Micropositioning device with multidegrees of freedom for piezoelectric actuators and associated method

Abstract

A micropositioning device for a piezoelectric actuator includes a means for controlling an electric field applied to the piezoelectric actuator so as to deform the piezoelectric material, and means for simultaneous measurement of a variation of electric charge accumulated on the piezoelectric actuator resulting from the deformation; and means for acquiring measurements of the variation of electric charge, for processing these acquisitions and for estimating a displacement (x, y, z) of the piezoelectric actuator and/or an applied force.

Claims

1. A micropositioning device for a piezoelectric actuator that includes at least one piezoelectric material capable of deforming when it is subjected to an electric field, the device comprising: means for controlling an electric field applied to the piezoelectric actuator so as to deform the piezoelectric material, and means for simultaneously measuring a variation in electric charge accumulated on the piezoelectric actuator as a result of the deformation; and means for acquiring measurements of the electric charge variation, for processing these acquisitions and for estimating a displacement (x, y, z) of the piezoelectric actuator and/or an applied force on the basis of the measurement of the variation in electric charge accumulated on the piezoelectric actuator, wherein the means for controlling and the means for measuring further comprises: a voltage generator connected in parallel to a divider bridge being composed of a first resistor and a second resistor in series, and to a first capacitor connected in series with the piezoelectric actuator, a charge amplifier having a first input connected to a node between the two resistors, forming a virtual ground at floating high potential, and a second input connected to a node between the first capacitor and the piezoelectric actuator, wherein the charge amplifier comprises an operational amplifier and a second capacitor connected between the second input and the output of the operational amplifier and the control and measurement means comprises a circuit for resetting to zero the charges present on the second capacitor.

2. The micropositioning device of claim 1, wherein the piezoelectric actuator comprises at least three electrodes, including at least one ground electrode and at least two active potential electrodes.

3. The micropositioning device of claim 2, wherein the piezoelectric actuator is a piezo tube comprising the piezoelectric material forming the tube, the ground electrode is arranged on an internal wall of the tube and at least two active potential electrodes are arranged on an external wall of the tube.

4. The micropositioning device of claim 1, wherein the means for controlling the electric field and the means for simultaneously measuring the variation in the electric charge comprise at least one circuit for controlling the electric field applied to the piezoelectric actuator and for measuring the variation of charge accumulated on the active potential electrodes.

5. The micropositioning device of claim 1, wherein the piezoelectric device, at rest, functions as a capacitor, the first capacitor and the first and second resistors are selected so that the ratio between the first and the second resistor is substantially equal to the ratio between the first capacitor and the capacitor of the piezoelectric device at rest.

6. The micropositioning device of claim 1, wherein the first capacitor is selected so as to have a leakage resistance of more than 100 Gohm.

7. The micropositioning device of claim 1, wherein the first capacitor is selected so that at least 50% of the voltage delivered by the voltage generator is found on the piezoelectric device.

8. The micropositioning device of claim 1, wherein the output of each charge amplifier is connected to an analog-digital converter connected to a computer via galvanic isolation circuits; the computer estimating the displacement of the actuator and/or the deformation of the materials and/or the applied force on the basis of the charge variation received from the charge amplifiers and the voltages applied by the voltage generators.

9. The micropositioning device of claim 1, wherein the piezoelectric device comprises at least three electrodes, including at least one ground electrode and at least two active potential electrodes, the device comprising as many control and measurement means as active electrodes, connected one by one, and the control and measurement means being connected to a single computer adapted to displace and measure the piezoelectric device in at least two dimensions.

