## 3 Application of the Proposed Method
### Determination of the Optimal Task Location Results  
Using the proposed methodology, the location of the tasks were derived. For comparison, a respective \"bad\" task location was also derived for both tasks, by simply interrupting the optimal search at the first non-singular task placement.  
For reference, the robot's joint angles for the pick and place task (cylindrical object) (in rad) for the optimal placement are presented in Table 2 and for the <<bad>> placement in Table 3.  
Accordingly, the best and bad locations for the path-following task are presented in Figure 4a,b, respectively. Given the fact a given end effector velocity would be utilized  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline \(\mathbf{q_{i}}\) & \(\mathbf{{}^{\rm B}}\)**p\({}_{1}\)** & \(\mathbf{{}^{\rm B}}\)**p\({}_{2}\)** & \(\mathbf{  
to execute the task, the total task execution time for both placements of the task was the same. Figure 5 presents the step by step process of the optimization process during the operation of the GA and the corresponding steps where the \"bad\" and best task locations were derived. As shown in Figure 5, the bad task location was determined utilizing the feasible solution of the GA derived at its first iteration, while the best task location was determined utilizing the optimal derived solution.  
Figure 4: Best (**a**) and bad (**b**) placement of the path-following task, as determined by the GA.  
Figure 5: The optimization process, showing the improvement of fitness function step by step and the steps where the “bad” and best task locations for the path-following task were derived.  
Moreover, a sensitivity analysis with respect to the task placement coordinates was conducted about the derived optimal task placement. The results are depicted in Figure 6, where Figure 6a presents the variance in the fitness function value with respect to a \(\pm\)5% variation in the x-position of the task around the best location, (Figure 6b) the according fitness function value variation to a respective variation of the y position, (Figure 6c) the according fitness function value variation to a respective variation of the z position and (Figure 6d) the according fitness function value variation to a respective variation of the task orientation. As seen, the variation of the position aspects of the task induces a negligible variation to the fitness function value derived (less than 1%).  
The effect that the number of intermediate points placed between consecutive task points has on the optimization process was also examined, with respect to the path-following task. Table 4 presents the optimization process duration and derived result with respect to the number of intermediate points inserted in each path segment. The task points set along the path and the population number were the same for all simulations. As seen, the increase in the number of points increases the required computational time, however the fitness function value is improved.  
Figure 6: Sensitivity analysis for the results of the best placement of the path-following task with respect to the considered variables: (**a**) x position, (**b**) y-position, (**c**) z-position, (**d**) orientation.