## 2 Mechanism Design  
In general, the reduction of displacement and the amplification of force in displacement reduction mechanisms can be achieved through centralized compliant lever mechanisms. However, large input displacements may lead to issues such as excessive deformation of the hinges and stress concentration. Therefore, this paper adopts a distributed compliant mechanism design. Additionally, an over-constrained design approach is employed to enhance the structural linear displacement range and output stiffness. This study is based on the design concept of distributed compliant mechanisms and utilizes the deformation behavior of compliant beams for structural design. Inspired by the theoretical findings on the input-output relationship of compliant mechanisms from [28; 30], a single-degree-of-freedom compliant displacement reducer with stable input-output characteristics has been successfully developed, as shown in Figure 1. This design adopts a double-layer over-constraint design method to enhance the range of the nearly constant reduction ratio and the load capacity of the displacement reduction mechanism. Figure 1a shows that \(DAD^{\prime}\) is a triangular mechanical amplification mechanism. Providing input displacement at position \(D\) and \(D^{\prime}\) can achieve a stable output displacement at point A, where the amplification ratio increases as the angle \(\alpha_{1}\) decreases. Conversely, when \(\alpha_{1}\) is sufficiently large, displacement reduction can be achieved [28]. After swapping the input and output of two identical triangular lever amplification mechanisms (triangles \(DEG\) and \(D^{\prime}E^{\prime}G^{\prime}\)), they are connected to the triangular mechanical amplification mechanism to achieve an external secondary reduction mechanism, as shown in Figure 1b. Based on the principles shown in the previous two steps, an internal secondary reduction mechanism has been designed, where \(CAC^{\prime}\) represents the triangular mechanical reduction mechanism and the reduction ratio is determined by \(\alpha_{2}\). Due to the adoption of a symmetric design concept, the external secondary reduction mechanism and the internal secondary  
Figure 1: Design procedure of the displacement reducer based on over-constraint: (**a**) A reducer designed based on triangular mechanical amplifier labeled by \(DAD^{\prime}\); (**b**) Added two triangular levers by \(DEG\) and \(D^{\prime}E^{\prime}G^{\prime}\); (**c**) Internal secondary reduction mechanism constructed on the same principle; (**d**) The final structure of the combined internal and external displacement reducer. reduction mechanism can ensure that, when input displacement is provided at points \(E\) and \(E^{\prime}\), a stable output displacement can be achieved at point \(A\)[28]. Additionally, to ensure an efficient use of space, the fixed constraints of the triangular lever mechanism (triangles \(CAB\) and \(C^{\prime}A^{\prime}B^{\prime}\)) in the internal secondary reduction mechanism are replaced by a symmetric structure, achieving effective spatial utilization, as shown in Figure 1c. The input and output positions of the internal and external secondary reduction mechanisms are connected. Additionally, to further enhance the reduction ratio and decrease the moving mass, a ground-mounted drive is implemented, and a third triangular mechanical reduction mechanism, \(EFE^{\prime}\) (with the reduction angle determined by \(\alpha_{3}\)), is added. Furthermore, due to geometric relationships, the deflection of \(Beam-DE\) and \(Beam-D^{\prime}E^{\prime}\) is excessive, resulting in nonlinear deformation, which causes inconsistencies in the linear range of the internal and external secondary reduction mechanisms. To improve the linear range, \(Beam-DE\) and \(Beam-D^{\prime}E^{\prime}\) are replaced with slightly curved \(Beam-DHE\) and \(Beam-D^{\prime}H^{\prime}E^{\prime}\), thereby increasing the range of the constant reduction ratio. The final structural design is shown in Figure 1d.