# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 3 Results
### Generic 3 DOF Planar Manipulator
\begin{table}
\begin{tabular}{c c|c c|c c} \hline \hline
**Path Segment** & **Position** & \(\hat{u}_{T}\) & \(\hat{u}_{R}\) & \(h\) \\ \hline (i) & X: 0.7720, Y: 0.3882 & [0.7720 & 0.3882] & [+1] & 1.3231 \\ (ii) & X: 1.0340, Y: 0.2374 & [1.0340 & 0.2374] & [+1] & 0.0493 \\ (iii) & X: 1.1640, Y: 0.3877 & [1.1640 & 0.3877] & [\(-1\)] & 62.4704 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Task requirements for three different path segments (3 DOF manipulator). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Path Segment** & \(V_{\text{max}}\) **(m/s)** & \(\Omega_{\text{max}}\) **(rad/s)** & \multicolumn{2}{c}{\(\hat{\theta}\) **(rad/s)**} \\ \hline (i) & 1.1439 & 0.8646 & \(\begin{bmatrix}-0.7471&0.6117&1.0000\\ (ii) & 0.0489 & 0.9935 & \(\begin{bmatrix}0.4469&-0.4534&1.0000\\ -0.1118&1.0000&-0.9034\end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Feasible directional kinematic capabilities for three different path segments (3 DOF manipulator). For each of the considered examples (i), (ii), and (iii), the twist polytope \(P^{tust}\) and the translational manipulability polytopes \(P^{T}_{\infty}\), \(P^{T}_{2}\), and \(P^{T}_{strong}\) are illustrated in Figure 7 in three different views, i.e., the 3D projection view, the 2D view in the \(v_{x}-\omega\) plane, and the 2D view in the \(v_{x}-v_{y}\) plane, respectively. The figure also depicts the maximum feasible linear and angular velocities, as determined by the proposed DTF method given by (30), (31). It can easily be verified that the translational manipulability polytope \(P^{T}_{\infty}\) fits to the orthogonal projection of the twist manipulability polytope \(P^{tust}\) onto the translational subspace, shown by the \(v_{x}\)-\(v_{y}\) plane in this case (Figure 7c,fj). For the synchronous linear and angular motion, considering the desired task requirements given by the synchronization parameter \(v_{\max}\)\(h\), the maximal linear velocity vector \(v_{\max}\) (see (32)) based on the proposed DTF method was calculated and shown as a vector in red. For the desired motion considered for the segment (i), the magnitude of the maximum linear velocity vector \(v_{\max}\) was smaller than the vector length of the manipulability polytope \(P^{T}_{\infty}\). When almost only angular motion was needed, as in the case of segment (ii), the maximum linear velocity \(v_{\max}\) was near zero. In the case of segment (iii), where the linear motion dominates the angular motion, the value of \(V_{\max}\) was equal to the vector length of the translational manipulability polytope \(P^{T}_{strong}\) in the strong sense, which was also the highest feasible linear velocity for this combination of \(\hat{u}_{T}\) and \(\hat{u}_{R}\). The \(v_{x}-\omega\) plane view (Figure 7b,c,e) shows the maximum feasible angular velocity capabilities of each path segment. The maximum feasible angular velocity for the specific task requirements of each considered segment is shown as a vector \(\omega_{\max}\) in blue. The maxi  
Figure 7: Graphical representation of \(v_{\max}\) and \(\omega_{\max}\) by the DTF method for a generic 3 DOF planar robot manipulator in relation to its manipulability polytopes \(P^{tust}\), \(P^{T}_{\infty}\), and \(P^{T}_{strong}\), respectively, for segment (i): (**a**) the 3D projection view; (**b**) the \(v_{x}-\omega\) plane view; (**c**) the \(v_{x}-v_{y}\) plane view; segment (ii): (**d**) the 3D projection view; (**e**) the \(v_{x}-\omega\) plane view; (**f**) the \(v_{x}-v_{y}\) plane view; segment (iii): (**g**) the 3D projection view; (**h**) the \(v_{x}-\omega\) plane view; (**j**) the \(v_{x}-v_{y}\) plane view. mum feasible angular velocity can be achieved for segment (ii), where the angular motion dominates the linear motion. A relatively high angular velocity can also be performed for the synchronous movement considered for segment (ii), while the lowest angular velocity can be performed in the case of segment (iii) since almost only linear motion was needed.