# A Piezoelectric MEMS Microgripper for Arbitrary XY Trajectory

## Abstract

**:**

## 1. Introduction

## 2. Analytical Model

## 3. Results and Discussion

## 4. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

b | out of plane thickness |

${E}_{M}$ | Young’s modulus of silicon |

${E}_{p}$ | Young’s modulus of PZT |

${h}_{M}$ | thickness of silicon |

${h}_{P}$ | thickness of PZT |

${M}_{a}$ | bending moment applied by PZTs |

u | displacement of the specimen along the x-axis |

v | displacement of the specimen along the y-axis |

${V}_{x},{V}_{y}$ | electrical voltages supplied to PZTs |

${x}_{p}\left(t\right),{y}_{p}\left(t\right)$ | parametric equations of the wanted trajectory |

## References

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**Figure 3.**Example of symmetrical and antisymmetrical loads with respect to the x-axis. ${M}_{x}\left(t\right)$ and ${M}_{y}\left(t\right)$ denote, respectively, the bending moments which produce only x-axis and y-axis motion of the tip.

**Figure 5.**Some possible trajectories applicable, just to mention a few examples, in the medical field for atherectomy operations: (

**a**) circular; (

**b**) elliptical; (

**c**) straight line; (

**d**) spiral; (

**e**) cycloid; (

**f**) infinity. All graphs were obtained by means of FEM simulations.

**Figure 6.**Some possible trajectories applicable, just to mention a few examples, for scanning methods in atomic force microscopes (AFM): (

**a**) star; (

**b**) cardioid; (

**c**) nephroid; (

**d**) four-leaf clover; (

**e**) Lisaajous-1; (

**f**) Lissajous 2. All graphs were obtained by means of FEM simulations.

**Figure 7.**Von Mises stress for (

**a**) circular, (

**b**) elliptical, (

**c**) cycloidal, and (

**d**) Lissajous 1 trajectories. All graphs were obtained by means of FEM simulations. Stress scale is in GPa.

**Figure 8.**Effect of the electrical voltage supplied to piezoelectric plates on different trajectories. _______: $\{{V}_{x}$ = 50 V, ${V}_{y}$ = 20 V}; _______: $\{{V}_{x}$ = 30 V, ${V}_{y}$ = 12 V}; _______: $\{{V}_{x}$ = 5 V, ${V}_{y}$ = 2 V}: (

**a**) circular; (

**b**) infinity; (

**c**) star; (

**d**) four-leaf clover; (

**e**) Lissajous 1; (

**f**) Lissajous 2. All graphs were obtained by means of FEM simulations.

**Figure 9.**Effect of dimension ${h}_{C}$ on different trajectories. _______: ${h}_{C}=36$ μm, _______: ${h}_{C}=72$ μm, _______: ${h}_{C}=108$ μm: (

**a**) circular; (

**b**) elliptical; (

**c**) straight line; (

**d**) spiral; (

**e**) cycloid; (

**f**) infinity. All graphs were obtained by means of FEM simulations.

**Figure 10.**Effect of dimension ${h}_{C}$ on different trajectories. _______: ${h}_{C}=36$ μm, _______: ${h}_{C}=72$ μm, _______: ${h}_{C}=108$ μm: (

**a**) star; (

**b**) cardioid; (

**c**) nephroid; (

**d**) four-leaf clover; (

**e**) Lisaajous-1; (

**f**) Lissajous 2. All graphs were obtained by means of FEM simulations.

**Figure 12.**Effect of dimension ${L}_{MeP}$ on different trajectories.

**_______**: ${L}_{MeP}=300$ μm,

**_______**: ${L}_{MeP}=225$ μm,

**_______**: ${L}_{MeP}=150$ μm: (

**a**) four-leaf clover; (

**b**) infinity; (

**c**) Lissajous 1; (

**d**) Lissajous 2. All graphs were obtained by means of FEM simulations.

**Figure 13.**Effect of dimension ${L}_{MeP}$ on $\stackrel{\wedge}{{A}_{x}}$ (in orange) and $\stackrel{\wedge}{{A}_{y}}$ (in green).

**Figure 14.**Effect of dimension ${h}_{M}$ on different trajectories.

**_______**: ${h}_{M}=10$ μm,

**_______**: ${h}_{M}=20$ μm, _______: ${h}_{M}=30$ μm: (

**a**) four-leaf clover; (

**b**) infinity; (

**c**) Lissajous 1; (

**d**) Lissajous 2. All graphs were obtained by means of FEM simulations.

**Figure 15.**Effect of dimension ${h}_{M}$ on $\stackrel{\wedge}{{A}_{x}}$ (in orange) and $\stackrel{\wedge}{{A}_{y}}$ (in green).

