## 3 Mathematical Model  
The compliance matrix method was used to calculate the mathematical model. First, the compliance matrix of a single flexure hinge was calculated. Second, the compliance model of the core part was calculated. Finally, the compliance model of the whole structure was obtained. The detailed formula derivation process of the compliance matrix of the rectangular beam hinges [34], the rotation matrix, and the translation matrix are given in Appendices A and B. If \(C_{p_{i}}\) represents the compliance matrix  
Figure 4: Working space description. Point a is the initial point in the working space; points b and d are the range limits when only one PZT is working; points c and e are the midpoint and range limit, respectively, when both PZTs are working. Figure 3: Motion form of Micro positioning stage. (**a**) Parallel units (**b**) Displacement transmission unit (**c**) The simplified schematic. of the flexure hinge in the local coordinate system, \(C_{w_{i}}\) represents its compliance matrix in the global coordinate system. The conversion relationship between the local and global compliance matrices is:  
\[C_{w_{i}}=R_{i}P(r_{i})C_{p_{i}}P(r_{i})^{T}{R_{i}}^{T}, \tag{1}\]  
where \(R_{i}\) is the local-to-global rotation matrix and \(P(r_{i})\) is the local-to-global translation matrix. While having \(m\) flexure hinges in series, the compliance matrix of \(m\) flexure hinges is:  
\[C_{p-m}=\sum_{i=1}^{m}C_{p_{i}}. \tag{2}\]  
When having \(n\) flexure hinges in parallel, the compliance matrix of \(n\) flexure hinges is:  
\[C_{p-n}=\left[\sum_{i=1}^{n}\left(C_{p_{i}}\right)^{-1}\right]^{-1}. \tag{3}\]  
The general mathematical model from (1)-(3) and the equations in Appendices A and B (Equations (A1), (B1)-(B3)) are applied here as the mathematical model of the core part and the whole structure. As shown in Figure 5, the local coordinate  
\(C_{1}\) is the point \(o_{p4}\) reached using rotation and translation from points \(o_{w1}\), \(o_{w2}\), and \(o_{w3}\) in Figure 6 as follows:  
\[C_{1}=\left[\left(C_{o_{w2-3}}\right)^{-1}+\left(C_{o_{w2^{\prime}-3}}\right)^{ -1}\right]^{-1}+C_{o_{w1-3}}, \tag{8}\]  
where the detailed process is in Appendix C (A13)-(A20). The point \(o_{p10}\) in displacement transmission units is given as:  
\[C_{o_{p10}}=\left[\left(C_{o_{p0}-10}\right)^{-1}+\left(C_{o_{p11-10}}\right)^ {-1}\right]^{-1}+C_{o_{p12-10}}. \tag{9}\]  
\(O_{o_{p10}}\) converts to the global coordinates in Figure 7 as follows:  
\[C_{2}=C_{o_{w4}}=R_{2}\left(\frac{\pi}{2}\right)R_{x}\left(\frac{\pi}{2} \right)C_{o_{p10}}R_{x}\left(\frac{\pi}{2}\right)^{T}R_{z}\left(\frac{\pi}{2} \right)^{T}, \tag{10}\]  
where the detailed process is in Appendix C (A21)-(A25). In Figure 8, \(C_{w1}\), \(C_{w2}\), \(C_{w3}\), and \(C_{w4}\) converts to the core of the output stage via rotation and translation to obtain \(C_{w}\). The compliance matrix of the whole structure is given as  
\[C_{w}=\left[\left(C_{w_{1}}\right)^{-1}+\left(C_{w_{2}}\right)^{-1}+\left(C_{w _{3}}\right)^{-1}+\left(C_{w_{4}}\right)^{-1}\right]^{-1}, \tag{11}\]  
where the detailed process is in Appendix C (A26)-(A29). Figure 6: Displacement transmission units. Figure 3:0 Single rectangular beam. As such, the compliance matrix of the whole structure \(C_{w}\) was obtained. The detailed formula derivation process is given in Appendix C. The structural parameters of the stage designed in the experiments are shown in Table 1 and Figure 3:1 \(a_{i}\), \(l_{i}\), and \(b_{i}\) represent the lengths of x-, y-, and \(z\)-axes, respectively. In the \(z\)-axis, all lengths were the same (\(b_{1}=b_{2}=b_{3}=b_{4}=b_{5}\)). These parameters in Table 1 were brought into the equations. MATLAB (MathWorks, Inc., version 2018) was used to calculate the compliance matrix. \(C_{w}\) is given as Equation (12). Its unit is micrometers per newton. Meanwhile, the equation between the input force and output displacement was obtained using Equation (13), according to the relationship between force and displacement (\(X=CF\)).