# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 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 [PERSON]'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 \(\mathbf{r}_{e}\) within element \(e\) can be represented using the shape function \(\mathbf{N}\) and the element displacement vector \(\mathbf{d}_{e}\) as follows:  
\[\mathbf{r}_{e}(s,t)=\mathbf{N}(s)\mathbf{d}_{e}(t)\ \text{for}\ s_{e}\leq s \leq s_{e+1} \tag{18}\]  
\[\mathbf{d}_{e}(t)=\{x_{e},\ y_{e},\ z_{e},\ x_{e+1},\ y_{e+1},\ z_{e+1}\}^{ \text{T}} \tag{19}\]Using the coordinate transformation given by  
\[\xi=\left(2s-s_{e}-s_{e+1}\right)/l_{e}\text{ for }-1\leq\xi\leq 1 \tag{20}\]  
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  
\[\mathbf{r}_{e}(\xi,t)=\mathbf{N}(\xi)\mathbf{d}_{e}(t)\text{ for }-1\leq\xi\leq 1 \tag{21}\]  
where  
\[\mathbf{N}(\xi)=\frac{1}{2} \begin{bmatrix}1-\xi&0&0&\xi-1&0&0\\ 0&1-\xi&0&0&\xi-1&0\\ 0&0&1-\xi&0&0&\xi-1\end{bmatrix}  \tag{22}\]  
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  
\[K=\sum_{e=1}^{N}\left(\frac{1}{2}\delta_{e1}m_{m}\mathbf{\dot{d}}_{e}^{\text{ T}}\mathbf{N}^{\text{T}}\mathbf{N}\mathbf{\dot{d}}_{e}\bigg{|}_{\xi=-1}+ \frac{1}{2}\delta_{eN}m_{n}\mathbf{\dot{d}}_{e}^{\text{T}}\mathbf{N}^{\text{T }}\mathbf{N}\mathbf{\dot{d}}_{e}\bigg{|}_{\xi=1}+\frac{l_{e}}{4}\int_{-1}^{1} \rho\dot{\mathbf{d}}_{e}^{\text{T}}\mathbf{N}^{\text{T}}\mathbf{N}\mathbf{ \dot{d}}_{e}\text{d}\xi\right) \tag{23}\]  
\[U=\sum_{e=1}^{N}\left(\frac{l_{e}}{4}\int_{-1}^{1}EA\xi^{2}\text{d}\xi-\left. \frac{\delta_{e1}GMm_{m}}{\sqrt{\mathbf{d}_{e}^{\text{T}}\mathbf{N}^{\text{T }}\mathbf{N}\mathbf{d}_{e}}}\right|_{\xi=-1}+\left.\frac{\delta_{eN}GMm_{n}}{ \sqrt{\mathbf{d}_{e}^{\text{T}}\mathbf{N}^{\text{T}}\mathbf{N}\mathbf{d}_{e}} }\right|_{\xi=1}+\frac{l_{e}}{2}\int_{-1}^{1}\frac{GM\rho A}{\sqrt{\mathbf{d}_ {e}^{\text{T}}\mathbf{N}^{\text{T}}\mathbf{N}\mathbf{d}_{e}}}\text{d}\xi\right) \tag{24}\]  
\[\delta W=\sum_{e=1}^{N}\delta\mathbf{d}_{e}^{\text{T}}\left[\frac{l_{e}}{2} \int_{-1}^{1}\mathbf{N}^{\text{T}}\mathbf{F}_{\text{f}}\text{d}\xi-\frac{EAal _{e}}{2}\int_{-1}^{1}\left(\frac{\partial\varepsilon}{\partial\mathbf{d}_{e} }\right)^{\text{T}}\dot{\varepsilon}\text{d}\xi\right] \tag{25}\]  
where \(\delta_{ij}\) is the Kronecker delta function. The discretized equations of motion are derived by substituting Equations (23)-(25) into the following [PERSON]'s equation:  
\[\frac{\text{d}}{\text{d}t}\left(\frac{\partial K}{\partial\mathbf{\dot{d}}_{e }}\right)^{\text{T}}-\left(\frac{\partial K}{\partial\mathbf{d}_{e}}\right)^{ \text{T}}+\left(\frac{\partial U}{\partial\mathbf{d}_{e}}\right)^{\text{T}}= \left(\frac{\partial\delta W}{\partial\mathbf{d}_{e}}\right)^{\text{T}}\text{ \ for \ }e=1,\ 2\,\cdots,\ N \tag{26}\]  
The derived discretized equations of motion for the EDT system can be expressed as  
\[\mathbf{m}_{e}\mathbf{\ddot{d}}_{e}+\mathbf{q}_{e}^{l}+\mathbf{q}_{e}^{S}+ \mathbf{q}_{e}^{d}=\mathbf{f}_{e}^{L},\text{ \ \ for }e=1,\ 2,\ \cdots,\ N \tag{27}\]  
where \(\mathbf{m}_{e}\) is the element mass matrix, \(\mathbf{q}_{e}^{l}\) is the element longitudinal elastic force vector, \(\mathbf{q}_{e}^{S}\) is the element gravitational force vector, \(\mathbf{q}_{e}^{d}\) is the element internal damping force vector, and \(\mathbf{f}_{e}^{L}\) is the element electrodynamic force vector. The matrix and vectors are given by  
Figure 3: Discretized model of the EDT system: (**a**) ANCF model and (**b**) two-node element.