Error-free strapdown inertial navigation method for transmitting system based on Lie group description

Abstract

The embodiment of the invention relates to a transmitting system error-free strapdown inertial navigation method and device based on Lie group description, equipment and a medium, and the method comprises the steps: obtaining a first Lie group updating equation and a second Lie group updating equation through a constructed first Lie group differential equation and a constructed second Lie group differential equation; approaching a first spiral Lie algebra and a second spiral Lie algebra based on a subsample algorithm, correspondingly obtaining an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra, obtaining a discretized first Lie group updating equation and a discretized second Lie group updating equation, and solving to obtain an attitude conversion matrix and a velocity vector; based on the attitude transformation matrix and the velocity vector, determining the position information of the cannonball relative to the earth, the attitude angle information of the cannonball and the velocity vector of the cannonball under the launching coordinate system, the rotation and translation differential equations of the carrier are incorporated into the special Euclidean group SE (3) to form a new differential equation, and motion solution is performed under the mathematical structure, so that the velocity of the cannonball is calculated. The purpose of improving the calculation precision of a numerical updating algorithm is achieved.

Description

Technical Field

[0001] The present application belongs to the field of inertial navigation, and in particular relates to a method, apparatus, device and medium for error-free strapdown inertial navigation of a transmitting system based on Lie group description. Background Art

[0002] The strapdown inertial navigation algorithm numerically integrates the output of gyroscopes and accelerometers to obtain attitude, velocity, and position information. It is the core algorithm of the strapdown inertial navigation system. The accuracy of next-generation inertial devices is expected to increase by one to two orders of magnitude compared to current-generation inertial devices, thus increasing the accuracy requirements for inertial navigation algorithms. Traditional algorithms describe angular motion using quaternions, while linear motion describes attitude information using direction cosine matrices. Attitude updates are performed using a series expansion method, while velocity updates are performed using a step-by-step integration method. The separate attitude and velocity update processes create difficulties in designing algorithms for higher-precision numerical updates. Summary of the Invention

[0003] In view of this, the embodiments of the present application propose a launch system error-free strapdown inertial navigation method, apparatus, device and medium based on Lie group description, aiming to improve the accuracy of the launch system strapdown inertial navigation.

[0004] To achieve the above-mentioned purpose, an embodiment of the present application provides a launch system error-free strapdown inertial navigation method based on Lie group description, comprising: setting a strapdown inertial navigation differential equation group of a projectile in a launch coordinate system, and constructing a first Lie group differential equation and a second Lie group differential equation according to the strapdown inertial navigation differential equation group; solving the first Lie group differential equation to obtain a first Lie group update equation, and solving the second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on the product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra; approximating the first Lie group update equation based on a sub-sample algorithm The spiral Lie algebra and the second spiral Lie algebra correspond to the approximate values ​​of the first spiral Lie algebra and the approximate values ​​of the second spiral Lie algebra, and based on the exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group, the discretized first Lie group update equation is obtained, and based on the exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group, the discretized second Lie group update equation is obtained, the first Lie group update equation after velocity compensation is solved to obtain the attitude transformation matrix, the discretized second Lie group update equation after velocity compensation is solved to obtain the velocity vector in the inertial system; based on the attitude transformation matrix and the velocity vector, the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system and the velocity vector of the projectile in the launch coordinate system are determined.

[0005] Optionally, before determining the second exponential mapping relationship based on the first exponential mapping relationship of the Lie group and the Lie algebra, the method further includes: setting a navigation coordinate system of the projectile, and determining the transformation relationship between each coordinate system based on the navigation coordinate system; wherein the navigation coordinate system includes an Earth-centered Earth-fixed coordinate system, a launch coordinate system, a launch inertial coordinate system and a carrier coordinate system; the transformation relationship includes a direction cosine matrix between the launch inertial coordinate system and the launch coordinate system, and an attitude transformation matrix from the launch coordinate system to the carrier coordinate system.

[0006] Optionally, constructing the first Lie group differential equation and the second Lie group differential equation based on the strapdown inertial navigation differential equation group includes: constructing the strapdown inertial navigation differential equation group according to the attitude, velocity and position of the carrier; constructing the Lie group differential equation of the carrier motion from the strapdown inertial navigation differential equation group; wherein, the first Lie group differential equation is constructed based on the attitude matrix belonging to the special orthogonal group and the first Lie algebra corresponding to the attitude matrix, and the second Lie group differential equation is constructed based on the velocity matrix belonging to the special orthogonal group and the second Lie algebra corresponding to the velocity matrix.

[0007] Optionally, after solving the first Lie group differential equation to obtain the first Lie group update equation and solving the second Lie group differential equation to obtain the second Lie group update equation, the method further includes: obtaining a differential expression of the indexed first spiral Lie algebra and a differential expression of the indexed second spiral Lie algebra, wherein the differential expression of the indexed first spiral Lie algebra includes a first differential term and an indexed first Lie algebraic representation, and the differential expression of the indexed second spiral Lie algebra includes a second differential term and an indexed second Lie algebraic representation; solving the first differential term in the differential expression of the indexed first spiral Lie algebra and the second differential term in the differential expression of the indexed second spiral Lie algebra based on the BCH algorithm, respectively corresponding to obtaining a differential representation matrix of the first exponential mapping and a differential representation matrix of the second exponential mapping; integrating the differential representation matrix of the first exponential mapping and the differential representation matrix of the second exponential mapping, respectively, to obtain an indexed first spiral Lie algebra matrix and an indexed second spiral Lie algebra matrix.

