Method and device for aiding the navigation of a carrier using an invariant Kalman filter and a navigation state of a second carrier

FIELD OF THE INVENTION

The present invention relates to a method for aiding the navigation of a carrier using an invariant Kalman filter.

STATE OF THE ART

The problem of estimating the state of a physical system is generally posed as follows. The state of the system at time n is represented by a vector X n , and an observation available at time n is represented by another vector Y n . The evolution of the system is written:

Xn + 1 P)

where / is a known function (generally called the propagation function), which may depend on sensor measurements. The observations Y n are linked to the state of the system by a known observation function:

Yn = KX n )

Building a good estimate X n of the state X n from the sequence (Y n n³0 is in general a difficult problem, which can nevertheless be simplified in certain cases.

One calls “linear systems” the particular case of the systems of the form:

Xn + i— FX n -F w n

Y n = HX n + V n

where F is a propagation matrix, H is an observation matrix, w n and V n are noise disturbing the predictions and the measurements.

In this linear case, a known method consists in constructing an estimator referred to as the Kalman filter. This Kalman filter implements the following calculations:

where the indices n + l | n and n + l | n + l respectively denote the estimate calculated at time n + 1 without taking into account the observation Y n + 1 and at time n + 1 taking into account account of the observation Y n + 1 . The matrix K n is called the “gain matrix”, it can be calculated using a Riccati equation. The estimation error is then defined as:

e n \ n = X n ~ Xn \ n (after taking into account the observation Y n ) e n + i \ n = Xn + 1 - Xn + i \ n (before taking into account the observation Y n + 1 )

We can easily verify that this error follows the following evolution:

e n + i \ n = F e n \ n (before taking into account the observation Y n + 1 )

(after taking into account the observation Y n + 1 ) where / designates the identity matrix.

The above equations do not depend on X n , so we can build an estimator that works for any real trajectory of the system, which is not the case for any nonlinear system.

In the case of a nonlinear system, an ordinary Kalman filter cannot be implemented. A variant of the Kalman filter called an “extended” Kalman filter has thus been proposed, which is suitable for a nonlinear system. However, when an extended Kalman filter is used, the simplifications observed in the linear case no longer occur, so that an error equation involving X n and X n is obtained . This problem is at the root of most of the discrepancies encountered when using an extended Kalman filter.

Nevertheless, a second particular case making the estimation problem easier is that of “group-affine” observation systems, that is to say the systems for which we have an internal composition law (i.e. that is to say an operation which will be noted in the following by a star * ) defined on the space of the state considered and such that the following two properties are verified:

at. The propagation function / verifies for any pair a, b of elements of the state space the relation:

/ (a * b) = f (a) * / (/ d) _1 * f (b)

Where Id is the identity element of the group induced by the law * .

b. The observation function h is of the form h (X) = Z (x, y 0 ), with y 0 an element of the observation space (to which the Y n belong ) and / (.,.) An action group, i.e. a function verifying

Under these two conditions, we can define an extended Kalman filter called “invariant” (generally called more simply “invariant Kalman filter”) which is governed by the following equations:

where exp (·) is the exponential map (this function is known as soon as the internal composition law is known, if it defines a Lie group) and K n is a “gain matrix” as in the linear case. We can then show that the estimation error will have there

also an autonomous evolution, as in the linear case. The state estimation problem is therefore simplified, even if the system considered is not linear.

When condition b. is not verified we can use a filter of the form

In practice, a Kalman filter is generally implemented on a system which does not satisfy condition a. , but which approaches a system which satisfies this condition well for it a.

Invariant Kalman filters have thus been used to aid in the navigation of a carrier. An invariant Kalman filter used in such a navigation aid context estimates a navigation state representative of a movement of the wearer considered.

