DOMAINE GENERAL

The invention relates to the field of inertial navigation.

The invention relates more particularly to a method for estimating the navigation state of a mobile carrier using an extended Kalman filter (EKF commonly abbreviated in the literature), and a suitable inertial unit to implement such process.

STATE OF THE ART

An inertial navigation system is a device for providing real-time information on the status of a carrier: its position, velocity, orientation, etc. This information can be used to drive the carrier.

A known solution for estimate is to be implemented in an inertial an extended Kalman filter that provides an estimated condition of the wearer by merging data from inertial sensors (accelerometers, gyroscopes), and non-inertial sensors suitable for the type of carrier (odometer, baro-altimeter, Doppler radar, GPS receiver, etc.). The inertial data act as controls, and non-inertial data that observation.

The Kalman filter is an algorithm typically comprising a prediction step and an update step repeated over time: the predicting step calculates a current state from a previous state and input commands; updating step refines the current status with comments. prediction calculations are based on a wave equation modeling the dynamics of the state based on the state and control. Update calculations are based on an observation equation modeling the observation function of the state.

In the Kalman filter, the propagation equations and observation are linear and take the form of matrices not depending on the estimated state.

In the extended Kalman filter, at least one of these functions is nonlinear. Data fusion is obtained by linearizing these functions to the propagation steps and update. The calculations are therefore carried out using dies that depend on the estimated state.

The extended Kalman filter can then be regarded as an algorithm making a share of non-linear calculations and also linear calculations in a Kalman filter within the extended Kalman filter. The state predicted by the nonlinear propagation equation will be in the following named "global state". The state estimated by the Kalman filter will be named "linearized state." The overall condition can be corrected by the information carried by the state linearized during stages of registration.

A Kalman filter (expanded or not) further calculates a margin of error associated with the estimation of the state taking into account an estimate of the statistical error due to many factors such as the initial errors and noise inertial sensors used. The statistical error at a given time can be represented as a point cloud in the state space, each point corresponding to a possible embodiment. To estimate the margin of error, the plant has a Gaussian statistical error model characterized at each moment with zero mean and a covariance matrix. The latter defines the envelope of an ellipsoidal volume in the state space, centered on zero error, and containing 99 97% of estimated errors in each direction in the state space. This envelope is then called "shell 3 sigma".

The real error statistical law is unknown and characterized by another envelope 3 sigma. Generally, this amount is not centered on zero and its shape is not ellipsoidal.

Figure 1 shows the variation of an estimated envelope 3 sigma and 3 sigma envelope true at different Kalman filter iterations.

The envelope true 3 sigma must be contained in the envelope 3 sigma estimated. This is a criterion of consistency to respect at all times and for all the achievements of the random variables of the system.

In general, the Kalman filter improves with comments its estimate in time, and envelopes 3 sigma tend to flatten out along certain lines of the so-called state space "observable", as shown in Figure 2. The axes are orthogonal to them are called "unobservable". The envelope 3 sigma estimated then assumes a shape elongated along the unobservable axes, which creates correlations between estimated errors.

Although the imperfection of the model used by the filter can play an important part, it was found that nonlinearities and system noise often prevent the criterion of consistency described above to be respected when a Kalman filter expanded is used (e.g., at time T3 in Figure 2, the casing 3 sigma true is no longer located within the casing 3 sigma estimated along an unobservable axis).

In the long term, these inconsistencies may become important and spread to other axes of the vector space of the state to estimate.

These inconsistencies usually appear only in particular conditions depending for example on the wearer's trajectory and initial conditions. These conditions are suitable for the observation and can be very difficult to predict. They are even stronger than the observation is nonlinear.

The cause inconsistencies is known: the linearization step inherent in an extended Kalman filter, is noisy by the estimation errors. Indeed, this linearization is performed at the last estimate produced by the filter, which is estimated by definition imprecise data. The second order changes produced on the linearized by functions that noise estimation while changing observations and cause inconsistencies.

In one case, an observable variable can be considered by the unobservable filter but this case is very unlikely; and in a second case, a non-observable variable may be considered observable by the filter. This reduces the envelope 3 sigma estimated along unobservable axes while it should maintain its original value. But this does not reduce the error true cloud along the same axis as the latter are actually unobservable. Part of increasing importance of true cloud mistake then may no longer be within the envelope 3 sigma estimated soon as it begins to shrink. The consequences of this inconsistency can be significant and depend on the initial allocation of long unobservable axes in default.

A solution of the state of the art for in most cases compensate for the reduction of the envelope 3 sigma estimated is to apply a model of noise. EKF algorithm has indeed an operation to increase the covariance prediction phase using a model of noise matrix to account for variables and dynamics that have not been modeled in the filter. In this issue of consistency, the noise matrix is ​​diverted from its function as the reduction of the estimated envelope is not due to a modeling problem but a problem of observability. In addition, for reasons of architecture and digital calculation accuracy, the sounds are often applied to the diagonal of the model noise matrix: As the covariance matrix, the diagonal of this matrix corresponds to the linearized state the components followed by the filter. Accordingly, noise is not applied only in the direction unobservable axes. As a result, the problem of consistency may not be fully resolved, and moreover it undermines the accuracy of estimates.

