# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 2 Swing Dynamics and Analysis
### Swing Analysis  
According to Equation (1), the in-plane swing angle \(\theta\) and the out-of-plane swing angle \(\varphi\) are coupled with each other, and the motion of the space tethered combination in the maneuver process is a two-dimensional swing, generally. Furthermore, as shown in Figure 1, the target motion relative to \(O_{0}\) is distributed on the sphere, with \(O_{0}\) as the center and \(l\) as the radius, which is called'spherical swing'. This is the first swing form of the tethered combination, which is also the most common case. When \(\varphi=0\), \(\dot{\varphi}=0\) it can be obtained that \(\ddot{\varphi}\)\(=0\), which means that the state \(\left(\varphi,\dot{\varphi}\right)\)\(=\)\((0,0)\) can be kept infinitely. Therefore, it can be seen that the motion of the space tethered combination is in-plane swing if there is no out-of-plane swing disturbance at the initial time. This is the second swing form of the tethered combination. The target motion will evolve into a circular motion when the initial tangential velocity of the target satisfies certain conditions. This is the third swing form of the tethered combination, called conical swing, named according to the cone shape of the tether's motion trajectory, which is another special case of spherical swing. In the conical swing form, the apex angle \(\alpha\) is constant. The dynamic model of the space towing tethered combination is expressed in the spherical coordinate as follows:  
Figure 1: Swing model of the space towing tethered combination. \[\left\{ \begin{array}{l}\ddot{\alpha}=(l\dot{\beta}^{2}\sin\alpha\cos\alpha-a_{z} \sin\alpha-2\dot{l}\dot{\alpha})/l\\ \ddot{\beta}=(-2\dot{l}\dot{\beta}\sin\alpha-2\dot{l}\dot{\alpha}\dot{\beta}\cos \alpha)/(l\sin\alpha)\\ \ddot{l}=l\dot{\alpha}^{2}+l\dot{\beta}^{2}\sin^{2}\alpha+a_{z}\cos\alpha-F_{ T}/m\end{array} \right. \tag{2}\]  
When the tether length \(l\) is constant, the problem can be simplified as follows:  
\[\left\{ \begin{array}{l}\ddot{\alpha}=(l\dot{\beta}^{2}\sin\alpha\cos\alpha-a _{z}\sin\alpha)/l\\ \ddot{\beta}=-2\dot{\alpha}\dot{\beta}\cos\alpha/\sin\alpha\end{array} \right. \tag{3}\]  
The notation \(\alpha_{0}\) is the initial apex angle, \(l_{0}\) is the initial tether length and \(v_{0}\) is the initial tangential velocity during the conical swing. Hence, the apex angle is constant with the value \(\alpha_{0}\), and the change rate of the apex angle and its second derivative yield \(\ddot{\alpha}=\dot{\alpha}=0\). By the integration of Equation (3), the angular velocity of the azimuth angle is obtained as follows:  
\[\dot{\beta}=\frac{C}{{l_{0}}^{2}\sin^{2}\alpha}=\frac{l_{0}\sin\alpha_{0}v_{0} }{l_{0}^{2}\sin^{2}\alpha}=\frac{v_{0}}{l_{0}\sin\alpha_{0}} \tag{4}\]  
Substituting Equation (4) into Equation (3), it can be obtained as follows:  
\[\frac{(l_{0}\sin\alpha_{0}v_{0})^{2}}{{l_{0}}^{4}\sin^{4}\alpha_{0}}\sin \alpha_{0}\cos\alpha_{0}-\frac{a_{z}\sin\alpha_{0}}{l_{0}}=0 \tag{5}\]  
Thus, the initial tangential velocity can be obtained:  
\[v_{0}=\sin\alpha_{0}\sqrt{\frac{a_{z}l_{0}}{\cos\alpha_{0}}} \tag{6}\]  
It can be seen that the formation condition of the conical swing motion is related to the initial value of apex angle \(\alpha_{0}\) and the initial tangential velocity \(v_{0}\) from Equation (6). According to Equations (3) and (6), the swing of the space tethered combination was simulated. The system parameters were set up as follows: the initial tether length \(l_{0}=3\) m, the initial internal swing angle \(\alpha_{0}=30^{\circ}=0.52\) rad, the acceleration \(a_{z}=-5\) m/s\({}^{2}\) and \(v_{0}=2.08\) m/s can be obtained by calculation from Equation (3). The simulation result of the target trajectory mapped to the \(XY\)-plane is shown in Figure 2a, which is a circle of 1.5 m radius. For comparison, the initial velocity of the target was set to \(v_{2}=0.83\) m/s; the simulation result of the target trajectory mapped to the \(XY\)-plane is shown in Figure 2b, which is composed of a series of precessional ellipses.