The use of an invariant Kalman filter requires finding an internal composition law * for which the conditions a. and B. are checked or nearly checked in order to make the estimation problem easier. There is no generic method for finding such an operation, and various publications have attempted to provide the correct operation for particular systems. By way of example:

a) The operation to be used for an orientation and speed estimation problem is described in the following document: Bonnable, S., Martin, P., & Salaün, E. (2009). Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC / CCC 2009. Proceedings of the 48th IEEE Conférence on (pp. 1297-1304). IEEE.

b) The operation to be used for inertial navigation with estimation of orientation, speed and position is described in the following document: Barrau, A., & Bonnabel, S. (2017). The invariant extended Kalman filter as a stable observer. IEEE Transactions on Automatic Control, 62 (4), 1797-1812.

c) The operation to be used for SLAM (acronym for Simultaneous Localization And Mapping), i.e. navigation using fixed landmarks from the wearer's environment, is described in the document Barrau, A ., & Bonnabel, S. (2015). An EKF-SLAM algorithm with consistency properties. arXiv preprint arXiv: 1510.06263.

d) A list of practical systems for which an operation is known is further described in Barrau, A., & Bonnabel, S. (2017). Linear observation Systems on groups (I).

In each of these applications, proprioceptive sensors of a carrier and an invariant Kalman filter are used to estimate the state of navigation of this carrier.

These applications are of two types. Some of them use kinematic data expressed in a reference frame in which the carrier is mobile. However, this kinematic data is not always available. The other applications do not use such kinematic data, but, on the other hand, do not make it possible to estimate a state of navigation of the carrier in the reference frame; these applications can only produce navigation states relating to an initial situation.

DISCLOSURE OF THE INVENTION

An aim of the invention is to overcome the drawbacks mentioned above.

There is therefore proposed, according to a first aspect of the invention, a method for aiding the navigation of a first carrier stationary relative to a second carrier which is itself mobile in a reference frame, the method comprising the steps following procedures implemented by the first carrier:

• receipt of a navigation status in the reference frame of the second carrier provided by the second carrier,

• acquisition of motion data of the first wearer by at least one proprioceptive sensor of the first wearer,

Estimation of a navigation state of the first carrier by an invariant Kalman filter using the navigation state of the second carrier as observation data, in which the navigation state of the first carrier comprises:

o first variables representative of a first rigid transformation linking a frame of reference attached to the first wearer to the reference frame of reference, and o second variables representative of a second rigid transformation linking a frame of reference attached to the second carrier to the frame of reference attached to the first carrier, and

in which the invariant Kalman filter uses as internal composition law a law comprising a term-to-term composition of the first rigid transformation and of the second rigid transformation.

The proposed method advantageously exploits the fact that the first carrier is stationary relative to a second carrier for which an estimated navigation state is already known, constituting information of an absolute nature, that is to say relative to the reference frame in which the second carrier is mobile.

This method has the advantage of operating even if the first carrier does not know his position relative to the second carrier (in other words when the first carrier and the second carrier are not harmonized).

It also happens that the law of internal composition used in this process makes it possible to approach the conditions a. and B. set out in the introduction. The calculations implemented by the invariant Kalman filter are therefore particularly simple to implement.

The method according to the first aspect of the invention may further comprise the following optional features, taken alone or in combination when technically possible.

Preferably, the first variables include a rotation matrix representing an orientation of the first carrier, and the second variables include a rotation matrix representing the orientation of the second carrier relative to the first carrier.

Preferably, the first variables comprise a position vector of the first carrier, and the second variables comprise a position vector of the second carrier relative to the first carrier.

Preferably, the navigation state of the estimated first carrier further comprises a speed vector of the first carrier in the reference frame.

Preferably, the internal composition law applies identical transformations to one of the position vectors and to the speed vector.

Preferably, the navigation state of the second carrier received comprises a position vector of the second carrier in the reference frame and a rotation matrix representing an orientation of the second carrier in the reference frame.

Preferably, the invariant Kalman filter uses an innovation comprising a vector Z x of the form:

Z = log (îf 1 oyof 21 )

or :

• f is an estimate of the first variables,

• f 21 is an estimate of the second variable, in which the second rigid transformation makes it possible to go from the first frame of reference to the second frame of reference,

• Y designates the observation data used by the invariant Kalman filter,

• o designates a composition,

• log (·) denotes a logarithm map of a Lie group.