Indeed, noise applied comprises a projection on unobservable axis and a projection on observable axes, these projections may change over time depending on changes unobservable axes. More projection according observable axes is large, and the asymptotic value of the error is large according to these directions, and hence less precision of the estimates is correct.

The value of noise to apply for its part is not known. An empirical value is chosen, usually large to accommodate all known problem cases. This compounds the problem of accuracy.

On the other hand, as pattern noise is not applied in the right direction, the estimated envelope elongates in a direction that moves away from the unobservable true axis. Accordingly, this ellipsoidal envelope, which certainly becomes more important as its main axes may no longer cover the cloud of real errors. So it may remain an inconsistency.

The problem of observability is treated according to the prior art with a solution that does not completely solve the problem, and that more affects the precision of the estimates.

PRESENTATION DE L'INVENTION

The invention aims to estimate the overall state of a movable carrier governed by a non-linear model, while minimizing the occurrence of inconsistencies defined in the introduction.

It is therefore proposed a method for estimating a navigation state to more of a mobile carrier vary according to the method of the extended Kalman filter, comprising the steps of:

- acquiring measurements of at least one variable,

- Extended Kalman filtering producing a current estimated state and a covariance matrix defining in the space of the navigation state a region error, from a previous estimated state, an observation matrix, d a transition matrix and the captured measurements, the method being characterized in that it comprises a step of transition of the matrix of adjustment and the observation matrix prior to use in the extended Kalman filter, so that the adjusted matrices satisfy an observability condition which depends on at least one of the variables of the state of the holder, the observability condition being adapted to prevent the Kalman filter to reduce the size of the region along at least one non-observable axis of the state space,

wherein the observability condition to be checked by the transition matrices and observation is adjusted kernel invalidity of observability matrix associated therewith, and

wherein adjusting comprises the steps of:

- calculating at least one primary base unobservable vectors from the preceding estimated state,

- for each array to be adjusted, calculation of at least one matrix spread associated with the matrix from the primary base vectors

offset of each matrix to adjust according to the standard matrix which is associated therewith so as to check the condition of observability.

The method according to the invention not only improves the consistency of the estimates, but also makes it possible to improve the accuracy of estimates, because the part model noise used according to the state of the art to solve the problems observability can be canceled.

Furthermore, this process does not change very little software architecture of the EKF filter, and can be implemented as a lightweight software update on material already in operation. This update is the addition of an adjustment based on 2 matrices, and a reduction in noise model settings.

The invention can also be supplemented with the following characteristics, taken alone or in any of their technically possible combinations.

The extended Kalman filtering step may comprise the substeps of:

- Propagation of the preceding estimated state in a state predicted by the adjusted transition matrix,

- the predicted state linearization of a nonlinear model to produce the observation matrix before adjustment,

- adjustment of the observation matrix produced by linearization.

The fact implement linearization after the propagation step allows to ensure that this linearization is performed at a point

the most likely vector space: the predicted state produced by the propagation step. Therefore, the adjustment of the observation matrix reflects this spread, and provides more accurate results than if linearization is implemented before the propagation step in the prior estimated state.

The adjustment may further comprise a step of orthogonalizing the primary database in a secondary basis vectors, the latter being stored from one cycle to the other, and the matrix spread associated with the observation matrix being calculated from the secondary base vectors.

The matrix spread associated with the observation matrix may be the sum of several independent matrix differences, each difference matrix being calculated from a vector of the sub-base of its own.

The method steps may be repeated in successive cycles. A given cycle, said current cycle, can then comprise the steps of:

- storing the secondary base vectors and orthogonalizing coefficients which are also produced by the step of orthogonalizing and

- transformation of the primary basis vectors in a tertiary base vectors by orthogonalizing coefficients stored during a previous cycle,

- the matrix spread associated with the transition matrix is ​​implemented from a secondary base stored in the previous cycle and of the tertiary base calculated in the current cycle.

The matrix spread associated with the transition matrix may also be the sum of several independent elementary matrix differences, each difference matrix being calculated from a secondary carrier from the secondary base stored in the previous cycle and a vector tertiary tertiary base calculated in the current cycle, the secondary and tertiary vectors being capable of elementary matrix gap.

For at least one matrix to be adjusted, one can further calculate:

- a plurality of candidate matrix differences, each candidate difference matrix being calculated from a primary base non- observable respective vectors, and

- a plurality of metric, each metric being representative of an amplitude adjustment induced by a matrix difference respective candidate, the matrix shift step of adjusting is implemented in a matrix difference elected among the candidates matrix deviations according to the computed metric.

