NAVIGATION INSTRUMENT WITH TILT COMPENSATION AND ASSOCIATED METHOD

Abstract

A navigation instrument including an orientation angle calculation unit, in particular the first and second orientation angles, wherein the orientation angle calculation unit is configured to be able to determine the first and second orientation angles in a given order and also in reverse order; and wherein the calculation unit is arranged to be able to choose between said given order and said reverse order, to calculate the first and second orientation angles, based on a comparison between an indicator of a risk of error, quantifying a risk of instability of said unit during the determination of the first and second orientation angles, and a predetermined threshold.

Claims

1. A method for determining the angles of orientation of a navigation instrument (4) with respect to a terrestrial reference frame (, , ), this method comprising the following steps: acquisition of components of a first vector, called first measured vector, of a first terrestrial force field in a reference frame (, , ) linked to said navigation instrument, called on-board reference frame, determination of a first angle of orientation and of a second angle of orientation of the on-board reference frame with respect to the terrestrial reference frame, these first and second orientation angles being defined by two successive rotations of the first measured vector, around a first reference axis of rotation for the first orientation angle and a second reference axis of rotation for the second orientation angle, allowing to align the first measured vector with a calculation vector having, in the on-board reference frame, three calculation components which are identical with three reference components, in the terrestrial reference frame, of a first reference vector which is predefined or predetermined in the terrestrial reference frame; said determination comprising, to calculate the first and second orientation angles, a step of choosing between a given order for the two successive rotations and a reverse order based on a comparison between an indicator of a risk of error, quantifying a risk of instability during the determination of the first and second orientation angles, and a predetermined threshold, said indicator of the risk of error being determined based on at least one of said components of the first measured vector.

2. The method according to claim 1, wherein said indicator of the risk of error is the norm of the projection of the first vector measured in a plane perpendicular to the first reference axis of rotation; and wherein, when said indicator of the risk of error is greater than said predetermined threshold, the first orientation angle is calculated based on said given order by firstly performing, among said two successive rotations, the rotation of the first vector measured around the first reference axis of rotation.

3. The method according to claim 2, wherein the value of the predetermined threshold is comprised between 0.15 and 0.45 of the norm of the first measured vector, preferably between 0.20 and 0.30 of this norm.

4. The method according to claim 1, wherein said indicator of the risk of error is the absolute value of the second orientation angle when, in said given order, the first orientation angle is calculated before the second orientation angle; and wherein, when said indicator of the risk of error is greater than said predetermined threshold, the first and second orientation angles are again calculated in said reverse order by performing, among said two successive rotations, firstly the rotation of the first vector measured around the second reference axis of rotation; the values of the recalculated first and second orientation angles then being retained for these first and second orientation angles.

5. The method according to claim 4, wherein the value of the predetermined threshold is comprised between 65° and 80°, preferably between 70° and 75°.

6. The method according to claim 1, wherein a predefined axis of the terrestrial reference frame is aligned with the first reference vector, preferably this first reference vector being aligned with a vertical axis of the terrestrial reference frame.

7. The method according to claim 1, wherein the first measured vector is a vector of the terrestrial gravitational force field.

8. The method according to claim 7, wherein the first measured vector is determined by means of an accelerometer (6).

9. The method according to claim 1, wherein the first orientation angle and the second orientation angle are respectively a roll angle and a pitch angle, the roll angle being associated with a rotation around the first reference axis of rotation, corresponding to the axis of the on-board reference frame when the roll angle is determined first, and the pitch angle being associated with a rotation around the second reference axis of rotation, corresponding to the axis of the on-board reference frame when the pitch angle is determined first.

10. The method according to claim 9, wherein the method further comprises a step of acquiring three components of a second vector, called second measured vector, of a second terrestrial force field in said on-board reference frame (, , ), this second measured vector being non-collinear with the first measured vector, and a step of determining a third orientation angle of the on-board reference frame, and thus of said navigation instrument, with respect to the terrestrial reference frame from the first and second calculated orientation angles and from the three components of the second measured vector.

11. The method according to claim 10, wherein the third orientation angle is determined by first applying to the second measured vector said first and second successive rotations, preferably in the order chosen between said given order and said reverse order, to obtain a second measured and reoriented vector, called second reoriented vector, in the on-board reference frame, then by calculating the angle between a projection of the second reoriented vector in a reference plane, comprising the first and second reference axes of rotation, and the first reference axis of rotation.

12. The method according to claim 10, wherein the third orientation angle is a yaw angle of the navigation instrument, this yaw angle being defined by a rotation of the on-board reference frame around a third reference axis of rotation, corresponding to the vertical axis in the terrestrial reference frame, after roll and pitch corrections.

13. The method according to claim 12, wherein the third orientation angle is considered as a heading of the navigation instrument.

14. The method according to claim 7, wherein the method further comprises a step of acquiring three components of a second vector, called second measured vector, of a second terrestrial force field in said on-board reference frame (, , ), this second measured vector being non-collinear with the first measured vector, and a step of determining a third orientation angle of the on-board reference frame, and thus of said navigation instrument, with respect to the terrestrial reference frame from the first and second calculated orientation angles and from the three components of the second measured vector; wherein the second vector of a second terrestrial force field is a vector of the terrestrial magnetic field; and wherein said second vector is measured by means of a magnetic sensor (10).

