Navigational aid method, computer program product and inertial navigation system therefor

The invention relates to a navigational aid method for an inertial navigation system including at least one inertial sensor (4) having a sensitive axis (X-X), each inertial sensor (4) comprising an ASG gyroscope (8) able to deliver an ASG signal representative of a rotation about the corresponding sensitive axis (X-X), and a MEMS gyroscope (10) able to deliver a MEMS signal representative of a rotation about the corresponding sensitive axis (X-X), the method including the steps of: between a first date and a subsequent third date, calculating a path from the MEMS signals; from the third date, calculating the path from the ASG signals; estimating a bias vector introduced by the MEMS gyroscopes (10), from the MEMS signals and ASG signals; at a fourth date subsequent to the third date, resetting the path.

TECHNICAL FIELD

The present invention relates to a navigational aid method for an inertial navigation system fixed with respect to a solid.

The invention also has the object to provide a computer program product and an inertial navigation system.

The invention is applicable to the field of the inertial navigation by gyroscopes, in particular by atomic spin gyroscopes (ASG), such as nuclear magnetic resonance gyroscopes, also called “NMR gyroscopes”, and co-magnetometers.

STATE OF PRIOR ART

The use of ASG gyroscopes as inertial sensors to make rotation measurements is known. Such ASG gyroscopes have generally a low angular random walk (ARW) and a low drift. Further, such gyroscopes are likely to be miniaturised (volume in the order of about ten cubic centimetres) and are likely to be produced at a low cost.

ASG gyroscopes are thus an interesting alternative for designing Inertial Navigation Systems (INS) which are reliable, of a small size and cheap, in particular for GPS (Global Positioning System)-free autonomous navigation applications.

However, such ASG gyroscopes are not fully satisfactory.

Indeed, the start-up time of such ASG gyroscopes, that is the duration, from the moment they are powered, at the end of which such gyroscopes are in an operational running phase, is likely to be too long for some GPS-free autonomous navigation applications.

The start-up time of an ASG gyroscope has two main limits:the first one is a limit of technical nature related to heating and stabilising closed loop controls on start-up;the second one, of a physical nature, depends on an alkaline gas/noble gas couple present in the ASG gyroscope, and corresponds to the minimum time necessary to polarise the noble gas by spin exchange with the alkaline one.

Thus, the start-up time of an ASG gyroscope is likely to reach one or more minutes.

But the implementation of an inertial navigation system requires to have available operational and high-performance inertial sensors a few seconds at most after the inertial navigation system is powered, so as to limit the duration of the initialisation phase of the inertial navigation system, also called an “alignment phase”, before switching to a so-called “navigation” mode during which the inertial system is operational and delivers position, velocity and attitude information to the users.

The alignment phase is comprised, for example, of the following steps:starting-up the inertial navigation system (a few seconds to a few tens of seconds);initialising position and velocity (a few tenths of a second); andorienting the navigation reference frame (a few minutes).

The durations of the different steps are given below by way of indicating purposes in the case of a gyrocompass type alignment phase used for air transport type applications for which the alignment phase thus has a duration of a few minutes.

It is understood that the use of ASG type gyroscopes in an inertial navigation system causes an increase in the duration of the starting-up step, thus in the duration of the alignment phase in the inertial system, because the start-up time of the ASG gyroscopes is one or more minutes.

This increase in the duration of the phase for initialising the inertial navigation system is not desirable. Indeed, it is generally desirable that this initialisation duration is as short as possible, in particular for a GPS-free navigation application.

One purpose of the invention is thus to provide an inertial navigation system using an ASG gyroscope which is reliable, of a small size and cheap while allowing a quick starting-up.

