# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 4 Dynamic Response Analysis of the EDT System
### 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. Figure 11: Comparison of the electrodynamic characteristics for different tether diameters: (**a**) average current and (**b**) Lorentz force. Figure 10: Comparison of the orbital motion of the EDT system for different tether diameters: (**a**) 0.5 mm, (**b**) 1 mm, and (**c**) 1.5 mm. The control conditions for the EDT system are expressed as follows:  
\[C_{l}=\mathbf{F}_{l}\cdot\mathbf{v}_{G} \tag{34}\]  
where \(\mathbf{F}_{l}\) is the Lorentz force vector mentioned above and \(\mathbf{v}_{G}\) 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  
\[I=\left\{ \begin{aligned} I(t)&\quad\text{for}\;\;C_{l} \leq 0\\ 0&\quad\text{for}\;\;C_{l}>0\end{aligned} \right. \tag{35}\]  
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.  
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\({}^{\circ}\). 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.