10. A method for micropositioning at least one piezoelectric actuator that includes at least one piezoelectric material capable of deforming when it is subjected to an electric field and includes: means for controlling an electric field applied to the piezoelectric actuator so as to deform the piezoelectric material, and means for simultaneously measuring a variation in electric charge accumulated on the piezoelectric actuator as a result of the deformation; and means for acquiring measurements of the electric charge variation, for processing these acquisitions and for estimating a displacement (x, y, z) of the piezoelectric actuator and/or an applied force on the basis of the measurement of the variation in electric charge accumulated on the piezoelectric actuator, wherein the means for controlling and the means for measuring further includes: a voltage generator connected in parallel to a divider bridge being composed of a first resistor and a second resistor in series, and to a first capacitor connected in series with the piezoelectric actuator, a charge amplifier having a first input connected to a node between the two resistors, forming a virtual ground at floating high potential, and a second input connected to a node between the first capacitor and the piezoelectric actuator, wherein the charge amplifier comprises an operational amplifier and a second capacitor connected between the second input and the output of the operational amplifier and the control and measurement means comprises a circuit for resetting to zero the charges present on the second capacitor, comprising the following steps: controlling, by the control means, the piezoelectric actuator by applying an electric field to the piezoelectric actuator so as to deform the piezoelectric material; and simultaneously measuring, by simultaneous measurement means, a variation in electric charge accumulated on the piezoelectric actuator as a result of the deformation wherein at least one step includes acquiring the measurements of the electric charge variation, processing these acquisitions and of estimating (i) a displacement (x, y, z) of the piezoelectric actuator and/or (ii) an applied force on the basis of the measurement of the variation in electric charge accumulated on the piezoelectric actuator.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other characteristics, details and advantages of the invention will become apparent on reading the following description with reference to the appended figures, in which: FIG. 1 shows a functional diagram of the means for control and simultaneous measurement, and the means for acquisition, processing and estimation, according to one embodiment;

(2) FIG. 2 shows an outline diagram of an actuator according to this embodiment;

(3) FIG. 3 shows a general structural diagram of a device for micropositioning a piezoelectric actuator according to this embodiment;

(4) FIGS. 4a and 4b show diagrams of an electrical circuit associated with an electrode according to this embodiment;

(5) FIG. 5 shows a diagram of an overall electrical circuit associated with a piezoelectric actuator of the piezo tube type according to this embodiment;

(6) FIG. 6 shows a detailed block diagram of an observer, or estimator, according to this embodiment.

(7) For greater clarity, elements which are the same or similar in the various embodiments are denoted by the same references throughout the figures. Their description will not be systematically repeated from one embodiment to another.

DETAILED DESCRIPTION

(8) FIG. 1 shows a functional diagram of the means for control and simultaneous measurement, and the means for acquisition, processing and estimation, according to one embodiment.

(9) More precisely, this figure represents a device 100 for micropositioning a piezoelectric actuator 200, said piezoelectric actuator 200 comprising a piezoelectric material 201 capable of deforming when it is subjected to an electric field.

(10) In this embodiment, the piezoelectric actuator 200 is a piezo tube comprising the piezoelectric material 201 forming the tube, a ground electrode E.sub.m arranged on an internal wall 202 of the tube and four active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4 arranged on different sectors of an external wall 203 of the tube.

(11) The piezo tube is defined in an orthogonal reference frame with axes x, y and z, the z axis being the longitudinal axis of the tubular shape of said piezo tube.

(12) In such a configuration, the application of a voltage to these active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4, and therefore the application of an electric field, makes it possible to impart a longitudinal deformation of the various sectors of the piezoelectric material 201 forming the tube, leading to deflection along the x and y axes as well as contraction or elongation along the z axis.

(13) More precisely, the micropositioning device 100 has two main functions here: a first main function FP1 making it possible to control and simultaneously measure a variation in the charge on the various sectors of the piezo tube; and a second main function FP2 ensuring the acquisition, processing and estimation of the x, y and z displacements.

(14) Furthermore, the main function FP1 is subdivided into four secondary functions FS(E.sub.1), FS(E.sub.2), FS(E.sub.3), FS(E.sub.4), each associated with one active potential electrode E.sub.1, E.sub.2, E.sub.3, E.sub.4.

(15) Furthermore, each of the secondary functions FS(E.sub.1), FS(E.sub.2), FS(E.sub.3), FS(E.sub.4) is subdivided into two secondary functions per active potential electrode E.sub.1, E.sub.2, E.sub.3, E.sub.4. Thus, each of the four active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4 respectively fulfills: a first secondary function FS11, FS12, FS13, FS14 fulfilled by means for controlling an electric field applied to said piezoelectric actuator 200 so as to deform the piezoelectric material, this first function ensuring said control of the electric field at said piezoelectric actuator (200); and a second secondary function FS21, FS22, FS23, FS24 fulfilled by means for simultaneously measuring a variation in electric charge accumulated on the piezoelectric actuator 200 as a result of the deformation, this second function ensuring said simultaneous electric charge measurement.

