# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 3 Results
### Collaborative Robot UR5e  
#### 3.3.1 Path Segment Feasiblity  
The proposed DTF method was also demonstrated and verified on the collaborative robot UR5e, which is of 6 DOF, thus enabling robot operation in a 3D space. In this case, the 3D curve on the workpiece surface was determined as the path that the robot tool had to follow. A set of waypoints was placed along the path, with the orientation normal to the surface. The robot end-effector was to simultaneously reach each of them with the desired position and orientation. In this experiment, the proposed DTF method will be used to determine the maximum feasible linear velocity at three different parts of the curved path, as shown in Figure 8, if we assume that the maximum kinematic capabilities of the robot are limited by the joint velocity \(\dot{\theta}_{\text{max}}=3.14\) rad/s for all joints. Similar to the 3 DOF planar case, example (i) represents a flat path segment of the surface, a strongly curved path segment was considered in example (ii), and a slightly curved path segment of the surface was analyzed in example (iii). At each of the segments under consideration, the directions of the linear and angular velocities were determined along the path. During the motion along path segment (ii), a significant change in the end-effector orientation was required, while a relatively small change in position was performed; therefore, the ratio between the magnitudes of the linear and angular velocities, denoted by \(h\), converged to 0. On the relatively flat path segment (i), the velocity ratio factor \(h\) was the highest since the linear motion dominated the angular motion. The direction of the linear velocity vector \(\dot{u}_{T}\), the direction of the angular velocity vector \(\dot{a}_{R}\), and the velocity ratio factor \(h\) of each segment are shown in Table 3.  
The maximum linear velocity \(V_{\text{max}}\) and maximum angular velocity \(\Omega_{\text{max}}\) based on the DTF method are shown in Table 4, together with the required joint velocities \(\dot{\theta}\). Figure 8: Collaborative robot UR5e with three different parts of the curved surface: (i) a flat path segment; (ii) a strongly curved path segment; (iii) a slightly curved path segment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Path Segment** & \multicolumn{2}{c}{\(\hat{\mathbf{u}}_{T}\)} & \multicolumn{2}{c}{\(\hat{\mathbf{u}}_{R}\)} & \multicolumn{2}{c}{\(\mathbf{h}\)} \\ \hline (i) & \(\left[0.9999\right.\) & \(0\) & \(0.0117\) \(\right]^{T}\) & \(\left[0.6209\) & \(0.7625\) & \(-0.1820\right]^{T}\) & 4.4632 \\ (ii) & \(\left[-0.9893\right.\) & \(0\) & \(0.1459\) \(\right]^{T}\) & \(\left[0.0865\) & \(-0.9761\) & \(-0.1994\) \(\right]^{T}\) & 0.0360 \\ (iii) & \(\left[0.9992\right.\) & \(0\) & \(0.0411\) \(\right]^{T}\) & \(\left[0.9997\) & \(-0.0249\) & \(0.0011\right]^{T}\) & 0.7454 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Task requirements for three different path segments (UR5e). For each of the considered examples (i), (ii), and (iii), the 3D translational manipulability polytope \(P_{\infty}^{T}\) in the \(L_{\infty}\) weak sense (magenta), the 3D translational manipulability polytope \(P_{strong}^{T}\) in the strong sense (green), the 3D rotational manipulability polytope \(P_{\infty}^{R}\) in the \(L_{\infty}\) weak sense (blue), and the 3D rotational manipulability polytope \(P_{strong}^{R}\) in the strong sense (red) are all illustrated in Figure 9. The maximum feasible linear velocity vector \(v_{\text{max}}\) (red arrow) and the maximum feasible angular velocity vector \(\omega_{\text{max}}\) (blue arrow) by the proposed DTF method are also illustrated. Note that the twist space in this case is of 6D, and thus inconvenient for a graphical presentation of the 6D twist polytope \(P^{\text{trist}}\).