Preferably, the navigation state of the second carrier received comprises a speed vector of the second carrier in the reference frame and a rotation matrix representing an orientation of the second carrier in the reference frame, the movement data of the first carrier, acquired. the proprioceptive sensor, include an angular velocity of the first wearer.

Preferably, the first variables include a rotation matrix representing an orientation of the first carrier, and the second variables include a rotation matrix representing the orientation of the second carrier relative to the first carrier, and the invariant Kalman filter uses an innovation comprising a vector Z v of the for

or

• Ri is an estimate of the rotation matrix representing an orientation of the first carrier.

• R 2 I is an estimate of the rotation matrix representing the orientation of the second carrier with respect to the first carrier,

• x 21 is an estimate of a translation vector defining the second rigid transformation,

• R 2 is the rotation matrix representing an orientation of the second carrier in the reference frame,

• v 2 is the speed vector of the second carrier in the reference frame,

• w is the angular speed of the first carrier,

• x denotes a cross product,

• log (·) denotes a logarithm map of the Lie group SE (3).

Preferably, the vector Z v consists of a first vector Z of size 3 representative of a rotation, and of a second vector Z of size 3 as well, and in which the invariant Kalman filter uses an innovation Z of the shape:

Preferably, the navigation state of the first estimated carrier further comprises at least one error state specific to the proprioceptive sensor, and the internal composition law is additive for this error variable specific to the proprioceptive sensor.

Preferably, the first carrier is an aircraft and the second carrier is an aircraft carrier, or the first carrier is a projectile and the second carrier is an aircraft which carries the projectile, or the first carrier is an inertial unit of a vehicle. and the second carrier is an exteroceptive sensor of the vehicle.

Preferably, the estimation step is implemented in parallel by several invariant Kalman filters, so as to obtain several estimates of the navigation state of the first carrier, and furthermore comprises steps of:

• for each estimate, determination of a likelihood metric of the estimate,

• production of a consolidated estimate of the navigation state of the first carrier based on the estimates and their associated likelihood metrics.

It is also proposed, according to a second aspect of the invention, a device for aiding the navigation of a first carrier stationary relative to a second carrier which is itself mobile in a reference frame, the device comprising:

• at least one proprioceptive sensor configured to acquire movement data from the first wearer,

• a communication interface for receiving a navigation status from the second carrier provided by the second carrier,

• an invariant Kalman filter configured to estimate a navigation state of the first carrier using the navigation state of the second carrier as observation data, in which the navigation state comprises:

o first variables representative of a first rigid transformation linking a frame of reference attached to the first wearer to the reference frame of reference, and o second variables representative of a second rigid transformation linking a frame of reference attached to the second carrier to the frame of reference attached to the first carrier, and

and wherein the invariant Kalman filter uses as its internal composition law a law comprising a term-to-term composition of the first rigid transformation and of the second rigid transformation.

DESCRIPTION OF FIGURES

Other characteristics, aims and advantages of the invention will emerge from the following description, which is purely illustrative and non-limiting, and which should be read with reference to the appended drawings in which:

• Figure 1 schematically illustrates two carriers and markers.

• Figure 2 schematically illustrates internal components of the carriers of Figure 1, according to one embodiment of the invention.

FIG. 3 is a flowchart of steps of a method of aiding navigation, according to one embodiment of the invention.

In all of the figures, similar elements bear identical references.

DETAILED DESCRIPTION OF THE INVENTION

1 / Description of the system

Referring to Figure 1, a system comprises a first carrier 1 and at least one second carrier 2 stationary relative to the first carrier 1.

In what follows, different marks are considered: a first mark attached to the first carrier 1, a second mark attached to the second carrier 2, and a reference mark in which the two carriers 1, 2 are movable. The reference frame is for example a celestial frame attached to stars or to the earth.