[0008] Optionally, the approximating the first spiral Lie algebra and the second spiral Lie algebra based on the sub-sample algorithm to obtain an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra accordingly includes: using second-order approximation and approximate replacement on the differential equations of the first spiral Lie algebra and the differential equations of the second spiral Lie algebra to obtain a first approximate differential expression and a second approximate differential expression accordingly; integrating the first approximate differential expression and the second approximate differential expression respectively within an update period to obtain a discretized first spiral Lie algebra and a discretized second spiral Lie algebra accordingly; wherein the discretized first spiral Lie algebra is determined based on the sum of the angular velocity increment and the first integral term; the discretized second spiral Lie algebra is determined based on the sum of the velocity increment and the second integral term; approximating the first integral term and the second integral term using sampling values ​​based on the sub-sample algorithm within the update period to obtain a first approximation and a second approximation; obtaining an approximate value of the first spiral Lie algebra based on the sum of the first approximation and the angular velocity increment; and obtaining an approximate value of the second spiral Lie algebra based on the second approximation and the sum of the velocity increment.

[0009] Optionally, determining the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system based on the attitude conversion matrix and the velocity vector includes: processing the velocity vector in the inertial system based on a simplified Lie group strapdown inertial navigation update algorithm to obtain the position information of the projectile relative to the earth in the launch coordinate system; obtaining the conversion matrix from the launch inertial coordinate system to the launch coordinate system based on the attitude conversion matrix, and obtaining the attitude angle information of the projectile in the launch coordinate system; and obtaining the velocity vector of the projectile in the launch coordinate system based on the position information in the launch coordinate system, the velocity vector in the inertial system, and the conversion matrix.

[0010] In order to achieve the above-mentioned purpose, the present application also provides a launch system error-free strapdown inertial navigation device based on Lie group description, comprising: an equation construction module, used to set the strapdown inertial navigation differential equation group of the projectile in the launch coordinate system, and construct a first Lie group differential equation and a second Lie group differential equation according to the strapdown inertial navigation differential equation group; an equation solving module, used to solve the first Lie group differential equation to obtain a first Lie group update equation, and solve the second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on the product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra; a parameter solving module, used to solve the first Lie group differential equation based on a sub-module. The method comprises the following steps: a first spiral Lie algebra and a second spiral Lie algebra are approximated by a sample algorithm, and an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra are obtained accordingly. The discretized first Lie group update equation is obtained based on the exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group. The discretized second Lie group update equation is obtained based on the exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group. The first Lie group update equation after velocity compensation is solved to obtain an attitude conversion matrix. The discretized second Lie group update equation after velocity compensation is solved to obtain a velocity vector in an inertial system. A projectile position determination module is used to determine the position information of the projectile relative to the earth in a launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system based on the attitude conversion matrix and the velocity vector.

[0011] To achieve the above-mentioned purpose, an embodiment of the present application also provides a server, comprising: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions that can be executed by the at least one processor, and the instructions are executed by the at least one processor so that the at least one processor can execute the above-mentioned error-free strapdown inertial navigation method of the transmitting system based on the Lie group description.

[0012] To achieve the above-mentioned purpose, an embodiment of the present application further provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, it can implement the above-mentioned error-free strapdown inertial navigation method of the transmitting system based on the Lie group description.

[0013] The embodiments of the present application propose an error-free strapdown inertial navigation method, device, equipment and medium for a launch system based on Lie group description. The method sets a strapdown inertial navigation differential equation group for a projectile in a launch coordinate system, and constructs a first Lie group differential equation and a second Lie group differential equation based on the strapdown inertial navigation differential equation group; solves the first Lie group differential equation to obtain a first Lie group update equation, and solves the second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on the product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra; approximates the first spiral Lie algebra and the second spiral Lie algebra based on a sub-sample algorithm, and obtains an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra. The approximate value of the number is obtained, and the discretized first Lie group update equation is obtained based on the exponential mapping relationship between the first spiral Lie algebra approximation and the Lie group, and the discretized second Lie group update equation is obtained based on the exponential mapping relationship between the second spiral Lie algebra approximation and the Lie group. The first Lie group update equation after velocity compensation is solved to obtain the attitude transformation matrix, and the discretized second Lie group update equation after velocity compensation is solved to obtain the velocity vector in the inertial system; based on the attitude transformation matrix and the velocity vector, the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system and the velocity vector of the projectile in the launch coordinate system are determined, and a new differential equation is formed by incorporating the carrier's rotation and translation differential equations into the special Euclidean group SE(3). The motion solution is performed under this mathematical structure to achieve the purpose of improving the calculation accuracy of the numerical update algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 This is a flow chart of a transmitting system error-free strapdown inertial navigation method based on Lie group description provided in one embodiment of the present application;

[0015] Figure 2 This is an algorithm structure diagram of a transmitting system error-free strapdown inertial navigation method based on Lie group description provided in one embodiment of the present application;

[0016] Figure 3 1. A schematic diagram of a launch coordinate system for a launch system error-free strapdown inertial navigation method based on a Lie group description, provided in one embodiment of the present application;

[0017] Figure 4 It is a Lie group / Lie algebra mapping diagram of a transmitting system error-free strapdown inertial navigation method based on a Lie group description provided in one embodiment of the present application. DETAILED DESCRIPTION

[0018] In order to make the purpose, technical solutions and advantages of the embodiments of the present application clearer, each embodiment of the present application will be described in detail below with reference to the accompanying drawings. However, it will be understood by those skilled in the art that in each embodiment of the present application, many technical details are proposed to enable the reader to better understand the present application. However, even without these technical details and various changes and modifications based on the following embodiments, the technical solutions claimed in the present application can be implemented. The division of the following embodiments is for convenience of description and should not constitute any limitation on the specific implementation of the present application. The various embodiments can be combined and referenced with each other under the premise of no contradiction.

