# Task-Oriented Evaluation of the Feasible Kinematic Directional Capabilities for Robot Machining
## 1 Introduction  
Collaborative robotic technologies enable humans and robots to work close together in a shared manufacturing environment. Contrary to classical industrial robots, which are primarily programmed to perform deterministic repetitive tasks, a new paradigm of collaborative robots encourages the development of flexible and intelligent robotic systems, especially for the needs of small and medium enterprises with low-volume, high-mix production [1, 2]. In order to help these companies to gain a competitive advantage, a rapid integration into the manufacturing process is needed in order for them to achieve improved flexibility, higher autonomy, and safety of a robotic system for the industrial floor. In this context, recent studies on intuitive motion planning algorithms [3, 4] and strategies toward safe and efficient human-robot collaboration [5, 6] have attracted growing attention. However, in order to further improve flexibility and the easy deployment of collaborative robots, efficient trajectory planning algorithms for complex robot machining tasks are also necessary [7]. To provide autonomous and singularity-free motion planning, a detailed understanding of available motion capabilities within a robot's workspace is key for the successful execution of complex robot machining tasks [8]. A robot's workspace is usually defined as the maximum reachable volume whereby the end-effector is capable of reaching each point in at least one orientation. The reachable workspace characteristics can be computed offline and visualized as a discretized map, which is called a reachability or a capability map [9; 10]. The manipulability of the robotic mechanism near the singular position can be largely reduced, although the robot can reach each point within a dexterous workspace--which is the subset of the reachable workspace--with any orientation. Consequently, there is still no guarantee for a successful and singularity-free Cartesian movement between two reachable poses in an operational task space. Therefore, kinematic and dynamic constraints such as positional joint limits, joint velocity, acceleration, and torque limits should also be considered when planning a feasible robotic task, for which the required kinematic and dynamic capabilities should be lower than the maximum allowable values of the robot. A detailed understanding of robot movement capability within its workspace is crucial for the successful planning and execution of more complex surface machining tasks that require the simultaneous tracking of the desired workpiece surface all the time along the path (robotic milling, welding, composite layup, hammer peening, etc.) [11; 12; 13; 14]. The execution of the complex robot task is not necessarily feasible anywhere in the robot workspace, especially in the case of collaborative robots, where the margin between their joint capabilities and task requirements is greatly reduced compared to conventional industrial robots [15]. To determine the optimal placement of a robot task within the robot workspace, it is necessary to propose a suitable objective function for optimization. In order to optimize a robotic machining trajectory, some existing kinematic performance indices are often taken into consideration [16; 17; 18]. A detailed analysis of these methods has shown that the most of them suffer from physical inconsistency (inhomogeneous Jacobian matrix [19], metric properties of twist [20]), do not take exact task requirements into account (such as task direction, synchronous linear and angular motion), do not include maximum kinematic constraints into consideration, or they are computationally intensive. If only the kinematic of the robot is considered, a qualitative measure of the ability to move the end-effector toward an arbitrary direction is most often based on analyzing the manipulability ellipsoid that was first defined by [PERSON] [21].