Method for determining a dynamic alignment path with a view to estimating by extended kalman filtering the state of a carrier

Abstract

Disclosed is a method for determining a dynamic alignment path (P) of a carrier by extended Kalman filtering, comprising the steps of: - (S3) determination of a first gradient of a scalar L(PF) with respect to drive commands, PF being a final covariance matrix, the determination comprising: - a calculation of a second gradient of the scalar with respect to transition matrices describing the transition from point n-1 to point n, - a calculation of a third gradient of the scalar with respect to action matrices of noise in the transition from point n-1 to point n, and - a calculation of a fourth gradient of the scalar with respect to an observation matrix of the state at point n, the determination of the first gradient comprising a backpropagation of the first, second, third and fourth gradients, - (S5) determination of a sequence of the commands that minimizes the scalar and operational constraints, and - (S6) determination of the path.

Description

[0001]Description Title: Method for determining a dynamic alignment trajectory for estimating the state of a carrier by extended Kalman filtering FIELD The invention relates to a method for determining a trajectory of a dynamic alignment to improve the estimation by extended Kalman filter of the state of a carrier, in particular a carrier equipped with an inertial measurement unit and other sensors. STATE OF THE ART The use of extended Kalman filtering is known for estimating the state of a carrier or a dynamic system equipped with an inertial measurement unit (also known as an IMU) and other sensors such as a satellite geo-positioning system (also known as a GPS) or odometers. The uncertainty of the estimates during navigation depends strongly on the initial uncertainty about the state of the system.In order to reduce this, it is possible to carry out a dynamic alignment, that is to say the realization of a maneuver of the carrier according to a particular trajectory during which relevant information is acquired. The known methods for determining the trajectory present different limitations such as for example the only application to terrestrial vehicles (that is to say flat systems), thus excluding more complex aerodynamic systems such as aeronautical systems. Another common limitation is a high computational complexity not allowing to solve an optimization problem on only a few discretization steps. There is therefore a need for a method for determining an optimized trajectory for a dynamic alignment applicable to an aeronautical system and not requiring too much computation time.PRESENTATIONOne aim of this presentation is to propose a method for determining an optimized trajectory for dynamic alignment applicable to an aeronautical system and requiring less computation time than in the prior art.The aim is achieved by a method for determining a dynamic alignment trajectory of a mobile carrier followed by extended Kalman filtering, the method comprising the following steps: - determining an initial covariance matrix ^0 of an initial estimate of a state of the movement of the carrier at a start of the trajectory, the initial covariance matrix representing an uncertainty of the initial estimate, - determining navigation equations of the carrier from measurements provided by an inertial unit and sensors of the carrier, - determining a first gradient of a scalar ^(^^) with respect to piloting commands of the carrier, the scalar ^(^^) being constructed from a final covariance matrix ^. ^of a final estimate of the state of the carrier's movement at an end of the trajectory, the scalar ^(^^) representing an uncertainty of the final estimate, the first gradient representing an influence of the piloting commands on the uncertainty of the final estimate, - determination of a gradient of a scalar of operational constraints of the carrier with respect to the carrier's commands, - determination of a sequence of commands which minimizes a sum of the scalar ^(^^) and the scalar of operational constraints, the determination using a projected gradient descent method, and - determination of the trajectory from the sequence of commands.Such a method is advantageously and optionally completed by the following different characteristics taken alone or in combination: the trajectory is discretized according to a plurality of points n, the determination of the first gradient comprising for each point n: - a calculation of a second gradient of the scalar ^(^^) with respect to transition matrices of an estimated value ^. ^−1 from the state of the carrier at point n-1 to an estimated value ^ ^ of the carrier state at point n, - a calculation of a third gradient of the scalar ^(^^) with respect to action matrices of a noise in the transition of the estimated value ^ ^−1 to the estimated value ^ ^, and- a calculation of a fourth gradient of the scalar ^(^^) with respect to an observation matrix of the state of the carrier at point n, the determination of the first gradient comprising a use of backpropagation equations of the first gradient, the second gradient, the third gradient and the fourth gradient.