# An Online Simultaneous Measurement of the Dual-Axis Straightness Error for Machine Tools
## 3 Uncertainty Analysis
### Verification of the Flatness Measurement System  
The flatness error verification of dual-axis measurement system was conducted by measuring the flatness of a surface-scraped cast-iron workpiece (300 mm \(\times\) 97 mm \(\times\) 33.9 mm) using a three-axis vertical milling machine tool (KSC-611, KENT, New Taipei city, Taiwan) equipped with a triangulation laser (LK-H020, Keyence Co., Osaka, Japan) in an air-conditioned laboratory, as shown in Figure 4.  
The triangulation laser was mounted on the spindle head (i.e., \(Z\)-axis) of the machine tool, which was utilized for ball-screw profile detection and flatness measurement in previous study [13; 14]. Up to the present, there is no commercial instrument capable of conducting dual-axis straightness measurements on the proposed system, therefore, the feasibility of the proposed system is determined according to the repeatability of the flatness measurement.  
Figure 16 shows the vertical straightness error model and corresponding flatness of X-Y plane on the moving working table detected by the proposed system, which could be regarded as the surface topography to a certain extent. The measurement interval of both axes is 50 mm, and the measurement ranges of the \(X\)- and \(Y\)-axes are 800 mm and 400 mm, respectively. The average flatness and repeatability (3\(\sigma\)) of the proposed system was 12.43 \(\upmu\)m and 0.65 \(\upmu\)m, respectively.  
Obviously, the vertical straightness errors measured along the \(X\)-axis with a varied \(Y\)-axis position were totally different, and their distribution were generally uneven. These finding prove that vertical straightness error cannot merely rely on the fitting results of individually-measured \(X\)- and \(Y\)-axis  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Result** & \multicolumn{2}{c}{**Instrument**} & \multicolumn{2}{c}{**Laser Interferometer**} & \multicolumn{2}{c}{**Proposed System**} \\ \hline Single-axis & Error (\(\upmu\)m) & 1.98 & 2.67 & 3.85 & 2.87 & 3.43 & 4.72 \\ Measuring & Repeatability (3\(\sigma\), \(\upmu\)m) & & 2.32 & & 2.32 & \\ \hline Dual-axis & Error (\(\upmu\)m) & 4.43 & 3.04 & 2.62 & 4.34 & 2.66 & 2.55 \\ Measuring & Repeatability (3\(\sigma\), \(\upmu\)m) & & 2.32 & & 2.46 & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Verification of vertical straightness measurements.  
Figure 15: Repeatability of dual-axis measurement verification.  
vertical straightness errors using the traditional single-axis method, for the \(X\)-axis vertical straightness errors are indeed varying with the \(Y\)-axis position.  
Before the workpiece flatness being measured, the upper surface of scrapped workpiece was placed on a blue-painted standard plate for marking high points on the scrapped surface. The workpiece was first placed in the detection range of the triangulation laser; thereupon, the proposed system was set on the three-axis machine tool. During the measurement process, the workpiece was moved by the working table according to the requested detection path; in addition, the flatness was obtained via the least-squares analysis.  
With the errors compensated by the proposed system, the matched high-points in the dash windows was conspicuously improved, evidencing the cancellation of vertical straightness errors originated from the moving working-table could be a significant benefit for surface topography measurements. Furthermore, both the average flatness and accuracy repeatability (3\(\sigma\)) were also improved from 64.65 \(\upmu\)m to 52.14 \(\upmu\)m and from 2.13 \(\upmu\)m to 1.46 \(\upmu\)m, respectively, as shown in Table 5, which proves the system's feasibility for