Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture

Abstract

This study focuses on the dynamic responses and libration angle control of an electrodynamic tether (EDT) system with a flexible tether that was designed for space debris capture. Utilizing the absolute nodal coordinate formulation (ANCF), we develop a comprehensive three-dimensional model to accurately capture the complex behavior of the EDT system, particularly the effects of tether deformation and electrodynamic forces. The analysis reveals that the behavior of the EDT system is strongly influenced by the Earth’s magnetic field in space, while the current generated in the system depends on the length and diameter of the tether—key parameters for optimizing performance. The analysis reveals that the current generated in the EDT system is influenced by tether length and diameter, which are critical parameters for optimizing system performance. Furthermore, the study examines the feasibility of using on–off switch control to manage the libration angles induced during debris capture. The results demonstrate that the EDT system can effectively stabilize libration angles across various debris masses and velocities, highlighting its potential for enhancing the efficiency and reusability of space debris removal missions. This research contributes to the understanding of flexible tether dynamics in EDT systems and provides insights into their application in sustainable space operations.

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 reaches an 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. Liou and Johnson [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. Shan et al. [5] categorized debris based on its characteristics and evaluated different capture and removal techniques. Zhang et al. [6] investigated the dynamics of a tether–net–robot system, proposing control strategies for effective debris capture. Ru et al. [7] demonstrated the capability of flexible nets to stably capture rotating debris of irregular shapes and remove them from orbit. Aslanov and Yudintsev [8] analyzed the dynamics of space tugs interacting with debris using tethers, presenting effective methodologies for large debris removal. Phipps et al. [9] proposed a laser-based system to generate plasma jets on debris, facilitating their deorbiting. Hu and Yang [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. Kumar et al. [11] explored the use of angular momentum during tether retrieval for payload repositioning. Williams et al. [12] developed tether control algorithms for payload rendezvous and capture. Zhao et al. [13] analyzed tether dynamics under chemical propulsion, demonstrating the effectiveness of tension control in suppressing libration angles. Jung et al. [14] and Chen 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. Ishige et al. [16] compared rigid and flexible models to analyze orbital descent efficiency. Li and Zhu [17] utilized the nodal position finite element method to accommodate large rotations and proposed improved numerical approaches for electrodynamic tether (EDT) systems. Lim and Chung [18] applied the absolute nodal coordinate formulation (ANCF) to model large deformations during debris capture. Si et al. [19] investigated the impacts of net self-collisions in tether–net systems. Shan and Shi [20] compared the behaviors in post-capture towing scenarios using the dumbbell model, the lumped-mass model, and the ANCF model.

EDT systems generate Lorentz forces through the interaction of tether currents with Earth’s magnetic field, enabling orbital control and debris removal. Sanmartin et al. [21] introduced an exposed tether system for efficient electron collection, applying the orbital motion limited (OML) theory. Ahedo et al. [22] studied orbital departure strategies, while Zhu and Zhong [23] modeled orbital dynamics considering perturbations. Luo et al. [24] proposed a current control method to stabilize EDT system vibrations. Li and Zhu [25] applied the finite element method to the OML theory for electron harvesting to establish discretized OML equations. Yao and Sands [26] demonstrated EDT system applications for microsatellites, achieving fuel-free orbit maintenance. Wang et al. [27] investigated the electrical current control of a spinning bare electrodynamic tether system during its spin-up process. Gao et al. [28] developed an advanced rigid–flexible coupling dynamic model for tethered satellite systems to improve space debris management. Li et al. [29] focused on designing collision-avoidance strategies for spinning electrodynamic tether systems utilizing electrodynamic forces.

Our investigation reveals that few studies have thoroughly analyzed the dynamic behavior of flexible tethers in EDT systems following the capture of space debris using electrodynamic forces. The majority of existing research assumes a rigid tether in EDT systems, thereby neglecting the impact of tether deformations. Moreover, prior studies on EDT systems have predominantly focused on stabilizing tether dynamics during deployment and facilitating the deorbiting of space debris. Although the dynamic behavior induced by debris capture has been explored for flexible tether systems, the extension of this research to stabilization strategies utilizing electrodynamic forces remains limited. Effective control of the tumbling motions induced during debris capture is crucial for the reusability of tether systems in debris removal missions. Consequently, a comprehensive study of the dynamic behavior of EDT systems in debris capture scenarios necessitates modeling that accounts for both tether flexibility and the electrodynamic forces acting on the system.

