# Improvement of Position Repeatability of a Linear Stage with Yaw Minimization
## 3 Factors Affecting the Yaw Motion
### Controllable Parameters: Design, Manufacturing Conditions
#### 3.2.2. Center of Mass & Center of Stiffness  
The moving platform has several midpoints: centers of geometry, mass, and stiffness. These midpoints are defined as  
\[\text{Center of geometry}=O(0,0) \tag{14}\]  
\[\text{Center of mass}=O^{\prime}\bigg{(}\frac{\int\!xdm}{m},\frac{\int\!ydm}{m} \bigg{)} \tag{15}\]  
\[\text{Center of stiffness}=O^{\prime}\bigg{(}\frac{\Sigma\,x_{i}k_{i}}{\sum k_{i }}\frac{\sum y_{i}k_{i}}{\sum k_{i}}\bigg{)} \tag{16}\]  
The midpoints are illustrated in Figure 17.  
Normalized y-coordinates of the platform mass center and LM guide stiffness center are defined as the ratios of the distances from the center of geometry (origin O) to each center point in the width direction to the half-width of the platform as  
\[CM(Normalizedcoordinateofplatformmasscenter)=\frac{y_{c}}{w/2}\times 100\ (\%) \tag{17}\]  
Figure 16: Results of the yaw angle of platforms. Figure 17: Midpoints of the platform. \[\text{CS}(\text{NormalizedcoodinateofLMguidestiffnesscenter})=\frac{y_{s}}{w/2} \times 100\ (\%) \tag{18}\]  
The center of mass can be modified by adding or removing weights for the given platform size. The center of stiffness can be modified by changing the positions of the LM Guides, installing additional guides, or replacing the LM Guide types. In general, two blocks on an LM Guide rail are identical, and thus \(k_{1}=k_{2}\), \(k_{3}=k_{4}\). Table 6 lists some of the \(CS\) values for the \(k_{3}/k_{1}\) ratios. The yaw motion dependence on the \(CS\) and \(CM\) for number four platform with zero parallelism at 80% of \(FO\) such as  
\[ \begin{array}{l}\theta=\left.\theta(PA:0,CS,CM)\right|_{FO:80\%,No4 Platform}\\ \lambda=\left.\lambda(PA:0,CS,CM)\right|_{FO:80\%,No4 Platform}\end{array}  \tag{19}\]  
is described in Figures 18 and 19. Figure 18 shows the value of the yaw angle for a range of _CS_ and _CM_, and Figure 19 shows the settling time. The yaw angle is greatly affected by the _CS_, while the effect of the _CM_ is minimal. The absolute value of the yaw angle is the smallest at a _CS_ of 80% for an _FO_ of 80%. Figure 19 shows that the settling time is also minimal when the _CS_ is the same as the _CM_. The lowest settling time is also achieved at a _CS_ of 80%. As for the yaw angle, Figure 18 indicates that it is governed primarily by the _CS_, indicating that the LM Guides are the most dominant elements constraining the yaw rotation of the platform. For the lower range of _CS_ values, the centers of stiffness and rotation are relatively close to each other. As the _CS_ increases closer to the value of _FO_, the constraints from guide rail number two are partially reduced, and thus, the center of rotation also moves closer to the _FO_ position. This will reduce the yaw moment by the driving force (_F_) about the center of rotation. Moreover, the rotational inertia of the platform is increased since the rotation center moves away from the center of mass. Settling time is a result of the complex interplay among the _CS_, _CM_, and _FO_, as indicated in Figure 19. It turns out that the settling time reaches its minimum at the values of _CS_ and _CM_ similar to a _FO_ of 80%. In this case, the centers of rotation, mass, and stiffness are closer to the _FO_ position, reducing the yaw moment and rotational inertia. The data for Figures 18 and 19 are listed in Tables 7 and 8. For the conditions where the _CS_ and _CM_ have an identical value, the effects of parallelism (_PA_) on the yaw angle and settling time, such as  
\[ \begin{array}{l}\theta=\left.\theta(PA,CS=CM)\right|_{F\text{080\%},No4 Platform} \\ \lambda=\left.\lambda(PA,CS=CM)\right|_{F\text{080\%},No4 Platform}\end{array}  \tag{20}\]  
may be investigated. Perfect parallelism between two LM Guide rails is impossible. The practical lower limit for _PA_ is 10 \(\upmu\)m for the moving platform. The maximum admissible settling time is 100 ms for the current motion profile of 1 g and 0.1 s for acceleration and deceleration.