The chosen matrix difference can be the difference matrix candidate associated with the metric representative of the amplitude of the lower adjustment.

The shift may only be implemented only if the metric of the chosen matrix difference is less than a predetermined threshold.

According to another aspect of the invention there is provided an inertial measurement unit comprising a plurality of sensors and an estimating module configured to estimate a navigational state of the inertial unit multivariate by implementation of the method according to the first aspect of the invention.

DESCRIPTION OF FIGURES

Other features, objects and advantages of the invention will become apparent from the following description, which is purely illustrative and not exhaustive, and should be read in conjunction with the accompanying drawings wherein:

- Figures 1 and 2, already discussed, are two envelopes 3 sigma changes over time, during the implementation of an extended Kalman filter.

- Figure 3 schematically shows a carrier embedding an inertial unit according to an embodiment of the invention.

- Figure 4 is a block diagram of an estimator according to an embodiment.

- Figure 5 is a block diagram of a linearization and adjusting block included in the estimator of Figure 2.

- Figure 6 is a block diagram of a sub-block shown in FIG 5.

- Figure 7 illustrates a geometric justification adjustment matrices used by the Kalman filter.

- Figure 8 shows a flow of data corresponding to the implementation of the steps illustrated in Figure 3.

Of all the figures, similar elements bear identical references.

DETAILED DESCRIPTION OF THE INVENTION

Referring to Figure 3, an inertial unit IN is embedded on a mobile carrier P, such as a land vehicle, a helicopter, an airplane, etc.

IN the inertial unit comprises several parts: Cl inertial sensors, additional sensors CC, and means E for implementing estimation calculations. These parts can be physically separated from each other.

The inertial sensors are typically Cl accelerometers and / or gyroscopes measuring respectively specific forces and rotational speeds experienced by the wearer relative to an inertial frame. The specific force corresponds to the acceleration of non-gravitational origin. When these sensors are fixed with respect to the carrier, the plant is called type "strap down".

Additional sensors CC are variable depending on the carrier type, its dynamics, and the intended application. The inertial systems typically use a GNSS (GPS for example). For a land vehicle, it is also one or more odometers. For a boat, it can be a "loch", giving the boat speed relative to that of water or the seabed. The cameras are another example of sensors.

The navigation E estimation means will subsequently designated as the "estimator".

The output data of the estimator E are a carrier of the navigation state and optionally internal states of the inertial unit.

The estimator E includes an extended Kalman filter EKF configured to merge the information provided by the additional sensors and inertial sensors so as to provide an optimal estimate of the navigation information.

The fusion is made by a continuous dynamic model system to predict the state at each instant with the aid of a function f nonlinear propagation, and how to observe it by means of an observation function h can also be non-linear (on this subject see Appendix 5 remembering some principles of dynamic systems), this function depending on the type of sensor used CC:

u (t) represents the input command e consists of the specific force and angular velocity. These two quantities are measured by the sensors Cl discretely in time according to a period 7 ^.

The overall status
includes inter alia the position coordinates, velocity, and orientation of the carrier in the form of one or more rotations, each represented for example by a matrix or quaternion attitude.

The estimator E operates a discrete version of these equations continuous time obtained according to the rules of the state of the art:

The EKF filter operates iteratively according to a prediction step taking into account the measures Cl sensors and an update step taking into account the measurements of DC sensors:

The predicting step comprises predicting an estimated global status, and the prediction of an associated covariance.

Prediction estimated overall state:

Prediction associated covariance:

which corresponds to the transition matrix dependent function

spread f, and Q model the noise matrix.

The update step uses the following variables:

Innovation :

Covariance innovation :

where H k is the observation matrix.

Gain de Kalman :

The update is implemented as follows:

Updated estimated overall state:

Update covariance associated:

theoretical transition matrix:

with:

observation matrix:

The estimator E although close to this algorithm is slightly different. The update of the overall state may not be fully additive manner, in particular to preserve the properties of rotation matrices, including also to reflect the fast pace action Cl sensors. The state prediction estimated overall can be so at a faster pace than other operations. The difference in rate can depend on the dynamics of the wearer. Other differences are possible. For example, an approximation can be made on the transition matrix.

Is illustrated in Figure 4 an estimator E according to an embodiment comprising various functional blocks corresponding to respective stages of its implementation.

The estimation is implemented in successive iterations, each iteration being identified by an index k.

Predicting the overall condition estimated using a rate 1 is performed in step 100.

A memory cycle 1 and a registration step operate unabated. This registration step takes into account the updated data at a rate 2 slower.

All other treatments work at the rate 2. Step 200 converts the rate of 1 states states cadence 2. cadence 2, the latter condition from the last update

Kalman filter 400, are in the order predicted the global state, the linearized state predicted, the overall status updated, the linearized state updated. The state x
may in some cases be always 0.