The innovative contributions of this study can be summarized as follows:

• This study develops a three-dimensional dynamic model that accurately represents the behavior of an EDT system in space debris capture. The model incorporates tether flexibility, electrodynamic forces, and the effects of Earth’s magnetic field, providing a more comprehensive and realistic analysis of EDT dynamics.
• This research establishes the influence of tether length and diameter on current generation, Lorentz forces, and system dynamics. The findings offer essential design guidelines to optimize the efficiency and effectiveness of EDT systems for space debris removal.
• The study demonstrates that applying Lorentz force control through an on–off switch mechanism can effectively suppress the libration angles induced during debris capture. This capability significantly enhances mission reliability and the reusability of EDT systems in active debris removal operations.
• By analyzing the complex interactions between tether deformation, electrodynamic forces, and orbital mechanics, this research expands the theoretical framework for EDT-based space missions. The insights gained contribute to improving stabilization strategies and advancing sustainable space operations.

The paper is organized as follows. Section 2 outlines the three-dimensional modeling of the EDT system. Section 3 describes the discretization using ANCF and the establishment of nonlinear motion equations using Lagrange’s equation, with responses calculated using the fourth-order Runge–Kutta method. Section 4 examines the electrodynamic characteristics and dynamic behavior of the tether, focusing on stabilization. Finally, Section 5 presents the conclusions.

2. Dynamics of a Flexible EDT System

As depicted in Figure 1, the EDT system, consisting of a main satellite, a sub-satellite, a bare tether, and an electron emitter, generates an electromotive force (EMF) as it orbits Earth, interacting with the magnetic field of Earth. This interaction induces a potential difference along the tether, enabling the collection of electrons from the surrounding plasma environment. These electrons are then transported through the tether and emitted via an electron emitter of an electric field emission type installed on the sub-satellite. The current flowing through the tether interacts with the magnetic field of Earth to generate a Lorentz force. In a passive tether system, this Lorentz force opposes the direction of the tether motion, contributing to the overall functionality of the system.

As shown in Figure 2, the main satellite has a mass $m_m$, the sub-satellite has a mass $m_n$, and the tether has a length $L$. Both satellites are treated as point masses, with the mass moment of inertia neglected. Due to the tether length being substantially greater than its diameter, it is modeled as a string. The position along the tether is denoted by the variable $s$, where $s=0$ corresponds to the location of the main satellite and $s=L$ corresponds to the location of the sub-satellite. A global coordinate system is established to describe the position of the EDT system in orbit, with the center of Earth as its origin $O$. The $X$ axis points towards the vernal equinox, the $Z$ axis towards the North Pole, and the $Y$ axis is determined by the right-hand rule. The position vector $r$, representing a point along the tether, is a function of arc length $s$ and time $t$, expressed as follows:

$$r\left(s,t\right)=x\left(s,t\right)i+y\left(s,t\right)j+z\left(s,t\right)k$$

(1)

where $i$, $j$, and $k$ are unit vectors along the $X$-, $Y$-, and $Z$-axis directions, respectively.

The equation of motion for the EDT system is derived using the Lagrange’s equation. First, the kinetic energy of the EDT system is the sum of the kinetic energies of the main satellite, the sub-satellite, and the tether. The total kinetic energy of the EDT system is given by

$$K={\displaystyle \frac{1}{2}}{m}_m{\dot{r}}_m\cdot {\dot{r}}_m+{\displaystyle \frac{1}{2}}{m}_n{\dot{r}}_n\cdot {\dot{r}}_n+{\displaystyle \frac{1}{2}}{{\displaystyle \int }}_0^L\rho A\dot{r}\cdot \dot{r}\mathrm{d}s$$

(2)

where the superposed dot represents differentiation with respect to time, $\rho$ and $A$ are the density and cross-sectional area of the tether, respectively, and $r_m$ and $r_n$ are the position vectors of the main satellite and sub-satellite. The first two terms in Equation (2) represent the kinetic energies of the satellites and the last term represents the kinetic energy of the tether.

