# Swing Analysis and Control Research of the Space Tethered Combination in the Maneuver Process
## 3 Swing Control and Simulation Study
### Approximate Linearization Control  
The nonlinear system can be degenerated into a second order gauge system by the linearization of the nonlinear terms. Based on the analysis of the second-order gauge system, the damping ratio and the frequency of the free oscillation were easily calculated, and a good transient response of the system was obtained. Compared with the exact linearization based on Li derivative analysis [25], the parameters of a nonlinear system model were required, and the Li derivative condition was satisfied. The approximate linearization control is widely used to deal with a class of state stability problems. In this section, an approximate linear control law is designed for the space towing tethered  
Figure 5: Simulation results with feedback linearization control. (**a**) Acceleration of the towing point; (**b**) Velocity of the towing point; (**c**) Displacement of the towing point; (**d**) Tether tension; (**e**) In-plane and out-of-plane swing angle of the target.  
combination model, and the control effect is compared with the feedback linearization control in the previous section.  
Approximate linearization ignores the higher order terms by Taylor approximation, and achieves decoupling. The trigonometric function can be expanded into a Taylor series near the stable position:  
\[\left\{\begin{array}{l}\sin\theta=\theta-\frac{\theta^{3}}{3!}+\cdots,\;\sin \varphi=\varphi-\frac{\varphi^{3}}{3!}+\cdots\\ \cos\theta=1-\frac{\theta^{2}}{2!}+\cdots,\;\cos\varphi=1-\frac{\varphi^{2}}{2!}+\cdots\end{array}\right. \tag{13}\]  
Ignoring the higher order items and truncating the first order item, the system's dynamical equation is obtained after the approximate linearization according to the Equation (1):  
\[\left\{\begin{array}{l}\ddot{\theta}=\frac{-\ddot{x}-g\theta}{l}\\ \ddot{\varphi}=\frac{-\ddot{y}-g\varphi}{l}\end{array}\right. \tag{14}\]  
where \(\theta\), \(\varphi\)\(\in(-0.2,0.2)\), the angle less than 10 degrees can be satisfied.  
Equation (14) is a typical specification system of the second-order, which can control swing based on a PD controller. The angle control law of the towing point (\(u_{ax2}\), \(u_{ay2}\)) is obtained:  
\[\left\{\begin{array}{l}u_{ax2}=\ddot{x}=k_{p1}\theta+k_{d1}\dot{\theta}\\ u_{ay2}=\ddot{y}=k_{p2}\varphi+k_{d2}\dot{\varphi}\end{array}\right. \tag{15}\]  
Substituting Equation (15) into Equation (14), we obtain:  
\[\left\{\begin{array}{l}\ddot{\theta}+\frac{k_{d1}}{l}\dot{\theta}+\frac{(k_ {p1}+g)}{l}\theta=0\\ \ddot{\varphi}+\frac{k_{d2}}{l}\dot{\varphi}+\frac{(k_{p2}+g)}{l}\varphi=0 \end{array}\right. \tag{16}\]  
Because of the parameters \(k_{ij}>0\), \(g>0\), \(l>0\), the coefficients of the characteristic equation for the typical second-order system in Equation (16) are positive. Thus, the approximate linearization control system is stable according to the Routh criterion.  
In the same way, the compound control inputs (\(u^{{}^{\prime}}_{x}\), \(u^{{}^{\prime}}_{y}\)) are obtained by the triple closed-loop control:  
\[\left\{\begin{array}{l}u^{\prime}{}_{x}=u_{vx}+u_{sx}+u_{ax2}\\ u^{\prime}{}_{y}=u_{vy}+u_{sy}+u_{ay2}\end{array}\right. \tag{17}\]  
where \(k_{ij}\)(\(i=p\), \(dj=1\),2) are control parameters.  
Setting control parameters as \(k_{p1}=1\), \(k_{d1}=1\), \(k_{p2}=1\), \(k_{d2}=1\), the simulation results with approximate linearization control are shown in Figure 6. Under the action of approximate linear control, the swing angle can finally be stable, and the velocity and displacement reach the demand value rapidly. The control process is closely related to parameters \(k_{ij}\). Thus, the parameters' optimization can be designed using the intelligent PID or parameter self-tuning PID control in the following research.