CN111060140B - Polar region inertial navigation error obtaining method under earth ellipsoid model - Google Patents

Polar region inertial navigation error obtaining method under earth ellipsoid model Download PDF

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CN111060140B
CN111060140B CN201911424397.6A CN201911424397A CN111060140B CN 111060140 B CN111060140 B CN 111060140B CN 201911424397 A CN201911424397 A CN 201911424397A CN 111060140 B CN111060140 B CN 111060140B
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张通
常晓飞
符文星
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Abstract

The invention relates to a polar region inertial navigation error obtaining method under an earth ellipsoid model.A calculation error source under the earth ellipsoid model is added into an attitude error equation; the speed error equation expresses the grid speed as a grid speed true value and a grid speed error, and each error source is added; and the position error equation adopts a rectangular coordinate system to solve the position, and the error source is added into a grid calculation value containing a grid position truth value and a grid position error. Compared with the prior art, the effect is as follows: according to the method, the error model of polar region inertial navigation calculation under the earth ellipsoid model is established, the influence of the earth ellipsoid model on the accuracy of the error propagation process is reduced, and the accuracy of the model is improved, so that the accuracy of combined navigation and transfer alignment in the polar region is improved.

Description

Polar region inertial navigation error obtaining method under earth ellipsoid model
Technical Field
The invention belongs to the field of inertial navigation, and relates to a polar region inertial navigation error obtaining method under a terrestrial ellipsoid model.
Background
The core of the inertial navigation system algorithm is mechanical arrangement, and the inertial navigation system algorithm mainly comprises two types: the mechanical arrangement of the north-pointing direction and the mechanical arrangement of the wandering direction. The north-pointing azimuth inertial navigation mechanical arrangement concept is simple and clear in physical significance, but the north-pointing arrangement has the problems of azimuth calculation overflow, error amplification and the like in high-latitude areas. When the calculation accuracy of the azimuth gyro command velocity term is gradually reduced along with the increase of the latitude until the calculation overflows, the calculation error of the horizontal velocity is seriously amplified, and even the overflow is generated. Therefore, the north-seeking system cannot normally work in high-latitude areas, is only suitable for navigation in middle-latitude and low-latitude areas, and the working area of the north-seeking system cannot exceed 75 degrees. The wandering azimuth inertial navigation mechanical programming has singular values when extracting position information from a position direction cosine matrix in a high-latitude area, so that the wandering azimuth inertial navigation system cannot realize navigation in a polar region range and only can finish resolving of an attitude direction cosine matrix and the position direction cosine matrix. In addition, the problems of longitude calculation error amplification and wandering azimuth calculation error amplification similar to those of a north-pointing azimuth inertial navigation system also exist in a high-latitude area, so that the wandering azimuth inertial navigation is mechanically organized to normally work but cannot complete the orientation and positioning tasks in the high-latitude area under the condition of not inputting position and azimuth information.
The Greenwich meridian is used as the heading reference, so that the problem of directional reference caused by meridian convergence can be avoided. The carrier is located the point department and is on a parallel with the plane of greenwich mean meridian as the graticule mesh plane, regard as the tangent plane with the horizontal plane that the carrier is located, graticule mesh plane and tangent plane's intersecting line define for graticule mesh north to, graticule mesh north is to the contained angle of same geographical direction for sigma, graticule mesh sky to coincide with geographical coordinate system sky, graticule mesh east is in tangent plane and constitutes right hand rectangular coordinate system with graticule mesh north is perpendicular, this is the graticule mesh coordinate system. The position information of the grid system under the output rectangular coordinate can be matched with the navigation map of the polar region for use, so that the Greenwich grid mechanical layout is a more suitable polar region inertial navigation layout scheme at present.
However, an error model for the conventional polar region grid inertial solution is derived under an earth sphere model, an error equation cannot accurately describe an error propagation process, a principle error is introduced into a mathematical model, and the accuracy of combined navigation and transfer alignment in a polar region is influenced.
The invention provides
Disclosure of Invention
Technical problem to be solved
In order to avoid the defects of the prior art, the invention provides a polar region inertial navigation error obtaining method under a terrestrial ellipsoid model.
Technical scheme
A polar region inertial navigation error obtaining method under an earth ellipsoid model is characterized by comprising an attitude error equation, a speed error equation and a position error equation;
the attitude error equation is:
Figure GDA0003692546790000021
the speed error equation is:
Figure GDA0003692546790000022
the position error equation is:
Figure GDA0003692546790000023
the method comprises the following steps:
step 1: calculating the relationship between the position error of the rectangular coordinate system and the longitude and latitude errors of the geographic coordinate system
Figure GDA0003692546790000024
