## 2 LMs-Driven X-Y-Y Stage  
The geometry of the LMs-driven X-Y-Y stage is given in Figure 1, in which one LM and two LMs are arranged on the \(x\) and \(y\) axes linear guides to yield displacements \(x,y_{1},\) and \(y_{2}\), respectively. As shown in Figure 1, the single LM in \(x\)-axis provides \(x\)-direction displacement. Moreover, the dual LMs in \(y\)-axis not only provide \(y\)-direction displacements but also can take the load, camera, or gripper along the guide. The two LMs in \(y\)-axis can move in synchronous or asynchronous ways according to the different requirements of the designed contouring control strategy. Accordingly, this X-Y-Y stage is able to perform two biaxial motions, which are the left contour motion in \(x\)-\(y_{1}\) axes and right contour motion in \(x\)-\(y_{2}\) axes. With this arrangement, working efficiency can be significantly improved.  
In the developed LMs-driven X-Y-Y stage, the electromagnetic thrust forces \(F_{mx}\) produced by the LM in \(x\)-axis and \(F_{myi}\), \(i=1,2\), produced by the LMs in \(y\)-axis are expressed as follows [9]:  
\[F_{mx}=K_{x}u_{x}, \tag{1}\]  
\[F_{myi}=K_{yi}u_{yi}, \tag{2}\]  
where \(K_{x}\) and \(K_{yi}\) denote the thrust coefficients in \(x\)-axis and \(y_{i}\)-axis, respectively; \(u_{x}\) and \(u_{yi}\) denote the control signals of three LMs. The resistance forces \(F_{rx}\) and \(F_{ryi}\) can be formulated as follows:(2)  
where \(B_{x}\) and \(B_{yi}\) denote the combined coefficients of the damping and viscous friction; \(\mathrm{Y}_{x}\) and \(\mathrm{Y}_{yi}\) denote the ripple forces and can be modeled by six terms as follows [10]:  
\[\mathrm{Y}_{x}=\sum_{j=1}^{6}\big{[}a_{xj}\sin(\omega_{xj}x)+\beta_{xj}\cos( \omega_{xj}x)\big{]}, \tag{5}\]  
\[\mathrm{Y}_{yi}=\sum_{j=1}^{6}\big{[}a_{yij}\sin(\omega_{yij}y_{i})+\beta_{yij }\cos(\omega_{yij}y_{i})\big{]}, \tag{6}\]  
where \(\alpha_{xj}\), \(\beta_{xj}\), \(\alpha_{yij}\), and \(\beta_{yij}\) denote the coefficients of the trigonometric series; \(\omega_{xj}\) and \(\omega_{yij}\) denote \(j\)th frequency; \(j=1\), \(2\), \(\ldots\), \(6\). Moreover, \(\chi_{x}\) and \(\chi_{yi}\) denote friction forces that can be expressed as follows [12]:  
\[\chi_{x}=\{F_{cx}+(F_{xx}-F_{cx})e^{-(|\frac{\ddot{x}}{x}|)^{6x}}+F_{xx}\big{[} \dot{x}\big{]}|\mathrm{sgn}(\dot{x}), \tag{7}\]  
\[\chi_{yi}=\{F_{cyi}+(F_{syi}-F_{cyi})e^{-(|\frac{\ddot{y}}{y_{m}}|)^{6y}}+F_{ yyi}\big{|}\dot{y}_{i}\big{|}\mathrm{sgn}(\dot{y}_{i}), \tag{8}\]  
where \(F_{cx}\) and \(F_{cyi}\) denote the Coulomb frictions; \(F_{xx}\) and \(F_{syi}\) denote the static frictions; \(F_{xx}\) and \(F_{syi}\) denote the viscous friction coefficients; \(x_{s}\) and \(y_{is}\) denote the lubricant; \(\delta_{x}\) and \(\delta_{yi}\) denote the additional empirical parameters; \(\mathrm{sgn}(\cdot)\) is a sign function. Consequently, the dynamics of the LMs-driven X-Y-Y stage are represented as follows:  
\[M_{x}\ddot{x}=F_{mx}-F_{rx}-F_{dx}, \tag{9}\]  
\[M_{yi}\ddot{y}_{i}=F_{myi}-F_{ryi}-F_{dyi}, \tag{10}\]  
where \(M_{x}\) and \(M_{yi}\) are the masses of LMs; \(F_{dx}\) and \(F_{dyi}\) are the unknown external disturbances. By substituting (1)-(4) into (9) and (10), the dynamic characteristics of the LMs-driven X-Y-Y stage can thus be derived as:  
\[\ddot{x}=-\frac{b_{x}}{M_{x}}\dot{x}-\frac{H_{x}}{M_{x}}+\frac{K_{x}}{M_{x}}u _{x}, \tag{11}\]  
\[\ddot{y}_{i}=-\frac{b_{yi}}{M_{yi}}\dot{y}_{i}-\frac{H_{yi}}{M_{yi}}+\frac{K_ {yi}}{M_{yi}}u_{yi}, \tag{12}\]  
where \(H_{x}\) and \(H_{yi}\) denote the lumped uncertainties, defined as follows:  
Figure 1: Geometry of the linear motors (LMs)-driven X-Y-Y stage.  
\[H_{x}=Y_{x}+x_{x}+F_{dx}, \tag{13}\]  
\[H_{yi}=Y_{yi}+x_{yi}+F_{dyi}. \tag{14}\]  
In general, accessing to the information of lumped uncertainties in advance is difficult in practical applications. Alternatively, an estimation scheme is employed in this study during the control process to enhance the contour tracking performance of the LMs-driven X-Y-Y stage.