## 5 Conclusions  
In order to achieve efficient solution goals for the dynamic optimization control and optimization constraint problems of underwater robots, this paper proposes a dynamic optimization control method for underwater robots based on an improved Gaussian pseudo-spectral method, which solves the contradiction between the increase in discrete points and the decrease in computational efficiency in the GPM method. In the dynamic modeling of industrial robots, the motion coordinate system of industrial robots and the corresponding kinematic coordinate transformation relationship in the coordinate system were established. Based on the force analysis of the robot, a six degree of freedom dynamic model of industrial robots was summarized. We designed performance indicators that balance energy and efficiency for the dynamic control of industrial robots. Then, within the Gauss pseudo-spectral control framework, we provided the Legendre-Gauss collocation solution formula and proposed a two-stage adaptive Gauss collocation reconstruction strategy for control variables. The comparative simulation results show that the proposed AGPM algorithm can effectively improve the dynamic optimization control effect of industrial robots. Compared with the CVP algorithm and the GPM algorithm, the driving range, driving attitude control stability, and optimization solution time under the AGPM algorithm can be optimized. Compared to the CVP and the GPM, the AGPM algorithm has improved energy consumption performance by 16.04% and 6.83%, trajectory tracking accuracy by 15.76% and 8.13%, and attitude control accuracy by 17.29% and 8.62%, respectively. In future research, it is expected to achieve new technological breakthroughs in model accuracy and environmental adaptability, as well as robustness and intelligence of control algorithms. The underwater robot dynamics model established in this study, although considering hydrodynamic effects, may still have limitations in simulating complex marine environments. Future research can further refine the model, improve its adaptability to extreme water quality conditions, and explore the integration of new sensor technologies and data processing algorithms, such as using machine learning and deep learning techniques to enhance perception capabilities. With the development of \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Trajectory Tracking Error**} & \multicolumn{3}{c}{**Attitude Control Error**} \\ \cline{2-7}  & **OXY (m)** & **OXZ (m)** & **OYZ (m)** & **Yaw Rate** & **Roll Rate** & **Pitch Rate** \\  & **OXY (m)** & **OXZ (m)** & **OYZ (m)** & **(deg/s)** & **(deg/s)** & **(deg/s)** \\ \hline CVP & 0.0286 & 0.0187 & 0.0388 & 0.1996 & 0.2901 & 0.3967 \\ GPM & 0.0237 & 0.0160 & 0.0216 & 0.1533 & 0.1649 & 0.2941 \\ AGPM & 0.0195 & 0.0135 & 0.0177 & 0.1171 & 0.1208 & 0.1536 \\ \hline \end{tabular}
\end{table} 
Table 4: \(e_{RMSE}\) of performance indicators under different control algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Trajectory Tracking Error**} & \multicolumn{3}{c}{**Attitude Control Error**} \\ \cline{2-7}  & **OXY (m)** & **OXZ (m)** & **OYZ (m)** & **Yaw Rate** & **Roll Rate** & **Pitch Rate** \\  & **OXY (m)** & **OXZ (m)** & **(deg/s)** & **(deg/s)** & **(deg/s)** \\ \hline CVP & 0.1912 & 0.0839 & 0.2177 & 0.3992 & 0.1217 & 0.0691 \\ GPM & 0.0865 & 0.0498 & 0.0587 & 0.2385 & 0.0932 & 0.0567 \\ AGPM & 0.0462 & 0.0305 & 0.0336 & 0.1679 & 0.0768 & 0.0409 \\ \hline \end{tabular}
\end{table} 
Table 3: \(e_{AVE}\) of performance indicators under different control algorithms. artificial intelligence technology, future research can explore more intelligent control strategies based on deep learning and reinforcement learning to improve the autonomous learning and adaptability of underwater robot control systems. In addition, in practical applications, research on the safety and robustness of robot control systems is also crucial.