## 3 Experiments
The average squareness error between the two respective CMM axes is then obtained by summing the negative pitch angles of both regression lines. To analyze the 21 items of geometric error of the CMM, the relationship between the CMM coordinate axes, the hole plate, and a homogeneous transformation matrix (HTM) was used. The equation \(\mathrm{T_{x}^{w}=T_{s}^{w}T_{z}^{z}T_{y}^{y}T_{x}^{w}}\), \(\mathrm{T_{x}^{w}}\) represents the HTM used to represent the relationship between adjacent coordinate axes. \(\mathrm{T_{s}^{w}}\) represents the homogeneous coordinate transformation matrix of the hole plate coordinate position relative to the column, which has three squareness errors. \(\mathrm{T_{z}^{s}}\) represents the homogeneous coordinate transformation of the column relative to the \(Z\)-axis, which has six geometric errors of the \(Z\)-axis. \(\mathrm{T_{y}^{z}}\) represents the homogeneous coordinate transformation matrix of the \(Z\)-axis moving system relative to the \(Y\)-axis, which has six geometric errors of the \(Y\)-axis. \(\mathrm{T_{x}^{y}}\) represents the homogeneous coordinate transformation matrix of the \(Z\)-axis moving system relative to the \(X\)-axis, including six geometric errors of the \(X\)-axis. To determine the geometrical error of the measuring machine at each coordinate point, the center coordinate position of each hole in the hole plate was measured. The difference between the coordinate position of the ideal hole and the actual hole was calculated, and the resulting difference value was used to determine the geometrical error. The HTM was utilized to investigate squareness error, positioning error, straightness error, and angle error. The measurement range of each error was between 0 mm and 550 mm, with a total of 12 points and a measurement interval of 50 mm. These errors can arise from a variety of factors, such as flawed assembly and geometry imperfections. Regular calibration of machine tools is necessary to maintain \begin{table}
\begin{tabular}{c c c} \hline \hline
**Axial** &
\begin{tabular}{c} **Symbols Used in** \\ **This Thesis** \\ \end{tabular} & **Definition of Geometric Errors** \\ \hline \multirow{6}{*}{\(X\)-axis} & Exx & Position error \\  & Eyx & Horizontal straightness error \\  & Exx & Vertical straightness error \\  & Exx & Roll error \\  & Ebx & Pitch error \\  & Ex & Yaw error \\ \hline \multirow{6}{*}{\(Y\)-axis} & Exy & Horizontal straightness error \\  & Eyy & Position error error \\  & Ezy & Vertical straightness error \\  & Easy & Pitch error \\  & Eby & Roll error \\  & Ecy & Yaw error \\ \hline \multirow{6}{*}{\(Z\)-axis} & Exz & Horizontal straightness error \\  & Eyz & Vertical straightness error \\  & Ezz & Position error \\  & Eaz & Pitch error \\  & Ebz & Yaw error \\  & Ecz & Roll error \\ \hline \multirow{6}{*}{Squareness between axes} & Ecox & Squareness of X to Y \\  & Eaoz & Squareness of Z to Y \\ \cline{1-1}  & Eboz & Squareness of Z to X \\ \hline \hline \end{tabular}
\end{table} 
Table 1: All geometric errors of linear axes of the three-axis machine tool. their accuracy and detect and rectify any geometric errors that may exist. By identifying and correcting the 21 geometric errors that can occur in these CMMs, manufacturers can ensure that their machine tools are operating at their maximum accuracy and producing high-quality products. To confirm the measurement accuracy of the hole plate, the results were compared with traditional standards. The measurements of the hole plate require a total of 2.5 h. When compared to traditional measurement methods such as the laser interferometer, autocollimator, and angle square, the laser interferometer takes 50 min, the autocollimator takes 50 min, and the angle square takes 20 min. Therefore, the traditional measurement methods require a total of 2 h to complete. The measurement procedure is followed by ISO 10360-2 [46]. Figure 6 shows the schematic representation of the experimental setup involving a laser interferometer, an autocollimator, and an angle square.