DISCLOSURE OF THE INVENTION

To that end, one object of the invention is to provide a navigational aid method of the aforementioned type, the inertial system including at least one inertial sensor having a sensitive axis, each inertial sensor comprising an ASG gyroscope and a MEMS gyroscope integral with each other, the ASG gyroscope being able to deliver an ASG signal representative of a rotation about the corresponding sensitive axis, the MEMS gyroscope being able to deliver a MEMS signal representative of a rotation about the corresponding sensitive axis, the method including the steps of:calculating, between a first date and a subsequent third date, a path and, for each inertial sensor, a corresponding biased path, from the MEMS signals, assuming, for the biased path, that the inertial sensor has a predetermined unit bias;calculating, from the third date, the path and each biased path from the ASG signals, assuming, for the biased path, that the inertial sensor has a predetermined unit bias;estimating a bias vector introduced by the MEMS gyroscopes, from the MEMS signals and ASG signals;resetting, at a fourth date subsequent to the third date, the path as a function of each biased path, the unit biases and the bias vector estimated, to obtain a nominal path which is not affected by the bias of the MEMS gyroscopes.

Indeed, MEMS gyroscopes have a short start-up time, which makes the inertial system quickly operational. Merging the signals collected from the MEMS gyroscopes and ASG gyroscopes contribute to compensate for errors related to the biases introduced by MEMS gyroscopes.

The nominal path thus obtained is no longer affected by the bias of the MEMS gyroscopes.

Further, unlike other types of instruments, MEMS and ASG gyroscopes are able to provide a measurement continuously, which makes it possible to use them in such an inertial system.

The navigational aid method subject matter of the invention is thus reliable and allows a quick starting-up.

According to other advantageous aspects of the invention, the inertial navigation system includes one or more of the following characteristics, taken alone or according to any technically possible combinations:for each MEMS gyrometer, the corresponding component of the bias vector is equal to the average, between a second date and the fourth date, of the difference between an angular velocity from the corresponding MEMS signal and an angular velocity from the corresponding ASG signal, the second date being included between the first date and the third date;the nominal trajectory is obtained by subtracting a reset from the path, the reset being a vectorial corrective term calculated according to:

where δXn is the reset;

b0iis the i-th component of the bias vector; and

the quantity

∂∂D⁢⁢0⁢i
is calculated according to:

where XnDi(trec) is the i-th biased path taken at the fourth date; and

D0i is the predetermined unit bias associated with the component i;the method comprises an overlap step, the overlap step including:between a second date and the third date, the second date being included between the first date and the third date, a first phase for calculating the path and each biased path from the MEMS signal;at the third date, a switching for calculating the path and each biased path from a corresponding angle increment, the angle increment being obtained, for each sensitive axis, by the relationship:
dθcom=θASG(tcom)−θMEMS(tcom−Te)−Δθwhere dθcomis the angle increment;θASG(tcom) is a quantity equal to the cumulation of rotation angle increments about the sensitive axis between the second date and the third date, calculated from the ASG signal upon switching;θMEMS(tcom−Te) is a quantity equal to the cumulation of rotation angle increments about the sensitive axis between the second date and a duration Tebefore the third date, which are calculated from the MEMS signal;each increment being equal to an integral, between two successive instants, of the angular velocity of rotation about a sensitive axis from the corresponding MEMS or ASG signal,

Δθ is a predetermined angular corrective term; andTeis a predetermined duration;between the third date and the fourth date, a second phase for calculating the trajectory and each biased trajectory from the ASG signal;the angular correction is equal to an average, between the second date and the third date, of the values taken over time by the quantity (θASG−θMEMS),

where θASGis a quantity equal, at a given instant, to the cumulation from the second date up to said given instant, of the rotation angle increments about the sensitive axis which are obtained from the ASG signal, for the inertial sensor4considered, and

θMEMSis a quantity equal, at a given instant, to the cumulation, from the second date up to said given instant, of the rotation angle increments about the sensitive axis, which are obtained from the MEMS signal;the method includes, from the fourth date, calculating the nominal path only from the ASG signals.

Further, one object of the invention is to provide a computer program product comprising program code instructions which, when executed by a computer, implement the method as defined above.

Further, one object of the invention is to provide an inertial navigation system, fixed with respect to a solid, the inertial system including at least one inertial sensor having a sensitive axis, each inertial sensor comprising an ASG gyroscope and a MEMS gyroscope integral with each other, the ASG gyroscope being able to deliver an ASG signal representative of a rotation about the corresponding sensitive axis, the MEMS gyroscope being able to deliver a MEMS signal representative of a rotation about the corresponding sensitive axis, the inertial system further comprising a calculator configured to implement the navigational aid method as defined above.

