# A Geometric Error Measurement System for Linear Guideway Assembly and Calibration
## 3 Uncertainty Analysis
### Pentaprism Setup  
The setup error originated from the pentaprism module used in the perpendicularity and parallelism measurement was analyzed, as depicted in Figure 11. Suppose the distance between the laser source and the fixed end is \(D_{pt}\), the error angle of the laser source in the vertical direction (\(\theta_{tv}\)) will generate an error \(\delta_{tv}\):  
\[\delta_{tv}=D_{pt}\ \tan\theta_{tv} \tag{7}\]  
The error analysis table of pentaprism in the vertical or horizontal direction is the same as Table 1.  
The measurement uncertainty of the proposed system can be obtained via partial differentiation of the linear displacements, \(\delta_{h}\) or \(\delta_{v}\), along the horizontal or vertical direction (\(x\)-axis or \(y\)-axis), which were, respectively, measured by the position deviation of the light spot along the \(x\)-axis or \(y\)-axis of PSD. The linear displacements (\(\delta_{h}\) and \(\delta_{v}\)) can be expressed as:  
\[\delta_{h}=f_{h}\Big{(}D_{lp},\theta_{ph},D_{pt}\Big{)} \tag{8}\]  
\[\delta_{v}=f_{v}\ \Big{(}D_{lp},\theta_{pv},D_{pt}\Big{)} \tag{9}\]  
Take the partial derivatives of \(\delta\) in Equations (8) and (9) with respect to the corresponding uncertain factors \(D_{l}\), \(\theta_{p}\), and \(D_{t}\), we get:  
\[ \begin{array}{l}d\delta_{h}=\frac{\partial f_{h}}{\partial D_{lp}}dD_{lp}+ \frac{\partial f_{h}}{\partial\theta_{ph}}d\theta_{ph}+\frac{\partial f_{h}} {\partial D_{pt}}dD_{pt}\\ =\tan\theta_{lh}\ dD_{lp}+F\tan\theta_{ph}\ \sec\theta_{ph}\ d\theta_{ph}+\tan \theta_{th}\ dD_{pt}\\ \\ d\delta_{v}=\frac{\partial f_{v}}{\partial D_{lp}}dD_{lp}+\frac{\partial f_{v }}{\partial\theta_{pv}}d\theta_{pv}+\frac{\partial f_{v}}{\partial D_{pt}}dD_ {pt}\\ =\tan\theta_{tv}\ dD_{lp}+F\tan\theta_{pv}\ \sec\theta_{pv}\ d\theta_{pv}+\tan \theta_{tv}\ dD_{pt}\\ \end{array}  \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{F_{v}}\) **or \(\mathbf{F_{h}}\) (\(\mathbf{\mu}\))** & \(\theta_{pv}\) **or \(\theta_{ph}\) (degree)** & \(\delta_{pv}\) **or \(\delta_{ph}\) (\(\mathbf{\mu}\)m)** \\ \hline
100 & 0.5 & 0.004 \\
100 & 1 & 0.015 \\
100 & 5 & 0.382 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error analysis of the PSD setup. Figure 11: Vertical setup error of the pentaprism. The horizontal and vertical straightness measurement uncertainties (\(\sigma_{\delta_{h}}\) and \(\sigma_{\delta_{p}}\)) of compound errors can be expressed as:  
\[\sigma_{\delta_{h}}=\pm\sqrt{\left(\frac{\partial f_{h}}{\partial D_{lp}}\sigma D _{lp}\right)^{2}+\left(\frac{\partial f_{h}}{\partial\theta_{ph}}\sigma\theta_{ ph}\right)^{2}+\left(\frac{\partial f_{h}}{\partial D_{pt}}\sigma D_{pt} \right)^{2}} \tag{12}\]  
\[\sigma_{\delta_{p}}=\pm\sqrt{\left(\frac{\partial f_{p}}{\partial D_{lp}} \sigma D_{lp}\right)^{2}+\left(\frac{\partial f_{v}}{\partial\theta_{pv}} \sigma\theta_{pv}\right)^{2}+\left(\frac{\partial f_{v}}{\partial D_{pt}} \sigma D_{pt}\right)^{2}} \tag{13}\]  
In the present study, for vertical vibration in the environment is always higher than horizontal vibration; therefore, the error angle of laser source at vertical direction can be assumed to three times of the horizontal direction as: The horizontal direction \(\theta_{lh}\): 20 arcsec and the vertical direction \(\theta_{lv}\): 60 arcsec; otherwise the travel distance of the laser beam passes through the pentaprism were much longer, therefore, the error angle of pentaprism can be assumed to be twice that of the laser source as: The horizontal direction \(\theta_{th}\): 40 arcsec, and the vertical direction \(\theta_{tv}\): 120 arcsec.