# A Novel Kinematic Directional Index for Industrial Serial Manipulators
## 1 Introduction  
Performance evaluation can provide useful information in the design of robots. For this reason, the first performance indexes were introduced as early as 1982. Manipulability [1], condition number [2], minimum singular value, kinematic isotropy [3], conditioning indexes [4] and dexterity analyses [5] provide a variety of information about a robot's performance throughout its workspace. Manipulability allows the robot to be kept far from the kinematic singularity while the condition number (and usually its reverse) is used to achieve a fast measurement of the workspace isotropy, i.e., the ratio between the maximum and the minimum performance of the robot. These indexes, as with many others, are based on the Jacobian matrix of the velocity kinematic problem. The use of these approaches brought up some problems [6] that have been resolved with the evolution of these methods. A modern performance index must provide information in a summarized form, such as a value, on the behavior that the manipulator can have in each position that it can reach within its workspace. The key features for a performance index are:  
* To be homogeneous/independent of the unit of measurement [7, 8, 9, 10, 11, 12]. * To be independent of the reference system [13, 14]. * Providing increasingly accurate information [15, 16, 17, 18]. A first sample of homogeneous indexes has been proposed in [7], where the main concept consists of directly relating the Cartesian and the actuator velocities. Another approach suggested introducing a novel dimensionally homogeneous Jacobian matrix [8]. In these methods, the velocities of three points of the flange have been considered and related with the motor velocities. A different approach was proposed in [10] where a proper performance index was developed to gather interesting information on a robot's behavior using a non-homogeneous Jacobian matrix. An interesting approach was proposed in [13, 14], where a motion/force transmission index was introduced. In these works, the use of proper virtual coefficients instead of the Jacobian matrix allowedintroducing a frame-free performance index. This means that the evaluation is completely independent from the choice of the absolute reference frame in which it is computed. Recent performance indexes [15; 16; 17; 18] give more information compared to previous ones and are usually not affected by the adopted units of measurement and the choice of the reference frame. Once a performance index can guarantee the above-mentioned features, it can be adopted for several purposes. In recent years, performance evaluation has been mainly conducted for the following uses:  
* Optimization of manipulator design [19; 20; 21]. * Comparison between different robot architectures [22; 23]. * Optimization of robot trajectory planning [24; 25; 26; 27]. * Optimization of task locating (or robot positioning) [28; 29]. The proposed performance index has been conceived to give useful information for the latter issue, i.e., finding the best location for a generic task, given the direction of the robot's main movements. The first performance index that took the direction of motion into account with respect to the actuator's movements was introduced in [30] and extended in [28]. Such an index, called a directional selective index (DSI), gives accurate information about the performance of parallel robots and allows finding the regions of the workspace where a parallel robot achieves its maximum and minimum performance. However, the DSI formulation cannot be easily extended to serial robots. For this reason, a novel approach, called the kinematic directional index (KDI), has been introduced in this paper. This method allows analyzing the behavior of a serial robot, taking into account the position of the robot and the direction of motion. As such, the index can be adopted for several purposes. In this paper, a performance analysis will be performed, by means of KDI, with two main goals:  
* Finding the direction of maximum velocity with respect to a point.