Transition matrices and observation are calculated in the linearization step 300. A sub-step of adjusting, specific to the invention and described in detail below, is added.

Innovation is calculated from the overall state and predicted from observation DC sensors shown at the bottom of Figure 4.

The other steps of the EKF are grouped in step 400 performing estimates using arrays, and constituting a Kalman filter, the estimates of linearized states and covariances. A special feature is the fact that step 400 does not include a prediction of linearized condition, which may optionally be incorporated into step 100 to rate 1.

Step 401 involves the prediction of the covariance associated with the predicted global state and operates the transition matrix. Step 402 includes calculating the covariance innovation, Kalman gains, the update of the covariance of the overall state and updating of the linearized state. This step exploits the observation matrix.

The points of linearisation in step 300 can be calculated based on the last state from the last

updating the Kalman filter 400. If, then and

A first linearization point X ^ k _ x calculates in step

300 the transition matrix to the period corresponding to Kalman

the cadence 2. This linearization point is the overall state estimated after the last resetting.

A second linearization point x H k is used to calculate the cycle of

Kalman k in step 300 the observation matrix This point of

linearization is obtained according to the calculations of step 100. It is the last before the next predicted state registration.

To simplify the notation, the transition matrix

will now be recorded and the observation matrix denoted

désormaisH
therefore each express in terms of a global state corresponding to their respective linearization points

As previously stated, the steps are repeated recursively to yet the next iteration k + 1 at the rate 2 rate of the Kalman filter.

Adjustment of the transition matrices and observation

The linearization step implemented in each Kalman cycle is a weak point of the estimator E.

Also, the estimation method comprises in step 300, an additional step of adjusting in accordance with a condition of observability, coming adjust before use by the Kalman filter, matrices generated by this step linearization : on the one hand the transition matrix for propagating the covariance matrix, and on the other hand the observation matrix.

The observability condition is a constraint on the core of the observability matrix (the explanation of this matrix is ​​shown in Appendix 6) through an adjustment of the transition matrices and observation to include in the core of a predetermined pattern unobservable subspace. Design of unobservability consists of an equation in the state space of a subspace represented by a base, using a global state, this subspace being updated at each cycle of Kalman. The inclusion of this sub-space in the core of the observability matrix makes no observable and decreases the risk of inconsistency if the model is relevant.

The overall state from which is calculated the subspace rendered unobservable is the point linearization of the observation function (see Appendix 6). An unobservable vector has indeed zero image through the observation matrix and vector corresponds to a linearized state in the vicinity of a global state corresponding to the point linearization of the observation function.

During the adjusting step, it will calculate, for each of the transition matrices and observation, a corresponding matrix gap.

By convention, we use the following notations stellate to designate the transition matrices and observation obtained by the adjustment.

The flow of data needed for the adjustment calculation is shown schematically in Figure 8, highlighting the fact that the adjustment of the observation matrix depends on the point of linearization of the observation function in the cycle of Kalman current, and that the adjustment of the transition matrix depends on the point of linearization of the observation function in the current cycle and the previous cycle.

It is therefore necessary to use a base B k generating the unobservable subspace at cycle k, and a base B k _ 1 generates the vector subspace unobservable in cycle k-1.

It is then possible, on a Kalman cycle, to memorize the linearization point and generate two times per cycle unobservable basis, as shown in Figure X. It is possible to equivalently a cycle Kalman , memorize the unobservable base so as not to generate it once per cycle. This second option is the block diagram of Figure 6. It details the adjustment functions described in Figure 5.

Figure 6 details the calculations leading to the production of two matrix differences relating to transition matrices and observation to the Kalman cycle time k.

The transition matrix of the Kalman cycle k-1 to k cycle and the cycle observation matrix k are adjusted according to a model of non-observability using the global state vectors X and linearization points of

observation function cycles k-1 and k. This model allows to estimate, in a step 301, the non-viewable space linearized statements as an unobservable primary database. Such a base may be calculated each

Kalman cycle using an analytic function obtained in the filter design according to the method presented in Annex 6 by means of a dynamic system continuously in time, and then applied to the dynamic system at discrete time used by the central navigation.

Two examples of risk situations that can be treated by the process of the invention are described in Appendix 1. An unobservable own basic model static alignment is developed in Appendix 2.

Simplifying assumptions are necessarily made in developing this model during the design phase because no equation containing a limited number of variables can not represent the real world. It is thus possible that the true error is reduced cloud slowly in all directions of the state space but EKF, due to nonlinearities and noise estimation, estimates a more rapid decrease according certain axes, creating a mismatch. It is then possible to model the slow decay at an axis with zero decay in time and consider this unobservable axis to be able to impose a constraint. This simplification slightly reduced accuracy but ensures consistency estimates. But as it n '

The static alignment described in Appendix 1 is an example of a situation in which the model does not take into account small movements

imperceptible which in the short term have no effect, but that can be observed in the long term, which causes a slow decay of the envelope 3 sigma real on some routes. It is suggested by the model simplification worries, not to take into account and model an unobservable basis considering that these small movements are zero, which is an example of simplification. In the second example of Annex 1, the wave motion can be neglected in the same way to model the unobservable basis.

