METHOD FOR CALIBRATING THE DIFFERENCE IN STIFFNESS AND/OR QUADRATURE OF A VIBRATING INERTIAL SENSOR

Abstract

A method for calibrating the stiffness mismatch ΔK or quadrature Kxy of a vibrating angular sensor includes a resonator extending about two axes x and y defining a sensor frame xy, comprising a vibrating proof mass comprising two parts configured to vibrate in phase opposition with respect to each other in a direction x′ defining a wave frame x′y′, the direction x′ making an electrical angle to the axis x; and detection, excitation, quadrature compensation and stiffness adjustment transducers; the resonator having a stiffness matrix K.sub.C in the sensor frame and a stiffness matrix K.sub.O in the wave frame; the method comprising steps of: A determining the electrical angle; B recovering a quadrature or stiffness term of the stiffness matrix K.sub.O in the wave frame, the term being a sum of functions in cos(iθ) and sin(iθ); steps A and B being reiterated either for a plurality of electrical angles (θ.sub.k), or for a duration during which the vibration wave continuously rotates through an electrical angle (θ(t)) varying as a function of time; C determining the amplitudes of the functions in cos(iθ) and sin(iθ); then D determining the stiffness mismatch ΔK or the quadrature Kxy, on the basis of the amplitudes.

Claims

1. A method for calibrating a stiffness mismatch ΔK and/or a quadrature Kxy of a vibrating inertial angular sensor, the inertial sensor comprising a resonator (Res) extending about two mutually perpendicular axes x and y defining a sensor frame xy and comprising: at least one vibrating proof mass (M1), said at least one proof mass comprising at least two parts configured to vibrate in phase opposition with respect to each other at a vibration angular frequency (ω) and in a direction x′ defining a wave frame x′y′, the vibration wave (OV) along x′ making an electrical angle (θ) to the axis x; a plurality of electrostatic transducers controlled by electrical voltages and operating along the two axes x or y, including at least: a pair of detection transducers (Dx, Dy) configured to detect the movements of the vibration wave along the axis x and the axis y; a pair of excitation transducers (Ex, Ey) to which excitation forces are applied along the axis x and the axis y, respectively, via a plurality of excitation commands determined by servos on the basis of the detected movements, these being configured to maintain the wave at a constant amplitude via an amplitude command (Ca) and, where appropriate, to rotate said vibration wave via a precession command (Cp) and a command (Cq) for controlling the quadrature of the wave; a pair of quadrature compensation transducers (Q+, Q−), controlled via a quadrature command (CTq); and a pair of stiffness adjustment transducers (Tx, Ty), controlled via a stiffness command (CTx) setting stiffness along the axis x and a stiffness command (CTy) setting stiffness along the axis y, respectively, forming a stiffness command (CTf); the resonator (Res) having a stiffness matrix K.sub.C in the sensor frame and a stiffness matrix K.sub.O in the wave frame; the calibrating method being applied when the inertial sensor is in operation with a vibration wave (OV) vibrating in the direction x′; the calibrating method comprising the steps of: A determining the electrical angle (θ.sub.k, θ(t)); B recovering at least one term of the stiffness matrix K.sub.O in the wave frame x′y′, which may be a quadrature term K.sub.O(2,1) or a stiffness term K.sub.O(1,1), said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated either for a plurality of electrical angles (θ.sub.k), where k is an integer varying between 1 and m, m being greater than or equal to 2, or for a duration (T) during which the vibration wave (OV) rotates continuously through an electrical angle (θ(t)) varying as a function of time (t); then C determining the amplitudes of the functions in cos(iθ) and sin(iθ); and D determining the stiffness mismatch ΔK and the quadrature Kxy, respectively, on the basis of the determined amplitudes.

2. The calibrating method as claimed in claim 1, the electrical angle describing a plurality of electrical angles (θ.sub.k) where k is an integer varying between 1 and m, m being greater than or equal to 2, step C of determining amplitudes comprising applying a least squares filter to the recovered term, the quantity (m) of electrical angles (θ.sub.k) being at least equal to the number of amplitudes to be determined.

3. The calibrating method as claimed in claim 1, the vibration wave (OV) continuously rotating through an electrical angle (θ(t)) varying as a function of time (t) for a duration (T), step C of determining amplitudes comprising demodulating the recovered term in cos(iθ) and sin(iθ) for each i varying between 1 and n, so as to determine the amplitudes of the functions in cos(iθ) and sin(iθ); and step D of determining the stiffness mismatch ΔK and the quadrature Kxy, respectively, is carried out on the basis of the determined amplitudes.

4. The calibrating method as claimed in claim 1, wherein the inertial sensor operates in gyrometer mode, the electrical angle (θ.sub.k, (θ(t)) determined in step A being equal to an imposed angle (θ.sub.k_imp, (θ(t).sub.imp) set via the precession command (Cp).

5. The calibrating method as claimed in claim 1, wherein the inertial sensor operates in gyroscope mode, the electrical angle (θ.sub.k, (θ(t)) resulting from a rotation of the inertial sensor being measured by said inertial sensor, potentially superimposed on a precession command (Cp), the electrical angle determined in step A being equal to said measured angle of rotation (θ.sub.k_m, (θ(t).sub.m).

6. The calibrating method as claimed in claim 1, further comprising an additional step of: E applying a stiffness command (CTf) and a quadrature command (CTq), respectively, on the basis of the stiffness mismatch ΔK and the quadrature Kxy determined in step D, respectively.

7. The calibrating method as claimed in claim 6, steps A to E being able to be included in a closed loop servo or being able to be implemented in open loop iteratively, and preferably twice.

