## 2 Materials and Methods
### Experiment Setup  
The proposed method was demonstrated and verified on a Universal Robots UR3 collaborative robot with 6 degrees of freedom (see Figure 2), first in a CoppeliaSim simulation configured according to the previous chapter, and finally also on a real physical robot. The values of the maximal allowed joint torque \(\tau_{m_{j}}\) and joint angular velocity \(\omega_{m_{j}}\) (5) for the UR3 robot are listed in Table 1.  
Five testing movement paths of the robot end-point were demonstrated, each in two variants with different velocities of the end-point (0.1 and 0.2 m/s), giving a total number of ten trajectories labeled A1, A2, B1, B2, \(\ldots\), E2; where A-E is the path type, 1 stands for 0.1 m/s, and 2 stands for 0.2 m/s.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(j\) & \(\tau_{m_{j}}\) [**Nm**] & \(\omega_{m_{j}}\) [**s\({}^{-1}\)**] \\ \hline
1 & 56 & \(\pi\) \\
2 & 56 & \(\pi\) \\
3 & 28 & \(\pi\) \\
4 & 12 & \(2\pi\) \\
5 & 12 & \(2\pi\) \\
6 & 12 & \(2\pi\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The joint parameters for the UR3 robot; \(\tau_{m_{j}}\) is the maximal permissible torque, and \(\omega_{m_{j}}\) is the maximal permissible angular velocity of the \(j\)-th joint.  
Figure 2: The UR3 robot. (**a**) The simulation model in the CoppeliaSim environment with a path representing the end-point trajectory and a coordinate system in the center of the robot base. (**b**) The real robot used in the experiments.  
The trajectories are a combination of real use-cases and artificially created simple trajectories that contain various elements (linear segments of different lengths, circle arcs, sections with frequent speed, direction changes, etc.), see Figure 3. Various velocities of the end-point cause a difference in the solution of possible robot base locations, because every robot has some velocity limits for individual joints (see Table 1), and they also represent different cases regarding the proposed wear factor calculation.  
The selected trajectories were of a smaller size, and there were no obstacles in the environment (except for the table that the robot was mounted onto), which leads to a larger number of different valid robot locations relative to the trajectory and, thus, also a larger data set of results.  
The chosen robot UR3 is a small robot with a reach of only 500 mm and although using larger trajectories combined with several obstacles in the environment would be possible, the valid locations of the robot would be very limited and it would be difficult to make a statistical evaluation. In general, the proposed optimization method does not depend on the size of the trajectory. A longer trajectory, requiring a longer time to traverse, would produce larger values of the discrete integral (4); however, this is true for all robot locations relative to the particular trajectory, and thus the optimization remains valid.  
In a real case, the simulation model would have to contain also all collision objects in the workplace, which could limit the valid robot locations considerably--the principle of the method is, however, still applicable. The concept of choosing the robot location is demonstrated on Figure 4 for the trajectory A1. The figure shows the grid of valid robot locations--each blue point represents a possible position of the robot base; the robot is able to reach the whole trajectory from each of these valid locations. For better clarity, the robot is shown in several selected locations.  
Figure 3: Visualization of the five testing robot end-point paths; the UR3 robot is shown as a scale reference.