## 1 Introduction  
The analysis of rigid-body motion and surface shape errors in mirror design problems plays an important role in evaluating the environmental adaptability, spatial position stability and imaging quality of opto-mechanical structures [1; 2; 3]. Usually, the root mean square value (RMS) of a mirror's surface shape error is used to verify whether the structural design meets the optical imaging requirements. It is worth noting that among the components of a mirror's surface shape error resulting from its own weight, rigid-body motion, which represents the translation and tilt of the mirror, could be eliminated by recalibrating the optical system. Therefore, rigid-body motion needs to be removed from a mirror's surface shape error while describing the mirror's surface shape quality. During the actual mirror design process, the structure is often modeled first and is subsequently simulated and analyzed in finite-element software for obtaining the node deformation on the mirror's surface, and then the results are exported for further processing to calculate the surface shape error. There are two main kinds of methods that are used to separate the rigid-body motion of the mirror from the surface shape error. One involves taking an accurate fitting for deformed surface shapes using Zernike polynomials [4]. The second involves the surface equation processing method proposed by [PERSON] et al. [5]. The traditional design relies more on experience. If the surface shape error exceeds the requirements, the designer needs to change the model parameters manually, and then the above simulation and processing procedure needs to be repeated several times until the results meet the requirements. As we can see, the whole design process always requires manual repetitive work, and, furthermore, the results often likely do not provide the optimal solution. With the development of finite-element software, many engineering designers have used this optimization design software to optimize the structure of mirrors, includingsize optimization, topology optimization and shape optimization [6]. There have been a lot of studies on about mirror optimization using different kinds of software. [PERSON] et al. [7] studied the optimization of the topology of a primary mirror under its own weight and polishing loading, where the measure of the Strehl ratio was used to perform the objective function, thus establishing the relationship between the optical surface quality and node deformation. The structural optimization of a 1.2 m diameter space mirror using the OptiStruct tool from HyperWorks 12 to minimize the mass and constrain optical aberrations was carried out by [PERSON] et al. [8]. [PERSON] et al. [9] also used OptiStruct to optimize the topology of a lightweight primary mirror assembly to determine its shape, and then the sensitivity analysis method and size optimization were adopted to determine the structural parameters. A parametric optimization method to design large, lightweight space mirrors based on the optimization of the topology of a basic mirror configuration was proposed by [PERSON] et al. [2], which combined the advantages of topology optimization and parametric optimization. The whole optimization process involved the Isight software, OptiStruct and the MATLAB software. Their study implies that optimization software could be effectively used to address mirror optimization. However, the following limitations exist for designers: (1) it takes a long time to learn and figure out the software's functions, and usually it is not easy to get started if you are not familiar with optimization design software; (2) the built-in optimization algorithms are fixed and are difficult to modify; (3) in optimization design software, usually only one objective function can be set. To achieve multi-objective optimization, the weights need to be introduced to transform the multi-objective optimization problem into a single-objective optimization problem. The choice of weights involves a subjective judgment from the designer.