# Improvement of Position Repeatability of a Linear Stage with Yaw Minimization
## 2 Dynamic Model for Moving Platform
### Equations of Motion  
A free body diagram is shown in Figure 4, presenting the reactions between the moving platform and two LM Guides. The platform guided by rail #1 and rail #2 is driven by force (\(F\)) offset from the platform's center of geometry by the distance (\(d\)). The blocks of rail #1 are numbered 1 and 2, respectively, while those of rail #2 are numbered 3 and 4, which are symmetric about the \(y\)-axis by distances of \(l\) and \(q\), respectively. The blocks have a horizontal stiffness of \(k_{n}\) and a damping coefficient of \(c_{n}\). (\(n=1\), \(2\), \(3\), \(4\)). The Cartesian coordinate system represents the movement direction and the width direction as \(x\) and \(y\) and the direction perpendicular to the \(x\)-\(y\) plane as \(z\). Figure 5 shows a schematic diagram of the platform rotated by an angle of \(\theta\) by the driving force (\(F\)). The coordinates (\(x_{c}\), \(y_{c}\)) and (\(x_{p}\), \(y_{p}\)) refer to the center of mass and the center of rotation, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Horizontal Stiffness** & **Vertical Stiffness** & **Effective Load (kN)** \\  & **(kN/mm)** & **(kN/mm)** & **Effective Load (kN)** \\ \hline HSR20 & 250 & 349 & 1.4 \\ HSR20L & 335 & 472 & 2.3 \\ HSR25L & 335 & 488 & 2.8 \\ HSR30L & 453 & 645 & 3.8 \\ HSR35L & 495 & 681 & 5.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Stiffness of HSR LM Guides. Figure 4: Free body diagram of the moving platform. Figure 3: LM Guide (**a**) components and (**b**) stiffness elements. \[ \begin{array}{l}\sum F_{x}=m\ddot{x}_{p}\\ \sum F_{y}=m\ddot{y}_{p}\\ \sum M=I\ddot{\theta}\end{array}  \tag{1}\]  
The force equilibrium equation for the platform is given as Equation (1), and the rotational inertia is given as Equation (2). \[I=\int\limits_{platform}r^{2}dm+m\Big{(}(x_{p}-x_{c})^{2}+(y_{p}-y_{c})^{2}\Big{)} \tag{2}\]  
The rotational inertia (\(I\)) is calculated with respect to the rotation center. The \(r\) is the distance from the center of rotation to the point mass (\(dm\)) of the moving platform. The position of the rotational center (Pole) is a function of time, and the lengths of the lines (\(s_{n}\)) from the pole to the block centers are given as,  
\[ \begin{array}{l}s_{1}(t)=\sqrt{\left(l/2+x_{p}(t)\right)^{2}+\left(w/2-y_{p }(t)\right)^{2}}\\ s_{2}(t)=\sqrt{\left(l/2-x_{p}(t)\right)^{2}+\left(w/2-y_{p}(t)\right)^{2}}\\ s_{3}(t)=\sqrt{\left(q/2+x_{p}(t)\right)^{2}+\left(w/2+y_{p}(t)\right)^{2}}\\ s_{4}(t)=\sqrt{\left(q/2-x_{p}(t)\right)^{2}+\left(w/2+y_{p}(t)\right)^{2}} \end{array}  \tag{3}\]  
while the angles (\(\tau_{n}\)) between the lines \(s_{n}\) and the \(x\)-axis are given as,  
\[ \begin{array}{l}\tau_{1}(t)=\tan^{-1}\left(\frac{w/2-y_{p}(t)}{x_{p}(2)+l/2}\right) \\ \tau_{2}(t)=\tan^{-1}\left(\frac{w/2-y_{p}(t)}{-x_{p}(t)+l/2}\right)\\ \tau_{3}(t)=\tan^{-1}\left(\frac{w/2+y_{p}(t)}{x_{p}(2)+l/2}\right)\\ \tau_{4}(t)=\tan^{-1}\left(\frac{w/2+y_{p}(t)}{-x_{p}(t)+l/2}\right)\end{array}  \tag{4}\]  
Figure 5: Schematic diagram of yaw motion. The angle (\(\rho_{n}\)) between the block center displacement vectors (\(\delta_{n}\)) and the \(x\)-axis can be derived as shown in Equation (5). \[ \begin{array}{l}\rho_{1}(t)=-\frac{\theta(t)+2\tau_{1}(t)}{2}+\frac{\pi}{2}\\ \rho_{2}(t)=\frac{\theta(t)-2\tau_{2}(t)}{2}+\frac{\pi}{2}\\ \rho_{3}(t)=\frac{\theta(t)-2\tau_{2}(t)}{2}+\frac{\pi}{2}\\ \rho_{4}(t)=-\frac{\theta(t)+2\tau_{4}(t)}{2}+\frac{\pi}{2}\\ \end{array}