8. An inertial angular sensor comprising a resonator (Res) extending about two mutually perpendicular axes x and y defining a sensor frame xy, and comprising: at least one vibrating proof mass (M1), said at least one proof mass comprising at least two parts configured to vibrate in phase opposition with respect to each other at a vibration angular frequency (ω) and in a direction x′ defining a wave frame x′y′, the vibration wave (OV) along x′ making an electrical angle (θ) to the axis x; a plurality of electrostatic transducers controlled by electrical voltages and operating along the two axes x or y, including at least: a pair of detection transducers (Dx, Dy) configured to detect the movements of the vibration wave along the axis x and the axis y; a pair of excitation transducers (Ex, Ey) to which excitation forces are applied along the axis x and the axis y, respectively, via a plurality of excitation commands determined by servos on the basis of the detected movements, these being configured to maintain the wave at a constant amplitude via an amplitude command (Ca) and, where appropriate, to rotate said vibration wave via a precession command (Cp) and a command (Cq) for controlling the quadrature of the wave; a pair of quadrature compensation transducers (Q+, Q−), controlled via a quadrature command (CTq); and a pair of stiffness adjustment transducers (Tx, Ty), controlled via a stiffness command (CTx) setting stiffness along the axis x and a stiffness command (CTy) setting stiffness along the axis y, respectively, thus forming a stiffness command (CTf), the resonator (Res) having a stiffness matrix K.sub.C in the sensor frame and a stiffness matrix K.sub.O in the wave frame; the inertial angular sensor further comprising: a means for determining the electrical angle (θ.sub.k, θ(t)); a means for recovering at least one term of the stiffness matrix K.sub.O in the wave frame x′y′, which may be a quadrature term K.sub.O(2,1) or a stiffness term K.sub.O(1,1); and a processing unit (UT) configured to implement at least steps A to D, and optionally step E, of the chosen calibrating method according to claim 1; the stiffness adjustment transducers (Tx, Ty) and quadrature bias compensation transducers (Q+, Q−), respectively, being configured to apply said stiffness command (CTf) and said quadrature command (CTq), respectively, to the resonator.

9. The inertial angular sensor as claimed in claim 8, said inertial sensor being axisymmetric.

10. The inertial angular sensor as claimed in claim 8, comprising at least two vibrating proof masses (M1, M2) forming the at least two parts configured to vibrate in phase opposition with respect to each other, one proof mass possibly being arranged about another proof mass.

11. A method for measuring an angular velocity or an angular orientation of a carrier on which is arranged an inertial sensor as claimed in claim 8, the measuring method comprising: calibrating a stiffness mismatch ΔK and/or a quadrature Kxy of said inertial sensor, the calibrating being applied when the inertial sensor is in operation with a vibration wave (OV) vibrating in the direction x′; the calibrating comprising the steps of: A determining the electrical angle (θ.sub.k, θ(t)); B recovering at least one term of the stiffness matrix KO in the wave frame x′y′, which may be a quadrature term KO(2,1) or a stiffness term KO(1,1), said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated either for a plurality of electrical angles (θ.sub.k), where k is an integer varying between 1 and m, m being greater than or equal to 2, or for a duration (T) during which the vibration wave (OV) rotates continuously through an electrical angle (θ(t)) varying as a function of time (t); then C determining the amplitudes of the functions in cos(iθ) and sin(iθ); and D determining the stiffness mismatch ΔK and the quadrature Kxy, respectively, on the basis of the determined amplitudes; and measuring angular velocity or angular orientation, the inertial sensor being used in gyrometer mode or in gyroscope mode.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0089] Other features, details and advantages of the invention will become apparent on reading the description given with reference to the appended drawings, which are given by way of example and which show, respectively:

[0090] FIG. 1 (already cited) illustrates the axisymmetric resonator of a MEMS sensor according to the prior art, this sensor being based on two vibrating proof masses arranged one about the other.

[0091] FIG. 2 (already cited) illustrates the structure of a MEMS sensor according to the prior art with a resonator that is axisymmetric about two axes x and y defining a sensor frame.

[0092] FIG. 3 (already cited) illustrates a MEMS sensor according to the prior art with transducers on both masses.

[0093] FIG. 4 (already cited) illustrates the operation of an inertial sensor according to the prior art.

[0094] FIG. 5 illustrates a first embodiment of a calibrating method according to the invention.

[0095] FIG. 6 illustrates a second embodiment of a calibrating method according to the invention.

[0096] FIG. 7 illustrates a third embodiment of a calibrating method according to the invention.

[0097] FIG. 8 illustrates a fourth embodiment of a calibrating method according to the invention.

[0098] In all of these figures, identical references may designate identical or similar elements.

[0099] In addition, the various parts shown in the figures are not necessarily shown to a uniform scale, to make the figures more readable.

DETAILED DESCRIPTION OF THE INVENTION

[0100] The calibrating method according to the invention applies to an inertial angular sensor comprising a resonator Res associated with means for vibrating the resonator and with means for detecting an orientation of the vibration (vibration wave) with respect to a frame of the sensor, for example excitation transducers E and detection transducers D, which are controlled with excitation commands (E), and trimming transducers TF and TQ controlled with trim commands (TF, TQ).

[0101] The invention may in particular be applied to one of the sensors presented above, in connection with FIGS. 1 to 3, or to sensors according to the variants also described above (at least one mass or at least two masses, axisymmetric or non-axisymmetric sensor, planar or non-planar structure, the MEMS sensor being one example of embodiment).

[0102] Furthermore, reference may also be made to the general operation of FIG. 4, the processing unit UT being configured to apply the steps of the method according to the invention. It may be a question of one or more modules added to the UT with a view to performing the steps of the calibrating method according to the invention.

