# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 1 Introduction
(2018) analyzed the dependence of deployment friction on the tether diameter, deployment velocity, and deployment angle. They also evaluated the effects of these deployment parameters on the dynamics of tethered satellites using both extended and unextended dumbbell models. In a study by [PERSON] et al. [PERSON] et al. (2018), the dynamic equations of a tethered system during orbital maneuvering were established based on a dumbbell model of a tethered satellite. Furthermore, considering the elastic strain of the tether in the dynamic model, the effects of the degree of slackness, the initial states of the librational angles, and variations in thrust acceleration on the tethered satellite were analyzed as the tether became slack. Among the studies analyzing the dynamic behavior of two-body tethered satellites, some modeled the tether as lumped masses to consider its mass and lateral deformation. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) investigated the relative dynamics and control of tethered satellites during the deployment process using a dumbbell model with tether mass and a lumped-mass model of a tethered satellite system. [PERSON] et al. [PERSON] et al. (2018) adopted a dumbbell model to analyze the dynamics of a tethered system, fully considering the mass variations in the base satellite. In addition, a lumped-mass model was introduced to reveal the strong coupling between the librational and vibrational characteristics. Recent studies have focused on the dynamic analysis of two-body tethered satellites, considering the lateral deformation of the tether through the absolute nodal coordinate formulation (ANCF) [PERSON] and [PERSON] (2018). [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) examined and compared modified dumbbell, lumped-mass, and ANCF models to describe a tethered post-capture system for space debris removal. [PERSON] et al. [PERSON] et al. (2018) used the ANCF to investigate the deployment of a tether in a tethered satellite system, highlighting complex dynamic properties such as the large deformation, slack, and rebound. [PERSON] et al. [PERSON] et al. (2018) employed variable-length elements based on ANCF to predict the length variations and flexibility of a tether during the retrieval and station-keeping phases. Various studies have examined the dynamics of three-body tethered satellite systems in which three satellites are interconnected by tethers. These analyses primarily focused on the dynamic behavior of the system during tether deployment or retrieval, as well as during the transportation of a payload along the tethers. For example, [PERSON] et al. [PERSON] et al. (2018) employed a two-piece dumbbell model in the dynamic analysis of a tethered satellite system with a moving mass. Similarly, [PERSON] (2018) utilized a two-piece dumbbell model to explore the nonlinear dynamics and control of such systems. [PERSON] et al. [PERSON] et al. (2018) investigated the dynamic characteristics of a three-body tethered satellite system in large elliptical orbits and developed a control strategy to mitigate liberation during orbital transfer. [PERSON] et al. [20] used a two-piece dumbbell model to investigate the dynamics and vibration suppression of a tethered system with a moving climber in a circular orbit. [PERSON] et al. [21] analyzed the nonlinear dynamic behavior of a three-body tethered satellite system during deployment and retrieval by considering the variable orbital radius of the instantaneous mass center of the system. [PERSON] et al. [22] employed a multiple dumbbell model to investigate the dynamics and liberation suppression of multibody-tethered satellites in elliptical orbits, where the system experiences periodic excitation owing to changes in the gravity gradient. The recent literature has shown significant interest in the control of tethered satellite systems, with a variety of approaches being developed to tackle the complexities of their dynamics. [PERSON] et al.