## 1 Introduction
Therefore, it is of great significance to conduct research on multi-objective optimization control algorithms while considering the improvement of their comprehensive performance [23; 24]. The GPM has been extensively studied and applied in solving optimal control problems in industrial control and other fields due to its advantages of high solving accuracy, low computational complexity, simple structure, and high solving efficiency [25; 26]. The idea of this method is to discretize the continuous optimal control problem at orthogonal collocation points and approximate the state variables and control variables through global interpolation polynomials, thereby transforming the optimal control problem into an NLP problem for solution [27; 28]. In the field of industrial robot motion control, nonlinear optimization problems have become an important research direction due to the complexity of underwater environments and the dynamic nature of robot systems. Compared to traditional linear optimization problems, nonlinear optimization problems are more complex, so they have stricter requirements for the efficiency of solving optimization control problems [29; 30]. However, traditional GPM methods use a globally unified point allocation approach to discretize state variables and control variables. In order to obtain accurate control results, a large number of point allocation points are usually required, which at the same time affects the optimization calculation efficiency and leads to a decrease in the control performance of industrial robots [31; 32]. On the other hand, unified positioning can easily cause oscillations in the control curve, thereby affecting the stability of the industrial robot's driving posture. Therefore, the coordination and smoothing of the problem between increasing discrete point allocation and computational efficiency in the GPM method is an important technological breakthrough direction for improving the dynamic optimization control problem of industrial robots [33; 34]. Therefore, in order to achieve an efficient solution to the dynamic optimization control and optimization constraint problems for industrial robots, this study combines the Gauss pseudo-spectral control theory and proposes a multi-objective optimal control method for industrial robots based on an adaptive Gauss point reconstruction pseudo-spectral method, which solves the problem of increasing discrete point allocation and computational efficiency in GPM and improves the comprehensive motion control performance of underwater industrial robots. Firstly, based on the kinematic coordinate transformation relationship and force analysis results of industrial robots, a mathematical model for the multi-degree of freedom dynamic control of industrial robots is established. Then, the Legendre-Gauss collocation solution formula was derived based on the Legendre polynomial, and a two-stage adaptive Gauss collocation reconstruction strategy was proposed for the problem of discrete collocation of control variables. By adaptively optimizing the collocation number of control variables, the solving efficiency was improved. Finally, a comparative simulation test was conducted on the dynamic optimization control problem of industrial robots, and the proposed control method was compared with the CVP algorithm and the traditional GPM method to effectively reflect the effectiveness of the proposed method. The rest of this paper is organized as follows. The dynamic modeling of industrial robots is presented in Section 2. The dynamic optimal control of industrial robots is designed in Section 3. The verification and analysis are provided in Section 4, and followed by the conclusion in Section 5.