## 3 Kinetostatic Modeling and FEA
### Modeling of the Compliant Reducer  
The compliant reducer is symmetrical at its midpoint; thus, it can be statically modeled using only on one half, as shown in Figure 3. The figure retains only the geometric relationships for the sake of analysis convenience. This structure is decomposed into ten compliant beams, namely, \(Beam-AC\), \(Beam-CB\), \(Beam-CE\), \(Beam-BE\), \(Beam-AD\), \(Beam-DH\), \(Beam-HE\), \(Beam-DG\), and \(Beam-GE\), and \(Beam-EF\). The coordinate system \(O-XY\) serves as the global reference coordinate system. Given the input displacement \(F_{in}\) and output displacement \(F_{out}\), the coordinates of each node are defined as shown in Figure 3. The coordinates of nodes \(A,B,C,D,E,F,G\) are in the coordinate system \(O-XY\). Assuming different widths for each beam, represented as \(w_{A}\), \(w_{B}\), \(w_{C}\), \(w_{D}\), \(w_{E}\), \(w_{F}\), \(w_{G}\), \(w_{H}\), and assuming the same material for each beam, we can derive the length of each beam, the angle of orientation, and the cross-sectional area. This half configuration can be decomposed into ten straight strip beams, as shown in Figure 4. Based on the geometric relationships, the rotation angles of each beam can be obtained, as listed in Table 1. The point \(e\) is taken as the coordinate origin. Figure 4: Local coordinate diagram. Figure 3: Model geometry analysis chart. In the local coordinate system, the global stiffness matrices of each beam are derived using geometric parameters from the previous section. These matrices incorporate beam orientation angles calculated according to Table 1 and Equation (6). The derived analytical model above can analyze half of the symmetric compliant displacement reducer. Hence, the rotations of the input and output rigid stages are constrained. Consequently, the static equilibrium equations for the seven unfixed nodes \(A,B,C,D,E,F\), and \(H\) under the action of input force \(F_{in}\) and output displacement \(F_{out}\) can be obtained, as shown in Equation (7). \[\left\{ \begin{array}{l}F_{D}^{t,DA}+F_{D}^{t,HD}+F_{D}^{t,DG}=0\\ F_{H}^{t,HD}+F_{H}^{t,EH}=0\\ F_{E}^{t,EH}+F_{E}^{t,EG}+F_{E}^{t,EB}+F_{E}^{t,EE}=0\\ F_{G}^{t,CA}+F_{G}^{t,CB}+F_{C}^{t,EC}=0\\ F_{E}^{t,CB}+F_{E}^{t,EB}=0\\ f_{F}^{t,EE}=\frac{f_{in}}{2}\\ f_{A,dy}^{t,DA}+f_{A,dy}^{t,CA}=\frac{f_{out}}{2}\end{array} \right., \tag{7}\]  
Further derivation yields Equation (8):  
\[\left\{ \begin{array}{l}K_{DA,1}^{t}U_{D}^{t}+K_{DA,2}^{t}(1,3,2)U_{A}^{t}+K_{ HD,4}^{t}U_{D}^{t}+K_{HD,3}^{t}U_{H}\\ +K_{DC,1}^{t}U_{D}^{t}+K_{DC,2}^{t}U_{G}^{t}=0\\ K_{HD,1}^{t}U_{H}^{t}+K_{HD,2}^{t}U_{D}^{t}+K_{EH,4}^{t}U_{H}+K_{EH,3}^{t}U_{E} ^{t}=0\\ K_{EH,1}^{t}U_{L}^{t}+K_{FLA,2}^{t}U_{H}^{t}+K_{EG,1}^{t}U_{E}^{t}+K_{EG,2}^{t }U_{G}^{t}+K_{EB,1}^{t}U_{E}^{t}\\ +K_{EB,2}^{t}(1,3,2)U_{B}^{t}+K_{E}^{t,E}U_{L}^{t}+K_{FEL,2}^{t}\langle 3,2 \rangle U_{E}^{t}=0\\ K_{CA,1}^{t}u_{A}^{t}+K_{CA,2}^{t}\langle 3,2\rangle U_{C}^{t}+K_{CB,1}^{t}U_{C}^{t}+K_{ CD,2}^{t}\langle 1,3,2\rangle U_{B}^{t}\\ +K_{EC,4}^{t}U_{C}^{t}+K_{EC,3}^{t}U_{C}^{t}=0\\ K_{CB,3}^{t}\langle 2,1\rangle U_{C}^{t}+K_{CB,4}^{t}\langle 2,2\rangle u_{b,y}^{t} +K_{EB,3}^{t}\langle 2,1\rangle U_{E}^{t}\\ +K_{EB,4}^{t}\langle 2,2\rangle u_{b,y}^{t}=0\\ K_{F,3}^{t}\langle 2,1\rangle U_{E}^{t}+K_{FEA}^{t}\langle 2,2\rangle u_{f,y}^{t}= \frac{f_{in}}{2}\\ K_{DA,3}^{t}\langle 2,1\rangle U_{D}^{t}+K_{DA,4}^{t}\langle 2,2\rangle u_{a,y}^{t} +K_{CA,3}^{t}\langle 2,1\rangle U_{C}^{t}\\ +K_{CA,4}^{t}\langle 2,2\rangle u_{A,y}^{t}=\frac{f_{out}}{2}\end{array} \right., \tag{8}\]  
From Equation (8), it is evident that, for a given input force \(F_{in}\), the node displacements can be determined through analysis. Consequently, the displacement reducer's reduction ratio can be determined as  
\[r=\frac{u_{A,y}^{t}}{u_{F,y}^{t}}, \tag{9}\]