A multi-layer electro elastic drive for micro and nano robotics

doi:10.15406/iratj.2024.10.00286

eISSN: 2574-8092

International Robotics & Automation Journal

Research Article Volume 10 Issue 2

A multi-layer electro elastic drive for micro and nano robotics

Afonin SM

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National Research University of Electronic Technology, MIET, Moscow, Russia

Correspondence: Afonin Sergey Mikhailovich, National Research University of Electronic Technology, MIET, 124498, Moscow, Russia, Tel 4997102233

Received: July 09, 2024 | Published: July 17, 2024

Citation: Afonin SM. A multi-layer electro elastic drive for micro and nano robotics. Int Rob Auto J. 2024;10(2):73-76. DOI: 10.15406/iratj.2024.10.00286

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Abstract

A multi-layer electro elastic drive of robotics is used in adaptive optics of compound telescope, scanning microscopy, interferometry and nanotechnology. For micro and nano robotics a multi-layer electro elastic drive is applied. The parametric model of a multi-layer electro elastic drive is determined. Its functions and matrix deformations are founded. The parameters of the multi-layer longitudinal PZT drive are determined.

Keywords: multi-layer electro elastic drive, multi-layer piezo drive, parametric model, micro and nano robotics.

Introduction

A multi-layer electro elastic drive is used promising for micro and nano robotics in the micro and nano displacement.¹⁻⁸ This drive based on the piezoelectric or electrostriction effects.^9s−19 A multi-layer electro elastic drive of robotics is applied in adaptive optics of compound telescope, scanning microscopy, micro surgery, interferometry, nano pump, nano stabilization and nanotechnology.²⁰⁻⁵³ The deformations of a multi-layer drive are described with the matrix equation. Its parametric model, scheme and functions are obtained by using of mathematical physics method.

Parametric model

The parametric model⁶⁻⁵³ of multi-layer piezo actuator with the voltage or current controlled are determined using the equation of inverse piezo effect in the form at the control of voltage

$S_{i} = d_{m i} E_{m} + s_{i j}^{E} T_{j}$ $S_{i} = d_{m i} E_{m} + s_{i j}^{E} T_{j}$

at the control of current

$S_{i} = g_{m i} D_{m} + s_{i j}^{D} T_{j}$ $S_{i} = g_{m i} D_{m} + s_{i j}^{D} T_{j}$

here $S_{i}$ $S_{i}$ , $E_{m}$ $E_{m}$ , $D_{m}$ $D_{m}$ , $T_{j}$ $T_{j}$ , $d_{m i}$ $d_{m i}$ , $g_{m i}$ $g_{m i}$ , $s_{i j}^{E}$ $s_{i j}^{E}$ , $s_{i j}^{D}$ $s_{i j}^{D}$ are the relative deformation, the strength of electric field, the electric induction, the strength of mechanic field, the piezo module, the piezo constant, the elastic compliances at $E = const$ $E = const$ and at $D = const$ $D = const$ , and $i, j, m$ $i, j, m$ are the indexes.

The equation of the direct piezo effect has the form⁶⁻⁵³

$D_{m} = d_{m i} T_{i} + ε_{m k}^{T} E_{k}$ $D_{m} = d_{m i} T_{i} + ε_{m k}^{T} E_{k}$

here $ε_{m k}^{T}$ $ε_{m k}^{T}$ is the dielectric constants at $T = const$ $T = const$ , k is the index

Then the electroelasticity equation of a multi-layer drive⁶⁻⁵³ has the form

$S_{i} = v_{m i} ψ_{m} + s_{i j}^{ψ} T_{j}$ $S_{i} = v_{m i} ψ_{m} + s_{i j}^{ψ} T_{j}$

here $Ψ = E, D$ $Ψ = E, D$ is control parameter at the control of voltage and the control of current.

