# Region-Based Approach for Machining Time Improvement in Robot Surface Finishing
## 4 Case Study
### Path Speed Validation
#### 4.4.2 Region-Based Design Speed Validation  
To validate the practicality of our region-based design path speed method combined with optimal workpiece placement, we also analyzed the infinity norm of the joint velocities representing the maximum value of the joint velocity vector at each step of all the generated trajectories. The primary objective was to confirm that the robot's movements adhered to the calculated machining speeds and that these velocities remained within the limits of the UR5e robot, specifically not exceeding its maximum joint speed limit \(\hat{\theta}_{\text{max}}\) of 3.14 rad/s across the four workpiece placements. Regions in the initial position for both workpieces can be seen in Figure 5, and regions in the optimal positions can be seen in Figure 7.  
The analysis revealed that the actual joint speeds in certain regions exceeded the maximum allowable limit slightly, as seen in Figure 10, for all workpiece placements. Still, the machining time was the lowest in the optimum position, confirmed in Table 5, which aligns with our previous observation regarding maximum feasible machining speeds. This finding indicates that, while the robot operated close to its kinematic threshold to achieve  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Path Directions** & **VVF Model** & **Machining Toolpath** & **Difference (\%)** \\ \hline \multicolumn{4}{c}{**Workpiece 1**} \\ \hline \hline \(X\)-direction & 82.3 & 81.9 & 0.5 \\ \(Y\)-direction & 72.2 & 71.2 & 1.3 \\ Optimal direction & 93.3 & 90.2 & 3.4 \\  & R1: 179.1, R2:213.4, & R1: 172.5, R2: 211.1, & R1: 3.8, R2: 1.1, R3: \\ Region-based & R3: 120, R4: 292.2, R5: & R3: 117.1, R4: 281, R5: & 2.4, **R4: 3.9**, R5: 0.6, \\ directions & 203.6, R6: 91.6, R7: & 202.4, R6: 89.6, R7: & R6: 2.2, R7: 2.5, R8: \\  & 104.5, R8: 228.4, R9: & 101.9, R8: 224, R9: 150 & 1.9, R9: 3.7 \\  & 155.7 & & \\ \hline \hline \multicolumn{4}{c}{**Workpiece 2**} \\ \hline \hline \(X\)-direction & 124.6 & 123 & 1.3 \\ \(Y\)-direction & 115.3 & 113.1 & 1.9 \\ Optimal direction & 155.1 & 151.4 & 2.4 \\  & R1: 233.1, R2: 271.2, & R1: 227.4, R2: 264.5, & R1: 2.5, R3: \\  & R3: 276.1, R4: 262.6, & R3: 264.9, R4: 261.2, & R1: 2.5, R2: 2.5, **R3:** \\ Region-based & R5: 327.4, R6: 240.4, & R5: 319.4, R6: 234.8, & **4.1**, R4: 0.5, R5: 2.5, \\ directions & R7: 454, R8: 415.3 R9: & R7: 443.8, R8: 404.1, & R6: 2.4, R7: 2.3, R8: \\  & 345, R10: 591.6, R11: & R9: 339.3, R10: 583.4, & 2.7, R9: 1.7, R10: 1.4, \\  & 216 & R11: 209.1 & R11: 3.2 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Validation results for maximum feasible speeds in mm/s between the initial design VVF model and machining toolpath for multiple machining cases.  
the highest possible linear velocities per our optimization, it occasionally surpassed the safety margins. This is because the VVF is designed in the tangent planes attached to the surface design points, whereas the maximum speeds on the path are calculated on the path point-to-point principle. Given these discrepancies and the minor overages in joint speeds, we recommend setting the surface finishing speeds at, e.g., 10% lower than those calculated by our method for future implementations in practice. However, here, we should consider that the safety margin depends on the density of the surface points considered within the design phase and the density of the generated path points. This adjustment will ensure that machining operations remain safely within the robot's mechanical capabilities, mitigating the risk of exceeding joint speed limits and enhancing the overall feasibility of the machining process.