# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 4 Discussion and Conclusions  
When planning a feasible robotic task, the required joint capabilities should be lower than the maximum capabilities available. To evaluate the feasible kinematic directional capabilities for robot machining, an approach called the Decomposed Twist Feasibility (DTF) method has been proposed in this paper. The basic idea behind the proposed DTF method is to find the maximal feasible directional kinematic capabilities in an operational space, individually for translational and rotational performances, under the robot joint velocity constraints. The method was algebraically derived and described by the manipulability polytopes as well. In contrast to the conventional manipulability methods, the proposed DTF method can provide accurate and dimensionally homogeneous information on how fast the robot end-effector can move under the constraint of joint velocities while maintaining the base task requirements. It can give us reliable information about the maximum linear and angular velocities when synchronized translational/rotational motion is required. A graphical representation of the proposed DTF method was presented by the twist manipulability polytope in the case of a generic 3 DOF planar mechanism. Although we gave individual consideration to the translational and rotational capabilities in the proposed DTF method, the presented results show that the tip of the resulting maximum feasible twist velocity vector obtained by (30)-(33) always ends on the surface of the well-known twist manipulability polytope at the given joint velocity constraint (8), which verifies the method. In comparison to the conventional analysis of the twist manipulability polytope--which requires a computation of its vertices, the construction of the corresponding convex hull, and a further computation analysis of the convex hull in order to determine the intersection point with the polytope surface in the desired direction--the computational complexity can be reduced significantly by the DTF method, and the accuracy of the resulting information is maintained as well. The usefulness was demonstrated, especially in the case of the robot machining task purpose on a generic 3 DOF planar mechanism and the collaborative robot UR5e. The maximum feasible linear velocity, considering the desired rotational motion,  
Figure 12: Graphical representation of \(v_{\text{max}}\) and \(\omega_{\text{max}}\) for the best path segment placement, determined by the DTF method for the 6 DOF UR5e robot manipulator in relation to its manipulability polytopes within the translational subspaces \(P_{\infty}^{T}\), \(P_{strong}^{T}\), and rotational subspaces \(P_{\infty}^{R}\) and \(P_{strong}^{R}\), respectively, for segment (i): (**a**) \(v_{\text{max}}\) and (**b**) \(\omega_{\text{max}}\); for segment (ii): (**c**) \(v_{\text{max}}\) and (**d**) \(\omega_{\text{max}}\); and for segment (iii): (**e**) \(v_{\text{max}}\) and (**f**) \(\omega_{\text{max}}\). was determined in order to follow a specific path segment on the workpiece surface. The regions with the highest and lowest maximum feasible linear velocities were determined and then illustrated by the colormap, which forms the so-called task-oriented kinematic capability map. In conclusion it should be noted that the proposed DTF method addresses a purely kinematic performance measure. To provide feasible and optimal workpiece placement, the dynamic performances regarding the given task parameters should also be considered. In order to find an optimal workpiece placement for the entire tool path, the maximum linear velocity should be calculated for each path segment, or at least for the most critical ones. Thus, the proposed DTF method could be used as a suitable optimization criterion for optimal task-oriented workpiece placement within the robot's workspace. The presented work was carried out in collaboration and with significant contributions from both authors.