(16) Furthermore, the second main function FP2 is fulfilled by means for acquiring the measurements of the electric charge variation, for processing these acquisitions and for estimating a displacement of the piezoelectric actuator 200 and/or an applied force on the basis of the measurement of the variation in electric charge accumulated on the piezoelectric actuator.

(17) FIG. 2 shows a diagram of an actuator according to this embodiment, here a piezo tube. This figure shows the piezoelectric material 201 forming the tube, the ground electrode E.sub.m arranged on the internal wall 202 of the tube and the active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4 arranged on different sectors of the external wall 203 of the tube (only three of the four active potential electrodes are represented here).

(18) Thus, in this figure, V.sub.1, V.sub.2, V.sub.3 and V.sub.4 respectively denote the electrical potentials applied to the active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4.

(19) FIG. 3 shows a general structural diagram of a device for micropositioning a piezoelectric actuator according to the same embodiment.

(20) More precisely represented in this electrical diagram of the micropositioning device are: an electrical circuit 210 making it possible to convert the charge variations Q.sub.i measured on the active potential electrode Ei into usable voltage V.sub.oi; an observer 220, or estimator, making it possible to provide the values x and y, which are the estimated values of the actual displacements x.sub.rel and Y.sub.rel at the end of the actuator, on the basis of the available signals, which are the usable voltages V.sub.oi and the control voltages V.sub.i; and a piezoelectric actuator 200;
where i is in this case an integer between 1 and 4, i being used to reference an electrode E.sub.i.

(21) In order to simplify the diagrams illustrated here, the calculation of the estimated value of the z displacement is not described in this case, but is of course estimated in a similar way to the x and y displacements.

(22) FIGS. 4a and 4b show diagrams of an electrical circuit associated with an electrode according to this embodiment.

(23) More precisely, FIG. 4a illustrates an electrical circuit of an electrode E.sub.i making it possible to fulfill both the first secondary function FS1i and the second secondary function FS2i, these two secondary functions being grouped as a secondary function associated with an electrode E and denoted by FS(E.sub.i).

(24) A partial electrical circuit is illustrated, which partially fulfills the main function FP2, here denoted by FP2p.

(25) Thus, FIG. 4a details the circuit diagram of a secondary function FS(E.sub.i) associated with an electrode E.sub.i, i being an integer between 1 and 4, applicable for a sector of the piezo tube.

(26) The index i representing the channel number will not be repeated in the rest of the explanation, in order to simplify the notation.

(27) Each sector of the piezo tube is denoted by “Piezo”, and is likened to a capacitor C.sub.p.

(28) A charge Q is applied to each sector of the actuator by means of the voltage V, via a capacitor C.sub.r. The latter is selected for its low losses. It is therefore considered as being linear and stable.

(29) Its value is such that most of the input voltage V is found on the piezoelectric actuator. A ratio of one tenth is satisfactory, although any other ratio is possible.

(30) The voltage V also supplies a divider bridge composed of R.sub.i1 and R.sub.i2 (i is again the channel number from 1 to 4), denoted below by R.sub.1 and R.sub.2, ensuring equilibrium between the two branches of the bridge if

(31) $\frac{R_1}{R_2}=\frac{C_p}{C_r}.$

(32) Any modifications of the capacitance C.sub.p, resulting from deformations of each sector, modify the equilibrium of the charges, which will then be measured by the charge amplifier composed of the circuit of the operational amplifier, denoted by AOP, and the capacitor C.

(33) The resistor R and the switch K, allow full resetting to zero of the charges present on the plates of the measurement capacitor C.

(34) The diagram of FIG. 4a may be likened to the diagram of FIG. 4b, where g.sub.i is a gain to be identified. This gain includes a gain in an opto-coupler, a gain in an analog-digital converter and possibly an adjustment gain in an integrator circuit.

