## 7 Conclusions  
In this paper, we design an optical angle nano-positioning mechanism based on CMs to achieve ultra-high resolution nano-displacement, as necessitated by the specific engineering requirements of the SSRF. This study focuses on the design parameters of the flexible hinge, using the driving force and the inherent frequency of the mechanism as the optimization objectives to conduct multi-objective optimization. The following conclusions can be drawn from this study:  
Figure 12: Inherent frequency test system.  
Figure 13: Spectrum diagram of the mechanism. The different colors of the lines represent data collected from different channels of the sensor. (**a**) All the results collected by the sensors. (**b**) Results before 250 Hz.  
By establishing the kinematic model based on the geometric dimensions of the mechanism, the relationship between the mechanism size and the output motion is determined. Analyzing the driving force required by the mechanism at the limit output angle begins with the mechanical characteristics of a single hinge. A mapping relationship between the stiffness of a single hinge and that of the entire mechanism is established. Utilizing the Lagrangian kinematics theory, the dynamic model of the mechanism is developed based on geometric size and mass data. The first-order inherent frequency is determined by solving the partial differential equation. The NSGA-II multi-objective optimization algorithm optimizes the design parameters and addresses the conflict inherent in multi-objective optimization for optical angle nano-positioning mechanisms. The theoretical model is validated through finite element calculations and prototype testing. The results indicate that the first-order inherent frequency of the theoretical model differs by 3.33% from the finite element analysis result and by 2.63% from the real object, substantiating the effectiveness of the proposed method. Future studies will target more complex CMs mechanisms, aiming to achieve higher motion response, higher output resolution, and enhanced mechanism stability as objectives for further optimization.  
Conceptualization, H.L.; Methodology, H.L.; Investigation, Z.Z., X.L., Y.W. and Y.G.; Writing--original draft, Y.D.; Writing--review & editing, Y.D. and B.J.; Project administration, B.J. All authors have read and agreed to the published version of the manuscript. This research wad funded by Shanghai Synchrotron Radiation Facility  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} CMs & Compliant mechanisms \\ SRF & Synchrotron radiation facility \\ APS & Argonne National Laboratory's advanced photon source \\ ESRF & European synchrotron radiation facility \\ SSRF & Shanghai synchrotron radiation facility \\ MOOP & Multi-objective optimization problem \\ PSO & Particle swarm optimization \\ NSGA-II & Nondominated sorting genetic algorithm-II \\ MOPSO & Multi-objective particle swarm optimization \\ PZT & Piezoelectric \\ QBe2 & Beryllium bronze \\ \end{tabular}