DETAILED DISCLOSURE OF PARTICULAR EMBODIMENTS

In what follows, vectorial quantities are noted in bold.

An inertial navigation system2according to the invention is represented inFIG. 1.

The inertial system2includes at least one inertial sensor4, a clock5and a calculator6.

Each inertial sensor4is able to detect a displacement, such as a rotation about a corresponding predetermined axis, also called a “sensitive axis”, or even a translation, or any combination of rotations and translations.

For example, as illustrated inFIG. 1, the inertial system2includes three inertial sensors4having respectively a sensitive axis X-X, Y-Y and Z-Z. The inertial sensors4are fixed with respect to each other.

Advantageously, the sensitive axes X-X, Y-Y and Z-Z of each of the inertial sensors4are not parallel by pairs, for example orthogonal to each other.

The clock5is configured to deliver a clock signal representative of the passage of time.

The calculator6is configured to calculate the path over time Xn(t) of a solid7fixed with respect to the inertial system2. Such a solid7is, for example, an aircraft taking the inertial system2on board.

In particular, the calculator6is configured to calculate the path Xn(t) of the solid7as a function of the clock signal and signals from the inertial sensors4and which are subsequently described.

By “path”, it is intended, for the purpose of the present invention, the datum of the position, velocity and attitude of the solid7in a reference axis system related to Earth.

By “attitude”, it is intended, for the purposes of the present invention, the datum of roll, pitch, and heading angles formed by predetermined axes of the solid7and the axes of the predetermined reference axis system. In this case, the path Xn(t) of the solid7is a nine-component vector, that is three position components, three velocity components and three attitude angles.

Each point of the path Xn(t) is associated with a date, also called an “instant”, given by the clock signal from the clock5.

Each inertial sensor4includes an ASG gyroscope8, a MEMS gyroscope10and an accelerometer11.

The ASG gyroscope8has a sensitive axis defining the sensitive axis of the inertial sensor4.

The ASG gyroscope8is able to deliver an ASG signal representative of a rotation of the inertial sensor4about the corresponding sensitive axis.

The ASG gyroscope8is, for example, an NMR gyroscope or a co-magnetometer, which are conventionally known.

The ASG gyroscope8is associated with a start-up time Td, also called “start-up duration”. From the powering, also called “starting-up”, of the ASG gyroscope8, the ASG gyroscope8is only operational at the end of a duration equal to the start-up duration Td.

For example, the start-up duration Tdis typically in the order of one minute.

A gyroscope is said “operational”, for the purposes of the present invention, when it is found in a nominal running mode.

The MEMS gyroscope10is integral with the ASG gyroscope8.

The MEMS gyroscope10has a sensitive axis identical to the sensitive axis of the ASG gyroscope8.

The MEMS gyroscope10is able to deliver a MEMS signal representative of a rotation of the inertial sensor4about to the corresponding sensitive axis.

The MEMS gyroscope10is operational at most a few seconds after being started-up.

For each inertial sensor4, the corresponding MEMS gyroscope10is supposed to have an average bias b0over the start-up duration Td. Such a bias is homogeneous at an angular velocity. The drift in the value of the bias b0over the start-up duration Tdis supposed to be negligible relative to the value of the bias b0and compatible with the needs of the inertial navigation system2.

The values of the biases for all three sensitive axes X-X, Y-Y, Z-Z form a three-component bias vector B0. The three components of the bias vector B0, noted b0x, b0y, b0z, are respectively associated with the sensitive axes X-X, Y-Y and Z-Z.

Further, for each inertial sensor4, the corresponding ASG gyroscope8is supposed to have, at the end of the start-up duration Td, a bias the value and the drift of which are negligible relative to the value of the bias b0of the associated MEMS gyroscope10.

The accelerometer11has a sensitive axis, preferably identical to the sensitive axis of the corresponding inertial sensor4.

The accelerometer11is able to deliver an acceleration signal representative of the non-gravitational acceleration, also called “specific force”, of the inertial sensor4along the corresponding sensitive axis.

Preferably, the accelerometer11is integral with the ASG gyroscope8and the MEMS gyroscope10.

The accelerometer11is operational at most a few seconds after being powered.

