# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 2 Analytical Guidance Law Design
### Guidance Law Design
The first part \(\mathbf{u}_{1}\) is the feedforward term, which makes the relative dynamics a second-order integration model and can be represented as follows:  
\[\mathbf{u}_{1}=-\frac{\mu}{R_{t}^{3}}r[-\mathbf{e_{r}}+3\mathbf{e_{R_{t}}}(\mathbf{e_{R_{t}}} \cdot\mathbf{e_{r}})]+2\mathbf{\omega}\times\dot{\mathbf{r}}+\mathbf{\omega}\times(\mathbf{\omega }\times\mathbf{r})+\dot{\mathbf{\omega}}\times\mathbf{r} \tag{6}\]  
The second part \(\mathbf{u}_{2}\) is the artificial gravity term, which has similar expressions to geocentric gravity. The center point of artificial gravity \(q\) is a fixed point relative to the LVLH coordinate system, and lies in the plane \(\Pi\). It is defined \(\delta\mathbf{r}\) as the relative position from the point \(q\) to the target, and expression \(\mathbf{r}=\underline{\mathbf{r}}+\delta\mathbf{r}\) holds. Then, expressions \(\dot{\mathbf{r}}=\dot{\mathbf{r}}=\mathbf{v}\) and \(\ddot{\mathbf{r}}=\ddot{\mathbf{r}}\) can be derived in the LVLH coordinate system. The artificial gravity \(\mathbf{u}_{2}\) can be represented as follows:  
\[\mathbf{u}_{2}=-\frac{\mu_{s}}{L^{3}}\dot{\mathbf{r}} \tag{7}\]  
where \(\mu_{s}\) is the artificial gravitational coefficient. The artificial gravity \(\mathbf{u}_{2}\) and relationship among the above vectors are illustrated in Figure 2.  
Where angle \(\theta\) is the angle from the \(x_{r}\)-axis to the vector \(\underline{\mathbf{r}}\). By defining the control input \(\mathbf{u}\) as \(\mathbf{u}=\mathbf{u}_{1}+\mathbf{u}_{2}\) and combining with Equations (3) and (4), the control input can be represented as follows:  
\[\mathbf{u}=-\frac{\mu}{R_{t}^{3}}r[-\mathbf{e_{r}}+3\mathbf{e_{R_{t}}}(\mathbf{e_{R_{t}}}\cdot \mathbf{e_{r}})]+2\mathbf{\omega}\times\dot{\mathbf{r}}+\mathbf{\omega}\times(\mathbf{\omega} \times\mathbf{r})+\dot{\mathbf{\omega}}\times\mathbf{r}-\frac{\mu_{s}}{L^{3}}\mathbf{\tau} \tag{8}\]According to Equation (2), the dynamic equation to obtain the desired nominal trajectory is as follows:  
\[\ddot{\mathbf{r}}+\frac{\mu_{s}}{\underline{r}^{3}}\mathbf{r}=0 \tag{6}\]  
According to Equation (6), the fly-around trajectory is a conic curve with point \(q\) as the focus, and lies in the plane \(\Pi\) determined by the relative position \(\mathbf{r}\) and relative velocity \(\mathbf{v}\). The type and shape of the fly-around trajectory are not only affected by the relative position and relative velocity at the initial time of the control action, but also by the parameters to be designed, such as the artificial gravitational coefficient \(\mu_{s}\) and the vector of artificial gravity center \(\delta\mathbf{r}\) or \(\underline{r}\). Under the control input shown in Equation (5), the apparent energy of the servicer relative to the target is defined as follows:  
\[\Im=\frac{1}{2}v^{2}-\frac{\mu_{s}}{\underline{r}} \tag{7}\]  
In order to make the relative fly-around trajectory a closed conic curve, when the artificial gravitational coefficient is designed, the apparent energy needs to be less than zero, so that the artificial gravitational coefficient meets the following constraint:  
\[\mathbf{u}_{2}=-\frac{\mu_{s}}{L^{3}}\dot{\mathbf{r}} \tag{4}\]0