[0103] The vibration wave OV vibrates at an angular frequency of vibration ω. The calibrating method according to the invention is applicable to an inertial sensor operating in gyrometer mode or in gyroscope mode, the excitation commands being servo-controlled in operation accordingly.

[0104] In a vibrating inertial angular sensor, in particular one that is axisymmetric, there are two frames: the sensor frame xy the axes x and y of which are the axes containing the excitation and detection transducers of the sensor, and the wave frame x′y′ in which the axis x′ is the axis of vibration of the wave OV and the axis y′ is the axis perpendicular to x′ in the plane of the sensor. The axis x′ makes to the axis x an angle θ denoted the “electrical angle” and the frame x′y′ is called the “wave frame”.

[0105] In a vibrating inertial angular sensor, a source of drift is generated by stiffness or quadrature errors multiplied by phase errors. It will be recalled that a stiffness error corresponds to a stiffness mismatch between the vibration axis and the axis perpendicular to the vibration, and that a quadrature error results from mechanical coupling between the axis of the vibration and the axis perpendicular to the vibration.

[0106] A vibrating inertial angular sensor including trimming transducers possesses actuators that allow stiffness and quadrature corrections to be made. However, there remain residual stiffness mismatches ΔK and/or residual quadratures Kxy, despite the stiffness and quadrature trim commands.

[0107] One objective of the invention is to eliminate, or at least decrease to values acceptable to the operator, stiffness mismatch and quadrature, so as to make the sensor less or even more sensitive to phase errors.

[0108] The calibrating method according to the invention may be applied to an inertial angular sensor operating in gyrometer mode, or may be applied to an inertial sensor operating in gyroscope mode. In the case of a gyroscope, each angle θ is an angle measured or experienced (as a result of the change in angle of the carrier), whereas with a gyrometer, the angle θ is set via a wave rotation command. Both in gyrometer mode and in gyroscope mode, the method according to the invention requires a plurality of different angles θ to be used.

[0109] A vibrating gyrometer possesses a stiffness matrix having the following form K.sub.C in the sensor frame (given here without/before use of trimming commands):

[00007]$\begin{array}{cc}{K}_C=\left[\begin{array}{cc}Kx& Kxy\\ Kxy& Ky\end{array}\right]& \left[\mathrm{Math}.11\right]\end{array}$

where Kx, Ky are the stiffnesses along the axes x and y in the sensor frame and Kxy is the quadrature in the sensor frame.

[0110] The stiffness matrix K.sub.O in the wave frame (making an angle θ to the sensor frame) is given by:

[00008]$\begin{array}{cc}{K}_O=\left[\begin{array}{ccc}K& +\left(\Delta K\right)\cos \left(2\theta \right)-\left(K\mathrm{xy}\right)\sin \left(2\theta \right)& \left(\Delta K\right)\sin \left(2\theta \right)+\left(K\mathrm{xy}\right)\cos \left(2\theta \right)\\ & \left(\Delta K\right)\sin \left(2\theta \right)+\left(K\mathrm{xy}\right)\cos \left(2\theta \right)& K-\left(\Delta K\right)\cos \left(2\theta \right)+\left(K\mathrm{xy}\right)\sin \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}.12\right]\end{array}$$\begin{array}{cc}K=\frac{Kx+Ky}{2}& \left[\mathrm{Math}.13\right]\end{array}$$\begin{array}{cc}\Delta K=\frac{Kx-Ky}{2}& \left[\mathrm{Math}.14\right]\end{array}$

[0111] With the trimming commands TFx, TFy (stiffness) and TQ (quadrature), the stiffness matrix K.sub.C is modified directly using a trimming matrix Kt in the sensor frame, where:

[00009]$\begin{array}{cc}{K}_t=\left[\begin{array}{cc}Ktx& Kq\\ Kq& Kty\end{array}\right]& \left[\mathrm{Math}.15\right]\end{array}$

[0112] The matrix K then becomes K.sub.mod in the sensor frame:

[00010]$\begin{array}{cc}{K}_{\mathrm{mod}}=K_C-K_t=\left[\begin{array}{cc}Kx-Ktx& Kxy-Kq\\ Kxy-Kq& Ky-Kty\end{array}\right]& \left[\mathrm{Math}.16\right]\end{array}$

where Ktx, Kty are the corrections of the stiffness trimming transducers, and Kq is the correction of the quadrature transducers. These corrections are expressed in the sensor frame. According to the invention, it is sought to eliminate stiffness mismatch and to decrease quadrature to zero; therefore, the matrix Kt is sought that will make K.sub.mod a matrix proportional to the identity matrix, i.e. such that:

[00011]$\begin{array}{cc}{K}_{\mathrm{mod}}=\left[\begin{array}{cc}K{x}_{\mathrm{mod}}& 0\\ 0& K{x}_{\mathrm{mod}}\end{array}\right]& \left[\mathrm{Math}.17\right]\end{array}$

[0113] When the sensor is in operation, i.e. when it is being servo-controlled, it is possible to measure the quadrature term in the wave frame, which is represented by the term K.sub.O(2,1), that is:

K.sub.O(2,1)=K.sub.O(1,2)=(ΔK)sin(2θ)+(Kxy)cos(2θ)   [Math. 18]

[0114] It is also possible to measure the stiffness term corresponding to the frequency along the wave axis x′, which is represented by the term K(1,1), that is:

K.sub.O(1,1)=K+(ΔK)cos(2θ)−(Kxy)sin(2θ)   [Math. 19]

[0115] It follows from the last two equations that if ΔK and Kxy were fixed terms, ΔK (and Kxy) could be determined simply through measurement of a value of the quadrature term K.sub.O(2,1) (and/or of the stiffness term K.sub.O(1,1)) for at least two electrical angles.