Property | Silicon | PZT-5A | Unit |
---|---|---|---|

Density | 2329 | 7750 | kg/m^{3} |

Poisson’s ratio | 0.28 | – | – |

Young’s modulus | 170 | – | GPa |

${\mathrm{d}}_{31}={\mathrm{d}}_{32}$ | – | −1.71 | ${10}^{-10}$ C/N |

${\mathrm{d}}_{33}$ | – | 3.74 | ${10}^{-10}$ C/N |

${\mathrm{d}}_{51}={\mathrm{d}}_{42}$ | – | 5.84 | ${10}^{-10}$ C/N |

Label | Value (μm) | Label | Value (μm) |
---|---|---|---|

${L}_{MeP}$ | 300 | ${h}_{M}$ | 10 |

${L}_{g}$ | 270 | ${h}_{C}$ | 36 |

${d}_{i}$ | 100 | ${h}_{g}$ | 119 |

${d}_{e}$ | 500 | ${s}_{g}$ | 120 |

Out-of-plane thickness | 10 |

**Table 3.**Voltage functions ${V}_{x}\left(t\right)$ and ${V}_{y}\left(t\right)$ required to generate the desired pathways.

Trajectory | Voltage Functions | |||
---|---|---|---|---|

Label | ${\mathit{f}}_{\mathit{x}}\left(\mathit{t}\right)$ | ${\mathit{f}}_{\mathit{y}}\left(\mathit{t}\right)$ | ${\mathit{V}}_{\mathit{x}}\left(\mathit{t}\right)$ | ${\mathit{V}}_{\mathit{y}}\left(\mathit{t}\right)$ |

Circular | $acos\left(t\right)$ | $asin\left(t\right)$ | $40cos\left(t\right)$ | $5.473sin\left(t\right)$ |

Elliptical | $acos\left(t\right)$ | $bsin\left(t\right)$ | $40cos\left(t\right)$ | $2sin\left(t\right)$ |

Straight line | $acos\left(t\right)$ | $b\left(t\right)$ | $40cos\left(t\right)$ | $5.473sin\left(t\right)$ |

Spiral | $a({e}^{\frac{t}{10}}-1)cos\left(5t\right)$ | $b({e}^{\frac{t}{10}}-1)sin\left(5t\right)$ | $40({e}^{\frac{t}{10}}-1)cos\left(5t\right)$ | $5.473({e}^{\frac{t}{10}}-1)sin\left(5t\right)$ |

Cycloidal | $\frac{a}{2\pi}(t+sin\left(10t\right))$ | $bcos\left(10t\right)$ | $\frac{40}{2\pi}(t+sin\left(10t\right))$ | $3cos\left(10t\right)$ |

Infinity | $asin\left(t\right)$ | $bsin\left(2t\right)$ | $40sin\left(t\right)$ | $5.473sin\left(2t\right)$ |

Star | $asin\left(12t\right)cos\left(t\right)$ | $bsin\left(12t\right)sin\left(t\right)$ | $40sin\left(12t\right)cos\left(t\right)$ | $5.473sin\left(12t\right)sin\left(t\right)$ |

Cardioid | $a(2\ast cos(t)+cos(2t\left)\right)$ | $b\ast (2\ast sin(t)+sin(2t\left)\right)$ | $15(2\ast cos(t)+cos(2t\left)\right)$ | $2\ast (2\ast sin(t)+sin(2t\left)\right)$ |

Nephroid | $a(4\ast cos(t)+cos(4t\left)\right)$ | $b\ast (4\ast sin(t)+sin(4t\left)\right)$ | $7.5(4\ast cos(t)+cos(4t\left)\right)$ | $1\ast (4\ast sin(t)+sin(4t\left)\right)$ |

Four-leaf clover | $asin\left(2t\right)cos\left(t\right)$ | $bsin\left(2t\right)sin\left(t\right)$ | $40sin\left(2t\right)cos\left(t\right)$ | $5.473sin\left(2t\right)sin\left(t\right)$ |

Lissajous 1 | $asin\left(2t\right)$ | $bsin\left(3t\right)$ | $40sin\left(2t\right)$ | $5.473sin\left(3t\right)$ |

Lissajous 2 | $asin\left(10t\right)$ | $bsin\left(7t\right)$ | $40sin\left(10t\right)$ | $5.473sin\left(7t\right)$ |

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**MDPI and ACS Style**

Botta, F.
A Piezoelectric MEMS Microgripper for Arbitrary XY Trajectory. *Micromachines* **2022**, *13*, 1888.
https://doi.org/10.3390/mi13111888

**AMA Style**

Botta F.
A Piezoelectric MEMS Microgripper for Arbitrary XY Trajectory. *Micromachines*. 2022; 13(11):1888.
https://doi.org/10.3390/mi13111888

**Chicago/Turabian Style**

Botta, Fabio.
2022. "A Piezoelectric MEMS Microgripper for Arbitrary XY Trajectory" *Micromachines* 13, no. 11: 1888.
https://doi.org/10.3390/mi13111888