15. The method according to claim 1, wherein the first vector of a first terrestrial force field is a vector of the terrestrial magnetic field; and wherein said first vector is measured by means of a magnetic sensor.

16. The method according to claim 15, wherein the method further comprises a step of acquiring three components of a second vector, called second measured vector, of a second terrestrial force field in said on-board reference frame (, , ), this second measured vector being non-collinear with the first measured vector, and a step of determining a third orientation angle of the on-board reference frame, and thus of said navigation instrument, with respect to the terrestrial reference frame from the first and second calculated orientation angles and from the three components of the second measured vector; and wherein the second vector of a second terrestrial force field is a vector of the terrestrial gravitational force field.

17. The method according to claim 16, wherein the second vector of a second terrestrial field is measured by means of an accelerometer (6).

18. The method according to claim 15, wherein said three reference components are determined from data coming from a location device of the navigation instrument, in particular of the GPS type.

19. The method according to claim 1, wherein said navigation instrument is an electronic compass (4).

20. The method according to claim 13, wherein said navigation instrument comprises means (20) for displaying data indicating the true heading; and wherein the method further comprises a step of determining the true heading according to said heading determined in one of the preceding steps, of a first parameter quantifying the declination and of a second parameter quantifying the deviation of the terrestrial magnetic field at the location where the navigation instrument is located.

21. The method according to claim 1, wherein the navigation instrument is configured to be mounted on or integrated into a wearable device (2).

22. The method according to claim 21, wherein the wearable device, in particular a wristwatch (2), is wearable on a user's wrist; and wherein the first reference axis of rotation () corresponds to the 6-12 o'clock axis, and the second reference axis of rotation () corresponds to the 9-3 o'clock axis.

23. A computer program comprising instructions which, when the program is executed by a computer, cause this computer to implement the steps of the method according to claim 1.

24. A computer-readable recording medium comprising instructions which, when executed by a computer, cause this computer to carry out the steps of the method according to claim 1.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0055] The invention will be described in more detail below using the appended drawings, given by way of non-limiting examples, wherein:

[0056] FIG. 1 shows a wristwatch wherein a navigation instrument, in particular an electronic compass is mounted, this FIG. 1 comprising a front view of the wristwatch on the left and a simplified perspective view of this watch on the right;

[0057] FIG. 2 illustrates an on-board reference frame linked to the wristwatch and a terrestrial reference frame;

[0058] FIG. 3 shows a succession of three rotations that a terrestrial reference frame should undergo so that it coincides with the on-board reference frame of a navigation instrument;

[0059] FIG. 4a illustrates a projection of a vector of the terrestrial gravitational field in a plane perpendicular to the axis of the on-board reference frame, defining a roll angle;

[0060] FIG. 4b shows a first rotation and a second rotation of the vector of the terrestrial gravitational field so as to make it coincide with a calculation vector whose components are proportional to those of the terrestrial gravitational field in the terrestrial reference frame;

[0061] FIG. 5 comprises two graphs illustrating the effect of the measurement noise on the calculation of the roll angle;

[0062] FIG. 6 shows an area of arbitration between the given order and the reverse order for two successive rotations relating to the roll and pitch of the navigation instrument;

[0063] FIGS. 7a to 7c show a series of graphs comparing the heading values to the pitch value for various given roll angles with the given order and the reverse order for the two roll and pitch rotations; and

[0064] FIG. 8 schematically shows a wearable object of the electronic type incorporating an electronic compass according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

[0065] FIG. 1 shows a first embodiment of the invention wherein the navigation instrument 4 (shown in FIG. 8) is an electronic compass incorporated in a wristwatch 2 which is shown in front view on the left and in perspective view on the right (schematic representation without the bracelet which will be used in the other figures). The navigation instrument comprises a unit 8 for calculating the orientation angles of the wristwatch which is incorporated in or formed by a central electronic unit (see FIG. 8). A reference frame ; ; , linked to the navigation instrument is shown in FIG. 1. This reference frame is also called on-board reference frame. Even if the on-board reference frame is linked to the navigation instrument, it can be defined with features of the wristwatch 2 wherein the navigation instrument is rigidly fixed. Consequently, advantageously, the on-board reference frame can therefore be defined with the axis and the axis aligned in direction with the 6 o'clock—12 o'clock axis and the 9 o'clock-3 o'clock axis of the wristwatch 2, defining the general plane of the digital display or analogue display (namely of the main dial) of this wristwatch 2. The axis is chosen so as to be perpendicular and preferably dextrorotatory with respect to the axes and and it is therefore directed towards the bottom of the wristwatch from said general plane of the display.