A multi-layer drive consist from the piezo layers connected the series mechanically and the parallel electrically [6 − 44]. We have the system of equations for T-form quadripole of k piezo layer

$F_{k inp} (s) = - (Z_{1} + Z_{2}) Ξ_{k} (s) + Z_{2} Ξ_{k + 1} (s)$ $F_{k inp} (s) = - (Z_{1} + Z_{2}) Ξ_{k} (s) + Z_{2} Ξ_{k + 1} (s)$

$- F_{k out} (s) = - Z_{2} Ξ_{k} (s) + (Z_{1} + Z_{2}) Ξ_{k + 1} (s)$ $- F_{k out} (s) = - Z_{2} Ξ_{k} (s) + (Z_{1} + Z_{2}) Ξ_{k + 1} (s)$

$Z_{1} = \frac{S_{o} γ t h (δ γ)}{s_{i j}^{ψ}}, Z_{2} = \frac{S_{o} γ}{s_{i j}^{ψ} s h (δ γ)}$ $Z_{1} = \frac{S_{o} γ t h (δ γ)}{s_{i j}^{ψ}}, Z_{2} = \frac{S_{o} γ}{s_{i j}^{ψ} s h (δ γ)}$

where $Z_{1}$ $Z_{1}$ , $Z_{2}$ $Z_{2}$ , $s$ $s$ , $δ$ $δ$ , $γ$ $γ$ , $F_{k inp} (s)$ $F_{k inp} (s)$ , $F_{k out} (s)$ $F_{k out} (s)$ , $Ξ_{k} (s)$ $Ξ_{k} (s)$ , $Ξ_{k + 1} (s)$ $Ξ_{k + 1} (s)$ are the resistances of quadripole k piezo layer, the transform parameter, the thickness, the coefficient wave propagation, the Laplace transform of the forces at the input and output ends of k piezo layer, the transforms of the displacements at input and output ends of k piezo layer.

The system of the equations for k piezo layer has the form

$- F_{k inp} (s) = (1 + \frac{Z_{1}}{Z_{2}}) F_{k_{o u t}} (s) + Z_{1} (2 + \frac{Z_{1}}{Z_{2}}) Ξ_{k + 1} (s)$ $- F_{k inp} (s) = (1 + \frac{Z_{1}}{Z_{2}}) F_{k_{o u t}} (s) + Z_{1} (2 + \frac{Z_{1}}{Z_{2}}) Ξ_{k + 1} (s)$

$Ξ_{k} (s) = \frac{1}{Z_{1}} F_{k out} (s) + (1 + \frac{Z_{1}}{Z_{2}}) Ξ_{k + 1} (s)$ $Ξ_{k} (s) = \frac{1}{Z_{1}} F_{k out} (s) + (1 + \frac{Z_{1}}{Z_{2}}) Ξ_{k + 1} (s)$

This system is founded in the matrix form

$[\begin{matrix} - F_{k inp} (s) Ξ_{k} (s) \end{matrix}] = [M] [\begin{matrix} F_{k out} (s) Ξ_{k + 1} (s) \end{matrix}]$ $[\begin{matrix} - F_{k inp} (s) \\ Ξ_{k} (s) \end{matrix}] = [M] [\begin{matrix} F_{k out} (s) \\ Ξ_{k + 1} (s) \end{matrix}]$

$[M] = [\begin{matrix} m_{11} & m_{12} m_{21} & m_{22} \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} 1 + \frac{Z_{1}}{Z_{2}} & Z_{1} (2 + \frac{Z_{1}}{Z_{1}}) \frac{1}{Z_{2}} & 1 + \frac{Z_{1}}{Z_{2}} \end{matrix} ⎤ ⎥ ⎦$ $[M] = [\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}] = [\begin{matrix} 1 + \frac{Z_{1}}{Z_{2}} & Z_{1} (2 + \frac{Z_{1}}{Z_{1}}) \\ \frac{1}{Z_{2}} & 1 + \frac{Z_{1}}{Z_{2}} \end{matrix}]$

$m_{11} = m_{22} = 1 + \frac{Z_{1}}{Z_{2}} = c h (δ γ),$ $m_{11} = m_{22} = 1 + \frac{Z_{1}}{Z_{2}} = c h (δ γ),$ $m_{12} = Z_{1} (2 + \frac{Z_{1}}{Z_{1}}) = Z_{0} s h (δ γ)$ $m_{12} = Z_{1} (2 + \frac{Z_{1}}{Z_{1}}) = Z_{0} s h (δ γ)$