If the basis
is orthogonal, there is a solution described in Annex

3 for adjusting the dies in a simple manner. However, this condition in the general case, is not satisfied. The proposed solution is to orthogonalizing the basis and then performing the adjustment. The treatments described below are justified by paragraph at the end of Annex 3 dealing with non-orthogonal basis.

In a step 302 orthogonalization, the primary database
is orthogonalized into a secondary base
according to a variant of the method of

Gramm-Schmitt, the distinction from the fact that the vectors are not made because standardization unit is implicitly included in the adjustment formula presented dies thereafter. This generates a coefficient matrix used in the orthogonalization.

Let q be the number of vectors of the unobservable primary database. We ask:

In cycle k, the orthogonalization step can implement the following calculation:

This base
as well as the orthogonalizing coefficients are
stored in step 303 during a Kalman cycle.

The base
used to calculate in step 308 a matrix difference? H k on the observation matrix and without inversion of one

matrix, which can reduce the computational load. This step can be omitted if the observation matrix is ​​independent of the estimated state. In this case, the difference matrix is ​​zero.

It is possible to decompose the matrix difference? H k on the observation matrix k , q in basic matrix differences:

This decomposition makes it possible to parallelize the calculation of each unit difference matrix and so shorten the calculation of step 308.

Each elementary matrix difference can, according to Appendix 3, be calculated as follows: For i between 1 and

Furthermore, at cycle k, the base
and the coefficients
(stored in the previous cycle Kalman k-1) can reconstruct an unobservable base
in a reconstruction step referenced 304. This base reconstructed at time k depends on the base primary at time k, but also orthogonalizing coefficients at time k-1. Its index is observed
corresponding to the time k knowing estimated at time k-1. Bases and are used to calculate the matrix inversion gap without

matrix.

On pose :

In cycle k, the reconstruction step 304 may implement the following calculation

In a step referenced 306, is calculated a matrix difference relating to the transition matrix.

It is possible to decompose the matrix gap on the matrix

of transition q basic matrix differences:

Each elementary matrix gap may, according to Annex 3, be calculated as follows:

For i between 1

This decomposition makes it possible to parallelize the calculation of each unit difference matrix and thus can help shorten the calculation of step 306.

To reduce memory usage and calculations can not be considered a sub-matrix of the transition matrix: This in correspondence with the non-zero values ​​unobservable vectors.

Both matrix deviations obtained are used to verify an important condition of observability condition is detailed in Appendix 7.

Furthermore, Annex 3 explains the origin of the method of calculation and Annex 8 gives a mathematical proof.

The step of calculating 306 further produces a metric representative of an amplitude adjustment induced by the matrix difference relating to the transition matrix.

The step of calculating 308 further produces a metric representative of an amplitude adjustment induced matrix gap on the observation matrix.

The same type of metric calculation can be implemented for the transition matrix and the observation matrix. For example, for the transition matrix, separately calculates the square of the norm of the matrix difference, and that of the transition matrix. The metric corresponds to the ratio of these two values, or the square root of the ratio of these two quantities.

The standard may be using weights to normalize the quantities involved in the state vector. For example, a position error may be in the order of a few meters, then an error of attitude perhaps on the order of a few milli-radians. Norm of the matrix may correspond to the norm of the vector that would be formed by keeping all the terms of the matrix in the form of a column vector (standard Froebenius).

Appendix 4 explains how to use weights in the calculation of metrics.

The calculation of metric called possibly the weighting functions and if the observation matrix is ​​independent of

the estimated state, the difference matrix are \. zero, and its metric is zero. It is

no need to calculate. Identically, if the transition matrix is ​​independent of the estimated state, the calculation of the metric associated may be omitted.

Returning to FIG 5, an offset step 330, the transition matrix is ​​summed to the associated matrix difference, so as to obtain an adjusted transition matrix; and the observation matrix is ​​summed to the associated matrix difference, so as to obtain an observation matrix adjusted.

The sequence 310 illustrated in Figure 6 produces a pair of matrix differences and a pair of metrics may be implemented n times in parallel.

We assume that we know, for n predetermined situations in the DC area, an unobservable base model. This model is derived from solving a system of equations representative of a respective type of observation and optionally a respective path of the state vector, including a method is given in Annex 6.

In a decision step 320, one of the n matrix is ​​eluted gap pairs (e.g. the pair obtained by the corresponding adjustment to the position i), and the shifting step is implemented only with this pair matrices to produce the adjusted matrices.

