# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 3 Typical Fly-Around Configuration  
### Spatial Ellipse with Target Spacecraft as the Focus  
According to Equation (6), if the focus of the fly-around ellipse is located at the position of the target, then formulas \(\delta\mathbf{r}=0\) and \(\underline{r}=\mathbf{r}\) are established. When the initial relative states are known, the fly-around period and the shape of the ellipse depend on the design of the artificial gravitational coefficient, which is determined by Equations (11), (12), and (15), respectively.  
A simulation example is conducted to verify the feasibility of the above-mentioned guidance law. The target is in elliptical orbit, which is a Molniya orbit, and the servicer is near the target. The relevant orbit parameters are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **a (m)** & **e** & **i (deg)** & **w (deg)** & **\(\Omega\) (deg)** & **f (deg)** \\ \hline Target spacecraft & 26,553,375 & 0.741 & 63.4 & 270 & \(-\)30.52 & 180 \\ \hline \end{tabular}
\end{table}
Table 1: Orbit parameters of the target spacecraft.  
The relative position and relative velocity from servicer to target in the LVLH coordinate systems are \(\mathbf{r}_{0}=\left[-26.11,16.13,8.07\right]^{\mathrm{T}}\) m and \(\mathbf{v}_{0}=\left[5,32.7,0\right]^{\mathrm{T}}\times 10^{-4}\) m/s, respectively. According to Equations (9) and (10), the angles that characterize the spatial position of the fly-around trajectory relative to the orbital plane of the target can be obtained. Then, the relative states obtained via rotation transformation are described in the RVRH coordinate systems as \(\mathbf{r}_{0}=\left[29.41,11.94,0\right]^{\mathrm{T}}\) m and \(\mathbf{v}_{0}=\left[0,33.1,0\right]^{\mathrm{T}}\times 10^{-4}\) m/s, respectively.  
Firstly, the influence of the artificial gravitational coefficient on the fly-around speed and eccentricity are analyzed through simulation. Parameter \(k\) is defined as the ratio of the average angular velocity of the fly-around trajectory \(\overline{\omega}_{s}\) relative to the average angular velocity reference orbit \(\overline{\omega}\).  
\[k=\frac{\overline{\omega}_{s}}{\overline{\omega}}\;\;k\in(-\infty,+\infty) \tag{19}\]  
The size of \(k\) determines the speed of the trajectory, and the sign of \(k\) determines whether the direction of the trajectory is clockwise or counterclockwise.  
The simulation results are illustrated in Figure 4.  
The simulation results verify the above analysis results, that is, as the artificial gravitational coefficient increases, the fly-around speed continues to increase, and at the same time, there is a trajectory with the minimum eccentricity.  
The shape of the fly-around trajectory corresponding to different artificial gravitational coefficients is simulated and analyzed, and Figure 5 presents the simulation results for \(\delta\)r = 0. Figure 6 shows the control inputs as a function of time.  
In Figure 5, the arrowed line segment points from the focus of the ellipse to the periapsis. The simulation results show that even if different artificial gravitational coefficients are designed, different elliptical trajectories with a target as the focus can still be obtained, which verifies the effectiveness of the designed guidance law. For any initial relative position and relative velocity, a series of fly-around trajectories can be achieved through the guidance law. When the artificial gravitational coefficient is different, the center of the elliptical trajectory, as well as the shape and size, are also different. Through the adjustment of the initial relative states, combined with the design of the artificial gravitational coefficient, the desired configuration can be achieved.  
Figure 4: Influence of artificial gravitational coefficient on fly-around speed (**a**) and eccentricity (**b**).  
Figure 5: Fly-around ellipses with typical artificial gravitational coefficients.