## 8 Simulation Analysis of the HCS of the MAR System  
To facilitate the simulation analysis of the feasibility of the HCS of the MAR system, a simulation model of the HCS is built using Simulink (10.1(R2020a)), as shown in Figure 16.  
The parameters of the MRC of the MAR system are \(J_{\alpha}\) = 1.49 \(\times\) 10\({}^{-2}\) kg\(\cdot\)m\({}^{2}\), the lever arm \(r\) = 0.1275 m, and the mass m = 1.83 kg. The parameter values of SYSTD are shown in 5.4. The other main parameters of the ADRC are shown in Table 2.  
Suppose at \(t\) = 0 s, when the Y-direction coordinate value of the MRC is 0, a sudden given value \(y_{\alpha}\)* = 1 mm and \(\psi_{\alpha}\)* = 0 rad/s are applied to detect the response characteristics of the MAR system when the MRC moves within one step during displacement detection.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Value** & **Name** & **Value** & **Name** & **Value** \\ \hline \(\beta_{e1}\) & 150 & \(\beta_{f1}\) & 27 & \(a_{3}\) & 0.5 \\ \(\beta_{e2}\) & 1078 & \(\beta_{f2}\) & 0.02 & & \\ \(\beta_{e3}\) & 819 & \(a_{2}\) & 0.25 & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Other main parameters of the ADRC.  
Figure 16: Simulation diagram of the HCS.  
In addition, at \(t\) = 0.5 s, a position disturbance of \(y_{b}\) = 0.005 mm is added, and, at \(t\) = 0.8 s, white-noise interference is added to simulate external interference. The simulation results are shown in Figure 17.  
Figure 17 is divided into four regions for detailed analysis:  
The partial enlarged view of Region 1 is shown in Figure 17. It can be seen that the MRC can reach the displacement given value of 1 mm (i.e., one step) in about 0.03 s after starting. Within one step, the MRC first accelerates from a stationary state, moves 0.5 mm, and then decelerates to a stop. Additionally, the displacement curve within one step is relatively smooth, which is conducive to improving the accuracy of the MAR system after subdivision.  
A partially enlarged view of Region 2 is shown in Figure 17. It can be seen that the MRC will have an overshoot at the end of each step. The overshoot does not exceed 6 \(\upmu\)m and is corrected within 0.2 s.  
A partially enlarged view of Region 3 is shown in Figure 17. It can be seen from the figure that the disturbance can be quickly eliminated.  
A partially enlarged view of Region 4 is shown in Figure 17. The white noise has little impact on the HCS.  
As can be seen from Figure 17, the HCS of the MAR system has strong robustness, and the positioning accuracy can reach \(\pm\)5 \(\upmu\)m.  
The following is true as is known from Reference [2]:  
\[m\frac{d^{2}S}{dt^{2}}=BI_{L}L \tag{67}\]  
By taking the integral of both sides of Equation (67) from 0 to \(t\) and ignoring the constant term, the following equation can be obtained:  
\[S=\frac{nBI_{L}L}{2m}t^{2} \tag{68}\]  
where \(n\) represents the number of turns, \(B\) represents the magnetic induction intensity in the air-gap, \(I_{L}\) represents the current of the horizontal control coil, \(L\) is the length of the horizontal control coil in the air-gap magnetic field, and \(m\) is the mass of the MRC. Assuming these quantities are constants, the relationship between the displacement S of the MRC and time \(t\) is a quadratic function, which is consistent with the simulation results in Figure 17.  
Figure 17: Temporal Evolution of \(y_{b}\) (mm) Measured Against Time (s).  
Figure 18 is the simulation diagram of the horizontal rotation angle of the MRC. It can be seen from the above two figures that, when the MRC starts, there is a slight angular deviation on the Z-axis. The angular deviation is corrected in about 0.2 s, and the movement distances at both ends are basically the same, indicating that the HCS has achieved decoupling control.