[0116] The problem is that a vibrating gyrometer generally uses electrostatic transducers that generate non-linearities, meaning that the terms K, Kxy and ΔK are not fixed but also dependent on the electrical angle θ, this making it difficult to identify the mismatch ΔK, and Kxy.

[0117] Thus, the stiffness matrix may be expressed in the following form, in the sensor frame:

[00012]$\begin{array}{cc}{K}_C=\left[\begin{array}{cc}\begin{array}{c}\underset{i}{.Math.}\mathrm{Aic}.\cos \left(i\theta \right)+\\ \mathrm{Ais}.\sin \left(i\theta \right)\end{array}& \begin{array}{c}\underset{i}{.Math.}\mathrm{Cic}.\cos \left(i\theta \right)+\\ \mathrm{Cis}.\sin \left(i\theta \right)\end{array}\\ \begin{array}{c}\underset{i}{.Math.}\mathrm{Dic}.\cos \left(i\theta \right)+\\ \mathrm{Dis}.\sin \left(i\theta \right)\end{array}& \begin{array}{c}\underset{i}{.Math.}\mathrm{Bic}.\cos \left(i\theta \right)+\\ \mathrm{Bis}.\sin \left(i\theta \right)\end{array}\end{array}\right]& \left[\mathrm{Math}.20\right]\end{array}$

where i is an integer varying between 1 and n, n being greater than or equal to 1, and Aic, Ais, Bic, Bis, Cic, Cis, Dic, Dis are values that vary with temperature slowly with respect to the corrections made and that may therefore be considered constants, some of which may be zero. The terms Kx, Ky and Kxy are therefore more complex and depend on the electrical angle θ.

[0118] In other words, each term of the stiffness matrix is composed of a sum of cosine and sine harmonics that are dependent on the electrical angle θ. In other words, these sinusoidal terms are modulated by multiples of the angle θ.

[0119] To pass to the wave frame, a change of basis is employed. The stiffness matrix in the wave frame is then:

[00013]$\begin{array}{cc}{K}_O=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]K_C\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]& \left[\mathrm{Math}.21\right]\end{array}$

[0120] The stiffness matrix K.sub.O has the same form as before, except that ΔK and Kxy and K also have more complex forms and depend on the angle θ.

[0121] As indicated above, when the sensor is in operation, before trimming, it is possible to measure quadrature in the wave frame, which is expressed by the term K.sub.O(2,1), and it is possible to measure stiffness along the wave axis x′, which is expressed by the term K.sub.O(1,1).

[0122] As indicated before, the excitation and trimming commands are determined in the wave frame but are applied in the sensor frame, but the aforementioned measurements are carried out in the wave frame.

[0123] However, the objective of the invention is to eliminate stiffness mismatch, i.e. to decrease K.sub.O(1,1)−K.sub.O(2,2) to zero, and to eliminate quadrature, i.e. to decrease the terms K.sub.O(1,2) and K.sub.O(2,1) to zero. The problem is that it is possible to measure only K.sub.O(1,1) and K.sub.O(2,1) and not K.sub.O(2,2). There are therefore two equations for three unknowns and it is not possible to use a plurality of electrical angles since the terms of the equations vary as a function of said angle. The terms must then be re-projected into the sensor frame to determine the trimming corrections Ktx, Kty and Kq to be applied to cancel ΔK and Kxy.

[0124] The inventor has observed that, when the stiffness matrix K.sub.C is projected from the sensor frame to the wave frame to obtain K.sub.O, the stiffness mismatch (Kx−Ky) appears in the term corresponding to the measurement of the quadrature in the wave frame and that it may be isolated. Likewise, the quadrature Kxy appears in the term corresponding to the measurement of stiffness along the axis x′ of vibration and may be isolated, allowing the quadrature Kxy to be deduced. This is illustrated below with a few simple examples.

[0125] First example: a first stiffness matrix K.sub.C1 in the sensor frame is considered, this matrix having the following form:

[00014]$\begin{array}{cc}{K}_{C1}=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]& \left[\mathrm{Math}.22\right]\end{array}$

where a corresponds to Kx, and b corresponds to Ky.

[0126] In the wave frame the following is obtained:

[00015]$\begin{array}{cc}{K}_{O1}=\left[\begin{array}{cc}\frac{a+b}{2}+\frac{a-b}{2}\cos \left(2\theta \right)& \frac{a-b}{2}\sin \left(2\theta \right)\\ \frac{a-b}{2}\sin \left(2\theta \right)& \frac{a+b}{2}-\frac{a-b}{2}\cos \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}.23\right]\end{array}$

[0127] It will be noted that the stiffness mismatch a−b appears in the term K.sub.O1(2,1) which represents the quadrature term in the wave frame, which term is:

[00016]$\begin{array}{cc}\frac{a-b}{2}\sin \left(2\theta \right)& \left[\mathrm{Math}.24\right]\end{array}$

[0128] The desired unknown is (a−b) since θ is known.

[0129] Second example: a second stiffness matrix K.sub.C2 in the sensor frame is now considered, this matrix having the following form:

[00017]$\begin{array}{cc}{K}_{C2}=\left[\begin{array}{cc}a\cos \left(2\theta \right)& 0\\ 0& b\cos \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}.25\right]\end{array}$

where a cos(2θ) corresponds to Kx, and b cos(2θ) corresponds to Ky.