The potential energy of the EDT system is derived as the sum of gravitational potential energy and strain energy. Using the Kelvin–Voigt model [30], the stress $\sigma$ within the tether can be expressed as a function of the strain ε and strain rate $\dot{\epsilon }$ as follows:

$$\sigma =E\left(\epsilon +\alpha \dot{\epsilon }\right)$$

(3)

where $E$ and $\alpha$ are Young’s modulus and the internal damping coefficient of the tether, respectively. The total potential energy of the EDT system may be written as

$$U={\displaystyle \frac{1}{2}}{{\displaystyle \int }}_0^{L}EA{\epsilon }^2\mathrm{d}s-GM\left({\displaystyle \frac{m_m}{\left|r_m\right|}}+{\displaystyle \frac{m_n}{\left|r_n\right|}}+{{\displaystyle \int }}_0^L{\displaystyle \frac{\rho A}{\left|r\right|}}\mathrm{d}s\right)$$

(4)

where $G$ is the gravitational constant and $M$ is the mass of the Earth. The values of $G$ and $M$ are given by $G=6.673\times {10}^{-11} \mathrm{N}\cdot {\mathrm{m}}^2/{\mathrm{kg}}^{-2}$ and $M=5.979\times {10}^{24} \mathrm{kg}$, respectively. Using the one-dimensional Green–Lagrange strain, the strain in Equation (4) is given by

$$\epsilon ={\displaystyle \frac{1}{2}}\left(r^{\prime }\cdot r^{\prime }-1\right)$$

(5)

where the prime ($\prime$) denotes differentiation with respect to $s$.

The work done by non-conservative forces such as atmospheric drag, solar radiation pressure, electrodynamic force, damping force, and so on should be considered; however, for simplicity, in this study only the work done by the electrodynamic and damping forces is considered. The virtual work done by these forces can be represented as

$$\delta W={{\displaystyle \int }}_0^L{F}_l\cdot \delta r\mathrm{d}s-{{\displaystyle \int }}_0^{L}EA\alpha \dot{\epsilon }\delta \epsilon \mathrm{d}s$$

(6)

where $\delta$ is the variation notation, $F_l$ is the electrodynamic force, and $\delta r$ the virtual displacement of the tether.

The electromagnetic force in the EDT system is generated by the interaction between the current flowing through the tether and the magnetic field of Earth as the tether moves through the magnetic field. In this study, the 13th IGRF Earth magnetic field model is used to compute electromagnetic force. The magnetic field vector of Earth, $B$, derived from the IGRF model, may be expressed using Legendre polynomials as follows:

$$B={\left\{B_r, B_{{\varphi }^{\prime }}, B_{\theta }\right\}}^{\mathrm{T}}$$

(7)

where $B_r$, $B_{{\varphi }^{\prime }}$, and $B_{\lambda }$ are the components of the magnetic field strength in spherical coordinates:

$$B_r={\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=0}^n{\left({\displaystyle \frac{R}{r}}\right)}^{n+2}\left(n+1\right)P_n^m\cos {\varphi }^{\prime }\left(G_n^m\cos m\lambda +H_n^m\sin m\lambda \right)$$

(8)

$$B_{{\varphi }^{\prime }}=-{\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=0}^n{\left({\displaystyle \frac{R}{r}}\right)}^{n+2}\left(G_n^m\cos m\lambda +H_n^m\sin m\lambda \right){\displaystyle \frac{\partial P_n^m\cos {\varphi }^{\prime }}{\partial \varphi \prime }}$$

(9)

$$B_{\theta }=-{\displaystyle \frac{1}{\sin {\varphi }^{\prime }}}{\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=0}^n{\left({\displaystyle \frac{R}{r}}\right)}^{n+2}{P}_n^m\cos {\varphi }^{\prime }\left(-G_n^m\sin m\lambda +H_n^m\cos m\lambda \right)$$

(10)

in which $r$ is the geocentric radius, $R$ is the radius of the Earth, ${\varphi }^{\prime }$ is the co-latitude, $\lambda$ is the longitude, $P_n^m$ is the Schmidt semi-normalized associated Legendre functions of degree $n$ and order $m$, and $G_n^m$ and $H_n^m$ are Gaussian coefficients.