[0066] The on-board reference frame allows to determine the orientation of the navigation instrument with respect to a predefined terrestrial reference frame as illustrated in FIG. 2. This orientation of the wristwatch 2 can vary significantly when the watch is worn by a user and when the latter changes the position of the wrist on which he wears this wristwatch. Note that the choice of an on-board reference frame ; ; just like that of a terrestrial reference frame ; ; is arbitrary. The choice is essentially related to conventions of use and/or the intended use of the navigation instrument. For example, the terrestrial reference frame is defined with an axis oriented in the vertical direction from top to bottom (according to the NED orientation, acronym of North East Down), so that the axis is aligned in direction with the local gravity field (at the navigation instrument). The local gravity field is characterised by a reference vector of the terrestrial gravitational field, represented by the vector . The axis of the reference frame is oriented according to the direction of the magnetic or geographic north pole, as illustrated in FIG. 2. Preferably, the axis results from the projection of a vector of the reference terrestrial magnetic field in a plane perpendicular to the vector , the vector of the terrestrial magnetic field pointing towards magnetic north, said vector being represented by the vector , In most inhabited regions on the surface of the earth, the vector is substantially perpendicular to the vector and consequently the axis , is substantially collinear with the vector . The third axis is defined so as to be perpendicular and dextrorotatory with respect to the axes and . The definition of the reference frame is not limited to the above example. For example the axis could have been aligned with the vector and the axis could have been obtained from the projection of the vector in a plane perpendicular to the axis . The orientation of the axis could also have been from bottom to top.

[0067] The orientation of the on-board reference frame with respect to the terrestrial reference frame can be defined by a succession of three rotations defined by three orientation angles characterising the movement that the terrestrial reference frame should undergo so that it coincides with the on-board reference frame, as illustrated in FIG. 3.

[0068] In a known manner, the orientation of the on-board reference frame relative to the terrestrial reference frame is defined by three reference angles called yaw-pitch-roll. These angles are commonly used especially in aeronautics. The invention is not however limited to this choice of angles because there are other conventions such as Euler angles which are used in other technical fields wherein the present invention is also applicable.

[0069] The first rotation called yaw rotation takes place around the axis with an amplitude ψ. This first rotation allows to determine the amplitude of the yaw movement of the wristwatch 2 (and consequently of the navigation instrument, in particular of the electronic compass). This amplitude also corresponds to the heading of the wristwatch 2 and therefore of the electronic compass. In a particular case where the wristwatch 2 is positioned so that the display plane of the wristwatch 2 is perpendicular to the field and therefore to the axis , the yaw rotation directly indicates the heading of the navigation instrument. Indeed, the determination of the heading of the navigation instrument, and therefore of the wristwatch 2, takes place by comparing the orientation of the real wristwatch 2 with respect to a fictitious positioning wherein the 6 o'clock-12 o'clock axis of the wristwatch 2 would be aligned with the axis implying that the 6 o'clock-12 o'clock axis is wedged with the vector of the terrestrial magnetic field . The yaw angle and therefore the heading is determined advantageously from trigonometric calculations on the coordinates of a vector of the measured terrestrial magnetic field . These coordinates are determined by means of a magnetic sensor 10 disposed in the electronic compass 4 (see FIG. 8). The coordinates B.sub.x,mes′, B.sub.y,mes′, B.sub.z,mes′ of the vector of the terrestrial magnetic field are expressed in the on-board coordinate system/reference frame.

[0070] The second rotation, called pitch rotation, takes place around the axis of the reference frame obtained after the yaw rotation. The axis corresponds to the axis of the terrestrial reference frame having undergone the yaw rotation. The pitch amplitude is symbolised by the Greek letter θ. When using a wristwatch 2, the pitch is generally similar to the rotational movement of the forearm that a person makes to look at the time on the display of the wristwatch 2 he is wearing.

[0071] The third rotation, called roll rotation, takes place around the axis which corresponds to the axis in the on-board reference frame. The amplitude of the roll is symbolised by the Greek letter ϕ. When using a wristwatch 2, the roll is generally similar to the rotational movement of the forearm relative to the horizontal.

[0072] The determination of the two roll and pitch angles can be carried out by comparing the position of the wristwatch 2 worn by the user with a virtual position wherein the plane of the display of the wristwatch 2 would be flat (that is to say horizontal) and therefore perpendicular to the vertical axis . To determine these two angles, the electronic compass includes a sensor 6 (see FIG. 8) which allows to measure the vector of the terrestrial gravitational field in the on-board coordinate system (linked to the watch), represented by . On the basis of the measurement of the components g.sub.x,mes′, g.sub.y,mes′, g.sub.z,mes of the vector in the on-board reference frame of the wristwatch 2 to which the magnetic compass 4 is rigidly fixed, the pitch and roll angles can easily be deduced from trigonometric calculations, as explained below.

[0073] With the knowledge of the amplitudes of the three rotations described above it is possible, as illustrated in FIG. 3, to bring a wristwatch 2 in a virtual manner for example from an initial virtual position wherein it occupies a reference position in the terrestrial reference frame to a final real position, when it is carried by the user. In the initial virtual position, the 6 o'clock-12 o'clock axis of the wristwatch 2 is aligned with the axis and the general plane of the display (formed by the 6 o'clock-12 o'clock axis and the 9 clock-3 o'clock axis) is perpendicular to the vertical axis . As indicated previously, the yaw angle determined during the first rotation allows to determine the heading of the wristwatch 2. The second and third rotation angles allow to determine the pitch angle and the roll angle of the wristwatch.

