## 2 Problem Statement
### Translational Relative Dynamics  
In LVLH frame, the relative position vector between the servicer and target is denoted as \(\mathbf{r}=\left[x,\ y,\ z\right]^{T}\), then the relative dynamics can be described as [41]  
\[\ddot{\mathbf{r}}=\mathbf{A}\mathbf{r}+\mathbf{B}\dot{\mathbf{r}}+f_{\zeta}+f_{d} \tag{1}\]  
where,  
\[\mathbf{A}=\left[ \begin{array}{cccc}{\omega_{e}}^{2}+2\frac{\mu}{r_{e}^{2}}& \dot{\omega}_{e}&0\\ -\dot{\omega}_{e}&{\omega_{e}}^{2}-\frac{\mu}{r_{e}^{2}}&0\\ 0&0&-\frac{\mu}{r_{e}^{2}}\end{array} \right],\ \mathbf{B}=\left[ \begin{array}{cccc}0&2\omega_{e}&0\\ -2\omega_{e}&0&0\\ 0&0&0\end{array} \right]. \tag{2}\]  
Figure 1: Mission scenario and three different coordinate frames. \(\mu~{}=~{}3.986\times 10^{14}\) m\({}^{2}\)/s\({}^{2}\) is gravitational constant of the Earth. \(r_{t}\) is orbit radius of target spacecraft. \(f_{c}\) is the controlled acceleration applied to the service spacecraft. \(f_{d}\) is the external disturbance. \(\omega_{e}\) and \(\dot{\omega}_{e}\) represent the angular velocity and angular acceleration of target spacecraft, which can be obtained by  
\[\omega_{e}=\frac{(1+e\cos f)^{2}}{(1-e^{2})^{\frac{3}{2}}}\sqrt{\frac{\mu}{a^{ 3}}}~{},\dot{\omega}_{e}=\frac{2\mu(1+e\cos f)^{3}e\sin f}{a^{3}(1-e^{2})^{ \frac{3}{2}}}, \tag{3}\]  
where, \(e\) is the eccentricity of the target, \(a\) is the semi-major axis, and \(f\) is the true anomaly. Equation (1) is established by [PERSON] (TH) [41], which offers better accuracy than the equation established by [PERSON] (CW) [42] in slightly eccentric orbits. The above analysis establishes the dynamic model of the two spacecraft in the frame \(\mathcal{F}_{l}\), without considering the attitude change of the target spacecraft. When the target spacecraft rotates on the orbit, its attitude change will cause a collision risk in the proximity operation process. To remedy this problem, the dynamic model in body-fixed frame \(\mathcal{F}_{b}\) needs to be considered, so as to depict the translational motion of servicer in rotating frame. Assuming the angle velocity of the target relative to the frame \(\mathcal{F}_{l}\) is \(\omega_{b}\), the vector conversion formulas between the two coordinate systems, \(\mathcal{F}_{l}\) and \(\mathcal{F}_{b}\), are introduced as  
\[\mathbf{r}^{l}=\mathbf{M}_{b}^{l}\mathbf{r}^{b}, \tag{4}\]  
\[\frac{d\mathbf{r}^{l}}{dt}=\frac{\delta\mathbf{r}^{b}}{\delta t}+\omega_{b}\mathbf{\times} \mathbf{r}^{b}, \tag{5}\]  
\[\frac{d^{2}\mathbf{r}^{l}}{dt^{2}}=\frac{\delta^{2}\mathbf{r}^{b}}{\delta t^{2}}+\frac {\delta\omega_{b}}{\delta t}\mathbf{\times}\mathbf{r}^{b}+2\omega_{b}\mathbf{\times}\frac {\delta\mathbf{r}^{b}}{\delta t}+\omega_{b}\mathbf{\times}\left(\omega_{b}\mathbf{\times} \mathbf{r}^{b}\right). \tag{6}\]  
Then, substituting Equations (4)-(6) into Equation (1) yields  
\[\ddot{\mathbf{r}}=\left(-\omega_{b}^{\times 2}+\mathbf{M}_{b}^{l}+\mathbf{B}\omega_{b}^{ \times}-\dot{\omega}_{b}^{\times}\right)\mathbf{r}+\left(\mathbf{B}-2\omega_{b}^{ \times}\right)\dot{\mathbf{r}}+\dot{f}_{c}^{b}+\dot{f}_{d}^{b}. \tag{7}\]  
Equation (7) can be written in the form of state equation as  
\[\dot{\mathbf{X}}=\mathbf{Q}\mathbf{X}+\mathbf{R}(\mathbf{u}+\mathbf{\gamma}), \tag{8}\]  
where, \(\mathbf{X}=\left[\mathbf{r}^{T},\dot{\mathbf{r}}^{T}\right]^{T}\), \(\mathbf{Q}=\left[ \begin{array}{cc}\mathbf{0}_{3}&\mathbf{I}_{3}\\ -\omega_{b}^{\times 2}+\mathbf{M}_{b}^{l}+\mathbf{B}\omega_{b}^{\times}-\dot{\omega}_{b}^{ \times}&\mathbf{B}-2\omega_{b}^{\times}\end{array} \right]\), \(\mathbf{R}=\left[ \begin{array}{c}\mathbf{0}_{3}\\ \mathbf{I}_{3}\end{array} \right]\), \(\mathbf{u}=\left[ \begin{array}{c}\mathbf{0}_{3}\\ \dot{f}_{c}^{b}\end{array} \right]\), and the \(\mathbf{\gamma}=\left[ \begin{array}{c}\mathbf{0}_{3}\\ \dot{f}_{d}^{b}\end{array} \right]\). Equation (8) is the relative dynamic model in target body-fixed frame \(\mathcal{F}_{b}\). Note that all the subsequent parameters are expressed in the frame \(\mathcal{F}_{b}\).