## 4 Simulation Results  
### Simulation Model  
This section presents the simulation results for the proposed control laws via the MMMS model. The kinematic models of the CMM and EMM are the same but with different link length parameters. The frames of the 7-DOF offset manipulator are shown in Figure 7, which are based on the modified D-H parameters in Table 1 where the subscripts \(B\) and \(E\) are the base and end of the manipulator, respectively. The CMM and EMM are connected in series by the MI shown in Figure 5. Due to the symmetry of the architecture, both manipulators can crawl on the space station, and the base or end of the manipulator can be reversed. Link 1 is physically formed by the combination of joint 3, lower arm boom, and joint 4. Link 2 is physically formed by the combination of the elbow joint, upper arm boom, and joint 5. We only consider the flexible booms of the CMM, and the link of EMM, and payload are approximated by rigid bodies. The viscous damping and joint torsional springs are included in all joints, which are modularized designed. The joint backlash of CMM is considered as Figure 4, where the backlash is \(L=0.2\) deg and stiffness and damping are listed in Table 2. The flexible boom is modeled as a hollow rod, with an elastic modulus of 90 GPa and a shear modulus of 34.6 GPa. The simulation model is built by ITI-SimulationX, where the inertial parameters and payload are listed in Table 3.  
The visual servo simulation of the MMMS with a payload is performed, where the pallet camera is in front of the payload, as shown in Figure 2. The task is to move the payload out of the cargo spaceship and away from the pallet with the support of the EVS. We assume that the light condition is ideal and the Assumption of 1 is considered, which means there is no external contact force. The desired berth place is 3.5 m away in the X-direction. The initial configurations of the CMM is \(\mathbf{q}_{b}=(-151,-20,-29,-37,-77,-72,-11)^{T}\) deg and EMM is \(\mathbf{q}_{m}=(160,-86.096,-56.871,-89.166,71.999,\,87.643,-167.113)^{T}\) deg with the joint order from base to end. The sampling time of the central controller and visual measure is 100 ms, the joint controller is 2 ms. Furthermore, an inherent time delay exists in the posture measuring and posture transferring due to the performance limitation of the EVS. A delay of 0.5 s is added in the visual feedback to simulate the time delay of the real physical system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Manipulator**} & \multicolumn{2}{c}{**CMM**} & \multicolumn{2}{c}{**EMM**} \\ \cline{2-5}  & **Stiffness** & **Damping** & **Stiffness** & **Damping** \\  & **(Nm/rad)** & **(Nms/rad)** & **(Nm/rad)** & **(Nms/rad)** \\ \hline \hline \(\mathrm{k}\) & 19 & 3 & \(5\times 10^{4}\) & 210 \\ \(\mathrm{k}\) & \(9\times 10^{5}\) & 3183 & \(5\times 10^{4}\) & 210 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Joint properties of the CMM and EMM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{Link}_{i}\) & \(\mathbf{a}_{i-1}\) & \(\mathbf{\kappa}_{i-1}\) (\({}^{\circ}\)) & \(\mathbf{d}_{i}\) & \(\mathbf{\theta}_{i}\) (\({}^{\circ}\)) \\ \hline Base & 0 & 0 & 0 & 90 \\
1 & 0 & 90 & \(l_{0}\) & 0 \\
2 & 0 & \(-90\) & \(l_{1}\) & 0 \\
3 & 0 & \(-90\) & \(l_{2}\) & \(-90\) \\
4 & \(l_{3}\) & 0 & \(l_{4}\) & 0 \\
5 & \(l_{5}\) & 0 & \(l_{6}\) & 90 \\
6 & 0 & 90 & \(l_{7}\) & 0 \\
7 & 0 & \(-90\) & \(l_{8}\) & 0 \\ End & 0 & 90 & 0 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: D-H Parameters of the CMM/EMM. Figure 7: The D-H Frame of the CMM/EMM. The velocity and acceleration limitations are imposed to avoid the sharp movement when the pose error is large at the beginning due to the fixed gain coefficient of \(k_{p}\). The max planned velocity and acceleration of the visual servo is (0.02 m/s, 1 deg/s) and (0.002 m/s\({}^{2}\), 0.1 deg/s\({}^{2}\)), respectively.