## 4 Verification and Analysis
The CVP method is widely used and has the advantage of a relatively simple solution, but it has the disadvantage of inaccurate optimization time nodes and control parameters due to uniform time grid partitioning. In order to intuitively reflect the effectiveness of the proposed control method, optimization constraint problem solving methods based on control vector parameterization (CVP), the Gaussian pseudo-spectral method, and the adaptive Gaussian pseudo-spectral method were carried out, and simulation results were compared and analyzed. The driving trajectories of industrial robots obtained under different control methods are shown in Figure 3. As shown in the figure, the overall trend of the trajectory curves of industrial robots under the three algorithms is relatively consistent. Among them, the trajectory curve under the CVP algorithm has relatively larger curvature changes and driving mileage, while the trajectory curve under the GPM algorithm has significantly improved in curvature changes and driving mileage compared to the CVP algorithm. In addition, the relevant indicators of the trajectory curve under the AGPM algorithm have been further improved compared to the GPM algorithm. In order to further demonstrate the comparison of the driving trajectories of industrial robots, the projected trajectories of industrial robot trajectories on different coordinate planes were compared, as shown in Figure 4. Combining the projected trajectories on the OXY plane, the OXZ plane, and the OYZ plane, it can be seen that the trajectory projection curve under the AGPM algorithm tends to be closer to the diagonal line from the starting point to the ending point, followed by the trajectory projection curve under the GPM algorithm, while the curvature change of the trajectory projection curve under the CVP algorithm is the most significant, and its overall driving mileage is relatively longer. Based on the comparison of trajectory projection curves, it can be concluded that the APGM algorithm proposed in this study can achieve optimal control of industrial robot dynamics. The optimization results show that the overall mileage of the driving trajectory is the smallest, and the curvature change is smoother, thus effectively achieving energy and efficiency optimization of overall control. the pitch angle under the GPM algorithm and the AGPM algorithm is generally stable in terms of variation amplitude, with the pitch angle obtained by the AGPM algorithm being the smallest in comparison. Based on the trajectory comparison results in Figures 2 and 3, as well as the attitude angle comparison results in Figure 4, it can be concluded that the AGPM algorithm can achieve trajectory optimization control while reducing the yaw angle during the industrial robot's movement, ensuring optimal driving efficiency from the starting point to the ending point. The control results of the industrial robot's driving attitude and angular velocity under different control methods are shown in Figure 6. In the control results of lateral angular velocity, the AGPM algorithm has the smallest overall lateral angular velocity and the fastest convergence speed, proving that the application of the AGPM algorithm effectively improves the lateral stability of industrial robots. According to the comparison results of roll angle velocity, it can be seen that there are severe fluctuations of different amplitudes and frequencies in the roll angle velocity under both the CVP algorithm and the GPM algorithm. Among them, the roll angle velocity fluctuation under the CVP algorithm is the most obvious, with the highest amplitude and frequency of the fluctuation. In contrast, the overall change in roll angle velocity under the AGPM algorithm is the smoothest, with almost no rapid amplitude fluctuations or shaking. The comparison results of pitch angular velocity obtained under the three control methods also showed similar trends of change.