## 3 Results and Discussion
### Simulation Results  
In order to validate the optimal workpiece placement method, the joint velocities along the entire trajectory were measured by the offline simulation tool URSim. The data were captured using the Real-Time Data Exchange (RTDE) protocol, which is based on the TCP/P communication, and allows reading data at a frequency of 500 Hz. Only the measured joint velocities were analyzed, since our goal was to find the kinematically optimal position of the workpiece. The joint velocities were measured for all four placements of the workpiece \"Init\", \"1\", \"2\", \"Optim\", and all the robotic trajectories were executed with the same machining parameters displayed in Table 3, where the path was followed at a constant TCP speed. \begin{table}
\begin{tabular}{c c} \hline \hline
**Machining Parameter** & **Value** \\ \hline TCP speed & 50 mm/s \\ TCP acceleration & 500 mm/s\({}^{2}\) \\ Blend radius & 1 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Machining parameters. Figure 6: Representation of the calculated maximum achievable tool velocity V\({}_{\text{max}}\) at each waypoint on the robot’s path, and minimum value of V\({}_{\text{max}}\) for four placements of the workpiece in the case of the machining paths: (**a**) 1A, (**b**) 1B, (**c**) 2A, (**d**) 2B, (**e**) 3A, (**f**) 3B. Red, blue, orange, and green colors refer to initial workpiece placement “Init”, intermediate workpiece placements “1” and “2” and optimal workpiece placement “Optim”, respectively. The infinity norm of the joint velocities, which represents the maximum value of the joint velocity vector, were calculated at each time step for all four placements of the workpiece. These values are shown by the swam chart plot in Figure 7. The maximum value of the entire trajectory is displayed for each placement. It can be seen that the lowest infinity norm of joint velocities was required for the optimal workpiece placement compared to the other workpiece placements, considering the same machining parameters. The percentage of joint velocity reduction was calculated by comparing the considered workpiece placement to its optimal placement in the form of the parameter of improvement _imp_:  
\[imp=100\cdot\left(1-\frac{\max(\dot{\mathbf{q}}_{optim})}{\max(\dot{\mathbf{q }})}\right) \tag{22}\]  
In the case of machining path 1A, the lowest value of the maximum achievable linear velocity along the entire path was calculated for the orange-colored workpiece placement. As a result, the highest measured values of the maximum joint velocities were measured for the same workpiece placement (2.66). A 52.8% reduction in maximum joint velocities was achieved in comparison to the optimal workpiece placement. Similar findings were found in the case of all the other machining paths as well. For machining path 1B, the lowest values of the maximum achievable linear velocity along the path were measured for the blue-colored workpiece placement (2.31). In this case, we reduced the maximum joint velocities by 41.3% for the optimal workpiece placement. In the case of machining path 2A, the lowest value of the maximum achievable linear velocity along the path was calculated for the blue-colored workpiece placement, as is shown in Figure 6c. The highest joint velocities were achieved for this workpiece placement as well (2.67), as can be seen from Figure 7c. For the optimal workpiece placement, the maximum joint velocities were reduced by 25.1%. Figure 7: Representation of the joint velocity infinity norm for four different placements of the workpiece in the case of the machining paths: (**a**) 1A, (**b**) 1B, (**c**) 2A, (**d**) 2B, (**e**) 3A, (**f**) 3B. Red, blue, orange, and green colors refer to initial workpiece placement “Init”, intermediate workpiece placements “1” and “2” and optimal workpiece placement “Optim”, respectively. As depicted in Figure 6d, the lowest value of the maximum achievable linear velocity along the path for machining path 2B was calculated for the placement of the workpiece shown in blue.