# Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption
## 5 Experimental Verification of Trajectory Optimization
### Basic Experiment  
With a group of pick-and-place endpoints taken and the pick-up height given, the trajectory planning above was adopted, i.e., the sleek gate-shaped path based on the Lame curve was selected, and the fifth- and sixth-order polynomial motion laws were respectively applied to carry out the optimization experiment based on the GWO algorithm for the lowest mechanical energy consumption.  
The coordinates of the starting pick-and-place endpoint were set at \(k_{0}=[-250,20,-543.5]\), the ending coordinates at \(k_{f}=[250,0,-543.5]\), and the pick-up height at \(H_{a}=100\) mm, so then the pick-up span could be obtained as \(B_{a}=500.4\) mm. The optimal values of parameter \(e\), parameter \(f\), and the minimum mechanical energy consumption (noted as \(E_{\min}\) for convenience) based on the GWO algorithm are shown in Table 2. The unit of parameters \(e\) and \(f\) is millimeters (mm), the unit of \(E_{\min}\) is joules (J). The units of these parameters in subsequent tables are the same. The numerical results in Table 2 reflect the fact that the mechanical energy consumption is relatively smaller under the law of quintic polynomial motion. It is worth mentioning that the optimization result of parameter \(e\) is the same (250.2 mm, \(B_{a}/2\)) regardless of the motion law of the quintic polynomial or the sextic polynomial. However, the optimization result of Parameter \(f\) is not the same. The optimal values in Table 2 were taken, and the trajectory planning method above was implemented. Then, the trajectories were obtained under the fifth-order and sixth-order polynomial motion laws, as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Length of big arm \(L_{i}\) & 273 mm \\ Length of small arm \(l_{i}\) & 600 mm \\ Radius of big arm revolute \(R\) & 190 mm \\ Length of moving platform along x axis \(d\) & 160 mm \\ Length of moving platform along y axis \(h\) & 160 mm \\ Distance between B\({}_{i}\) and \(C_{i}\) along y axis \(h_{i}\) & 20 mm \\ Distance between B\({}_{i}\) and \(C_{i}\) along x axis \(d_{i}\) & 20 mm \\ Load mass of moving platform \(m_{c}\) & 0 kg \\ Mass of big arm \(m_{b}\) & 0.08 kg \\ Mass of small arm \(m_{s}\) & 0.07 kg \\ Mass of moving platform \(m_{d}\) & 2.6 kg \\ Mass of big arm end connections \(m_{end}\) & 0.59 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Par4 parallel robot-related parameters.  
Figure 5: The Par4 parallel robot platform.  
The iteration trajectory of mechanical energy consumption \(E_{\min}\) is shown in Figure 7. From the results in Figure 7, the overall \(E_{\min}\) value of the quintic polynomial motion law (left figure) is smaller than that of the sextic polynomial motion law (right figure). When the \(E_{\min}\) reading is at a higher value, the left graph displays a thin thread-like peak in Figure 7, which shows that only a few wolves in the GWO algorithm failed to find the best prey in time. In other words, in the iterative process, GWO has higher efficiency and faster convergence under the law of quintic polynomial motion. The rotation angles, angular velocities, and corresponding drive torques of each joint of the Par4 parallel robot were obtained based on the optimum parameters \(e\) and \(f\), which are shown in Figure 8.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Motion Law** & \(e/\)**mm** & \(f/\)**mm** & \(E_{\min}/\)**J** \\ \hline Quintic polynomial & 250.2 & 32.3 & 8.209 \\ Sextic polynomial & 250.2 & 38.2 & 8.500 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of parameter optimization based on GWO.  
Figure 6: Motion trajectory.  
Figure 7: Iterative results of the minimum mechanical energy consumption based on GWO.