## 6 Experimentation
### Experimental Results  
_Case 1. Circular contour command tracking_  
In order to investigative the control performance, the LMs-driven X-Y-Y stage is first commanded to track a dual circular contour simultaneously with the maximum frequency \(f\) of 1 Hz, as given by:  
\[x_{d}=r\cos(2\pi ft)\, \tag{41}\]  
\[y_{1d}=\left[r\sin(2\pi ft)\ +70\ \right]\, \tag{42}\]  
\[y_{2d}=\left[r\sin(2\pi ft)\ +210\right]\, \tag{43}\]  
where the radius of each circular contour is \(r=0.05\) m. In the experimentation, three movers of the LMs-driven X-Y-Y stage were concurrently controlled to carry out the \(x\)-\(y_{1}\) and \(x\)-\(y_{2}\) biaxial motions. The constant control parameters for the 2 PID control system were selected as \(K_{px}=0.0139\), \(T_{lx}=6.2195\), \(T_{Dx}=0.00094\), \(\alpha_{x}=0.0020\), \(\beta_{x}=8.35\times 10^{-7}\), \(K_{py1}=0.0018\), \(T_{ly1}=0.1308\), \(T_{Dy1}=6.63\times 10^{-6}\), \(\alpha_{y1}=0.00012\), \(\beta_{y1}=2.73\times 10^{-7}\), \(K_{py2}=0.007\), \(T_{ly2}=0.2308\), \(T_{Dy2}=9.19\times 10^{-6}\), \(\alpha_{y2}=5.81\times 10^{-5}\), and \(\beta_{y2}=7.36\times 10^{-7}\). These control parameters were selected on the basis of several trials to achieve the favorable transient response considering the occurrence of uncertainties and the requirement of steady-state stability. However, it is difficult to choose all the control parameters simultaneously due to the dependence among parameters. Moreover, it cannot be ensured that the 2 PID control system can achieve the best control performances by adopting the manually selected control parameters. The experimental results of the LMs-driven X-Y-Y stage for the circular contouring control using 2 PID control system are shown in Figure 6. From Figure 6a, the end-effectors of the LMs driven-X-Y-Y stage were controlled by the 2 PID control system to track the contour command certainly. However, both the tracking error and contour error were not favorable due to the constant control gains used in the 2 PID control system. Figure 6: \(Cont.\)  
In the A2 PID control system, the system parameters of the LMs driven X-Y-Y stage were firstly identified by the IABC algorithm as \(K_{\rm{X}}\)/\(M_{x}\) = 97.01, \(b_{\rm{X}}\)/\(M_{x}\) = 39.21, \(H_{\rm{X}}\)/\(M_{x}\) = 1.57, \(K_{\rm{Y1}}\)/\(M_{\rm{Y1}}\) = 881.36, \(b_{\rm{y1}}\)/\(M_{\rm{y1}}\) = 142.324, \(H_{\rm{Y1}}\)/\(M_{\rm{y1}}\) = 0.17, \(K_{\rm{y2}}\)/\(M_{\rm{y2}}\) = 1949.706, \(b_{\rm{y2}}\)/\(M_{\rm{y2}}\) = 179.56, \(H_{\rm{y2}}\)/\(M_{\rm{y2}}\) = 1.71. Meanwhile, the control parameters were optimized by the IABC algorithm in which the constant parameters for the IABC algorithm were chosen as \(V\) = 20, \(D\) = 15, and \(\Phi\) = 200. In the evolutions, all the bees had random values initially and searched for the optimal solutions individually. The upper bounds for the IABC parameters were designed as \(\psi_{\rm{1,max}}\) = 0.61, \(\psi_{\rm{2,max}}\) = 10, \(\psi_{\rm{3,max}}  
The proposed ACC2 PID control system was applied to the LMs-driven X-Y-Y stage, finally, in which two contour error compensators were added to directly suppress the contour errors. The parameters for the contour error compensators were selected as: \(K_{c1x}\) = 0.0895, \(K_{c2x}\) = 0.0488, \(K_{c1y}\)= 0.001, and \(K_{c2y}\)  
Figure 7: Experimental results of the LMs-driven X-Y-Y stage using auto-tuning 2 PID (A2 PID) control system for circular contour tracking: (**a**) contour tracking responses, (**b**) tracking errors in \(x\), \(y_{1}\), and \(y_{2}\) axes, and (**c**) contour errors of left and right circular contours. _Case 2. Three-leaf contour command tracking_  
The proposed ACC2 PID method is further tested for extreme three-leaf reference contour with large curvature and high speed. The reference contour can be expressed by  
\[x_{d}=r\cos(6\pi ft)\ \cos(2\pi ft)\, \tag{44}\]  
\[y_{1d}=[r\cos(6\pi ft)\sin(2\pi ft)\ +30]\, \tag{45}\]  
\[y_{2d}=[r\cos(6\pi ft)\sin(2\pi ft)\ +90]\, \tag{46}\]  
with \(r=0.025\) m and the maximum frequency \(f\) up to 1 Hz.