## 2 Materials and Methods
### Feasible Kinematic Directional Capabilities
However, both methods could lead to a performance  
Figure 2: Graphical representation of the Yoshikawa manipulability index \(w\); maximum feasible tool velocity \(\mu_{2}\) (ellipsoid approach); maximum feasible tool velocity \(\mu_{\infty}\) (polytope approach); maximum linear velocity \(\
u_{a}\) (DTF method); and maximum angular velocity \(\omega_{a}\) (DTF method) in the case of a 3-DOF planar robotic mechanism. measure that combines linear and angular movement capabilities into a single scalar value, which makes the calculation physically inconsistent [33]. Different methods have been proposed to address this issue and overcome the problem of physical inconsistency. Some authors have introduced new performance measures [34; 35], which consider only translational directional capabilities. Another method, proposed in [36], involves power manipulability as a fully homogeneous and physically consistent performance measure. Unfortunately, establishing a direct relationship between the power vector and particular task requirements could be challenging in this case. Since elements of the twist vector space do not share the same physical unit, the linear and angular velocity capabilities should be addressed independently. Therefore, [PERSON] [37] separated the total manipulability measure into translational and rotational manipulability measures. He introduced the translational and rotational manipulability measures in both weak and strong senses. Similarly, the separated translational and rotational capabilities can be defined based on the manipulability polytope approach [38]. In this case, two different types of weak-sense manipulability polytope can be obtained, i.e., the \(L_{\infty}\) velocity polytope and the \(L_{2}\) velocity polytope, respectively. The weak-sense manipulability polytope can be used only in the case where the angular velocity is irrelevant to the task execution [38; 39; 40], which is not suitable in the case of robot machining applications. The set of all linear (angular) velocities that are feasible under the constraints that the tool orientation is kept constant form the translational (rotational) manipulability polytope in the strong sense [37; 41; 42]. However, robot surface machining applications with purely translational (rotational) motion exist rarely in actual industrial cases. To avoid the shortcomings of the described methods in the case of simultaneous linear and angular motion analysis, a method called Decomposed Twist Feasibility (DTF) was presented in [25], to evaluate accurate and physically meaningful information about a feasible linear and angular velocity in a specific task direction for robotic machining purposes. The kinematic capabilities of the tool in a task space are limited by the joint velocity limit. To determine the maximal velocity capabilities in the task space, we also need take the desired motion synchronization into consideration. Therefore, the maximum linear and angular velocity can be derived from the condition:  
\[\left\|\widetilde{J}_{T}^{\intercal}v+\widetilde{J}_{R}^{\intercal}\omega \right\|_{\infty}\leq 1 \tag{4}\]  
and as a result, it can be calculated based on the DTF method as:  
\[v_{a}=\left\|\widetilde{J}_{T}^{\intercal}\hat{a}_{T}+h^{-1}\cdot\widetilde{J} _{R}^{\intercal}\hat{a}_{R}\right\|_{\infty}^{-1}, \tag{5}\]  
\[\omega_{a}=\left\|h\cdot\widetilde{J}_{T}^{\intercal}\hat{a}_{T}+\widetilde{J }_{R}^{\intercal}\hat{a}_{R}\right\|_{\infty}^{-1}, \tag{6}\]  
where \(\widetilde{J}_{T}\) denotes a 3 xn translational Jacobian submatrix, \(\widetilde{J}_{R}\) denotes a 3 xn rotational Jacobian submatrix, \(\hat{a}_{T}\in\mathbb{R}^{3}\) is the unit vector in the desired linear velocity direction and \(\hat{a}_{R}\in\mathbb{R}^{3}\) is the unit vector in the desired angular velocity direction.