Method for determining the temporal invariances of the degree of linear polarization in the sky

Abstract

A method is proposed for determining the temporal invariances of the DOLP linear polarization degree of light from a celestial body in a celestial vault, implemented in a positioning and navigation information determination system, said vault being centered on the geographical position of said system. The method comprises the following steps: acquiring (102) a plurality of successive polarized images of said vault; determining (104), for each acquired image, a DOLP representation; determining (106) at least one grouping of DOLP invariances from a pair of representations selected from said DOLP representations; determining (110) a measurement point P from the grouping of invariances, the point P corresponding to the celestial pole in said vault.The process further includes a step of determining the altitude and / or azimuth of said celestial pole from point P, said azimuth corresponding to the direction of the geographic pole. Figure for the abbreviation: Fig. 3.

Description

Title of the invention: Method for determining the temporal invariances of the degree of linear polarization in the sky Technical field

[0001] The present invention relates generally to geolocation, and in particular to a method for determining the temporal invariances of the degree of linear polarization in the sky for determining the celestial pole.

[0002] Geolocation systems, such as GNSS, are conventionally used to determine the position of an object in the Earth system.

[0003] A GNSS (Global Navigation Satellite Systems) system is a satellite positioning system configured to provide the position of an object (device or system) with good accuracy, across the entire Earth and in real time. Such a system is today considered one of the most accurate geolocation systems. However, such a system is susceptible to multiple signal reflections introduced by urban canyons (e.g. skyscrapers, buildings, foliage, etc.), jamming and identity theft.

[0004] To improve satellite positioning systems, it is known to combine them with devices using geopositioning approaches based on measurements of polarized light from the sky. These approaches are biomimetic approaches or bioinspired approaches. For example, in the article “Desert navigator: the joumey of an ant” by Rüdiger Wehner, Harvard University Press, 2020, the proposed biomimetic approach is inspired by the navigation mechanism implemented by desert ants that are sensitive to the polarization pattern of the sky. It should be noted that known bioinspired approaches require the use of conventional localization modalities (e.g., the solar astronomical calendar or solar ephemeris) combined with an estimation of the position of the sun obtained via the polarization pattern of the sky.Estimating the position of the sun involves calculating the angles of polarization (AoLP) and degrees of linear polarization (DoLP). This solution is for example addressed in the article “Global autonomous positioning in gnss challenged environments: A bioinspired strategy by polarization pattern” by J. Yang et al., IEEE Trans. on Industrial Electronics, 68(7), pages 6308-6317, 2020. However, using conventional localization modalities in combination with measurements of polarized light from the sky can generate a significant memory cost as well as a computational cost for the system and the heading determination method.

[0005] There is thus a need for an improved method based solely on measurements of polarized light from the sky to determine the positioning and heading of a system relative to the geographic pole. Summary of the invention

[0006] The present invention improves the situation by proposing a method for determining the temporal invariances of the degree of linear polarization of light coming from a star in a celestial vault, implemented in a system for determining positioning and navigation information, the celestial vault being centered on the geographical position of the system. The method comprises at least the following steps consisting of: - acquiring a plurality of successive polarized images In of at least part of the celestial vault; - determine, for each image In acquired, a representation of the degrees of linear polarization Idn; - determining at least one grouping of invariances of degrees of linear polarization of the light coming from the star, a grouping of invariances being determined from a pair of representations selected from the plurality of representations of the degrees of linear polarization Idn; - determine a measurement point P from at least one group of invariances, the measurement point P corresponding to the celestial pole in the celestial vault of the system.

[0007] The method further comprises a step of determining the altitude and / or azimuth of the celestial pole from the measurement point P obtained. The azimuth corresponds to the direction of the geographic pole associated with the geographic position of the system.

[0008] Advantageously, the system for determining positioning and navigation information can be geographically positioned at a latitude coordinate ¢), and the method can further comprise a step of determining the latitude coordinate (j)0 of the system from the determined altitude.

[0009] Each polarized image In can be acquired at an instant L, the minimum time difference between two acquisition instants of two successive polarized images being predefined.

[0010] According to certain embodiments, each polarized image In may comprise a set of pixels Pxy, each pixel being associated with coordinates (x, y) and with a set of detection intensities according to distinct polarization angles such that p = (Jæ, I90Oj I43O> Ij35°)xv- Each representation Idn may comprise a set of linear polarization degrees dxy, each degree dxy being calculated from the Stokes vector, such that: [°011] __ct dxy_ [00121 (s0)xy= (1,,.),,+ (Wxy • (mxy= (VV (W)xy. (S2) xy = (Ï45J) Xy + (1135°) Xy'

[0013] Advantageously, each grouping of invariances can be determined from a differential representation Dm comprising a set of differences Axy calculated from the pair of representations of the degrees of linear polarization Idn selected.

[0014] Each grouping of invariances can be determined from a binary representation Bm determined from the differential representation Dm and a predetermined threshold of binarization SP of the difference in degrees of linear polarization Axy.

[0015] In embodiments, the plurality of polarized images In may comprise at least three images. At least two groupings of invariances of degrees of linear polarization of the light coming from the star may be determined. The measurement point P may be determined from the intersection between the at least two groupings of invariances.

[0016] The intersection between the at least two groups of invariances can be carried out from the determination of an intersection representation H comprising a set of intersection values ​​hxy calculated from at least two of the binary representations Bm. The measurement point P can correspond to the coordinates for which the intersection value hxy is maximum.

[0017] In embodiments, the plurality of polarized images In may comprise at least two images, the grouping of invariances comprising a radial invariance and a plane invariance of degrees of linear polarization of the light coming from the star. The measurement point P may be determined from a point Pi of intersection between the radial invariance and the plane invariance, and from information on the offset of the celestial pole relative to the point of intersection P;.

[0018] Advantageously, the method may comprise a step consisting of determining at least one simulation of the pattern of degrees of linear polarization of the light coming from the star over the entire celestial vault, each simulation being determined from one of the images of the plurality of polarized images In of at least one part of the celestial vault.

[0019] The method may further comprise a step of acquiring displacement and rotation data of the operated system between the successive acquisition of at least two of the polarized images In. The determination of the at least one grouping of invariances may comprise taking into account the displacement and rotation data.

[0020] The geographical position of the system may further be associated with a coordinate longitude f0. The method may comprise a step of determining the co ordinate l0 of longitude of the system from data from a referenced clock and the direction of polarization at the measurement point P of the system positioned successively at at least two distinct geographical positions.