[0074] The on-board reference frame relative to the terrestrial reference frame can therefore be defined by a transformation of reference frame to describe a “solid” movement, wherein the 3 orientation angles, namely yaw, pitch and roll, together define the angular displacements of the terrestrial reference frame axes (; ; ) so as to make them coincide with the corresponding axes of the on-board reference frame (; ; ); . This reference frame transformation corresponds, for a point (x, y, z) in the terrestrial reference frame, to a change of coordinates which is expressed by the matrix product given in equation 1 below, this equation 1 providing the coordinates (x′, y′, z′) from this point in the on-board reference frame.

[00001]$\begin{array}{cc}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=R_{\phi }^{-1}.Math.R_{\theta }^{-1}.Math.R_{\psi }^{-1}\left(\begin{array}{c}x\\ y\\ z\end{array}\right)\mathrm{with}{R}_{\phi }=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \phi & -s\mathrm{in}\phi \\ 0& \sin \phi & \cos \phi \end{array}\right)\mathrm{and}{R}_{\phi }^{-1}=R_{\phi }^T{R}_{\theta }=\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -s\mathrm{in}\theta & 0& \cos \theta \end{array}\right)\mathrm{and}{R}_{\theta }^{-1}=R_{\theta }^T{R}_{\psi }=\left(\begin{array}{ccc}\cos \psi & -s\mathrm{in}\psi & 0\\ \sin \psi & \mathrm{\psi cos}\psi & 0\\ 0& 0& 1\end{array}\right)\mathrm{and}{R}_{\psi }^{-1}=R_{\psi }^T& \mathrm{Equation}1\end{array}$

where R.sub.φ.sup.T, R.sub.θ.sup.T, R.sub.ψ.sup.T are the transposed matrices of R.sub.φ, R.sub.θ and R.sub.ψ.

[0075] As the electronic compass is incorporated in the wristwatch and that, for this electronic compass, only the on-board reference frame is initially defined and known, and as the planned method consists in bringing the on-board reference frame virtually to coincide with the terrestrial reference frame to determine the orientation angles of the wearable device that is the wristwatch, in particular a heading of this wearable device, attention is then paid to the transformation of reference frame reverse to that described above. Starting from a vector or more vectors having coordinates (x′, y′, z′) in the on-board reference frame (the vectors having their origin at point (0,0,0) in the on-board reference frame and also in the terrestrial reference frame), it is sought to determine the coordinates (x, y, z) of this or these vector(s) in the terrestrial reference frame. On the basis of equation 1 given previously, the mathematical relationship, in matrix form, between a vector (x′, y′, z′) expressed in the on-board reference frame and this same vector (x, y, z) expressed in the terrestrial reference frame is given by equation 2 below. This property will be exploited in the remainder of the description of the invention for the calculation of the angles of orientation of the wearable device incorporating the electronic compass. The matrix relation of equation 2 defines a passage matrix allowing, for a point or a vector, to switch from the on-board reference frame to the terrestrial reference frame (that is to say for the expression of its coordinates).

[00002]$\begin{array}{cc}\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)& \mathrm{Equation}2\end{array}$

[0076] Given that the coordinates of a point can define the end of a vector whose origin coincides with the common origin of the on-board and terrestrial reference frames, as indicated previously, equation 2 thus allows to establish a relation between the measured vector defined in the on-board reference frame and the same vector defined in the terrestrial reference frame. This relation is as follows:

[00003]$\begin{array}{cc}\left(\begin{array}{c}{g}_{x,\mathrm{mes}}\\ g_{y,\mathrm{mes}}\\ g_{z,mes}\end{array}\right)=R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{g}_{x,\mathrm{mes}}^{\prime }\\ g_{y,\mathrm{mes}}^{\prime }\\ g_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \mathrm{Equation}3\end{array}$

[0077] Also, the relation between the measured vector defined in the on-board reference frame and the same vector defined in the terrestrial reference frame is as follows:

[00004]$\begin{array}{cc}\left(\begin{array}{c}{B}_{x,\mathrm{mes}}\\ B_{y,\mathrm{mes}}\\ B_{z,mes}\end{array}\right)=R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{\prime }\\ B_{y,\mathrm{mes}}^{\prime }\\ B_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \mathrm{Equation}4\end{array}$

[0078] However, the calculation of the angles of orientation starts from the assumption that the measured vectors , in the on-board reference frame are collinear with reference vectors , which correspond to them and whose components in the terrestrial reference frame are predefined or predetermined. Thanks to equations 3 and 4, it is possible to determine the components of each of the measured vectors , in the terrestrial reference frame, as follows:

[00005]$\begin{array}{cc}\mathrm{Equations}5a\mathrm{and}5b& \\ \left(\begin{array}{c}{g}_{x,\mathrm{ref}}\\ g_{x,ref}\\ g_{z,ref}\end{array}\right)=k1.Math.\left(\begin{array}{c}{g}_{x,\mathrm{mes}}\\ g_{x,mes}\\ g_{z,mes}\end{array}\right)=k1.Math.R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{g}_{x,\mathrm{mes}}^{\prime }\\ g_{y,\mathrm{mes}}^{\prime }\\ g_{z,\mathrm{mes}}^{\prime }\end{array}\right)\left(\begin{array}{c}{B}_{x,\mathrm{ref}}\\ B_{y,\mathrm{ref}}\\ B_{z,ref}\end{array}\right)=k2.Math.\left(\begin{array}{c}{B}_{x,\mathrm{mes}}\\ B_{y,\mathrm{mes}}\\ B_{z,mes}\end{array}\right)=k2.Math.R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{\prime }\\ B_{y,\mathrm{mes}}^{\prime }\\ B_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \end{array}$

[0079] It is therefore possible to calculate the angles of orientation by solving equations 5a and 5b. However, this approach to finding the angles of orientation is far from being advantageous in all circumstances, as there may be singular solutions (for example gimbal lock). For this reason, according to the invention, a more sophisticated approach has been developed, as discussed below.

