# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 5 Conclusions  
In this study, we conducted a detailed analysis of the dynamic behavior of a three-body tethered satellite system, where a payload is transported along a flexible tether connecting two satellites. By employing the Absolute Nodal Coordinate Formulation (ANCF) to model tether flexibility, we captured important dynamics such as tether deformation, slack, and rebound--factors often overlooked in traditional dumbbell models. The findings and contributions of this study are summarized as follows:  
1. We developed a comprehensive model of the three-body tethered satellite system using the ANCF. This model successfully illustrated the complex dynamics of the system, revealing how large deformations and the effects of tether slack and rebound are essential for realistic simulations of satellite behavior.
2. The discretized nonlinear equations of motion were derived using [PERSON]'s equation, yielding a global nonlinear matrix-vector form. We solved these equations using the Newmark time integration method, computing dynamic responses that reflect the intricate behavior of the tethered system.
3. Our analysis highlighted the significant role of Coriolis effect in the trajectory of the payload. As the mass and speed of the payload increased, deviations from the straight line connecting the two satellites became pronounced, resulting in substantial tether deformation. Quantitatively, we observed that at higher payload masses (\(\geq\)400 kg) and speeds (\(\geq\)20 m/s), the tether deformation led to potential tumbling, inducing slack and rebound, and thereby risking system destabilization.
4. Initially, as the payload transitioned along the tether, axial force fluctuations were minimal. However, as it reared the other satellite, these fluctuations increased markedly, transitioning between tensile and compressive states. The quantitative results indicated that fluctuations in axial force could reach magnitudes exceeding 400 N, which has critical implications for tether design and operational strategies.  
While this study provides valuable insights, several limitations exist. The ANCF model, while effective in capturing tether flexibility, may not fully account for other dynamic factors, such as nonlinear material properties or the effects of environmental disturbances (e.g., atmospheric drag). Future work should explore enhancements to the model by integrating these elements and conducting more extensive simulations that encompass a wider range of operational scenarios. Additionally, validating the model against experimental data could strengthen its applicability in real-world scenarios and improve its robustness.  
Conceptualization, J.C. and J.L; methodology: W.J. and J.L; software programming, Y.A. and W.J.; validation, Y.A. and J.L.; formal analysis, J.C. and W.J.; investigation, Y.A., W.J. and J.L.; resources, Y.A., W.J. and J.L.; data curation, Y.A.; writing--original draft preparation, Y.A. and W.J.; writing--review and editing, J.L. and J.C.; visualization, Y.A. and W.J.; supervision, J.C.; project administration, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by a grant from the National Research Foundation of Korea (NRF), funded by the Korean Government (MEST) (NRF-2021R1A2C2007979).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.