## 1 Introduction  
In applications requiring high-precision control and positioning, including micro-grippers, biological manipulation, and micro-alignment mechanisms, there is a growing demand for actuators with high resolution [1; 2; 3; 4; 5; 6; 7; 8]. Compliant mechanisms, known for their advantages in operational accuracy and compactness, can be employed to design displacement reducers that enhance the motion resolution of actuators [9; 10; 11]. Compliant mechanisms leverage the elastic deformation and potential energy of materials to convert force and energy [12; 13; 14]. Compared to traditional rigid structures, compliant mechanisms offer benefits such as ease of fabrication, reduced assembly errors, improved operational precision, and a more compact structure [15; 16; 17; 18]. In the previous literature, numerous designs, analyses, and optimization methodologies for the compliant-based displacement amplification mechanism have been presented [19; 20; 21]. In theory, resolution can be improved by swapping the input and output positions of a compliant amplification mechanism. However, this simple swap overlooks different standards in the design of amplifiers and reducers, such as drive compatibility, structural design specifications, and variations in mechanical performance [19; 22; 23; 24; 25]. [PERSON] and [PERSON] [26] designed a reducer that achieves the precise positioning of probe tips by combining springs of different stiffness levels. [PERSON] [19] proposed a series design method for reducers, integrating two bridge amplification mechanisms to design the reducer. However, the use of flexible hinge connections in these designs may lead to stress concentration issues. [PERSON] [18] designed a distributed compliant displacement reducer that replaces the flexible hinge of the bridge mechanism with distributed compliant beams. However, this will lead to a decrease in output stroke. The design of the reducer necessitates the careful consideration of the constant reduction ratio range of the driving components [27; 28; 29]. A narrow linear interval between input and output strokes can lead to nonlinear issues, especially when subjected to significant input displacements, resulting in output deviations from expected performance standards and presenting challenges for practical implementation. This study references existing research on displacement reducers and defines the reduction ratio as the ratio of input displacement to output displacement [18; 19], facilitating theoretical calculations and performance evaluation. This study incorporates the large constant reduction ratio into the design of a compliant displacement reducer, employing an over-constrained structure to ensure the constant reduction ratio. Compared to the previous literature on reducers, this compliant mechanism maintains a nearly constant reduction ratio. Additionally, it features a small moving lumped mass, easing manufacturing at both macro- and micro-scales. The main contributions of this paper can be encapsulated as follows:  
* An over-constrained distributed compliant reducer is proposed, which provides a nearly constant reduction ratio across a wide range of input displacement. Simultaneously, this mechanism significantly reduces the moving lumped mass;
* The mathematical model of the displacement reducer is established based on the compliance matrix method, to predict the input-output relationship and to perform a detailed analysis of the structural sensitivity;
* A general optimization method for distributed compliance mechanisms based on the PSO algorithm is proposed. This optimization approach is applicable to the sizing optimization of most distributed compliant mechanisms. The optimization objectives under three different scenarios were considered separately. The organization of this paper is as follows: Section 2 outlines the design concept of the mechanism. Section 3 establishes the analytical model of the mechanism. Finite element analysis (FEA) is employed to validate the analytical model.