In FIG. 1, the second carrier is very schematically represented by a parallelepiped, and the first carrier by an object of smaller dimensions placed on the first carrier. This representation is however only schematic, and the two carriers can be of various types.

In one embodiment, the first carrier is an aircraft, for example an airplane or a helicopter, and the second carrier is an aircraft carrier, for example a ship of the aircraft carrier type on which the aircraft is placed.

In another embodiment, the first carrier is a projectile, for example a missile, and the second carrier is an aircraft which carries the projectile, for example an airplane or a helicopter.

In another embodiment, the first carrier is an inertial unit of a vehicle, for example terrestrial, and the second carrier is an exteroceptive sensor of the same vehicle.

With reference to FIG. 2, the first carrier 1 comprises a data communication interface with the second carrier 2. The communication interface 4 is of the wireless radio type (Wi-Fi, bluetooth) and comprises for example an antenna. As a variant, this interface is wired.

The first wearer 1 comprises at least one proprioceptive sensor 6. Each proprioceptive sensor used is configured to acquire movement data from the first wearer in the first frame of reference. These data typically include angular velocities, and accelerations.

The proprioceptive sensor comprises for example an inertial unit which comprises a plurality of inertial sensors such as gyrometers and accelerometers.

As a variant or complement, the proprioceptive sensor comprises at least one odometer.

The first carrier 1 also comprises a data processing unit 10. The processing unit 10 is arranged to process data received by the communication interface 4.

The data processing unit 10 typically comprises at least one processor configured to implement a navigation aid method which will be described below, by means of a Kalman filter of invariant type. The invariant Kalman filter typically takes the form of a computer program executable by the processor of the data processing unit. The general operation of an invariant Kalman filter is known per se. However, it will be seen below that the internal composition law used to configure the invariant Kalman filter implemented by the processing unit 10 is chosen in a particular way.

Preferably, the processing unit 10 is suitable for implementing several Kalman filters in parallel.

Furthermore, the second carrier 2 comprises means for estimating a navigation state of this second carrier in the reference frame. These means are also known. For example, the means described in the document entitled “Aided navigation: GPS with high rate sensors” by Jay Farell, published in 2008, could be used.

The second carrier 2 further comprises a communication interface with the first carrier 1, of the same type as the communication interface X.

2 / Configuration of the invariant Kalman filter

The invariant Kalman filter implemented by the processing unit 10 is configured to estimate a navigation state of the first carrier 1 in the reference frame.

The navigation state of the first carrier comprises first variables representative of a first rigid transformation linking the first frame of reference (attached to the first carrier 1) to the reference frame of reference, and second variables representative of a second rigid transformation binding the second frame of reference (attached to the second carrier 2) at the first mark.

As is well known, a rigid transformation (also known under the name of affine isometry), is a transformation which preserves the distances between each pair of points of a solid. Thus, each of the first and second rigid transformations can be characterized by the composition of a rotation and a translation.

In the following, an embodiment will be described in detail in which the navigation state of the first carrier, denoted X l t, comprises the following elements:

Xi— (R 1 , V 1 , X 1 , R 21 , X 21 )

or:

• R ± is a rotation matrix representing the orientation of the first carrier in the reference frame of reference (component of rotation of the transformation which makes it possible to go from coordinates of a point in the first frame of reference to coordinates of the same carrier point in the reference mark),

• is a speed vector representing the speed of the first carrier in the reference frame,

• x 1 is a position vector representing the position of the first carrier in the reference frame (it is the translation component of the transformation which makes it possible to go from coordinates of a point in the first frame to coordinates of the same point in the reference frame),

• R is a matrix of rotation representing a relative orientation between the first reference mark and the second reference mark. We will take as an example here a matrix making it possible to go from the first reference mark (attached to the first carrier 1) to the second reference mark (attached to the second carrier 2), the reverse being perfectly possible.

• x 21 is a lever arm between the first carrier 1 and the second carrier 2. It is a translation vector making it possible to go from the coordinates of a point in the first frame of reference to the coordinates of the same point in the second landmark.

