## 1 Introduction
In the case of collaborative robots, authors often search for the kinematically optimal placement of the workpiece in order to ensure the feasibility of the robotic task [11; 17], to perform the task as quickly as possible without exceeding the robot's maximum kinematic capabilities [18], or to identify optimal workpiece placement that results in the maximum possible manipulability and consequently requires the lowest possible joint torques [19]. Different objective functions can be chosen in order to determine the optimal placement of the workpiece. For kinematically optimal placement of the workpiece, most authors use well-accepted and widely used methods, such as the Yoshikawa manipulability index [20], the minimum singular value, or the condition number. These quantitative measures represent approximation of the general motion ability, which are not sufficiently reliable if we have to improve the directional motion capabilities along the required task direction. In this case, the manipulator velocity ratio is usually used to estimate the ability to move in the desired direction. Some other authors also include additional criteria in the optimization process, such as constraints related to joint limits, singularities, collisions with the environment [21], etc. To determine the optimal placement of a known robotic path composed of a set of waypoints, it is necessary to evaluate the motion capabilities of each waypoint on the robotic path and formulate an appropriate objective function that characterizes the placement of the entire path in the form of a single scalar value. Some authors maximize the mean value of the velocity performance measure to find the optimal path placement [22; 23; 24]. The mean value is simple to calculate and understand. However, sometimes the mean result can be misleading if the spread of results is uneven. Since the average is computed considering all the data values in a data set, it is influenced highly by the outliers, especially if they are located on only one side of the data set. In the case of workpiece placement, a high average value may indicate a workpiece placement where the selected kinematic index is extremely high, while, for other waypoints, the movement capabilities may be considerably worse, and vice versa. In [22; 24], the minimum value of kinematic performance measures is selected, which characterizes the worst kinematic abilities throughout a path to determine the best worst-case performance along the path. In this paper, we will optimize workpiece placement by maximizing the motion capabilities along the entire path, whereby the evaluation of the maximum achievable tool velocity will be based on the DTF (Decomposed Task Feasibility) method presented in [25]. Unlike other methods, the DTF method offers a simple and accurate solution for the determination of the total directional kinematic capabilities of each waypoint on the path, and it is specifically designed for robotic surface machining purposes as we show in this paper via a short analysis, which incorporates differential geometric properties of the machining path on the workpiece surface. The optimization results will determine the workpiece placement that requires the lowest possible joint velocities for task execution. The numerical results will confirm the effectiveness of the DTF method in the case of the optimal placement of a predefined machining task within the robot's workspace. Thereby, the feasibility and increased flexibility of the robotic machining task can be achieved in combination with intuitive collaboration between human and robot. The remainder of this paper is structured as follows. In Section 2, the intuitive human-robot collaboration is presented in the case of the definition of the robotic machining task. The related methods for evaluation of the feasible kinematic directional capabilities are also outlined.