- the determination of the first gradient comprises the use of the following equations: in which ^ ^ denotes the transition matrix of an estimated value ^ ^−1 from the state of the carrier at point n-1 to an estimated value ^ ^ of the state of the carrier at point n, ^^ denotes the action matrix of a noise ^^ on the estimated value ^ ^ , the noise ^ ^ being a white Gaussian noise of zero expectation and covariance matrix ^^, ^^|^ is the covariance matrix of the estimated value ^ ^ which takes into account an observation made at point n, ^ ^|^−1 is the covariance matrix of the estimated value ^^ which does not take into account the observation made at point n, ^ ^ denotes an observation matrix of the estimated value ^^ and ^^ denotes a Gaussian measurement noise of zero expectation and covariance matrix ^^; - the determination of the first gradient of the scalar ^(^^) includes a step of determining for each point n a fifth gradient of the scalar ^(^^) with respect to the components ^ of the estimated value ^^ and noted ^ ^ ^ ;- the determination of the fifth gradient includes a backpropagation of the values ​​of the partial derivative of the scalar ^(^^) as a function of the components ^ of the estimated value ^^ and noted ^ ^ ^ ;- the determination of the partial derivatives of the scalar ^(^^) as a function of the matrices ^ ^|^ includes a backpropagation of the covariance matrices^^|^ from the partial derivative of the scalar ^(^^) as a function of the matrix ^ ^;- the determination of the first gradient includes a step of determining, for each point n of the trajectory, partial derivatives ^^(^ ^ ) ^^ ^ ^ of the scalar ^(^^) as a function of components ^ ^ ^ of the order ^ ^ at point n according to the following relation in which ^ ^ ^ is a Jacobian of a propagation function ^^ with respect to the commands ^ ^ so that ^ ^ ^ = the propagation function ^^ allowing to estimate the orientation, the speed and the position of the carrier at point n+1, from the orientation, the speed and the position of the carrier at point n, the commands at time n and the noise at time n, and ^ ^ is a vector whose components are all zero except the j-th which is 1; and- the scalar ^(^^) is the trace or the largest eigenvalue of the matrix^ ^. The presentation further relates to a method for navigating a carrier using an extended Kalman filter estimating a navigation state of the carrier, the method comprising the following steps: - determining a trajectory of a dynamic alignment as just presented, - piloting the carrier so that the carrier follows the trajectory, - recording measurements from the inertial unit and the sensors of the carrier during the trajectory, - determining the covariance matrix ^^ of the states of the carrier at the end of the trajectory, and - replacing the initial covariance matrix ^0 of the states of the carrier by the matrix ^ ^The disclosure also relates to a computer program product comprising program code instructions for executing the steps of one of the methods just described, when this program is executed by a computer.The disclosure also relates to a device for tracking the trajectory of a mobile carrier, the device comprising: - a reception interface configured to receive measurements acquired by an inertial unit and sensors, - a processor configured to implement a determination of a dynamic alignment trajectory of the carrier, the device being configured to implement the following steps: - determination of an initial covariance matrix ^0 of an initial estimate of a state of the movement of the carrier at a start of the trajectory, the initial covariance matrix representing an uncertainty of the initial estimate, - determination of navigation equations of the carrier from measurements provided by an inertial unit and sensors of the carrier, - determination of a first gradient of a scalar ^(^^) with respect to piloting commands of the carrier, the scalar ^(^^) being constructed from a final covariance matrix ^. ^of a final estimate of the state of the carrier's movement at an end of the trajectory, the scalar ^(^^) representing an uncertainty of the final estimate, the first gradient representing an influence of the piloting commands on the uncertainty of the final estimate, - determination of a gradient of a scalar of operational constraints of the carrier with respect to the carrier's commands, - determination of a sequence of commands which minimizes a sum of the scalar ^(^^) and the scalar of operational constraints, the determination using a projected gradient descent method, and - determination of the trajectory from the sequence of commands.Such a device can advantageously and optionally be configured to implement the determination of the first gradient by carrying out the following sub-steps, the trajectory being discretized according to a plurality of points n, the determination of the first gradient comprising for each point n: - a calculation of a second gradient of the scalar ^(^^) with respect to transition matrices of an estimated value ^. ^−1 from the state of the carrier at point n-1 to an estimated value ^ ^ of the carrier state at point n, - a calculation of a third gradient of the scalar ^(^^) with respect to action matrices of a noise in the transition of the estimated value ^ ^−1 to the estimated value ^ ^, and- a calculation of a fourth gradient of the scalar ^(^^) with respect to an observation matrix of the state of the carrier at point n, the determination of the first gradient comprising a use of backpropagation equations of the first gradient, the second gradient, the third gradient and the