# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 3 Results
### Collaborative Robot UR5e
#### 3.3.2 Workspace Analysis  
An important issue in robot task design is to place the robot tool path in such a location in the robot's workspace where the robot will be capable of performing a desired movement utilizing the lowest possible joint velocities, or, in other words, performing a desired movement with maximum feasible tool velocity. In order to demonstrate that we can apply the proposed DTF method, we generated a colored task-oriented kinematic capability map, which represents the maximum directional kinematic capabilities for the desired movement. The kinematic capabilities were computed for equidistantly distributed sets of points in the UR5e's horizontal x-y plane (z = 0) of the base robot frame {B}, as shown in Figure 10, where \(n\) = 0.05 m denotes the discretization step for both directions, and \(R\) = 1 m and \(r\) = 0.2 m represent the inner and the outer workspace boundaries, respectively. The maximum linear velocity was calculated at each discretized point along a specified task direction (i), (ii), and (iii) (see Figure 8), which required synchronous translational-rotational motion. The robot configuration was chosen, as shown in Figure 10, for all discretized points. At each point, the inverse kinematic solution was firstly computed for the considered robot configuration. Only those points for which the inverse kinematic solution exists were included in further workspace analysis; the others were disregarded. The maximum linear velocity \(V_{\text{max}}\) based on the proposed DTF method (30) was calculated for those with a valid inverse kinematic solution; then each of them was assigned to a different color, ranging from magenta to green (see the colorbar in Figure 11 for the selected value), which formed a so-called task-oriented kinematic capability map, illustrated in Figure 11 for each path segment. The magenta-colored areas represent the placement with the lowest feasible linear velocity (the worst robot kinematic capabilities for the desired task), and the green-colored areas represent locations with the maximum value of the feasible linear velocity (the best robot kinematic capabilities for the desired task). Since \(V_{\text{max}}\) is related to \(\Omega_{\text{max}}\) by \(\Omega_{\text{max}}=h^{-1}\cdot V_{\text{max}}\), the distribution of the colormap would be the same for \(\Omega_{\text{max}}\), only scaled by \(h^{-1}\). The randomly chosen initial placements of each considered segment (see Figure 8) are denoted by the white point on the generated task-oriented kinematic capability maps, while the locations with the highest maximum linear velocity \(V_{\text{max}}\) are represented by the black point. Figure 10: Discretization of the robot workspace. As shown in Figure 11, the distribution of the maximum linear velocity \(V_{\text{max}}\) based on the proposed DTF method varied greatly according to the task requirements. Although the considered direction of the linear velocity vector \(\hat{u}_{T}\) was pointing in almost the same direction in the case of path segments (i) and (iii) (see Table 3 and Figure 9), the best locations to perform a required movement were not located in the same place in the robot workspace since the requirements for rotational movement were quite different. An even greater difference in the distribution of the maximum linear velocity can be seen for path segment (ii), where the angular motion dominated the linear. The feasible directional kinematic capabilities for the best placement of each path segment are presented in Table 5. In comparison to the maximum kinematic capabilities from Table 4, much higher linear velocity can be achieved under the same joint velocity constraints (8). By the optimal placement of the workpiece with the considered machining path, a great improvement in the linear velocity amplitude can be achieved, while keeping the same desired base task parameters (\(\hat{u}_{T}\), \(\hat{u}_{R}\), \(h\)).