# Dynamic Behavior Analysis and Stability Control of Tethered Satellite Formation Deployment
## 2 TSS Equations of Motion Using Lagrangian Method  
Since the 1990s, TSS theory has developed rapidly from rigid-rod models to bead models and from continuous models to discrete models, and model accuracy is constantly improving [30; 31]. However, other models' dynamics equations are more complex and the quantity of computation is larger when compared with rigid-rod models [32]. The classical rigid-rod model is widely used. As shown in Figure 1, a rigid tethered satellite system consists of a mother satellite, \(m_{1}\), and a subsatellite, \(m_{2}\), (mass points), respectively. Both satellites are connected by a rigid tether. In this model, regardless of tether flexibility and tether elasticity, tethers released into outer space are considered as straight rods of infinite stiffness that do not bend or twist. Tether wounds on a pool of a deployment device in the mother satellite and tether length can be effectively controlled by the device. The inertial geocentric frame, \(O-XYZ\), the orbital frame, \(o-xyz\), and the tether body frame, \(o-x_{\text{b}}y_{\text{b}}z_{\text{b}}\), are all radial-transversal out-of-plane frames, and these are established in order to describe TSS position and attitude in Figure 1. Orbital radius and tether length are expressed as \(R,~{}l\), respectively; right ascension and declination to the center of mass are expressed as \(\alpha,~{}\delta\), respectively; and the tether in-plane angle and tether out-of-plane angle are expressed as \(\theta,~{}\phi\), respectively. The radius vector with respect to the center of mass is written in inertial coordinates as follows. \[\mathbf{R}=R\cos\delta\cos\alpha\mathbf{i}+R\cos\delta\sin\alpha\mathbf{j}+R\sin \delta\mathbf{k}. \tag{1}\]  
Total kinetic energy, \(T_{k}\), consists of the translation of the center of mass, \(T_{t}\), system rotation, \(T_{r}\), and tether deployment, \(T_{e}\):  
\[T_{t}=\frac{1}{2}m\bigg{(}\dot{R}^{2}+R^{2}\dot{\delta}^{2}+R^{2}\dot{\alpha}^{ 2}\cos^{2}\delta\bigg{)} \tag{2}\]  
where \(m=m_{1}+m_{2}+m_{t}\) is the total system mass, \(m_{t}=\rho l=m_{1}^{0}-m_{0}\) is the tether mass released into the external environment, \(\rho\) is the tether linear density and \(m_{1}^{0}\) is the mass of the mother satellite before tether deployment that includes the tether mass:  
\[T_{r}=\frac{1}{2}\{\omega\}^{T}[I]\{\omega\} \tag{3}\]  
where \(I\) is the tensor matrix of moment of inertia of the tethered satellite system, and \(\omega\) is the inertial angular velocity of the tether in the inertial frame [33], which can be expressed as follows:  
\[ \begin{array}{c}\omega=\Big{(}\dot{\alpha}\sin\delta\cos\theta\cos\phi+\dot {\theta}\sin\phi-\dot{\delta}\sin\theta\cos\phi+\dot{\alpha}\cos\delta\sin \phi\Big{)}\mathbf{i}\\ \quad-\Big{(}\dot{\phi}+\dot{\alpha}\sin\delta\sin\theta+\dot{\delta}\cos\theta \Big{)}\mathbf{j}\\ \quad+(\dot{\theta}\cos\phi-\dot{\alpha}\sin\delta\cos\theta\sin\phi+\dot{ \delta}\sin\theta\sin\phi+\dot{\alpha}\cos\delta\cos\phi)\mathbf{k}\\ \end{array}  \tag{4}\]  
\[ \begin{array}{c}T_{r}=\frac{1}{2}m^{*}l^{2}\big{[}\Big{(}\dot{\phi}+\dot{ \alpha}\sin\delta\sin\theta+\dot{\delta}\cos\theta\Big{)}^{2}\\ \quad+(\dot{\delta}\sin\theta\sin\phi-\dot{\alpha}\sin\delta\cos\theta\sin\phi +\dot{\alpha}\cos\delta\cos\phi+\dot{\theta}\cos\phi)^{2}\big{]}\end{array}  \tag{5}\]  
where \(m^{*}=\left(m_{1}+m_{t}\,/2\right)\left(m_{2}+m_{t}\,/2\right)/m-m_{t}/6\) is the reduced mass of the system. \[T_{e}=\frac{1}{2}\frac{m_{1}\left(m_{2}+m_{t}\right)}{m}\dot{l}^{2}. \tag{6}\]  
The tether is assumed to be stationary relative to the mother satellite within the winch control mechanism, and its speed is provided during deployment. When TSS systems are active in space, they are still within Earth's gravitational field, and their potential energy is caused by Earth's attraction. Potential energy is obtained from the mother satellite, the subsatellite and the tether, which is simplified by taking the  
Figure 1: Rigid tether dynamical model. first term of [PERSON]'s series expansion.