# Maximum Correntropy Unscented Kalman Filter for Spacecraft Relative State Estimation
## 4 Illustrative Examples
### Example 2
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,  
\[ \begin{array}{l}q_{1}\sim N(0,(10^{-7}\;\text{km/s}^{2})^{2}),\;r_{1}\sim N (0,(10^{-3}\;\text{km})^{2}),\\ q_{2}\sim N(0,(10^{-7}\;\text{km/s}^{2})^{2}),\;r_{2}\sim N(0,(0.05\pi/180)^{2}),\\ q_{3}\sim N(0,(10^{-7}\;\text{km/s}^{2})^{2}),\;r_{3}\sim N(0,(0.05\pi/180)^{2}).\end{array} \]  
Figure 3 describes the relative motion of the deputy spacecraft in the Hill coordinate frame, Figures 4 and 5 demonstrate the MSD\({}_{\text{p}}\) and MSD\({}_{\text{v}}\), defined in Equations (42) and (43), with different filters in Gaussian noises, and Table 4 summarizes the corresponding TAMSD\({}_{\text{p}}\) and TAMSD\({}_{\text{v}}\), defined in Equations (44) and (45) (the parameter is set to \(k_{1}=1000,\;k_{2}=7200\)). 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. \begin{table}
\begin{tabular}{c c} \hline \hline
**Orbital Elements** & **Chief Spacecraft** \\ \hline Semi-major axis & 8000 km \\ Eccentricity & 0.150 \\ Orbit inclination & \(\pi/6\) rad \\ Argument of perigee & \(\pi/6\) rad \\ Right ascension of the ascending node & \(\pi/18\) rad \\ True anomaly & 0 rad \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial orbital elements of chief spacecraft. Figure 4: MSD\({}_{\rm p}\) with different filters in Gaussian noises. Figure 5: MSD\({}_{\rm v}\) with different filters in Gaussian noises. Figure 3: Relative motion of the deputy spacecraft in the Hill frame. Second, we consider the case in which the measurement noises are heavy-tailed, with mixed-Gaussian distributions  
\[q_{1} \sim N(0,(10^{-7}\ \mathrm{km}/\mathrm{s}^{2})^{2}),\] \[q_{2} \sim N(0,(10^{-7}\ \mathrm{km}/\mathrm{s}^{2})^{2}),\] \[q_{3} \sim N(0,(10^{-7}\ \mathrm{km}/\mathrm{s}^{2})^{2}),\] \[r_{1} \sim 0.9N(0,(10^{-3}\ \mathrm{km})^{2})+0.1N(0,(10^{-2}\ \mathrm{km})^{2}),\] \[r_{2} \sim 0.9N(0,(0.05\pi/180)^{2})+0.1N(0,(0.5\pi/180)^{2}),\] \[r_{3} \sim 0.9N(0,(0.05\pi/180)^{2})+0.1N(0,(0.5\pi/180)^{2}).\]  
Figures 6 and 7 reveal the MSD\({}_{\mathrm{p}}\) and MSD\({}_{\mathrm{v}}\) with different filters in non-Gaussian noises, and Table 5 lists the corresponding TAMSD\({}_{\mathrm{p}}\) and TAMSD\({}_{\mathrm{v}}\). As one can observe again, the UKF type filters give a smaller TAMSD\({}_{\mathrm{p}}\) and TAMSD\({}_{\mathrm{v}}\) 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 \(\sigma=2.0\), the MCUKF exhibits the smallest TAMSD\({}_{\mathrm{p}}\) and TAMSD\({}_{\mathrm{v}}\). Moreover, to compare the computational cost, the computation times of different filters in this example are shown in Figure 4:0 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.