The calculator6is connected to the ASG gyroscope8to receive the ASG signal. The calculator6is also connected to the MEMS gyroscope10to receive the MEMS signal. The calculator6is, further, connected to the accelerometer11to receive the acceleration signal. The calculator6is also connected to the clock5to receive the clock signal.

The calculator6includes a memory12and a processor14.

The memory12includes a configuration location16and a record location18.

The memory12is further configured to store a navigation software20, a calculation software22and a correction software24.

The configuration location16is configured to store the start-up duration Td, an overlap duration Trec, and a switching duration Tcom.

For example, the switching duration Tcomis typically in the order of a few seconds. The switching duration Tcomis lower than or equal to the overlap duration Trec.

The configuration location16is also configured to store, for each MEMS gyroscope10, a predetermined arbitrary constant unit bias. For each sensitive axis X-X, Y-Y and Z-Z, the unit biases are respectively noted D0x, D0y and D0z. The value of the unit biases D0x, D0y, D0z is, preferably lower than a few tenths of a degree per hour (°/h), for example D0x=D0y=D0z=0.01°/h. Such a value minimises linearisation errors, as will be subsequently described.

The record location18is configured to store the bias vector B0.

The record location18is also configured to store the path Xn(t) of the solid7.

The record location18is further configured to store three biased paths XnDi(t) (i being x, y or z) of the solid7and an offset δXn, which are subsequently defined.

Each biased path XnDi(t) is a path calculated assuming that the inertial system2is, for the axis i (i being x, y or z), affected by the corresponding unit bias D0i.

The navigation software20is configured to calculate, for each sensitive axis X-X, Y-Y, Z-Z, the change over time of the angular velocity w about the sensitive axis, illustrated by the curve26inFIG. 3.

The navigation software20is also configured to calculate over time, and for each sensitive axis X-X, Y-Y, Z-Z, the value of a corresponding angle increment dθ. For each of the sensitive axes X-X, Y-Y, Z-Z, in the case of a discretisation of the numerical calculations enabling Xn(t) to be calculated, the increment dθ is equal to the integral, between two successive instants, of the angular velocity ω from the corresponding MEMS or ASG signal. The increment is noted dθMEMS, respectively dθASG, if it is obtained from the MEMS signal, respectively from the ASG signal.

The navigation software20is also configured to calculate the path over time Xn(t) of the solid7, from the MEMS signal and/or the ASG signal and from the acceleration signal provided by each inertial sensor4. In particular, the navigation software20is configured to calculate the path Xn(t) from the increments dθMEMSand/or dθASG, and of each acceleration signal.

Further, the navigation software20is configured to calculate the three biased paths XnDi(t) over time of the solid7from the unit biases D0x, D0y, D0z stored in the configuration location16, and of the MEMS signal and/or of the ASG signal, and the acceleration signal provided by each inertial sensor4. In particular, the navigation software20is configured to calculate the biased paths XnDi(t) from the unit biases D0x, D0y, D0z, of the increments dθMEMS, dθASG, and of each acceleration signal.

The calculation software22is configured to calculate the bias vector B0.

The calculation software22is also configured to calculate an angular correction Δθ between the MEMS signal and the ASG signal, which correction is subsequently defined.

The correction software24is configured to calculate the offset δXn.

The processor14is adapted to execute each among the navigation software20, the calculation software22and the correction software24stored in the memory12of the calculator6.

The operation of the inertial navigation system2will now be described in reference toFIG. 3.

During a start-up step, the ASG gyroscope8, the MEMS gyroscope10and the accelerometer11of each inertial sensor4of the inertial system2are started-up, that is powered, at an instant t=0 corresponding to the beginning of the start-up step.

The start-up step has a duration equal to the start-up duration Tdstored in the configuration location16. During the start-up step, the ASG gyroscope8is not operational.

During the start-up step, the navigation software20calculates the path over time Xn(t) of the solid7from the MEMS signal and the acceleration signal from each inertial sensor4, that is the navigation software20calculates the path Xn(t) as a function of the increments dθMEMSand of each acceleration signal. Such a calculation is conventionally known.

Further, the navigation software20calculates the biased paths over time XnDi(t) of the solid7.

