# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 2 Materials and Methods
### Proposed Method
The example (i) represents the case where the curvature of the path is high, and the change in the end-effector orientation is much bigger than the change in position (the highest value of \(h^{-1}\)). Since the example path curve is of a concave function shape, the direction of angular velocity is \(\hat{u}_{R}=+1\), and the tips of the resultant vectors are located on the upper part of the twist manipulability polytope \(P^{trust}\)(see Figure 5a)). The set of maximum feasible linear velocity vectors form a circle; thus, the maximum directional kinematic capabilities are the same in all directions (see Figure 5b). Since almost only angular motion is needed in this case, the maximum feasible linear velocities based on the proposed DTF method (\(v_{\text{max}}\)) are quite low. Note that in the case of the translational manipulability polytope \(P_{\infty}^{T}\) (reddish) or \(P_{2}^{T}\) (yellowish), the maximum linear velocity vector would end on the surface of each polytope (see Figure 5b) since the analysis of linear velocity capabilities are evaluated separately from the rotational subspace in this case. An example of the path segment with a convex function shape (\(\hat{u}_{R}=-1\)) is demonstrated in the case (ii). The tips of the resultant vectors are located at the bottom of the manipulability polytope (see Figure 5c) in this case. The desired combination of linear and angular motion can be performed faster in the \(+y\) direction than in the \(-y\) direction if the same direction of angular velocity is considered (see Figure 5d). The magnitude of the maximum feasible linear velocity based on the manipulability polytope \(P_{\infty}^{T}\) (reddish) is the same as in Figure 5b; an even more flattened path segment is considered in this case, and different synchronization between the linear and angular velocities is needed (more linear motion). In the example (iii), almost only translational motion is considered. The magnitudes of the linear velocity vectors are equal to the vector lengths of the translational manipulability polytope in the strong sense \(P_{strong}^{T}\) (see Figure 5e). The resultant vectors lie in the \(v_{x}-v_{y}\) plane since there is no rotational motion (see Figure 5f). Although the results based on the manipulability polytope \(P_{strong}^{T}\) (greenish) can give the same solution as the proposed method, the analysis of this polytope is not suitable for other cases, where synchronization between the linear and angular motion is needed. In contrast to (iii), example (iv) represents a more flattened path segment. The magnitudes of the feasible linear velocity vectors are larger than the magnitudes of the angular velocity vectors. Especially in Figure 5g, it can clearly be seen that the maximal directional kinematic capabilities are strongly dependent on the direction of the linear velocity vector \(\hat{u}_{T}\), considering that the velocity ratio factor \(h\) and the direction of angular velocity \(\hat{u}_{R}\) for all directions are the same. As it can be seen from Figure 5g, the geometry obtained by the proposed DTF method touches the translational manipulability polytope \(P_{\infty}^{T}\) (reddish) only in two directions, which means that only for those directions, the same solution can be obtained based on the translational manipulability polytope in the weak sense. For all other directions, the linear velocity evaluated by \(P_{\infty}^{T}\) can be feasible only if suitable synchronization to rotational motion is considered, as can be seen in Figure 5h. The angular velocity vector should not exceed the dimension of the twist manipulability polytope. Note that Figure 5b,d,e,g beside the weak and the strong translational polytopes show regions of feasible linear velocities under the constraint of the task-dependent velocity ratio factor \(h\), which are introduced as subregions within the \(L_{\infty}\) weak polytopes.