$m_{21} = \frac{1}{Z_{2}} = \frac{s h (δ γ)}{Z_{0}},$ $m_{21} = \frac{1}{Z_{2}} = \frac{s h (δ γ)}{Z_{0}},$ $Z_{0} = \frac{S_{0} γ}{s_{i j}^{ψ}}$ $Z_{0} = \frac{S_{0} γ}{s_{i j}^{ψ}}$

The equation of forces at the boundary between two layers is obtained in the form

$F_{k out} (s) = - F_{k + 1 inp} (s)$ $F_{k out} (s) = - F_{k + 1 inp} (s)$

For a multi-layer electro elastic drive with n layers and l length its system has the matrix form

$[\begin{matrix} - F_{1 inp} (s) Ξ_{1} (s) \end{matrix}] = {[M]}^{n} [\begin{matrix} F_{n out} (s) Ξ_{n + 1} (s) \end{matrix}]$ $[\begin{matrix} - F_{1 inp} (s) \\ Ξ_{1} (s) \end{matrix}] = {[M]}^{n} [\begin{matrix} F_{n out} (s) \\ Ξ_{n + 1} (s) \end{matrix}]$

${[M]}^{n} = [\begin{matrix} c h (n δ γ) & Z_{0} s h (n δ γ) \frac{s h (n δ γ)}{Z_{0}} & c h (n δ γ) \end{matrix}]$ ${[M]}^{n} = [\begin{matrix} c h (n δ γ) & Z_{0} s h (n δ γ) \\ \frac{s h (n δ γ)}{Z_{0}} & c h (n δ γ) \end{matrix}]$

then

${[M]}^{n} = [\begin{matrix} c h (l γ) & Z_{0} s h (l γ) \frac{s h (l γ)}{Z_{0}} & c h (l γ) \end{matrix}]$ ${[M]}^{n} = [\begin{matrix} c h (l γ) & Z_{0} s h (l γ) \\ \frac{s h (l γ)}{Z_{0}} & c h (l γ) \end{matrix}]$

The equations of forces a multi-layer drive we have in the form

at $x = 0, T_{j} (0, s) S_{0} = F_{1} (s) + M_{1} s^{2} Ξ_{1} (s)$ $x = 0, T_{j} (0, s) S_{0} = F_{1} (s) + M_{1} s^{2} Ξ_{1} (s)$

at $x = l, T_{j} (l, s) S_{0} = - F_{2} (s) - M_{2} s^{2} Ξ_{2} (s)$ $x = l, T_{j} (l, s) S_{0} = - F_{2} (s) - M_{2} s^{2} Ξ_{2} (s)$

The transform of its force causes deformation has the equation in the form

$F (s) = \frac{v_{m i} S_{0} ψ_{m} (s)}{s_{i j}^{ψ}}$ $F (s) = \frac{v_{m i} S_{0} ψ_{m} (s)}{s_{i j}^{ψ}}$

Then the parametric model and scheme on Figure 1 of a multi-layer electro elastic drive are obtained in the form

$Ξ_{1} (s) = (1 / (M_{1} s^{2})) {- F_{1} (s) + (1 / χ_{i j}^{ψ}) [\begin{matrix} v_{m i} ψ_{m} (s) - (γ / s h (l γ)) \times \times (c h (l γ) Ξ_{1} (s) - Ξ_{2} (s)) \end{matrix}]}$ $Ξ_{1} (s) = (1 / (M_{1} s^{2})) {- F_{1} (s) + (1 / χ_{i j}^{ψ}) [\begin{matrix} v_{m i} ψ_{m} (s) - (γ / s h (l γ)) \times \\ \times (c h (l γ) Ξ_{1} (s) - Ξ_{2} (s)) \end{matrix}]}$

$Ξ_{2} (s) = (1 / (M_{2} s^{2})) {- F_{2} (s) + (1 / χ_{i j}^{ψ}) [\begin{matrix} v_{m i} ψ_{m} (s) - (γ / s h (l γ)) \times \times (c h (l γ) Ξ_{2} (s) - Ξ_{1} (s)) \end{matrix}]}$ $Ξ_{2} (s) = (1 / (M_{2} s^{2})) {- F_{2} (s) + (1 / χ_{i j}^{ψ}) [\begin{matrix} v_{m i} ψ_{m} (s) - (γ / s h (l γ)) \times \\ \times (c h (l γ) Ξ_{2} (s) - Ξ_{1} (s)) \end{matrix}]}$