[0019] One embodiment of the present application proposes a transmission system error-free strapdown inertial navigation method based on Lie group description, which is applied to an electronic device, wherein the electronic device can be a terminal or a server. This embodiment and the following embodiments are described using a server as an example. The implementation details of the transmission system error-free strapdown inertial navigation method based on Lie group description proposed in this embodiment are described in detail below. The following content is only provided for ease of understanding and is not required for implementing this solution.

[0020] A Lie group is an algebraic structure consisting of sets and operations, where the group operations satisfy the property of continuous differentiability. Lie groups are mathematical objects that possess both the structure of an algebraic group and the properties of a differential manifold. The combination of their smooth manifold properties and continuous group operations gives Lie groups a unique advantage in solving differential equations.

[0021] The special orthogonal group and the special Euclidean group are defined as follows

[0022]

[0023] A Lie algebra consists of a set A number field and a binary operation [·,·], where the binary operation is also called Lie bracket and is represented as: [X,Y]. Based on the above conditions, the Lie algebras so(3) and se(3) are defined as follows:

[0024]

[0025] where ξ∈se(3), where φ represents the Lie algebra corresponding to the rotational motion of the rigid body, is the angular velocity during rotational motion, and ρ represents the Lie algebra corresponding to the translational motion of the rigid body, is the velocity during translational motion.

[0026] The relationship between Lie groups and Lie algebras can be expressed as Figure 4 To express, Lie algebra T E M (grid plane) is the tangent space of the Lie group manifold M (here represented by a light blue sphere) at the unit point ε. By exponential mapping, every red straight line path ζ from the origin on the Lie algebrai Generates yellow mapping paths exp(ζ along their respective paths around the manifold i ). Every element in a Lie group has a completely equivalent linear vector space in a Lie algebra.

[0027] Among them, the exponential mapping of the matrix is:

[0028]

[0029] φ=θa, where a ∧ Represents the antisymmetric matrix of the unit vector a, satisfying the following two properties:

[0030] a ∧ a ∧ =aa T -I a ∧ a ∧ a ∧ =-a ∧

[0031] Then the direction cosine matrix and the rotation vector have the following conversion relationship:

[0032] R=exp(θa ∧ )=cosθI+(1-cosθ)aa T +sinθa ∧

[0033] Thus we get the exponential mapping from so(3) to SO(3). On this basis, we can deduce that for T = [ωv] T The exponential mapping of ∈se(3) is:

[0034]

[0035] Where Ω=cosθI+(1-cosθ)aa T +sinθa ∧ .

[0036] Another embodiment of the present application proposes a launch system error-free strapdown inertial navigation method based on Lie group description. The details of the launch system error-free strapdown inertial navigation method based on Lie group description proposed in this embodiment are described in detail below. The following content is only the implementation details provided for the convenience of understanding and is not necessary for the implementation of this embodiment. Figure 1 : is a schematic diagram of the launch system error-free strapdown inertial navigation method based on Lie group description proposed in this embodiment. The specific process of the launch system error-free strapdown inertial navigation method based on Lie group description proposed in this embodiment may include the following execution process:

[0037] S101, setting a strapdown inertial navigation differential equation group of the projectile in a launch coordinate system, and constructing a first Lie group differential equation and a second Lie group differential equation based on the strapdown inertial navigation differential equation group;

[0038] Before step S101, the transmitting system error-free strapdown inertial navigation method based on Lie group description further includes:

[0039] Set the navigation coordinate system of the projectile and determine the conversion relationship between the coordinate systems based on the navigation coordinate system;

[0040] Among them, the navigation coordinate system includes the Earth-centered Earth-fixed coordinate system, the launch coordinate system, the launch inertial coordinate system and the carrier coordinate system;

[0041] The transformation relationship includes the direction cosine matrix between the launch inertial coordinate system and the launch coordinate system, and the attitude transformation matrix between the launch coordinate system and the carrier coordinate system.

[0042] refer to Figure 3 For example, in this step, the processor may select a navigation coordinate system and establish a navigation kinematics model in the corresponding coordinate system. The navigation coordinate system may include:

[0043] (1) Earth-centered Earth-fixed coordinate system (e system)

[0044] The Earth-centered, Earth-fixed coordinate system, or e-system, has its origin at the center of the Earth, the x-axis in the equatorial plane pointing to the prime meridian, the z-axis being the Earth's rotation axis and pointing to the North Pole, and the y-axis in the equatorial plane forming a right-handed rectangular coordinate system with the x-axis and the z-axis.

