## 4 Simulation Results
### The Mantec R3 Model 1  
The Mantec r3 model 1 (R3M1) is a kind of six-axis robotic manipulator constructed by [PERSON] and [PERSON]. However, the dynamics of the model are not presented in the literature. The proposed method in this work fills this gap. The exact boundary conditions and constraints are displayed in Table 1 to solve this problem.  
The terminal time \(t_{f}\), the number of nodes, the discrete initial states and the discrete initial controls are set as usual. The loads are set to 0 kg, 7 kg, and 15 kg, respectively. The results of the proposed method are displayed in Figure 5 and Table 2, and the impact of different loads is analyzed.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model(R3M2)**} & **Nodes of LPM** & **Total Cost Time of** &
\begin{tabular}{c} **Performance Index** \\ **(Terminal Time \(t_{f}\))** \\ \end{tabular} \\  & **Method** & **CPU (s)** & **(s)** \\ \hline REF & 20 & 2.221 & 0.498 \\  & 100 & 196.933 & 0.495 \\ \hline FMU & 20 & 2.144 & 0.498 \\  & 100 & 125.136 & 0.495 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of results of the improved method with FMU and the reference research.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Loads of R3M1**} & **Nodes of LPM** & **Total Cost Time of** &
\begin{tabular}{c} **Performance Index** \\ **(Terminal Time) (s)** \\ \end{tabular} \\  & **Method** & **CPU (s)** & **(Terminal Time) (s)** \\ \hline
0 kg & 20 & 16.644 & 0.476 \\  & 100 & 468.226 & 0.475 \\
7 kg & 20 & 8.144 & 0.497 \\  & 100 & 383.118 & 0.496 \\  & 20 & 15.319 & 0.522 \\
15 kg & 100 & 469.191 & 0.521 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of results of the improved method with different loads.  
Figure 5: Cont.  
Figure 5 not only has the same content as Figure 4, but also displays the results of the R3M1 manipulators with different loads. It is found that, with the increase in the load, the optimized terminal time \(t_{f}\) increases. Joints 4 and 5 are influenced by loads greatly.  
What is more, the oscillating behavior of the controls caused by the discrete property is common for a collocation method. Therefore, a better approximation of the controls can be found by applying a huge number of nodes in the LPM method, or by taking  
Figure 5: The solutions of the R3M1. Subgraphs **(a–c)** are the solutions using LPM method with 20 nodes; subgraphs **(d–f)** are the solutions using LPM method with 100 nodes. Subgraphs **(a,d)** and print the curve of the input voltage of joint actuators. Subgraphs **(b,e)** print the curve of the joint positions. Subgraphs **(c,f)** print the curve of the joint velocities. The blue curves with circles \((-\infty-)\) refer to the solutions with 0 kg load. The green curves with squares \((-\square-)\) refer to the solutions with 7 kg load. The orange curves with crosses \((-\mathrm{x}-)\) refer to the solutions with 15 kg load.  
the switching structure into account to transform the original one-stage problem to a multi-stage problem [38]. According to Table 2, the calculation time of the first plan increases exponentially. However, the switching structure can be generally figured out even with the LPM method with fewer nodes. Therefore, the second plan has broader application prospects.