Figure 1 Parametric scheme multi layer electro elastic drive.

here $v_{m i} = {\begin{matrix} d_{33}, d_{31}, d_{15} g_{33,} g_{31}, g_{15} \end{matrix},$ $v_{m i} = {\begin{matrix} d_{33}, d_{31}, d_{15} \\ g_{33,} g_{31}, g_{15} \end{matrix},$ , $Ψ_{m} = {\begin{matrix} E_{3}, E_{1} D_{3}, D_{1} \end{matrix}$ $Ψ_{m} = {\begin{matrix} E_{3}, E_{1} \\ D_{3}, D_{1} \end{matrix}$ , $s_{i j}^{ψ} = {\begin{matrix} s_{33}^{E} & s_{11}^{E} & s_{55}^{E} s_{33}^{D} & s_{11}^{D} & s_{55}^{D} \end{matrix}$ $s_{i j}^{ψ} = {\begin{matrix} s_{33}^{E} & s_{11}^{E} & s_{55}^{E} \\ s_{33}^{D} & s_{11}^{D} & s_{55}^{D} \end{matrix}$ , $c^{Ψ} = {\begin{matrix} c^{E} c^{D} \end{matrix}$ $c^{Ψ} = {\begin{matrix} c^{E} \\ c^{D} \end{matrix}$ , $γ = {\begin{matrix} γ^{E} γ^{E} \end{matrix}$ $γ = {\begin{matrix} γ^{E} \\ γ^{E} \end{matrix}$ , $χ_{i j}^{ψ} = s_{i j}^{ψ} / S_{0}$ $χ_{i j}^{ψ} = s_{i j}^{ψ} / S_{0}$ .

The matrix function of a multi-layer drive has the form

$[\begin{matrix} Ξ_{1} (s) Ξ_{2} (s) \end{matrix}] = [\begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} ψ_{m} (s) F_{1} (s) F_{2} (s) \end{matrix} ⎤ ⎥ ⎦$ $[\begin{matrix} Ξ_{1} (s) \\ Ξ_{2} (s) \end{matrix}] = [\begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) \\ W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix}] [\begin{matrix} ψ_{m} (s) \\ F_{1} (s) \\ F_{2} (s) \end{matrix}]$

then

$[Ξ (s)] = [W (s)] [P (s)]$ $[Ξ (s)] = [W (s)] [P (s)]$

$[Ξ (s)] = [\begin{matrix} Ξ_{1} (s) Ξ_{2} (s) \end{matrix}]$ $[Ξ (s)] = [\begin{matrix} Ξ_{1} (s) \\ Ξ_{2} (s) \end{matrix}]$ , $[P (s)] = ⎡ ⎢ ⎣ \begin{matrix} Ψ_{m} (s) F_{1} (s) F_{2} (s) \end{matrix} ⎤ ⎥ ⎦$ $[P (s)] = [\begin{matrix} Ψ_{m} (s) \\ F_{1} (s) \\ F_{2} (s) \end{matrix}]$

$[W (s)] = [\begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix}]$ $[W (s)] = [\begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) \\ W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix}]$

The functions of a multi-layer drive are determined

$W_{11} (s) = Ξ_{1} (s) / ψ_{m} (s) = v_{m i} [M_{2} χ_{i j}^{ψ} s^{2} + γ t h (l γ / 2)] / A_{i j}$ $W_{11} (s) = Ξ_{1} (s) / ψ_{m} (s) = v_{m i} [M_{2} χ_{i j}^{ψ} s^{2} + γ t h (l γ / 2)] / A_{i j}$