[0045] (2) Emission coordinate system (g system)

[0046] Emission coordinate system, g system, coordinate origin O g The launch point is the launch point. The x-axis points to the launch aiming direction in the horizontal plane of the launch point. The y-axis is perpendicular to the horizontal plane of the launch point and points upward. The z-axis, x-axis, and y-axis form a right-handed rectangular coordinate system. The launch system is fixed to the earth. The geographic latitude B0, longitude λ0, altitude h0, and launch azimuth A0 of the launch point determine the relationship between the launch coordinate system and the earth, as shown in the following example: Figure 3 shown.

[0047] (3) Launch inertial coordinate system

[0048] The launch inertial coordinate system, or a-frame, coincides with the launch coordinate system at the moment of launch. Due to the Earth's rotation, the launch coordinate system, fixed to the Earth, changes its orientation in inertial space. Both the launch inertial coordinate system and the Earth-centered inertial coordinate system are inertial coordinate systems, but their origins and axes point in different directions.

[0049] (4) Carrier coordinate system

[0050] The projectile coordinate system, b system, has its origin at the center of mass of the guided projectile, the x-axis coincides with the longitudinal axis of the projectile, the y-axis is located in the longitudinal symmetry plane of the projectile and is perpendicular to the x-axis, and the z-axis, x-axis, and y-axis form a right-handed rectangular coordinate system.

[0051] (5) Rotation Matrix

[0052] The rotation matrices for rotating α around the x-axis, y-axis, and z-axis are

[0053]

[0054] (6) Launch coordinate system and launch inertial coordinate system

[0055] The direction cosine matrix between the launch inertial coordinate system and the launch coordinate system is As shown in the following formula. Let the time interval from the launch moment to the moment in question be t, then the launch coordinate system rotates around the earth's axis ω ie t angle.

[0056]

[0057] (7) Launch coordinate system and carrier coordinate system

[0058] Transformation matrix from launch coordinate system to carrier coordinate system as follows

[0059]

[0060] Transformation matrix from carrier coordinate system to launch coordinate system for

[0061]

[0062] In one embodiment of the present application, constructing a first Lie group differential equation and a second Lie group differential equation according to the strapdown inertial navigation differential equation group includes:

[0063] Construct a set of strapdown inertial navigation differential equations based on the attitude, velocity and position of the carrier;

[0064] Because the gyroscope and accelerometer data outputs are relative to the inertial frame, the launch inertial coordinate system and the launch coordinate system are initially coincident. When using Lie groups / Lie algebras for strapdown calculations, the attitude matrix and velocity information form an SE(3) group. The differential equations of the launch inertial coordinate system precisely meet the conditions for forming an SE(3) group. The strapdown differential equations for the launch coordinate system are shown below.

[0065]

[0066] A strapdown inertial navigation differential equation group is constructed to construct the Lie group differential equations of the carrier motion; wherein, the first Lie group differential equation is constructed based on the attitude matrix belonging to the special orthogonal group and the first Lie algebra corresponding to the attitude matrix, and the second Lie group differential equation is constructed based on the velocity matrix belonging to the special orthogonal group and the second Lie algebra corresponding to the velocity matrix.

[0067] For example, Lie groups are continuous groups. Special orthogonal groups and special Euclidean groups are both time-continuous groups and are a type of Lie group. Here we take the attitude transformation of an aircraft as an example. Suppose the attitude rotation matrix of the coordinate system g relative to the reference system a can be expressed as Since the attitude matrix is an orthogonal matrix with determinant 1, satisfying RR T =I, then Pose Matrix The differential equation is:

[0068]

[0069] in, is the angular velocity of system b relative to system g, for The antisymmetric matrix, That is, the posture matrix that satisfies the Lie group conditions Similarly, we can obtain the second Lie group differential equation based on the velocity matrix belonging to the special orthogonal group and the second Lie algebra corresponding to the velocity matrix.

[0070] S102, solving a first Lie group differential equation to obtain a first Lie group update equation, and solving a second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on the product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra;

[0071] In one embodiment of the present application, after solving the first Lie group differential equation to obtain the first Lie group update equation and solving the second Lie group differential equation to obtain the second Lie group update equation, the method further includes:

[0072] Obtaining a differential expression of the indexed first spiral Lie algebra and a differential expression of the indexed second spiral Lie algebra, wherein the differential expression of the indexed first spiral Lie algebra includes a first differential term and the indexed first Lie algebraic representation, and the differential expression of the indexed second spiral Lie algebra includes a second differential term and the indexed second Lie algebraic representation;

[0073] Based on the BCH algorithm, the first differential term in the differential expression of the exponential first spiral Lie algebra and the second differential term in the differential expression of the exponential second spiral Lie algebra are solved, and the differential representation matrix of the first exponential mapping and the differential representation matrix of the second exponential mapping are obtained respectively;

[0074] The differential representation matrix of the first exponential mapping and the differential representation matrix of the second exponential mapping are integrated respectively to obtain an indexed first spiral Lie algebra matrix and an indexed second spiral Lie algebra matrix respectively.

[0075] For example, the following description is made by taking the first Lie group differential equation as an example. is a first-order homogeneous differential equation. Solving it gives the first Lie group update equation, which can be expressed as:

[0076]

[0077] in, is the initial posture matrix within a preset update period, is the first spiral Lie algebra indexed within a preset update period. For the discrete time For example, it means that during this period of time The increase in It corresponds to the rotation matrix and describes the mapping relationship between angular velocity and attitude transformation matrix. It is the Lie algebra so(3) corresponding to the special orthogonal group SO(3).