In this particular embodiment, the first variables are R l f x 1 and v 1 ; and the second variables are R 21 , x 21 .

In the following we consider that the first rigid transformation is T = (R l y x ± ) and that the second rigid transformation is Tl1 (R 21 ) x 2l

The invariant Kalman filter is further configured to use as observation data a navigation state X 2 of the second carrier 2, expressed in the reference frame.

The navigation state of the second carrier 2 typically comprises

• a rotation matrix R 2 representing the orientation of the second carrier in the reference frame (component of rotation of the rigid transformation which makes it possible to go from coordinates of a point in the second frame of reference to coordinates of the same carrier point in the reference mark),

• x 2 a position vector representing the position of the second carrier in the reference frame (it is about the component of translation of the rigid transformation which makes it possible to pass from coordinates of a point in the second reference mark to coordinates of the same bearing point in the reference frame).

It is possible to include a speed vector v 2 representing the speed of the second carrier in the reference frame, in this state. Nevertheless, it will be considered in the following that this state does not include such a speed.

The observation of the invariant Kalman filter is then written

Y = T 2 = (R 2 I X 2 )

The invariant Kalman filter is configured to use as internal composition law, denoted * , a term-to-term composition of the first rigid transformation and of the second rigid transformation.

This composition law can be extended to the speed vector of the first carrier, when the latter is also included in the navigation state of the first carrier. In this case, the law of internal composition * applies identical transformations to one of the position vectors and to the speed vector.

The law of internal composition is applied to states (R ll v ll x ll R 21l x 21 ) and

It can be seen here that the second term and the third term of the product of the law of internal composition * are of the same form due to the fact that the positions and the speeds are treated by this law in the same way.

3 / Procedure for assisting the navigation of the first carrier

With reference to FIG. 3, a method for aiding the navigation of the first carrier 1 according to one embodiment, and implementing an invariant Kalman filter configured as indicated in section 2 /, comprises the following steps.

It is assumed that an estimate X of the navigation state of the first carrier 1 has been estimated by the invariant Kalman filter.

It is also assumed that the second carrier has estimated a navigation state X 2 of this second carrier in the reference frame, using its internal means 12. This navigation state X 2 comprises the rigid transformation T 2 formed by the couple (R 2 , x 2 ).

In a step 102, the communication interface 4 of the first carrier 1 receives from the communication interface 14 of the second carrier 2 the data representative of the rigid transformation T 2 . These data are then transmitted to the processing unit 10.

In a step 104, the processing unit 10 calculates the innovation Z of the invariant Kalman filter, as follows

Z = Z x = log (îf 1 oyof 21 )

or :

• Y designates the observation data used by the invariant Kalman filter, which, we recall, correspond to T 2 )

• o indicate the operator of composition of the rigid transformations,

• log () denotes a logarithm map within the meaning of the theory of Lie groups, known to those skilled in the art.

This innovation calculation is particularly advantageous because it makes it possible to satisfy the conditions a. and B. set out in the introduction.

In a correction step 106, the data processing unit 10 multiplies the innovation Z by a so-called “gain” matrix K, which translates Z into a linear correction dX 1 = KZ to be applied to the state of the system. X 1 .

The choice of earnings is a classic question common to most estimation methods (see below).

In a retraction step 108, the processing unit 10 transforms the linear correction dX 1 into a non-linear correction C of the same nature as X (state X is not a vector because it contains rotations). The transformation used is any function taking as argument a vector of the dimension of state 1 (15 in our case) and returning an object of the same nature as X but a particularly powerful choice is the term-to-term exponential of the group of Bind couples of rigid transformations.

A nonlinear update step 110 is then implemented by the processing unit 10. In this step 110, the processing unit 10 combines the estimate X of the state of the system with the nonlinear correction C to build a corrected estimate:

cΐ = X 1 * C 1

The gain matrix K is chosen so as to stabilize the nonlinear estimation error e defined by:

e = X 1 ~ 1 * 1

Where the symbol r 1 is the usual inversion associated with the law of internal composition *. In this embodiment, an invariance to the left of the estimation error e is obtained. It is of course possible to envisage modifying the preceding equations in order to obtain a right-hand invariance (the left-hand invariance being, however, a preferred embodiment).