[0021] The invention also provides a system for determining positioning and navigation information. The system comprises:

[0022] - a polarized light acquisition module, the light coming from a star in the celestial vault centered on the geographical position of the system, the acquisition module being configured to carry out the acquisition of a plurality of successive polarized images In of at least part of the celestial vault;

[0023] - a processing module configured to determine, for each acquired image In, a representation of the degrees of linear polarization Idn, the processing module being further configured to determine at least one grouping of invariances of degrees of linear polarization of the light coming from the star, each grouping of invariances being determined from a pair of representations Idn selected from the plurality of representations of the degrees of linear polarization Idn, the processing module being further configured to determine a measurement point P from the at least one grouping of invariances, the measurement point P corresponding to the celestial pole in the celestial vault of the system.

[0024] The system further comprising a positioning and navigation module configured to determine the altitude and / or the azimuth of the celestial pole from the measurement point P obtained, the azimuth corresponding to the direction of the geographic pole associated with the geographic position of the system.

[0025] The method and system according to the embodiments of the invention make it possible to detect a celestial pole solely from the polarized light of the sky and more precisely from the degree of linear polarization (DoLP) pattern, without requiring the use of other conventional location modalities.

[0026] They thus make it possible to determine the direction of geographic north and the latitude of the system from the estimation of the position of the celestial pole, that is to say, positioning and heading information relevant for improving geopositioning and navigation systems. Description of figures

[0027] Other characteristics, details and advantages of the invention will emerge from reading the description given with reference to the appended drawings given by way of example.

[0028] [Fig-1] [Fig.l] is a diagram representing a system of determination positioning and navigation information, according to embodiments of the invention.

[0029] Figures [Fig.2(a)], and

[0030] [Fig.2(b)] are schematic representations of the geographical position of the celestial pole determination system, according to embodiments of the invention.

[0031] [Fig.3] [Fig.3] is a flowchart representing the method of determining the temporal invariances of the degree of linear polarization of light coming from a star in the sky, according to embodiments of the invention.

[0032] Figure [Fig.4(a)] is a schematic representation of the pixels of a sensor of a polarized light acquisition module of the celestial pole determination system, according to embodiments of the invention.

[0033] Figure [Fig.4(b)] is a schematic representation of a polarized light acquisition module, according to embodiments of the invention.

[0034] [Fig.5] [Fig.5] illustrates the results of steps of the method for determining the temporal invariances of the degree of linear polarization of light coming from a star in the sky, according to embodiments of the invention.

[0035] [Fig.6] [Fig.6] is a flowchart representing sub-steps of the method for determining invariance groups, according to embodiments of the invention.

[0036] [Fig.7] [Fig.7] is a diagram representing the axes of plane and radial invariance, via a half-sphere visualization of a differential representation of two representations of the degree of linear polarization, according to embodiments of the invention.

[0037] The figures [Fig.8(a)],

[0038] [Fig.8(b)], and

[0039] [Fig.8(c)] are flowcharts representing sub-steps of the method of determining mination of invariance groups, according to embodiments of the invention.

[0040] [Fig.9] [Fig.9] is a schematic representation of the geographical position of the system for determining the celestial pole in a reference frame (E,N,Z), according to embodiments of the invention.

[0041] [Fig. 10] [Fig. 10] is a flowchart showing substeps of determining longitude coordinates of the celestial pole determination system, according to embodiments of the invention.

[0042] [Fig. 11] [Fig. 11] is a diagram showing a celestial pole determination system, according to embodiments of the invention.

[0043] [Fig. 12] [Fig. 12] illustrates the results of steps of binarization and intersection of the differences in degrees of linear polarization, according to exemplary embodiments of the invention.

[0044] [Fig. 13] [Fig. 13] is a graph illustrating the results of the geographic pole direction values ​​and latitude values ​​determined by the method of determining mination of the temporal invariances of the degree of linear polarization of light coming from a star in the sky, according to embodiments of the invention.

[0045] Identical references are used in the figures to designate identical or similar elements. For reasons of clarity, the elements shown are not to scale. Detailed description

[0046] [Fig.l] represents a system for determining positioning and navigation information 1, according to embodiments of the invention.

[0047] The system for determining positioning and navigation information 1 comprises a device 10 for determining the temporal invariances of the degree of linear polarization of the light coming from a star in the sky, also called hereinafter a celestial pole determination device, according to certain embodiments of the invention.

[0048] As used herein, the term "light" refers to one or more optical waves, for example and without limitation, in the ultraviolet, visible and / or infrared ranges. Such optical waves may for example be associated with a spectral (or multi-spectral) band predefined by the system 1.

[0049] The celestial pole determination device 10 is configured to determine the coordinates of a celestial pole relative to the system 1, on the surface of a planet such as Earth. The positioning and navigation information determination system 1 further comprises a positioning and navigation information determination unit 20 configured to determine the latitude coordinate of the system 1 and the direction of the geographic pole, from the determination of the coordinates of the celestial pole. The positioning and navigation information determination system 1 can then use the latitude coordinate of the system 1 and the direction of the geographic pole for geopositioning and / or navigation of the system 1.

[0050] As used herein, the term "celestial pole" refers to a virtual point on the celestial vault of a planet, such as Earth, corresponding to the intersection between the planet's rotational axis and its associated celestial sphere. A celestial pole thus corresponds to a projection onto the celestial sphere of a geographic pole (also called a 'true pole' in opposition to the magnetic pole). In other words, a celestial pole refers to a virtual point on the celestial vault around which the stars rotate from the point of view of an observer on the surface of the Earth, for example, due to the rotation of the Earth on its axis.

[0051] Figure 2(a) schematically represents the Earth, in the form of a circle with center C composed of two hemispheres including the Northern hemisphere and the Southern hemisphere South. There are two celestial poles as shown in Figure 2(a): the North Celestial Pole (NCP) associated with the Northern Hemisphere, and the South Celestial Pole (SCP) associated with the Southern Hemisphere. The positioning and navigation information determination system 1 is characterized by its geographical position, denoted O, on the surface of the Earth for example. In particular, the geographical position of system 1, for example in the Northern Hemisphere, comprises the coordinate longitude lo (not shown in Figure 2(a)) and the coordinate latitude $o defined from the Zenith axis Z passing through the center C of the earth and point O. In Figure 2(a), the angle 0' corresponds to the co-latitude coordinate of the geographical position of system 1. The celestial vault, shown in Figure 2(b), is centered at point O and is also called 'sky'.In Figure 2(b), the angle F called 'diffusion angle' corresponds to the angle between the vector 'observer-sun' and the vector 'observer-observation point'.