The calculation of the biased paths XnDi(t) differs from the calculation of the path Xn(t) only in that the increments dθMEMSobtained from the MEMS signal are raised by an angle increment δθ.

For example, in the case where the MEMS signal and ASG signal are each discrete signals obtained by sampling, at a sampling frequency fe, of a corresponding continuous signal, the angle increment δθ is equal, for each sensitive axis X-X, Y-Y, Z-Z, to the result of the division of the corresponding unit bias D0x, D0y, D0z by the sampling frequency fe, expressed in the suitable unit.

During the start-up step, the navigation software20writes, in the record location18, the path Xn(t) and the biased paths XnDi(t) calculated.

The step following the start-up step is an overlap step.

During the overlap step, for each inertial sensor4, each of both ASG8and MEMS10gyroscopes is an operational running phase, both ASG8and MEMS10gyroscopes being used together.

During the overlap step, the ASG and MEMS signals are compared to each other in order to switch from the MEMS gyroscope10to the ASG gyroscope8.

By “switching”, it is intended, for the purposes of the present invention, switching from one calculation of the path Xn(t) from the MEMS signal to a calculation of the path Xn(t) from the ASG signal.

Further, during the overlap step, both ASG and MEMS signals are also used to estimate the bias vector B0associated with each MEMS gyroscope10. The beginning of the overlap step corresponds to an instant t=Td, also noted td.

The overlap step has a duration equal to the overlap duration Trecstored in the configuration location16, such that the overlap step is completed at the instant t=Td+Trec, also noted trec.

The overlap step is comprised of a first phase, called a switching phase, and a second phase.

The first phase has a duration equal to the switching duration Tcom, stored in the configuration location16. The first phase starts as soon as the overlap step begins, at the instant td, and is completed at the instant t=Td+Tcom, also noted tcom.

The switching occurs at the instant tcom.

The second phase begins at the instant tcom, and is completed at the end of the overlap step, that is at the instant trec.

During the overlap step, the calculation software22calculates, for the sensitive axis i of each inertial sensor4, a corresponding bias b0i, equal to the average, preferably on the entire overlap step, of the difference between the angular velocity from the MEMS signal and the angular velocity from the corresponding ASG gyroscope8. Since the bias of the ASG gyroscope8is assumed to be low with respect to the bias of the MEMS gyroscope, the bias deviation between both MEMS10and ASG8gyroscopes is ascribed to the MEMS gyroscope10.

Then, for each component of the bias vector B0associated with a sensitive axis X-X, Y-Y, Z-Z, the calculation software22writes, in the record location18, the bias b0i(i being x, y or z), calculated for the MEMS gyroscope10of the corresponding inertial sensor4. The duration Trecis chosen to allow an ARW white noise filtering of both MEMS10and ASG8gyroscopes in order to estimate the bias vector B0at best.

The estimation accuracy of each component of the bias vector B0is given by the formula (1):

where σ(δ) is the standard deviation of the estimation error of the bias of the MEMS gyroscope10(in °/h);

qARWm is the drift white noise power spectral density of the MEMS gyroscope10(in °/√h); and

qARWr is the drift white noise power spectral density of the ASG gyroscope8(in °/√h).

For example, with a power spectral density qARWm of the drift white noise of the MEMS gyroscope10being 10−3°/√h, a power spectral density qARWr of the drift white noise of the ASG gyroscope8being 10−3°/√h, an overlap duration Trecbeing 60 sec, the standard deviation of the error on the estimation σ(δ) of the bias of the MEMS gyroscope10is 0.011°/h.

Further, during the overlap step, the navigation software20calculates the path Xn(t) and the biased paths XnDi(t) of the solid7.

More precisely, during the first phase, the navigation software20calculates the path Xn(t) of the solid7from the MEMS signal and the acceleration signal from each inertial sensor4. In particular, the navigation software20calculates the path Xn(t) from the angle increment dθMEMSfrom each MEMS signal, and the acceleration signal from each inertial sensor4.

Further, during the first phase, the navigation software20calculates the biased paths XnDi(t) of the solid7from the MEMS signal and the acceleration signal from each inertial sensor4, and unit biases. More precisely, the navigation software20calculates the biased paths XnDi(t) from the angle increment dθMEMSfrom each MEMS signal, of the angle increment δθ determined from the unit biases D0i and from each acceleration signal.

