# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 2 Analytical Guidance Law Design
### Guidance Law Design  
Under the action of dynamic Equation (2), the trajectory of the servicer relative to the target is defined as the nominal trajectory. The principle of low-thrust fly-around formation design is to make the nominal trajectory form the desired configuration through the control input \(\mathbf{u}\)[28]. Based on the principle of spacecraft orbital motion under geocentric gravity, the trajectory of the spacecraft relative to the center of Earth is a conic curve, and when the apparent energy of the spacecraft is less than zero, the trajectory is a closed conic curve, such as an ellipse or a circle [29]. The control input of the guidance law is designed to contain two parts. 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:  
\[\mathbf{u}_{1}=-\frac{\mu}{R_{t}^{3}}r[-\mathbf{e_{r}}+3\mathbf{e_{R_{t}}}(\mathbf{e_{R_{t}}} \cdot\mathbf{e_{r}})]+2\mathbf{\omega}\times\dot{\mathbf{r}}+\mathbf{\omega}\times(\mathbf{\omega }\times\mathbf{r})+\dot{\mathbf{\omega}}\times\mathbf{r} \tag{3}\]  
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:  
\[\mathbf{u}_{2}=-\frac{\mu_{s}}{L^{3}}\dot{\mathbf{r}} \tag{4}\]  
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:  
\[\mathbf{u}=-\frac{\mu}{R_{t}^{3}}r[-\mathbf{e_{r}}+3\mathbf{e_{R_{t}}}(\mathbf{e_{R_{t}}}\cdot \mathbf{e_{r}})]+2\mathbf{\omega}\times\dot{\mathbf{r}}+\mathbf{\omega}\times(\mathbf{\omega} \times\mathbf{r})+\dot{\mathbf{\omega}}\times\mathbf{r}-\frac{\mu_{s}}{L^{3}}\mathbf{\tau} \tag{5}\]  
where \(\mathbf{e_{R_{t}}}\) and \(\mathbf{e_{r}}\) are the unit vectors of \(\mathbf{R_{t}}\) and \(\mathbf{r}\), respectively, \(\mathbf{\omega}\) and \(\dot{\mathbf{\omega}}\) are the target's orbital angular velocity and angular acceleration, respectively. Equation (2) can describe not only the relative motion of a circular reference orbit, but also an elliptical reference orbit. This linearized model is applicable for short distances. Additionally, since the J2 perturbation and differential drag effects are not incorporated, the model is valid for a limited time frame. ### Guidance Law Design  
Under the action of dynamic Equation (2), the trajectory of the servicer relative to the target is defined as the nominal trajectory. The principle of low-thrust fly-around formation design is to make the nominal trajectory form the desired configuration through the control input \(\mathbf{u}\)[28]. Based on the principle of spacecraft orbital motion under geocentric gravity, the trajectory of the spacecraft relative to the center of Earth is a conic curve, and when the apparent energy of the spacecraft is less than zero, the trajectory is a closed conic curve, such as an ellipse or a circle [29]. The control input of the guidance law is designed to contain two parts.