[0052] [Fig. 3] represents a method for determining positioning and navigation information, according to embodiments of the invention. The method for determining positioning and navigation information can be implemented by the system 1. Steps 102 to 108 correspond more specifically to the method for determining the temporal invariances of the degree of linear polarization of the light coming from a star in the sky (or method for determining the celestial pole), implemented by the celestial pole determination device 1.

[0053] As shown in Figure 3, the method comprises a step 102 of acquisition by the system 1 of a number N of successive polarized images, denoted In. The index ' / l' associated with different polarized images is an integer between 1 and N, and the value of N is an integer greater than or equal to 2.

[0054] Each polarized image In represents at least a portion of the celestial vault. For the N different successive polarized images, the acquired light comes from the same star. For example and without limitation, for a system positioned on Earth, this star may be the sun or the moon, while for a system positioned on Mars, this star may be the sun. Thus, the N successive polarized images ln may be acquired during the day. Alternatively, the N successive polarized images In may be acquired at night. In particular, each polarized image In is acquired at an instant (or acquisition time) The minimum time difference between two instants 0 / and tn+\ of image acquisition may be defined according to the technological capabilities of the polarized light acquisition module used to acquire the plurality of images In. For example and without limitation, the minimum time difference Ôtmm may be between a few milliseconds and a few minutes.The maximum time difference ôtmax between two instants and ^w+i of image acquisition can advantageously be . several hours, for example.

[0055] Furthermore, each acquired image In can be characterized by a set of acquisition points denoted Pxy, such that each acquisition point (also called pixel in the remainder of the description) is associated with coordinates (x, y) • For example and without limitation, a set of pixels of an image In can be distributed according to a circular distribution, as shown in figure [Fig.4(a)]. Each coordinate (X, y) then represents a projection of a point of the celestial vault in an XY plane perpendicular to the axis U of the system 1 (XY plane shown in figure [Fig.4(b)]). The axis U shown in figure [Fig.4(b)] corresponds to the optical axis associated with the system 1 during the acquisition of an image In. Such an axis U can be for example equivalent to the Zenith axis Z.

[0056] The images being polarized, each pixel is associated with a plurality of detection intensities according to distinct polarization angles. For example and without limitation, each pixel Pxy can be associated with 4 detection intensity values ​​corresponding respectively to distinct polarization angles according to the following equation (01):

[0057] Pv= (Ww> Ww),,®»

[0058] It should be noted that according to the Rayleigh scattering model of light coming from a star propagating in the atmosphere (especially terrestrial), the interactions between the photons of light and the molecules of the atmosphere cause a partial linear polarization of this light from the celestial vault according to a pattern called a "polarization pattern". This polarization pattern depends on the relative position of the star with respect to the position O of system 1, on the surface of the Earth, for example.This polarization pattern can be mathematically described from two quantities: the angle between the direction of linear polarization of light and an axis of a predefined reference frame (i.e. the angle of linear polarization or AoLP) and the degree of linear polarization of light (or DoLP) as described for example in the article “Polarization and Intensity of Light in the Atmosphere” by KL Coulson et al., Deepak Pub., 1988. In particular, the DoLP (or “DoLP pattern”) can be mathematically modeled as a function of the scattering angle F shown in Figure [Fig.2(b)]. The DoLP can be described according to the following equation (02): . V / l+cos-(y)

[0060] In step 104, for each image In acquired, a representation Id„ of the DoLP pattern of at least one part of the sky is determined (or calculated).

[0061] Each determined DoLP Idn representation can be characterized by a set of calculated linear polarization degrees noted dxy associated with the coordinates (x, y) Pxy pixels.

[0062] In embodiments, for each coordinate (x, y), the degree of linear polarization dxy can be calculated from the Stokes vector defined from the Rayleigh scattering model and described according to the following equation (03):

[0063] (03) c^y ~ üô • xy

[0064] In the preceding equation (03), the coefficients (s0), (,yl) and (y2) are defined as a function of the values ​​of the plurality of detection intensities of each pixel Pxy of the image ln, and can be described respectively according to the following equations (04), (05) and (06):

[0065] ( / ^+(W„ <04'

[0066] (sl)^ (“„-( / „),, (“S)

[0067] (s2)„ = (Z45.),ï+( / kw.),v (“O)

[0068] A set of calculated linear polarization degrees dxy of a DoLP representation Idn can be distributed, according to a circular distribution, as illustrated by the set of pixels in figure [Fig.4(a)].

[0069] Note that the values ​​ax> of linear polarization angles AoLP can also be calculated from the Stokes vector, according to the following equation (07): [00™] „ -l^t«n4('2M<07) dxy - TX Idn 7———

[0071] In other embodiments, for each coordinate (x, y), the degree of linear polarization dxy can be calculated from Malus's law for example.

[0072] [Fig.5] represents an example of implementation of steps of the method for determining positioning and navigation information, according to certain embodiments of the invention.

[0073] In particular, in Figure 5, the DoLP representations Id^ Id2, Id3 and IdA (representing the result of the application of step 104) come from the polarized images, respectively 1\, l2, / 3 and / 4, acquired at the acquisition times t2, and / 4. In this example, the time differences Ôt between two polarized image acquisition times In are between 30 minutes and 60 minutes. Furthermore, the DoLP representations ld„ are illustrated according to a gray level, such that for each coordinate ( X, y ), a value dxy can be between a minimum value of linear polarization degrees DoLPtnin and a maximum value of linear polarization degrees DoLPmax.

[0074] According to equation (02), the minimum value DoLPmin is equal to 0. According to the Rayleigh scattering model, the minimum value DoLPmbl corresponds to a scattering angle 1'' equal to 0, associated with the light rays coming from the projection of the star (i.e. the sun or the moon for example) on the celestial vault, i.e. point A in figure 2(b). Thus, the coordinates (x, y) associated with the minimum value DoLPmin correspond to the position of the projection of the star on the celestial vault on the XY plane perpendicular to the U axis of system 1. According to equation (02), the maximum value DoLPmax is equal to 1. According to the Rayleigh scattering model, the maximum value DoLPmax corresponds to a scattering angle Fequal to 71 j 2, associated with the light rays coming from point A' in figure 2(b). It should be noted that due to the phenomenon of Mie scattering and multiple scattering for example, the maximum value DoLPmax is in practice significantly less than 1. As illustrated in Figure 5, a DoLP Idn representation includes a first radial symmetry with respect to this projection of the star (i.e.i.e. relative to the center of the black circles on the DoLP Idn representations), as well as a second so-called plane (or axial) symmetry. Such a plane symmetry of the DoLP pattern is defined on the celestial vault as a function of the plane perpendicular to a vector OA passing through the point A associated with the star and the point O (i.e. position of system 1). On the DoLP Idn representation, such a plane symmetry is represented by a curve passing through the maximum values ​​IdoLPmax- .