Wherein: (x, y, z) represents the position of a rectangular coordinate system, (L, λ, h) represents the longitude, latitude and altitude of a geographic coordinate system, e is the earth oblateness, R N Principal radius of curvature, R, of earth-unitary fourth of twelve earthly branches M The radius of curvature of the meridian of the earth;
geographic coordinate system error: delta P e ＝A -1 ·δR e
Figure GDA0003692546790000031
Figure GDA0003692546790000032
Step 2, calculating a speed error equation: and expressing the grid speed as grid speed and grid speed error, adding each error source to obtain a speed error equation:
Figure GDA0003692546790000033
wherein: v. of G Is the grid velocity, δ v G The grid speed error is obtained;
velocity differential equation in grid coordinate system:
Figure GDA0003692546790000034
wherein:
Figure GDA0003692546790000035
representing a rotation matrix between the carrier coordinate system and the grid coordinate system, f b Representing accelerometer output, g G The gravity acceleration under the grid coordinate system is represented,
Figure GDA0003692546790000036
the rotational angular velocity of the earth under the grid coordinate system is represented,
Figure GDA0003692546790000037
representing the rotation angular rate of the grid coordinate system relative to the earth;
differential equation of grid velocity error
Figure GDA0003692546790000038
Wherein: f. of G Is the output of the accelerometer under the grid coordinate system,
Figure GDA0003692546790000039
is the earth rotation angular velocity error under the grid coordinate system,
Figure GDA00036925467900000310
is the error of grid coordinate system with respect to the rotation angular rate of the earth, δ g G Is the gravity acceleration error, phi G Is the error in the attitude of the object,
Figure GDA00036925467900000311
is the accelerometer error;
step 3, calculating an attitude error equation: adding a calculation error source under an earth ellipsoid model into an attitude differential equation in grid inertial navigation solution:
Figure GDA00036925467900000312
wherein:
Figure GDA00036925467900000313
is the relative inertia of grid coordinate systemThe angular velocity of rotation of the linear coordinate system,
Figure GDA00036925467900000314
it is the error of the gyroscope that is,
Figure GDA00036925467900000315
is the east-direction velocity of the grid coordinate system,
Figure GDA00036925467900000316
is the north velocity of the grid coordinate system.
Figure GDA0003692546790000041
Figure GDA0003692546790000042
Figure GDA0003692546790000043
Figure GDA0003692546790000044
Figure GDA0003692546790000045
Figure GDA0003692546790000046
Step 4, calculating a position error differential equation:
Figure GDA0003692546790000047
wherein,
Figure GDA0003692546790000048
is a rotation matrix of the grid coordinate system relative to the earth coordinate system,
Figure GDA0003692546790000049
is a rotation matrix of the geographic coordinate system relative to the grid coordinate system,
Figure GDA00036925467900000410
a rotation matrix of the terrestrial coordinate system with respect to the geographic coordinates.
Figure GDA00036925467900000411
Figure GDA0003692546790000051
Figure GDA0003692546790000052
Advantageous effects
The invention provides a polar region inertial navigation error obtaining method under a terrestrial ellipsoid model.
Attitude error equation: a calculation error source (a gyroscope error, an earth rotation angular rate error and a grid relative earth rotation angular rate error) under an earth ellipsoid model is added into an attitude differential equation in grid inertial navigation resolving. And the relationship between the position error of the core rectangular coordinate system of the earth ellipsoid model and the longitude and latitude error of the geographic coordinate system. According to the position relation of two coordinate systems, a first-order full increment is established and expressed in a matrix form, and is brought into a position error to complete an attitude error equation.
The velocity error equation: and expressing the grid speed as a grid speed true value and a grid speed error, and adding various error sources (attitude calculation error, table addition error, gravity vector error, earth rotation angular rate error and grid coordinate system relative earth rotation angular rate error) to obtain a speed error equation.
Position error equation: and solving the position by adopting a rectangular coordinate system, adding an error source (grid speed error and grid attitude matrix error) into a grid calculated value containing a grid position true value and a grid position error, and calculating to obtain a position error equation.
Compared with the prior art, the invention has the following effects: according to the method, the error model of polar region inertial navigation calculation under the earth ellipsoid model is established, the influence of the earth ellipsoid model on the accuracy of the error propagation process is reduced, and the accuracy of the model is improved, so that the accuracy of combined navigation and transfer alignment in the polar region is improved.
Drawings
FIG. 1: definition of grid coordinate system
A plane parallel to the meridian plane of Greenwich mean is taken as a grid plane at the P point where the carrier is located, a horizontal plane where the carrier is located is taken as a tangent plane, an intersection line of the grid plane and the tangent plane is defined as the north direction of the grid, the east direction of the grid and the north direction of the grid are perpendicular to each other in the tangent plane, the sky direction of the grid and the sky direction of a geographic coordinate system coincide to form a right-hand rectangular coordinate system, and a grid coordinate system (e) is represented by G (G) E 、e N 、e U Representing the east, north and sky directions of a geographic coordinate system;
Figure GDA0003692546790000061
representing grid coordinate system east, north and sky)
FIG. 2: rotation relationship between grid north and geographical north
