A Parametric Study of Mechanical Cross-Coupling in Parallel-Kinematics Piezo-Flexural Nano-Positioning Systems

Piezo-electric nano-positioning stages are being widely used in applications in which precision and accuracy in the order of nano, and high scanning speeds are paramount. This paper presents a Finite Element Analysis (FEA) of the parallel piezo-flexural nano-positioning (PPNP) stages to investigate motion interference between their different axes. Cross-coupling is one of the significant contributors to undesirable runouts in the precision positioning of PPNP actuators. Using ABAQUS/CAE 2018 software, a 3D model of a PPNP stage was developed. The model consists of a central elastic body connected to a fixed frame through four flexural hinges. A cylindrical stack of multiple piezoelectric disks is placed between the moving central body and the fixed frame. Extensive simulations were carried out for three different friction coefficients in the piezoelectric disks’ contact surfaces, different frame materials, and different geometrical configurations of the stage and the hinges. As a result, it was observed that the primary root cause of the mechanical crosscoupling effect could be realized in the combination of the slip and rotation of the piezoelectric disks due to their frictional behavior with the stage moving in the tangential direction, concurrent with changes in the geometry of the stage.


Introduction
Parallel piezo-flexural nano-positioning (PPNP) stages are used in a variety of *Corresponding Author. # The second and third authors have the same contribution.
How to cite this paper: Shafiee, A., Ahmadian, A. and Akbari, A. (2021) A Parametric Study of Mechanical Cross-Coupling in Parallel-Kinematics Piezo-Flexural Nano-advanced technologies such as precision metrology, scanning probe microscopy [1]- [7], energy harvesting [8] [9] [10] [11], medical robotics [12] [13], biomechanics research apparatus [14]- [20], orthodontics [21] [22] [23], material [24] [25] [26], structural engineering [27] [28], microfluidic devices [29] [30] [31] [32] [33] and other instruments which need precise actuation. Several different factors limit the precision positioning of PPNP stages, including mechanical vibrations, external disturbances, hysteresis and creep effects, temperature variation, and the cross-coupling of the different axes. Understanding the underlying mechanism of cross-coupling in the PPNP stages is the main focus of this effort. In this work, a finite element model for a double-axis PPNP stage is developed to investigate this problem and propose potential solutions for its mitigation. Elmustafa and Lagally [34] conducted an FEA to examine the flexural hinge behavior with controlled motion in a nano-positioning stage used for precision machining. They changed the stage's dimensions and realized that the natural frequency, displacement, stress, and the applied force to the hinges could be controlled and optimized for more convenient and accurate positioning of the stage. Sun et al. [35] designed a 2 DOF nano-positioning system and analyzed it through FEA. Their results show that the excitation force, stiffness, and the natural frequency of the system would increase by decreasing the length and increasing the hinges' width. Li and Xu [36] presented the process of designing and manufacturing a nano-positioning platform with compound parallelogram flexures and bridge-type displacement amplifiers. They derived analytical models for the mechanical performance evaluation of the stage in terms of stiffness, load capacity, kinematics, and dynamics and verified their results through FEA.
Shi et al. [37] designed a novel parallel kinematic XY positioner with the large stroke, decoupled motions, compact size, and large out-of-plane stiffness. The three parts of the positioner are the thermal actuator, displacement amplifier, and a guidance mechanism. They showed that an asymmetrical parallel guide mechanism could be implemented to reduce the cross error in the x and y directions while increasing the out-of-plane stiffness. Nagel et al. [38] predicted the dual-stage, three-axis hybrid parallel-serial-kinematic nano-positioner's parasitic motion by introducing a new parallel-kinematic mechanism that can eliminate the effects of planar coupling. Their results also indicated the potential of the non-orthogonal mechanical amplifier design in minimizing the widespread cross-coupling nonlinearity of the parallel-kinematic designs. Zhang et al. [39] suggested a distributed multi-channel Hammerstein model, composed of a cascaded connection of a static nonlinearity and dynamic linearity to approximate the nonlinear spatial/temporal cross-coupling hysteresis. To reduce the crosscoupling hysteresis nonlinearities' adverse effect, Kang et al. [40] proposed a new modeling and control method for a six-axis parallel piezo-flexural micropositioning stage for bio micromanipulation application. A novel frictional- The remainder of the paper is organized as follows: In Section 2, the experimental observation of the cross-coupling effect in the PPNP stage from Bashash et al. [41] is reviewed. Section 3 presents model development and finite element analysis of the stage. Section 4 presents the FEA results and discussions. Finally, Section 5 summarizes the paper's concluding remarks.