[0075] In step 106, at least one grouping of temporal invariances of degrees of linear polarization of the light coming from a celestial body are determined. Each grouping of invariances is determined from a pair of Idn representations of the DoLP determined in step 104.

[0076] A grouping of invariances includes a time invariance by radial symmetry and a time invariance by plane symmetry of the degree of linear polarization of the light between the two Idn representations of the pair.

[0077] [Fig.6] represents sub-steps of step 106 of determining one or more groups of invariances of the method of determining positioning and navigation information, represented in [Fig.5], according to certain embodiments of the invention.

[0078] In substep 1062, at least one pair (or couple) of DoLP representations may be selected. Referring to the example of FIG. 5, a first pair C। of DoLP representations may be Cj ( Id^, Id2 ), a second pair C2 of DoLP representations may be, for example, C2 ( hd, Id^ ), and a third pair C3 of DoLP representations may be selected as, for example, Id^ ■

[0079] Those skilled in the art will easily understand that other pairs of DoLP representations can be selected from representations having non-successive acquisition times such as for example C4( Id^, Id^)-

[0080] Referring again to Figure 6, in substep 1064, for each distinct pair Cm, a differential representation Dm of the two selected DoLP representations can be determined. The index 'm' associated with the different pairs of selected DoLP representations (and therefore with different determined differential representations) is an integer between 1 and M, and the value of M is an integer greater than or equal to 1.

[0081] In particular, each determined differential representation Dm can be characterized by a set of calculated polarization degree differences, denoted 4vy, associated with the coordinates (x, y). For each coordinate (y, y), the difference 4Xy can be determined between the polarization degrees dxy of the two selected DoLP representations. For example, for a couple Cm (Id„, Idn+i), the differential representation Dm can thus be defined according to the following equations (08):

[0082] Dm^{Axy],Axy^dxy(Idn)-dxy(ldn+i) (08)

[0083] A set of calculated polarization degree differences 4 <y d’une représentation différentielle Dm peut être distribué, selon une répartition circulaire, comme par exemple l’ensemble de pixels de la figure [Fig.4(a)] et l’ensemble de degrés de polarisation linéaire calculés dxy.

[0084] Advantageously, step 104 of determining the representation Idn of the DoLP pattern may comprise a sub-step of applying a filter making it possible to limit the noise inherent in the acquisition of the images. For example and without limitation, such a filter may be a low-pass spatial filter and / or a temporal filter applied to an element n taking into account the elements n-1 of the previous acquisitions.

[0085] In particular, in Figure 5, the differential representations D2 and D3 represent the result of the application of the sub-steps 1062 and 1064 and are derived respectively from the DoLP representations of the selected pairs Cj(Id^ Id2C2(Id^ and C3(Jd3, Id^). The differential representations Dm are illustrated according to a gray level such that, for each coordinate (x, y), a value 4« can be included, in absolute value, between a minimum value of difference in degrees of linear polarization Ami„ and a maximum value of difference in degrees of linear polarization A„MX.

[0086] In the case where the maximum and minimum values ​​of the calculated degree of linear polarization are respectively 1 and 0, the maximum difference value Amax can be equal to or can tend towards 1 and the minimum difference value Amin can be equal to or can tend towards 0. These maximum and minimum difference values ​​depend on the time difference 5t between the two acquisition times of the images associated with the pair Cm. Thus, the different coordinates (.y, y ) associated with a set of values ​​equal to or close to the minimum difference value Amin correspond to the pro projections of points of the celestial vault where the degrees of linear polarization between the two DoLP representations of the selected couple Cm are equal or approximate values. Such a set of values ​​corresponds to the projections of points where the DoLP pattern remains invariant or approximately invariant. Thus, for a differential representation Dm, the set of values ​​equal to or close to the minimum difference value An™ then corresponds to the grouping of invariances of the degree of linear polarization of the light coming from a celestial body.

[0087] As illustrated by the two black bands on the images of the middle line of Figure 5, each grouping of invariances of a differential representation Dm can schematically comprise a first axis of invariance and a second axis of invariance. As illustrated in Figure 5, the first axis, noted ip (or iPm), then corresponds to an axis called the “plane invariance axis”. The second axis, noted ir (or inr\ corresponds to an axis called the “radial invariance axis” positioned equidistant between the two projections of the star of the two DoLP representations, noted respectively Al and A2, corresponding to a half-sphere visualization of a differential representation Dm as represented in [Fig.7].

[0088] In sub-step 1066 of figure 6, a binarization of each difference in calculated polarization degrees Av associated with the coordinates (.y, y) can be applied so as to obtain a binary representation Bm of the differential representation Dm, which makes it possible to improve the highlighting of the grouping of invariances, and in particular the highlighting of the plane invariance axis ip and the radial invariance axis ir associated with the pair Cm.

[0089] As used herein, the term "binarization" refers to the conversion of each difference value Ay associated with the coordinates (xy) into binary values. Thus, the binary representation Bm can be characterized by a set of determined binary values, denoted 0XV and associated with the coordinates (y, y). For each coordinate (X, y), the binary value Pxy can take one of two possible values, such as 0 or 1. Furthermore, the binary value 0xy can be chosen from the absolute value of the difference value Ay and a predetermined threshold of binarization SP of the difference in degrees of linear polarization. For example and without limitation, a binary representation Bm can thus be defined according to the following equations (09):

[0090] Bm= (A,,)- 0 (09) 1

[0091] A set of differences of determined binary values ​​Pxx of a binary representation Bm can be distributed, according to a circular distribution, such as for example the set of pixels of the figure [Fig.4(a)].