[0078] because Indicates that within a single sampling period The increment, its differential, that is, the differential of the first exponential spiral Lie algebra, the differential of the first exponential spiral Lie algebra can include the first differential term and the exponential first Lie algebra representation, and the expression is defined as:

[0079]

[0080] According to the BCH (Baker-Campbel-Hausdorff) formula,

[0081]

[0082] It can be solved

[0083]

[0084] Where φ represents the cumulative amount of ω during this period of time, that is Similarly, ρ represents the cumulative amount of v during this period of time, that is,

[0085] In order to facilitate the subsequent formula expression, the following formulas are simply expressed

[0086] α=scβ=s 2

[0087] So for a given T = [ωv] T ∈se(3), the differential of its exponential mapping, that is, the differential representation matrix of the first exponential mapping can be expressed as

[0088]

[0089] in,

[0090]

[0091] Then, the first spiral Lie algebra matrix is

[0092]

[0093] in,

[0094]

[0095] From this we can get the solution method of the differential equation corresponding to the special orthogonal group SO(3). Since the special Euclidean group and the special orthogonal group are similar in form, their solutions are also similar. The differential equation corresponding to the special Euclidean group is:

[0096]

[0097] Among them, Y′ represents the result after updating the solution, V=(v;ω) T ∈se(3), then It is the Lie algebra of the Lie group Y.

[0098] Then the solution of the above differential equation is:

[0099] Y(t)=Y(0)exp(φ(t))

[0100] Among them, the Lie algebra φ(t) satisfies the differential equation

[0101]

[0102] By We can get:

[0103]

[0104] According to the formula have

[0105]

[0106] In summary, there are

[0107]

[0108] Pass-through You can get the formula The solution to the differential equation, and finally its discretization, yields the Lie group update represented by the equation Y(t) = Y(0)exp(φ(t)). This completes the basic concepts of Lie groups / Lie algebras and the solution of Lie group differential equations. Next, we will design a Lie group / Lie algebra strapdown inertial navigation algorithm based on the launch coordinate system.

[0109] S103, approximating the first spiral Lie algebra and the second spiral Lie algebra based on a sub-sample algorithm, obtaining an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra respectively, obtaining a discretized first Lie group update equation based on an exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group, and obtaining a discretized second Lie group update equation based on an exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group, solving the velocity-compensated first Lie group update equation to obtain an attitude transformation matrix, and solving the velocity-compensated discretized second Lie group update equation to obtain a velocity vector in an inertial system;

[0110] In one embodiment of the present application, approximating the first spiral Lie algebra and the second spiral Lie algebra based on the sub-sampling algorithm to obtain the approximate value of the first spiral Lie algebra and the approximate value of the second spiral Lie algebra may include the following execution process:

[0111] The differential equations of the first spiral Lie algebra and the differential equations of the second spiral Lie algebra are both subjected to second-order approximation and approximate substitution, and the first approximate differential expression and the second approximate differential expression are obtained respectively;

[0112] Integrating the first approximate differential expression and the second approximate differential expression in an update period respectively, and obtaining a discretized first spiral Lie algebra and a discretized second spiral Lie algebra respectively;

[0113] The discretized first spiral Lie algebra is determined based on the sum of the angular velocity increment and the first integral term;

[0114] The discretized second spiral Lie algebra is determined based on the sum of the velocity increment and the second integral term;

[0115] Specifically, the navigation solution information of the guided artillery shell is provided by the IMU, and the IMU outputs the angular velocity and specific force of the carrier. Therefore, for the navigation solution of the guided artillery shell, the corresponding Lie algebra is replaced by the velocity v of the rigid body and the specific force f b That's it.

[0116] The attitude differential equation and velocity differential equation are established as differential equations of special Euclidean group SE(3):

[0117]

[0118] The following is an example of the differential equation of the first spiral Lie algebra. The differential equation of the first spiral Lie algebra in the differential equation of the special Euclidean group SE(3) can be simplified as:

[0119]

[0120] in, represents a special Euclidean group, By solving the above differential equations, we can obtain the attitude transformation matrix of the carrier system b relative to the inertial coordinate system a: and the velocity of the carrier system relative to the inertial system Finally, convert it to the navigation system.

[0121] Since the information collected by the sensor in navigation is discrete information, the computer cannot directly process the differential equation, so it is necessary to discretize the Lie group differential equation. The key to solving the Lie group differential equation is to calculate the Lie algebra within the update time interval. According to the formula It can be seen that the Lie algebraic differential equation is:

[0122]

[0123] General Substitute into the above formula and use the second-order approximation, that is, D = 0, Available

[0124]

[0125] Usually the update time is very short, and Φ∈se(3) is a small quantity during the update period. The first approximate differential expression of the above formula can be:

[0126]

[0127] in Mode The integral within the update period, that is, the discretized first spiral Lie algebra can be expressed as:

[0128]

[0129] Approximating the first integral term and the second integral term using the sampled values ​​within the update period based on the sub-sampling algorithm, and obtaining a first approximation and a second approximation respectively;

[0130] Based on the first approximation and the sum of the angular velocity increments, the first spiral Lie algebra approximation is obtained;

[0131] Based on the second approximation and the sum of the velocity increment, the second spiral Lie algebra approximation is obtained.