The north direction of the grid coordinate system and the direction of the geographic coordinate system form an included angle sigma. It can be seen from the figure that the rotation relationship is coordinate rotation performed by taking the day coordinate axis as a rotation axis
Detailed Description
The invention will now be further described with reference to the following examples and drawings:
1. relationship between position error of rectangular coordinate system and longitude and latitude errors of geographic coordinate system
The relationship between the position of the rectangular coordinate system and the longitude, latitude and height of the geographic coordinate system is as follows: (x, y, z) represents the position of a rectangular coordinate system, (L, λ, h) represents the longitude, latitude and altitude of a geographic coordinate system, and e is the earth oblateness.
Figure GDA0003692546790000062
Considering the main curvature radius R of the earth-unitary fourth of twelve earthly branches in a short time N And the radius of curvature R of the meridian of the earth M The variation is small and constant.
First order full increment of the rectangular coordinate system error (δ x, δ y, δ z):
Figure GDA0003692546790000063
written in matrix form as:
Figure GDA0003692546790000064
δR e ＝A·δP e
rewriting to geographical coordinate system error representation:
δP e ＝A -1 ·δR e
Figure GDA0003692546790000071
2. equation of speed error
The velocity differential equation in the grid coordinate system is:
Figure GDA0003692546790000072
v G the velocity in the grid coordinate system is represented,
Figure GDA0003692546790000073
presentation carrierRotation matrix between body coordinate system and grid coordinate system, f b Representing accelerometer output, g G The gravity acceleration under the grid coordinate system is represented,
Figure GDA0003692546790000074
the rotational angular velocity of the earth under the grid coordinate system is represented,
Figure GDA0003692546790000075
representing the angular rate of rotation, g, of the grid coordinate system relative to the earth G Is the grid coordinate system gravitational acceleration, when there is an error:
Figure GDA0003692546790000076
wherein (delta v) G The error in the speed of the grid is,
Figure GDA0003692546790000077
is the earth rotation angular velocity error under the grid coordinate system,
Figure GDA0003692546790000078
is the error of grid coordinate system with respect to the rotation angular rate of the earth, δ g G Is the gravity acceleration error, phi G Is the error in the attitude of the object,
Figure GDA0003692546790000079
is accelerometer error):
Figure GDA00036925467900000710
Figure GDA00036925467900000711
substituted into the velocity differential equation and neglecting the second order small quantity to obtain (
Figure GDA00036925467900000712
Accelerometer error under grid coordinate system):
Figure GDA00036925467900000713
here, the rotation angle rate of the earth under the grid coordinate system is expressed as follows (the included angle between the north direction of the grid coordinate system and the direction of the geographic coordinate system is σ):
Figure GDA00036925467900000714
then, the rotational angular velocity error of the earth in the grid coordinate system can be expressed as:
Figure GDA0003692546790000081
Figure GDA0003692546790000082
Figure GDA0003692546790000083
Figure GDA0003692546790000084
Figure GDA0003692546790000085
it can be abbreviated as:
Figure GDA0003692546790000086
the grid coordinate system is expressed in relation to the angular rate of rotation of the earth as
Figure GDA0003692546790000087
The north-direction speed of the grid coordinate system is represented,
Figure GDA0003692546790000088
representing the grid coordinate system moving direction speed):
Figure GDA0003692546790000089
here:
Figure GDA00036925467900000810
Figure GDA00036925467900000811
the error of the grid coordinate system with respect to the angular rate of rotation of the earth is expressed as:
Figure GDA0003692546790000091
specifically, unfolding:
Figure GDA0003692546790000092
Figure GDA0003692546790000093
Figure GDA0003692546790000094
wherein:
Figure GDA0003692546790000095
Figure GDA0003692546790000096
if, order:
Figure GDA0003692546790000097
Figure GDA0003692546790000098
and:
Figure GDA0003692546790000099
the coefficient matrix for δ h is Q
Therefore, the error of the grid coordinate system with respect to the rotation angular rate of the earth is collated as:
Figure GDA0003692546790000101
order:
Figure GDA0003692546790000102
irrespective of δ g G Then the velocity error equation under the grid coordinate system is:
Figure GDA0003692546790000103
3. equation of attitude error
The attitude error equation under the grid coordinate system is as follows (
Figure GDA0003692546790000104
Gyroscope error):
Figure GDA0003692546790000105
wherein the error of the grid coordinate system with respect to the angular rate of rotation of the earth
Figure GDA0003692546790000106
Error of grid coordinate system relative to rotation angular rate of earth
Figure GDA0003692546790000107
While the velocity differential equation has been described, the attitude error differential equation in the grid system is as follows.
Figure GDA0003692546790000108
4. Differential equation of position error
Position error equation of grid coordinate system:
Figure GDA0003692546790000109
the representation under the geographic coordinate system is:
Figure GDA00036925467900001010
FIG. 1: a plane parallel to the meridian plane of Greenwich mean is taken as a grid plane at the P point where the carrier is located, a horizontal plane where the carrier is located is taken as a tangent plane, an intersection line of the grid plane and the tangent plane is defined as the north direction of the grid, the east direction of the grid and the north direction of the grid are perpendicular to each other in the tangent plane, the sky direction of the grid and the sky direction of a geographic coordinate system coincide to form a right-hand rectangular coordinate system, and a grid coordinate system (e) is represented by G (G) E 、e N 、e U Representing the east, north and sky directions of a geographic coordinate system;
Figure GDA0003692546790000111
representing the grid coordinate system east, north, and sky).