The decision step 320 sends a plausibility check for each pair of matrix differences, inspecting the value of the corresponding metrics. The test is to take the decision to recognize a given situation.

The chosen deviation matrix is ​​the matrix associated with the candidate difference metric representative of the amplitude of the lower adjustment. Thus, the adjustment is made with the matrix spreads the most "plausible". The likelihood criterion may relate to the adjustment of the transition matrix and / or the observation matrix.

Furthermore, the offset is implemented only if the metric of the chosen matrix difference is less than a predetermined threshold: it can indeed be preferable not to operate shift transition matrices and observation, if this difference is not considered sufficiently reliable and introduces an additional noise in the system rather than it corrects.

The estimation method above can be implemented by a computer program executed by the means for calculating the inertial IN. Furthermore, the card 1 identification comprises a computer program product comprising program code instructions

for executing the steps of the method described, when this program is executed by the card 1 identification.

ANNEX 1: Examples of situations at risk of an inertial navigation operating an EKF filter

A first example relates to an inertial navigation phase static alignment the ground. In this phase, the local vertical is estimated by measuring the gravity vector using accelerometers, and orientation of the control unit is estimated by measuring the angular velocity vector of Earth's rotation using gyroscopes. The observation used in this phase is typically the ZUPT (Zero Velocity Update) is an observation of a virtual sensor indicating that the speed of the carrier is zero with respect to the Earth. The measurement error is modeled by white noise. However, it happens that the inertial sensors record small non-random movements due for example to wind or expansion effects due to heat. This corresponds to an error of deterministic action not taken into account in the non-inertial sensor model producing the ZUPT observations. Then there is inconsistency between the true and dynamic system modeled dynamic system. As the duration of the static alignment is short, the inconsistency is negligible. There is, against numerous applications requiring an alignment of several hours to several days, weeks or months. In this case, it is possible that the inconsistency becomes predominant. There is, against numerous applications requiring an alignment of several hours to several days, weeks or months. In this case, it is possible that the inconsistency becomes predominant. There is, against numerous applications requiring an alignment of several hours to several days, weeks or months. In this case, it is possible that the inconsistency becomes predominant.

A second example relates to an inertial navigation system mounted on a vessel surface a projection operator in speed observation in the boat mark. This non-inertial measurement is generally made using a "Loch" sensor and this is more a relative measure compared to water. The fact that it performed in projection in the boat landmark addictive measurement vis-à-vis the attitude of the boat. It is then possible, assuming that only this observation is used, an inconsistency appears between the estimates and the real world EKF filter, and that this inconsistency produces significant errors after several hours when the ship sails to steady course .

APPENDIX 2: Model axes unobservable static alignment

This appendix modeling unobservable basis for resolving the inconsistency of the first example in Schedule 1 using the proposed method.

Assumptions: The Kalman filter operates the error model PHI and mechanization works self azimuth.

The following notations are used:

Components of states involved in the equations are presented in an arbitrary order that is:

The measurements are acquired speeds. The carrier is assumed to be perfectly immobile in translation and rotation. Sensors faults are assumed to be constant over time.

These equations form a stationary system of the form

search method unobservable vectors

Unobservable axes are

APPENDIX 3: filter layout constraint

Matrix gaps are applied to the transition matrices and observation each Kalman cycle after a pattern of q unobservable axes obtained at each cycle according to k

functions of the overall state x H k corresponding to the point of linearization of the observation matrix to the period of Kalman.

is replaced by:

and such that for i between 1 and q:

Equation 1 :

minimum vis-à-vis all the possible solutions.

is replaced by near

and such that for i between 1 and q:

Equation 2: Minimum vis-à-vis all

possible solutions.

The standard matrices is considered standard Froboenius.

The matrices are then used in

the propagation step of the covariance matrix and calculation of the Kalman gain.

We now adopt the following notations:

Solving Equations 1 and 2 if the modeled unobservable vectors are mutually orthogonal to each

instant k :

The total matrix away from the transition matrix is ​​the sum of q independent matrix differences. It is the same for the matrix away from the observation matrix.

And, for i between 1 and q:

The proof of these relationships is given in Annex 8 in a geometric method.

Solving equations 1 and 2 for a non-orthogonal base, and for any q:

This base defines a model of the vector space unobservable: Any linear combination of base vectors is not observable.

The image under observation matrix adjusted any linear combination of these vectors is zero. It is therefore possible to orthogonalize the base, then apply the above method to calculate the matrix away from the observation matrix.

On the other hand, the image adjusted by the transition matrix of a linear combination of vectors is unobservable to the same linear combination applied to the images by the same transition matrix of each of the vectors starting unobservable. It is therefore possible to determine the linear combination to orthogonalize the base unobservable vectors in cycle k-1, and using the same combination for converting the unobservable basic to k cycle. This allows to apply the above method to calculate the matrix away from the transition matrix.