Further, during the first phase, the calculation software22calculates, for each inertial sensor4, a corresponding angular correction Δθ. The angular correction Δθ is equal to the average, over the switching duration Tcom, of the values taken over time by the quantity (θASG−θMEMS), where θMEMSis a quantity calculated from the MEMS signal and equal, at a given instant, to the cumulation, from the instant tdto said given instant, of the increments dθMEMS, and where θASGis a quantity calculated from the ASG signal and equal, at a given instant, to the cumulation, from the instant tdto said given instant, of the increments dθASG, for the inertial sensor4considered. The angular correction Δθ is intended to correct the error induced, upon switching, by the angular white noise on the measurements from the ASG and MEMS gyroscopes.

Then, upon switching, the calculation software22transmits to the navigation software20the angular correction Δθ obtained at the end of the first phase, so as to ensure continuity between the measurements based on the MEMS gyroscopes10and the measurements based on the ASG gyroscopes8.

Further, the navigation software20calculates the point of the path Xn(tcom) of the solid7, at the instant tcom, from an angle increment dθcom, and of each acceleration signal.

For a given sensitive axis, the corresponding angle increment dθcomis obtained by the relationship (2):
dθcom=θASG(tcom)−θMEMS(tcom−Te)−Δθ  (2)

where θASG(tcom) is the value taken by θASGat the instant tcom;

θMEMS(tcom−Te) is the value taken by θMEMSone sampling period before the instant tcom; and

Teis the sampling period, equal to the inverse of the sampling frequency.

Further, upon switching, the correction software24rewrites, in the configuration location16, the value of each unit drift D0x, D0y, D0z to ascribe it a zero value. This is due to the fact that, from switching, the calculation of the path Xn(t) and the biased paths XnDi(t) is made from the ASG signals, the drift of the ASG gyroscopes8being assumed to be negligible relative to the drift of the MEMS gyroscopes10.

For each sensitive axis, the navigation software20calculates the point of the biased path XnD(tcom) of the solid7, at the instant tcom, from the angle increment dθcomand from each acceleration signal, the value of each unit drift D0x, D0y, D0z having been set to zero upon switching.

Switching to a calculation of the path Xn(t) (and of the biased paths XnDi(t)) from the ASG signal instead of the MEMS signal is possible because the error related to switching mainly depends on the angular white noise on the measurements from the ASG8and MEMS10gyroscopes.

The standard deviation of the angular error made and related to switching is given by the relationship (3):

where qBAm is the angular white noise power spectral density of the MEMS gyroscope10(in μrad/√Hz);

qBAr is the angular white noise power spectral density of the ASG gyroscope8(in μrad/√Hz);

σ(θ) is the standard deviation of the angular error due to switching.

For example, for a power spectral density of the angular white noise being 1 μrad/√Hz for each of both ASG8and MEMS10gyroscopes, and a switching duration Tcomequal to 5 s, the standard deviation of the angular error σ(θ) related to switching is 0.63 μrad.

Then, during the second phase, the navigation software20calculates the path Xn(t) of the solid7only from the ASG signal and the acceleration signal from each inertial sensor4.

The navigation software20also calculates the biased paths XnDi(t) of the solid7only from the ASG signal and the acceleration signal from each inertial sensor4, the value of each unit drift D0x, D0y, D0z having been set to zero upon switching.

During the overlap step, the navigation software20writes, in the record location18, the path Xn(t) and the biased paths XnDi(t) calculated.

In summary, during the first phase, the navigation software20uses the angle increments dθMEMSfrom the MEMS signal; at the switching instant, the navigation software20uses the increment dθcom; then, during the second phase, the navigation software20uses the increments dθASGfrom the ASG signal.

The step following the overlap step is a correction step intended to correct the angular errors introduced by the bias of the MEMS gyroscopes during the use of the MEMS gyroscopes during a start-up step and the first phase of the overlap step.

The correction step occurs at the date trec.

During the correction step, the navigation software20calculates the path Xn(t) of the solid7from the ASG signal and the acceleration signal from each inertial sensor4.

