## 6 Conclusions  
To define the tip-over stability constraint of WMDAR, the forces and moments between wheel-terrain and body-arm have been analyzed by kinematics and force analysis of robot. A new stability criterion for the overturning moment is proposed, which is obtained by improving the WMDAR. A new stability constraint for calculating TOSM is derived based on the improved stability criterion. Then, an optimal joint trajectory planning model based on time and energy is proposed. The optimization model is solved according to the quintic B-spline interpolation algorithm and an improved NSGA-II optimization algorithm, and the robot motion simulation experiment is conducted. The trajectory planning and optimization can improve the efficiency and stability of the WMDAR system through the analysis of two optimization cases. Some main conclusions are drawn through graphical and theoretical analysis, as follows:  
1. When the joint angular velocity and angular acceleration are reduced, the energy reduction is much greater than the time increase, and the stability of the robot is clearly improved, which has obvious practical significance for the battery-powered mobile robot. 2. This trajectory planning method can be applied to other mobile robots, and can also provide guidance for other mobile robots' trajectory path planning. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \begin{tabular}{c} **Joint Serial** \\ **Number** \\ \end{tabular} & \(|\mathbf{v}_{mean\_case1}|\) & \(|\mathbf{v}_{mean\_case2}|\) & \
\begin{tabular}{c} **Percentage** \\ **Change** \\ \end{tabular} & \(|\mathbf{v}_{max\_case1}|\) & \(|\mathbf{v}_{max\_case2}|\) &
\begin{tabular}{c} **Percentage** \\ **Change** \\ \end{tabular} \\ \hline \multirow{10}{*}{Left} & Joint 1 & 2.8910 & 2.6190 & \(\downarrow\)9.41\% & 6.4963 & 5.7032 & \(\downarrow\)12.21\% \\  & Joint 2 & 4.2417 & 3.8249 & \(\downarrow\)9.83\% & 9.4964 & 8.9211 & \(\downarrow\)6.06\% \\  & Joint 3 & 6.0687 & 5.4717 & \(\downarrow\)9.84\% & 11.9745 & 11.3003 & \(\downarrow\)5.63\% \\ arm & Joint 4 & 9.0835 & 8.2043 & \(\downarrow\)9.68\% & 19.3534 & 17.9745 & \(\downarrow\)7.12\% \\  & Joint 5 & 9.4287 & 8.5083 & \(\downarrow\)9.76\% & 21.5095 & 19.8789 & \(\downarrow\)7.58\% \\  & Joint 6 & 11.5040 & 10.4060 & \(\downarrow\)9.54\% & 21.6774 & 19.2030 & \(\downarrow\)11.41\% \\  & Joint 1 & 9.0771 & 8.1354 & \(\downarrow\)10.37\% & 14.3601 & 12.9474 & \(\downarrow\)9.84\% \\  & Joint 2 & 10.8972 & 8.8202 & \(\downarrow\)19.06\% & 21.7574 & 17.4806 & \(\downarrow\)19.66\% \\ Right & Joint 3 & 5.0876 & 4.5726 & \(\downarrow\)10.12\% & 10.1924 & 8.2633 & \(\downarrow\)18.93\% \\ arm & Joint 4 & 9.7424 & 8.7512 & \(\downarrow\)10.17\% & 22.2041 & 16.3423 & \(\downarrow\)26.4\% \\  & Joint 5 & 10.3016 & 9.2776 & \(\downarrow\)9.94\% & 17.6119 & 14.4317 & \(\downarrow\)18.06\% \\  & Joint 6 & 5.4250 & 4.8701 & \(\downarrow\)10.23\% & 11.0025 & 9.0673 & \(\downarrow\)17.59\% \\  & Left wheel & 191.7866 & 173.2072 & \(\downarrow\)9.69\% & 361.3142 & 351.4745 & \(\downarrow\)2.72\% \\  & Right wheel & 320.1776 & 288.9103 & \(\downarrow\)9.77\% & 608.9494 & 598.4260 & \(\downarrow\)1.73\% \\ \hline \hline \end{tabular} Note: The unit of angular velocity is (\({}^{\circ}\)/s), “\({}_{1}\)” represents the percentage decline in case 2 over case 1.  
\end{table} 
Table 6: The effects of different stability constraints on joint velocity.