## 3 Results  
For each of the ten trajectories, the values of \(w_{0},w_{1},\ldots,w_{6}\) (5), and then \(f_{f}\) (8) were calculated based on the CoppeliaSim simulation for each particular possible robot location in the grid. The important values are summarized in Table 2, particularly the lowest fitness function value \({f_{f}}^{B}\) in the best robot location, which should, thus, be the selected location for the robot, provided it is physically possible. The table also shows the percentage of improvement that the best location offers compared to the worst location,  
\[imp_{W}^{B}=100\times\left(1-\frac{{f_{f}}^{B}}{{f_{f}}^{W}}\right), \tag{9}\]  
which theoretically represents the greatest possible improvement in robot wear reduction. If we consider a random robot location versus the best one, the average improvement can be calculated compared to the average value  
\[imp_{A}^{B}=100\times\left(1-\frac{{f_{f}}^{B}}{{f_{f}}^{A}}\right). \tag{10}\]  
Figure 5 shows the distribution of \(f_{f}\) values in all possible robot locations for each trajectory in the form of a standard box-plot diagram. The height of each shaded rectangle represents the interquartile range (third quartile minus first quartile), which indicates that 50% of all \(f_{f}\) values lie inside the rectangle. The small circles in each column represent the outliers--the topmost one corresponds to the worst location with \({f_{f}}^{W}\). Arrangements of the robot locations in the 3D space around the corresponding trajectories are displayed in Figure 6--the color coding indicates the \(f_{f}\) values using the gradient red-yellow-green, where green is the best location (\({f_{f}}^{B}\)), and red is the worst (\({f_{f}}^{W}\)). This image also shows the positions of the best (\"B\"), worst (\"W\"), and three other robot locations (\"1\", \"2\", and \"3\") that will be used later in the experiments on a real robot. Note that the small cubes rendered in Figure 6 as a visual representation of the possible robot locations are shifted away from the ideal grid positions by a small random offset (less than half the grid spacing). This is to achieve better visual clarity by preventing moire patterns and reducing the concealment of more distant cubes. Figure 4: Example of a grid of valid robot locations for a given trajectory (A1). Each blue point represents a possible position of the robot base center point; the robot is displayed in four distinct sample locations. GCS represents the global coordinate system. To better demonstrate the meaning of the proposed optimization criterion, Figure 7 shows, for the two trajectories A1 and B1, the individual values of the relative joint wear factors \(w_{1},w_{2},\ldots,w_{6}\) (5) for the best (\(w_{j}^{B}\)) and worst (\(w_{j}^{W}\)) robot location, together with the corresponding total fitness function value \(f_{f}\) (8). In the first example (trajectory A1), it is clear that, in the worst location, the third joint was extremely stressed and the six \(w_{i}^{W}\) values differ quite considerably. In the best location, joints 2, 3, and 6 have almost the same \(w_{i}^{B}\) values and, although the relative wear factor of the 6 th joint increased, the overall \(f_{f}^{B}\) value was much lower. A similar situation can be seen in the second image (trajectory B1). Here, the relative wear factor values in the best location are not so well balanced-the overall fitness function value was reduced considerably nonetheless because the average value \(A\) (Equation (6)) lowered. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Traj.