In an acquisition step 1 12, the proprioceptive sensor 3 also acquires movement data from the first wearer 1 in the first frame. These movement data typically include accelerations and / or speeds, for example angular. These acquired movement data are transmitted to the processing unit 10.

Step 1 12 can be implemented before, during or after any one of steps 102, 104, 106, 108, 110.

In a propagation step 114, known per se to those skilled in the art, the processing unit 10 generates a propagated navigation state, from the state Xf. For this, the processing unit 10 applies, in a manner known per se, an evolution model resulting from an integration of the data acquired by the proprioceptive sensor 6.

The steps described above form an iteration of the invariant Kalman filter.

Thanks to the invariant Kalman filter, we obtain a property that we would also obtain in a linear case: the evolution of the estimation error is autonomous (it depends neither on X nor on ¾).

The navigation state transmission from the second carrier is repeated over time, so that these states are received by the first carrier 1.

The processing unit 10 repeats these same steps, 104, 106, 108, 1 10, 1 12, 1 14 in new iterations of the invariant Kalman filter, for each new state of the second carrier received. The state estimated during the propagation step 1 12 of a given iteration is used as input data for the innovation calculation 104 and nonlinear update 110 steps of a following iteration.

Ultimately, thanks to the method, the first carrier 1 can obtain an aid on his own navigation using the data which the second carrier 2 already has on his own navigation.

Several Kalman filters are advantageously implemented in parallel by the processing unit 10, so as to obtain several estimates of the navigation state of the first carrier 1.

It should be noted that certain processing operations can be carried out only once for all the Kalman filters in play. In particular, one of these processing operations is the resolution of a Ricatti equation known to those skilled in the art.

In this case, the processing unit determines, for each estimate, a deviation metric from the data of the estimate.

This metric L is written for example:

L = Z T S ~ 1 Z

Where S is the covariance of innovation Z as conventionally calculated in the step of updating a conventional Kalman filter.

The metric L can be calculated for a given measurement. As a variant, the metric L is the sum of the values ​​obtained over a set of past measurements.

In a merging step 116, the processing unit 10 produces a consolidated estimate of the navigation state of the first carrier as a function of the estimates and their associated likelihood metrics L.

In one embodiment, the estimate obtained by one of the filters which has the metric which reflects the smallest deviation from the data is selected as the consolidated estimate.

In another embodiment, the consolidated estimate is an average of the estimates determined by the different filters, which is weighted by the metrics.

Since the estimated states X are not vectorial in nature, those skilled in the art will be able to use an average adapted to the case of varieties.

In other variants, it is possible to supplement the estimated state X with other interesting variables, for example an error state specific to the proprioceptive sensor (bias, scale factor, drift, etc.).

In this case, the law of internal composition * is additive for the proprioceptive sensor error state.

Let B be the proprioceptive sensor error state considered.

The law of internal composition then becomes:

Moreover, as already indicated above, the matrix R 21 can be replaced by a matrix R 12 which makes it possible to go from the second mark (attached to the second carrier 2) to the first mark (attached to the first carrier 1). Likewise, the vector x 21 can be replaced by a vector x 12 , which is a translation vector making it possible to pass from the coordinates of a point in the second frame of reference to the coordinates of the same point in the first frame of reference.

We also set T12— (R l2 ' x 12 ) - T 21 ·

In this case the law of internal composition, expressed in the new variables, becomes:

In this case, innovation Z becomes:

Z = log (f 1 _1 ° Z ° ffz 1 )

In another embodiment, the proprioceptive measurements of the carrier 1 include an angular velocity w. In addition, the navigation state X 2 of the second carrier 2 received comprises a speed vector of the second carrier 2 in the reference frame and a rotation matrix representing an orientation of the second carrier in the reference frame. The invariant Kalman filter then uses an innovation Z of the form:

or :

• R t is an estimate of the rotation matrix R t contained in the rigid transformation 7. More precisely, R t is such that the coordinates u Ë IR 3 of a point in the frame of reference attached to the carrier 1 become R u + x 1 in the reference frame, where x 1 is the position of the first carrier in the frame of reference.

