# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods
### Proposed Method
Unlike the existing methods [32; 49] that separately evaluate maximal end-effector capabilities based on the desired velocity direction for the translational and the rotational subspaces, the proposed DTF method, although based on twist decomposition, links both individual subspace constraints by the inclusion of the specific robot task requirements with the velocity ratio factor \(h\). In terms of the robot machin  
Figure 3:4 Graphical representation of the proposed DTF method with the twist manipulability polytope \(P^{\it twist}\) (**left**), and in a detailed view (**right**). complex surface geometric properties, the task requirements depend on the variation in the surface curvature and the defined tool path, which is then reflected in the velocity ratio factor \(h\). Thus, for a given translational direction \(\hat{u}_{T}\), rotational direction \(\hat{u}_{R}\), and velocity ratio factor \(h\), we seek the maximum linear \(V_{\text{max}}\) and rotational velocity \(\Omega_{\text{max}}\) that could be achieved. To provide an illustrative and intuitive demonstration of the proposed DTF method with respect to different task requirements, the maximal linear and angular end-effector velocities are illustrated graphically by the task-space twist polytope in Figure 3:5  
In the middle of Figure 5, we show the twist polytope with the curves on its surface, described by the tips of the resultant twist velocity vectors at different values of the velocity ratio factor \(h\), while the translational direction vector interpolates as \(\hat{u}_{T}=\left[\cos(\alpha)\quad\sin(\alpha)\right]\), \(\alpha=0\dots 2\pi\). Note that the different values of the velocity ratio factor \(h\) are related to different task requirements, which, in our case, are chosen as: (i) Concave path segment with \(h\) = 0.21, (ii) Convex path segment with \(h\) = 0.42, (iii) Flat path segment with \(h\) = 1000, (iv) Flat convex path segment with \(h\)=1.25. Each of the polytope iso-curves is related to the evaluation of the twist vector tips with the same task requirements at the specific value of the synchronization parameter \(h\) (also see the colorbar for the selected value). It can be noted that each iso-curve lies entirely on the surface of the manipulability polytope, thus satisfying the constraint given by (8). If the reverse direction of angular velocity is considered, the set of resultant tips are mirrored to the opposite side of the twist manipulability polytope. The maximum linear and angular velocity vectors of each example (i)-(iv) is additionally shown in detail by two different projection views, i.e., the \(v_{x}-\omega\) plane and the \(v_{x}-v_{y}\) plane, respectively. In the \(v_{x}-v_{y}\) plane, the maximum feasible linear velocity vectors  
Figure 3:6 Twist polytope \(P^{twist}\) with the iso-curves of the same task requirements (\(h\) = const. ); (**a,c,f,h**) a polytope-based representation of the feasible angular velocity (\(\omega_{\text{max}}\) —blue) for (i) a concave path segment, (ii) a convex path segment, (iii) a flat path segment, and (iv) a flat convex path segment, respectively; (**b,d,e,g**) a polytope-based representation of the feasible linear velocity (\(v_{\text{max}}\) —red) for (i) a concave path segment, (ii) a convex path segment, (iii) a flat path segment, and (iv) a flat convex path segment, respectively. (plotted in red) can be obtained if the unit direction vector \(\hat{u}_{T}\) is interpolated as described above. Note that this projection view also depicts the translational polytopes \(P_{\infty}^{T}\), \(P_{2}^{T}\), and \(P_{strong}^{T}\) in reddish, greenish, and yellowish colors, respectively. The corresponding angular velocity vectors are plotted in blue in the \(v_{x}-\omega\) plane for the directions \(\hat{u}_{R}=+1\) or \(\hat{u}_{R}=-1\). The maximum feasible magnitude of the linear and angular velocity vectors for the same combination of \(\hat{u}_{T}\) and \(\hat{u}_{R}\) obviously depends on the value of the synchronization parameter \(h\).