# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 1 Introduction
Yoshikawa's manipulability index is proportional to the volume of the manipulability ellipsoid and should give information about the proximity to a singular configuration, which can also be estimated by the condition number or minimal singular value [22]. These indices can provide general evaluations of the robot's capabilities in order to optimize the design of the robot [23] or any other global characteristic of a robot. However, such performance measures are often of limited practical usefulness since they are not reliable enough if the directional kinematic capabilities have to be improved along the predefined robotic task. A suitable performance metric to determine the maximum directional kinematic capabilities of the robot end-effector was proposed in [24], that being the vector length from the center of the manipulability ellipsoid to the surface intersection point in the desired direction. It is also frequently referred to as the manipulator velocity ratio, first introduced in [25]. An alternative approach has been proposed based on the manipulability polytope [26], which is derived from the exact joint velocity limits. The ellipsoid approach, which involves the Euclidian norm, can provide an estimation of a robot's capabilities in a task space relative to the effort in a joint space. However, the manipulability polytope appears to be more appropriate for the estimation of the robot's feasible capabilities in the task space constrained by the joint velocity limits. Both methods may result in a combination of translational and angular motion capabilities in a single scalar performance measure, which makes them physically inconsistent [27]. Several approaches have been suggested to overcome the drawback of physical inconsistency. One of the most common weights the translation capacity against the rotational [25], or handles them in a separate manner [28]. Different methods exist for the determination of the weights. However, arbitrariness in the choice of the weights is unavoidable [29]. The choice is usually made based on the task specifications [30] or based on the minimal principal axes of the translational and rotational subspace [31]. Some authors attempt to avoid this problem by introducing the new performance indices [32; 33]; however, they only include translational directional capacities in their consideration. A new approach based on power manipulability and considered to be fully homogeneous and physically consistent has also been proposed in [34], but a direct relation between available power and specific task requirements could not be established clearly. The more objective approach seems to be the partition on the translational and rotational motion capabilities. Through the decomposition of the task space into a translational and rotational subspace, the translational and rotational manipulability polytope in a weak and strong sense was derived in [35], similarly to [PERSON]'s approach [36] in the case of the manipulability ellipsoids. When the translational (rotational) manipulability polytope in a weak sense is taken into consideration, the maximum possible linear (angular) velocity can be evaluated separately from the rotational (translational) subspace, whereby the angular (linear) velocity may not be zero and possibly uncontrolled. That means that the length of the translational (rotational) manipulability polytope represents the maximal feasible linear (angular) velocity in the specified direction, generating a simultaneous movement in the rotational (translational) subspace, which cannot be arbitrarily selected in accordance with the task requirements. Thus, this approach can be useful only for the applications where the angular velocity is irrelevant to the task performance [28; 35; 37]. However, in the case of robot machining applications, the desired synchronization between both subspaces according to the task requirements must be taken into account.