[0080] In the case of switching from the on-board reference frame (that of the measurement) to the terrestrial reference frame (that of the reference vectors whose expression is at least partially known in this terrestrial reference frame) for a point or a measured vector, it is observed in equation 2 that the succession of the reference rotations is: first the roll (angle φ), then the pitch (angle θ) and finally the yaw (angle ψ). Preferably, the roll and pitch angles are determined, in a first step, on the basis of the measurement of a vector of the gravitational field (having a norm g), this first step being followed by the determination of the yaw and therefore the heading during a second step, the calculation of the yaw being based on the measurement of a vector of the terrestrial magnetic field .

[0081] The first step therefore consists in determining the roll and pitch angles from equation 5a. The choice of an appropriate terrestrial reference frame with a vector of the gravitational field aligned with the vertical axis allows to simplify the calculations and better anticipate a risk of dial lock, as shown in the equations below.

=k.Math.R.sub.ψ.Math.R.sub.θ.Math.R.sub.φ.Math.

according to equation 5a
with

[00006]$\stackrel{.Math.}{g_{ref}}=\left(\begin{array}{c}0\\ 0\\ g_{z,\mathrm{ref}}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ g_z\end{array}\right)$$\mathrm{and}$$\stackrel{.Math.}{g_{\mathrm{mes}}^{\prime }}=\left(\begin{array}{c}{g}_{x,\mathrm{mes}}^{\prime }\\ g_{y,\mathrm{mes}}^{\prime }\\ g_{z,\mathrm{mes}}^{\prime }\end{array}\right)=\left(\begin{array}{c}{g}_x^{\prime }\\ g_y^{\prime }\\ g_z^{\prime }\end{array}\right)$

[0082] Equation 5a can be rewritten and simplified as follows:

R.sub.ψ.sup.−1.Math.=R.sub.ψ.sup.T.Math.==k.Math.R.sub.θ.Math.R.sub.φ.Math.

And thus:

[00007]$\left(\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -s\mathrm{in}\psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right).Math.\left(\begin{array}{c}0\\ 0\\ g_z\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ g_z\end{array}\right)=k.Math.\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -s\mathrm{in}\theta & 0& \cos \theta \end{array}\right).Math.\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \phi & -s\mathrm{in}\phi \\ 0& \sin \phi & \cos \phi \end{array}\right).Math.\left(\begin{array}{c}{g}_x^{\prime }\\ g_y^{\prime }\\ g_z^{\prime }\end{array}\right)$

[0083] The roll angle resulting from equation 5a, in the case of a first measured vector which is collinear with a first vertical reference vector, is given by the following trigonometric relation:

φ=arctan(g′.sub.y/g′.sub.z)  Equation 6

[0084] The pitch angle is then given by:

θ=−arctan(g′.sub.x/g*.sub.z)  Equation 7

with

g*z=g′.sub.y−sin φ+g′.sub.z.Math.cos φ

[0085] Advantageously, it follows from equation 6 that the roll angle φ corresponds to the angle between the projection of the terrestrial gravitational field vector , in the reference plane Y′-Z′ (perpendicular to the axis ) and the axis , as illustrated in FIG. 4a.

[0086] Then, it follows from equation 7 that the pitch angle θ corresponds to the angle between a work vector, resulting from the rotation of the terrestrial gravitational field vector in the reference plane X′-Z′ (perpendicular to the axis ) and the axis , as illustrated in FIG. 4b.

[0087] FIG. 4b illustrates that the roll and pitch angles are defined by two successive rotations that the measured terrestrial gravitational force field vector should undergo, around first the axis for the roll rotation and then around the axis for the pitch rotation, to align this measured vector on the axis .

[0088] The vector of the gravitational field measured in the on-board coordinate system can be determined by an accelerometer which provides output signals representative of the three components of this vector. However, it has been observed that the three signals corresponding to the three components each have noise, the amplitude of which is not negligible and can greatly disturb the measurement of the angles of orientation if no suitable measures are taken. Note that, for each of the three components of the measured vector, the noise is a random effect that does not depend on the amplitude of the measurement signal. Noise is therefore problematic when calculating the attitude of the wearable device. Indeed, the calculation of the roll angle is done on a projection, and therefore potentially on a low amplitude vector. As the noise is statistically invariant on each measurement axis, the measurement uncertainty increases when the amplitude of the measured and projected vector decreases. In the example shown in FIG. 5 where the roll rotation is performed before the pitch rotation, it is observed in the plane Y′-Z′ (defined by the axes/vectors , and of the on-board reference frame) that the noise generates a much larger roll angle error when the norm is low (see the image on the right relative to the image on the left).

