## 2 Mathematical Model
### Dynamic Model of TB-TSS
 \tag{6}\]  
Since vector Equation (4) is formulated in the tether coordinate system, the forces must be transformed from the inertial coordinate to the tether coordinate frame. The following transformations are employed:  
\[\mathbf{T}_{2}=A_{y}(\alpha_{1})A_{z}(\theta_{1})A_{z}(-\theta_{2})A_{y}(-\alpha_ {2}) \begin{bmatrix}T_{2}\\ 0\\ 0\end{bmatrix}  \tag{7}\]  
\[\mathbf{r}_{0}=A_{y}(\alpha_{1})A_{z}(\theta_{1}) \begin{bmatrix}r_{0}\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}\cos\theta_{1}\cos\alpha_{1}\\ -\sin\theta_{1}\\ \cos\theta_{1}\sin\alpha_{1}\end{bmatrix} r_{0} \tag{8}\]  
By substituting Equations (3) and (6)-(8) into Equation (4), after a set of complex but routine algebraic manipulations, we obtain the dynamic equations of tether L\({}_{1}\):  
\[ \begin{array}{ll}I_{1}^{\prime\prime}=&l_{1}\Big{[}\alpha_{1}^{\prime 2}+ \cos^{2}(\alpha_{1})(\omega+\theta_{1}^{\prime})^{2}\Big{]}-\omega^{2}l_{1} \big{(}1-3\cos^{2}\theta_{1}\cos^{2}\alpha_{1}\big{)}+\\ &(\cos(\alpha_{1})\cos(\alpha_{2})\cos(\theta_{1}-\theta_{2})+\sin(\alpha_{1} )\sin(\alpha_{2}))\frac{T_{2}}{m_{1}}-\frac{T_{1}}{m_{0}}-\frac{T_{1}}{m_{1}} \end{array}  \tag{9}\]  
\[ \begin{array}{ll}\theta_{1}^{\prime\prime}=&-3\omega^{2}\cos\theta_{1}\sin \theta_{1}-2\big{(}\omega+\theta_{1}^{\prime}\big{)}\Big{[}\frac{l_{1}^{ \prime\prime}}{l_{1}}-\tan(\alpha_{1})\alpha_{1}^{\prime}\Big{]}\\ &-\frac{1}{\cos(\alpha_{1})l_{1}}\Big{\{}\big{(}\cos(\alpha_{2})\sin(\theta_{ 1}-\theta_{2})\big{)}\frac{T_{2}}{m_{1}}\Big{\}}\end{array}  \tag{10}\]  
\[ \begin{array}{ll}\alpha_{1}^{\prime\prime}=&-2\frac{l_{1}^{\prime\prime}}{l _{1}}\alpha_{1}^{\prime}-\cos(\alpha_{1})\sin(\alpha_{1})(\omega+\theta_{1}^{ \prime})^{2}-\omega^{2}\big{(}3\cos^{2}\theta_{1}\sin\alpha_{1}\cos\alpha_{1} \big{)}\\ &-(\cos(\alpha_{2})\sin(\alpha_{1})\cos(\theta_{1}-\theta_{2})-\cos(\alpha_{1} )\sin(\alpha_{2}))\frac{T_{2}}{m_{1}l_{1}}\end{array}  \tag{11}\]  
Figure 2: Schematic diagram of tether 1. \[ \begin{array}{ll}\alpha_{2}^{\prime\prime}=&-\cos(\alpha_{2})\sin(\alpha_{2})( \omega+\theta_{2}^{\prime})^{2}-\omega^{2}\big{(}3\cos^{2}\theta_{2}\sin\alpha_ {2}\cos\alpha_{2}\big{)}\\ &-(\cos(\alpha_{1})\sin(\alpha_{2})\cos(\theta_{1}-\theta_{2})-\cos(\alpha_{2}) \sin(\alpha_{1}))\frac{T_{1}}{m_{1}l_{2}}\end{array}  \tag{18}\]  
where \(T_{1}\) and \(T_{2}\) can be solved by the follow equations:  
\[ \begin{array}{ll}0=&l_{1}\Big{[}\alpha_{1}^{\prime 2}+\cos^{2}(\alpha_{1})( \omega+\theta_{1}^{\prime})^{2}\Big{]}-\omega^{2}l_{1}\big{(}1-3\cos^{2} \theta_{1}\cos^{2}\alpha_{1}\big{)}+\\ &(\cos(\alpha_{1})\cos(\alpha_{2})\cos(\theta_{1}-\theta_{2})+\sin(\alpha_{1}) \sin(\alpha_{2}))\frac{T_{2}}{m_{1}}-\frac{T_{1}}{m_{0}}-\frac{T_{1}}{m_{1}} \end{array}  \tag{19}\]  
\[ \begin{array}{ll}0=&l_{2}\Big{[}\alpha_{2}^{\prime 2}+\cos^{2}(\alpha_{2})( \omega+\theta_{2}^{\prime})^{2}\Big{]}-\omega^{2}l_{2}\big{(}1-3\cos^{2} \theta_{2}\cos^{2}\alpha_{2}\big{)}+\\ &(\cos(\alpha_{1})\cos(\alpha_{2})\cos(\theta_{1}-\theta_{2})+\sin(\alpha_{1}) \sin(\alpha_{2}))\frac{T_{1}}{m_{1}}-\frac{T_{2}}{m_{1}}-\frac{T_{2}}{m_{2}} \end{array}  \tag{20}\]  
In this case, the length of the tether is fixed and the Coriolis acceleration is zero. According to the principle of gravity gradient force on subsatellites increasing as the orbital radius decreases [3], the gravitational acceleration on subsatellites increases as the length of the tethers increases. The effect of the gravity gradient will cause the system to stabilize along the local vertical, and centripetal acceleration will speed up the whole system. Thus, it is necessary to study how to arrive at equilibrium and how the coupling libration motion influences the dynamic characteristics of the TB-TSS with fixed-length tethers.