## 2 Mechanical Design  
The proposed 2-DOF flexure-based XY micropositioning stage is shown in Figure 1 (the scale for the geometrical dimensions is 1:30 (mm)). The overall structure of the stage is symmetrical along the \(Y\)-axis. Two piezoelectric actuators located at the frame are respectively mounted along the X-axis (piezoelectric actuator-1 (PZT-1) and piezoelectric actuator-2 (PZT-2) in Figure 1). The contact condition between the piezoelectric actuator and the driving point was assumed to be Hertzian contact. The actuators are preloaded by a thread and in contact with flexure hinges through the ball-screw (part G in Figure 1). The power amplifier controlled by a personal computer (PC) is used to supply control voltage signals for the expansion and retraction of the piezoelectric actuator. The core part of the positioning stage that consists of an output stage (part D in Figure 1) and chains (X- and Y-axes), as shown in Figure 1.  
In the stage, the flexure hinge of the chains is a rectangular beam hinge with different scale parameters. Compared with other types of flexure hinges, a rectangular flexure hinge (Figure 2a) has more advantages, such as ease of processing and a larger motion displacement. In the \(X\)-axis chain, the double parallel rectangular (Figure 2b) beam was chosen as the guide link between the input and the output stage (part D in Figure 1). Parts A and E are symmetrical along the \(Y\)-axis to avoid the parasitic movement in the vertical direction and greatly increase the motion stability. In the \(Y\)-axis chain, the part B and F links are the structure composed of a single rectangular beam (Figure 2c), while it reduces the structural static stiffness to obtain a larger motion displacement. The displacement input from the piezoelectric actuators is transferred by displacement transmission units (part C in Figure 1). Two units are arranged in parallel (Figure 3a) such that it greatly increases the stage stiffness and makes motion transmission more stable. The structure of the displacement transmission unit is shown in Figure 3b. In the simplified schematic (Figure 3c), the input displacement is transferred separately to the X- and Y-axes, while their values are dependent on the angle \(\alpha\). The value of different angles brings different transmission effects. This issue is not discussed in detail herein. The appropriate value was chosen based on actual experiences, discussed in Section 3, for analyzing the proposed 2-DOF flexure-based micropositioning stage. Figure 1: Micropositioning stage. PZT: piezoelectric actuator. Figure 2: Core part of the micropositioning stage. The two aforementioned piezoelectric actuators were arranged symmetrically along the \(X\)-axis to realize the differential drive for the stage. Herein, (1) if two actuators are given the same input displacement, the output stage achieves excellent single-axis positioning performance along the \(Y\)-axis; (2) if given different displacements, the output stage achieves \(X\)-axis and \(Y\)-axis displacements to realize 2-DOF positioning. The positioning range depends on the difference value between the input displacements of PZT-1 and PZT-2 (Figure 1). The workspace is an arc, as shown in Figure 4. (1) When only one PZT (PZT-1 or PZT-2) is working, the motion trajectory of the stage is the outer contour of the working space. The point b or d, which are reached depending on the chosen PZT actuator, are at the limits of the travel range. (2) When the two PZT actuators are not working, point a in Figure 4 is the initial point of the working space. (3) When the input of the two PZT actuators is the same, the center of the stage is on the line ae. Point c is half the travel range of the two chosen PZT actuators. Point e is at the limit of the travel range of the two chosen PZT drivers. The working space of the stage depends on the stroke of the chosen PZT actuators' travel range. When only one PZT is working, the inactive one acts as the blocking barrier and decreases the positive \(x\)-axis.