[0132] Mode Where α = (Δv; Δθ) is the angular increment and velocity increment during the update period. The angular increment and velocity increment can be directly measured by the gyroscope and accelerometer. If the update time interval contains N samples, the relationship between the angular velocity increment α and the inertial navigation device output is

[0133]

[0134] Where Δv i and Δα i Represents the i-th sampling data of the inertial device within the update cycle. The second term in is caused by the non-commutativity error. Unlike the traditional algorithm, which needs to calculate two non-commutativity terms, namely, cone and rowing, the Lie group / Lie algebra algorithm only needs to calculate one. The integral term in is represented by ΔΦ s =(Δη s ;Δφ s ) is expressed as follows. The sub-sampling algorithm uses the sampling values ​​within the update period to update ΔΦ s Approximation is performed, and the first approximate expression is:

[0135]

[0136] S104 , based on the attitude conversion matrix and the velocity vector, determine the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system.

[0137] In one embodiment of the present application, based on the attitude conversion matrix and the velocity vector, determining the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system includes:

[0138] The velocity vector in the inertial system is processed based on the simplified Lie Group strapdown inertial navigation update algorithm to obtain the position information of the projectile relative to the earth in the launch coordinate system.

[0139] Based on the attitude conversion matrix, a conversion matrix from the launch inertial coordinate system to the launch coordinate system is obtained, and attitude angle information of the projectile in the launch coordinate system is obtained;

[0140] Based on the position information in the launch coordinate system, the velocity vector in the inertial system and the transformation matrix, the velocity vector of the projectile in the launch coordinate system is obtained.

[0141] For example, by solving the above differential equation and performing gravity compensation on the velocity, the attitude transformation matrix of the carrier system b relative to the inertial coordinate system a can be obtained: and the velocity v in the inertial system a The launch inertial system position update adopts the simplified Lie Group strapdown inertial navigation update algorithm, which is expressed as follows:

[0142]

[0143] The transformation matrix from the launch inertial coordinate system to the launch coordinate system can be obtained by the formula Y(t+ΔT)=Y(t)exp(Φ(ΔT)) Then there is

[0144]

[0145] According to the posture transformation matrix The information of the three attitude angles can be obtained. Similarly, the velocity vector in the launch coordinate system can be obtained by the following formula

[0146]

[0147] Therefore, the position of the inertial system P is obtained by Lie Group strapdown inertial navigation update a , we can get the latitude and longitude of the aircraft relative to the earth (B, λ, h), and also get the position P of the aircraft in the earth-fixed system. e , and the position P in the launch system can be obtained g .

[0148] In summary, this application is based on the characteristics of Lie group differential manifolds. The IMU output angular increment and velocity increment can be solved in the form of exponential mapping to obtain attitude and velocity information. The algorithm's compensation for coning error and paddling error is the same as the equivalent rotation vector method, while the equivalent rotation vector method needs to discard high-order terms, and usually only retains third-order to fourth-order terms. This method can avoid the errors caused by discarding high-order terms.

[0149] In the specific implementation process, a launch system error-free strapdown inertial navigation method based on Lie group description is used for numerical integration of the inertial navigation system. The algorithm solves the navigation attitude, velocity and position change information based on the inertial device information, and integrates to obtain the navigation information of the carrier. Figure 1 and Figure 2 shown.

[0150] S1. Establish a kinematic model. Define the navigation coordinate system and the carrier coordinate system, and establish the motion differential equation.

[0151] S2. Lie group representation of carrier motion. A Lie group-based description of the aircraft motion state is established, and the attitude and velocity differential equations in the launch coordinate system are incorporated into the Lie group to construct the SE(3) model of carrier motion in the launch coordinate system.

[0152] S3. Establish the Lie group differential equation. According to the formula The angular velocity and specific force within an update period are recorded as a Lie algebra and mapped to the Lie group.

[0153] S4. Derive the discretization update formula. According to the formula The discretization update formula for Lie group differential equations is given.

[0154] S5. Complete the solution of the differential equation. Update the discretized Lie group differential equation according to the formula Y(t+ΔT)=Y(t)exp(Φ(ΔT)). Solve the spiral Lie algebra from the carrier system to the launch system. Update the Lie group variables (including attitude, velocity, and position) using the spiral Lie algebra. Substitute the IMU measurements, while compensating for Earth's gravity, angular velocity, and the drag velocity caused by Earth's rotation. Complete the incremental solution of attitude, velocity, and position, and complete the numerical update of attitude, velocity, and position information in the launch system.

[0155] The step division of the above various methods is only for the purpose of clear description. During implementation, they can be combined into one step or some steps can be split and decomposed into multiple steps. As long as they include the same logical relationship, they are all within the scope of protection of this application; adding insignificant modifications or introducing insignificant designs to the algorithm or process without changing the core design of the algorithm and process are all within the scope of protection of this application.

[0156] On the basis of the above-mentioned method embodiment, the present application also provides a launch system error-free strapdown inertial navigation device based on Lie group description, which is used to solve the same technical problem as the method embodiment. The device may include an equation construction module, an equation solving module, a parameter solving module and a projectile position determination module, wherein the equation construction module is used to set the strapdown inertial navigation differential equation group of the projectile in the launch coordinate system, and construct a first Lie group differential equation and a second Lie group differential equation based on the strapdown inertial navigation differential equation group; the equation solving module is used to solve the first Lie group differential equation to obtain a first Lie group update equation, and solve the second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an indexed first spiral Lie algebra, and the second Lie group update equation is determined based on an initial velocity matrix within a preset update period and an indexed first spiral Lie algebra. The product of the digitized second spiral Lie algebra is determined; a parameter solving module is used to approximate the first spiral Lie algebra and the second spiral Lie algebra based on the sub-sample algorithm, and obtain the approximate value of the first spiral Lie algebra and the approximate value of the second spiral Lie algebra accordingly, and obtain the discretized first Lie group update equation based on the exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group, and obtain the discretized second Lie group update equation based on the exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group, solve the first Lie group update equation after velocity compensation to obtain the attitude conversion matrix, and solve the discretized second Lie group update equation after velocity compensation to obtain the velocity vector in the inertial system; a projectile position determination module is used to determine the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system based on the attitude conversion matrix and the velocity vector.