APPENDIX 4: Weighted Metrics

Scalar ιη φ and m H called "metric" accompany the calculated adjustment matrices and give an indication of the relative magnitude of the adjustment, in order to modify as little as possible the transition matrices and observation.

Different methods are possible to compare

and has

the elemental constituents that s standards which

be calculated in the same manner as a vector by collecting all coefficients of the matrix in the column.

Examples are given below:

Variations are possible by using weighting functions and to balance the weight of the various components in

calculating matrices standards. The same examples as the above can be taken in the light of these functions.

A possible realization of these weight functions is to first build a weighting function of a state vector, then

to deduce

Thus by this method, the vector
is associated with a weighting function
vector

where real coefficients l

form the weighting coefficients. These allow then

weighting the transition matrix as follows:

Car and vice versa. These are 2 ways

equivalent to write the relationship between U and V, the second standards for calculating a balanced manner.

Indeed, asking

D

APPENDIX 5: Notion of dynamic system and application navigation

This is a deterministic system of equations with an evolution equation and an observation equation. The evolution equation describes a state vector based on its previous state and a control. The observation equation provides a scalar or vector dependent state vector. These equations can be nonlinear. There are continuous-time systems and discrete-time systems.

Continuous time system:

discrete-time system:

Where
and x k denote the state of the system, u (t) and u k denote the input command, and the observation means to a continuous or instant time t

discrete k, depending on the mode of continuous or discrete representation.

The equations of classical Newtonian mechanics can describe the state of a moving frame according to its acceleration and its angular velocity at any time. These equations are an example of evolution function continues, the state vector collecting the position information, speed, and orientation of the moving frame, and the control vector consisting of the acceleration and angular velocity.

In the state of the art of an inertial navigation calculations can only be carried out discreetly. As a result, this evolution function continues there is modeled as a discrete evolution function. Various known methods are used to convert the continuous evolution equations in discrete evolution equations.

The observation function modeled in the dynamic system in discrete time the inertial unit described measurements made by a particular non-inertial sensor. Several observational functions can then be implemented successively in time depending on the sensors used.

The discrete-time dynamic system used in inertial navigation is therefore the most accurate possible model of another dynamic system, continuous time, corresponding to the real world. The observability properties of the two systems are different.

The observability properties of a dynamic system characterize the system status information at a time that can be achieved in the light of subsequent observations made and evolutionary functions and observation. These properties depend in particular on the control, that is to say the wearer's movements in the case of the inertial navigation.

APPENDIX 6: Observability of a continuous time system

Consider a continuous non-linear dynamic system, at time 0 is in a state x (o). Subsequently, the propagation function and the input control switch are in the system of statements and defining

a particular path in the state space.

This path is known, assume that the initial state is now away from a small value called linearized state and the

input commands are the same as in the previous situation. The following states are then
The observation function to collect every step information on the current state. The problem is whether this information from the beginning to the end of the scenario considered possible to know Space linearized states at time 0 is

then composed of a subspace observable and unobservable subspace. The non-observable space at time 0 characterized by δχ state (θ) can then be deduced according to a new dynamic system for describing states linearized by knowing the tra ectoire

where F and H are matrices

variable in time corresponding to the evolution matrix and the observation matrix of the dynamic system linearized for a given trajectory.

The unobservable space then consists of the linearized 5X states (t) such

for

The solutions form a linearized vector space of states. Then there exists a base B (O) of the space forming unobservable vectors at time 0.

APPENDIX 7: observability of a discrete-time system

Consider a nonlinear discrete dynamic system which, at the instant 0 is in a state X 0 . A different successive discrete times, the propagation function and the input control switch system are in states
Knowing these successive states, assume that the initial state is now deviates from X 0 to a small value δΧ 0 called linearized state and the input commands are the same as in the previous situation. The following successive states are then function

observation collects at each step information on the current state. The problem is whether this information from the beginning to the end of the scenario considered possible to know δΧ 0 . The state space linearized at time 0 is then composed of a subspace observable and unobservable subspace. An observer operating on the principle of linearization, as EKF will then never estimate an error in unobservable space linearized states.

The non-observable space at time 0 characterized by the state X can then

be deduced by considering the dynamic system linearized states
implementing the evolutionary function and linearized observation function to the various states The evolution function linearized

at time k is called "transition matrix" noted function

linearized observation at time k is denoted by H

A first observation of the gap δΧ 0 around X 0 is made using the observation matrix
corresponding to the observation function linearized X 0 . A prediction of the difference δΧ ι around X l is made from the standard δΧ 0 around X 0 using the transition matrix
corresponding to the changing function linearized in a second

Observing δΧ gap ι around
is made using the observation matrix
etc ..