Cross-Coupling Effect in Piezo-Flexural Stages
According to Bashash et al., the coupling of perpendicular axes in the PPNP stages imposes a significant limitation on their positioning accuracy [42]. Figure   1 shows a double-axis Physic Instrument (PI) piezo-flexural stage with two piezoelectric stack actuators and two capacitive position sensors [42]. By applying a voltage to one of the piezoelectric stacks, the stage moves along the stack's expansion/contraction direction. However, due to flexural coupling between the axes, a small portion of the motion is transferred to the perpendicular direction, inducing an undesirable motion in the other axis.
An earlier study has shown by moving the stage in one direction; the corresponding position sensor picks a slight displacement in the other direction [42]. In Figure 2, the PI stage's cross-coupling effect for two low and high-frequency sinusoidal inputs with the magnitude of 80 V applied to axis x of the PI stage, with no voltage applied to axis y, can be seen. The resultant motions are about 75 μm for axis x and about 0.22 μm for axis y. This coupling includes a hysteresis effect and excites the system's natural frequency if the excitation frequency is high, e.g., 50 Hz. Although the cross-coupling magnitude is shown as small as about 0.3%, its impact on nanoscale positioning can be detrimental [42]. Hence, it is essential to take a step beyond the traditional control-oriented analysis of the cross-coupling effect and understand its underlying mechanics to minimize Open Journal of Applied Sciences its undesirable impacts potentially. In the remainder of this effort, it focuses on the finite element analysis of the PI stage's cross-coupling effect, where the finite element model is corroborating with Bashash's work [42].

Finite Element Analysis (FEA)
In this paper, ABAQUS FEA software is used to carry out the process of mechanical cross-coupling simulation. In the simulation, the loading steps are modeled based on the closed-curve of the experimental data. This curve is mainly due to hysteresis and is modeled virtually in this study.
Due to the piezoelectric actuators' asymmetrical arrangements, the simulation is applied in a 3D geometry instead of the 2D geometry. Figure 3 and Figure 4 show the dimensions of the frame and the cylindrical piezoelectric disks, respectively. Table 1 shows the mechanical properties of the frame and the piezoelectric disk. In all simulations, seven cylindrical piezoelectric elements have been used, as shown and numbered in Figure 5.    The surface contact between different piezoelectric elements is modeled through Coulomb's law of Friction. This model can prevent the piezoelectric surfaces from unwanted penetration by selecting the hard contact penetration option in ABAQUS. These surface contacts include contacts of the piezoelectric elements with each other, the contact of piezoelectric disk #1 with the moving part of the frame, and the contact of the piezoelectric disk #7 with the fixed part of the frame. Friction coefficients for the three different cases used in this study are tabulated in Table 2. These values have been adopted from the reference [43].
The first, second, and third cases are related to the different friction coefficients between the piezoelectric disks, whereas the friction coefficients between the top and bottom piezoelectric disks and the frame are kept the same for all cases.
Finite element analysis can be used to investigate the effects of different materials for the frame and the piezoelectric ceramics on the cross-coupling effect.
The effects of geometrical perturbations, such as changes in the square hole's dimensions inside the frame and the flexural hinges, on the cross-coupling effect, could be investigated.
Therefore, the following cases to this investigation are added:  Sixth case: Changing the geometry of the frame by changing the dimensions of the inner square hole. These changes include changing the length from 35 mm to 26.25 mm and the radius corner from 4 mm to 3 mm, as shown in Figure 6. In this case, the material properties are similar to Table 1, and the frictional conditions are similar to the first case.
Seventh case: Changing the frame's geometry by decreasing the thickness of one of the hinges in the y-direction from 1.5 mm to 1.25 mm, as shown in Figure 7. In this case, the material properties are similar to Table 1, and the frictional conditions are similar to the first case.
In all cases, all the six degrees of freedom of the frame edges are constrained. Actuation by piezoelectric stacks for the moving part of the frame is modeled with a left-side pressure and pre-pressures in the x and y directions, respectively. These pressures are applied in lieu of the voltage applied to the piezoelectric actuators (see Figure 5). Magnitudes of these loadings are adjusted in a way that the same displacements are gained from both the voltage and the pressure excitation methods.
This simulation includes the following four important steps: 1) Pre-pressure loading step; 2) Left side pressure loading step; 3) Pre-pressure unloading step; 4) Left side pressure unloading step. In other words, the loading process consists of two primary steps, and the third and fourth steps are the unloading process of the model. At the first step, only a pre-pressure with a magnitude of 25 KPa is applied. This pressure is applied smoothly and statically within 0.05 seconds. At the second step, another pressure is applied to the moving frame's left side without changing the prepressure applied in the first step. This pressure is equal to 148 KPa, which is applied smoothly and statically within 1 second. So, the loading process takes 1.05 seconds to be completed. In the third step, the pre-pressure is unloaded within 0.05 seconds while no changes are applied to the left side pressure. In the fourth step, the left side pressure is unloaded at 1 second. So, the unloading process, which is completed in the third and fourth steps, takes 1.05 seconds, the same as the loading process, resulting in a total simulation time of 2.1 seconds. Simulation parameters are tabulated in Table 3. Note that due to the low acceleration of the system, all simulations are considered to be static.
The type of element used for meshing is C3D8R, an eight-node linear brick element with 24 DOFs. This type of element is capable of traversing shear deformation. The meshing of different parts of the frame, hinges, and the piezoelectric stack is shown in Figure 8.

