# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 1 Introduction  
Satellite systems have undergone significant advancements over the past few decades, driven by the increasing demand for robust and efficient space exploration and communication technologies. Among various configurations, tethered satellite systems have garnered substantial attention due to their potential applications in space debris removal, intersatellite communication, and satellite service missions. The tethers used in these systems can be classified into four types: momentum exchange tethers, gravity gradient stabilization tethers, space elevator tethers, and electrodynamic tethers. A typical tethered satellite system comprises two or more satellites connected by a flexible tether, allowing for unique dynamics and control opportunities absent in free-flying satellite formations. A comprehensive review [1] has recently been published, detailing the dynamics of tethered systems in various space applications. It encompasses theoretical foundations, modeling techniques, control strategies, and practical implementations, highlighting advancements and challenges in the field. Research on tethered satellite systems can be classified according to the number of satellites connected by tethers. Systems with two satellites connected by a tether are referred to as two-body-tethered satellites [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16], those with three connected satellites are termed three-body tethered satellites [17; 18; 19; 20; 21], and those with multiple connected satellites are known as multibody tethered satellites [22]. Furthermore, the research can be divided between studies on dumbbell models, which do not consider the transverse deformation of the tether [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 17; 18; 19; 20; 21; 22], and studies on flexible tether models, which account for this deformation [14; 15; 16]. Numerous studies have explored the dynamics of tethered satellite systems using a two-body dumbbell model, where the tether linking two satellites is assumed to be massless and undeformed. [PERSON] and [PERSON] [PERSON] and [PERSON] (1973) investigated the controllability of a tethered geosynchronous satellite solely through tether length adjustments in a central inverse squared gravity field. [PERSON] et al. [PERSON] et al. (2018) determined the in-plane periodic solutions of a dumbbell-shaped satellite system in elliptical orbits via bifurcation with respect to orbital eccentricity. [PERSON] et al. [PERSON] et al. (2018) derived analytical first-order solutions for the librational angles of a tethered satellite subjected to small, continuous, and constant thrust in the transverse, tangential, radial, and binormal directions. [PERSON] et al. [PERSON] et al. (2018) developed dynamic equations for a tethered system during orbital maneuvers and analyzed the equilibrium position stability for in-plane angles during omnidirectional maneuvering in the orbital plane. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) addressed the deployment control of a tethered satellite system to rendezvous a sub-satellite with a target object orbiting at a lower altitude. [PERSON] et al. [PERSON] et al. (2018) formulated dynamic equations for a tethered satellite system during orbital maneuvers and designed a controller using the small-gain theorem based on a control-oriented model. [PERSON] et al. [PERSON] et al. (2018) investigated the chaotic behavior of an in-plane tethered satellite system influenced by atmospheric drag and the Earth's oblateness. Some studies treat two-body tethered satellites as dumbbell models, where the tether has no mass but is longitudinally elastic. [PERSON] et al. [PERSON] et al. (2018) studied the integrability of an elastic satellite model whose center of mass moved in a circular Keplerian orbit around the center of gravity. They modeled the satellite as two-point masses connected by an extensible massless spring that obeyed [PERSON]'s law. [PERSON] et al. [PERSON] et al.