## 4 Experimental Validation of the Method
### Results  
#### 4.3.1 Benchmark Trajectories  
The zigzag and figure-eight considered benchmark trajectories were evaluated for execution times of \(t_{f}=10.5\) and \(t_{f}=19.0\) s, respectively. Given the path and time of execution, the Pareto optimal sets were computed. Eleven discrete values of the weighting function \(\alpha\) and therefore eleven separate optimizations were carried out considering the composite objective function and a logarithmic distribution used for a well-approximated Pareto front,  
\[\alpha=\left\{0.999,0.998,0.996,0.992,0.984,0.968,0.937,0.874,0.749,0.499,0.0 01\right\}. \tag{39}\]  
For the two sets of via-points, the optimization procedure utilized as a first guess \(\boldsymbol{\tau}_{0}=t_{f}/N\boldsymbol{1}\). To reduce computational effort, each subsequent optimization started at the previous optimum. In this study, the number of evaluations was hereby able to be reduced by approximately 30% for both cases. Further, as both sets of starting points led to the exact same results, we assumed that gradient-based optimization was effective (without commenting on the convexity of the problem). The eleven optimization runs resulting in the Pareto front for the zigzag trajectory found in Figure 3 (left) needed 98 evaluations and 72 analytical sensitivity evaluations, requiring 7.37 s on a four core Intel i5-7200U CPU with 2.50 GHz.  
Physically, this solution lengthened the time on the short ends of the zigzag trajectory and shortened the time spent on the longer ends. The \(L^{2}\) norm of the jerk was reduced, while increasing the \(L^{2}\) norm of the velocity.  
An 11 point Pareto front was also calculated for the figure-eight trajectory, and this can be seen in Figure 3 (right).Due to the higher number of via-points and therefore design variables, the complexity of the optimization was increased. Here, four-hundred thirty-eight system evaluations and 179 analytical sensitivity evaluations were required and, in turn, 65 s on the same four core Intel i5-7200U CPU with 2.50 GHz.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{5}{c}{**(a)** Picking trajectory.} \\ \hline  & J 1 & J 2 & J 3 & J 4 & J 5 & J 6 \\ \(q_{1}\) & \(-1.60\) & \(-1.90\) & \(-1.76\) & \(-2.57\) & \(-1.55\) & 0.00 \\ \(q_{2}\) & \(-0.96\) & \(-1.96\) & \(-1.76\) & \(-2.58\) & \(-1.74\) & 0.00 \\ \(q_{3}\) & 0.78 & \(-2.23\) & \(-1.95\) & \(-2.09\) & \(-2.36\) & 0.00 \\ \(q_{4}\) & 0.81 & \(-2.24\) & \(-1.93\) & \(-2.10\) & \(-2.32\) & 0.00 \\ \(q_{5}\) & 0.87 & \(-2.26\) & \(-1.87\) & \(-2.14\) & \(-2.27\) & 0.00 \\ \(q_{6}\) & 0.92 & \(-2.29\) & \(-1.80\) & \(-2.18\) & \(-2.22\) & 0.00 \\ \hline \hline  & \multicolumn{5}{c}{**(b)** Passing trajectory.} \\ \hline  & J 1 & J 2 & J 3 & J 4 & J 5 & J 6 \\ \(q_{1}\) & 0.92 & \(-2.29\) & \(-1.80\) & \(-2.18\) & \(-2.22\) & 0.00 \\ \(q_{2}\) & 0.92 & \(-2.23\) & \(-1.79\) & \(-2.25\) & \(-2.22\) & 0.00 \\ \(q_{3}\) & 0.92 & \(-2.1 Being that the figure-eight path had no edges, the optimization had less effect on the arc-length and jerk values with respect to the previous benchmark trajectory. However, the jerk values can be greatly modified according to the selection of the weighting parameter.  
To complement these results, in Figures 4 and 5, the numerical results obtained for the two trajectories with the weighting factor \(\alpha\to 0\), i.e., min-jerk trajectory in red, and with \(\alpha\to 1\), i.e., arc-length minimization in red, are reported. As one might have expected, the red curve shows less overshoots with respect to the red one, in particular in the case of an \"edgy\" path.  
Figure 4: Numerical results for the benchmark trajectories in the operational space: (red) trajectory for \(\alpha\to 1\) and (blue) for \(\alpha\to 0\).  
Figure 3: Pareto fronts for the benchmark trajectories: (**left**) zigzag trajectory; (**right**) figure-eight trajectory.