# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 3 Swing Control and Simulation Study  
### Feedback Linearization Control  
The essence of feedback linearization control is the state of a closed-loop feedback. In the control strategy design process, the actual control law is obtained indirectly from the virtual control law, and the virtual control law can be obtained directly by the state differential equation constructed. The desired value of the in-plane swing angle and out-of-plane swing angle were set to \(\theta_{d}=\varphi_{d}=0\); the stabilizing control strategy of the space tethered combination has been designed as follows. Based on Equation (1), the virtual control law of the in-plane and out-of-plane swing angle is obtained:  
\[\left\{ \begin{array}{l}\ddot{\theta}=u_{\theta}=\ddot{\theta}_{d}+k_{10}( \dot{\theta}_{d}-\dot{\theta})+k_{20}(\theta_{d}-\theta)\\ \ddot{\varphi}=u_{\varphi}=\ddot{\varphi}_{d}+k_{1\varphi}(\dot{\varphi}_{d}- \varphi)+k_{2\varphi}(\varphi_{d}-\varphi)\end{array} \right. \tag{7}\]  
where \(u_{\theta}\) is the virtual control of \(\theta\), \(u_{\varphi}\) is the virtual control of \(\varphi\) and \(k_{j}\) (\(i=1,2;j=\theta\),\(\varphi\)) are the control parameters. This can be written as  
\[\left\{ \begin{array}{l}\ddot{e}_{\theta}+k_{10}\dot{e}_{\theta}+k_{20}e_{ \theta}=0\\ \ddot{e}_{\varphi}+k_{1\varphi}\dot{e}_{\varphi}+k_{2\varphi}e_{\varphi}=0 \end{array} \right. \tag{8}\]  
where \(e_{\theta}\) is the in-plane swing angle error and \(e_{\varphi}\) is the out-of-plane swing angle error. The control parameters \(k_{ij}>0\) mean that the coefficients of the characteristic equation for the typical second-order system in Equation (8) are positive. Thus, the feedback linearization control system is stable according to the [PERSON] criterion, and both the in-plane swing angle \(\theta\) and out-of-plane swing angle \(\varphi\) are finally stable at the expected value \(\theta_{d}\). According to Equations (2) and (7), the angle control law of the towing point (\(u_{ax1}\), \(u_{ay2}\)) is obtained:  
\[\left\{ \begin{array}{l}u_{ax1}=a_{x}=\frac{l\cos\varphi(k_{10}\dot{\theta}+k _{20}\theta)+2l\dot{\varphi}\dot{\theta}\sin\varphi-a_{z}\sin\theta}{\cos \theta}\\ u_{ay1}=a_{y}=\frac{u_{x}\sin\varphi\sin\theta-a_{z}\sin\varphi\cos\theta-l \dot{\theta}^{2}\sin\varphi\cos\varphi+l(k_{10}\dot{\varphi}+k_{20}\varphi)}{ \cos\varphi}\end{array} \right. \tag{9}\]  
where \(u_{x}\) and \(u_{y}\) are the anti-swing control inputs of the towing point in the \(x\) direction and \(y\) direction, respectively. Due to the integral relationship between the acceleration, velocity and displacement, the acceleration and deceleration processes will mean that the velocity and displacement of the towing point are stable at a certain constant value over time. The PID controller was used to realize the velocity and displacement control of the towing point. Based on the feedback linearization control, a triple closed-loop anti-swing control strategy that includes angle closed-loop, velocity closed-loop and displacement closed-loop was designed, as shown in Figure 3.  
Angle closed-loop: The swing angle error serves as the angle's closed-loop input, and the control output (\(u_{ax1}\), \(u_{ay2}\)) is obtained from the feedback linearization controller defined in Equation (9). The purpose of this loop is to stabilize the in-plane and out-of-plane swing angle. Velocity closed-loop: The velocity error serves as the closed-loop input, and the feedback velocity value (\(\dot{x}\), \(\dot{y}\)) is obtained from the integral of the towing point acceleration (\(\ddot{x}\), \(\ddot{y}\)) = (\(u_{x}\),\(u_{y}\)). The feedback velocity value is considered to be approximately equal to the actual measured velocity value in the simulation. Displace closed-loop: The displacement error serves as the displacement closed-loop input, and the feedback displacement value (\(x\), \(y\)) is obtained from the integral of the towing point velocity value (\(\dot{x}\), \(\dot{y}\)).