As the EDT system traverses the magnetic field of Earth, the EMF induced in the tether is determined by

$$E_m=\left(v\times B\right)\cdot e_t$$

(11)

where $v$ denotes the relative velocity vector between the tether and the magnetic field and $e_t$ is the unit vector directed from the sub-satellite to the main satellite. This induced EMF causes the EDT system to develop partial positive and negative charges relative to the surrounding ionospheric plasma. The positively charged section of the tether attracts free electrons, while the negatively charged section attracts ions. Ultimately, electrons flowing along the tether are emitted through an electron emitter positioned at the tether end, resulting in an induced current within the EDT system. The current $I$ and voltage $\mathsf{\Phi }$ generated in the EDT are characterized by the OML theory as follows:

$${\displaystyle \frac{\mathrm{d}I}{\mathrm{d}s}}=\left\{\begin{array}{cc}{q}_e{N}_{e}d\sqrt{2q_e\mathsf{\Phi }/m_e}& \mathrm{for} \mathsf{\Phi }>0\\ -\mu q_e{N}_{e}d\sqrt{-2q_e\mathsf{\Phi }/m_e}& \mathrm{for} \mathsf{\Phi }<0\end{array}\right.$$

(12)

$${\displaystyle \frac{\mathrm{d}\mathsf{\Phi }}{\mathrm{d}s}}={\displaystyle \frac{I}{{\sigma }_{c}A}}-E_m$$

(13)

where $q_e$ is the electron charge, $N_e$ is the ionospheric plasmas density, $d$ is the tether diameter, $m_e$ is the mass of electron, $m_i$ is the mass of ion, $\mu$ is the mass ratio of electrons to ions, and ${\sigma }_c$ is the electrical conductivity of the tether. Equations (12) and (13) are solved numerically by the golden section method with the following boundary conditions:

$$I=0 \mathrm{at} s=s_A=0$$

(14)

$$\mathsf{\Phi }=0 \mathrm{at} s=s_B$$

(15)

$$V_{CC}+Z_T{I}_C=E_m\left(L-s_B\right)-{{\displaystyle \int }}_{s_B}^L{\displaystyle \frac{I}{\sigma A_t}}\mathrm{d}s \mathrm{at} s=s_C=L$$

(16)

where $I_C$ and $V_{CC}$ represent the current and voltage of the cathode at end of the tether and $Z_T$ is the equivalent impedance of the emitter electric circuit. Once the current and magnetic field are established, the electrodynamic force can be expressed as

$$F_l={{\displaystyle \int }}_0^L\left(e_t\times B\right)I\left(s\right)\mathrm{d}s$$

(17)

3. Discretized Equations of Motion

To derive the discretized equations of motion for the EDT system, the kinetic and potential energies must be discretized and substituted into Lagrange’s equations. The EDT system is discretized into $N$ two-node elements between the main satellite and the sub-satellite using the ANCF, as depicted in Figure 3a. Figure 3b illustrates the $e$-th two-node element. The position vector $r_e$ within element $e$ can be represented using the shape function $N$ and the element displacement vector $d_e$ as follows:

$$r_e\left(s,t\right)=N\left(s\right)d_e\left(t\right) \mathrm{for} s_e\le s\le s_{e+1}$$

(18)

$$d_e\left(t\right)={\left\{x_e, y_e, z_e, x_{e+1}, y_{e+1}, z_{e+1}\right\}}^{\mathrm{T}}$$

(19)

Using the coordinate transformation given by

$$\xi =\left(2s-s_e-s_{e+1}\right)/l_e \mathrm{for} -1\le \xi \le 1$$

(20)

where $\xi$ is the parametric coordinate and $l_e$ is the constant element length defined by $l_e=s_{e+1}-s_e$. The position vector of the $e$-th element can be rewritten as

$$r_e\left(\xi ,t\right)=N\left(\xi \right)d_e\left(t\right) \mathrm{for}-1\le \xi \le 1$$

(21)

where

$$N\left(\xi \right)={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}\begin{array}{ccc}1-\xi & 0& 0\\ 0& 1-\xi & 0\\ 0& 0& 1-\xi \end{array}& \begin{array}{ccc}\xi -1& 0& 0\\ 0& \xi -1& 0\\ 0& 0& \xi -1\end{array}\end{array}\right]$$

(22)