fourth gradient. The disclosure finally relates to a navigation unit for a mobile carrier, comprising:- an inertial unit,- sensors, and- at least one trajectory tracking device such as has just been presented. DESCRIPTION OF THE FIGURES Other characteristics and advantages will emerge from the following description, which is purely illustrative and non-limiting, and must be read in conjunction with the appended drawings in which:[Fig. 1] Figure 1 is a schematic representation of a trajectory tracking device; and[Fig. 2] Figure 2 is a schematic representation of a trajectory tracking method. DETAILED DESCRIPTION OF THE INVENTION A mobile carrier can be a land vehicle, a ship or an aircraft. The mobile carrier comprises a navigation unit 1 which is fixed to the structure of the carrier so that the navigation unit 1 is stationary relative to the mobile carrier. With reference to FIG. 1, the navigation unit 1 comprises an inertial unit 11 and sensors 12. An inertial unit 11 is also referred to by the English expression "Inertial Measurement Unit" abbreviated to IMU. The inertial unit comprises three gyrometers measuring the three components of an angular acceleration (in radians per square second) of the mobile carrier and three accelerometers configured to estimate the three components of a metric acceleration (in meters per square second) of the carrier.The sensors 12 or complementary sensors 12 are for example a satellite geo-positioning system (also known as the English expression “Global Positioning System” abbreviated to GPS) or an odometer. The navigation unit 1 comprises a trajectory tracking device 2 of the mobile carrier. The trajectory tracking device 2 comprises a reception interface 21 configured to receive measurements acquired by the inertial unit 11 and the sensors 12. The trajectory tracking device 2 comprises at least one processor 20 configured to implement an extended Kalman filter (generally designated by the acronym EKF in the literature). The extended Kalman filter EKF is an algorithm capable of being coded in the form of a computer program executable by the processor 20. The trajectory tracking device 2 further comprises an output 23 for delivering output data calculated by the processor 20.Extended Kalman Filter In a known manner, an extended Kalman filter EKF is a recursive estimator of a state representative of the navigation of the carrier, called hereinafter navigation state. This state is a state of movement of the mobile carrier. This navigation state can comprise at least one navigation variable of the carrier (position, speed, acceleration, orientation, etc.). The navigation state can in any case be represented in the form of a vector, each component of which is a navigation variable of the carrier. We consider in the following an embodiment in which the navigation state comprises in particular the following navigation variables: 1. a position vector ^ of the carrier of dimension 3, 2. a velocity vector ^ of the carrier of dimension 3, 3. An orientation matrix Ω of the carrier, defined as the rotation matrix allowing the change of reference frame from the reference frame of the carrier to an inertial reference frame.The inertial reference frame can for example be a terrestrial reference frame centered on the center of the Earth whose axis points towards the north pole, whose x axis points towards the intersection of the Greenwich meridian and the equator at time t = 0 (the point thus defined will then move in our reference frame because of the rotation of the Earth) and whose y axis points in the direction of the vector z × x, × designating the vector product). The navigation state can also include additional variables which can be for example the noise associated with the operation of the accelerometers and gyrometers (vectors of dimension 3), lever arms between the IMU and the complementary sensors (vectors of dimension 3). Method for determining a dynamic alignment trajectory of the mobile carrier A trajectory tracking device as just presented makes it possible to implement a method P to determine a dynamic alignment trajectory of the mobile carrier.In relation to Figure 2, the steps of the method P are described. A dynamic alignment trajectory, when followed by the carrier, allows measurements to be made along this trajectory. These measurements are sufficiently relevant to reduce the estimation uncertainty of the extended Kalman filter. The maneuvering of the carrier along this particular trajectory allows acquiring relevant information and reducing the estimation uncertainty of the extended Kalman filter. Such maneuvers can be performed at any time during the carrier's mission. In a first step S1, the localization is initialized with an estimation of the navigation state to which a covariance matrix P0 is associated. The covariance matrix represents the uncertainty associated with the estimation of the initial state of the carrier. The extended Kalman filter EKF is initialized with an initial state, which will serve as an input for a first iteration of the filter.Each subsequent iteration of the filter takes as input a state estimated by a previous iteration of the filter, and provides a new estimate of the carrier state. The extended Kalman filter is therefore implemented by successive iterations so as to determine ^^ which is an estimated value of a carrier state at a step n (or point n or iteration n) from. which