## 6 Conclusions  
A 2-DOF flexure-based differential XY micropositioning stage driven by piezoelectric actuators was proposed in this study. The stage showed great performance in terms of accuracy and motion. The main contributions of this study were the establishment of the analytical model and the use of the finite element analysis method and experiments to validate its performance. The following conclusions were drawn:  
1. The stage was symmetrical along the \(Y\)-axis, and consisted of an output stage, double parallel rectangular beams, single rectangular beams, and displacement transmission units, while they were all on a rectangular beam hinge base. The two piezoelectric actuators were arranged symmetrically along the \(X\)-axis to realize the displacement input. The input displacement was transferred by displacement transmission units. Different working ranges were achieved via the cooperation of two piezoelectric actuators. Along the \(X\)-axis, double parallel rectangular beams and displacement transmission units added individual stiffness to improve the transmission stability of displacement input.
2. The compliance matrix method was used to establish an analytical model of the proposed positioning stage. The output stiffness in the X- and Y-directions were 1.34 \(\upmu\)m/N and 0.438 \(\upmu\)m/N. Finite element analysis was used to validate the theoretical equation. The error between the theoretical calculation and simulation analysis was within 2.2 and 5.3% in the X- and Y-directions, respectively. The frequency of the X- and Y-directions were 1088.9 and 2240.1 Hz, respectively. When given only a single direction input, the displacement average error was 3.83% and the displacement ratio of the X- to Y-directions was 10.07%. When given double direction inputs, in the \(Y\)-axis, the stage could realize a better displacement magnification, where the magnification ratio was 2.77 and the coupling error was \(1\times 10^{-5}\%\). Using this input method, the positioning stage could be used as a great direction amplifier, while also increasing the positioning range.  
Figure 24: Displacement error of the \(y\)-axis.  
Figure 25: Trajectory curve.  
3. The performance evaluation of the mechanism was tested via experiments. The actual output signal tended to be flat after approximately 0.02 s. Along the \(X\)-axis, the initial position offset was approximately 0.084 um and the maximum displacement difference caused by hysteresis was 1.11 um. Along the \(Y\)-axis, the initial position offset was about 0.03 um and the maximum displacement error was 0.3 um. Under a certain safety factor and the stroke limit of the piezoelectric actuator, the maximum displacement in the X-direction was about 14.5 um; the maximum displacement in the Y-direction was 6.97 um. The experimental tests showed that the working range had an elliptical trajectory.  
All the results indicated that the proposed stage possessed an ideal positioning property. Considering the practical requirements, more sizes and structures should be researched. Optimizations through algorithms should be carried out to improve the positioning property in the future.  
Conceptualization, [PERSON] and Z.W.; methodology, [PERSON] and C.J.; software, B.L.; validation, W.Y. and C.J.; formal analysis, C.J.; investigation, C.J. and B.L.; resources, [PERSON]; writing--original draft preparation, C.J.; writing--review and editing, [PERSON] and C.J.; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, G.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China, grant number 2017 YFB1303502; the National Natural Science Foundation of China, grant number 51405196; the Natural Science Foundation of Tianjin, grant number 18 JCYBJC20100; National Undergraduate Innovation and Entrepreneurship Training Program (201810060017).  
The authors declare no conflict of interest.