# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 3 Swing Control and Simulation Study
### Variable Structure Control  
Variable structure control (VSC) is a typical control strategy used in nonlinear systems for its various attractive features, such as its rapid response, good transient performance and robustness with respect to uncertainties and external disturbance.  
The theoretical idea of the variable structure control is to construct the state hyperplane, such as by constructing the \(s=c^{T}x\) about the state variable set \(x\) and designing the state approach law, which satisfies the derivative negative definite of the Lyapunov function \(V=1/2\)\(s^{2}\) and stabilizes the state variable set.  
By applying the VSC to control the swing of the space towing tethered combination, the sliding mode control law is designed as two subsystems; \(\left(\theta,\dot{\theta}\right)\) and \(\left(\varphi,\dot{\varphi}\right)\). As a usual method in VSC theory, the linear switching function is chosen as follows: \(s_{1}=a_{1}\theta+a_{2}\dot{\theta}\) and \(s_{2}=b_{1}\varphi+b_{2}\dot{\varphi}\), and the approach law is \(\dot{s}=-\lambda s-\eta sgn(s)\), where \(\lambda\), \(\eta\) are parameters of the sliding mode approach law.  
Assuming that \((a_{x},a_{y})=(u_{\alpha\alpha 3},\,u_{\alpha\beta 3})\) and the tether length is a constant, the dynamics of the space towing tethered combination can be recast into the following state-space form:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f_{1}+f_{2}\alpha_{\alpha\alpha 3}\\ \dot{x}_{3}=x_{4}\\ \dot{x}_{4}=f_{3}+f_{4}\alpha_{\alpha\alpha 3}+f_{5}\alpha_{\alpha\beta 3} \end{array}\right. \tag{18}\]  
Figure 6: Simulation results with approximate linearization control. (**a**) Acceleration of the towing point; (**b**) Velocity of the towing point; (**c**) Displacement of the towing point; (**d**) In-plane and out-of-plane swing angle of the target.  
where \(f_{1}=\frac{2l\dot{\theta}\dot{\varphi}\sin\varphi-g\sin\theta}{l\cos\varphi}\), \(f_{2}=\frac{\cos\theta}{l\cos\varphi}\), \(f_{3}=\frac{l\dot{\theta}^{2}\cos\varphi\sin\varphi-g\sin\varphi\cos\theta}{l}\), \(f_{4}=\frac{\sin\theta\sin\varphi}{l}\), \(f_{5}=\frac{\cos\varphi}{l}\).  
The angle control inputs (\(u_{ax3}\), \(u_{ay3}\)) are obtained:  
\[\left\{\begin{array}{l}u_{ax3}=\frac{-\lambda s_{1}-\eta sgn(s_{1})-a_{1}x_{ 2}-a_{2}f_{1}}{a_{2}f_{2}}\\ u_{ay3}=\frac{-\lambda s_{2}-\eta sgn(s_{2})-b_{1}x_{4}-b_{2}f_{2}-b_{2}f_{4}u _{1}}{b_{2}f_{5}}\end{array}\right. \tag{19}\]  
The approach law of subsystem 1 is  
\[\dot{s}_{1}=-\lambda s_{1}-\eta sgn(s_{1}) \tag{20}\]  
and the Lyapunov function of subsystem 1 can be written as  
\[V=\frac{1}{2}{s_{1}}^{2} \tag{21}\]  
We obtain:  
\[\dot{V}=s_{1}\dot{s}_{1}=s_{1}(-\lambda s_{1}-\eta sgn(s_{1}))=(-\lambda{s_{1} }^{2}-\eta|s_{1}|) \tag{22}\]  
because parameters \(\lambda\), \(\eta\) are postive, therefore  
\[\dot{V}=(-\lambda{s_{1}}^{2}-\eta|s_{1}|)<0 \tag{23}\]  
Therefore, subsystem 1 is stable according to Lyapunov stability theory. The stability of subsystem 2 can be analysed in the same way.  
Based on the design of the velocity closed-loop and the position closed-loop, the compound control inputs (\(u^{{}^{\prime}}_{x}\), \(u^{{}^{\prime}}_{y}\)) are obtained:  
\[\left\{\begin{array}{l}u^{\prime}{}_{x}=u_{vx}+u_{sx}+u_{ax3}\\ u^{\prime}{}_{y}=u_{vy}+u_{sy}+u_{ax3}\end{array}\right. \tag{24}\]  
Setting control parameters as \(\lambda=1\), \(\eta=0.1\), \(a_{1}=0.7\), \(a_{2}=0.1\), \(b_{1}=0.7\), \(b_{2}=0.1\), the simulation results are shown in Figure 7. Under the variable structure control, the swing angle can finally be stable, and the velocity and displacement reach the demand value slowly. Compared with the motion response of the two control strategies discussed above, the control inputs have obvious high-frequency oscillation after the stabilization of the swing angle, and the oscillation of the swing angle is more severe during the control process.