Substituting Equation (21) into Equations (2) and (4) and (6), the discretized kinetic energy, potential energy, and the virtual work done by the non-conservative forces can be represented by

$$K={\displaystyle \sum }_{e=1}^N\left({\displaystyle \frac{1}{2}}{\delta }_{e1}{m}_m{\left.{\dot{d}}_e^{\mathrm{T}}{N}^{\mathrm{T}}N{\dot{d}}_e\right|}_{\xi =-1}+{\displaystyle \frac{1}{2}}{\delta }_{eN}{m}_n{\left.{\dot{d}}_e^{\mathrm{T}}{N}^{\mathrm{T}}N{\dot{d}}_e\right|}_{\xi =1}+{\displaystyle \frac{l_e}{4}}{{\displaystyle \int }}_{-1}^1\rho A{\dot{d}}_e^{\mathrm{T}}{N}^{\mathrm{T}}N{\dot{d}}_e\mathrm{d}\xi \right)$$

(23)

$$U={\displaystyle \sum }_{e=1}^N\left({\displaystyle \frac{l_e}{4}}{{\displaystyle \int }}_{-1}^{1}EA{\epsilon }^2\mathrm{d}\xi -{\left.{\displaystyle \frac{{\delta }_{e1}GM{m}_m}{\sqrt{d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e}}}\right|}_{\xi =-1}+{\left.{\displaystyle \frac{{\delta }_{eN}GM{m}_n}{\sqrt{d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e}}}\right|}_{\xi =1}+{\displaystyle \frac{l_e}{2}}{{\displaystyle \int }}_{-1}^1{\displaystyle \frac{GM\rho A}{\sqrt{d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e}}}\mathrm{d}\xi \right)$$

(24)

$$\delta W={\displaystyle \sum }_{e=1}^N\delta d_e^{\mathrm{T}}\left[{\displaystyle \frac{l_e}{2}}{{\displaystyle \int }}_{-1}^1{N}^{\mathrm{T}}{F}_l\mathrm{d}\xi -{\displaystyle \frac{EA\alpha l_e}{2}}{{\displaystyle \int }}_{-1}^1{\left({\displaystyle \frac{\partial \epsilon }{\partial d_e}}\right)}^{\mathrm{T}}\dot{\epsilon }\mathrm{d}\xi \right]$$

(25)

where ${\delta }_{ij}$ is the Kronecker delta function. The discretized equations of motion are derived by substituting Equations (23)–(25) into the following Lagrange’s equation:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\left({\displaystyle \frac{\partial K}{\partial {\dot{d}}_e}}\right)}^{\mathrm{T}}-{\left({\displaystyle \frac{\partial K}{\partial d_e}}\right)}^{\mathrm{T}}+{\left({\displaystyle \frac{\partial U}{\partial d_e}}\right)}^{\mathrm{T}}={\left({\displaystyle \frac{\partial \delta W}{\partial d_e}}\right)}^{\mathrm{T}} \mathrm{for} e=1, 2 ,\cdots , N$$

(26)

The derived discretized equations of motion for the EDT system can be expressed as

$$m_e{\ddot{d}}_e+q_e^l+q_e^g+q_e^d=f_e^L, \mathrm{for} e=1, 2, \cdots , N$$

(27)

where $m_e$ is the element mass matrix, $q_e^l$ is the element longitudinal elastic force vector, $q_e^g$ is the element gravitational force vector, $q_e^d$ is the element internal damping force vector, and $f_e^L$ is the element electrodynamic force vector. The matrix and vectors are given by

$$m_e={\delta }_{e1}{m}_m{\left.N^{\mathrm{T}}N\right|}_{\xi =-1}+{\delta }_{eN}{m}_n{\left.N^{\mathrm{T}}N\right|}_{\xi =1}+{\displaystyle \frac{l_e}{2}}{{\displaystyle \int }}_{-1}^1\rho A{N}^{\mathrm{T}}N\mathrm{d}\xi$$

(28)

$$q_e^l=\left({\displaystyle \frac{4EA}{l_e^3}}{{\displaystyle \int }}_{-1}^1\left(d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e\right)N^{\mathrm{T}}N\mathrm{d}\xi -{\displaystyle \frac{EA}{l_e}}{{\displaystyle \int }}_{-1}^1{N}^{\mathrm{T}}N\mathrm{d}\xi \right)d_e$$

(29)

$$q_e^g=\left({\displaystyle \frac{l_e}{2}}{{\displaystyle \int }}_{-1}^1{\displaystyle \frac{GM\rho A{N}^{\mathrm{T}}N}{{\left(d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e\right)}^{3/2}}}\mathrm{d}\xi +{\left.{\displaystyle \frac{{\delta }_{e1}GM{m}_s{N}^{\mathrm{T}}N}{{\left(d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e\right)}^{3/2}}}\right|}_{\xi =-1}+{\left.{\displaystyle \frac{{\delta }_{eN}GM{m}_d\left.N^{\mathrm{T}}N\right|}{{\left(d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{d}_e\right)}^{3/2}}}\right|}_{\xi =1}\right)d_e$$

