# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 2 Swing Dynamics and Analysis  
### Dynamical Model  
The schematic diagram of the space tethered combination in the maneuver process after target capture is shown in Figure 1. The mass center of mission platform \(O_{0}\) is equivalent to the towing point, which can move freely. The mass of the target is \(m\) and the coordinate of the mass center is \(O_{1}\). The target is connected to the mission platform by a tether. The tether's length is \(l\), and the tension force of the tether is \(F_{T}\). The orbital reference frame \(O_{0}XYZ\) is established, with its origin attached to the towing point \(O_{0}\), its \(X\)-axis pointing to the Earth's center \(O_{e}\), its \(Y\)-axis perpendicular to the \(x\)-axis in the orbital plane and contrary to the motion direction, and its \(Z\)-axis along the positive norm of the orbital plane. It is considered that the orbit motion of the mission platform is linear, along with the \(Z\)-axis when it runs on high orbit in a short time. Hence, the Earth-centered inertial reference frame \(O_{e}xyz\) is established, with its origin attached to the Earth's center, and the \(x\)-axis, \(y\)-axis and \(z\)-axis parallel to \(X\)-axis, \(Y\)-axis and \(Z\)-axis, respectively. The angles \(\theta\) and \(\varphi\) are named 'in-plane' and 'out-of-plane', relative to the frame \(O_{0}XYZ\). For convenience of analysis, the spherical coordinate frame \(O_{0}\)-\(\alpha\beta l\) is defined. Let \(\alpha\) and \(\beta\) be the apex angle and azimuth angle of the spherical coordinate frame. It is assumed that the mission platform is completely controllable by thrust force, and the acceleration components provided by the thrust force are \(a_{x}\), \(a_{y}\), \(a_{z}\), respectively. The \(z\) acceleration component \(a_{z}\) is constant, in order to realize the maneuver of the tethered combination, and \(a_{x}\), \(a_{y}\) are controllable in order to realize the in-plane and out-of-plane swing control of the tethered combination.  
The following assumptions are made:  
1. The mission platform is completely controllable, so the influences of its mass and attitude motion are neglected.
2. The target is treated as a mass point, regardless of its attitude motion.
3. The tether is taut and straight at all times, regardless of its mass and elasticity.
4. There are no other external forces affecting the combination, such as atmospheric drag, solar light pressure and lunisolar perturbation, etc.  
By a straightforward application of [PERSON]'s equations, with the generalized coordinate vector \(\boldsymbol{q}=\{\theta,\rho,l\}^{T}\boldsymbol{\in}R^{n}\), substituting \(a_{x}=\bar{x}\) into equations, the swinging dynamical model of the space tethered combination can be obtained as follows:  
\[\left\{\begin{array}{l}\ddot{\theta}=\frac{-a_{x}\cos\theta+a_{z}\sin\theta+2 \dot{l}\dot{\theta}\phi\sin\varphi-2\dot{l}\dot{\theta}\cos\varphi}{l\cos\varphi} \\ \ddot{\varphi}=\frac{a_{x}\sin\theta\sin\varphi-a_{y}\cos\varphi+a_{z}\sin \varphi\cos\theta-l\dot{\theta}^{2}\cos\varphi\sin\varphi-2\dot{\rho}\dot{l}}{ l}\\ \ddot{l}=-\frac{F_{T}}{m}-a_{x}\cos\varphi\sin\theta-a_{y}\sin\varphi-a_{z} \cos\varphi\cos\theta+l\dot{\theta}^{2}\cos^{2}\varphi+l\dot{\varphi}^{2} \end{array}\right. \tag{1}\]