Claims (1)

1. A polar region inertial navigation error obtaining method under an earth ellipsoid model is characterized by comprising an attitude error equation, a speed error equation and a position error equation;
the attitude error equation is:
Figure FDA0003692546780000011
the speed error equation is:
Figure FDA0003692546780000012
the position error equation is:
Figure FDA0003692546780000013
the method comprises the following steps:
step 1: calculating the relationship between the position error of the rectangular coordinate system and the longitude and latitude errors of the geographic coordinate system:
Figure FDA0003692546780000014
wherein: (x, y, z) represents the position of a rectangular coordinate system, (L, λ, h) represents the longitude, latitude and altitude of a geographic coordinate system, e is the earth oblateness, R N Principal radius of curvature, R, of earth fourth prime circle M The radius of curvature of the meridian of the earth;
geographic coordinate system error: delta P e ＝A -1 ·δR e
Figure FDA0003692546780000015
Figure FDA0003692546780000016
Step 2, calculating a speed error equation: and expressing the grid speed as grid speed and grid speed error, adding each error source to obtain a speed error equation:
Figure FDA0003692546780000017
wherein: v. of G Is the grid velocity, δ v G The grid speed error is obtained;
velocity differential equation in grid coordinate system:
Figure FDA0003692546780000021
wherein:
Figure FDA0003692546780000022
representing a rotation matrix between the carrier coordinate system and the grid coordinate system, f b Representing accelerometer output, g G The gravity acceleration under the grid coordinate system is represented,
Figure FDA0003692546780000023
representing the rotation angular rate of the earth under the grid coordinate system,
Figure FDA0003692546780000024
representing the rotation angular rate of the grid coordinate system relative to the earth;
grid velocity error differential equation:
Figure FDA0003692546780000025
wherein: f. of G Is the output of the accelerometer under the grid coordinate system,
Figure FDA0003692546780000026
is the earth rotation angular velocity error under the grid coordinate system,
Figure FDA0003692546780000027
is the error of grid coordinate system relative to the rotation angular rate of the earth, δ g G Is the gravity acceleration error, phi G Is the error in the attitude of the object,
Figure FDA0003692546780000028
is the accelerometer error;
step 3, calculating an attitude error equation: adding a calculation error source under an earth ellipsoid model into an attitude differential equation in grid inertial navigation solution:
Figure FDA0003692546780000029
wherein:
Figure FDA00036925467800000210
is the rotation angular velocity of the grid coordinate system relative to the inertial coordinate system,
Figure FDA00036925467800000211
Figure FDA00036925467800000212
it is the error of the gyroscope that is,
Figure FDA00036925467800000213
is the east-direction velocity of the grid coordinate system,
Figure FDA00036925467800000214
is the grid coordinate system north velocity;
Figure FDA00036925467800000215
Figure FDA00036925467800000216
Figure FDA00036925467800000217
Figure FDA00036925467800000218
Figure FDA0003692546780000031
Figure FDA0003692546780000032
step 4, calculating a position error differential equation:
Figure FDA0003692546780000033
wherein,
Figure FDA0003692546780000034
is a rotation matrix of the grid coordinate system relative to the terrestrial coordinate system,
Figure FDA0003692546780000035
Figure FDA0003692546780000036
is a rotation matrix of the geographic coordinate system relative to the grid coordinate system,
Figure FDA0003692546780000037
a rotation matrix of the terrestrial coordinate system with respect to the geographic coordinates.
Figure FDA0003692546780000038
Figure FDA0003692546780000039
Figure FDA00036925467800000310
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