[0089] Mathematically, the problems appear when one approaches a case of gimbal lock which occurs when the axis of rotation of the roll becomes substantially collinear with the axis of rotation of the pitch. Note that the roll value has a direct impact on the calculated yaw value. It is seen that if the calculated roll value is unstable, the yaw value and therefore the heading indicated by a hand or other display will also be unstable.

[0090] According to the invention, the order of the calculation of the orientation angles can be reversed in order to avoid this instability which appears when the projection of the vector measured in the plane Y′-Z′ is too small. Thanks to this measurement, the norm of the projection of the measured vector in the plane X′-Z′ (defined by the axes/vectors and of the on-board reference frame), used to calculate the pitch angle θ, then remains within an acceptable range. This phenomenon is explained because, as the acceleration is of constant norm, if its projection is of low norm in one plane, it is necessarily large in the other perpendicular planes.

[0091] The order of calculation of the roll and pitch angles is therefore either in a given order or in a reverse order. The given order involves first a roll rotation followed by a pitch rotation. For the reverse order, an inversion of the given order is provided with a pitch rotation which is calculated first and then a roll rotation. It will also be noted that these rotations applied to a vector measured in the on-board reference frame correspond to a correction of the roll and of the pitch involved in the wearable device incorporating or provided with the electronic compass according to the invention and therefore to rotations in the opposite direction (−φ, −∝) of the on-board reference frame, so that this on-board reference frame is virtually reoriented with its axis collinear with the vertical axis/vector of the terrestrial reference frame.

[0092] For the order given above, the roll and pitch angles are determined by equations 6 and 7 given above. For the reverse order, the simplified equation 5a is replaced by the following equation 5c:

R.sub.ψ.sup.−1.Math.=R.sub.ψ.sup.T.Math.==k.Math.R.sub.φ.Math.R.sub.θ.Math.

[0093] The pitch angle resulting from the equation 5c, in the case of a first measured vector which is collinear with a first vertical reference vector, is given by the following trigonometric relation:

θ=−arctan(g′.sub.x/g′.sub.z)  Equation 8

[0094] The roll angle is then given by:

[00008]$\begin{array}{cc}\phi =\arctan \left(\frac{g_y^{\prime }}{-g_x^{\prime }.Math.\sin \theta +g_z^{\prime }.Math.\cos \theta }\right)& \mathrm{Equation}9\end{array}$

[0095] Note that, in general, R.sub.φ−R.sub.θ is not equal to R.sub.θ.Math.R.sub.φ, so that the inversion of the two successive considered rotations has an influence on the yaw value calculated subsequently to obtain a heading of the wristwatch 2. This will be discussed later.

[0096] The choice of the order in which the roll and pitch angles are calculated can be based on various criteria. For example, a first criterion exploits the norm (or the square of the norm) of the projection of the vector in the plane Y′-Z′. If this norm is less than a given value, the order of calculating the angles is reversed (calculating the pitch first, then the roll). Alternatively, a second proposed criterion concerns the value of the pitch angle when this value is determined according to the given order (calculating the roll first, then the pitch). If the pitch turns out to be greater than a given angle, the calculation of the two angles considered here is performed a second time with the pitch being calculated first.

[0097] The first criterion, known as the basic criterion, aims at ensuring that the limit of the norm projected on the plane Y′-Z′ (defined by the axes and , of the on-board reference frame) is much larger than the maximum noise norm. In this way, a certain precision is ensured for the calculation of the roll angle. Advantageously, this condition can be carried out by comparing the projection of the measured vector in the plane Y′-Z′ and a predetermined threshold value (limit radius as illustrated in FIG. 6). A condition for switching into pitch first would be that the projection of the projected vector has a norm less than a given threshold value. For example, the value of the predetermined threshold is comprised between 0.15 and 0.45 of the norm of the measured vector , preferably between 0.20 and 0.30 of this norm. These values are particularly favourable for a navigation instrument mounted on or in a wristwatch. Indeed, it turns out that the user tends to look at his watch with his forearm positioned horizontally, with a wrist turned approximately 30° to 60° in pitch. Thus, in an advantageous embodiment of the invention, the choice to use the reverse order occurs when the pitch exceeds a threshold value comprised between 70° and 75° (see alternative criterion). Indeed, it is advantageous to prevent the order from being reversed within a useful range of 30° to 60° for the pitch when the user looks at his watch, hence the choice of an angle of the above-mentioned threshold value.

[0098] Other ranges of values can be chosen for the threshold value depending on the field of application of the navigation instrument, for example a range centred around cos (45°), that is to say 0.707. This selection allows to optimise the reduction in the influence of the measurement noise for the calculated values, but it risks creating instability in the display of the heading for an electronic compass incorporated in an object wearable on the wrist, and possibly more repetitions of the sequence for calculating the roll and pitch angles due to this instability. Indeed, as the given order and the reverse order for the roll and pitch rotations applied to the measured vector leads to a different yaw/heading value, if the pitch angle varies around the threshold value during use of the electronic compass, the indicated heading will be, without other measurements, recalculated at each change in the result of the comparison defined by the criterion for the choice of the order of the two rotations in question. The second criterion, called alternative criterion, is substantially equivalent to the first criterion. It is based on the comparison between the pitch angle calculated according to the given order and a threshold value comprised between 65° and 80°, preferably between 70° and 75°.

