## 4 Verification and Analysis
The pitch angular velocity under the CVP algorithm exhibits significant and severe fluctuations in amplitude during the initial start-up stage. Although the fluctuation amplitude decreases in the subsequent process, it remains the largest among the three control algorithms. In contrast, the pitch angular velocity under the GPM and the AGPM algorithms is significantly smaller overall, with the AGPM algorithm having the smallest pitch angular velocity fluctuation and the smoothest variation curve. The above comparison results reflect that the AGPM algorithm can significantly improve the attitude angular velocity control results of industrial robots, making them more stable overall and effectively ensuring the driving stability of industrial robots. The resultant control force results of an underwater industrial robot under different control methods are shown in Figure 7. As shown in the figure, the trend of the comparison results with other state variables is consistent. The control force under the  
Figure 5: The control results of the driving posture angle of industrial robot. (**a**) Yaw angle, (**b**) roll angle, and (**c**) pitch angle. CVP algorithm is correspondingly the largest, while the control force of the underwater robot under the AGPM algorithm is significantly the smallest, which helps to achieve the energy-saving goal of the underwater industrial robot. In order to further demonstrate the optimal control performance of the underwater industrial robot dynamics under different control algorithms, a comparative analysis of optimization solution capability performance indicators was conducted, and the energy consumption performance of an underwater industrial robot under different algorithms was also compared, as shown in Table 2. All three control algorithms use the interior point method for solving optimization constraint problems. The CVP algorithm uses the unified grid method to discretize the state and control variables, while the  
Figure 6: The control results of the driving attitude angular rate of industrial robot. (**a**) Yaw rate, (**b**) roll rate, and (**c**) pitch rate. Figure 7: The resultant control force of industrial robot. GPM algorithm and the AGPM algorithm use the LG collocation method and adaptive LG collocation method for discretization, respectively. The three control algorithms are optimized and solved under different allocation points of control variables. Among them, the number of allocation points under the CVP algorithm is 30, the number of allocation points under the GPM algorithm is 38, and the number of allocation points under the AGPM algorithm is 17. Meanwhile, based on the optimization solution times under the three control algorithms, it can be concluded that the optimization solution times for CVP, GPM, and AGPM are 19.83 s, 11.65 s, and 5.33 s, respectively. By comparison, it can be seen that the CVP algorithm has the longest optimization solution time. Compared with the CVP algorithm, the GPM algorithm can effectively improve optimization solving time, but its number of allocation points is relatively large. In contrast, the AGPM algorithm achieves a significant reduction in optimization solution time with fewer allocation points. The comparative results show that after adopting the two-stage adaptive Gauss collocation algorithm, the number of control variable collocation points are significantly reduced, thereby effectively reducing its optimization solution time. In addition, the energy consumption in maneuvering cruise of underwater industrial robot under the CVP, the GPM, and the AGPM algorithms is 206.79 J, 186.34 J, and 173.61 J, respectively, indicating that the proposed optimization control algorithm can achieve the control goal of energy-saving for underwater industrial robots. Based on the above data, it can be concluded that compared to the CVP and the GPM algorithms, the AGPM algorithm has improved energy consumption performance by 16.04% and 6.83%, respectively.