[0130] In the wave frame the following is obtained:

[00018]$$\begin{gathered} \begin{array}{cc}{K}_{O2}= \\ \left[\begin{array}{cc}\begin{array}{c}\frac{a+b}{2}\cos \left(2\theta \right)+\frac{a-b}{4}+\\ \frac{a-b}{4}\cos \left(4\theta \right)\end{array}& \frac{a-b}{4}\sin \left(4\theta \right)\\ \frac{a-b}{4}\sin \left(4\theta \right)& \begin{array}{c}\frac{a+b}{2}\cos \left(2\theta \right)-\frac{a-b}{4}-\\ \frac{a-b}{4}\cos \left(4\theta \right)\end{array}\end{array}\right]& \left[\mathrm{Math}.26\right]\end{array} \end{gathered}$$

[0131] It may be seen that the stiffness mismatch (a−b)cos(2θ) (the desired unknown being (a−b) since θ is known) appears in the term K.sub.O2(2,1), which is:

[00019]$\begin{array}{cc}\frac{a-b}{4}\sin \left(4\theta \right)& \left[\mathrm{Math}.27\right]\end{array}$

[0132] Third example: a similar result may be observed for a third stiffness matrix K.sub.C3 in the sensor frame of the following form:

[00020]$\begin{array}{cc}{K}_{C3}=\left[\begin{array}{cc}a\sin \left(2\theta \right)& 0\\ 0& b\sin \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}.28\right]\end{array}$

where a sin(2θ) corresponds to Kx, and b sin(2θ) corresponds to Ky.

[0133] In this case, the stiffness mismatch (a−b)sin(2θ) appears in the term K.sub.O3(2,1), which is:

[00021]$\begin{array}{cc}\frac{a-b}{4}-\frac{a-b}{4}\cos \left(4\theta \right)& \left[\mathrm{Math}.29\right]\end{array}$

[0134] It is possible to determine (a−b) since θ is known then to determine (a−b)sin(2θ).

[0135] Fourth example: a fourth stiffness matrix K.sub.C4 in the sensor frame is now considered, this matrix having the following form:

[00022]$\begin{array}{cc}{K}_{C4}=\left[\begin{array}{cc}0& c\\ c& 0\end{array}\right]& \left[\mathrm{Math}.30\right]\end{array}$

where c corresponds to Kxy.

[0136] In the wave frame the following is obtained:

[00023]$\begin{array}{cc}{K}_{O4}=\left[\begin{array}{cc}-c.\sin \left(2\theta \right)& c.\cos \left(2\theta \right)\\ c.\cos \left(2\theta \right)& c.\sin \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}.31\right]\end{array}$

[0137] It may be seen that the quadrature c appears in the term K.sub.O4(1,1), namely (−c⋅sin(2θ)), which represents the stiffness term in the wave frame; and c may be determined since θ is known.

[0138] Fifth example: a fifth stiffness matrix K.sub.C5 in the sensor frame is now considered, this matrix having the following form:

[00024]$\begin{array}{cc}{K}_{C5}=\left[\begin{array}{cc}0& c.\cos \left(2\theta \right)\\ c.\cos \left(2\theta \right)& 0\end{array}\right]& \left[\mathrm{Math}.32\right]\end{array}$

where c⋅cos(2θ) corresponds to Kxy.

[0139] In the wave frame the following is obtained:

[00025]$\begin{array}{cc}{K}_{O5}=\left[\begin{array}{cc}-\frac{c}{4}\sin \left(4\theta \right)& \frac{c}{4}+\frac{c}{4}\cos \left(4\theta \right)\\ \frac{c}{4}+\frac{c}{4}\cos \left(4\theta \right)& \frac{c}{4}\sin \left(4\theta \right)\end{array}\right]& \left[\mathrm{Math}.33\right]\end{array}$

[0140] It may be seen that the quadrature error c⋅cos(2θ) appears in the term K.sub.O5(1,1), which is

[00026]$\begin{array}{cc}-\frac{c}{4}\sin \left(4\theta \right)& \left[\mathrm{Math}\text{.34}\right]\end{array}$

[0141] It is possible to determine c since θ is known then to determine c⋅cos(2θ).

[0142] Sixth example: a sixth stiffness matrix K.sub.C6 in the sensor frame is now considered, this matrix having the following form:

[00027]$\begin{array}{cc}{K}_{C6}=\left[\begin{array}{cc}0& c.\sin \left(2\theta \right)\\ c.\sin \left(2\theta \right)& 0\end{array}\right]& \left[\mathrm{Math}\text{.35}\right]\end{array}$

where c⋅sin(2θ) corresponds to Kxy.

[0143] In the wave frame the following is obtained:

[00028]$\begin{array}{cc}{K}_{O6}=\left[\begin{array}{cc}-\frac{c}{4}+\frac{c}{4}\cos \left(4\theta \right)& \frac{c}{4}\sin \left(4\theta \right)\\ \frac{c}{4}\sin \left(4\theta \right)& \frac{c}{4}-\frac{c}{4}\cos \left(4\theta \right)\end{array}\right]& \left[\mathrm{Math}\text{.36}\right]\end{array}$

[0144] It may be seen that the quadrature error c⋅sin(2θ) appears in the term K.sub.O6(1,1), which is

[00029]$\begin{array}{cc}-\frac{c}{4}+\frac{c}{4}\cos \left(4\theta \right)& \left[\mathrm{Math}\text{.37}\right]\end{array}$

[0145] It is possible to determine c since θ is known then to determine c⋅sin(2θ).

[0146] In all the examples, it may be seen that it is thus possible to exploit the term K.sub.O(1,1) to determine the stiffness mismatch and/or to exploit the term K.sub.O(2,1) to determine the quadrature. More generally, it is possible to exploit one of the terms of the stiffness matrix in the wave frame to determine the stiffness mismatch and/or quadrature.

