# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 3 Typical Fly-Around Configuration
### Planar Fly-Around Trajectory  
According to the spacecraft's relative dynamics, when neglecting the perturbations, the relative uncontrolled motion in the \(z\)-axis direction of the target's orbital plane will not diverge. Therein, the relative motion in the orbital plane is focused. The above discussion focuses on the motion in the plane determined by the initial relative states, and the obtained conclusions can be transplanted to the motion in the orbital plane. Accordingly, only the relative states in the orbital plane need to be considered, and also the control input in this plane. The planar fly-around trajectory also includes ellipses with the target spacecraft as the center or focus.  
A simulation of a planar fly-around ellipse with the target spacecraft as the center is conducted. The orbit parameters of the two spacecraft are also the same as the parameters in Section 3.1. The parameter in Equation (23) is designed as \(\rho=3\). There is no active control in the \(z\)-axis direction. The simulation results are shown in Figure 11. Obviously, the fly-around trajectory is a projected ellipse, and there is a stable 3D configuration.  
Figure 10: Control inputs of fly-around trajectories where parameters \(\underline{r}\) are 25 m (**a**), 30 m (**b**), 35 m (**c**), and 40 m (**d**), respectively.  
Particularly, when the target spacecraft is in a circular orbit, the circular fly-around configuration in the orbital plane centered on the target spacecraft is considered. Referring to Equation (24), the control input can be expressed as follows:  
\[\begin{pmatrix}u_{x}\\ u_{y}\end{pmatrix}=\omega^{2}\begin{pmatrix}-3x\\ 0\end{pmatrix}+\omega\begin{pmatrix}-2\dot{y}\\ 2\dot{x}\end{pmatrix}-\left(k\omega\right)^{2}\begin{pmatrix}x\\ y\end{pmatrix} \tag{29}\]  
where initial relative state constraints \(\dot{y}_{0}=k\omega x_{0}\) and \(\dot{x}_{0}=-k\omega y_{0}\) need to be satisfied. During a fly-around period, the fuel consumption \(\Delta V_{x}\) in the \(x\)-axis direction is defined as follows:  
\[\Delta V_{x}=\int_{0}^{\frac{2\pi}{|k|}\omega}|u_{x}|dt \tag{30}\]  
Referring to Equation (29), the total fuel consumption \(\Delta V\) can be expressed as follows:  
\[\begin{array}{ll}\Delta V&=\Delta V_{x}+\Delta V_{y}\\ &=4\frac{\omega}{|k|}r(k^{2}+2k+3+\left|k^{2}+2k\right|)\end{array} \tag{31}\]  
According to Equation (31), the fuel consumption required in a fly-around period satisfies the following laws:  
1. When the orbital angular velocity of the target and the fly-around speed are determined, that is, \(\omega\) and \(k\) are determined, the greater the fly-around radius, the greater the fuel consumption required to complete the fly-around;
2. When the ratio of the fly-around period relative to the orbital period of the target and the fly-around radius are determined, that is, \(k\) and \(r\) are determined, as the orbital height of the target increases, the orbital angular velocity \(\omega\) decreases, and the fuel consumption also decreases;
3. When the orbital period of the target and the fly-around radius are determined, that is, \(\omega\) and \(r\) are determined, the magnification \(|k|>1\) needs to be satisfied to achieve a fast fly-around. If expression \(1<k<\sqrt{3/2}\) or \(-2<k<-1\) holds, as \(|k|\) increases, the fuel consumption decreases accordingly. If expression \(\sqrt{3/2}<k\) or \(k<-2\) holds, as \(|k|\) increases, the fuel consumption also increases.  
The simulations are conducted to illustrate the relationship between the fuel consumption, ratio \(k\), and fly-around radius, as shown in Figure 12.  
Figure 11: Three-dimensional fly-around trajectory.  
Additionally, it can be seen from Figure 12 that there is a trade-off between fuel consumption and fly-around speed and radius. If the fuel consumption is certain, the fly-around speed and the fly-around radius relative to the target can be negotiated, which is mainly dependent on the mission requirements.