[0092] As illustrated in Figure 5, the binary representations B^ B2 and B3 represent the result of the application of sub-step 1066 and are derived respectively from the differential images D2 and D3. The grouping of invariances of each binary representation (represented by two white bands according to equations (09)) schematically comprises the first axis (or curve) of radial invariance (respectively iry 6-2 and 6-3) and the second axis (or curve) of plane invariance (respectively ipi, ip2 and Îp3)-

[0093] Reference is again made to the method of Figure 3. After step 106, the method for determining positioning and navigation information may comprise a step 108, in which a measurement point P is determined between one or more determined invariance groups each associated with a pair Cm of selected DoLP representations. The coordinates (x, y) of the measurement point P correspond to the projection of the celestial pole (NCP or SCP) of the celestial vault onto the plane perpendicular to the axis U of the system 1.

[0094] Indeed, the measurement point P corresponds to a calculated polarization degree difference value Ay equal to or close to the minimum value Anin, for each couple Cm considered. The measurement point P then corresponds to an invariant or approximately invariant DoLP pattern (and therefore a constant diffusion angle value P) for each couple Cm considered, that is to say for each calculated DoLP representation Idn.

[0095] In embodiments, the determination of the measurement point P can be carried out from an addition operation, applied to each coordinate (y, y), of the different binary values ​​Pxy associated with at least two binary representations Bm of the pairs Cm. In this case, the plurality of polarized images I„ comprises at least three images used to determine at least two binary representations Bm and obtain at least two invariance groupings. The determination of the measurement point P can then comprise a sub-step corresponding to the determination of an intersection representation H illustrated in FIG. 8(a). Such an intersection representation H is characterized by a set of superposition values ​​noted ty associated with the coordinates (x, y) and defined according to the following equation (10):

[0096]

[0097] A set of superposition values ​​hxy of an intersection representation H can be distributed, like the set of pixels in figure [Fig.4(a)], according to a circular distribution.

[0098] For example, in Figure 8(a), the intersection representation H comes from the binary representations B^ B2 and B^ and from a binary representation B4 not shown in Figure 8(a). The invariance groupings of each binary representation, represented by the different white bands, then intersect at an intersection point corresponding to the measurement point P.

[0099] The determination of the coordinates (a; y) associated with the measurement point P may also comprise a sub-step consisting of applying a predetermined intersection threshold sh to the superposition values ​​hxy of the intersection representation H. In this case, the intersection point Pa may then correspond to a set of coordinates (x, y) defined according to the following equation (11):

[0100] p«- | (hxy > sh) (11)

[0101] For example and without limitation, for an intersection representation H determined from binary values ​​Pxv associated with two binary representations Bm, the predetermined intersection threshold sh may be equal to 1.

[0102] Alternatively, the determination of the coordinates (x, y) associated with the measurement point P may also comprise a sub-step consisting of determining maximum superposition values ​​hxy of the intersection representation H. In this case, the measurement point P may then correspond to a set of coordinates (.X, y) defined according to the following equation (12):

[0103] p^ - (max hxy) (12)

[0104] In other embodiments, the intersection representation H can be generated directly from the differential representations Dm of the pairs Cm. The set of superposition values ​​hxy can thus be determined from the calculated polarization degree differences Axy as defined by the following equation (13):

[0105] H={l^,lhv = ïy u (c)1(13)

[0106] In this case, the application of a predetermined intersection threshold sh to the superposition values ​​hxy of the intersection representation H to determine the coordinates (.X, y) associated with the measurement point P can be defined according to the following equation (14):

[0107] pa = {xy}\(hxy <Sh) (14)

[0108] For example and without limitation, for an intersection representation H determined from binary values ​​Pxv, associated with two differential representations Dfn, the predetermined intersection threshold sh may be a value substantially greater than 0.

[0109] Alternatively, the determination of the minimum hxy superposition values ​​of the intersection representation H to determine the coordinates (x, y) associated with the measurement point P can be defined according to the following equation (15):

[0110] p« = |xy}| (min hxy) (15)

[0111] It should be noted that the superposition point is positioned at the intersection radial invariance axes (or curves) irm. Thus, the determination of the measurement point P as a function of an intersection representation H can be carried out from a prior operation of determining the radial invariance axis irm of at least two groupings among the plurality of invariance groupings generated.

[0112] In this case, the intersection representation H comes from the superposition (i.e. an addition operation for example) of the radial invariance axes (or curves) irm of each invariance grouping, which then intersect at an intersection point corresponding to the measurement point P. In figure 8(b), the radial invariance curves irm of each invariance grouping on the intersection representation H are represented by the black curves.

[0113] In particular, the determination of the radial invariance axis im of an invariance grouping associated with a pair of DoLP representations Cm{ld„, Id may comprise a prior sub-step of modeling a first curve defined in the XY plane and a second curve defined in the XY plane, distinct from the first curve. For example and without limitations, the first and second curves C(x, y) may be modeled from the maximum binary values ​​fixy of the representation binary representation Bm associated with the couple Cm, or the minimum values ​​of the differences, in absolute value, of degrees of polarization calculated zlxy of the differential representations Dm associated with the couple Cm for example.

[0114] Furthermore, the determination of the radial invariance axis of a group of invariances may comprise a sub-step of evaluating the coordinate points (x, y) of the two minimum DoLPinin values ​​corresponding respectively to the position of An and An+i on the XY plane (i.e. position of the projection of the star on the celestial vault of the representations Idn and W„+i) - In this case, for relatively small time differences ø between two polarized image acquisition times In (i.e. differences of a few hours maximum, such as two hours for example) the curve of the radial invariance axis zr(x, y) corresponds to the curve C(x, y) cutting (i.e. crossing or intercepting) perpendicularly the line y) formed by the coordinates (x, y) of the points An and An+i.

[0115] The determination of the radial invariance axis of a group of invariances can also comprise a sub-step of evaluating one or more curves called 'plane symmetry curves', with coordinates (x, y), each plane symmetry curve being relative to the maximum values ​​DoLPmax of one of the two DoLP representations (in particular Idn and / or Idn+A) and corresponding to the associated plane symmetry. In this case, the determination of the radial invariance axis may comprise a sub-step of distance measurement (for example Euclidean distance) of a first distance value between the first curve C(x, y) and the one or two plane symmetry curves, and of a second distance value between the second curve C(x, y) and the one or two plane symmetry curves. Thus, for time differences 5t between two relatively small polarized image acquisition times In (i.e. differences of a few hours maximum, such as for example two hours), the curve of the radial invariance axis ir(x, y) may correspond to the curve C(x, y) associated with the highest distance value.