Further, during the correction step, the correction software24offsets the path Xn(t) of the solid7from the value δXn at the instant trec. The path thus offset is the path which would have been calculated by the navigation software20if the ASG gyroscopes8had been operational as soon as the inertial system2had started up.

As previously described, the correction software24calculates, during the start-up step and the overlap step, the three biased paths XnDi(t) (with i taking the value x, y or z) corresponding to the output data of the navigation algorithm when the data of the MEMS10are biased by a constant unit bias D0i(with i taking the value x, y or z). For example, XnDx(t) is the path calculated by the navigation algorithm of the inertial navigation system when the nominal measurements of the inertial sensor4with the sensitive axis X-X are offset by a further unit bias D0x. This bias D0i(stored in the configuration location16) takes two values depending on the instant considered:from t=0 to tcom, the value of this unit bias is set to a value which has to be low to minimise linearisation errors. Typically, D0x=D0y=D0z=0.01°/h;then, from tcomto trec, the biases are set to 0 because switching results in continuing navigation using data from the ASG gyroscopes considered as bias-free.

At the beginning of the correction step, the correction software24calculates partial derivatives

∂Xn∂Di
(with i taking the value x, y or z). Each partial derivative

∂Xn∂Di
(with i taking the value x, y or z) is the derivative of the path Xn(t) with respect to the unit drift D0iof the corresponding MEMS gyroscope10, calculated with the following relationship (4):

where D0iis the unit bias associated with the axis i.

The correction software24then calculates the vectorial offset δXn from the estimate of the partial derivatives

∂∂Di
and from the estimate of the bias vector B0of the MEMS gyroscope10according to the formula (5):

b0ibeing the component i of the bias vector B0.

δXn is thus a nine-dimension vector.

Then, the correction software24resets the path Xn(t) at the instant trecby subtracting the correction term δXn from the path Xn(t) according to the relationship (6):
Xn(after resetting)=Xn(before resetting)−δXn(6).

In this way, the initial error due to the use of the MEMS gyroscopes10is corrected.

The path Xn(t) reset is called a “nominal path”.

Once the path Xn(t) is reset, the calculation of the biased paths is interrupted, because useless. In the following, the navigation software20continues the calculation of the path Xn(t) of the solid7from the angle increment only from the ASG signal and the acceleration signal from each inertial sensor4.

The bias correction method of the MEMS gyroscope10set forth above results in linearising Xn with respect to the three unit biases D0x, D0y and D0z. This imposes a bias value b0i(with i taking the value x, y or z) which does not exceed a few tenths of a degree per hour, to avoid too high a navigation error resulting into strong non-linearities making the above correction formula 5 invalid.

The path Xn(t) calculated by the navigation software20at the end of switching is affected by the bias of the MEMS gyroscopes10, which introduces navigation errors in calculating the path during the use of the MEMS gyroscopes10from starting-up of the inertial system (t=0) until the end of the first phase (tcom), these errors being propagated until the end of the second phase of the overlap step (trec). By virtue of such an inertial system2, such navigation errors are compensated for and a quick start-up of the inertial system is possible.

The method for correcting the errors induced by the bias in the MEMS gyroscopes10which has been described above has the advantage, unlike a method which would consist in recalculating the entire path from the beginning with measurements from the MEMS gyroscopes corrected by B0, of being simple to implement in real time, and of not requiring storing a significant data volume in a very short time.

Such a method enables, at the instant trec, the errors induced during navigation from the instant t=0 to the instant tcomto be corrected by the drift of the MEMS gyroscope, and without requiring replay (that is recalculation) of navigation from the beginning with gyroscope measurements corrected by the bias value.

Resorting to MEMS and ASG gyroscopes allows a continuous operation over time, such gyroscopes being capable of providing a continuous measurement over time. This property makes possible the use of such gyroscopes in an inertial system. Indeed, specially for security reasons, a discontinuity over time in the rotation angle or rotational velocity measurements cannot be tolerated. The use of such gyroscopes is thus advantageous in comparison with the use, for example, of matter wave gyroscopes, which have the drawback of having a low passband and of providing discontinuous measurements over time.

Further, the low dimensions and production costs of ASG and MEMS gyroscopes make the inertial system2cheap.