## 1 Introduction  
With the advantages of simple structure, high positioning accuracy, and fast dynamic response, the parallel mechanism has been extensively used in the fields of advanced manufacturing equipment, national defense and military, automobile manufacturing, aerospace, and motion simulation [1; 2; 3]. In previous studies, researchers have explored parallel mechanisms from various aspects, especially in structural synthesis and design theory [4; 5; 6], kinematics and dynamics studies [7; 8; 9], singularity analysis [10; 11; 12], operation mode studies [13; 14], and efficient control strategies [15; 16]. These studies provide a deep foundation for the development of parallel mechanisms. The main body of the 6-UPS parallel mechanism is composed of a mobile platform, a fixed platform, and six drive branch chains, where U stands for universal joint, P stands for prismatic pair, and S stands for spherical hinge. The 6-UPS parallel mechanism performance and its structural parameters are closely related. When the structural parameters are unreasonably designed, the mechanism's performance will be significantly reduced. Therefore, researchers have also been working on the optimization design of the structure and performance of parallel mechanism, where the most commonly used optimization methodsare divided into two categories: optimization methods based on objective functions and optimization methods based on performance atlas [17; 18; 19]. The optimization method based on objective function is to build up the objective function and constraints according to the optimization index and then use the optimization algorithm to search for the optimal solution. The optimization methods based on objective function are mostly used in the case of many parameter variables and too complex objective function. When there are two or more objective functions, it is called a multi-objective optimization problem, and solving a multi-objective optimization problem is a process of finding the Pareto optimal solution [20; 21; 22]. [PERSON] et al. [23] conducted a multi-objective optimization of a typical parallel tracking mechanism under comprehensive consideration of the kinematics, stiffness, workspace, and dynamic performances, while considering manufacturing and assembly errors. [PERSON] et al. [24] used the PSO algorithm to perform multi-objective optimization of the workspace, dexterity, stiffness, energy efficiency, motion/force transfer efficiency, and inertial coupling index of the established 2 RPU-2 SPR super-constrained redundantly driven parallel mechanism. [PERSON] et al. [25] studied the effects of different parameters, such as the rotation joint angle and spherical joint position of the 6-RUS parallel robot mobile platform on the workspace, kinematics and dynamics indices, and they used the bees algorithm approach to optimize the manipulator structure. The optimization method based on performance atlas refers to intuitively expressing the relationship between design indices and design parameters in a limited design space, so as to obtain the performance atlas of the mechanism. [PERSON] and [PERSON] [26] optimized the 3 PRS mechanism and the spherical 5R parallel mechanism with 2-DOF by using the performance atlas method, so that they had better kinematic performance and force transmission performance in the workspace. [PERSON] and [PERSON] [27] conducted a comprehensive study on a mobile complex multi-body system, through mechanism analysis and identification, introducing sensitivity analysis and extracting key design variables from global variables for multi-objective optimization design. [PERSON] et al. [28] took the motion/force transmissibility and the workspace range as the optimization objectives, respectively, expressing the relationship between the corresponding motion indices and structural parameters by using the performance atlas method and realized the optimal design of 3-PUU mechanism.