# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 2 Analytical Guidance Law Design
### Fly-Around Formation Analysis  
According to the designed guidance law in Section 2.2, the spatial configuration of the obtained fly-around formation is illustrated in Figure 3.  
Where line \(l\) is the intersection of the plane \(\Gamma\) and the plane \(\Pi\), and \(l\) passes through point \(o\). \(\alpha\) is defined as the angle from \(x\)-axis to the line \(l\), similar to the concept of ascending node right ascension in absolute orbital elements. Furthermore, \(\varphi\) is the angle between the plane \(\Gamma\) and \(\Pi\), similar to the concept of orbital inclination. According to \(\alpha\) and \(\varphi\), the spatial position of the formation can be determined. Figure 2: Artificial gravity and vector relationship. \(h\) is defined as the moment of momentum of the fly-around trajectory. In the LVLH coordinate system, it can be expressed as follows:  
\[\mathbf{h}=\mathbf{\upvarepsilon}\times\mathbf{v}= \begin{bmatrix}\mathbf{h}_{x}\\ \mathbf{h}_{y}\\ \mathbf{h}_{z}\end{bmatrix}  \tag{9}\]  
where  
\[ \begin{cases}\cos\varphi=\frac{h_{z}}{h}\\ \tan\alpha=\frac{h_{x}}{-h_{y}}\end{cases}  \tag{10}\]  
Then, the size and shape of the fly-around trajectory are analyzed. The semi-major axis of the fly-around trajectory can be obtained from Equations (6) and (7). \[a=-\frac{\mu_{s}}{2\Im}=-(\frac{v^{2}}{\mu_{s}}-\frac{2}{\mathbf{\upvarepsilon}})^ {-1} \tag{11}\]  
Furthermore, the fly-around period \(T\) can be obtained as follows:  
\[T=2\pi\sqrt{\frac{a^{3}}{\mu_{s}}}=2\pi\sqrt{-\frac{1}{8}(\frac{v^{2}}{2\mu_{ s}{}^{2/3}}-\frac{\mu_{s}{}^{1/3}}{r})^{-3}} \tag{12}\]  
It can be seen from Equations (11) and (12) that under the condition that \(\delta\mathbf{r}\) and the initial relative states have been determined, the design of \(\mu_{s}\) can be used to control the semi-major axis and the period of the fly-around trajectory. In order to evaluate the relationship between the artificial gravitational coefficient and the fly-around period, the variable \(\xi=\mu_{s}{}^{1/3}\) and function \(f(\xi)\) are defined. \[f(\xi)=\frac{v^{2}}{2\xi^{2}}-\frac{\xi}{\xi}\quad\xi>(\frac{1}{2}\xi v^{2})^ {1/3} \tag{13}\]  
The derivation of the function \(f(\xi)\) with respect to \(\xi\) can be obtained. \[\ddot{f}(\xi)=-\frac{v^{2}}{\xi^{3}}-\frac{1}{t} \tag{14}\]  
Figure 3: Spatial configuration of the fly-around formation. It can be seen that when expression \(\dot{f}(\xi)=0\) is satisfied, that is \(\xi=\left(-\varepsilon v^{2}\right)^{1/3}\), the function \(f(\xi)\) takes the extreme value. When expression \(\xi>\left(-\varepsilon v^{2}\right)^{1/3}\) is obtained, the function \(f(\xi)\) decreases. According to the relationship between variable \(\xi\) and the artificial gravitational coefficient \(\mu_{s}\), as \(\mu_{s}\) increases, the fly-around period \(T\) decreases at the same time. Therefore, increasing the artificial gravitational coefficient can increase the fly-around speed and reduce the fly-around period. By referring to the calculation expression of eccentricity in the two-body problem [30], according to the initial relative states and the designed artificial gravitational coefficient, the eccentricity of the fly-around trajectory can be obtained. \[\mathbf{e}=\frac{1}{\mu_{s}}\mathbf{v}\times\left(\underline{r}\times\mathbf{v}\right)- \frac{\underline{r}}{\underline{r}} \tag{15}\]  
Defining the following variables such as \(E=\left\|\mathbf{e}\right\|^{2}\), \(\mathbf{m}=\mathbf{v}\times\left(\underline{r}\times\mathbf{v}\right)\), \(\mathbf{p}=\underline{r}/\underline{L}\), and \(b=1/\mu_{s}\), Equation (15) can be expressed as follows:  
\[E=m^{\mathsf{T}}mb^{2}-2m^{\mathsf{T}}pb+\mathbf{p}^{\mathsf{T}}\mathbf{p} \tag{16}\]  
The curve described using Equation (16) is a parabola with an upward opening. When expression \(b=\mathbf{m}^{\mathsf{T}}\mathbf{p}/\mathbf{m}^{\mathsf{T}}\mathbf{m}\) is met, the formula \(\mu_{s}=\underline{r}v^{2}\) holds, and the eccentricity of the fly-around trajectory takes the minimum value.