is an estimated value of a state of the carrier at the previous step n-1 (or previous point n-1 or previous iteration n-1). In a second step S2, we determine the navigation equations of the inertial unit. The dynamics of the IMU can be locally linearized around zero according to the following equations: ^ ^ is a transition matrix of ^ ^−1 towards ^ ^,^^ is an action matrix of the noise ^^ on ^^, with ^^~^(0, ^^) i.e. the noise ^^ is a white Gaussian noise of zero expectation and covariance matrix ^^,^^ is a measure of the carrier's navigation state at step n,^^ is an observation matrix of ^^, and^ ^ is a measurement noise which is a white Gaussian noise with zero expectation and covariance matrix ^^, ^^~^(0, ^^). It should be noted that a white noise corresponds to an independent character of the noise with respect to time: two noises at two different times are independent. The first equation numbered (1) represents the transition from ^^−1 to^ ^.The second equation numbered (2) represents the observation ^^made during the trajectory at step n. This observation depends on the estimated value ^^ of the state of the system at this step n. Both the transition and the observation are noisy by Gaussian white noises ^^ and ^^ respectively which are all of zero expectation. It should be noted that, given the linearization, the matrices ^^ and ^^ depend on ^^−1 and the command ^^. The command ^^ is the carrier piloting command which is sent to the carrier propulsion system at point n. The command ^ ^ is a vector whose each component corresponds to a specific control signal to be transmitted to the carrier to ensure its control, for example the command ^ ^is a vector comprising three metric accelerations and three angular accelerations. Like the other quantities, the command is discretized. At each step, the uncertainty in the estimate of the carrier state is evaluated. For this, the matrix ^ is used ^ of covariance of the estimated value ^ ^ of the carrier state at step n. The matrix ^ ^ is called the posterior covariance matrix ^^|^ when it takes into account the observation ^^ made at point n. It is called the prior covariance matrix ^^|^−1 when it does not take into account the observation ^^ made at point n. The propagation and correction equations of the covariance matrix in extended Kalman filtering are: The term ^ ^ ^ denotes the transpose matrix of the matrix ^ ^ .Equation (3) allows us to calculate the a priori covariance matrix ^^|^−1 without taking into account the observation ^ ^. This covariance estimates the uncertainty on the state a priori. Knowing this uncertainty, equation (4) allows us to calculate the observation covariance matrix ^^ at step n by combining the a priori uncertainty ^^|^−1 on the state and the covariance ^^ of the measurement noise. Knowing both the a priori uncertainty ^^|^−1 on the state and the uncertainty^ ^ on the observation, equation (5) allows to calculate the matrix ^ ^ , called the Kalman gain. This gain matrix ^ ^is notably used to update the estimated value of the system state by adding the product of the gain matrix with the difference between the actual observation and the observation estimated from the a priori estimate of the state (i.e. without knowledge of the observation). Following this update of the estimated value of the state, equation (6) makes it possible to calculate a new covariance ^^|^ of the estimated value of the updated state, taking into account the previous covariance and the gain. The trajectory sought to perform the dynamic alignment is discretized according to a plurality of points n (or steps n). 0 denotes the initial point of the curve and F denotes the end point of the trajectory. In a third step S3, the gradient of a scalar ^(^^) is determined with respect to the carrier piloting commands ^^, this gradient quantifying the variation of ^(^^) when the ^^ vary. The scalar ^(^^) being constructed from a matrix ^ ^of covariance of an estimated value of the carrier state at the end of the trajectory and a scalar function ^. The scalar function ^ can notably be the trace or the largest eigenvalue of the matrix ^^. The quantity ^(^^) makes it possible to represent the uncertainty of an estimate of the carrier state by Kalman filtering extended by a scalar. The gradient of the scalar ^(^^) is calculated with respect to the commands ^^ for piloting the carrier throughout the trajectory, that is to say that for each step n of the trajectory and for each component of the order ^ ^ , we determine the partial derivative The gradient of the scalar ^(^^) with respect to the ^^ commands for piloting the carrier is a first gradient of the scalar ^(^^). The determination of this first gradient may comprise for each point n a calculation of a second gradient of the scalar ^(^^) with respect to the transition matrices from the estimated value ^^−1 of the state of the carrier at point n-1 to the estimated value ^^ of the state of the carrier at point n. The transition matrices are the matrices ^ ^ . For each step n of the trajectory, we determine the term .The determination of the first gradient may comprise for each point n a calculation of a third gradient of the scalar ^(^^) with respect to action matrices of a noise in the transition from the estimated value ^^−1 to the estimated value ^^. The action matrices of the noise are the matrices^^(^ ^ ) ^ ^ . For each step n of the trajectory, we