(35) More precisely, the various signals and variables of FIG. 4b are listed below: V.sub.i is the control potential, V.sub.pi is the potential found on the electrode E.sub.i, V.sub.qi is the potential at the output of the operational amplifier AOP.sub.i, V.sub.oi is the usable potential found at the output of the electrical circuit, Q.sub.i indicates the charges appearing on the electrode E.sub.i, C.sub.ri is a capacitor for a divider bridge, C.sub.pi is an equivalent static capacitance of the piezoelectric actuator in its electrode E.sub.i part, C.sub.i is a capacitor for measuring variations in charges, R.sub.i1 and R.sub.i2 are two resistors for the divider bridge at the input of the operational amplifier AOP.sub.i, R.sub.i is a discharge resistance, g.sub.i is the gain.

(36) In this embodiment, the operation of the device for micropositioning the piezoelectric actuator is governed by electrical equations and piezoelectric equations. These equations are subsequently used in order to derive the observer, or the estimator.

(37) We recall that, in FIG. 4b which illustrates the electrical diagram for the electrode i, V.sub.pi is the voltage across the terminals of the actuator, electrode i part, V.sub.ci is the voltage across the terminals of the capacitor C.sub.i, i.sub.i is the current which flows through the capacitor C.sub.ri, i.sub.ci is the current which flows through the capacitor C.sub.i, and i.sub.pzti is the current which flows through the actuator via the electrode i.

(38) The electrical equations are then more precisely: the equations of the input voltages; the equation of the voltage c; and the equation of the usable output voltage V.sub.oi.

(39) More precisely, the equations of the input voltages are defined by the relations between the voltages at the input of the operational amplifier, including the resistive divider, which are as follows:

(40) $\begin{array}{cc}\{\begin{array}{c}{V}_{\mathrm{pi}}=V_{+}\\ V_{+}=V_{-}\\ V_{-}=\frac{R_{i2}}{\left(R_{i1}+R_{i2}\right)}{V}_i\end{array}& \left[1\right]\end{array}$
where V.sub.+ and V.sub.− are the voltages at the input of the operational amplifier.

(41) Furthermore, the equation of the voltage V.sub.ci is linked with the charge Q.sub.ci on the capacitor C.sub.i by the following relation:

(42) $\begin{array}{cc}{V}_{\mathrm{ci}}=\frac{1}{C_i}{Q}_{\mathrm{ci}}=\frac{1}{C_i}{\int }_0^t{i}_{\mathrm{ci}}dt& \left[2\right]\end{array}$
Now:
i.sub.ci=i.sub.i−i.sub.pzti−i.sub.biasi   [3]
where i.sub.biasi is the leakage current in the operational amplifier.

(43) Furthermore, the current in the relation linking the current i.sub.pzt in the actuator and the charges thereon is:
∫.sub.0.sup.ti.sub.pztidt=Q.sub.defi+G.sub.DAi+∫.sub.0.sup.ti.sub.leakidt   [4]
where Q.sub.defi is the charge generated by the application of the voltage V.sub.pi and by the deformation of the actuator, Q.sub.DAi is the charge due to the dielectric absorption of the material (201) and i.sub.leaki is the leakage current. These quantities may have positive or negative gains. The leakage current i.sub.leaki is linked with the leakage resistance R.sub.fpi in the actuator as follows:

(44) $\begin{array}{cc}{i}_{\mathrm{leaki}}=\frac{V_{\mathrm{pi}}}{R_{\mathrm{fpi}}}& \left[5\right]\end{array}$

(45) By using Equations 2, 3, 4 and 5, it is possible to deduce therefrom the voltage V.sub.ci:

(46) $\begin{array}{cc}{V}_{\mathrm{ci}}=\frac{1}{C_i}\left({\int }_0^t{i}_{i}dt-Q_{\mathrm{defi}}-Q_{\mathrm{DAi}}-{\int }_0^t\frac{V_{\mathrm{pi}}}{R_{\mathrm{fpi}}}dt-{\int }_0^t{i}_{\mathrm{biasi}}dt\right)& \left[6\right]\end{array}$

(47) Now, according to FIG. 4a, the currents i.sub.i are as follows:

(48) $\begin{array}{cc}{i}_i=C_{\mathrm{ri}}\frac{d\left(V_i-V_{\mathrm{pi}}\right)}{dt}\iff {\int }_0^t{i}_{i}dt=C_{\mathrm{ri}}\left(V_i-V_{\mathrm{pi}}\right)& \left[7\right]\end{array}$
and according to equation 1 the voltage V.sub.pi is:

(49) $\begin{array}{cc}{V}_{\mathrm{pi}}=\frac{R_{i2}}{\left(R_{i1}+R_{i2}\right)}{V}_i& \left[8\right]\end{array}$
which, using Equations 7 and 8, leads to:

(50) $\begin{array}{cc}{\int }_0^t{i}_{i}dt=\frac{C_{\mathrm{ri}}{R}_{i1}}{\left(R_{i1}+R_{i2}\right)}{V}_i& \left[9\right]\end{array}$

(51) By using Equations 6, 8 and 9, it is possible to deduce the final equation of the voltage V.sub.ci:

(52) $\begin{array}{cc}{V}_{\mathrm{ci}}=\frac{1}{C_i}\left(\frac{C_{\mathrm{ri}}+R_{i1}}{\left(R_{i1}+R_{i2}\right)}{V}_i-Q_{\mathrm{defi}}-Q_{\mathrm{DAi}}-\frac{R_{i2}}{R_{\mathrm{fpi}}\left(R_{i1}+R_{i2}\right)}{\int }_0^t{V}_{i}dt-{\int }_0^t{i}_{\mathrm{biasi}}dt\right)& \left[10\right]\end{array}$

(53) Furthermore, the equation of the usable output voltage V.sub.oi may be deduced from the equations linking the voltages V.sub.oi, V.sub.ci and V.sub.pi, which are as follows:

(54) 0$\begin{array}{cc}\{\begin{array}{c}\frac{V_{\mathrm{oi}}}{g_i}=V_{\mathrm{qi}}\\ V_{\mathrm{qi}}+V_{\mathrm{ci}}-V_{+}=0\end{array}.Math.V_{\mathrm{oi}}=g_i\left(V_{+}-V_{\mathrm{ci}}\right)& \left[11\right]\end{array}$

(55) By using Equations I and II, the voltage V.sub.oi can be defined as follows:

(56) $\begin{array}{cc}{V}_{\mathrm{oi}}=g_i\left(\frac{R_{i2}}{\left(R_{i1}+R_{i2}\right)}{V}_i-V_{\mathrm{ci}}\right)& \left[12\right]\end{array}$

(57) By combining Equations 10 and 12, the following equation of the usable voltage V.sub.oi is obtained:

(58) $\begin{array}{cc}{V}_{\mathrm{oi}}=g_i\left[\frac{R_{i2}}{\left(R_{i1}+R_{i2}\right)}{V}_i-\frac{1}{C_i}\left(\frac{C_{\mathrm{ri}}{R}_{i1}}{\left(R_{i1}+R_{i2}\right)}{V}_i-Q_{\mathrm{defi}}-Q_{\mathrm{DAi}}-\frac{R_{i2}}{R_{\mathrm{fpi}}\left(R_{i1}+R_{i2}\right)}{\int }_0^t{V}_{i}dt-{\int }_0^t{i}_{\mathrm{biasi}}dt\right)\right]& \left[13\right]\end{array}$

(59) As regards the piezoelectric equations, these are more precisely the equations making it possible to calculate the charges on the electrodes.

(60) The relation which links the x deflection at the end of the piezoelectric actuator, here of the piezo tube type, and the voltages V.sub.1 and V.sub.3 respectively applied to the electrodes 1 and 3 is given by the following equation:

(61) $\begin{array}{cc}x={\mathrm{aV}}_1-{\mathrm{aV}}_3\iff \left(V_1-V_3\right)=\frac{x}{a}& \left[14\right]\end{array}$
where a is the piezoelectric coefficient for a unipolar control, that is to say U.sub.1 is not necessarily equal to −U.sub.3. This coefficient is available in numerous articles, such as the article “Introduction to Scanning Tunneling Microscopy” by C. J. Chen and published in the review Oxford University Press in 1993. The same relation is obtained for the y axis:

(62) $\begin{array}{cc}y={\mathrm{aV}}_2-{\mathrm{aV}}_4\iff \left(V_2-V_4\right)=\frac{y}{a}& \left[15\right]\end{array}$

