## 2 Mathematical Model
### Dynamic Model of TB-TSS
#### 2.3.2 Coupling Characteristic of Libration Angles  
The dynamic coupling characteristic of libration angles was investigated using Equations (15)-(18), as described in this section. Here, the changes in tether length were ignored, and the focus was solely on the angular coupling. Two distinct cases were considered in order to analyze and discuss the mechanical characteristics of libration motion. Case 1: This case assumes there is no initial in-plane roll motion and studies the effect of out-of-plane pitch motion on in-plane roll motion. The in-plane libration angles and velocities are defined as \(\theta_{1}=\theta_{2}=\theta_{1}^{\prime}=\theta_{2}^{\prime}=0\), and Equations (15)-(18) can be simplified as:  
\[\theta_{1}^{\prime\prime}=2\omega\tan(\alpha_{1})\alpha_{1}^{\prime} \tag{27}\]  
\[ \begin{array}{rl}\alpha_{1}^{\prime\prime}=&-\omega^{2}\cos(\alpha_{1})\sin (\alpha_{1})-\omega^{2}(\sin\alpha_{1}\cos\alpha_{1})\\ &-(\cos(\alpha_{2})\sin(\alpha_{1})-\cos(\alpha_{1})\sin(\alpha_{2}))\frac{T_ {2}}{m_{1}I_{1}}\end{array}  \tag{28}\]  
\[ \begin{array}{rl}\theta_{2}^{\prime\prime}=2\omega\tan(\alpha_{2})\alpha_{2 }^{\prime}\end{array}  \tag{29}\]  
\[ \begin{array}{rl}\alpha_{2}^{\prime\prime}=&-\omega^{2}\cos(\alpha_{2})\sin (\alpha_{2})-\omega^{2}(3\sin\alpha_{2}\cos\alpha_{2})\\ &-(\cos(\alpha_{1})\sin(\alpha_{2})-\cos(\alpha_{2})\sin(\alpha_{1}))\frac{T_ {1}}{m_{1}I_{2}}\end{array}  \tag{30}\]  
Equations (27) and (30) indicate that, in this scenario, the acceleration of the in-plane angle is influenced by the Coriolis acceleration generated by the relative motion of the in-plane and out-of-plane angles. Additionally, the velocity of the in-plane angle is affected by the motion of the out-of-plane angle. Energy transfer between in-plane and out-of-plane angular motion occurs via centripetal angular acceleration, expressed as \(\omega^{2}\cos(\alpha_{i})\sin(\alpha_{i}),i=1,2\). In more detail, in this case, we assume that the initial out-of-plane angles are any nonzero angles, and the effect of the gravity gradient causes the subsatellites to be \"pulled\" to the position along the local vertical. This effect causes out-of-plane motion to occur and produces out-of-plane angular velocity. Thus, Equations (27) and (29) are nonzero for \(2\omega\tan(\alpha_{i})\alpha_{i}^{\prime}\
eq 0\) and lead to in-plane motion. Finally, the transfer of energy from out-of-plane motion to in-plane motion is realized. Case 2: In this case, the dynamic characteristics of in-plane roll motion are analyzed by assuming there is no initial out-of-plane motion. The out-of-plane libration angles and velocities are defined as \(\alpha_{1}=\alpha_{2}=\alpha_{1}^{\prime}=\alpha_{2}^{\prime}=0\), and Equations (15)-(18) can be simplified as:  
\[\theta_{1}^{\prime\prime}=-3\omega^{2}\cos\theta_{1}\sin\theta_{1}-\frac{1}{ l_{1}}\bigg{\{}\sin(\theta_{1}-\theta_{2})\frac{T_{2}}{m_{1}}\bigg{\}} \tag{31}\]  
\[\alpha_{1}^{\prime\prime}=0 \tag{32}\]  
\[\theta_{2}^{\prime\prime}=-3\omega^{2}\cos\theta_{2}\sin\theta_{2}+\frac{1}{ l_{2}}\bigg{\{}\sin(\theta_{1}-\theta_{2})\frac{T_{1}}{m_{1}}\bigg{\}} \tag{33}\]  
\[\alpha_{2}^{\prime\prime}=0 \tag{34}\]  
Figure 3: Equilibrium configurations: (**a**) parallel to x-axis; (**b**) parallel to y-axis; (**c**) parallel to z-axis. Equations (32) and (34) demonstrate that the out-of-plane angular acceleration is constantly zero in this situation. The in-plane motion is solely influenced by gravitational and tensional acceleration, with no coupling relationship with out-of-plane motion. Accordingly, the out-of-plane motion does not receive any energy injection; as a result, the out-of-plane libration angles remain constant at all times, as given by \(\alpha_{1}=\alpha_{2}=0\). These conclusions are also verified in the simulation in Section 3.2. Furthermore, in both cases, tension is not the main factor affecting the energy transfer between in-plane and out-of-plane motion, although there is tension in libration motion.