determine the term ^^ ^The determination of the first gradient may include for each point n a calculation of a fourth gradient of the scalar ^(^^) with respect to an observation matrix of the state of the system at point n. The observation matrices are the matrices ^ ^ . For each step n of the trajectory, we determine the term Determining the first gradient may include using backpropagation equations from the partial derivatives of the scalar ^(^^) as a function of the ^^ control commands at the end of the trajectory. In other words, at step n of the trajectory, the terms ^^(^ ^ ) ^^ ^ ^ indexed by i based on the terms^^(^ ^ ) ^^ ^ ^+1indexed by i. The determination of the first gradient can also include the use of backpropagation equations of the second gradient, the third gradient and the fourth gradient from the end of the trajectory. In other words, at step n of the trajectory, we determine the ^^(^ ) ^^(^ ) ^^(^ ) term ^ ^ on the ba ^ ^ ^^ of the term ^^ ^+1 , the term ^^ ^ based on the term t erme ^^(^ ^ ) ^^(^ based on the term ^ ) ^^ ^ e ^^ ^+1The use of backpropagation equations corresponds here for each point n to a calculation for each point n of the first gradient at point n as a function of the first gradient at point n+1, of the second gradient at point n+1, of the third gradient at point n+1 and of the fourth gradient at point n+1. By proceeding in this way and in particular by using backpropagation, it is possible to obtain the first gradient of ^(^^) with a less significant calculation time than in the prior art when taking into account the a priori uncertainties of the state of the system noted ^ ^|^−1 . Advantageously, to determine the first gradient of the scalar ^(^^), we use the following equations Equation (7) corresponds to the second gradient of the scalar ^(^^ ) par with respect to the transition matrices. Equation (8) corresponds to the third gradient of the scalar ^(^^) with respect to the action matrices of a noise. Equation (9) corresponds to the fourth gradient of the scalar ^(^^) par with respect to the observation matrices. The determinations of the expressions of the partial derivatives of the function^(^^) with respect to the matrices ^^, ^^ and ^^, as given in equations (7), (8) and (9), are intermediate calculations which allow the first gradient of the function ^(^^) to be established. The matrices ^^ and ^^ depend on the commands ^ ^ according to a propagation function noted ^^. The Jacobian of ^^ with respect to the orders ^ ^ (i.e. the derivative e ^ is noted ^ ^ . An example of a propagation function ^^ can be given as follows. The commands ^^ are represented by the vector where ^^ represents the angular velocity given as a setpoint (vector ^^ is of size 3), and ^^ the acceleration given as a setpoint (vector ^ ^ is size 3). So, the noise ^ ^ is represented by the vector where ^ ^^ is the noise on the angular velocity (the vector ^^^ is size 3) and ^^ ^is the noise on the acceleration (the vector ^ ^ ^ is of size 3). An example of a ^^ function can then be written as follows: In this equality, Ω ^+1 represents the orientation of the carrier at time n+1, ^ ^+1 represents the carrier's speed at time n+1, ^ ^+1 represents the position of the carrier at time n+1, Ω ^ represents the orientation of the carrier at time n, ^ ^ represents the carrier's speed at time n, and ^ ^represents the position of the carrier at time n. The propagation function ^^ allows us to estimate the orientation Ω^+1, the velocity ^^+1 and the position ^^+1 of the carrier at time n+1, from the orientation Ω^, the velocity ^^ and the position ^^ of the carrier at time n, the commands ^^ at time n and the noise ^^ at time n. This equality (Ω^+1, ^^+1, ^^+1) = ^^(Ω^, ^^, ^^; ^^; ^^) can be written in the form of the following equations: where ^ represents gravity and ^^ the time elapsed between point n-1 and point n. For a vector ^ of size 3, the operator ^( ) is defined as where ^^ is the identity matrix, and ^ = ‖^‖^ is the norm of the vector ^. The term (^)× denotes the matrix of size 3*3 written: ^ ^ where the scalars ^1, ^2, ^3 are the components of the vector ^ : ^ = ^^ ^ ^. ^ ^The determination of the first gradient of the scalar ^(^^) may comprise a step S3a of determining partial derivatives of the scalar ^(^^) as a function of components ^ of the estimated value ^^ (we note ^ ^ ^ these components). In other words, the determination of the first gradient of the scalar ^(^^) includes the step (S3a) of determining for each point n a fifth gradient of the scalar ^(^^) with respect to the components^ of the estimated value ^^ and noted ^ ^ ^ .The partial derivatives, noted^^(^ ^ ) ^^^ , can be determined ^ ^^(^ ) ^ notably using the relation ^ ^^ ^ = ^^ ^ ^^(^ ^ ) ^^^ ^ and one ^ ^^ ^ ^^ ^ ^ backpropagation of the gradient of ^(^^) with respect to the state of the carrier ^^. In other words, at step n of the trajectory, we determine the term ^^(^ ^ ) ^^ ^ ^ based on the term ^^(^ ^ ) ^^ ^ . ^ +1 ^^(^ ^) Partial derivatives, noted ^^ ^ ^ , can be determined using backpropagation of the intermediate computations (7), (8) and (9). Similarly, the determination of the partial derivatives of the scalar ^(^^) with respect to the matrices ^^|^ and ^^|^−1 uses backpropagation of the gradient of ^(^^) with respect to the covariance matrices ^^|^ and ^^|^−1 from the partial derivative of the scalar ^(^^) with respect to the matrix^ ^ For this