# Maximum Correntropy Unscented Kalman Filter for Spacecraft Relative State Estimation
## 4 Illustrative Examples
### 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  
\[\vec{r}_{c}=-\frac{\mu}{r_{c}^{3}}\vec{r}_{c}, \tag{48}\]  
\[\vec{r}_{d}=-\frac{\mu}{r_{d}^{3}}\vec{r}_{  
The position vector of the deputy spacecraft relative to the chief spacecraft is  
\[\mathbf{\ae}_{dc}=\mathbf{\mathrm{r}}_{d}-\mathbf{\mathrm{r}}_{c}. \tag{50}\]  
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  
\[\ddot{x}=2\omega\dot{y}+\dot{\omega}y+\omega^{2}x+\frac{\mu}{r_{c}^{2}}-\frac{ \mu(r_{c}+x)}{\left[\left(r_{c}+x\right)^{2}+y^{2}+z^{2}\right]^{3/2}}+q_{1}, \tag{51}\]  
\[\ddot{y}=-2\omega\dot{x}-\dot{\omega}x+\omega^{2}y-\frac{\mu y}{\left[\left(r_ {c}+x\right)^{2}+y^{2}+z^{2}\right]^{3/2}}+q_{2}, \tag{52}\]  
\[\dot{z}=-\frac{\mu z}{\left[\left(r_{c}+x\right)^{2}+y^{2}+z^{2}\right]^{3/2} }+q_{3}, \tag{53}\]  
with  
\[\ddot{r}_{c}=r_{c}\omega^{2}-\frac{\mu}{r_{c}^{2}}, \tag{54}\]  
\[\dot{\omega}=-\frac{2\dot{r}_{c}\omega}{r_{c}}, \tag{55}\]  
where \(x\), \(y\) and \(z\) are the radial, in-track and cross-track coordinates of the Hill frame, and \(\omega\) 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  
\[\rho=\sqrt{(x^{2}+y^{2}+z^{2})}+r_{1}, \tag{56}\]  
\[\vec{r}_{c}=-\frac{\mu}{r_{c}^{3}}\vec{r}_{c}, \tag{48}\]0  
\[\vec{r}_{c}=-\frac{\mu}{r_{c}^{3}}\vec{r}_{c}, \tag{48}\]1  
where \(\rho\) is the relative range between the chief spacecraft and deputy spacecraft, \(\theta\) is the azimuth angle, and \(\phi\) is the elevation angle. \[\vec{r}_{c}=-\frac{\mu}{r_{c}^{3}}\vec{r}_{c}, \tag{48}\]2 Measurement coordinate system. 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 \(\mathbf{x}(k)=\left[ \begin{array}{cccc}x(k)&y(k)&z(k)&\hat{x}(k)&\hat{y}(k)&\hat {z}(k)\end{array} \right]^{T}\) contains the relative position and velocity components in Hill frame, respectively. The initial true state is  
\[ \begin{array}{l}x(0)=31.9262\;\text{km},\;\hat{x}(0)=-0.005583\;\text{km/s}, \\ y(0)=-7.1384\;\text{km},\;\hat{y}(0)=-0.071774\;\text{km/s},\\ z(0)=33.4729\;\text{km},\;\hat{z}(0)=0.026249\;\text{km/s},\end{array} \]  
and the initial estimates and covariance matrix of the states are chosen as  
\[ \begin{array}{l}\widehat{x}(0|0)=31.9262\;\text{km},\;\widehat{\hat{x}}(0|0)= -0.004416\;\text{km/s},\\ \widehat{y}(0|0)=-8.1384\;\text{km},\;\widehat{y}(0|0)=-0.061774\;\text{km/s}, \\ \widehat{z}(0|0)=32.4729\;\text{km},\;\widehat{z}(0|0)=0.036249\;\text{km/s}, \\ \mathbf{P}(0|0)=diag([1,1,1,0.01^{2},0.01^{2},0.01^{2}]).\end{array} \]  
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 \(K=7200,\;M=100\). 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.