# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 3 Swing Control and Simulation Study
### Feedback Linearization Control
The feedback displacement value is considered to be approximately equal to the actual measured displacement value in the simulation. The complex control law (\(u^{{}^{\prime}}_{x}\), \(u^{{}^{\prime}}_{y}\)) is obtained from the interaction of the angle closed-loop law (\(u_{ax}\), \(u_{ay}\)), the velocity closed-loop law (\(u_{vx}\), \(u_{yy}\)) and the displacement closed-loop law (\(u_{sx}\), \(u_{sy}\)), which can control the swing angles of the target, and the velocity and displacement of the towing point. Assume that the initial state of the towing point is (\(\dot{x}_{0}\), \(\dot{y}_{0}\)) = (\(x_{0}\), \(y_{0}\)) = (0, 0), and the velocity is set at (\(\dot{x}_{d}\), \(\dot{y}_{d}\)) = (\(x_{d}\), \(y_{d}\)) = (0, 0). The control law (\(u_{xx}\), \(u_{yy}\)) and the displacement control law (\(u_{sx}\), \(u_{sy}\)) of the towing point are obtained by PID controller:  
\[\left\{ \begin{array}{l}u_{xx}=k_{pxx}\Delta v_{x}+k_{ixx}\int\Delta v_{x}dt+K _{dxx}d\Delta v_{x}/dt\\ u_{yy}=k_{pxy}\Delta v_{y}+k_{iyy}\int\Delta v_{y}dt+K_{dyy}d\Delta v_{y}/dt\\ u_{sx}=k_{pxx}\Delta x+k_{isx}\int\Delta xdt+K_{dsx}d\Delta x/dt\\ u_{sy}=k_{psy}\Delta y+k_{isy}\int\Delta ydt+K_{dsy}d\Delta y/dt\end{array} \right. \tag{10}\]  
where \(\Delta v_{x}\), and \(\Delta v_{y}\) are the velocity component errors of the towing point, and \(\Delta x\) and \(\Delta y\) are the displacement component errors of the towing point. According to Equations (9) and (10), the complex control law (\(u^{{}^{\prime}}_{x}\), \(u^{{}^{\prime}}_{y}\)) can be obtained:  
\[\left\{ \begin{array}{l}u^{\prime}{}_{x}=u_{ux}+u_{sx}+u_{ax1}\\ u^{\prime}{}_{y}=u_{yy}+u_{sy}+u_{ay1}\end{array} \right. \tag{11}\]  
The anti-swing control law expressed in Equation (11) is applied to the dynamical model of the space tethered combination in the maneuver process expressed in Equation (1). The tether length \(l\) is constant, and the case of short tether is considered. The physical parameters and initial conditions used in simulation are listed in Table 1.  
Figure 3: Schematic diagram of the anti-swing control strategy, based on feedback linearization control. First, the case without control was studied as a comparison. With the initial parameters given in Table 1, the simulation results are shown in Figure 4. The trajectory of the target mapped to the \(XY\)-plane is composed of a series of precessional ellipses, as shown in Figure 4a. The precession angle of the elliptical trajectory is related to the major axis and minor axis of the ellipse, as follows:  
\[\Delta\theta=\frac{3}{4}\pi abl^{2} \tag{12}\]  
where \(\Delta\theta\) is the angle between the major axes of adjacent elliptical trajectories, and \(a\) and \(b\) are respectively the major axis length and minor axis length of the elliptical trajectory. The time history of the in-plane swing angle and out-of-plane swing angle are shown in Figure 4b. When the swing elimination control is not applied, the response of the in-plane and out-of-plane swing is varied amplitude periodical vibration, and there is a phase difference between them. Under the same initial conditions, the anti-swing control of the space towing tethered combination was carried out. The expected velocity demand of the towing point was set to (\(\dot{x}_{d}\), \(\dot{y}_{d}\)) = (0, 0) and the displacement of the towing point was set to (\(x_{d}\), \(y_{d}\)) = (0, 0); the simulation results with triple closed-loop control are shown in Figure 5. It can be seen that the control inputs \(u\) range between \(-\)1 m/s\({}^{2}\) and 1 m/s\({}^{2}\). The towing point returns to the origin with zero speed after control, which can be seen from the velocity response and the displacement response. The tether is always taut with nonnegative tension, meaning that the anti-swing control is physically achievable. The in-plane swing angle and out-of-plane swing angle decay dramatically to 0\({}^{\circ}\) under the feedback linearization control strategy.