purpose, we can use the following backpropagation equations: with ^ ^ which corresponds to the vector whose components are all zero except the j-th which is 1. Equations (10) to (12) correspond to the backpropagation of the second gradients of ^(^^) with respect to the covariance matrices (covariance matrix ^^|^−1 a priori and covariance matrix ^^|^ aposteriori) and the state of the system. Equations (10) to (11) link the second gradient of the scalar ^(^^) par relative to covariance ^ ^|^ estimated at time n with the second gradient of the scalar ^(^^) with respect to the covariance ^^−1|^−1 estimated at time n-1. This allows a recursive calculation. Equation (12) links the gradient of the scalar ^(^^) with respect to the component ^ ^^ at time n with the gradient of the scalar ^(^^) with respect to the component This allows a recursive calculation. Equation (12) is enriched with terms taking into account the impact of the indentation and whose calculation is allowed by equations (7), (8) and (9). The determination of the gradient of the scalar ^(^^) with respect to the carrier commands can include a step S3b of determination, for each point n of the trajectory, of partial derivatives ^^(^ ^ ) ^^ ^ ^ of the scalar ^(^^) as a function of components i of the command ^^ (noted^ ^ ^ ) according to the following relation where ^ ^^ is the Jacobian of the propagation function ^^ with respect to the command ^ ^. In a fourth step S4, a gradient of a scalar of operational constraints of the carrier is determined with respect to the carrier commands. The operational constraints designate practical conditions for achieving the optimized trajectory. They can, for example, correspond to a maximum energy consumption to achieve the trajectory, or the regularity of the controls applied to achieve the trajectory. These constraints are functions of the commands along the trajectory, that is to say that each constraint can be expressed as a function of the variables ^1, ^2, … ^^ . We can denote the set of constraints ^^(^^, … , ^^), where ^ denotes a column vector listing the different constraints and ^ denotes a row vector which contains the weightings of each constraint so that ^^(^^, … , ^^) denotes a scalar which is a linear combination of the different constraints or in other words a sum of the weighted constraints.^^(^^, … , ^^) corresponds to a scalar of operational constraints of the carrier. We assume that the gradient of the term ^^(^^, … , ^^) is fast to calculate. In a fifth step S5, we determine a sequence of commands that minimizes a sum of the scalar ^(^^) and the operational constraints, the determination using a projected gradient descent method applied to the gradient of the scalar ^(^^) and to the gradient of the operational constraints. This involves solving the following optimization problem S: find the ^′1, … , ^′^ that minimize ^(^^) + ^^(^^, … , ^^) with ∀^, ^^ ∈ ^. The problem S can also be written in the form:. ^ corresponds to the set of controls admissible by the mobile carrier, such as for example an acceleration lower than a maximum acceleration between two iterations, a speed lower than a maximum speed, etc. The determination of the minimum of the sum of the function ^(^^) and the operational constraints as a function of the commands does not require a significant computation time because the gradient of the function ^(^^) and the gradient of the operational constraints can be easily calculated at each gradient descent iteration from the previously determined equations. In a sixth step S6, the trajectory is determined from the sequence of commands determined in step S5. Knowledge of the ^′1, … , ^′^ which minimize ^(^^) + ^^(^^, … , ^^) with ∀^, ^^ ∈ ^ allows the searched optimized trajectory to be found.Thus, if the carrier follows this trajectory, it is possible to obtain an estimated value ^^ of the final state and the covariance matrix ^^ of the estimated value ^. ^ . By construction, the uncertainty associated with the matrix ^ ^of covariance is less than the uncertainty associated with the initial covariance matrix ^0 of the estimated value ^0 of the initial state of the mobile carrier. In this way, one can, by following the optimized trajectory, carry out a dynamic alignment which reduces the estimation uncertainty of the extended Kalman filter. The method described allows determination of an optimized trajectory for a dynamic alignment requiring less computation time than in the prior art. Furthermore, the method is applicable to a larger class of systems. Method for navigating a carrier In relation to Figure 2, a method Q for navigating a carrier comprises all of steps S1 to S6 of the method P. The method Q further comprises the following steps. During a seventh step S7, the carrier is piloted so that the carrier follows the optimized trajectory determined in step S6.In an eighth step S8, the measurements of the inertial unit and the carrier's sensors are recorded during the trajectory. In a ninth step S9, the matrix ^ is determined. ^ decovariance of the carrier states at the end of the trajectory optimized using an extended Kalman filter. In a tenth step S10, the initial matrix ^0 decovariance of the carrier states is replaced by the matrix ^^. This concludes the alignment procedure and allows the mission to begin with an initial uncertainty on the state of the system that is reduced compared to the previous situation.

