## 2 Mathematical Model
### Dynamic Model of TB-TSS  
In this subsection, we develop an advanced three-dimensional dynamic model of TB-TSS, specifically focusing on describing the libration motion of the tether during deployment or retrieval. The system is represented using a two-piece dumbbell model, neglecting the lateral and transversal vibrations caused by the tether's flexibility. The two-piece dumbbell model is divided into two components in order to derive the dynamic equation. The schematic diagram in Figure 2 illustrates the connection between the main satellite (m\({}_{0}\)) and the subsatellite (m\({}_{1}\)) via tether L\({}_{1}\). To formulate the equations of motion for this section using [PERSON]'s law, vector \(I_{1}\) is introduced, which represents the displacement from m\({}_{0}\) to m\({}_{1}\) and has a magnitude of \(-l_{1}\). Thus, the position of tether L\({}_{1}\) can be expressed in inertial coordinates as:  
\[I_{1}=r_{0}-r_{1} \tag{1}\]  
where vectors \(r_{0}\) and \(r_{1}\) represent the position of main satellite m\({}_{0}\) and subsatellite m\({}_{1}\), respectively. The orbit angular velocity \(\omega_{1}\) of tether L\({}_{1}\) in the tether coordinate system, relative to the Earth-centered inertial coordinate system, is given by:  
\[\omega_{1}=A_{y}(\alpha_{1})\times \begin{bmatrix}0\\ 0\\ (\theta_{1}^{\prime}+\omega)\end{bmatrix} + \begin{bmatrix}0\\ \alpha_{1}^{\prime}\\ 0\end{bmatrix} = \begin{bmatrix}-\sin\alpha_{1}\times(\theta_{1}^{\prime}+ \omega)\\ \alpha_{1}^{\prime}\\ \cos\alpha_{1}\times(\theta_{1}^{\prime}+\omega)\end{bmatrix}  \tag{2}\]  
where \(A_{y}\) denotes the rotation matrix of the o\({}_{y}\)-axis relative to the inertial coordinate system (described in Appendix A); \(\theta_{1}^{\prime}\) and \(\alpha_{1}^{\prime}\) represent the in-plane and out-of-plane angular velocity of tether L\({}_{1}\); \(\omega=\sqrt{\frac{\mu}{r}}\) is the angular velocity of the orbit; \(\mu\) is the gravitational constant of Earth. The second derivative of Equation (1) with respect to time is obtained as follows:  
\[\begin{split} I_{1}^{\prime\prime}&=r_{0}^{\prime\prime}-r_{1}^{ \prime\prime}\\ &=(-l_{1}^{\prime}\overset{\prime}{\underset{1}{\overset{\prime}{ \underset{1}{\overset{\prime}{\underset{1}{\overset{\prime}{\underset{1}{ \overset{\prime}{\underset{1}{\overset{\prime}{\underset{1}{\overset{ \prime}{\underset{\overset{\overset{\overset{\overset{\overset{\overset{\overset{\leftleftleftleftleftleftleftleft( \left(\left(\left\left(\left\left(\left\left((( \left\left( \left( \leftleftleftleftleftleftleftleft( \leftleftleftleftleftleftleftleft( \leftleftleftleftleftleftleftleftleftleft( \leftThe dynamic equations of tether L\({}_{1}\) in the inertial coordinate frame is then derived by [PERSON]'s law:  
\[r_{0}^{\prime\prime}-r_{1}^{\prime\prime}=\left(\mathbf{T}_{1}+\mathbf{G}_{0}\right)/m_{0 }-\left(\mathbf{T}_{2}-\mathbf{T}_{1}+\mathbf{G}_{1}\right)/m_{1} \tag{4}\]  
where \(\mathbf{T}_{1}\) and \(\mathbf{T}_{2}\) are the tension vectors acting on mass m\({}_{0}\) and m\({}_{1}\), respectively, and \(\mathbf{G}_{0}\) and \(\mathbf{G}_{1}\) denote the gravitational force vectors acting on mass m\({}_{0}\) and m\({}_{1}\), respectively. \(\mathbf{G}_{0}\) and \(\mathbf{G}_{1}\) can be written as:  
\[G_{i}=-\frac{\mu m_{i}}{\left|\mathbf{r}_{0}+\mathbf{I}_{i}\right|^{3}}(\mathbf{r}_{0}+\bm {I}_{i}),i=0,1 \tag{5}\]  
Furthermore, by expanding Equation (5) using the [PERSON] formula, and retaining only the first-order terms, we obtain:  
\[ \begin{array}{ll}G_{i}&\approx-\frac{\mu m_{i}}{\left(\mathbf{r}_{0}^{2}+2\mathbf{ r}_{0}\cdot\mathbf{I}_{i}\right)^{3/2}}(\mathbf{r}_{0}+\mathbf{I}_{i})\\ &=-\frac{\mu m_{i}}{\left|\mathbf{r}_{0}\right|^{3}}\left(1-\frac{3\mathbf{r}_{0}\cdot \mathbf{I}_{i}}{\left|\mathbf{r}_{0}\right|^{2}}\right)(\mathbf{r}_{0}+\mathbf{I}_{i})\\ &=-\omega^{2}m_{i}\bigg{(}\mathbf{r}_{0}+\mathbf{I}_{i}-\frac{3(\mathbf{r}_{0}\cdot\mathbf{I} _{i})\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|^{2}}\bigg{)}\end{array}