[0147] Generally, and as indicated above, the stiffness matrix is of more complex form in the sensor frame, i.e. it takes the form of a sum of cos(iθ) and sin(iθ) harmonics, i being an integer varying between 0 and n, n being greater than or equal to 1, as expressed in the formula Math. 20. Typically, n may be comprised between 2 and 4. It is then possible to decompose the stiffness matrix into a sum of various simpler matrices such as those of the four previous examples (K.sub.C1, K.sub.C2, K.sub.C3, K.sub.C4, K.sub.C5, K.sub.C6 . . . ), these matrices generally being weighted by values that may be considered to be constants. It is thus possible to express the stiffness error and/or quadrature in the form of a sum of terms, as explained below.

[0148] In each of the cases presented, the inventor has determined that by carrying out an analysis of the cosine and sine harmonics, which are dependent on the electrical angle θ, of the terms representing stiffness and/or quadrature in the stiffness matrix in the wave frame, it is possible to deduce therefrom the stiffness mismatch and quadrature related to non-linearities in the sensor frame, by isolating the amplitudes of these terms.

[0149] According to the invention, a plurality of electrical angles are determined (applied and/or measured depending on whether the sensor is operating in gyrometer or gyroscope mode), to thus obtain an angle modulation. This may be done using continuous angle values θ(t) (the wave rotates continuously) or discontinuous angle values θ(t) (a plurality of angles θ.sub.k).

[0150] When the electrical angles of the wave have discontinuous values, the terms representative of quadrature and/or stiffness in the stiffness matrix in the wave frame are filtered, for example using a least squares filter, so as to isolate and recover the constants (amplitudes) of these terms.

[0151] When the wave rotates continuously, an anglewise demodulation may then be carried out—more precisely, the terms representative of quadrature and/or stiffness in the stiffness matrix in the wave frame are demodulated, the demodulation comprising applying a filter, for example a low-pass filter, allowing the constants (amplitudes) of the quadrature and/or of stiffness terms to be isolated and recovered. Said amplitudes allow the stiffness mismatch and quadrature to be deduced.

[0152] FIG. 5 illustrates a first embodiment of the calibrating method of the invention, in which embodiment the following operations are carried out: [0153] A: determining (applying or measuring) the electrical angle (θ.sub.k); [0154] B: recovering the quadrature term K.sub.O(2,1) of the stiffness matrix K.sub.O in the wave frame x′y′, said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated for a plurality of k electrical angles θ.sub.k, where k is an integer varying between 1 and m, m being greater than or equal to 2, [0155] C: determining the amplitudes of the functions in cos(iθ) and sin(iθ); [0156] D: determining the stiffness mismatch ΔK on the basis of the determined amplitudes; and [0157] E: applying stiffness trimming Ktx, Kty depending on the determined stiffness mismatch ΔK.

[0158] Step C of determining amplitudes may be carried out by applying a least squares filter to the quadrature term K.sub.O(2,1). In this case at least as many angles θ.sub.k are required as there are amplitudes to be determined.

[0159] FIG. 6 illustrates a second embodiment of the calibrating method of the invention, in which embodiment the following operations are carried out: [0160] A: determining (applying or measuring) the electrical angle (θ.sub.k); [0161] B: recovering the stiffness term K.sub.O(1,1) of the stiffness matrix K.sub.O in the wave frame x′y′, said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated for a plurality of k electrical angles θ.sub.k, where k is an integer varying between 1 and m, m being greater than or equal to 2, [0162] C: determining the amplitudes of the functions in cos(iθ) and sin(iθ); [0163] D: determining the quadrature Kxy on the basis of the determined amplitudes; and [0164] E: applying quadrature trimming Kq depending on the determined quadrature Kxy.

[0165] Step C of determining amplitudes may be carried out by applying a least squares filter to the stiffness term K.sub.O(1,1). In this case at least as many angles θ.sub.k are required as there are amplitudes to be determined.

[0166] Obviously, the calibrating methods according to the two embodiments of FIGS. 5 and 6 may be combined with each other in order to correct both the stiffness mismatch and the quadrature

[0167] FIG. 7 illustrates a third embodiment of the calibrating method of the invention, in which embodiment the vibration wave rotates continuously and the following operations are carried out: [0168] A: determining (applying or measuring) the electrical angle θ(t); [0169] B: recovering the quadrature term K.sub.O(2,1) of the stiffness matrix K.sub.O in the wave frame x′y′, said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated for a duration T, it thus being possible to: [0170] C: demodulate the quadrature term in cos(iθ) and sin(iθ) for each i varying between 1 and n, the demodulation including application of a low-pass filter, so as to determine the amplitudes of the functions in cos(iθ) and sin(iθ); [0171] D: determine the stiffness mismatch ΔK on the basis of the determined amplitudes; and [0172] E: apply stiffness trimming Ktx, Kty depending on the determined stiffness mismatch ΔK.

[0173] The duration T may correspond to one revolution if the terms are in cos(iθ) and sin(iθ) as indicated above, or to half a revolution if alternatively the terms are in cos(2iθ) and sin(2iθ). The wave may for example rotate at a speed of 1° per second, although this is non-limiting.

[0174] The quadrature term K.sub.O(2,1) may be recovered as follows. The command Cq corresponds to the quadrature force Fq used to control the quadrature of the wave to within a known gain (called the scale factor) which corrects Ko(2,1).Math.x.sub.0, where x.sub.0 is the amplitude of the wave. Thus, it is possible to obtain K.sub.O(2,1) via the formula:

K.sub.O(2,1)=Fq/x.sub.0   [Math. 38]

[0175] FIG. 8 illustrates a fourth embodiment of the calibrating method of the invention, in which embodiment the vibration wave rotates continuously and the following operations are carried out: [0176] A: determining (applying or measuring) the electrical angle θ(t); [0177] B: recovering the stiffness term K.sub.O(1,1) of the stiffness matrix K.sub.O in the wave frame x′y′, said term taking the form of a sum of functions in cos(iθ) and sin(iθ), i being an integer varying between 1 and n, n being greater than or equal to 1; steps A and B being reiterated for a duration T, it thus being possible to: [0178] C: demodulate the quadrature term in cos(iθ) and sin(iθ) for each i varying between 1 and n, the demodulation including application of a low-pass filter, so as to determine the amplitudes of the functions in cos(iθ) and sin(iθ); [0179] D: determine the quadrature Kxy on the basis of the determined amplitudes and [0180] E: apply quadrature trimming Kq depending on the determined quadrature Kxy.