(63) Furthermore, by assuming the charges Q.sub.def1 and Q.sub.def3: on the electrodes to be identical (a.sub.1=a.sub.3=a) and antagonistic 1 and 3 for the x axis; and on the electrodes to be identical (β.sub.2=β.sub.4=β) and antagonistic 2 and 4 for the y axis;
written parametrically with respect to the voltages U.sub.i (i here being an integer between 1 and 4), and taking the symmetry of the actuator into account, we obtain:

(64) $\begin{array}{cc}\{\begin{array}{c}{Q}_{\mathrm{def}1}=\alpha V_1-\alpha V_3+\beta V_2-\beta V_4=\alpha \left(V_1-V_3\right)+\beta \left(V_2-V_4\right)\\ Q_{\mathrm{def}3}=-\alpha V_1-\alpha V_3+\beta V_2-\beta V_4=-\alpha \left(V_1-V_3\right)+\beta \left(V_2-V_4\right)\end{array}& \left[16\right]\end{array}$

(65) The same equation can be written for the charges Q.sub.def2 and Q.sub.def4 for the y axis:

(66) $\begin{array}{cc}\{\begin{array}{c}{Q}_{\mathrm{def}2}=\alpha V_2-\alpha V_4+\beta V_1-\beta V_3=\alpha \left(V_2-V_4\right)+\beta \left(V_1-V_3\right)\hspace{1em}\\ Q_{\mathrm{def}4}=-\alpha V_2+\alpha V_4+\beta V_1-\beta V_3=-\alpha \left(V_2-V_4\right)+\beta \left(V_1-V_3\right)\end{array}\hspace{1em}& \left[17\right]\end{array}$

(67) By introducing Equations 14 and 15 into Equations 16 and 17, we obtain the equations of the charges on the electrodes:

(68) $\begin{array}{cc}\{\begin{array}{c}\{\begin{array}{c}{Q}_{\mathrm{def}1}=\alpha \frac{x}{a}+\beta \frac{y}{a}\\ Q_{\mathrm{def}3}=-\alpha \frac{x}{a}+\beta \frac{y}{a}\end{array}\\ \{\begin{array}{c}{Q}_{\mathrm{def}2}=\alpha \frac{y}{a}+\beta \frac{x}{a}\\ Q_{\mathrm{def4}}=-\alpha \frac{y}{a}+\beta \frac{x}{a}\end{array}\end{array}& \left[18\right]\end{array}$

(69) By combining the electrical and piezoelectric equations, that is to say by introducing the equations of the charges on the electrodes (Equation 18) into the equation of the usable voltage V.sub.oi (Equation 13), we obtain:

(70) $\begin{array}{cc}\frac{V_{o1}}{g_1}=\frac{R_{12}}{\left(R_{11}+R_{12}\right)}{V}_1-\frac{C_{r1}{R}_{11}}{C_1\left(R_{11}+R_{12}\right)}{V}_1+\frac{1}{C_1}\left(\alpha \frac{x}{a}+\beta \frac{y}{a}+Q_{\mathrm{DA}1}+\frac{R_{12}}{R_{\mathrm{fp}1}\left(R_{11}+R_{12}\right)}{\int }_0^t{V}_{1}dt+{\int }_0^t{i}_{\mathrm{bias}1}dt\right)\frac{V_{o3}}{g_3}=\frac{R_{32}}{\left(R_{31}+R_{32}\right)}{V}_3-\frac{C_{r3}{R}_{31}}{C_2\left(R_{31}+R_{32}\right)}{V}_3+\hspace{1em}\frac{1}{C_3}\left(-\alpha \frac{x}{a}+\beta \frac{y}{a}+Q_{\mathrm{DA}3}+\frac{R_{32}}{R_{\mathrm{fp}3}\left(R_{31}+R_{32}\right)}{\int }_0^t{V}_{3}dt+{\int }_0^t{i}_{\mathrm{bias}3}dt\right)\frac{V_{o2}}{g_2}=\frac{R_{22}}{\left(R_{21}+R_{22}\right)}{V}_2-\frac{C_{r2}{R}_{21}}{C_3\left(R_{21}+R_{22}\right)}{V}_2+\hspace{1em}\frac{1}{C_2}\left(\alpha \frac{y}{a}+\beta \frac{x}{a}+Q_{\mathrm{DA}2}+\frac{R_{22}}{R_{\mathrm{fp}3}\left(R_{21}+R_{22}\right)}{\int }_0^t{V}_{2}dt+{\int }_0^t{i}_{\mathrm{bias}2}dt\right)\frac{V_{o4}}{g_4}=\frac{R_{42}}{\left(R_{41}+R_{42}\right)}{V}_4-\frac{C_{r4}{R}_{41}}{C_4\left(R_{41}+R_{42}\right)}{V}_4+\frac{1}{C_4}\left(-\alpha \frac{y}{a}+\beta \frac{x}{a}+Q_{\mathrm{DA}4}+\frac{R_{42}}{R_{\mathrm{fp}4}\left(R_{41}+R_{42}\right)}{\int }_0^t{V}_{4}dt+{\int }_0^t{i}_{\mathrm{bias}4}dt\right)& \left[19\right]\end{array}$

