## 2 Materials and Methods
### Feasible Kinematic Directional Capabilities  
To ensure the feasibility of the predefined machining task, workpiece placement should be such that all waypoints on the surface are reachable by the robot, and the tool tip can move between them with the desired tool speed. To find such placement of a robotic machining trajectory, motion capabilities have to be analyzed for each segment of the desired path. Some existing kinematic performance indices are typically used for this purpose [28]. A detailed review of the related studies has found that most of them overlook the exact task requirements, suffer from physical inconsistency, do not include the maximum kinematic limits in the consideration, or are computationally intensive. When evaluating a robot's ability to move its end-effector toward an arbitrary direction, a qualitative measure \(w\) based on analyzing the manipulability ellipsoid is most often used, which was first introduced by [PERSON] [20]. It is shown in Figure 2 and defined as:  
\[w=\sqrt{\det(J\bar{J}^{T})} \tag{1}\]  
Figure 1: Intuitive definition of the robotic surface machining area through human–robot collaboration: (**a**) the kinesthetic guidance of the robot, (**b**) the definition of the machining area through a holographic interface, (**c**) setting up the technological parameters through a holographic interface. where \(J(\theta)\in\mathbb{R}^{mxn}\) is a Jacobian matrix of the robot mechanism, with elements of \(J_{ij}=\partial f_{i}(\theta)/\partial\theta_{j}\) and \(i=1\dots m\), \(j=1\dots n\). [PERSON]'s manipulability index \(w\) aims to provide information regarding the proximity to a singular configuration. Additionally, the condition number, or minimum singular value [29], can also be used to estimate the proximity to a singular configuration. These indices offer a general evaluation of a robot's capabilities, which can be useful for optimizing the design [30] or other general characteristics of a robot. But it is important to note that these performance measures are often of limited practical usefulness, particularly when the focus is on improving the directional kinematic capabilities within a predefined robotic task. For evaluation of the directional kinematic capabilities of the robot's end-effector, researchers have proposed various performance metrics. One such metric, presented in [31], involves calculating the distance from the center to the surface of the manipulability ellipsoid along the desired directional vector. This metric is also named the manipulator velocity ratio, which was first presented in [32], and defined as:  
\[\mu_{2}=\left[\hat{u}^{T}(JJ^{T})^{-1}\hat{u}\right]^{-1/2} \tag{2}\]  
where \(\hat{u}\) denotes the direction of the end-effector velocity vector, and \(\mu_{2}\) is the scalar, which represents the velocity transmission ratio in direction \(\hat{u}\), as is shown in Figure 2. By utilizing the Euclidian norm, this approach provides estimation of a robot's motion capabilities in the task space relative to the robot effort required in the joint space. Another approach, outlined in [32], suggests using the manipulability polytope as an alternative metric, which is obtained considering the exact joint velocity limits. The magnitude \(\mu_{\infty}\) of the maximum feasible tool velocity along \(\hat{u}\) under the velocity constraints, as can be seen from Figure 2, can be expressed as:  
\[\mu_{\infty}=\left(\max_{i}\left|\hat{u}_{i}^{*}\right|\right)^{-1} \tag{3}\]  
where \(\hat{\theta}^{*}=\left[\hat{\theta}_{1}^{*},\hat{\theta}_{2}^{*},\dots,\hat{ \theta}_{n}^{*}\right]^{T}\) is the minimum infinity norm solution of \(\hat{u}=j\hat{\theta}\). In comparison to the manipulability ellipsoid, the manipulability polytope seems to be more appropriate for the calculation of the robot's feasible task-space capabilities, while it considers the exact joint velocity limits.