# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 4 Numerical Simulation  
In this paper, the characteristics of the collision process for the above SMS-FDCDM system are analyzed in the ADAMS workspace, and the vibration absorption of collision experiments performed in MATLAB environment. According to the whole-body model of SMS-FDCDM in Section 2, the collision process can be simulated in ADAMS. The dynamic parameters of the SMS-FDCDM model are shown in Table 1; the initial position of the end effector is [250, 250, 250] mm, and the target is set as a sphere with spin motion and initial velocity. The motion parameters of the target are shown in Table 2, the collision parameters in Adams are shown in Table 3, and the impact force that occurs at the time of SMS-FDCDM contact with the space target is shown in Figure 5.  
Figure 4: The block diagram of the developed controller. From the results shown in Figure 5, we can see that the maximum normal pressure is 45.2 N. The maximum impact torque is 0.38 N \(\cdot\) m. The duration is about 0.2 s, so the contact force is the impact force in a short time. The results obtained in ADMAS are used as input into MATLAB, and we can establish a MATLAB/Simulink-based simulation environment to verify the optimal results of coordinated control, which combine the quick non-singular integral terminal sliding mode control (NFITSMC) and FDCDM proposed in Section 3. Without loss of generality, spatial 3-D cases have demonstrated to be a suitable platform to test control algorithms. Therefore, a 2-DOF manipulator and 4-DOF full-dimensional controllable damping mechanism (FDCDM) mounted on a free-floating (no position and attitude control) base (6-DOF) is used, which totals 12-DOF (6-DOF controllable) platform on spatial 3-D. A PID controller, traditional SMC, and a non-singular terminal sliding mode controller (NTSMC) are used as the comparison method in this paper. The selected parameters of the controller are reported in Table 4. The contrast results are shown in Figures 6 and 7. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Rigid Bodies \(k\)**} & \multirow{2}{*}{\(m_{k}/\text{kg}\)} & \multicolumn{3}{c}{\(\mathbf{I_{k}/kg\cdot m^{2}}\)} & \multirow{2}{*}{\(d_{k}/\text{m}\)} & \multirow{2}{*}{\(r_{k}/\text{m}\)} \\ \cline{5-6}  & & **X** & **Y** & **Z** & \\ \hline
1 & 100 & 40 & 45 & 40 & 5.4 & 1.6 \\ \hline
2 & 10 & 1 & 1 & 1 & 0 & 0 \\
3 & 10 & 1 & 1 & 1 & 0 & 0 \\
4 & 1.2 & 0.02 & 0.04 & 0.02 & 0 & 0 \\
5 & 1.2 & 0.02 & 0.06 & 0.05 & 0 & 0 \\
6 & 1.2 & 0.012 & 0.08 & 0.06 & 0.32 & 0.28 \\
7 & 6 & 0.4 & 0.4 & 0.5 & 0.15 & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dynamic parameters of the SMS-FDCDM model. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Orientation**} & **Linear Velocity** & **Angular Velocity** & **Inertia** & **Mass** \\  & **mm/s** & **mm/s** & **kg \(\cdot\) m\({}^{2}\)** & **kg** \\ \hline \(X\) & 400 & 50 & 4000 & \\ \(Y\) & 400 & 50 & 4000 & 5 \\ \(Z\) & 400 & 50 & 4000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The motion parameters of target. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Type**} & **Title 2** & **Title 3** & **Unit** \\ \hline \multirow{4}{*}{Normal force} & Damping & 15 & N \(\cdot\) s/mm \\  & Stiffness & 50,000 & N/mm \\ \cline{1-1}  & Force index & 1.5 & / \\ \cline{1-1}  & Penetration depth & 0.1 & mm \\ \cline{1-1}  & Static friction coefficient & 0.4 & / \\ \cline{1-1} Friction & Static translation velocity & 100 & mm/s \\ \cline{1-1}  & Dynamic friction coefficient & 0.2 & / \\ \cline{1-1}  & Friction translation velocity & 1200 & mm/s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The contact model parameters. Figure 5: **(a)** The impact force when the SMS-FDCDM contact with space target; **(b)** the impact torque when the SMS-FDCDM contact with space target. Figure 6: Attitude tracking error for different controllers of joint 1. **Remark 1. ** In this part, the sliding surface of traditional SMC and NTSMC is \(s=ce+\dot{e}\) and \(s=e+(1/\beta)\dot{e}^{p/q}\), respectively.