(71) Assuming the following equations:

(72) $\begin{array}{cc}\{\begin{array}{c}{C}_1=C_2=C_3=C_4=C\\ g_1=g_2=g_3=g_4=g\\ {\gamma }_i=\frac{R_{i2}}{\left(R_{i1}+R_{i2}\right)}-\frac{C_{\mathrm{ri}}{R}_{i1}}{C_i\left(R_{i1}+R_{i2}\right)}\\ {\lambda }_i=\frac{R_{i2}}{R_{\mathrm{fpi}}\left(R_{i1}+R_{i2}\right)}\end{array}& \left[20\right]\end{array}$
we obtain the equations given below:

(73) 0$\begin{array}{cc}\frac{V_{o1}}{g}={\gamma }_1{V}_1+\frac{1}{C}\left(\alpha \frac{x}{a}+\beta \frac{y}{a}+Q_{\mathrm{DA}1}+{\lambda }_1{\int }_0^t{V}_{1}dt+{\int }_0^t{i}_{\mathrm{bias}1}dt\right)\frac{V_{o3}}{g}={\gamma }_3{V}_3+\frac{1}{C}\left(-\alpha \frac{x}{a}+\beta \frac{y}{a}+Q_{\mathrm{DA}3}+{\lambda }_3{\int }_0^t{V}_{3}dt+{\int }_0^t{i}_{\mathrm{bias}3}dt\right)\frac{V_{o2}}{g}={\gamma }_2{V}_2+\frac{1}{C}\left(\alpha \frac{y}{a}+\beta \frac{x}{a}+Q_{\mathrm{DA}2}+{\lambda }_2{\int }_0^t{V}_{2}dt+{\int }_0^t{i}_{\mathrm{bias}2}dt\right)\frac{V_{o4}}{g}={\gamma }_4{V}_4+\frac{1}{C}\left(-\alpha \frac{y}{a}+\beta \frac{x}{a}+Q_{\mathrm{DA}4}+{\lambda }_4{\int }_0^t{V}_{4}dt+{\int }_0^t{i}_{\mathrm{bias}4}dt\right)& \left[21\right]\end{array}$

(74) Lastly, on the basis of Equation 21, it is possible to calculate the differences (V.sub.o1−V.sub.o3) and (V.sub.o2−V.sub.o4) as follows:

(75) $\begin{array}{cc}\{\begin{array}{c}{V}_{o1}-V_{o3}=g\left({\gamma }_1{V}_1-{\gamma }_3{V}_3\right)+\frac{2g\alpha }{\mathrm{aC}}x+\frac{g}{C}\left(Q_{\mathrm{DA}1}-Q_{\mathrm{DA}3}\right)+\\ \frac{g}{C}{\int }_0^t\left(i_{\mathrm{bias}1}-i_{\mathrm{bias}3}\right)dt+\frac{g}{C}{\int }_0^t\left({\lambda }_1{V}_1-{\lambda }_2{V}_2\right)dt\\ V_{o2}-V_{o4}=g\left({\gamma }_2{V}_2-{\gamma }_4{V}_4\right)+\frac{2g\alpha }{\mathrm{aC}}y+\frac{g}{C}\left(Q_{\mathrm{DA}2}-Q_{\mathrm{DA}4}\right)+\\ \frac{g}{C}{\int }_0^t\left(i_{\mathrm{bias}2}-i_{\mathrm{bias}4}\right)dt+\frac{g}{C}{\int }_0^t\left({\lambda }_2{V}_2-{\lambda }_4{V}_4\right)dt\end{array}& \left[22\right]\end{array}$

