## 1 Introduction  
Compliant mechanisms (CMs) represent a significant area in modern mechanism design, characterized by high precision, high reliability, and absence of clearance, friction, or wear [1; 2; 3; 4]. By enhancing the reduction ratio of CMs, it is possible to avoid friction and backlash errors typically associated with gear reducers, thereby effectively improving the resolution while maintaining accuracy [5]. Recently, mechanisms based on flexible designs have garnered considerable attention for applications in microscopic imaging [6; 7] and micromanipulation [8; 9; 10; 11; 12; 13]. The synchrotron radiation facility (SRF) marks a significant milestone in the development of human light sources, providing a new, high-performance, highly collimated, pure, and bright light source [14; 15; 16]. A monochromator, a core component of the synchrotron radiation device, separates the polychromatic light emitted by the source into the required monochromatic light [17], as depicted in Figure 1.  
To achieve angular positioning of the monochromator splitter crystal, extensive research has been conducted. The Argonne National Laboratory's advanced photon source (APS) has developed a series of nano-positioning instruments [18; 19; 20; 21; 22; 23; 24]. Among these, CMs utilized in the artificially slotted crystal monochromator and the hard X-ray polarizer can attain angular resolutions of 20-40 nrad and 80 nrad, respectively, in angular adjustments. The European synchrotron radiation facility (ESRF) employs traditional slow-walking silk-cutting technology to fabricate the four-link straight circular flexhing bending mechanism in a single piece, achieving a focused light spot of less than 50 nm [25]. Utilizing CMs, the Shanghai synchrotron radiation facility (SSRF) has demonstrated the feasibility of creating micro displacement adjustment mechanisms for synchrotron radiation devices using 3D printing technology [26]. The multi-objective optimization problem (MOOP) aims to address scenarios with multiple conflicting objectives by finding a solution that meets all optimization criteria. This is crucial for both theoretical research and practical engineering applications. Traditional methods for multi-objective optimization include the weighted sum method, constraint method, objective programming method, distance function method, and minimax method [27]. Traditional optimization techniques often struggle to guarantee excellent outcomes when solving MOOP's due to their inherent limitations. Consequently, the study of intelligent optimization algorithms, which are essentially stochastic search algorithms that mimic the behavior of natural biological groups, has garnered significant interest. These algorithms are particularly effective for tackling complex problems characterized by discontinuity, nonlinearity, and multiple variables. In the application research of multi-objective optimization, [PERSON] et al. developed a hybrid algorithm combining the random frog leaping algorithm (RFLA) and particle swarm optimization (PSO), achieving high solution quality [28]. [PERSON] et al. utilized the nondominated sorting genetic algorithm-II (NSGA-II) to optimize the design of a compliant linear guide mechanism with a high-precision feed drive, achieving a closed-loop control error in output displacement of less than 0.02 \(\upmu\)m [29]. Their team also applied NSGA-II to the design of micromachining feed drive mechanisms, producing and evaluating the optimization results [9]. Additionally, [PERSON] et al. implemented the multi-objective particle swarm optimization (MOPSO) algorithm to design a planar parallel 3-DOF nano-positioner for scanning probe microscopy (SPM). The effectiveness of the proposed modeling and optimization methods was corroborated by both simulation and experimental results [30]. From the investigations discussed, it is evident that as the requirements for the monochromatic beam of synchrotron radiation devices continue to increase, so too do the technical demands on nano-scale micro-displacement tables.