# Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption
## 5 Experimental Verification of Trajectory Optimization
### Experiments with Different Pick-and-Place Endpoints and the Same Pick-Up Height and Span  
For the sake of comparison and analysis, here, five groups of different pick-and-place endpoints were taken for planning and comparison in the Par4 robot's spatial feasible region. The X and Y axis coordinates of the former two groups of pick-and-place endpoints were both small and far away from the boundary of the robot's feasible region, while the X or Y axis coordinates of the last three groups of pick-and-place endpoints were relatively larger and closer to the boundary of the robot's feasible region. In the experiments, the pick-up height was the same, set at 120 mm; the pick-up spans were all set at 500.4 mm. Table 4 shows the optimization results based on GWO, and the minimum mechanical energy consumptions are given accordingly. Since all the \(Z\)-axis coordinates (\(-\)543.5) were the same, for simplicity, only \(X\)-axis and \(Y\)-axis coordinates are listed in Table 4 for the pick-and-place endpoints.  
From Table 4, the mechanical energy consumptions under the former two sets of coordinates were relatively lower than those under the last three sets of coordinates, and in particular, the \(E_{\text{min}}\) values for the fourth set of coordinates were the highest. Moreover, in the experiments, the mechanical energy consumption based on the quintic polynomial was less than that based on the sextic polynomial under the same conditions. Besides, it is worth mentioning that almost all of the optimal values of parameter \(e\) were 250.2 mm (this value is \(B_{a}/2\)), while the optimal values of parameter \(f\) were relatively smaller in the former two sets of coordinates than those in the last three sets of coordinates, which were close to the pick-up height \(H_{a}\) (120 mm). Hence, the optimal value of parameter \(f\) is related to the spatial coordinates of the pick-and-place endpoints.  
Experiments with Different Pick-Up Spans, Different Pick-and-Place Endpoints and the Same Pick-Up Height  
The pick-up height was taken as 120 mm, and six groups of different pick-and-place endpoints were selected, of which the corresponding pick-up spans are shown in Table 5 as \(B_{a}/2\). Since all the Z axis coordinates are \(-\)543.5, only X axis and Y axis coordinates of the pick-and-place endpoints are listed in Table 5, for simplicity. In Table 5, the difference between the two motion laws of quintic and sextic polynomials is not obvious when the pick-up height and the pick-and-place coordinates are the same. If the average mechanical energy consumption (denoted as average \(E_{\text{min}}\)) in the six different groups of coordinates is taken for comparison, the value under the quintic polynomial motion law is  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Motion Law/Pick-and-Place Points**} & \multicolumn{3}{c}{**Quintic Polynomial**} & \multicolumn{3}{c}{**Sextic Polynomial**} \\ \cline{2-7}  & \(e\)**/mm** & \(f\)**/mm** & \(E_{\text{min}}\)**/J** & \(e\)**/mm** & \(f\)**/mm** & \(E_{\text{min}}\)**/J** \\ \hline
100 mm & 250.2 & 32.3 & 8.209 & 250.2 & 38.2 & 8.500 \\
110 mm & 250.2 & 33.7 & 8.757 & 250.2 & 40.2 & 9.054 \\
120 mm & 250.2 & 35.4 & 9.323 & 250.2 & 42.3 & 9.628 \\
130 mm & 250.2 & 37.1 & 9.912 & 2 lower than the one under the sextic polynomial motion law. In addition, regardless of the movement law of the quintic or sextic polynomial, almost all the optimization values of parameter \(e\) are equal to \(B_{a}/2\), but the optimization feature of parameter \(f\) is not so obvious, and its value is not only related to the movement law but also related to the pick-and-place endpoints.