# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
## 2 Analytical Guidance Law Design  
### Relative Dynamics  
Earth-Centered Inertial (ECI) Coordinate System. The origin of the ECI coordinate system \(O_{I}-X_{I}Y_{I}Z_{I}\) is centered on Earth, the \(Z_{I}\)-axis is along the rotational axis, and the \(X_{I}\)-axis points toward the vernal equinox. The \(Y_{I}\)-axis completes the right-handed orthogonal [26].  
Local Vertical-Local Horizontal (LVLH) Coordinate System. The origin of the LVLH coordinate system \(o-xyz\) lies on the center of mass (c.m.) of the target spacecraft. The \(x\)-axis is in the opposite direction to Earth's center, the \(y\)-axis is along the direction of flight, and the \(z\)-axis completes the right-handed orthogonal. The orbital plane of the target is defined as plane \(\Gamma\)[27].  
Relative Vertical-Relative Horizontal (RVRH) Coordinate System. The origin of the RVRH coordinate system \(o-x_{r}y_{r}z_{r}\) also lies on the c.m. of the target spacecraft. Assuming that the relative position vector \(\mathbf{r}\) and the relative velocity vector \(\mathbf{v}\) from the servicer to the target are non-collinear, the above two vectors determine the plane \(\Pi\), and the norm vector of plane \(\Pi\) is \(\mathbf{n}\), described as \(\mathbf{n}=\mathbf{r}\times\mathbf{v}\). The \(z_{r}\)-axis is along the direction of norm vector \(\mathbf{n}\), the \(y_{r}\)-axis is along the direction of relative velocity vector \(\mathbf{v}\), and the \(x_{r}\)-axis completes the right-handed orthogonal. Obviously, since the servicer is constantly moving relative to the target, the RVRH coordinate system is time-varying. Define the time when the continuous low thrust is initially applied to the servicer as the initial time, and the RVRH coordinate system of the initial time is uniquely determined.  
The relative geometrical relationship and coordinate systems are illustrated in Figure 1.  
As shown in Figure 1, \(\mathbf{R}_{t}\) and \(\mathbf{R}_{s}\) denote vectors directed from the Earth center to the c.m. of the target and the servicer, respectively. The relative position \(\mathbf{r}\) can be represented as \(\mathbf{r}=\mathbf{R}_{s}-\mathbf{R}_{t}\). Assuming that the target has no active control and the  
Figure 1: Geometrical relationship and coordinate systems.  
ignored, the relative translational dynamics in the ECI coordinate system can be obtained as follows:  
\[\frac{d^{2}\mathbf{r}}{dt^{2}}=-\mu(\frac{R_{s}}{R_{s}^{3}}-\frac{R_{t}}{R_{t}^{3}})+ \mathbf{u} \tag{1}\]  
where \(\mu\) is the geocentric gravitational coefficient, \(\mathbf{u}\) is the active control input of the servicer, and the first term at the right of Equation (1) is the differential spherical Earth gravity. Ignoring the second-order small satellite of the differential spherical Earth gravity, the relative translational dynamics in the LVLH coordinate system can be obtained as follows:  
\[\ddot{\mathbf{r}}=\frac{\mu}{R_{t}^{3}}r[-\mathbf{e_{r}}+3\mathbf{e_{R_{t}}}(\mathbf{e_{R_{t}}} \cdot\mathbf{e_{r}})]-2\mathbf{\omega}\times\dot{\mathbf{r}}-\mathbf{\omega}\times(\mathbf{\omega }\times\mathbf{r})-\dot{\mathbf{\omega}}\times\mathbf{r}+\mathbf{u} \tag{2}\]  
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.