Claims

CLAIMS 1. Method for determining a dynamic alignment trajectory (P) d’un porteur mobile suivi par filtrage de Kalman étendu, le procédé comprenant les étapes suivantes : - (S1) détermination d’une matrice initiale de covariance ^0 d’une estimation initiale d’un état du mouvement du porteur à un début de la trajectory, the initial covariance matrix representing an uncertainty of the initial estimate, - (S2) détermination d’équations de navigation du porteur à partir de mesures fournies par une centrale inertielle et des capteurs du porteur, - (S3) détermination d’un premier gradient d’un scalaire ^(^ ^ ) par rapport à des commandes de pilotage du porteur, le scalaire ^(^ ^ ) étant construit from a final covariance matrix ^ ^ of a final estimate of l’état du mouvement du porteur à une fin de la trajectoire, le scalaire ^(^^ ) représentant une incertitude de l’estimation finale, le premier gradient representing an influence of the pilot controls on the uncertainty of the final estimate, - (S4) détermination d’un gradient d’un scalaire de contraintes operational of the carrier in relation to the carrier's orders, - (S5) détermination d’une séquence des commandes qui minimise une sum of the scalar ^(^^ ) et du scalaire des contraintes opérationnelles, la determination using a projected gradient descent method, and - (S6) détermination de la trajectoire à partir de la séquence des commands.

2. Method according to claim 1 in which the trajectory is discretized according to a plurality of points n, la détermination du premier gradient comprenant pour chaque point n : - un calcul d’un deuxième gradient du scalaire ^(^ ^ ) par rapport à des transition matrices of an estimated value ^ ^−1from the state of the carrier at point n-1 to an estimated value ^ ^ of the state of the carrier at point n, - un calcul d’un troisième gradient du scalaire ^(^ ^ ) par rapport à des action matrices of a noise in the transition of the estimated value ^ ^−1 towards the estimated value ^ ^ , And - un calcul d’un quatrième gradient du scalaire ^(^ ^ ) par rapport à une observation matrix of the state of the carrier at point n, the determination of the first gradient comprising a use of backpropagation equations of the first gradient, the second gradient, the third gradient and the fourth gradient.

3. Procédé selon la revendication 2 dans lequel la détermination du premier gradient comprend l’utilisation des équations suivantes : in which ^ ^ denotes the transition matrix of an estimated value ^ ^−1 of l’état du porteur au point n-1 vers une valeur estimée ^^ de l’état du carrier at point n, ^ ^ désigne la matrice d'action d’un bruit ^^ sur la valeur estimée ^^, le bruit ^^ étant un bruit blanc gaussien d’espérance nulle et de matrice de covariance ^^, ^ ^|^ is the covariance matrix of the estimated value ^ ^ which takes into account an observation made at point n, ^ ^|^−1 est la matrice de covariance de la valeur estimée ^^ qui ne does not take into account the observation made at point n, ^ ^ désigne une matrice d'observation de la valeur estimée ^^ et ^^ désigne un bruit de mesure gaussien d’espérance nulle et de matrice de covariance ^^.