[0116] In certain embodiments, the determination of the measurement point P can be carried out from the point P, of intersection between the curve of the radial invariance axis ir(x, y) and the curve of the plane invariance axis ip(x, y) of a grouping of invariances resulting from a pair Cm, as illustrated in FIG. 8(c). In this case, the determination of the measurement point P can then comprise a step of applying drift corresponding to the shift of the coordinates y) of the intersection point Pj on the radial invariance axis ir(x, y) according to a drift value r. The drift value z can for example be predefined according to the calendar date (i.e. the declination of the star) of the acquisition of the images associated with the pair Cm.

[0117] The positioning and navigation method further comprises a step 110 of determining the altitude of the celestial pole and / or the azimuth of the celestial pole from the obtained measurement point P. In particular, step 110 may consist of determining the latitude coordinate and / or the direction of the geographic pole of the system 1 from the obtained measurement point P. This step may be implemented by the positioning and navigation information determination unit 20 of [Fig.l].

[0118] The elevation angle or altitude of the celestial pole (NCP or SCP) on the celestial vault centered at point O is equivalent to the angle 3O defined with respect to the XY plane perpendicular to the U axis of system 1. In the case where the U axis is equivalent to the Zenith Z axis, as illustrated in figure 9 where point O is represented at the center of a reference (E,N,Z) (meaning East, North, Zenith according to the English name), the altitude of the celestial pole corresponds to the latitude coordinate ÿ of the geographical position of system 1.

[0119] Furthermore, the direction axis passing through point O and the projection of the celestial pole onto the XY plane perpendicular to the U axis of system 1 corresponds to the direction of the geographic pole (e.g., the North Pole, NP, or the South Pole, SP). As shown in Figure 9, the direction of the geographic pole can be described by the angle aP relative to the reference direction of the acquisition module used to acquire the plurality of images. It should be noted that aP is also called the azimuth of the pole. celestial and corresponds to the N axis of the (E,N,Z) reference frame as illustrated in [Fig.9].

[0120] For example and without limitation, the determination of the latitude coordinate of the system 1 may comprise a sub-step consisting of determining the length Lxy between the coordinates (x, y) associated with the measurement point P and the coordinates (X, y) of the center of the image ln, as well as the application of a calculation function to the length Lxy associated with the measurement point P. The celestial vault being considered as a half-sphere, the latitude coordinate of the system 1 may then be defined according to the following equation (16):

[0121] <pa=M(Lxy) (16)

[0122] The calculation function can be predefined according to the parameters associated with the step 102 of acquiring the polarized images ln. For example and without limitations, in the case of using an “equidistant fisheye” lens of focal length f in this acquisition step, as shown in figure [Fig.4(b)], the calculation function M of the latitude coordinate $o of the system 1 can be defined according to the following equation (17):

[0123] ^=2.^)(17)

[0124] In embodiments, a step of determining one or more simulations frf of the pattern of degrees of linear polarization of the light of the entire celestial vault centered at point O can be carried out from the DoLP representation Idn of only a part of the sky and from equation (02) of the DoLP pattern according to the Rayleigh scattering model. In this case, the step 106 of determining the grouping of invariances can be applied directly to the simulations / J* of the DoLP pattern.

[0125] Such Jdsfi simulations are useful when one or more In images include meteorological and environmental obstacles (such as clouds, buildings, etc.). Such hf simulations make it possible to supplement DoLP Idn representations truncated by these obstacles, the truncated parts potentially including the desired celestial pole position. Such obstacles can also significantly modify the degree of linear polarization of the surrounding light, the simulations then making it possible to correct the deviations due to these modifications.

[0126] Such simulations are useful when one or more images In are acquired from a localized sensor. Advantageously, an image can also be produced from an image In comprising few acquisition points Pxy, i.e. an image In comprising two to a few tens of pixels.

[0127] In embodiments where the U axis is not equivalent to the Zenith Z axis, the Determining the latitude coordinate of the system 1 may comprise a sub-step consisting of correcting the length Lxy to take into account the inclination of the plane associated with the pixels of the image ln relative to the surface of the Earth. In particular, the method may comprise implementing a calibration of the orientation of the system 1. Such a calibration may be implemented from the measurement of the gravity vector at the geographical position of the system 1, for example. Such a calibration may also be implemented beforehand at a position of known latitude and heading to determine the celestial pole and deduce the orientation of the U axis. Finally, such a calibration may further be implemented from the use of ephemerides.

[0128] In embodiments, a step of acquiring displacement data, i.e. rotation and translation data, of the system 1, can be implemented, during the acquisition of successive polarized images In. In this case, step 106 of determining the invariance groupings and / or step 108 of determining the measurement point P can be applied by taking into account these rotation data and these translation data according to a maximum value of a few kilometers. For example and without limitation, for two selected DoLP representations of the same pair Cm(jdn, Idn+i), if the system 1 has undergone a rotation of + 90° relative to the axis U between the acquisition times and tn+i, the sub-step 1064 of determining a differential representation Dm can be carried out after applying a rotation of - 90° of the DoLP representation Idn+ï in the plane associated with the pixels of the image In.

[0129] In embodiments, the method for determining positioning and navigation information may further comprise a step of determining the longitude of the system 1 from the data of a referenced clock and determining the coordinates y) of the measurement point P for a system 1 positioned successively at at least two distinct geographical positions Oi and O 2, on the surface of the Earth.

[0130] [Fig. 10] represents sub-steps of the step of determining the longitude of the system 1, according to certain embodiments.

[0131] The referenced clock may be a “Polar Wheatstone Clock” type clock. In this case, the step of determining the longitude may comprise, in sub-steps 124 and 128 respectively, the calculation of the value of the direction of linear polarization of light (AoLP) at the celestial pole (coordinates (x, y) of the measurement point P) then the determination of the solar time for the system 1 at the geographical position Oi (positioning carried out in step 122) and of the solar time wo2 for the system 1 at the geographical position O2 (positioning carried out in step 126). It should be noted that solar time corresponds to the angle between the arcs of the great circles connecting the star to the celestial pole and the celestial pole to the zenith (i.e. the 'zenith-celestial pole' vector), which is equivalent to the angle between the polarization vector at the celestial pole and the great circle arc connecting the celestial pole and the zenith.