[0099] The approach proposed by the invention is in particular based on the observation that, despite and regardless of the noise, it is always possible to bring the gravitational vector measured on the axis with relatively good precision in determining the roll and pitch angles. Indeed, despite the noise in the measurement of the gravitational vector, the calculation of the roll of a measured vector coplanar with the plane Y′-Z′ (defined by the vectors and ) is always relatively precise and the calculation of the pitch of a measured vector coplanar to the plane X′-Z′ (defined by the vectors and ) is always relatively precise. Then, after the roll is calculated first and the corresponding rotation applied to the measured vector, the measured vector thus reoriented is in the plane X′-Z′ so that the pitch can then be calculated with good precision. Likewise, once the pitch is calculated first and the corresponding rotation applied to the measured vector, the measured vector thus reoriented is in the plane Y′-Z′ so that the roll can then be calculated with good precision. It is therefore understood that it is the first calculation of an orientation angle of the electronic compass which is critical since it is based on a projection in a reference plane and that the norm of this projection can be relatively small, so that the noise occurring during the measurement can strongly distort the result of the first angle calculated subsequently by the calculation unit 8 of the electronic compass 4, as explained previously. Thanks to the invention, good precision or sufficient precision for this first calculation is ensured by selecting between ‘roll first’ and ‘pitch first’.

[0100] The second step consists in determining the yaw to obtain the heading. This step consists in solving the equation 5b given previously. The choice of a judicious terrestrial reference frame allows to simplify the calculation steps, in particular when the projection of the reference terrestrial magnetic field vector , corresponding to a second reference vector, in a plane perpendicular to the axis is collinear with the axis ; which is the case for the terrestrial reference frame NED defined at the start. In this case the following equation is obtained for the magnetic vector measured by the magnetic sensor:

[00009]$\begin{array}{cc}\left(\begin{array}{c}{B}_{x,\mathrm{ref}}\\ B_{y,\mathrm{ref}}\\ B_{z,ref}\end{array}\right)=\left(\begin{array}{c}{B}_{x,\mathrm{ref}}\\ 0\\ B_{z,\mathrm{ref}}\end{array}\right)=k2.Math.\left(\begin{array}{c}{B}_{x,\mathrm{mes}}\\ B_{y,\mathrm{mes}}\\ B_{z,mes}\end{array}\right)=k2.Math.R_{\psi }.Math.R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{\prime }\\ B_{y,\mathrm{mes}}^{\prime }\\ B_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \mathrm{Equation}10a\end{array}$

[0101] Equation 10a corresponds to the ‘roll first’ alternative and for the ‘pitch first’ case there is an equation 10b similar to equation 10a but with an inversion of the rotation matrices R.sub.φ and R.sub.θ. The yaw angle ψ can be determined by first applying to the vector of the measured terrestrial magnetic field , in the on-board frame, two successive rotations corresponding to a roll rotation and a pitch rotation according to an order chosen between said given order and said reverse order, as explained previously, so as to obtain a measured and reoriented vector of the terrestrial magnetic field , then by calculating the angle between a projection, in the reference plane X′-Y′ defined by the axis , and the axis , of the vector of the reoriented terrestrial magnetic field and the axis . Starting from equations 10a and 10b, the following simplified equation is obtained:

[00010]$\left(\begin{array}{c}{B}_{x,\mathrm{ref}}\\ 0\\ B_{z,\mathrm{ref}}\end{array}\right)=k2.Math.R_{\psi }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{*}\\ B_{y,\mathrm{mes}}^{*}\\ B_{z,\mathrm{mes}}^{*}\end{array}\right)$

with

[00011]$\begin{array}{cc}\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{*}\\ B_{y,\mathrm{mes}}^{*}\\ B_{z,\mathrm{mes}}^{*}\end{array}\right)=R_{\theta }.Math.R_{\phi }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{\prime }\\ B_{y,\mathrm{mes}}^{\prime }\\ B_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \mathrm{Equation}11a\end{array}$

in the given order:
in the reverse order:

[00012]$\begin{array}{cc}\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{*}\\ B_{y,\mathrm{mes}}^{*}\\ B_{z,\mathrm{mes}}^{*}\end{array}\right)=R_{\phi }.Math.R_{\theta }\left(\begin{array}{c}{B}_{x,\mathrm{mes}}^{\prime }\\ B_{y,\mathrm{mes}}^{\prime }\\ B_{z,\mathrm{mes}}^{\prime }\end{array}\right)& \mathrm{Equation}11b\end{array}$

[0102] From equation 10a or 10b, the yaw/heading angle is therefore given by the following equation:

ψ=−arctan(B.sub.y,mes*/B.sub.x,mes*)  Equation 12

[0103] All the calculation steps described above can be carried out by the central processing unit 14 of the electronic compass 4. The determination of the heading is based on the yaw angle calculated with the equation 12 by a unit 12 for determining a heading, which can be formed by the same electronic circuit as the unit 8 for calculating the roll and pitch angles. In a variant, the electronic compass comprises a magnetic north or geographic north indicator 18 and a device 16 for controlling this indicator (see FIG. 1 and FIG. 8). The control device 16 may be arranged to control the indicator 18 so that it has an angle relative to the axis equal to the opposite of the yaw angle, preferably the correction of the deviation of the terrestrial magnetic field at the location of the navigation instrument for the indication of the magnetic north and, for the indication of the geographic north, in addition to the declination of the terrestrial magnetic field.