[0181] The duration T may correspond to one revolution if the terms are in cos(iθ) and sin(iθ) as indicated above, or to half a revolution if alternatively the terms are in cos(2iθ) and sin(2iθ). The wave may for example rotate at a speed of 1° per second, although this is non-limiting.

[0182] The stiffness term K.sub.O(1,1) may be recovered through evaluation of the angular frequency of vibration ω of the wave:

KK.sub.O(1,1)=M×ω.sup.2   [Math. 39]

where M is the mass and is known.

[0183] Obviously, the calibrating methods according to the two embodiments of FIGS. 7 and 8 may be combined with each other in order to correct both the stiffness mismatch and the quadrature.

[0184] In any one of these four embodiments, and generally in the calibrating method according to the invention, it is also possible to use the stiffness term K.sub.O(1,1) of the stiffness matrix in the wave frame to determine the stiffness mismatch and/or to use the quadrature term K.sub.O(2,1) of the stiffness matrix in the wave frame to determine the quadrature. More generally, it is possible to exploit any one of the terms of the stiffness matrix in the wave frame to determine the stiffness mismatch and/or quadrature.

[0185] In any one of these four embodiments, and generally in the calibrating method according to the invention, said method may be implemented while the inertial sensor is operating in gyrometer mode. The electrical angle θ.sub.k or θ(t) determined in step A is in this case equal to an angle θ.sub.k_imp or θ(t).sub.imp imposed on the vibration via the precession command Cp. It is possible to use various values of θ.sub.k_imp or θ(t).sub.imp to average the errors, either via continuous rotation θ(t).sub.imp, for example over one revolution or half a revolution (or a plurality of revolutions or half revolutions), or via discontinuous rotation, measurements for example being carrying out for θ.sub.k_imp equal to 30° then 60° then 90°. Steps A to B are implemented successively for each electrical angle θ.sub.k_imp or θ(t).sub.imp.

[0186] Alternatively, the calibrating method may be implemented while the inertial sensor is operating in gyroscope mode. The electrical angle θ.sub.k or θ(t) then results from a rotation of the inertial sensor and is measured by the latter. The electrical angle determined in step A is equal to the measured angle of rotation θ.sub.k_m or θ(t).sub.m. Steps A to B are implemented successively for each electrical angle θ.sub.k_m or θ(t).sub.m. The angle θ.sub.k_m may result from modifications associated with movements of the carrier, but also from modifications associated with a precession command.

[0187] Also alternatively, the calibrating method according to the invention may be implemented in a hybrid gyrometer/gyroscope mode.

[0188] Thus, the trimming commands are carried out by modifying the values of Ktx, Kty and Kq depending on the stiffness mismatch and quadrature determined depending on the electrical angle. This makes it possible to correct non-linearities.

[0189] It is thus possible to continuously determine the stiffness mismatch and quadrature, and to correct them by trimming. A closed loop may update the trimming matrix, i.e. the values Ktx, Kty and Kq, so as to continuously correct non-linearity errors. The corrections may also be applied in a set manner or even be applied only if they are higher than a predetermined threshold, this allowing the noise associated with application of closed-loop control to be decreased.

[0190] Steps A to E mentioned above may be implemented in a servo until ΔK and Kxy have decreased to zero.

[0191] Steps A to D of the calibrating method will now be illustrated for each of the first four examples given above. In the first three illustrated examples given, stiffness mismatch is determined, and in the fourth illustrated example given, quadrature is determined.

First Example

[0192] For the stiffness matrix K.sub.O1 in the wave frame x′y′ and for a plurality of electrical angles θ: [0193] A the electrical angle θ is determined; [0194] B the term K.sub.O1(2,1) (which is equal to the term K.sub.O1(1,2)) of the stiffness matrix in the wave frame x′y′, i.e. Math. 24

[00030]$\frac{a-b}{2}\sin \left(2\theta \right),$

is recovered; [0195] (N.B. in the stiffness matrix K.sub.C1 in the sensor frame, a corresponds to Kx and b corresponds to Ky); then [0196] C the term K.sub.O1(1,2) in sin(2θ) is demodulated and the following is obtained:

[00031]$\begin{array}{cc}\frac{a-b}{4}-\frac{a-b}{2}\sin \left(4\theta \right);& \left[\mathrm{Math}\text{.40}\right]\end{array}$

then, with a low-pass filter, the constant (the amplitude) is obtained: (a−b)/4; [0197] D therefore, Kx−Ky, which is equal to 4 times (a−b)/4, is deduced.