(30)

$$q_e^d=\left[{\displaystyle \frac{8EA\alpha }{l_e}}{{\displaystyle \int }}_{-1}^1\left(d_e^{\mathrm{T}}{N}^{\mathrm{T}}N{\dot{d}}_e\right){N^{\prime }}^{\mathrm{T}}{N}^{\prime }\mathrm{d}\xi \right]d_e$$

(31)

$$f_e^L={\displaystyle \frac{l_e}{2}}{{\displaystyle \int }}_{-1}^1{N}^{\mathrm{T}}{F}_l\mathrm{d}\xi$$

(32)

The equations of motion for each element, given by Equation (27), are assembled to form the global matrix–vector equation. Subsequently, the fourth-order Runge–Kutta time integration method is applied to this equation to obtain the dynamic response of the EDT system.

4. Dynamic Response Analysis of the EDT System

4.1. Comparison of the ANCF Model with a Dumbbell Model

Before analyzing the dynamic behavior of the EDT system, the dynamic responses of the proposed ANCF model are compared with those of the simplified dumbbell model. In the dumbbell model, the tether is idealized as massless and straight, with no deformation, resulting in a uniform Lorentz force acting along the entire tether length. Conversely, the ANCF model accounts for large tether deformations, causing variations in the Lorentz force direction on each tether element. These differences significantly impact the forces and behaviors observed in the EDT system. Therefore, a comparative analysis of the dumbbell and ANCF models is conducted to highlight the unique insights provided by the ANCF model.

The parameters used for both models are as follows: the main satellite and sub-satellite masses are $m_m=10,000 \mathrm{kg}$ and $m_n=50 \mathrm{kg}$, respectively; the tether length and diameter are $L=10 \mathrm{km}$ and $d=0.5 \mathrm{mm}$. The main satellite operates at an altitude of 450 km. The ANCF model has a density of $\rho =1450 \mathrm{kg}/{\mathrm{m}}^3$ and Young’s modulus of $E=131$ GPa. The tether system undergoes circular orbital motion, with the initial velocity calculated using the following equation:

$$v_0\left(s\right)=\left|r\left(s\right)\right|\sqrt{GM/r_c^3}$$

(33)

where $r_c$ is the distance between the center of mass of the EDT system and the center of the Earth.

During circular orbital motion, the dynamic behavior and electrodynamic characteristics of the ANCF model are compared with those of the dumbbell model. Figure 4 illustrates the orbital motion of both models over one orbital period in an equatorial orbit, with the main satellite and sub-satellite represented as green and red circles, respectively. To clearly illustrate the change in attitude, the images are created by enlarging the tether length relative to the orbital radius of the satellites. The ANCF model displays tether deformation due to the electrodynamic forces acting on the EDT system, whereas the dumbbell model represents the tether as a straight line. This distinction arises because the ANCF model exhibits a more subtle behavior than the dumbbell model by taking into account the flexibility and deformation of the tether.

Figure 5 presents the average current generated by both models during circular orbital motion in an equatorial orbit. Despite minor discrepancies, the average current over one orbital period is similar for both models. The variations in current during orbital motion are attributed to changes in the magnetic field, which influence current generation as the system progresses along its orbit. The total computational time required to perform the simulations for Figure 4 and Figure 5 is 631.7 min for the ANCF model and 315.8 min for the dumbbell model. As expected, the ANCF model requires significantly more computation time than the dumbbell model due to its higher complexity.

The libration angles generated by the ANCF model and the dumbbell model are also compared. As depicted in Figure 6, the libration angle is defined as the smallest angle between the local vertical axis, which connects the main satellite to the center of Earth, and an imaginary line linking the main satellite and sub-satellite. Figure 7 compares the libration angles from both models, showing that the ANCF model exhibits larger amplitude variations. This is due to the ANCF model accounting for tether deformation effects induced by electrodynamic forces, unlike the dumbbell model, which treats the tether as a rigid body and neglects deformation effects. These comparisons demonstrate that the ANCF model provides a more accurate analysis of the dynamic responses of the EDT system, highlighting the importance of incorporating tether flexibility in the model.