[0157] It is not difficult to find that this embodiment is a system embodiment corresponding to the above-mentioned method embodiment, and this embodiment can be implemented in conjunction with the above-mentioned method embodiment. The relevant technical details and technical effects mentioned in the above-mentioned embodiments are still valid in this embodiment, and to reduce repetition, they are not repeated here. Accordingly, the relevant technical details mentioned in this embodiment can also be applied to the above-mentioned embodiments.

[0158] It is worth mentioning that all modules involved in this embodiment are logical modules. In actual applications, a logical unit can be a physical unit, a part of a physical unit, or a combination of multiple physical units. In addition, to highlight the innovation of this application, this embodiment does not introduce units that are not closely related to solving the technical problems proposed by this application. However, this does not mean that other units do not exist in this embodiment.

[0159] Another embodiment of the present application proposes an electronic device, comprising: at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to execute the transmission system error-free strapdown inertial navigation method based on Lie group description in the above-mentioned method embodiments.

[0160] The memory and processor are connected using a bus, which can include any number of interconnected buses and bridges. The bus connects various circuits of one or more processors and memories. The bus can also connect various other circuits such as peripheral devices, voltage regulators, and power management circuits. These are all well known in the art and therefore will not be described further in this article. The bus interface provides an interface between the bus and the transceiver. The transceiver can be a single component or multiple components, such as multiple receivers and transmitters, providing a unit for communicating with various other devices on a transmission medium. Data processed by the processor is transmitted on a wireless medium via an antenna. Furthermore, the antenna also receives data and transmits it to the processor.

[0161] The processor is responsible for managing the bus and general processing, and may also provide various functions, including timing, peripheral interfacing, voltage regulation, power management, and other control functions. Magnetic memory can be used to store data used by the processor while performing operations.

[0162] Another embodiment of the present application relates to a computer-readable storage medium storing a computer program, which implements the above method embodiment when executed by a processor.

[0163] That is, those skilled in the art will understand that all or part of the steps in the above-mentioned embodiments can be implemented by instructing the relevant hardware through a program, which is stored in a storage medium and includes a number of instructions for causing a device (which may be a single-chip microcomputer, chip, etc.) or a processor to execute all or part of the steps of the methods described in each embodiment of the present application. The aforementioned storage medium includes: a USB flash drive, a mobile hard drive, a ROM (Read-Only Memory), a RAM (Random Access Memory), a magnetic disk, or an optical disk, etc., various media that can store program code.

[0164] Those skilled in the art will appreciate that the above embodiments are specific embodiments for implementing the present application, and that in actual applications, various changes may be made thereto in form and detail without departing from the spirit and scope of the present application.

Claims

1. A method for error-free strapdown inertial navigation of a transmitting system based on Lie group description, characterized in that: include: Setting a strapdown inertial navigation differential equation group of the projectile in the launch coordinate system, and constructing a first Lie group differential equation and a second Lie group differential equation based on the strapdown inertial navigation differential equation group; Solving the first Lie group differential equation to obtain a first Lie group update equation, and solving the second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on the product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on the product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra; Approximating the first spiral Lie algebra and the second spiral Lie algebra based on a sub-sample algorithm, obtaining an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra correspondingly, and obtaining a discretized first Lie group update equation based on an exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group, and obtaining a discretized second Lie group update equation based on an exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group, solving the velocity-compensated first Lie group update equation to obtain an attitude transformation matrix, and solving the velocity-compensated discretized second Lie group update equation to obtain a velocity vector in an inertial system; Based on the attitude conversion matrix and the velocity vector, the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system are determined.

2. The error-free strapdown inertial navigation method for a transmitting system based on Lie group description according to claim 1, wherein: Before determining the second exponential mapping relationship according to the first exponential mapping relationship between the Lie group and the Lie algebra, the method further includes: Set the navigation coordinate system of the projectile and determine the conversion relationship between the coordinate systems based on the navigation coordinate system; Among them, the navigation coordinate system includes the Earth-centered Earth-fixed coordinate system, the launch coordinate system, the launch inertial coordinate system and the carrier coordinate system; The transformation relationship includes the direction cosine matrix between the launch inertial coordinate system and the launch coordinate system, and the attitude transformation matrix between the launch coordinate system and the carrier coordinate system.

3. The error-free strapdown inertial navigation method for a transmitting system based on Lie group description according to claim 2, wherein: The constructing of the first Lie group differential equation and the second Lie group differential equation according to the strapdown inertial navigation differential equation group includes: Construct a set of strapdown inertial navigation differential equations based on the attitude, velocity and position of the carrier; Construct the strapdown inertial navigation differential equations to construct the Lie group differential equations of the carrier motion; Among them, the first Lie group differential equation is constructed based on the attitude matrix belonging to the special orthogonal group and the first Lie algebra corresponding to the attitude matrix, and the second Lie group differential equation is constructed based on the velocity matrix belonging to the special orthogonal group and the second Lie algebra corresponding to the velocity matrix.