• R 2 I and x 21 are the estimates of the rotation matrix R 21 and of the translation vector * 21 defining the rigid transformation T 21 = (R 2i , ^ 2i) · More precisely, R 21 and x 2l are such that the coordinates u Ë IR 3 of a point in the frame attached to carrier 1 become R 2l u + x 21 in the frame attached to carrier 2,

• R 2 is the rotation matrix representing the orientation of wearer 2 (available as an observation, as indicated above). More precisely, R 2 is such that the coordinates u Ë IR 3 of a point in the frame attached to the carrier 2 become R 2 u + x 2 in the reference frame, where x 2 is the position of the carrier 2 in the frame reference (which it does not need to be observed for this claim).

• v 2 is the speed of carrier 2 in the reference frame (available as an observation)

• is the estimated speed of carrier 1 in the reference frame.

• x denotes the classical cross product

• log (·) denotes the logarithm function (reciprocal of the exponential function) of the Lie group SE 3). It should be noted that the object passed to the input of the log function is an object belonging to the Lie group SE 3) and written as a couple made up of a rotation matrix and a vector of size 3.

This other embodiment makes it possible to come very close to the conditions a. and B. stated in the introduction, and to satisfy them only under the assumption that the second carrier is “flat”, that is to say under the assumption that the angular speed measurements w are always on the same axis, and that this axis is also that of rotation T? 21 . This hypothesis is verified especially if all considered rotations (7? ! And R 2 ) have a vertical axis, as is usually the case on a land vehicle. This is the reason why the embodiment in which we have Z = Z X is more advantageous.

The vector Z v consists of a first vector Z of size 3 representative of a rotation, and of a second vector Z% of size 3 as well.

It is also possible, in another embodiment, to combine the data described above to form a more complex innovation Z of the following form:

CLAIMS

1. Method for aiding the navigation of a first carrier (1) stationary relative to a second carrier (2) which is itself mobile in a reference frame, the method comprising the following steps implemented by the first carrier (1) ):

• reception (102) of a navigation state (X 2 ) in the reference frame of the second carrier (2) supplied by the second carrier (2),

• acquisition (1 12) of movement data of the first wearer (1) by at least one proprioceptive sensor (6) of the first wearer (1),

• estimation of a navigation state (of the first carrier (1) by an invariant Kalman filter using the navigation state (X 2 ) of the second carrier (2) as observation data, in which the navigation state of the first carrier (1) comprises: o first variables representative of a first rigid transformation (TJ linking a frame of reference attached to the first carrier (1) to the reference frame, and o second variables representative of a second rigid transformation (T 21 ) linking a marker attached to the second carrier (2) to the marker attached to the first carrier (1), and

in which the invariant Kalman filter uses as internal composition law a law comprising a term-to-term composition of the first rigid transformation and of the second rigid transformation.

2. Method according to the preceding claim, wherein the first variables comprise a rotation matrix representing an orientation of the first carrier (1), and the second variables comprise a rotation matrix representing the orientation of the second carrier (2) relative to the first carrier (1).

3. Method according to one of the preceding claims, wherein the first variables comprise a position vector of the first carrier (1), and the second variables comprise a position vector of the second carrier (2) relative to the first carrier (1). )

4. Method according to one of the preceding claims, wherein the estimated navigation state of the first carrier (1) further comprises a speed vector of the first carrier (1) in the reference frame.

5. The method of claim 4 in its dependence on claim 3, wherein the internal composition law applies identical transformations to one of the position vectors and to the speed vector.