[0132] The sub-step 124 of determining the longitude may also comprise the initialization of an internal clock in the system 1, in the step of determining the coordinates of the measurement point P, for a system 1 positioned at the geographical position Op. Such an internal clock makes it possible to determine the time difference tx between the determination of the coordinates of the measurement point P, for a system 1 positioned at the geographical position Ob and the determination of the coordinates of the measurement point P, for a system 1 positioned at the geographical position O2.

[0133] In response to the movement of the system 1 from the geographical position Oi to the geographical position O2 and to the determination of the hour angle M<>2, the step of determining the longitude may further comprise a comparison value Am of the solar times and ^2 respectively at the geographical positions Oi and O2. The comparison value may be for example the difference between the solar time (V«2 determined at the geographical position O2 and the sum between the time difference tx and the solar time determined at the geographical position Op The comparison value Aw may then be defined according to the following equation (18):

[0134] Am=^2-(m^ + a) (18)

[0135] In embodiments, the time difference tx may be, for example and without limitations, less than or substantially equal to one week. In this case, the variations in the equation of time are small and the solar hours aLi and a't>2 may be compared according to equation (18) without taking into account the variation in duration between the two measurements.

[0136] In other embodiments, the maximum time difference between coordinate determinations of the measuring point P does not verify the approximation of small variations of the equation of time. In this case, the solar hours and (0o2 can be transformed into mean solar hours and w'^2 from the equation of time given by the ephemeris. Thus, the mean solar hours and û / o2 can be compared according to equation (18).

[0137] In sub-step 130, the longitude difference A4 between the coordinate longitude toi of system 1 of the geographical position Oi and the coordinate longitude / en of system 1 of the geographical position O2 can be determined according to the comparison value Am of the hour angles.

[0138] In sub-step 132, in the case where the coordinate longitude (o\ of system 1 at geographic position Oi is a known datum of system 1, the coordinate longitude of system 1 at geographic position O2 can be determined by function of the value of longitude Z 01 and the difference in longitude A Z.

[0139] Figure 11 represents an example of structure of the system 1 for determining positioning and navigation information positioned at the coordinate longitude lo and at the coordinate latitude according to certain embodiments of the invention.

[0140] The system 1 for determining positioning and navigation information, shown in FIG. 11, may comprise a polarized light acquisition module 210 configured to carry out the acquisition of successive polarized images In. The polarized light acquisition module 210 notably comprises a mathematical model called 'camera optical model' associated with the projection of the celestial sphere in the 2D plane, that is to say in the XY plane perpendicular to the optical axis of the system 1. The polarized light acquisition module 210 may, for example and without limitation, be a digital single-lens reflex camera comprising a so-called "fisheye" lens making it possible to ensure acquisition over a 180° viewing angle.In this case, the polarized light acquisition module 210 may comprise an objective 212 and a detector 214 such that the optical model for determining the coordinate (x, y) of a projection point of the vault on the detector corresponds to the ratio between the focal length of the objective and the angle of the projection point with the optical axis. Furthermore, the detector 214 of the polarized light acquisition module 210 may have a generally rectangular or square geometry. However, for such a so-called circular "fisheye" objective, the useful area of ​​the detector 214 generally forms a circular area as shown in the figure [Fig.4(a)]. In addition, each pixel of the sensor of the digital camera may comprise a 4-pixel grouping sensitive to a specific polarization at 0°, 90°, 45° and 135° for example.Alternatively, the camera may include a polarizing filter and the acquisition of each polarized image corresponds to the acquisition of 4 images taken for 4 different polarizer angles at 0°, 90°, 45° and 135° for example.

[0141] The celestial pole determination device 10 may comprise a processing module 220 configured to perform the calculation and determination of the coordinates (X, y) of the measurement point P and the different representations necessary during the method, comprising the DoLP representations Idn, differential representations Dm, binary representations Bm, intersection representation H.

[0142] The unit 20 for determining the positioning and navigation information may comprise a module 230-1 for determining the latitude coordinate of the geographic position O of the system 1 and a module 230-2 for determining the azimuth of the celestial pole.

[0143] In embodiments, the position information determination system 1 operation and navigation may also include a correction module 240 configured to generate simulations of the pattern of degrees of linear polarization of the light of the entire celestial vault centered at point O from DoLP Idn representations of only a part of the sky and from equation (02) of the DoLP pattern according to the Rayleigh scattering model.

[0144] The correction module 240 can also be configured to detect an offset between the optical axis U associated with the polarized light acquisition module 210 and the Zenith axis Z. In the case of the detection of such an offset, the correction module 240 can be configured to apply a correction of the lengths Lxy and £ to take into account the inclination of the plane associated with the pixels of the image In relative to the surface of the Earth.

[0145] In embodiments, the positioning and navigation information determination system 1 may also include a movement and movement acquisition module 250 configured to move the system 1 to different geographic positions on Earth.

[0146] The displacement and displacement acquisition module 250 can thus be configured to carry out the acquisition of translation and rotation data of the system 1 during and / or after the displacement of the system 1. For this, the module 250 can comprise physical measurement units such as accelerometers, a gyroscope, a gravity sensor, etc.

[0147] Advantageously, the system 1 can be used to perform geopositioning in civil or military applications of autonomous navigation on unmanned platforms such as terrestrial drones, aerial drones positioned at high altitude or low altitude, or submarines positioned at a maximum depth defined by the spectral band predefined by the system 1. Such unmanned platforms can have automatic or remote-controlled piloting. Such use in an application of the invention to geopositioning combining an inertial navigation system and celestial pole determination (or heading determination) makes it possible to compensate for positioning and orientation errors of the unmanned platforms generated over time by exclusive inertial navigation. Furthermore, the system 1 can be used to measure the orientation of an object on earth relative to the geographic pole.

[0148] In embodiments, the system 1 for determining positioning and navigation information may also comprise a memory module 260 configured to save all of the acquisitions, measurements, and calculations performed. The memory module 260 may also comprise the referenced clock enabling the determination of the longitude of the system 1.

[0149] In embodiments, the system 1 may include one or more data input / output interfaces, such as human-machine interfaces 270 (HMI). These human-machine interfaces may comprise one or more means of acquiring or means of transmitting information, such as input and control devices (e.g. microphone, loudspeaker, keyboard) and / or one or more display devices (e.g. video screen, touch screen, etc.). For example and without limitation, the system 1 may comprise a human-machine interface 270 configured to generate a display of the longitude ( o and latitude ÿo coordinates at point O, intended for an operator of the system 1.

[0150] Figures 12 and 13 illustrate the results obtained by implementing the method for determining positioning and navigation information, according to certain embodiments of the invention.