Second Example

[0198] For the stiffness matrix K.sub.O2 in the wave frame x′y′ and for a plurality of electrical angles θ: [0199] A the electrical angle θ is determined; [0200] B the term K.sub.O2(2,1) (which is equal to the term K.sub.O2(1,2)) of the stiffness matrix in the wave frame x′y′, i.e. Math. 27

[00032]$\frac{a-b}{4}\sin \left(4\theta \right),$

is recovered; [0201] (N.B. in the stiffness matrix K.sub.C2 in the sensor frame, a cos(2θ) corresponds to Kx and b cos(2θ) corresponds to Ky); then [0202] C the term K.sub.O1(1,2) in sin(2θ) is demodulated and the following is obtained:

[00033]$\begin{array}{cc}\frac{a-b}{8}-\frac{a-b}{8}\sin \left(8\theta \right);& \left[\mathrm{Math}\text{.41}\right]\end{array}$ [0203] then, with a low-pass filter, the constant (the amplitude) is obtained: (a−b)/8, [0204] D therefore, Kx−Ky, which is equal to 8 times (a−b)/8 multiplied by cos(2θ), is deduced.

Third Example

[0205] For the stiffness matrix K.sub.O3, the result differs in that it is necessary to multiply by sin(2θ) and not by cos(2θ).

Fourth Example

[0206] For the stiffness matrix K.sub.O4 in the wave frame x′y′ and for a plurality of electrical angles θ: [0207] A the electrical angle θ is determined; [0208] B the term K.sub.O4(1,1) of the stiffness matrix in the wave frame x′y′ is recovered, namely: −c⋅sin(2θ); [0209] (N.B. in the stiffness matrix K.sub.C4 in the sensor frame, c corresponds to Kxy); then [0210] C the term K.sub.O4(1,1) in sin(2θ) is demodulated and the following is obtained:

[00034]$\begin{array}{cc}-\frac{c}{2}+\frac{c}{2}\sin \left(4\theta \right);& \left[\mathrm{Math}\text{.42}\right]\end{array}$ [0211] then, with a low-pass filter, the constant (the amplitude) is obtained: −c/2; [0212] D therefore, Kxy, which is equal to −c/2 times (−2), is deduced.

[0213] The same logic is applicable to the fifth and sixth examples.

[0214] In the fifth example, the term K.sub.O5(1,1) of the stiffness matrix in the wave frame x′y′, i.e. Math. 34

[00035]$-\frac{c}{4}\sin \left(4\theta \right)$

is recovered; [0215] this term is demodulated in sin4θ and filtered to extract the constant (the amplitude) c, which may then be multiplied by cos2θ.

[0216] In the sixth example, the term K.sub.O6(1,1) of the stiffness matrix in the wave frame x′y′, i.e. Math. 37

[00036]$-\frac{c}{4}+\frac{c}{4}\cos \left(4\theta \right)$

is recovered, demodulated in cos4θ and filtered to extract the constant (the amplitude) c, which may then be multiplied by sin2θ.

[0217] As indicated above, when the stiffness matrix is of more complex form, i.e. when it takes the form of a sum of cos(iθ) and sin(iθ) harmonics, i being an integer varying between 1 and n, n being greater than or equal to 1, it is possible to decompose the stiffness matrix K.sub.C into a sum of various simpler matrices such as those (K.sub.C1, K.sub.C2, K.sub.C3, K.sub.C4, K.sub.C5, K.sub.C6 . . . ) described above, which sum is generally weighted by values that may be considered to be constants. Thus, the stiffness error and/or quadrature error corresponds to the sum of the terms determined in the various step Ds for each of the simple matrices.

[0218] One alternative is to exploit the term K.sub.O(1,1) to determine the stiffness mismatch, and/or to exploit the term K.sub.O(2,1) to determine the quadrature, as should be clear from the various examples. For example:

[0219] In the second example:

[00037]$\begin{array}{cc}{K}_{C2}=\left[\begin{array}{cc}a\cos \left(2\theta \right)& 0\\ 0& b\cos \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}\text{.25}\right]\end{array}$

converts to:

[00038]$\begin{array}{cc}{K}_{O2}=\left[\begin{array}{cc}\begin{array}{c}\frac{a+b}{2}\cos \left(2\theta \right)+\\ \frac{a-b}{4}+\frac{a-b}{4}\cos \left(4\theta \right)\end{array}& \frac{a-b}{4}\sin \left(4\theta \right)\\ \frac{a-b}{4}\sin \left(4\theta \right)& \begin{array}{c}\frac{a+b}{2}\cos \left(2\theta \right)-\\ \frac{a-b}{4}-\frac{a-b}{4}\cos \left(4\theta \right)\end{array}\end{array}\right]& \left[\mathrm{Math}\text{.26}\right]\end{array}$

[0220] It may be seen that it is also possible to use the term K.sub.O2(1,1) to determine (a−b).

[0221] In the fourth example:

[00039]$\begin{array}{cc}{K}_{C4}=\left[\begin{array}{cc}0& c\\ c& 0\end{array}\right]& \left[\mathrm{Math}\text{.30}\right]\end{array}$

converts to:

[00040]$\begin{array}{cc}{K}_{O4}=\left[\begin{array}{cc}-c.\sin \left(2\theta \right)& c.\cos \left(2\theta \right)\\ c.\cos \left(2\theta \right)& c.\sin \left(2\theta \right)\end{array}\right]& \left[\mathrm{Math}\text{.31}\right]\end{array}$

[0222] It is also possible to use the term K.sub.O4(2,1) to determine c.

[0223] Thus, the invention uses trimming transducers to correct stiffness mismatch and quadrature and above all the invention exploits the fact that harmonics are transformed on passing from the sensor frame to the wave frame, and that thus the stiffness mismatch and quadrature appear in one or more of the terms of the stiffness matrix in the wave frame, in some form. As quadrature and stiffness are determinable without having to apply any disturbance, it is possible to observe and correct in real time both quadrature and stiffness mismatch, without disturbing the sensor with a disturbance that could end up in the measurement delivered by the sensor.

[0224] The various described embodiments may be combined with one another.

[0225] Furthermore, the present invention is not limited to the embodiments described above but encompasses any embodiment falling within the scope of the claims.