## 5 Proposed Contour Tracking Control Systems
### Contour Tracking Using ACC2 PID Control System  
To improve the contour tracking performance levels and robustness of the 2 PID control system, the ACC2 PID contour tracking control system is further proposed. The control block diagram of the LMs-driven X-Y-Y stage using ACC2 PID control system is shown in Figure 4. Regarding the biaxial structure of the LMs-driven X-Y-Y stage, two contour errors \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are in the \(x\)-\(y_{1}\) and \(x\)-\(y_{2}\) contours, respectively. Recall from Figure 2, each contour error can be further decomposed to the component in the \(x\)-axis as (20) and the component in the \(y\)-axis as (21). To suppress the contour errors, two contour error compensators \(G_{cix}\)(s) and \(G_{ciy}\)(s), \(i=1,2\), are further designed to confront the contour errors in the \(x\)-axis and \(y\)-axis, respectively. Now, design the contour error compensator as follows:  
\[u_{cix}=G_{cix}(s)\varepsilon_{ix}=K_{cix}\varepsilon_{ix}, \tag{36}\]  
\[u_{ciy}=G_{ciy}(s)\varepsilon_{iy}=K_{ciy}\varepsilon_{iy}, \tag{37}\]  
where \(G_{cix}\)(s) and \(G_{ciy}\)(s) are selected as P gains \(K_{cix}\) and \(K_{ciy}\) for the \(x\)-axis and \(y\)-axis, respectively. Thus, as shown in Figure 4, three 2 PID controllers with two compensators were constructed for the proposed ACC2 PID control system to concurrently perform contours tracking and contour error compensation.  
To improve the contour tracking performance and system robustness, the IABC algorithm was used to tune the 2 PID controller parameters for each axis LM as in (34) and (35), and one control parameter of each compensator as in (36) and (37) during the contour tracking process. Thus, each bee of the IABC algorithm can be formulated as a 19-dimensional vector \(\psi_{i}=[K_{px}\ T_{lx}\ T_{Dx}\ \alpha_{x}\ \beta_{x}\ K_{py1}\ T_{ly1}\ T_{Dy1}\ \alpha_{y1}\ \beta_{y1}\ K_{py2}\ T_{Dy2}\ \alpha_{y2}\ \beta_{y2}\ K_{c1x}\ K_{c1y}\ K_{c2x}\ K_{c2y}]\). According to the different bees represented by the distinct vectors, different control signals \(u_{x}\), \(u_{y1}\), and \(u_{y2}\) under the same tracking error and contour error will be generated. To avoid performance degradation due to the use of immature parameters, the control signals \(u_{x}\), \(u_{y1}\), and \(u_{y2}\) with the parameters in the evolution were sent to the mover positions observer for observing the virtual mover positions \(\hat{x}\), \(\hat{y}_{1}\), and \(\hat{y}_{2}\) as shown in Figure 4. Moreover, the observation errors \(\tau_{x}\), \(\tau_{y1}\), and \(\tau_{y2}\), which were defined as the differences between \(\hat{x}\) and \(x\), \(\hat{y}_{1}\) and \(y_{1}\), \(\hat{y}_{2}\) and \(y_{2}\), were used in the parallel system parameters identification approach. On the other hand, the virtual mover positions \(\hat{x}\), \(\hat{y}_{1}\), and \(\hat{y}_{2}\) are used to evaluate the fitness values as follows:  
\[\begin{split} fit^{-1}(I)=\sqrt{\frac{1}{n}\sum\limits_{l=1}^{n} \left[x_{d}(I)-\hat{x}(I)\ \right]^{2}}\ +\sum\limits_{i=1}^{2}\sqrt{\frac{1}{n}\sum\limits_{l=1}^{n}\left[y_{id}(I)- \hat{y}_{i}(I)\ \right]^{2}}+\sum\limits_{i=1}^{2}\sqrt{\frac{1}{n}\sum\limits_{k=1}^{n}\hat{ \varepsilon}_{i}(I)\ ^{2}}\\ =\sqrt{\frac{1}{n}\sum\limits_{l=1}^{n}\hat{\varepsilon}_{x}(I) \ ^{2}}+\sum\limits_{i=1}^{2}\sqrt{\frac{1}{n}\sum\limits_{l=1}^{n}\hat{ \varepsilon}_{y}(I)\ ^{2}}+\sum\limits_{i=1}^{2}\sqrt{\fracthe online IABC optimization mechanism can concurrently reduce the tracking error in each axis and the contour errors of biaxial motions simultaneously.