## 4 Experimental Verification
### Experimental Results  
#### 4.2.1 High-Speed Circular Contouring Trajectory  
A high-speed circular contouring trajectory with a radius of 5 \(\upmu\)m and a frequency of 200 Hz was chosen to test the effectiveness of the proposed control method first. To get a circular trajectory, sinusoidal signals were adopted as input signals in the \(X\)- and \(Y\)-axes with the phase quadrature. As each axis tracks a sinusoidal trajectory when the micro-motion stage tracks a circular trajectory, the tracking performance in one direction was considered first. Figure 4 shows the tracking errors in the \(X\)-axis over 0.2 s using four control methods. In addition, two performance indexes, namely the root mean square value of the tracking error (\(e_{RMS}\)) and the maximum value of the tracking error (\(e_{RMS}\)), were introduced to evaluate the performances of different controllers mentioned above, with the performance indexes of the tracking error in Figure 4 being compared in Table 1.  
It can be observed from Figure 4 that the tracking performance of PI controller in the \(X\)-axis is the worst, due to the phase lag problem of PI controller under high-speed motion conditions. In comparison, the other controllers can achieve good tracking performances. However, the dynamics of the stage in \(X\)- and \(Y\)-axes are identical theoretically due to the symmetrical structure of the micro-motion stage. As a result, the phase lag in two directions is the same for the circular trajectory, which will lead to a relatively good contouring performance. As for the other three controllers, the \(e_{RMS}\) of the ILC and DCARC controllers were above 0.13 \(\upmu\)m, while the \(e_{RMS}\) of the proposed LODCAC controller is around 0.08 \(\upmu\)m. It also can be seen from the \(e_{M}\) in Table 1 that the proposed LODCAC controller performs best.  
Following this, the contouring performance of the micro-motion stage for tracking circular trajectory was analyzed. The contour tracking results during one circular motion period are depicted in Figure 5, with the corresponding contouring errors over 0.2 s being illustrated in Figure 6. The experimental results in terms of performance indexes are given in Table 2, where \(e_{RMS}\) is the root mean square value of the contouring error and \(e_{M}\) is the maximum value of the contouring error.  
Figure 3: Experiment setup.  
Figure 4: Axis tracking errors of circular trajectory in the \(X\)-axis.  
Figure 5: Contouring performances of circular trajectory using (**a**) PI (proportional-integral) controller; (**b**) ILC (iterative learning control) controller; (**c**) DCARC (desired compensation adaptive robust control) controller; and (**d**) LODCARC (learning-based optimal desired compensation adaptive robust control) controller.  
It is clearly demonstrated from Figure 5 that all controllers can achieve good contouring performance. According to the contouring errors of all four controllers plotted in Figure 6 and the performance indexes given in Table 2, the contouring error of the PI controller is much larger than that of the other three controllers, with the \(\varepsilon_{RMS}\) of PI controller being over 0.2 \(\upmu\)m. The proposed LODCARC controller achieved an \(\varepsilon_{RMS}\) of 0.066, which is much smaller than the others. It can be subsequently obtained that the \(\varepsilon_{RMS}\) of the proposed LODCARC controller is 29% of the \(\varepsilon_{RMS}\) of PI controller, 56% of the \(\varepsilon_{RMS}\) of ILC controller and 69% of the \(\varepsilon_{RMS}\) of the DCARC controller. Furthermore, the \(\varepsilon_{M}\) of all four controllers illustrated in Table 2 also shows that the proposed LODCARC controller outperforms the other controllers with respect to contouring performance.