# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 1 Introduction  
The sustainability of the space environment is increasingly threatened by the proliferation of space debris. The rapid expansion of space development activities has led to a significant increase in the number of debris items in Earth's orbit, posing substantial risks to the operation of satellites and the safety of space missions. Currently, over 9000 man-made objects, with a total mass of approximately 5 million kg, are present in Earth's orbit [1]. A significant portion of these objects, particularly those in low Earth orbit (LEO), are defunct and no longer in use (these types of objects constitute approximately 80% of the total number of objects in LEO). The collision of debris fragments could generate additional fragments, exacerbating the issue and elevating the threat level [2, 3]. Therefore, the removal of space debris has become an imperative task to ensure the continued safe operation of space systems. The process of capturing space debris consists of several steps. First, the tether system approaches the space debris and moves to an optimal position for capture. Second, various capture methods, such as robotic arms, tether nets, and harpoon systems, are used to capture the debris. Third, the system gradually deorbits the debris, and once it reachesan appropriate orbit, it releases the debris, allowing it to re-enter the atmosphere and be removed. Finally, the tether system re-boosts to the target orbit for the removal of additional debris. Numerous researchers have explored various methods for the removal of space debris. [PERSON] and [PERSON] [4] assessed the number of debris removals necessary to maintain a stable LEO environment using the long-term orbital debris projection model of NASA, highlighting the effectiveness of active debris removal (ADR) methods. [PERSON] et al. [5] categorized debris based on its characteristics and evaluated different capture and removal techniques. [PERSON] et al. [6] investigated the dynamics of a tether-net-robot system, proposing control strategies for effective debris capture. [PERSON] et al. [7] demonstrated the capability of flexible nets to stably capture rotating debris of irregular shapes and remove them from orbit. [PERSON] and [PERSON] [8] analyzed the dynamics of space tugs interacting with debris using tethers, presenting effective methodologies for large debris removal. [PERSON] et al. [9] proposed a laser-based system to generate plasma jets on debris, facilitating their deorbiting. [PERSON] and [PERSON] [10] suggested a CubeSat-based system for efficient fuel usage in debris removal operations. Among the various technologies, space tether systems offer advantages in efficiency and sustainability for debris removal. As a result, research in this area has intensified. [PERSON] et al. [11] explored the use of angular momentum during tether retrieval for payload repositioning. [PERSON] et al. [12] developed tether control algorithms for payload rendezvous and capture. [PERSON] et al. [13] analyzed tether dynamics under chemical propulsion, demonstrating the effectiveness of tension control in suppressing libration angles. [PERSON] et al. [14] and [PERSON] et al. [15] examined the dynamic behavior and stability of dumbbell model tether systems under varying conditions. The dumbbell model assumes a massless and rigid tether, which limits its applicability. To address tether deformations, advanced models have been developed. [PERSON] et al. [16] compared rigid and flexible models to analyze orbital descent efficiency. [PERSON] and [PERSON] [17] utilized the nodal position finite element method to accommodate large rotations and proposed improved numerical approaches for electrodynamic tether (EDT) systems. [PERSON] and [PERSON] [18] applied the absolute nodal coordinate formulation (ANCF) to model large deformations during debris capture. [PERSON] et al. [19] investigated the impacts of net self-collisions in tether-net systems.