4. Procédé selon la revendication 3 dans lequel la détermination du premier gradient du scalaire ^(^ ^ ) comprend une étape (S3a) de détermination pour chaque point n d’un cinquième gradient du scalaire ^(^^ ) par rapport aux composantes ^ de la valeur estimée ^^ et notées ^ ^ ^ .

5. Procédé selon la revendication 4 dans lequel la détermination du fifth gradient includes a backpropagation of the values ​​of the partial derivative of the scalar ^(^^ ) en fonction des composantes ^ de la estimated value ^ ^ and noted ^ ^ ^ .

6. Procédé selon l’une quelconque des revendications 3 à 5 dans lequel la determination of the partial derivatives of the scalar ^(^^ ) en fonction des matrices ^ ^|^ includes backpropagation of covariance matrices ^^|^ à partir de la dérivée partielle du scalaire ^(^^) en fonction de la matrix ^ ^ .

7. Procédé selon l’une quelconque des revendications 3 à 6, dans lequel la détermination du premier gradient comprend une étape (S3b) de determination, for each point n of the trajectory, of partial derivatives ^^(^ ^ ) ^^ ^ ^ du scalaire ^(^^) en fonction de composantes ^ ^ ^ of the commande ^^ au point n selon la relation suivante in which ^ ^ ^ est une jacobienne d’une fonction de propagation ^ ^ relative to the commands ^ so that ^ ^ ^ ^ = the propagation function ^^ allowing to estimate the orientation, the speed and the position of the carrier at point n+1, from the orientation, speed and position of the carrier at point n, commands at time n and bruit à l’instant n, et ^^ est un vecteur dont toutes les composantes sont null except the j-th which is equal to 1.

8. Method according to any one of claims 1 to 7, in which the scalar ^(^^ ) est la trace ou la plus grande valeur propre de la matrice ^ ^ .

9. Procédé de navigation (Q) d’un porteur à l’aide d’un filtre de Kalman étendu estimant un état de navigation du porteur, le procédé comprenant les étapes suivantes : - détermination d’une trajectoire d’un alignement dynamique selon l’une des revendications 1 à 8, - (S7) pilotage du porteur de sorte que le porteur suit la trajectoire, - (S8) relevé de mesures de la centrale à inertie et des capteurs du porteur during the trajectory, - (S9) détermination de la matrice ^^ de covariance des états du porteur en fin de trajectoire, et - (S10) remplacement de la matrice initiale ^0 de covariance des états du porteur par la matrice ^^.

10. Computer program product comprising program code instructions for executing the steps of the method according to one of the preceding claims, when this program is executed by a computer.

11. Dispositif de suivi de trajectoire d’un porteur mobile, le dispositif including: - une interface de réception configurée pour recevoir des mesures acquises par une centrale à inertie et des capteurs, - un processeur configuré pour mettre en œuvre une détermination d’une trajectoire d’alignement dynamique du porteur, le dispositif étant configuré pour mettre en œuvre les étapes suivantes : - (S1) détermination d’une matrice initiale de covariance ^0 d’une initial estimate of a state of the carrier's motion at the start of the trajectory, the initial covariance matrix representing an uncertainty of the initial estimate, - (S2) détermination d’équations de navigation du porteur à partir de mesures fournies par une centrale inertielle et des capteurs du porteur,- (S3) détermination d’un premier gradient d’un scalaire ^(^ ^ ) par rapport to carrier piloting commands, the scalar ^(^^ ) étant construit from a final covariance matrix ^ ^ from a final estimate of the state of the carrier's motion at an end of the trajectory, the scalar ^(^^ ) représentant une incertitude de l’estimation finale, le premier gradient representing an influence of the pilot controls on the uncertainty of the final estimate, - (S4) détermination d’un gradient d’un scalaire de contraintes operational of the carrier in relation to the carrier's orders, - (S5) détermination d’une séquence des commandes qui minimise une sum of the scalar ^(^^ ) et du scalaire des contraintes opérationnelles, la determination using a projected gradient descent method, and - (S6) détermination de la trajectoire à partir de la séquence des orders.

12. Centrale de navigation pour porteur mobile, comprenant : - une centrale inertielle, - des capteurs, et - au moins un dispositif de suivi de trajectoire selon la revendication previous.

Images