## 1 Introduction  
Since [PERSON] proposed the concept of a tethered satellite system (TSS) in 1895 [1], it has been extended to address many interesting space issues, such as artificial gravity, space debris removal, orbit transfer, and so on. The focus on TSSs has increased due to space missions that have been undertaken to verify this concept. For instance, the Young Engineers' Satellite 2 (YES2) was carried out by Delta-Utec, and the Tethered Satellite System-1R (TSS-1R) was developed by NASA and the Italian Space Agency (ASI) [2, 3]. Furthermore, numerous previous works [4, 5, 6, 7, 8, 9] have addressed the dynamic analysis and control of TSSs. These successful space missions inspire researchers to explore the dynamic modeling, analysis, and control of multitethered satellite formations, which are made up of numerous satellites connected by space tethers [4, 10, 11]. One of the multitethered satellite formations is the three-body tethered satellite system (TB-TSS), which has potential applications in mission scenarios like high atmospheric observations and space transportation [4, 10, 11, 12]. A TB-TSS is generally formed by one main satellite and two substellites connected by tethers, where the main satellite can transport the two substellites to the desired position by deploying and retrieving the tether without thrusters [13]. During both the deployment and retrieval phases, one difficulty associated with the TB-TSS is the libration motion of the substellites (or climber) caused by the Coriolis force due to their movement [14, 15]. Research shows that the movement that occurs may lead to the tumbling of the whole system [16]. Thus, the dynamic characteristics analysis of such systems when deploying, station-keeping, and transporting the climber is pivotal for a successful operation. Due to the fact that the dynamic characteristics of the TB-TSS play a critical role in the process of the successful completion of a scheduled mission, a lot of research has been devoted to investigating the complex dynamics and control of this system in recent years. Based on the assumption of a rigid tether, [PERSON] et al. [17] examined the dynamics of a double-pendulum TB-TSS for the cases of fixed-length and variable-length tethers. Because of its simplicity and adaptable accuracy, the classical two-piece dumbbell model (TPDM) proposed in Ref. [17] is frequently utilized to investigate the dynamic responses and control of TB-TSS. [PERSON] and [PERSON] [18] further studied the libration frequencies and moving effects of climbers in a TB-TSS. [PERSON] et al. [19] analyzed the dynamic responses of a tethered satellite system while the middle body was moving along the tether. In addition, the same authors surveyed the dynamic characteristics of a TB-TSS during the deployment and retrieval phases [20]. For the TB-TSS with moving subsatellites, [PERSON] et al. [21] used [PERSON]'s method to derive the system dynamics equations. An active force is required to drive the middle subsatellite to suppress the potential wobble of the tether. In order to investigate the libration motion of a partial space elevator with a climber moving at a constant speed in a circular orbit, [PERSON], [PERSON], and [PERSON] [22] established the convenient nonlinear dynamics of a TB-TSS. The dynamics of a TB-TSS were investigated and analyzed by taking into account the effects of the climber's climbing velocity, the mass ratio of the climber over the end body, and the end body's deployment and retrieval. Furthermore, for various climbing speeds, a parametric dynamic analysis of the TB-TSS that transfers the middle subsatellite from the lower orbit to the upper orbit at a constant speed was performed in Ref. [13]. Some works further studied the unstable dynamic characteristics of a TB-TSS in an elliptic orbit. [PERSON] et al. [23] presented the chaotic motion of a TB-TSS in an elliptic orbit. [PERSON] et al. [24] investigated the oscillation motion of a TB-TSS with constant-length tethers in a large elliptic orbit.