We then know that the unobservable space describing the system disruption is the core of the matrix:

The space unobservable and observable space therefore depend on the initial reference state X 0 , the input control (inertial data in the case of inertial), and evolutionary functions and observation.

Assume that the vector is part of the core ()

Let:

At a given moment, a condition of observability is then formed to an unobservable management ar the system of equations:

Where unobservable vector form a particular discrete time k.

This applies at every moment and is linked to the path

In EKF, this trajectory is noisy by the estimation errors. This noise tends to reduce the core of the observability matrix, and thus reduce the size of the unobservable space.

Moreover the transition functions and observation are linearized at the same point. Following is replaced by a series

The condition of observability is then:

APPENDIX 8: Demonstration of the die adjustment formula to an orthogonal basis vectors

Hypothesis: p column matrices Ui such as: | |

Let p ¼ column matrices same size as the Ui.

Let the matrix A.

Problem: Wanted δΑ the matrix minimum standard such as:

The matrix may be formed into p line matrices

column, and the vector V k in the form of scalar q column:

Therefore sought 6L k minimum standard such as

geometric interpretation of problem: The column and line matrices are associated with the vectors in a vector space of dimension greater than q. We keep the same notation by adding an arrow above variables.

Condition (1) results in:

The condition (2) is expressed by the following inner product:

The projection of the vector L k on the oriented axis defined by the unit vector

Its abscissa on this axis. This is the direction cosine of

along this axis.

6L Wanted k such that the projection of this axis is

can be decomposed as Ui vectors which generate a vector space of dimension q, and according to a ¾ residual component orthogonal to this space

And so

Figure 7 shows that there are infinitely many solutions Solution

minimum corresponds to a vector 6L kl collinear with

From the figure, we have:

So we
, and

Back in matrix notation, we thus

In reforming the matrices A and δΑ, after some manipulations, we obtain the relation:

CLAIMS

1. Estimation method for a navigation state in several variables of a movable carrier according to the method of the extended Kalman filter, comprising the steps of:

- acquiring measurements of at least one variable,

- filtering (400) extended Kalman producing a current estimated state and a covariance matrix defining in the space of the navigation state a region error, from a previous estimated state, a matrix observation of a transition matrix and the captured measurements,

the method being characterized in that it comprises a step of adjusting (310, 330) of the transition matrix and the observation matrix prior to use in the extended Kalman filter, so that the adjusted matrices satisfy a observability condition which depends on at least one of the variables of the state of the holder, the observability condition being adapted to prevent the Kalman filter to reduce the size of the region along at least one axis non- observable state space, wherein the observability condition to be checked by the transition matrices and observation is adjusted kernel invalidity of observability matrix associated therewith, and

wherein adjusting comprises the steps of:

- calculating (301) at least one primary base unobservable vectors from the preceding estimated state,

- for each array to be adjusted, calculation (306, 308) of at least one matrix spread associated with the matrix from the primary base vectors

- shifting (330) each matrix to adjust according to the standard matrix which is associated therewith so as to check the condition of observability.

2. The method of claim 1, wherein the extended Kalman filtering step comprises the substeps of:

- Propagation of the preceding estimated state in a state predicted by the adjusted transition matrix,

- the predicted state linearization of a nonlinear model to produce the observation matrix before adjustment,

- adjustment of the observation matrix produced by linearization.

3. Estimation method according to one of the preceding claims, further comprising an orthogonalizing step (302) of the primary database in a secondary basis vectors, the matrix spread associated with the observation matrix being calculated from the secondary base vectors.

4. Estimation method according to one of the preceding claims, wherein the matrix spread associated with the observation matrix is ​​the sum of several elementary matrix differences, each elementary matrix difference is calculated from a vector secondary base of its own.

5. An estimation method according to one of the preceding claims whose steps are repeated in successive cycles, and wherein a given cycle, said current cycle, comprises the steps of:

- storing (303) the secondary base vectors and orthogonalizing coefficients which are also produced by the step of orthogonalizing and

- processing (304) of the primary basis vectors in a tertiary base vectors by orthogonalizing coefficients stored during a previous cycle,

calculating (306) the matrix spread associated with the transition matrix is ​​implemented from a secondary base stored in the previous cycle and of the tertiary base calculated in the current cycle.

6. An estimation method according to the preceding claim, wherein the matrix spread associated with the transition matrix is ​​the sum of several elementary matrix differences, each elementary matrix deviation being calculated from a stored vector of the secondary base during the previous cycle and clean elementary matrix difference, and a vector of the tertiary base calculated in the current cycle and clean elementary matrix gap.

7. Inertial (IN) comprising a plurality of sensors (C, CC), and means (E) for estimating a navigational state of the inertial unit by implementation of the estimation method according to one of the preceding claims.

8. A computer program product comprising program code instructions for executing steps of the method according to one of claims 1 to 6 when this program is executed by means (E) for data processing.