparts in the Gaussian noises. Moreover, it is worth noting that when the kernel bandwidth is small, the MCUKF may achieve a worse performance; whereas, when the kernel bandwidth becomes larger, it has similar results to the UKF.
Second, we consider the case in which the measurement noises are heavy-tailed, with mixed-Gaussian distributions
Figure 6 and
Figure 7 reveal the
and
with different filters in non-Gaussian noises, and
Table 5 lists the corresponding
and
. As one can observe again, the UKF type filters give a smaller
and
than the EKF type counterparts. All the robust filters are superior to their non-robust counterparts in impulsive noises. It is noted that when the kernel bandwidth is very large, MCUKF achieves almost the same performance as the UKF. However, with a proper kernel bandwidth, the MCUKF can outperform the UKF significantly. Particularly, when
, the MCUKF exhibits the smallest
and
.
Moreover, to compare the computational cost, the computation times of different filters in this example are shown in
Table 6. We can see that the computation time of the MCUKF is moderate compared with the UKF, and is superior to that of the HEKF and NRUKF.