## 2 Problem Formulation
### Controller Design
\tag{13}\]  
For x1, we designed a negative proportional feedback controller for the inclination \(\alpha\) as Equation (14):  
\[\mathrm{v}_{\mathrm{x1}}=\mathrm{K}_{\mathrm{P}}\big{(}\mathrm{e}_{\alpha( \mathrm{n})}\big{)}\cdot\mathrm{f}_{\mathbf{e}_{\alpha(\mathrm{n})}} \tag{14}\]  
In Equation (14), the coefficient \(\mathrm{K}_{\mathrm{P}}\) was set according to the function given in method 2 so that the actuator was difficult to start for adjustment when the error was very small, and a low-pass filter with a cutoff frequency of 2 Hz was to be used for \(\mathrm{e}_{\alpha(\mathrm{n})}\). There were two actuators in the Y direction, \(\mathrm{y}_{\mathrm{s}}\) and \(\mathrm{y}_{\mathrm{P}}\mathrm{:}\mathrm{y}_{\mathrm{s}}\) was characterized by a shorter range of motion but could adapt to a greater acceleration; In contrast, \(\mathrm{y}_{\mathrm{b}}\) was characterized by a long range of motion but could adapt to a smaller acceleration. When designing the controller, we fed forwards part of the input of \(\mathrm{y}_{\mathrm{b}}\) to \(\mathrm{y}_{\mathrm{s}}\), and the existence of this part of the input could prevent the motion range of \(\mathrm{y}_{\mathrm{s}}\) from being excessively large. The expression of speed input \(\mathrm{v}_{\mathrm{ys}}\) is shown in Equation (15):  
\[\mathrm{v}_{\mathrm{ys}(\mathrm{n})}=(1-\mathrm{k})\Big{(}(1-\mathrm{a}) \mathrm{v}_{\mathrm{yscmd}(\mathrm{n})}+\mathrm{a}^{\mathrm{l}}\mathrm{v}_{ \mathrm{ys}(\mathrm{n}-1)}\Big{)}+\mathrm{k}\Big{(}(1-\mathrm{b})\mathrm{v}_ {\mathrm{ysmes}(\mathrm{n}-1)}+\mathrm{bv}_{\mathrm{ysmes}(\mathrm{n}-2)} \Big{)} \tag{15}\]  
where \(\mathrm{v}_{\mathrm{yscmd}}\) represents the speed command, \(\mathrm{v}_{\mathrm{ys}}\) represents the speed input processed by method 4, and \(\mathrm{v}_{\mathrm{ysmes}}\) represents the measured value of the actuator speed. The speed command consists of two parts, the filtered PID controller output and \(\mathrm{v}_{\mathrm{ysb}}\), as in Equation (16):  
\[\mathrm{v}_{\mathrm{yscmd}(\mathrm{n})}=\mathrm{f}\Bigg{(}\mathrm{K}_{ \mathrm{P}}\Big{(}\mathrm{e}_{\mathrm{F}(\mathrm{n})}\Big{)}\mathrm{e}_{ \mathrm{F}(\mathrm{n})}+1\Bigg{(}\mathrm{K}_{\mathrm{i}}\Big{(}\mathrm{e}_{ \mathrm{F}(\mathrm{n})}\Big{)}\sum_{0}^{\mathrm{n}}\mathrm{e}_{\mathrm{F}( \mathrm{n})}\mathrm{dt}\Bigg{)}+1\Bigg{(}\mathrm{K}_{\mathrm{d}}\Big{(}\mathrm{ e}_{\mathrm{F}(\mathrm{n})}\Big{)}\frac{\mathrm{f}\Big{(}\mathrm{e}_{ \mathrm{F}(\mathrm{n})}-\mathrm{e}_{\mathrm{F}(\mathrm{n}-1)}\Big{)}}{\mathrm{ dt}}\Bigg{)}\Bigg{)}+\mathrm{K}_{\mathrm{F}}\mathrm{F}^{1}\mathrm{v}_{\mathrm{ys}(\mathrm{n})} \tag{16}\]  
The PID coefficients in Equation (16) are a function of the force error F according to method 2, and the variation in the coefficients is shown in Figure 4. The output amplitude of the integral action and the differential action are limited by method 1. The amplitude limiting function of the integral can keep the integral within the controllable range and avoid excessive integral output, which requires a long time of reverse error to clear. The input of the differential action was filtered by method 3 to reduce the noise impact caused by the difference. The existence of \(\mathrm{v}_{\mathrm{ysb}}\) was used to adjust \(\mathrm{v}_{\mathrm{yscmd}}\), which could keep \(\mathrm{v}_{\mathrm{yscmd}}\) within the effective range; see Equation (17):  
\[\mathrm{v}_{\mathrm{ysb}(\mathrm{n})}=\mathrm{K}_{\mathrm{bs}}(\mathrm{y}_{ \mathrm{s}(\mathrm{n}-1)})\mathrm{y}_{\mathrm{s}(\mathrm{n}-1)} \tag{17}\]  
Figure 4: Coefficient variation tendency. (**a**) \(\mathrm{K}_{\mathrm{P}}\); (**b**) \(\mathrm{K}_{\mathrm{i}}\); (**c**) \(\mathrm{K}_{\mathrm{d}}\).