# Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption
## 6 Conclusions  
The main work of this paper is as follows:  
1. The Lame curve was employed in the trajectory planning of the Par4 parallel robot, and the piecewise design method was adopted. Detailed analysis and precise planning were elaborated, focusing on displacement, velocity, and acceleration in the Par4 parallel robot end operating space.  
2. To minimize the total mechanical energy consumption of the driving joints, asymmetric displacement planning was introduced based on the quintic and sextic polynomial motion laws. A series of experiments were performed to compare and analyze the mechanical energy consumption of the Par4 parallel robot under the quintic and sextic polynomial motion laws. The results reflect that the mechanical energy consumption is lower when the motion law of the quintic polynomial is adopted.  
3. To find the optimal trajectory, the recently proposed GWO algorithm was used. After a series of experiments to minimize the total mechanical energy consumption, it turned out that the optimal value of parameter \(e\) of the Lame curve can be set as half of the pick-up span in the Par4 parallel robot's trajectory planning, while the optimal value of parameter \(f\) depends on different pick-up heights and different pick-and-place endpoints.  
In addition, the mechanical energy consumption is related not only to the pick-up height and pick-up span of the trajectory, but also to the coordinate position of the pick-and-place endpoints. The closer the pick-and-place endpoints to the boundary of the Par4 parallel robot's feasible region, the higher the mechanical energy consumption will be.  
The trajectory planning method proposed in this paper is also suitable for other parallel robots, such as Delta parallel robots. Although the trajectory is based on the Lame curve in this paper, the design method is well suited to two-dimensional curves whose coordinate variables can both be expressed as functions of an intermediate variable that changes over time, such as the circle, the transitional parametric spline curve, and so on.  
[PERSON], as the corresponding author, takes primary responsibility for this article. [PERSON] drafted the manuscript. All co-authors reviewed the manuscript. This research was funded by the National High-Tech Research and Development Program (863 Torch Program) (grant no. 2015 AA7041003), the Educational Reform Project of Xianyang Normal University (grant no. 2017Y004), and the Scientific Research Plan Projects of Shaanxi Education Department (grant no. 15 JK1784).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Motion**} & & & & & & \\ \multicolumn{2}{c}{**Law/Pick-and-Place**} & & & & & & \\ \multicolumn{2}{c}{**Points**} & & & & & & \\ \hline \(k_{0}\) & \(k_{f}\) & **Ba/2/mme/mm** & \(f\)/mm** & \(E_{\text{min}}\)/J** & \(e\)/mm & \(f\)/mm & \(E_{\text{min}}\)/J \\ \hline
[80,30] & [30,\(-\)20] & 35.4 & 35.4 & 81.5 & 7.283 & 35.3 & 76.8 & 7.287 \\
[75,55] & [\(-\)35,150] & 72.7 & 72.7 & 120 & 7.746 & 72.7 & 120 & 7.744 \\
[150,50] & [\(-\)50,20] & 101.1 & 101.1 & 120 & 7.883 & 101.1 & 120 & 7.878 \\
[300,0] & [\(-\)10,75] & 159.5 & 159.5 & 120.0 & 7.721 & 159.5 & 120 & 7.711 \\
[\(-\)210,50] & [150,235] & 202.4 & 202.4 & 37.0 & 8.852 & 202.4 & 40.8 & 8.890 \\
[0,\(-\)305] & [0,396.7] & 350.9 & 350.9 & 63.3 & 11.048 & 350.9 & 76.5 & 11.124 \\ average \(E_{\text{min}}\) & & & & & & & & 8.439 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimized results with the same \(H_{a}\), different \(B_{a}\) and different endpoints.