4.2. Influence of Tether Length and Diameter on EDT System Performance

We examine the effects of the tether length and diameter on the response and electrodynamic characteristics of the EDT system. These parameters significantly impact the mechanical and electrical properties of the system, including the electron collection area, stiffness, and mass. The reference values of the EDT system properties used in the analysis are summarized in

Table 1. Reference values of the EDT system properties.

	Symbols	Values
Mass of main satellite	$m_m$	10,000 kg
Mass of sub-satellite	$m_n$	50 kg
Length of tether	$L$	10 km
Diameter of tether	$d$	0.5 mm
Mass density of tether	$\rho$	1450 kg/m3
Young’s modulus of tether	$E$	131 GPa

. The analysis considered tether lengths of 5, 10, and 20 km, and tether diameters of 0.5, 1, and 1.5 mm.

Figure 8 illustrates the orbital motion of the EDT system for different tether lengths. In Figure 8a, the 5 km tether shows minimal deformation, with the main satellite and sub-satellite maintaining a stable circular orbit. As depicted in Figure 8b, increasing the tether length to 10 km results in noticeable deformation and the onset of libration angles within the system. Figure 8c shows that at a tether length of 20 km, the deformation is more pronounced, with longer tethers exhibiting enhanced interactions with the magnetic field of Earth, generating higher currents and Lorentz forces.

Figure 9 presents the average current and Lorentz force for the three tether lengths. In Figure 9a, the 5 km tether exhibits limited interaction with the magnetic field of Earth, generating low current. The 10 km tether produces a higher current, with an increasing trend as the tether length extends. The 20 km tether generates the highest current due to the expanded electron collection area. Similarly, Figure 9b shows the Lorentz force increasing with tether length. The difference in Lorentz force between the 10 and 20 km tethers is more pronounced than the current difference, as the Lorentz force is proportional to both the current and tether length.

The effects of tether diameter on dynamic behavior and electrodynamic characteristics are shown in Figure 10, which depicts the orbital motion of the EDT system for tether diameters of 0.5, 1, and 1.5 mm. The tether diameter influences the mechanical stiffness and mass, affecting tether deformation and dynamic response. As the diameter increases, more pronounced tether behavior is observed (Figure 10), leading to greater instability in the orbit of the sub-satellite and increased tether deformation. For a 1.5 mm diameter, the Lorentz force becomes significantly large, causing the orbit of the sub-satellite to deviate completely from its circular trajectory.

Figure 11 illustrates the average current and Lorentz force for varying tether diameters. Figure 11a shows that the average current increases with larger diameters. For diameters of 0.5 mm and 1 mm, the current remains relatively stable. However, at a diameter of 1.5 mm, significant tether deformation leads to intervals of reduced current. In Figure 11b, a similar trend can be observed for the Lorentz force. While it grows with the current, tether deformation at larger diameters causes intervals of sharply reduced Lorentz force. At 1.5 mm, pronounced tether vibrations toward the end of the orbit cause more significant variations in the Lorentz force. In conclusion, both tether length and diameter substantially impact the dynamic response and electrodynamic characteristics of the EDT system. Longer and thicker tethers increase the current generation and Lorentz force but also lead to greater tether deformation and dynamic instability.

4.3. Analysis of Libration Angle Stabilization in EDT Systems

In this section, we analyze libration angle stabilization using electrodynamic force. Various factors encountered during space missions can affect the stability of the EDT system. For example, in space debris collection missions, the impact of capturing fast-moving debris can induce libration angles in the system, which in turn influence the dynamic behavior of the tether and current generation.

To effectively mitigate the libration angles generated in the EDT system, electrodynamic forces are utilized in conjunction with a simple on–off switch control strategy. This method adjusts the libration angles by enabling or disabling current flow under specific conditions, providing a straightforward and effective approach to improving system stability. The on–off switch control regulates the electrodynamic force, ensuring that it acts in the opposite direction to the libration of the tether system. This regulation is achieved by leveraging the velocity of the center of mass for the system and the Lorentz force.

