## 1 Introduction
The formulated problem is then solved utilizing an optimal search method. Some of the most recent and characteristic examples from the relevant literature are the following. As already stated, the chosen index is the cornerstone of any proposed optimal task placement problem. In the case of kinematic tasks, such as those addressed in this work, a kinematic index is used. A local kinematic index provides quantifiable information of a selected aspect of the manipulator at a given point within the manipulator's workspace. For example, [PERSON] et al. in [6] used [PERSON]'s manipulability index as the index for their proposed method of optimal task placement within the workspace of a surgical robot. One example of a task based kinematic index was the Approximation of the Minimum Manipulator Velocity Ratio (AMMVR) presented in [7]. The index is formed by calculating the value of the Manipulator Velocity Ratio (MVR) [5] at several end effector frames along a 3D curve representing a trajectory to be followed by the robot's tool frame and selecting their minimum value as the task-based index value. The \"approximation\" description was used because the local index value (MVR) was calculated at a given number of frames along the trajectory. In a similar fashion, [PERSON] et al. [8] proposed a task-based index consisting of the minimum time required by the end effector of a manipulator to move between two task points within its workspace. The index was formed locally as the maximum value of the time required to traverse from one point to another. The index was then transformed to a task-based index by selecting the minimum value of all the local index valued determined for the given task. [PERSON] and [PERSON] in [9] used a task-based index based on the manipulability index to determine the optimal placement of a patient for the optimal execution of a robotic minimal evasive surgery task. The task-based index resulted from the integration of the local index value versus time along the required task trajectory divided by the total task execution time. The problem of determining the optimal task placement may be reversed to the problem of determining the optimal robot base placement when the task(s) location(s) is(are) given. In such cases, usually a global kinematic index is proposed, in order to determine the overall best position for the robot's placement, such as the reachability measure proposed in [10]. The index was a measure of the reachability of different tasks in a given workspace that the manipulator must be able to reach. [PERSON] et al. on the other hand utilized a task-based index for the determination of the optimal location of a manipulator's base for a given kinematic task [11]. Their index was the sum of the mean manipulability index throughout the possible robot base locations. In some cases, multi-criteria indices may be presented for providing a measurable function for the determination of the optimal location of the robot's base. [PERSON] and [PERSON] in [12] presented such an index formed by four criteria, all of which were expressed as a function of the robot's kinematic and structural properties and constraints. The first one depicted the joint's limitations, the second one was the possibility of the robot reaching a singular configuration, the third one depicted the possibility for the requirement of collision avoidance and the fourth was defined as the ratio of the distance between the robot's base frame origin to the commanded target frame of the end effector by the theoretically reachable range from the origin of the robot's base frame. Four weighting factors were introduced for each of the criteria and the index was defined either as the sole value of the fourth criterion or the maximum value of the other three criteria. In the context of the proposed methods, it can be said that the approaches used are similar.