4. The error-free strapdown inertial navigation method for a transmitting system based on Lie group description according to claim 1, wherein: After solving the first Lie group differential equation to obtain the first Lie group update equation and solving the second Lie group differential equation to obtain the second Lie group update equation, the method further includes: Obtaining a differential expression of the indexed first spiral Lie algebra and a differential expression of the indexed second spiral Lie algebra, wherein the differential expression of the indexed first spiral Lie algebra includes a first differential term and the indexed first Lie algebraic representation, and the differential expression of the indexed second spiral Lie algebra includes a second differential term and the indexed second Lie algebraic representation; Based on the BCH algorithm, the first differential term in the differential expression of the exponential first spiral Lie algebra and the second differential term in the differential expression of the exponential second spiral Lie algebra are solved, and the differential representation matrix of the first exponential mapping and the differential representation matrix of the second exponential mapping are obtained respectively; The differential representation matrix of the first exponential mapping and the differential representation matrix of the second exponential mapping are integrated respectively to obtain an indexed first spiral Lie algebra matrix and an indexed second spiral Lie algebra matrix respectively.

5. The error-free strapdown inertial navigation method for a transmitting system based on Lie group description according to claim 4, wherein: The method of approximating the first spiral Lie algebra and the second spiral Lie algebra based on the sub-sample algorithm, and correspondingly obtaining an approximate value of the first spiral Lie algebra and an approximate value of the second spiral Lie algebra, includes: The differential equations of the first spiral Lie algebra and the differential equations of the second spiral Lie algebra are both subjected to second-order approximation and approximate substitution, and the first approximate differential expression and the second approximate differential expression are obtained respectively; Integrating the first approximate differential expression and the second approximate differential expression in an update period respectively, and obtaining a discretized first spiral Lie algebra and a discretized second spiral Lie algebra respectively; The discretized first spiral Lie algebra is determined based on the sum of the angular velocity increment and the first integral term; The discretized second spiral Lie algebra is determined based on the sum of the velocity increment and the second integral term; Approximating the first integral term and the second integral term using the sampled values ​​within the update period based on the sub-sampling algorithm, and obtaining a first approximation and a second approximation respectively; Based on the first approximation and the sum of the angular velocity increments, the first spiral Lie algebra approximation is obtained; Based on the second approximation and the sum of the velocity increment, the second spiral Lie algebra approximation is obtained.

6. The error-free strapdown inertial navigation method for a transmitting system based on Lie group description according to claim 1, wherein: The method of determining the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system based on the attitude conversion matrix and the velocity vector includes: The velocity vector in the inertial system is processed based on the simplified Lie Group strapdown inertial navigation update algorithm to obtain the position information of the projectile relative to the earth in the launch coordinate system. Based on the attitude conversion matrix, a conversion matrix from the launch inertial coordinate system to the launch coordinate system is obtained, and attitude angle information of the projectile in the launch coordinate system is obtained; Based on the position information in the launch coordinate system, the velocity vector in the inertial system and the transformation matrix, the velocity vector of the projectile in the launch coordinate system is obtained.

7. A transmitting system error-free strapdown inertial navigation device based on Lie group description, characterized in that: include: An equation construction module is used to set a strapdown inertial navigation differential equation group of the projectile in the launch coordinate system, and to construct a first Lie group differential equation and a second Lie group differential equation based on the strapdown inertial navigation differential equation group; an equation solving module, configured to solve a first Lie group differential equation to obtain a first Lie group update equation, and solve a second Lie group differential equation to obtain a second Lie group update equation, wherein the first Lie group update equation is determined based on a product of an initial attitude matrix within a preset update period and an exponential first spiral Lie algebra, and the second Lie group update equation is determined based on a product of an initial velocity matrix within a preset update period and an exponential second spiral Lie algebra; a parameter solving module, configured to approximate the first spiral Lie algebra and the second spiral Lie algebra based on a sub-sample algorithm, obtain approximate values ​​of the first spiral Lie algebra and the second spiral Lie algebra, obtain a discretized first Lie group update equation based on an exponential mapping relationship between the approximate value of the first spiral Lie algebra and the Lie group, and obtain a discretized second Lie group update equation based on an exponential mapping relationship between the approximate value of the second spiral Lie algebra and the Lie group, solve the velocity-compensated first Lie group update equation to obtain an attitude transformation matrix, and solve the velocity-compensated discretized second Lie group update equation to obtain a velocity vector in an inertial system; The projectile position determination module is used to determine the position information of the projectile relative to the earth in the launch coordinate system, the attitude angle information of the projectile in the launch coordinate system, and the velocity vector of the projectile in the launch coordinate system based on the attitude conversion matrix and the velocity vector.

8. A terminal / electronic device / server, characterized in that: include: at least one processor; and, a memory communicatively coupled to the at least one processor; The memory stores instructions that can be executed by the at least one processor, and the instructions are executed by the at least one processor so that the at least one processor can execute the error-free strapdown inertial navigation method of the transmitting system based on the Lie group description as described in any one of claims 1 to 6.

9. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the error-free strapdown inertial navigation method for a transmitting system based on Lie group description as claimed in any one of claims 1 to 6 can be implemented.

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