$χ_{i j}^{ψ} = s_{i j}^{ψ} / S_{0}$ $χ_{i j}^{ψ} = s_{i j}^{ψ} / S_{0}$

$A_{i j} = \begin{matrix} M_{1} M_{2} {(χ_{i j}^{ψ})}^{2} s^{4} + {(M_{1} + M_{2}) (χ_{i j}^{ψ}) / [c^{ψ} t h (l γ)]} s^{3} + + [(M_{1} + M_{2}) χ_{i j}^{ψ} α / t h (l γ) + 1 / {(c^{ψ})}^{2}] s^{2} + 2 α s / c^{ψ} + α^{2} \end{matrix}$ $A_{i j} = \begin{matrix} M_{1} M_{2} {(χ_{i j}^{ψ})}^{2} s^{4} + {(M_{1} + M_{2}) (χ_{i j}^{ψ}) / [c^{ψ} t h (l γ)]} s^{3} + \\ + [(M_{1} + M_{2}) χ_{i j}^{ψ} α / t h (l γ) + 1 / {(c^{ψ})}^{2}] s^{2} + 2 α s / c^{ψ} + α^{2} \end{matrix}$

$W_{21} (s) = Ξ_{2} (s) / ψ_{m} (s) = - v_{i j}^{ψ} M_{1} χ_{i j}^{ψ} s^{2} + γ t h (l γ / 2) / A_{i j}$

$W_{12} (s) = Ξ_{1} (s) / F_{1} (s) = - χ_{i j}^{ψ} [M_{2} χ_{i j}^{ψ} s^{2} + γ t h (l γ)] / A_{i j}$

$\begin{array}{l} W_{13} (s) = Ξ_{1} (s) / F_{2} (s) = \\ = W_{22} (s) = Ξ_{2} (s) / F_{1} (s) = [χ_{i j}^{ψ} γ / s h (l γ)] / A_{i j} \end{array}$

$W_{23} (s) = Ξ_{2} (s) / F_{2} (s) = - χ_{i j}^{ψ} [M_{1} χ_{i j}^{ψ} s^{2} + γ / t h (l γ)] / A_{i j}$

The function of the multi-layer longitudinal piezo drive with one fixed face is determined at the elastic-inertial load and the control of voltage in the form

$W (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{d_{33} n}{(1 + C_{e} / C_{33}^{E}) (T_{t}^{2} s^{2} + 2 T_{t} ξ_{t} s + 1)}$

$T_{t} = \sqrt{M_{2} / (C_{e} + C_{33}^{E})}, ξ_{t} = α l^{2} C_{33}^{E} / [3 c^{E} \sqrt{M_{2} (C_{e} + C_{33}^{E})}]$

Its transient response has the form

$ξ (t) = ξ_{m} {(1 - \frac{e^{- \frac{ξ_{t} t}{T_{t}}}}{\sqrt{1 - ξ_{t}^{2}}} sin (ω_{t} t + ϕ_{t}))}_{}$

$ξ_{m} = \frac{d_{33} n U_{m}}{1 + C_{e} / C_{33}^{E}}$ , $ω_{t} = \sqrt{1 - ξ_{t}^{2}} / T_{t}$ , $ϕ_{t} = arctg (\sqrt{1 - ξ_{t}^{2}} / ξ_{t})$

For the multi-layer longitudinal PZT drive at $U_{m}$ =60 V, $d_{33}$ = 4∙10^-10 m/V, n= 8, M= 1 kg, $C_{33}^{E}$ = 5.8∙10⁷ N/m, $C_{e}$ = 0.6∙10⁷ N/m its parameters $ξ_{m}$ = 174 nm and $T_{t}$ = 1.25∙10^-4 s are obtained with error 10%.

Discussion

We have the parametric model and scheme of a multi-layer electro elastic drive of robotics by using of mathematical physics method. The system of equations for T-form quadripole of piezo layer is used. For a multi-layer electro elastic drive its system has the matrix form. By using the equations of forces on ends of a multi-layer drive and the equation of force causes deformation its parametric model and scheme, functions are determined.

Conclusion

A multi-layer electro elastic drive of robotics is used in adaptive optics, scanning microscopy, micro surgery, interferometry, nanotechnology, nano pump, nano stabilization, compensation of deformations. The parameters of the multi-layer longitudinal PZT drive at the control of voltage are obtained.

The characteristics in of the multi-layer piezo drive are obtained by applied of mathematical physics method. Future works are planned to explore the multi-layer piezo drive for adaptive optics of compound telescope.