**2. Precondition**

This section is divided into two parts. The first part designs a nanopositioning stage with a double-parallel guiding mechanism, including its mechanical structure selection and sti ffness calculation. The second part is the voltage–displacement experiment process and the conclusion obtained.

#### *2.1. Mechanical Design and Calculation of Nanopositioning Stage*

The nanopositioning stage uses a completely flexible mechanism. This flexible mechanism has a microscale range for the elastic deformation motion. It can achieve the transmission of the motion and force without friction [25,26]. Considering the thinness of the flexible hinge, the wire cutting method is adopted. Therefore, the slit that is easier for the machine is selected as a flexible hinge of a rectangular cross-section, as shown in Figure 1a. The flexible hinge, with four rectangular sections as the base members, is the double-parallel guiding mechanism used in the design, as shown in Figure 1b.

**Figure 1.** Flexible hinge selection: (**a**) rectangular cut section; (**b**) double-parallel guide mechanism; (**c**) flexible hinge main parameter identification

The mechanical model for the rectangular section of the flexible hinge is shown in Figure 1c. Since the movement causes the flexible hinge to elastically deform, the stiffness in each direction must be calculated. According to the equations of material mechanics [27], the stiffness *kx* and torsional stiffness *k*θ*x* motion in the direction x are:

$$k\_x = \frac{48EI\_{yy}}{l^3} = \frac{4Et^3b}{l^3} \tag{1}$$

$$k\_{\oplus x} = \frac{Etb^3}{l^3}B^2\tag{2}$$

where *E* is the modulus of elasticity, *t* is the thickness of the flexible hinge, *b* is the width of the flexible hinge, *l* is the length of the flexible hinge, *Iyy* is the bending section coefficient of the y-axis, and *B* is the width of the nanopositioning stage.

The stiffness *ky* and torsional stiffness *<sup>k</sup>*θ*y* in the vertical direction y are:

$$k\_y = \frac{12EI\_{zz}}{l^3} = \frac{Etb^3}{l^3} \tag{3}$$

$$k\_{\oplus y} = \frac{Etb}{l}D^2\tag{4}$$

where *Izz* is the bending section coefficient of the z-axis and *D* is the distance between the flexible hinges of the two parallel rectangular sections.

The stiffness *kz* and torsional stiffness *k*θ*z* in the vertical direction *z* are:

$$k\_z = \frac{4EA}{l} = \frac{Etb}{l} \tag{5}$$

$$k\_{\theta z} = \frac{Etb^3}{3l} \left(\frac{D^2}{2l^2} + \frac{t^2}{5b^2}\right) \tag{6}$$

where *A* is the flexible hinge cross-sectional area.

The designed flexible hinge has a length of 14.5 mm, a thickness of 0.3 mm, a width of 15 mm, and a material elastic modulus of 72 GPa. The maximum equivalent stress is 7.8 MPa. The sti ffness of the double-parallel guiding positioning platform at six degrees of freedom is obtained, as shown in Table 1.

**Table 1.** Sti ffness of double-parallel-oriented positioning platform under six degrees of freedom (unit: *kx*, *ky*, *kz*: N/mm; *k*θ*<sup>x</sup>*, *<sup>k</sup>*θ*y*, *k*θ*z*: N·mm/rad).


In order to strictly guarantee the accuracy of the nanopositioning stage, a microlevel precision slow wire cutting technology is adopted [28]. At the same time, a material with a small thermal expansion coe fficient is selected [29]. In order to prevent the wire mechanism from oxidizing, the surface of the flexible hinge needs to be nickel plated. Figure 2a shows a schematic diagram of the nanopositioning stage. Figure 2b is the actual diagram of the nanopostioning stage with double-parallel guiding mechanism.

**Figure 2.** Double-parallel guiding mechanism nanopositioning stage: (**a**) the schematic diagram (**b**) and the actual diagram.
