## 4 Conclusions  
In this paper, a two-piece dumbbell mathematical model with deployment/retrieval is developed based on [PERSON]'s law. The proposed model extends the existing dynamic models by incorporating out-of-plane roll motion and in-plane pitch motion with deployment for a TB-TSS. The dynamic characteristics of the TB-TSS under both fixed and varying tether lengths due to release are investigated. Additionally, a deployment strategy is proposed to suppress the libration motion during deployment. Numerical simulations are conducted to examine the influence of various conditions on the system's dynamics. The simulation results yield several important conclusions, as follows:  
(1) Three equilibrium configurations can be acquired for the fixed-length tethers of a TB-TSS in circular orbit. One configuration is stable, with both tethers aligned along the local vertical direction of the Earth. The other two configurations are unstable, parallel to the y-axis and z-axis of the LVLH frame.  
Figure 12: (a) Phase diagram of \(\theta_{1}-\theta_{2}\). (b) Phase diagram of \(\alpha_{1}-\alpha_{2}\).  
Figure 13: Tension of two tethers.  
(2) For a fully deployed TB-TSS, the simulation results indicate a close relationship between the peak value of tether tension and the initial attitude angles. Furthermore, the simulation shows that libration energy can be delivered from out-of-plane motion to in-plane motion when out-of-plane libration occurs, while libration energy does not transfer from in-plane to out-of-plane when only in-plane libration occurs.  
(3) The predesigned deployment strategy for length-rate control can suppress the libration of the system in the process of effective deployment. Furthermore, the out-of-plane motion rapidly converges to zero, while the in-plane motion retains a purely sinusoidal oscillation.  
In summary, the coupling dynamic responses of the TB-TSS are analyzed and discussed with a six-DOF dynamic model. The results have practical applications for guiding the deployment and suppressing the libration of a TB-TSS in circular orbit, as well as for designing stability controllers. However, it is assumed that the system follows a planar circular orbit and ignores the influence of external space perturbations, such as solar radiation pressure, atmospheric drag, and J2 perturbation. The dynamic characteristics of the TB-TSS can be further discussed for some complex practical scenarios, such as considering the influence of the orbital motion of the main satellite and external space perturbations. In addition, the proposed deployment strategy is an open-loop strategy, which may produce unstable output and cause mission failure. An improved closed-loop control scheme could be further designed to guarantee the robustness and control performance of the TB-TSS.  
Conceptualization, T.H. and Z.Z.; methodology, T.H.; software, T.H.; validation, T.H. and Z.Z.; formal analysis, T.H.; investigation, T.H.; resources, Z.Z.; data curation, T.H.; writing--original draft preparation, T.H.; writing--review and editing, T.H. and Z.Z.; visualization, T.H.; supervision, Z.Z.; project administration, Z.Z.; funding acquisition, Z.Z. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflicts of interest.