# Region-Based Approach for Machining Time Improvement in Robot Surface Finishing
## 4 Case Study
### Path Speed Validation  
#### 4.4.1 Path Point--Wise Design Speed Validation  
Our methodology for calculating the maximum feasible robot surface machining speed \(V_{\text{max}}\) (17) in the optimal direction was developed in the surface tangent plane. Note that, in this approach, the \(V_{\text{max}}\) was calculated for all surface points, essentially creating  
Figure 7: Final region-based machining directions in the optimal workpiece position: a perspective view for WP1 (**a**) and WP2 (**c**), and top view with maximum region-wise feasible speeds in mm/s for WP1 (**b**) and WP2 (**d**). \begin{table}
\begin{tabular}{l r r r r r} \hline \hline  & **Ran. Pos. 1** & **Ran. Pos. 2** & **Initial Pos.** & **Optimal Pos.** & **Impr. \%** \\ \hline \multicolumn{6}{c}{**Workpiece 1**} \\ \hline Position [mm; mm; \({}^{\circ}\)] & \(-200\)\(\times\) & \(-525\); \(2\); \(30\) & \(125\); \(1\)\(\times\)\(504\); \(5\)\(147.5\) & \(0\); \(-570\); \(0\)\(-17.4\); \(-564.6\); \(223.1\) & \\ MTI [s] & 69.04 & 71.24 & 66.82 & 60.98 & \\ \multicolumn{6}{c}{**Machine time [mins]**} \\ \hline \multicolumn{6}{c}{**Workpiece 2**} \\ \hline Position [mm; mm; \({}^{\circ}\)] & \(55.4\); \(-566.3\); \(-165.6\); \(-566.3\) & \(0\); \(-570\); \(0\) & \(-104.9\); \(-407.9\) & \\ MTI [s] & \(48.18\) & \(43.45\) & \(42.06\) & \(34.81\) & \\ \multicolumn{6}{c}{**Machine time [mins]**} \\ \hline \multicolumn{6}{c}{**Machine time [mins]**} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of minimal (maximum feasible speeds) and machining times for different region-based workpiece positions \(x[mm];y[mm];\gamma[^{\circ}]\). a velocity vector field (VVF), as seen in the left columns of Figures 8 and 9. The surface machining paths seen in the right columns of Figures 8 and 9 were created based on these directions. The maximum feasible speeds for those machining paths were calculated by the DTF method [19], which computes the robot's maximum feasible linear speed between consecutive points along the machining path:  
\[V_{\max}^{path}=\frac{1}{\left\|\hat{\hat{T}}_{T}^{t}\hat{a}_{T}^{path}+h^{-1} \hat{\hat{T}}_{R}^{t}\hat{a}_{R}^{path}\right\|_{\infty}} \tag{27}\]  
where \(\hat{a}_{T}^{path}\) and \(\hat{a}_{R}^{path}\) are linear and angular directions along the path, and \(h=V/\Omega\) is a task-dependent velocity ratio factor that reflects the synchronization between the translational and rotational motion, calculated as the ratio of the linear velocity to the angular velocity. In this validation process, we wanted to test if the actual machining paths and their maximum feasible speeds corresponded to the initial design calculations. In Figure 8, there is a comparison for WP1 in the \(X\)-direction (Figure 8a,b), \(Y\)-direction (Figure 8c,d), optimal direction (Figure 8e,f), and region-based directions (Figure 8g,h). This applies similarly to WP2, which is depicted in Figure 9, where the \(X\)-direction is shown in Figure 9a,b, the \(Y\)-direction in Figure 9c,d, the optimal direction in Figure 9e,f, and region-based directions in Figure 9g,h. Figure 8: Speed colormap comparison between the initial VVF model and the actual generated toolpath with marked critical points for multiple WP1 scenarios: (**a**) VVF model in \(X\)-direction, (**b**) toolpath in \(X\)-direction, (**c**) VVF model in \(Y\)-direction, (**d**) toolpath in \(Y\)-direction, (**e**) VVF model in the optimal direction, (**f**) toolpath in the optimal direction, (**g**) VVF model in the region-based optimal directions, (**h**) the region-based optimal toolpath. Figure 9: \(Cont.\)  
The mapping from the model-based VFF to the actual machining path can result in deviations in the maximum feasible speeds. These deviations arise due to the disparity in the density of model points and generated path points and their different positions.