The control conditions for the EDT system are expressed as follows:

$$C_l=F_l\cdot v_G$$

(34)

where $F_l$ is the Lorentz force vector mentioned above and $v_G$ represents the velocity vector of the center of mass. The sign of $C_l$ is determined based on the angle between the Lorentz force vector and the velocity vector of the center of mass. Depending on the sign of $C_l$, the current control law is defined as

$$I=\left\{\begin{array}{cc}I\left(t\right)& \mathrm{for} C_l\le 0\\ 0& \mathrm{for} C_l>0\end{array}\right.$$

(35)

The mass and velocity of debris traveling parallel to the EDT system in the same orbital plane are varied to analyze libration angle control under various scenarios. The momentum exchange occurring during space debris collection is referenced from a previous study [18]. The system parameters of the EDT system remain consistent with those used in prior simulations, with a tether diameter of $d=1$ mm and tether length $L=10$ km. The mass and velocity of the space debris for three simulation cases are provided in

Table 2. Mass and velocity of space debris for three simulation cases.

	Mass	Velocity
Case 1	50 kg	7800 m/s
Case 2	100 kg	7800 m/s
Case 3	50 kg	7900 m/s

.

The libration angles of the EDT system are analyzed for varying debris masses and velocities. Figure 12a presents the dynamic responses of the libration angle for Case 1. In this figure, the black line represents the dynamic response without control (i.e., without current), while the red line represents the response with current-based control. Immediately after debris capture, the amplitudes of the libration angle in both cases (with and without control) are similar. However, over time, a clear distinction emerges. In the absence of current, the amplitude of the libration angle remains constant, whereas with current applied, the amplitude decreases and eventually stabilizes at a lower value. The oscillation is damped, and the amplitude converges to below 1°. These results demonstrate that when the EDT system is controlled with current, the amplitude of the libration angle decreases over time, ultimately achieving dynamic stabilization of the system.

To investigate the effect of debris mass on the dynamic response of the libration angle, simulations are conducted for Case 2, in which the debris mass is increased from 50 kg in Case 1 to 100 kg. The dynamic responses of the libration angle for Case 2 are shown in Figure 12b. Similar to Figure 12a, the black and red lines in Figure 12b represent the cases with and without current-based control, respectively. The amplitude of the libration angle in Case 2 without control is nearly identical to that of Case 1. However, an interesting observation is that the time required for the amplitude to decrease when current-based control is applied in Case 2 is longer than in Case 1.

Figure 12c illustrates the dynamic response of the libration angle for Case 3, where the debris velocity is increased to 7900 m/s, while the mass remains unchanged compared to Case 1. In comparison to Cases 1 and 2, the amplitude of the libration angle response without current-based control is significantly larger in Case 3, indicating that debris velocity has a significant impact on the amplitude of the response. Furthermore, the time required for amplitude reduction with current-based control in Case 3 is longer than in Cases 1 and 2. In conclusion, for all three cases, the libration angle of the EDT system can be reduced by applying current-based control.

5. Conclusions

This study investigates the dynamic responses and libration angle control capabilities of an EDT system for space debris capture. By employing the ANCF, we develop a three-dimensional model that accurately captures the complex behavior of the EDT system, particularly the effects of tether deformation and electrodynamic forces. Additionally, the study explores the electrodynamic characteristics of the system and examines the feasibility of on–off switch control for stabilizing the libration angles induced by debris capture. The findings demonstrate that the EDT system effectively manages libration angles across varying debris masses and velocities, highlighting its potential for enhancing the efficiency and repeatability of space debris removal missions.

The key conclusions are as follows: (1) The developed three-dimensional EDT system model, based on ANCF, accurately represents the intricate dynamics of the system, emphasizing the crucial role of tether deformation and electrodynamic forces. (2) The current generated in the EDT system is directly influenced by tether length and diameter, with longer and thicker tethers producing higher currents, underscoring the importance of these parameters in system performance. (3) Tether length significantly affects both current generation and Lorentz force distribution, where an increase in length leads to higher current and a more extensive force distribution, thereby enhancing the total Lorentz force. (4) Tether diameter plays a key role in optimizing system efficiency, as a larger diameter increases the electron collection capability, resulting in higher current generation. (5) In the absence of electrodynamic forces, the libration angle remains unchanged after debris capture. However, applying the Lorentz force through the EDT system and implementing on–off switch control effectively dampens and stabilizes libration angles across various debris masses and velocities, demonstrating the feasibility of this approach for mission stability.

These findings contribute to a deeper understanding of flexible tether dynamics in EDT systems and provide valuable insights for optimizing space debris removal strategies using electrodynamic forces.