# Dynamic Behavior Analysis and Stability Control of Tethered Satellite Formation Deployment
## 2 TSS Equations of Motion Using Lagrangian Method
We assume that a spherical earth is considered as follows:  
\[V=-\frac{\mu_{e}m}{R}+\frac{\mu_{e}m^{*}l^{2}}{2R^{3}}(1-3\cos^{2}\phi\cos^{2}\theta) \tag{7}\]  
where \(\mu_{e}=398,600\) km\({}^{3}\)/s\({}^{2}\) is Earth's gravitational coefficient. The Lagrange function can be formed as follows. \[L=T_{t}+T_{r}+T_{e}-V. \tag{8}\]  
By substituting Equation (8) into [PERSON]'s equations and by assuming that the system's center of mass is running in a constant orbital plane (\(\delta=0\)), the system's equations of motion can be obtained as follows:  
\[m\ddot{R}-mR\dot{\alpha}^{2}+\frac{\mu m}{R^{2}}-\frac{3\mu m^{*}l^{2}}{2R^{4} }(1-3\cos^{2}\theta\cos^{2}\phi)=Q_{R} \tag{9}\]  
\[ \begin{array}{l}2 mR\dot{R}\dot{\alpha}+\dot{m}^{*}l^{2}(\dot{\alpha}+\dot{ \theta})\cos^{2}\phi+2m^{*}l[(\dot{\alpha}+\dot{\theta})\cos^{2}\phi]\\ +m^{*}l^{2}[(\ddot{\alpha}+\ddot{\theta})\cos^{2}\phi-2(\dot{\alpha}+\dot{ \theta})\dot{\phi}\sin\phi\cos\phi]+mR^{2}\ddot{\alpha}=Q_{\alpha}\\ m^{*}l^{2}[(\ddot{\alpha}+\ddot{\theta})\cos^{2}\phi-(\dot{\alpha}+\dot{ \theta})\dot{\phi}\sin 2\phi]+2m^{*}l\dot{l}(\dot{\alpha}+\dot{\theta})\cos^{2} \phi\\ +\dot{m}^{*}l^{2}(\dot{\alpha}+\dot{\theta})\cos^{2}\phi+\frac{3m^{*}l^{2}}{2R^ {3}}\sin 2\theta\cos^{2}\phi=Q_{\theta}\\ \\ \dot{m}^{*}l^{2}\dot{\phi}+2m^{*}l\dot{l}\dot{\phi}+m^{*}l^{2}\ddot{\phi}+ \frac{1}{2}m^{*}l^{2}(\dot{\alpha}+\dot{\theta})^{2}\sin 2\phi+\frac{3\mu m^{*}l^{2}}{2R^ {3}}\sin 2\phi\cos^{2}\theta=Q_{\phi}\\ \\ \ddot{m}^{*}\ddot{l}l+m^{*}\ddot{l}-\frac{1}{2}(m^{*})^{\prime}l^{2}[\dot{ \phi}^{2}+(\dot{\alpha}+\dot{\theta})^{2}\cos^{2}\phi]-m^{*}l[\dot{\phi}^{2}+ (\dot{\alpha}+\dot{\theta})^{2}\cos^{2}\phi]\\ -\frac{1}{2}(m^{\#})\dot{l}^{2}+\frac{\mu(m^{*})l^{2}}{2R^{3}}(1-3\cos^{2} \theta\cos^{2}\phi)+\frac{\mu m^{*}l}{R^{3}}(1-3\cos^{2}\theta\cos^{2}\phi)=Q_ {l}\end{array}  \tag{13}\]  
where \((\ )^{\prime}=\text{d}(\ )/\text{d}l,\ m^{*}=m_{t}(3m_{1}-3m_{2}-m)/(6m)\), \(m^{\#}=m_{t}(2m_{1}-m)/m\), \(Q_{l}=-T\) is the tension control and the generalized forces, \(Q_{\theta}\) and \(Q_{\phi}\), are typically assumed to be negligible as a result of distributed forces along the tether. It should be noted that tether length can be controlled by deployment/retrieval of the winch control mechanism in the mother satellite; therefore, \(m_{1},\ m_{t}\) in Equations (9)-(13) are functions of tether length. Based on the premise of a Keplerian reference orbit for the center of mass as the independent variable, the orbit's true anomaly, \(\
u\), is used to replace the generalized coordinate, \(\alpha\), in order to withdraw the premise, which can be expressed as follows:  
\[\dot{\
u}=\sqrt{\frac{\mu_{e}}{a^{3}(1-e^{2})^{3}}}\kappa^{2},\ R=\frac{a(1-e^ {2})}{\kappa} \tag{14}\]  
where \(e\) is the orbital eccentricity, and \(a\) is the semi-major axis of the orbit, \(\kappa=1+e\cos\
u\). \(\frac{\text{d}(i)}{\text{d}t}=\frac{\text{d}(i)}{\text{d}\
u}\times\frac{\text {d}\
u}{\text{d}t}\Rightarrow i=i^{\prime}\dot{\
u}\), \(\ddot{i}=i^{\prime\prime}\dot{\
u}^{2}+\frac{\ddot{\
u}\dot{\
u}^{\prime}}{ \dot{\
u}}\), \(i=\theta\), \(\phi\), \(l\) is utilized by Equations (9)-(13). Nondimensional equations of motion can be written as follows:  
\[\theta^{\prime\prime}=2(\theta^{\prime}+1)[\frac{e\sin\
u}{\kappa}+\phi^{ \prime}\tan\phi-\Phi_{1}\frac{\Lambda^{\prime}}{\Lambda}]-\frac{3}{2\kappa} \sin 2\theta \tag{15}\]  
\[\phi^{\prime\prime}=\frac{2e\sin\
u}{\kappa}\phi^{\prime}-2\Phi_{1}\frac{ \Lambda^{\prime}}{\Lambda}\phi^{\prime}-\frac{1}{2}[(\theta^{\prime}+1)^{2}+ \frac{3}{\kappa}\cos^{2}\theta]\sin 2\phi \tag{16}\]  
\[\Lambda^{\prime\prime}=\frac{2e\sin\
u}{\kappa}\Lambda^{\prime}-\Phi_{2}\frac{ \Lambda^{\prime 2}}{\Lambda}+\Phi_{3}\Lambda[\phi^{\prime 2}+(\theta^{\prime}+1)^{2}\cos^{2}\phi\]  
\[+\frac{1}{\kappa}(3\cos^{2}\theta\cos^{2}\phi-1)]-\frac{mT}{m_{1}\dot{\
u}^{2}L( m_{2}+m_{t})} \tag{17}\]where \((\ )^{\prime}=\mathrm{d}(\ )/\mathrm{d}\
u\), \(\Lambda=l/L\) is the nondimensional tether length, and \(L\) is the reference tether length.