# Design and Accuracy Analysis of a Micromachine Tool with a Co-Planar Driving Mechanism
## 3 Error Model and Analysis
### Error Model
To analyze the sensitivity of the ballscrew motion and the tilt-drive angle on total machine errors, the differences between the ideal tool-end position and the actual tool-end position were substituted into Equation (11) under the condition that all other error sources were zero; consequently, the total position error of the machine can be expressed as follows. \[\Delta\text{E}_{\text{P}}=\left[ \begin{array}{cccc}1&0&0&\text{x}-\delta c ^{\prime}\cdot\cos(\theta+\delta\theta_{3})\\ 0&1&0&\text{y}-\delta d^{\prime}\cdot\tan(\theta+\delta\theta_{4})\\ 0&0&1&\text{z}-\delta bd^{\prime}\cdot\cos\theta+\delta\theta_{2}\\ 0&0&0&1\end{array} \right] \tag{14}\]  
where \(\delta b^{\prime},~{}\delta c^{\prime},~{}\delta d^{\prime}\) represent ballscrew motion errors along the X-, Y-, and Z-axes, respectively; \(\delta\theta_{2},~{}\delta\theta_{3},~{}\delta\theta_{4}\) represent the tilt-drive angle errors along X-, Y-, and Z-axes, respectively. According to Equation (14), it is noted that the motion error of each ballscrew and the assembly error of the tilt-drive angle cause position errors at each axis. The position error at the X-axis can be derived from Equation (14), as  
\[\left(\Delta\text{E}_{\text{P}\times}\right)_{\delta~{}c^{\prime},~{}\delta \theta_{3}}=\text{x}-\delta~{}\text{c}~{}\prime\times\cos(\theta+\delta\theta _{3}) \tag{15}\]  
The sensitivity parameter of the ballscrew motion and the assembly error of the tilt-drive angle can be obtained by the differential of Equation (15), as follows. \[d\left(\Delta\text{E}_{\text{P}\times}\right)_{\delta~{}c^{\prime},\delta \theta_{3}}=\cos(\theta+\delta\theta_{3})d\delta~{}\text{c}~{}\prime-\delta~{} \text{c}~{}\prime~{}\sin(\theta+\delta\theta_{3})~{}d\delta\theta_{3} \tag{16}\]  
The first sensitivity parameter, \(\frac{\partial\left(\Delta\text{E}_{\text{P}\times}\right)_{\delta c^{\prime} \delta\theta_{3}}}{\partial\delta c^{\prime}}\), is \"\(\cos(\theta+\delta\theta_{3})\)\". This shows that this parameter will not change when the position of the platform changes, but it will change when the tilt-drive angle changes. Figure 8 shows the relationship between the parameter and the tilt-drive angle. The sensitivity parameter \(\frac{\partial\left(\Delta\text{E}_{\text{P}\times}\right)_{\delta c^{\prime} \delta\theta_{3}}}{\partial\delta c^{\prime}}\) will decrease when the tilt-drive angle increases. That is, the larger the tilt-drive angle error, the lower the influence of this parameter. Another sensitivity parameter, \(\frac{\partial\left(\Delta\text{E}_{\text{P}\times}\right)_{\delta c^{\prime} \delta\theta_{3}}}{\partial\delta\theta_{3}}\), is \"\(-\delta~{}\text{c}^{\prime}~{}\sin(\theta+\delta\theta_{3})\)\", in which \(\delta\text{c}^{\prime}\) changes when the ballscrew position varies. It can be seen from Figure 9 that the value of the sensitivity parameter decreases when either of the tilt-drive angles increases. Figure 8: Relationship between the ballscrew motion error \(\delta c^{\prime}\) and parameter sensitivity. By substituting the condition of the prototype machine--where the tilt-drive angle \(\theta\) is 25.84\({}^{\circ}\), the assembly error of the tilt-drive angle \(\delta\theta\)3 is 0.05\({}^{\circ}\), and the ballscrew motion error is 0.5\(\upmu\)m--into Equation (17), it is found that machine accuracy can be controlled within a range of 1 \(\upmu\)m. Additionally, it can be concluded that the ballscrew motion error source has a greater impact on the overall X-axis error. Similarly, the error source of the Z-axis ballscrew motion has a greater impact on the overall Z-axis error. \[\frac{\left|\cos(\theta+\delta\theta_{3})\right|}{\left|\delta c^{\prime}\, \sin(\theta+\delta\theta_{3})\right|}>1 \tag{17}\]  
The position error of the Y-axis can be calculated using Equation (18). The sensitivity parameter of the ballscrew motion and the assembly error of the tilt-drive angle can be obtained by means of the differential of Equation (18), as follows.