(76) It is furthermore assumed that the charge Q.sub.DAi(t) due to the dielectric absorption can be approximated by a first-order system. In Laplace space, this gives:

(77) $\begin{array}{cc}{Q}_{\mathrm{DAi}}\left(s\right)=\frac{k_{\mathrm{DAi}}}{\left(1+{\tau }_{\mathrm{DAi}}s\right)}{V}_i\left(s\right)=Q_{\mathrm{fDAi}}\left(s\right)V_i\left(s\right)& \left[23\right]\end{array}$

(78) where s is the Laplace variable, k.sub.DAi is the static gain (which may be positive or negative in our case) and τ.sub.DAi is the time constant. Q.sub.tfDAi(s) is the transfer function which links the charge Q.sub.DAi(s) with the input voltage V.sub.i(s).

(79) Consequently, by virtue of all these equations, it is thus possible to deduce the equations of the observer, or estimator. In particular, on the basis of Equation 22, the x and y displacements are governed by the following equations:

(80) $\begin{array}{cc}\{\begin{array}{c}x=\frac{a}{2\alpha }[\frac{C}{g}\left(V_{o1}-V_{o3}\right)-C\left({\gamma }_1{V}_1-{\gamma }_3{V}_3\right)-\left(Q_{\mathrm{DA}1}-Q_{\mathrm{DA}3}\right)-\\ {\int }_0^t\left(i_{\mathrm{bias}1}-i_{\mathrm{bias}3}\right)dt-{\int }_0^t\left({\lambda }_1{V}_1-{\lambda }_3{V}_3\right)dt]\\ y=\frac{a}{2\alpha }[\frac{C}{g}\left(V_{o2}-V_{o4}\right)-C\left({\gamma }_2{V}_2-{\gamma }_4{V}_4\right)-\left(Q_{\mathrm{DA}2}-Q_{\mathrm{DA}4}\right)-\\ {\int }_0^t\left(i_{\mathrm{bias}2}-i_{\mathrm{bias}4}\right)dt-{\int }_0^t\left({\lambda }_2{V}_2-{\lambda }_4{V}_4\right)dt]\end{array}& \left[24\right]\end{array}$

(81) FIG. 5 shows a diagram of an overall electrical circuit associated with a piezoelectric actuator of the piezo tube type according to this embodiment.

(82) Specifically, this figure represents four circuit diagrams of the secondary functions FS(E.sub.1), FS(E.sub.2), FS(E.sub.3), FS(E.sub.4) as illustrated in FIG. 4a, each respectively associated with the active potential electrodes E.sub.1, E.sub.2, E.sub.3, E.sub.4 arranged on various sectors of the external wall 203 of the piezoelectric material 201 forming the tube.

(83) Furthermore, each of these electrical circuits fulfilling said secondary functions FS(E.sub.1), FS(E.sub.2), FS(E.sub.3), FS(E.sub.4) is connected in series with another electrical circuit which, for its part, associated with a control unit, fulfills the second main function FP2, in particular ensuring the acquisition, processing and estimation of the x, y displacements, the values x and y being the estimated values of the actual displacements x.sub.rel and y.sub.rel.

(84) In order to simplify the diagrams illustrated here, the calculation of the estimated value of the z displacement is not described, but is of course estimated in a similar way.

(85) FIG. 6 shows a detailed block diagram of an observer, or estimator, according to this embodiment, by which the mathematical operations carried out as described above make it possible to measure at the output the x and y displacements which are governed by Equation 24.

(86) Numerous modifications may be made to the particular embodiment described above without departing from the scope of the invention.

(87) For instance, the number of electrodes may vary as a function of the number of measurements desired.

(88) Furthermore, the electrical diagrams may differ without departing from the scope of the invention, so long as the electrical circuits are suitable for fulfilling the same functions and aim to achieve equivalent results.