6. Method according to one of the preceding claims, wherein the navigation state (X 2 ) of the second carrier (2) received comprises a position vector of the second carrier (2) in the reference frame and a rotation matrix. representing an orientation of the second carrier in the reference frame.

7. Method according to one of the preceding claims, in which the invariant Kalman filter uses an innovation comprising a vector Z x of the form:

Z = log (îf 1 oyof 21 )

or :

• T is an estimate of the first variables,

• f 21 is an estimate of the second variable, in which the second rigid transformation makes it possible to go from the first frame of reference to the second frame of reference,

• Y designates the observation data used by the invariant Kalman filter,

• o designates a composition,

• log (·) denotes a logarithm map of a Lie group.

8. Method according to one of the preceding claims, wherein the navigation state (X 2 ) of the second carrier (2) received comprises a speed vector of the second carrier (2) in the reference frame and a rotation matrix representing an orientation of the second carrier in the reference frame, and in which the motion data of the first carrier (1), acquired by the proprioceptive sensor (6), comprises an angular speed of the first carrier (1).

9. Method according to the preceding claim, wherein the first variables comprise a rotation matrix representing an orientation of the first carrier (1), and the second variables comprise a rotation matrix representing the orientation of the second carrier (2) with respect to the. first carrier (1), and in which the invariant Kalman filter uses an innovation comprising a vector Z v of the form:

or

• R t is an estimate of the rotation matrix representing an orientation of the first carrier (1).

• R 21 is an estimate of the rotation matrix representing the orientation of the second carrier (2) relative to the first carrier (1),

• x 21 is an estimate of a translation vector defining the second rigid transformation,

• R 2 is the rotation matrix representing an orientation of the second carrier in the reference frame,

• v 2 is the speed vector of the second carrier (2) in the reference frame,

• w is the angular speed of the first carrier (1),

• x denotes a cross product,

• log (·) denotes a logarithm map of the Lie group SE (3).

10. The method of claims 7 and 9 taken in combination, wherein the vector Z v consists of a first vector Z R V of size 3 representative of a rotation, and a second vector Z% of size 3 also. , and in which the invariant Kalman filter uses an innovation Z of the form:

11. Method according to the preceding claims, in which the estimated navigation state of the first carrier (1) further comprises at least one error state specific to the proprioceptive sensor, and in which the internal composition law is additive for this variable of error specific to the proprioceptive sensor.

12. Method according to one of the preceding claims, wherein:

• the first carrier (1) is an aircraft and the second carrier (2) is an aircraft carrier, or

• the first carrier (1) is a projectile and the second carrier (2) is an aircraft which carries the projectile, or

• the first carrier (1) is an inertial unit of a vehicle and the second carrier (2) is an exteroceptive sensor of the vehicle.

13. Method according to one of the preceding claims, in which the estimation step is implemented in parallel by several invariant Kalman filters, so as to obtain several estimates of the navigation state of the first carrier (1). , and also includes steps of:

• for each estimate, determination of a likelihood metric of the estimate,

• production of a consolidated estimate of the navigation state of the first carrier (1) based on the estimates and their associated likelihood metrics.

14. Device for aiding the navigation of a first carrier (1) stationary relative to a second carrier (2) itself mobile in a reference frame, the device comprising:

• at least one proprioceptive sensor (6) configured to acquire movement data from the first wearer (1),

• a communication interface (4) for receiving a navigation state (X 2 ) from the second carrier (2) supplied by the second carrier (2),

• an invariant Kalman filter (10) configured to estimate a navigation state (¾) of the first carrier (1) using the navigation state of the second carrier (2) as observation data, in which the state of navigation includes:

o the first variables (7 ^) representative of a first rigid transformation linking a frame of reference attached to the first carrier (1) to the reference frame, and

o second variables (T 21 ) representative of a second rigid transformation linking a frame of reference attached to the second carrier (2) to the frame of reference attached to the first carrier (1), and

and wherein the invariant Kalman filter uses as its internal composition law a law comprising a term-to-term composition of the first rigid transformation and of the second rigid transformation.