## 4 Results  
In order to avoid the influence of kinematics parameter errors excepting joint angles on the pose error, the kinematics parameters of the robot, including 24 DH parameters and the default \({}^{\rm W}\)\(T_{\rm T}\) are calibrated by the least squares method in Ref. [13]. The calibrated DH parameters are listed in Table 1.  
Figure 8: Experimental setup of the robotic drilling system. Figure 9: Robotic drilling task. (**a**) Region division of robotic drilling task; (**b**) Sampling area division of robotic drilling task. It can be seen that the maximum difference of the \({}^{\rm W}\,T_{\rm T}\) appears in the \(Y\) direction, and the maximum deviation of \(Y\) from \(\Omega_{1}\) to \(\Omega_{4}\) is 1.39 mm, but the maximum deviation of \(Y\) from working regions to \(\Omega_{0}\) is 9.92 mm, which is caused by the large change of the orientation in the \(YOZ\) plane of the robot base frame. Simultaneously, it can be seen from Table 3 that the distances between the origin of four \({}^{\rm W}\,T_{\rm T}\) in the drilling regions from \(\Omega_{1}\) to \(\Omega_{4}\) (up to 4.44 mm) are much smaller than those between the drilling regions and \(\Omega_{0}\) (up to 9.98 mm), which shows that the positioning error of the robot with large orientation variation seriously affects the positioning accuracy of the actual wrist frame, so that the \({}^{\rm W}\,T_{\rm T}\) calibrated by the least square method is no longer applicable. However, the distance of the ipsilateral regions (\(\Omega_{1}\) vs. \(\Omega_{2}\) and \(\Omega_{3}\) vs. \(\Omega_{4}\)) is 1.08 mm and 1.1 mm, which indicates that even if the positions are close, \({}^{\rm W}\,T_{\rm T}\) will also be affected by the orientation variation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Joint Frame** & \(\theta\) (\({}^{\circ}\)) & \(a\) (mm) & \(d\) (mm) & \(\alpha\) (\({}^{\circ}\)) \\ \hline \(\{\beta\}\) & 0 & 0 & 0 & 0 \\ \(\{\Omega_{1}X_{1}Y_{1}Z_{1}\}\) & \(\theta_{1}\) & 500.5065 & 1045 & \(-\)90 \\ \(\{\Omega_{2}X_{2}Y_{2}Z_{2}Z_{2}\}\) & \(\theta_{2}\) & \(-\)1300.4225 & 0 & 0 \\ \(\{\Omega_{3}X_{3}Y_{3}Z_{3}\}\) & \(\theta_{3}\) & 54.39 & 0 & \(-\)90 \\ \(\{\Omega_{4}X_{4}Y_{4}Z_{4}\}\) & \(\theta_{4}\) & 0 & \(-\)1025.1913 & 90 \\ \(\{\Omega_{5}X_{5}Y_{5}Z_{5}\}\) & \(\theta_{5}\) & 0 & 0 & \(-\)90 \\ \(\{\Omega_{6}X_{6}Y_{6}Z_{6}\}\) & \(\theta_{6}\) & 0 & 0 & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calibrated DH parameters of KUKA KR500-2830. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Region** & \(\Omega_{0}\) & \(\Omega_{1}\) & \(\Omega_{2}\) & \(\Omega_{3}\) & \(\Omega_{4}\) \\ \hline \(x_{\rm T}\) (mm) & 453.31 & 452.23 & 452.03 & 451.24 & 450.50 \\ \(y_{\rm T}\) (mm)  
As shown in Figure 9b, according to the spatial distribution of the drilling task, the length, width and height of sampling space \(\Omega_{1}\), \(\Omega_{2}\), \(\Omega_{3}\) and \(\Omega_{4}\) are 300 mm \(\times\) 300 mm \(\times\) 500 mm. Latin hypercube sampling [25] is used to plan 100 sampling points in each region, and the positioning errors of \(\{TR\}\) with the \({}^{\rm W}\)T\({}_{\rm T}\) calibrated in \(\Omega_{0}\), \(\Omega_{1}\), \(\Omega_{2}\)\(\Omega_{3}\) and \(\Omega_{4}\) are collected respectively. After that, the positioning error of the robot is calibrated by the method proposed by Ref. [28] and this paper for comparison. The robot positioning accuracy comparison results are shown in Figure 10, where all 64 desired drilling poses in \(\Omega_{1}\), 27 poses in \(\Omega_{2}\), 45 poses in \(\Omega_{3}\) and 50 poses in \(\Omega_{4}\) are tested according to the part drawing. Figure 10: _Cont._  
The experimental results demonstrate that for any region, the proposed method and the method in Ref. [28] both effectively reduce the positioning error. In detail, the differences between the method in Ref. [28] and the proposed method with \({}^{W}\)\(T_{\text{T}}\) calibrated in the corresponding regions (blue lines vs. gray lines) are caused by the difference between compensation methods.