# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 2 Modeling of a Tethered Satellite System  
Consider tethered satellites with moving payloads, as shown in Figure 1. For simplicity, we modeled the two satellites and the moving payload as point masses without moments of inertia. Since the tether length in typical tethered satellite systems is very large compared to the dimensions of the satellites or payloads, they can be assumed to be point masses. However, if the tether length is relatively short, the positions of the centers of mass and the mass moments of inertia for the satellites and payloads must be taken into account. InFigure 1, \(m_{A}\) and \(m_{B}\) represent the masses of the two satellites and \(m_{v}\) denotes the mass of the payload. The satellites and payload were assumed to be in the same orbital plane. The tether connecting the two satellites, with the total length \(L\), was modeled as a flexible string with mass. The payload moved along the tether with a constant speed \(V\). Since the tether's length is much greater than its cross-sectional dimensions, its bending rigidity can be ignored, allowing us to model the tether as a string. Position vectors with their origin at the center of the Earth were employed to track the movements of the satellites and payload. In Figure 1, the positions of satellites A and B, along with the payload, are denoted by position vectors \(\mathbf{r}_{A}\), \(\mathbf{r}_{B}\), and \(\mathbf{r}_{v}\), respectively, using the \(xy\) absolute coordinate system with the origin \(O\). The position of a particle on the tether is expressed as \(\mathbf{r}\), which can alternatively be described by its orbital radius \(r\) (the magnitude of \(\mathbf{r}\)) and the true anomaly \(\psi\). Since the position of the tether depends on both time \(t\) and arc length \(s\), its coordinates \(x\) and \(y\) are functions of \(t\) and \(s\). Therefore, the position vector of the tether can be represented as  
\[\mathbf{r}(s,t)=\genfrac{\{}{\}}{0.0 pt}{}{x(s,t)}{y(s,t)}. \tag{1}\]  
Since the coordinate \(s\) of satellite \(A\) is \(0\) and that of satellite \(B\) is \(L\), the position vectors of satellites A and B, \(\mathbf{r}_{A}\) and \(\mathbf{r}_{B}\), can be expressed as  
\[\mathbf{r}_{A}(t)=\genfrac{\{}{\}}{0.0 pt}{}{x(0,t)}{y(0,t)},\ \mathbf{r}_{B}(t)= \genfrac{\{}{\}}{0.0 pt}{}{x(L,t)}{y(L,t)}. \tag{2}\]  
In contrast, the payload \(m_{v}\) moves along the tether, so the arc length \(s\), which represents the position of \(m_{v}\), is a function of time \(t\). Therefore, the position vector of the moving payload can be expressed as  
\[\mathbf{r}_{v}(s,t)=\genfrac{\{}{\}}{0.0 pt}{}{x(s(t),t)}{y(s(t),t)}. \tag{3}\]  
Before deriving the equations of motion, we need to represent the kinetic and potential energies of the tethered satellite system in terms of the above position vectors. The kinetic energy of the system is given by  
\[K=\frac{1}{2}\int_{0}^{L}\rho A\dot{\mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d }s+\frac{1}{2}m_{A}\dot{\mathbf{r}}_{A}\cdot\dot{\mathbf{r}}_{A}+\frac{1}{2}m _{B}\dot{\mathbf{r}}_{B}\cdot\dot{\mathbf{r}}_{B}+\frac{1}{2}m_{v}\dot{ \mathbf{r}}_{v}\cdot\dot{\mathbf{r}}_{v}, \tag{4}\]  
where the dot represents differentiation with respect to time, \(\rho\) is the mass density of the tether, and \(A\) is the cross-sectional area of the tether. Assuming a uniform tether, \(\rho\) and  
Figure 1: Dynamic model of tethered satellites with a moving payload, assuming the satellite to be a point mass. \(\mathbf{r}\) is the position vector of the tether, \(\mathbf{r}_{A}\), and \(\mathbf{r}_{B}\) are the position vectors of satellites A and B, and \(\mathbf{r}_{v}\) is the position vector of the payload. \(A\) are constant along the length. The first term represents the kinetic energy of the tether, whereas the last three terms represent the kinetic energies of the satellites and payload.