In general, when designing a flexure stage two types of motion amplification structures are employed: the lever type shown in Fig. 2-(a) and the bridge type shown in Fig. 2-(b). The lever type allows for a large amplification ratio (b/a) that is proportional to the length of the structure. However, the structural stress also increases proportionally. This renders the need for stronger materials. Overall, the lever-type is more appropriate for in-plane motion amplification. The bridge-type has a smaller symmetric form-factor allowing for a denser design and has better vibrational characteristics making it better for out-of-plane motion structures. In this paper, we are implementing out-of-plane motion, therefore, the bridgetype hinge is employed.

Fig. 2 

Topology of amplification mechanism

Two types of flexure hinges have been designed. First, as shown in Fig. 3-(a), the bridge transforms the x-axis motion of the piezo-element into y-axis motion. Second, as shown in Fig. 3-(b), a half-bridge transforms the y-axis bridge motion into z-axis motion.

Fig. 3 

Bridge mechanism

These two bridges have been used to design a dense structure which can transform x-axis motion into z-axis motion. Furthermore, stress concentration is avoided by avoiding a notch type hinge. Although this can lead decreased flexibility, it is beneficial for performance with regards to stress.

A kinematic model of bridge-type amplification mechanism can be expressed as Figure 4. Defining the velocity of point A and B,

Fig. 4 

Kinematic model of bridge-type amplification mechanism

Where ∂x is the displacement of the piezo-element and ∂y is the displacement of the bridge. Expressing equations (1) and (2) as an amplification ratio

Where ∂x is the displacement of the piezo-element and ∂y is the displacement of the bridge. Expressing equations (3) and (4) as an amplification ratio,M5

Defining O (Figure 4) as the instantaneous center of rotation of rigid body AB, as the instantaneous angular velocity, velocity of A and B can be expressed as,

Expressing (6) and (7) as amplification ratio, the following equation is obtained,M8

It is evident from the above equation that determines the amplification ratio, therefore is the critical design factor.

The optimization method employed in this paper is Response surface analysis. For Design of Experiment (DOE) the Central Composite Design (CCD) was used. The design factor used is α (Fig. 4) of the Bridge-type flexure hinge, which has been shown to have a critical effect on the amplification ratio of the motion amplification structure.

Design variables θ₁, θ₂ (Fig. 5) which correspond to the critical design factor α (Eq. 8) for each bridge structure are chosen as the optimization variables. Optimization is divided into a two-step process where θ₁ of the bridge moving in the y-axis is optimized first, then θ₂ of the bridge moving in the z-axis is

Fig. 5 

Optimization Factor

optimized next. The range of variables θ₁, θ₂ are set to,M9M10

Where the maximum of θ₁ and the minimum of θ₂ were each set near angles that render a flat bridge structure. The minimum of θ₁ and the maximum of θ₂ were set heuristically so that the total range would be approximately 20 degrees. The output variable is the maximum displacement of the stage structure. The input value is set at 20 μm according to the specification of the piezo-element to be used in the experiment (Table 1).

Fig. 6 and Fig. 7 show the response surface analysis for the variables θ₁, θ₂ where the results are almost identical for both Al 7075-T6 and ABS. Both materials have a motion amplification ratio of 4.0. Also for both materials, the maximum displacement for θ₁ occurs between 165° and 170° and for θ₂ occurs between 100° and 105°. Based on these simulation results, the values chosen for θ₁, θ₂ are 168.1 and 101.0 respectively for designing the motion stage.

Fig. 6 

Response surface of Al 7075-T6

Fig. 7 

Response surface of ABS

Table 2 shows the results of the response surface analysis. Although the displacements for Al 7075-T6 and ABS are almost identical, note that the stress level of the ABS is approximately 31 times lower. The yield stress for Al 7075-T6 and ABS are respectively 570 MPa, 40 MPa. Therefore, the simulation results for stress are each 4.1 vs 9 times smaller than the yield stress for Al 7075-T6 and ABS. This shows that ABS material compared to Al 7075-T6 is less restricted due to stress in designing flexure structures.

Fig. 8 and Fig. 9 each show the maximum displacement and maximum stress simulation results for the ABS material.

Fig. 8 

Maximum Displacement

Fig. 9 

Maximum Stress

Based on the optimization results presented in Section 3, a Vertical micro-positioning stage has been fabricated using a 3D printer with ABS material. The stage is shown in Fig. 10. The 3D printer used is the Projet® 5000(3D Systems), which uses the MJM (Multi-Jet Modeling) method. MJM method sprays a photopolymer resin and wax from the printer head followed by a UV light which hardens the structure in layers. This 3D printer has maximum fabrication volume of this device is 550 × 393 × 300 mm, 0.025-0.05 mm precision, 375× 375 × 790 DPI (xyz) in HD mode, a minimum layer thickness of 32 μm. This resolution is sufficient to manufacture our motion stage.

Fig. 10 

Image of 3D printed Motion Stage (ABS material)

The measurement of the vertical motion was performed using an interference microscope as shown in Figure 11. In order to measure vertical motion, the method developed by Kim et. al ^[13,14] was employed. The measurement method measures the movement of interference fringes using a CCD camera to estimate the vertical motion of the motion stage. Fig. 12 shows the flow diagram of the measurement process which is implemented in Labview. First, the fringe image model M_i(x, y) is defined. As the stage moves, the image of the stage I(x, y) is obtained. The model is found in the image using the correlation function,M11

Fig. 11 

Microscope used for displacement measurement

Fig. 12 

Microscope used for displacement measurement

and the position (x^1, y^1) of the model is found where,M12

Further details of the measurement method can be found in the literature ^[13].

The input voltage to the piezo-element was varied from 0 to 40 V in 9 steps while the displacement of the end-effector, i.e. the half bridge, was measured. The experimental results are shown in Table 3. Amplification ratio is approximately 4 which agree with the simulation results. Fig. 13 plots the applied input voltage vs piezoelement displacement while Fig. 14 plots input voltage vs end effecter displacement.

Fig. 13 

Piezo displacement

Fig. 14 

End effector displacement

In order to test the performance of the motion stage, 100 nm step inputs were applied to the stage (see Fig. 15). Measurement noise in the interference microscope is currently approximately 10 nanometers which attribute to the noise at each step. Furthermore, a gradual drift can also be seen at each step. It is thought that in the future improving the CCD camera performance and introducing further feedback control can improve motion control performance. Utilizing the motion measurement as feedback for close-loop control could especially correct for drift and non-linearity.

Fig. 15 

100nm Stepping