# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 3 Results
### Generic 3 DOF Planar Manipulator  
Path Segment Feasiblity  
The proposed DTF method was firstly applied to a generic 3 DOF planar manipulator, which enables an easier graphical representation of the manipulability polytope in a twist space. The 2D curve shown in Figure 6 represents the path that the manipulator end-effector followed. A set of waypoints was placed along the path, with the orientation normal to the curve. The end-effector was to simultaneously reach each of them at the desired position and orientation. In this experiment, the proposed DTF method was used to determine the maximum feasible linear velocity at three different parts of the curved path, as shown in Figure 6. Example (i) represents a slightly curved path segment, whereas a strongly curved path segment was considered in example (ii), and a flat path segment was analyzed in example (iii). We assume that the maximum kinematic capabilities of the robot are limited by the joint velocity \(\dot{\theta}_{\text{max}}=1\) (rad/s) for all joints. Figure 6: Three different parts of the curved path: (i) Slightly curved path segment; (ii) Strongly curved path segment; (iii) Flat path segment. At each of the path segments under consideration, the direction of linear and angular velocity was determined along the path. Since simultaneous synchronous linear and angular motion was required along the path, the relative importance of each motion type plays a crucial role in the accurate determination of the maximal end-effector capabilities. In the case of following the curved path, the ratio between the translations and the rotations depends on its curvature. During the motion along the strongly curved path segment, as in case (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 the synchronization parameter \(h\), converged to 0. For the flat path segment (iii), the synchronization parameter \(h\) value was the highest since the linear motion dominated over the angular. The values for the considered TCP position, the direction of the linear velocity vector \(\hat{u}_{T}\), the direction of the angular velocity vector \(\hat{u}_{R}\), and the velocity ratio factor \(h\) of each path segment are summarized in Table 1.  
Based on the proposed DTF method, the maximum feasible directional kinematic capabilities (\(V_{\text{max}}\) denotes the maximal magnitude of linear velocity evaluated by (30), whereas \(\Omega_{\text{max}}\) denotes the maximal magnitude of angular velocity evaluated by (31)) was calculated for each path segment. The required joint velocities were also computed to confirm that condition (8) was satisfied. From Table 2, it can be noted that the computed combination of the maximal linear and angular velocity reaches the joint velocity limit in at least one joint in all cases. Although the velocity limit was achieved in all path segments, at least at a single joint, the maximum linear velocity varied, depending on the considered robot configuration, the linear and angular velocity directions, and the specific task requirements linked to the curvature of the path that determines the velocity ratio factor \(h\). The highest maximum feasible linear velocity was achieved in the case of segment (i), where the required change in position was approximately the same in relation to the change in the orientation (\(h\approx 1\)). The lowest maximum feasible linear velocity was obtained for segment (ii), where almost only the angular motion was needed. For a better understanding of the computed results, the graphical representation of the proposed DTF method is presented in Figure 7 in relation to the manipulability polytopes defined above.