[0104] The electronic compass 4 preferably comprises a device 20 for displaying data indicating the true heading (see FIG. 1 and FIG. 8), the unit 12 for determining a heading being configured to calculate the true heading based on the heading provided by the equation 12 given above, a first parameter quantifying the declination of the terrestrial magnetic field and a second parameter quantifying the deviation of the terrestrial magnetic field at the location where the navigation instrument is located. In particular, for determining the declination and deviation of the terrestrial magnetic field, the wearable object 2 further comprises a geolocation device. The electronic compass 4 may have a non-volatile memory wherein are stored information relating to the declination and the deviation and/or a program provided for determining them in particular on the basis of geolocation. In addition, the wearable object can have wireless communication means allowing in particular to acquire at the declination and the deviation of the magnetic field depending on the location where the wearable object incorporating the electronic compass is located.

[0105] A comparative study was carried out to demonstrate that the choice of the order of rotation between the given order (roll then pitch) or the reverse order (pitch then roll) has relatively little importance on the result of the calculation of the yaw angle, and therefore of the heading, in the context of the application of a wristwatch 2. Indeed, the order of the roll and pitch rotations normally influences the calculation of the roll angle and the pitch angle because, as already indicated, the product of matrices is not commutative (R.sub.θ.Math.R.sub.φ≠R.sub.φ.Math.R.sub.θ). FIG. 7a shows that in the absence of roll (the forearm wearing the wristwatch provided with the electronic compass being horizontal), the permutation between the given and reverse order has no impact. This finding is important because it turns out that a user tends to look at his watch on his wrist with his forearm positioned substantially horizontal, with a wrist turned in particular 30° to 60° in pitch. Thus, in an advantageous embodiment of the invention, the choice to use the reverse order occurs when the pitch exceeds 70°, but one could also choose another threshold value, for example 60°. Indeed, it is advantageous, as already explained, to prevent the order from being reversed in a usual pitch range in order to avoid instability in the succession of sequences for calculating the angles of orientation and/or an increase in the number of these calculation sequences, hence the (arbitrary) choice of an angle greater than 60°, in particular 70°. As long as the roll remains low, which is practically the case in general, even when operating above this value, the user of the watch will not notice any difference when switching from the given order to the reverse order (FIG. 7a). As a reminder, this inversion allows to reduce the influence of the measurement noise on the determination of the orientation angles of the navigation instrument (in particular of an electronic compass), to increase the precision of the heading measurement and also to reduce any possible instability of the successive results given by a succession of calculation sequences, such instability resulting in an oscillation of the hand indicating the north and/or of the heading value displayed.

[0106] However, it will be noted that, for a relatively low roll (see FIG. 7b: 10° of roll), there is already a significant deviation of maximum of 10° within the range 0°-90°, in particular about 5° to 8° within the range 40°-60° which corresponds to the current values. Thus, preventing the predetermined threshold value from being within or corresponding to the usual pitch range is advantageous. FIG. 7c shows that as the roll angle increases, the difference in the result for the heading between the given order and the reverse order increases. Thus, for a roll angle of 45° and within a pitch range of 0° to 30°, the difference for the heading value can reach 20°.

[0107] The invention is not limited to the wearable device of the first embodiment described above. Indeed, the inversion of the order of the rotations not only allows to advantageously reduce the sensitivity to measurement noise during the prediction of two orientation angles but also to no longer suffer from the gimbal locking effect for many objects, such as a transport vehicle or a flying object. In general, the inversion of the order of the rotations according to the invention operates as a filter aiming at reducing the effect of the noise of the measurement of at least one vector of a terrestrial force field and at removing the gimbal lock when determining the orientation angles of an object relative to a terrestrial reference frame.

[0108] In a second embodiment, the navigation instrument allows to determine two angles of orientation of a system within the framework of spherical coordinates. The navigation instrument therefore allows to determine the two angles of orientation of the axis of the on-board reference frame on the basis of the measurement of a vector of the terrestrial magnetic field determined via a geolocation means.

[0109] The first embodiment proposes to first calculate the roll and pitch angles according to the measurement, in a reference frame linked to the navigation instrument, of a gravitational field vector , then the yaw angle from the measurement in the same on-board reference frame of a vector of the terrestrial magnetic field . This order can be reversed with a first step of calculating the yaw and roll angle from a measurement, still in a reference frame linked to the navigation instrument, of a vector of the terrestrial magnetic field , then the pitch angle from the measurement, in this on-board reference frame, of a terrestrial gravitational field vector . This example shows that the choice of the axes of the considered reference frames as well as the given order of the rotations is arbitrary.