[0151] In particular, figure 12 illustrates the results of step 1066 of binarization of the differences in degrees of polarization calculated to obtain the binary representations Bm and to highlight the groupings of invariances of degrees of linear polarization between the representations DoLP of couple Cm. Figure 12 also illustrates the results of step 108 of determination of the measurement point P carried out from the generation of an intersection representation H by operation of addition of the binary representations Bm.

[0152] Figure 13 is an example of graphs illustrating the results of the values ​​of direction of the geographic pole aP and of the values ​​of latitude 0 determined by the method of [Fig.3] for a system 1 positioned at point O, as a function of the rotation of the system 1 around the optical axis U.

[0153] Those skilled in the art will understand that the method, according to the embodiments of the invention, can be implemented in various ways by hardware, software, or a combination of hardware and software, in particular in the form of program code that can be distributed as a program product, in various forms. The program code can be distributed using computer-readable media, which can include computer-readable storage media and communication media. The methods described in the present description can be implemented in particular in the form of computer program instructions executable by one or more processors in a computer computing device. These computer program instructions can also be stored in a computer-readable medium.

[0154] The invention is not limited to the embodiments described above as a non-limiting example. It encompasses all the variant embodiments that may be envisaged by those skilled in the art. In particular, those skilled in the art will understand that the invention is not limited to the different steps of the method and to the different modules of the system described as non-limiting examples.

Claims

Claims

1. Method for determining the temporal invariances of the degree of linear polarization of light coming from a star in a celestial vault, implemented in a system for determining positioning and navigation information, said celestial vault being centered on the geographical position of said system, the system having an optical axis (U), characterized in that the method comprises at least the following steps consisting in: • acquiring (102) a plurality of successive polarized images ln of at least a part of said celestial vault; • determining (104), for each image / „ acquired, a representation of the degrees of linear polarization Idn;• determining (106) at least one grouping of temporal invariances of degrees of linear polarization of the light coming from said star, a grouping of invariances being determined from a pair of representations selected from said plurality of representations of the degrees of linear polarization Idn; • determining (108) a measurement point P from said at least one grouping of invariances, the measurement point P corresponding to the celestial pole in the celestial vault of said system, the method further comprising a step of determining the altitude and / or the azimuth of said celestial pole from the measurement point P obtained, said azimuth corresponding to the direction of the geographic pole associated with the geographic position of said system.;

2. The method of claim 1, wherein said positioning and navigation information determining system is geographically positioned at a coordinate latitude and wherein the method further comprises a step of determining the latitude coordinate $ of said system from said determined altitude.

3. Method according to one of the preceding claims, in which each polarized image In is acquired at an instant the minimum time difference between two instants of acquisition of two successive polarized images being predefined.

4. Method according to one of the preceding claims, in which each polarized image I„ comprises a set of pixels Pxy, each pixel being associated with coordinates (y, y) and comprises a set of detection intensities according to distinct polarization angles such that p^ = / [ 35O), and in which each representation Idn comprises a set of degrees of linear polarization dxy, each degree dxy being calculated from the Stokes vector, such that: , 'and dx>~ ( ) xy ( 'œ ) + ( ^9œ ) xv ' (51 ) xy ~ ) vv ' ( '

5. Method according to one of the preceding claims, in which each grouping of invariances is determined from a differential representation Dm comprising a set of differences Ay calculated from said pair of representations of the degrees of linear polarization Idtl selected.

6. Method according to claim 5, in which each grouping of invariances is determined from a binary representation Bm determined from said differential representation Dm and a predetermined threshold of binarization SP of the difference in degrees of linear polarization Æy.

7. Method according to one of claims 1 to 6, wherein said plurality of polarized images ln comprises at least three images, wherein at least two groupings of invariances of degrees of linear polarization of the light coming from said star are determined, and wherein said measurement point P is determined from the intersection between said at least two groupings of invariances.

8. Method according to claims 6 and 7, wherein said intersection is determined from an intersection representation H characterized by a set of superposition values hxy, and calculated from at least two of said binary representations Bm, and wherein said measurement point P corresponds to the coordinates for which the superposition value hxy is maximum.

9. Method according to one of claims 1 to 6, wherein said plurality of polarized images ln comprises at least two images, said grouping of invariances comprising a radial invariance by radial symmetry and a plane invariance by plane symmetry of the degree of linear polarization of the light coming from said star, the radial invariance and the plane invariance each being represented by a curve, and in which said measurement point P is determined from a point P, of intersection between said curve representing the radial invariance and the curve representing the plane invariance, and from information on the offset of said celestial pole relative to said point of intersection P^

10. Method according to one of the preceding claims, in which the method comprises a step of determining at least one simulation of the pattern of degrees of linear polarization of the light coming from said star over the whole of said celestial vault, each simulation being determined from one of the images of said plurality of polarized images I„ of at least a part of said celestial vault.

11. Method according to one of the preceding claims, in which the method comprises a step consisting of carrying out the acquisition of displacement and rotation data of said operated system between the successive acquisition of at least two of said polarized images In, and in which the determination of said at least one grouping of invariances comprises taking into account said displacement and rotation data.

12. Method according to one of the preceding claims, in which said geographical position of said system is further associated with a longitude of coordinate l 0, and in which the method comprises a step consisting of determining the coordinate (0 of longitude of said system from data of a referenced clock, and from the direction of polarization at the level of the measurement point P of said system positioned successively at the level of at least two distinct geographical positions.

13. System (10) for determining positioning and navigation information, characterized in that the system comprises: • a polarized light acquisition module (210), said light coming from a star in the celestial vault centered on the geographical position of said system, the system having an optical axis (U), said module (210) being configured to carry out the acquisition of a plurality of successive polarized images I„ of at least a part of said celestial vault; • a processing module (220) configured to determine, for each image In acquired, a representation of the degrees of linear polarization hln, said module (220) being further configured to determine at least one grouping of temporal invariances of degrees of linear polarization of the light coming from said star, each grouping of invariances being determined from a pair of representations Idn selected from said plurality of representations of the degrees of linear polarization Idn, said module (220) being further configured to determine a measurement point P from said at least one grouping of invariances, the measurement point P corresponding to the celestial pole in the celestial vault of said system; the system further comprising a positioning and navigation module (230) configured to determine the altitude and / or azimuth of said celestial pole from the measurement point P obtained, said azimuth corresponding to the direction of the geographic pole associated with the geographic position of the system.

Images