## 1 Introduction
However, there is a problem in the optimization method based on performance atlas, that is, when there are many characteristic parameters to be optimized, it cannot completely represent the performance atlas in a limited space. In recent years, the approximate model technology has developed rapidly in multi-objective optimization [29; 30]. By fitting the mathematical relationship between input and output, the approximate model technology can replace the real value with the predicted value of the approximate model within a certain error range, so as to effectively improve the calculation efficiency and simulation accuracy [31; 32]. The widely used approximate models in engineering mainly include response surface method surrogate model, Kriging surrogate model, and radial basis function surrogate model [33; 34]. [PERSON] et al. [35] developed a Latin hypercube design, Kriging interpolation, and neural network training (LKN) method based on the response surface method, taking into account both kinematic and dynamic performance indices, carrying out the robustness design of the 4 PUS-1 RPU parallel mechanism. The Taguchi method was initially created for quality engineering to assess and produce better robustness, tolerance specification, and quality management of production processes [36]. The Taguchi method does not rely on complex probabilistic or statistical analysis. It can mine the information of the whole parameter space through a little number of experiments to obtain the optimal solution of the experimental design [37; 38]. [PERSON] et al. [39] performed kinematic optimization of a redundantly driven parallel mechanism using the Taguchi method with the optimization objective of maximizing the total of energy efficiency and workspace. [PERSON] et al. [40] used the Taguchi method and Monte Carlo simulation method to optimize the 3-RRR parallel micro-movement platform iteratively and found the optimal geometric size of the flexure hinge. [PERSON] et al. [41] proposed an optimization method using the Taguchi method to optimize the redundantly driven parallel mechanism by studying the kinematic parameters of a planar 2-DOF parallel robot; the optimal parameter combination between the link length and stiffness was obtained. The multi-objective optimization design problem is driven by many competitive standards, and although optimization algorithm can provide designers with a large number of non-dominated optimal solutions, designers still need to use engineering knowledge to independently choose the best compromise solution [42]. Gray correlation analysis (GRA) has been widely used in various decision-making problems. Its basic idea is to use the linear interpolation between adjacent points of the sequence to map the discrete data to the geometry of the space and to determine the correlation between the sequences by calculating the distance between the reference sequence and the comparison sequence [43; 44]. [PERSON] and [PERSON] [45; 46] used optimization techniques, such as the Taguchi method, response surface method, GRA, and entropy weight measurement, to carry out a multi-objective optimization design for the 2-DOF flexible mechanism and the 2-DOF flexible mechanism with a modified double-lever amplification mechanism, respectively. [PERSON] et al. [47] used the GRA method to design the mechanical structure of the conical guide mechanism to solve the interference misalignment problem during excavator attachment assembly. However, traditional optimization methods are generally only suitable for linear and simple systems, and when they are applied to multi-objective optimization problems, there will be some limitations, such as difficulty in solving and global optimal solutions being ignored [48]. Although intelligent optimization methods outperform traditional optimization methods in solving multi-objective problems, they also have their own limitations, such as being time consuming and having poor convergence [49].