Results
The simulation results for the seven cases mentioned in the previous section are discussed here. The simulation results of the first case for the displacement and stress distributions in the stage at the end of loading are shown in Figure 9(a) and Figure 9(b), respectively. Table 3. Simulation steps.
Step number Step Step     Figure 2. In Figure 10, point D to A represents the first step, and point A to B represents the second step. Hence, point D to B represents the loading process's total time, i.e., 1.05 seconds. Also, point B to C represents the third step, and point C to D represents the fourth step. Hence, point B to D represents the total time of unloading, i.e., 1.05 seconds.
Note that the mismatch between the two diagrams in Figure 10 is due to excluding hysteresis and creep models in the simulation. Due to good agreement between the experimental and simulation results, these data and figures are used as a reference for other simulation cases. As seen from Figure 10   In order to understand the effects of the frame's material stiffness, the FEA simulations for two different frames Young's modulus values are run and the results are plotted in Figure 13.
As can be seen from Figure 13, increasing Young's modulus of the frame leads to decreasing the displacement of the stage in both directions, while the friction coefficient is fixed like the first case. Increasing Young's modulus causes  [46] showed that the stiffness of the hinge hinge K is proportional to Young's modulus and the third power of the hinge thickness in the XY plane: where E is Young's modulus and d is the thickness of the hinge in the XY plane.
Therefore, due to an identical increase of the stage stiffness in both directions, the cross-coupling effect does not undergo any changes. This result is in good agreement with the experimental results.
To evaluate the effects of the piezoelectric material's stiffness on the mechanical cross-coupling effect, the FEA simulation for two different piezoelectric Young's modulus values are rerun and the results are plotted in Figure 14.   In order to investigate the impacts of changing the stage's dimension on the mechanical cross-coupling effect, the FEA simulation for two cases with different frame and hinge geometries (sixth and seventh case) are re-run and the results are plotted in Figure 15 and Figure 16.
It can be seen from Figure 15 that by decreasing the dimensions of the inner hole and maintaining the material properties and frictional conditions of the first case, the stage's stiffness does not change in the x-direction, but it increases in the y-direction. Hence, as the stage has no displacement in the x-direction comparing to the first case, its displacement decreases in the y-direction. In total, the mechanical cross-coupling effect decreases.
It can be concluded from Figure 16 that by decreasing the hinges' thickness in the y-direction, the stiffness of the hinge decreases as well. However, the stiffness of the stage in the y-direction decreases more than that in the x-direction. Hence, the cross-coupling effect is amplified. The amplification of the cross-coupling effect is due to the decrease in the stage's stiffness, which can also be concluded from Equation (1).
By analyzing Figure 15 and Figure 16, it is evident that one of the other ways to change the mechanical cross-coupling effect between the two axes is changing the stage's geometry. In other words, in some cases where the stiffness ratio between the two axes is changed, the mechanical cross-coupling effect is altered.

Conclusions
In this study, FEA simulations are carried out to investigate the effects of differ- stick-slip frictional behavior. This sticking leads to some degree of rotation in the piezoelectric disks, resulting in a nonlinear cross-coupling behavior. Another observation shows that changing the frame's material causes the stage's stiffness to be altered in both directions. Hence, changing the material of the frame has no impact on the cross-coupling effect. Moreover, changing the piezoelectric disks' material does not cause the crosscoupling effect to be changed, as it does not affect the stage's stiffness. Different geometries of the stage are also considered by changing both the dimensions of the inner hole of the stage and the hinges' thickness. The result is that changing the dimensions of the inner hole of the stage only causes the stiffness to be changed in the y-direction, and consequently, the cross-coupling effect is changed. FEA simulations also show that the proportion of stiffness between two axes is changed by choosing a different thickness for the hinge. This consequently causes the mechanical cross-coupling effect to be changed. Finally, the main parameters that affect the mechanical cross-coupling are the combination of slip and rotation of the piezoelectric disks concurrent with a change in the stage's geometry. The results obtained in this paper can provide a better insight into the mechanics of the PPNP stages for potential improvement of the cross-coupling effect in the product design and development phase.