CN113051757B - Strapdown inertial navigation generalized PSI angle error model construction method - Google Patents

Strapdown inertial navigation generalized PSI angle error model construction method Download PDF

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CN113051757B
CN113051757B CN202110312139.XA CN202110312139A CN113051757B CN 113051757 B CN113051757 B CN 113051757B CN 202110312139 A CN202110312139 A CN 202110312139A CN 113051757 B CN113051757 B CN 113051757B
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generalized
error equation
attitude
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CN113051757A (en
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常路宾
高端阳
李开龙
覃方君
查峰
胡柏青
田佳玉
吕旭
赵仁杰
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Naval University of Engineering PLA
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/10—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration
    • G01C21/12—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning
    • G01C21/16—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning by integrating acceleration or speed, i.e. inertial navigation
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C25/00—Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass
    • G01C25/005—Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass initial alignment, calibration or starting-up of inertial devices
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/02—Reliability analysis or reliability optimisation; Failure analysis, e.g. worst case scenario performance, failure mode and effects analysis [FMEA]

Abstract

The invention discloses a strapdown inertial navigation generalized PSI angle error model construction method, which decomposes a carrier attitude matrix into a carrier angular motion excitation item and a linear motion excitation item through coordinate system decomposition, wherein an error equation corresponding to the angular motion excitation item is an attitude error equation in the generalized PSI angle error model, and an error equation corresponding to the linear motion excitation item is a position error equation in the generalized PSI angle error model. On the premise of not introducing an approximate speed error term, the method has the advantage that an attitude angle error model is independent of speed and position errors; meanwhile, all error equation deductions strictly follow a chain multiplication rule of an attitude matrix (quaternion), so that the deduction process is clear on one hand, and the deduced error equation is more directly corrected in the inertial navigation calculation result in the filtering application on the other hand; in addition, the velocity error equation takes into account
Figure DDA0003595075100000011
The error contained in the method improves the accuracy of the error model.

Description

Strapdown inertial navigation generalized PSI angle error model construction method
Technical Field
The invention relates to the technical field of error analysis of a strapdown inertial navigation system, in particular to a strapdown inertial navigation generalized PSI angle error model construction method based on coordinate system decomposition.
Background
The inertial navigation system is an autonomous navigation system which does not depend on any external information and does not radiate energy to the outside, has the characteristics of good concealment and can work in various complex environments such as air, ground, underwater and the like, and is mainly divided into a platform type inertial navigation system and a strapdown type inertial navigation system. The strapdown inertial navigation system is developed on the basis of a platform type inertial navigation system, is a frameless system and consists of three rate gyros, three linear accelerometers and an airborne computer. The gyroscope and the accelerometer are directly and fixedly connected to the carrier and are respectively used for measuring angular velocity information and linear motion information of the carrier, and the onboard computer calculates the course, the attitude, the speed and the position of the carrier according to the measurement information. Because a complex electromechanical platform is omitted, the strapdown inertial navigation system has the advantages of simple structure, small volume, light weight, low cost, simple maintenance, high reliability and the like, and the fault-tolerant capability of the strapdown inertial navigation system can be improved through a redundancy technology; meanwhile, with the appearance of solid-state inertial devices such as laser gyroscopes and fiber-optic gyroscopes, the rapid development of computer technology and the increasing perfection of computing theory, the superiority of strapdown inertial navigation systems is gradually revealed.
The PSI angle error model of the strapdown inertial navigation system is a concept introduced from a platform type inertial navigation error equation and mainly relates to a carrier coordinate system b, a local real horizontal geographic coordinate system t, a calculation geographic coordinate system c and a calculation platform coordinate system p. In fact, the calculated geographic coordinate system c and the calculated platform coordinate system p are both calculated local horizontal coordinate systems, except that the excitation errors are not consistent, and the so-called PSI angle is the error between the c-system and the p-system. Due to the special definition of the c system and the p system, the physical meaning of the PSI angle is not clear, so that the attitude correction process in the filtering application is not intuitive. In addition, the introduction of the c system in the PSI angle error model ensures that the derivation process of the speed error equation assumes
Figure GDA0003595075090000011
The method does not contain errors, although the form of a speed error equation is simpler, the result is not strictly established, so that a PSI angle error model exists compared with a disturbance angle PHI error modelLess accurate problem.
Disclosure of Invention
Aiming at the problems, the invention provides a strapdown inertial navigation generalized PSI angle error model construction method based on coordinate system decomposition.
The invention protects a strapdown inertial navigation generalized PSI angle error model, the attitude error equation of which is
Figure GDA0003595075090000021
The position error equation is
Figure GDA0003595075090000022
Wherein θ [ - δ L δ λ cosL δ λ sinL] T The velocity error equation is
Figure GDA0003595075090000023
Wherein
Figure GDA0003595075090000024
Figure GDA0003595075090000025
Defining state vectors
Figure GDA0003595075090000026
Then its corresponding state space model is
Figure GDA0003595075090000027
Wherein the content of the first and second substances,
Figure GDA0003595075090000028
Figure GDA0003595075090000029
Figure GDA00035950750900000210
the invention also protects a construction method of the strapdown inertial navigation generalized PSI angle error model, which comprises the following steps:
the attitude error equation derivation comprises the following steps:
step A1, defining attitude quaternion error
Figure GDA00035950750900000211
Step A2, differentiating the above formula to obtain
Figure GDA0003595075090000031
Wherein
Figure GDA0003595075090000032
Figure GDA0003595075090000033
Step A3, obtainable according to step A2
Figure GDA0003595075090000034
Step A4, combining the relation of quaternion and three-dimensional vector multiplication to obtain
Figure GDA0003595075090000035
Wherein the content of the first and second substances,
Figure GDA0003595075090000036
in step A5, it is assumed that the angle between the calculated carrier coordinate system b' and the real carrier coordinate system b is a small angle, that is, the included angle is
Figure GDA0003595075090000037
The error quaternion may be approximated by δ q 1 ＝[1 α T /2] T Derivation ofThe equation of attitude error can be obtained
Figure GDA0003595075090000038
The position error equation derivation comprises the following steps:
step B1, defining error quaternion
Figure GDA0003595075090000039
Step B2, differentiating the above formula to obtain
Figure GDA00035950750900000310
Wherein the content of the first and second substances,
Figure GDA00035950750900000311
step B3, obtained according to step A2
Figure GDA00035950750900000312
Step B4, defining an error vector
Figure GDA00035950750900000313
Further obtain
Figure GDA00035950750900000314
Step B5, under the assumption of small angle, the error quaternion can be approximated to be δ q 2 ＝[1 θ T /2] T Derived position error equation of
Figure GDA00035950750900000315
The derivation of the speed error equation comprises the following steps:
step C1, defining a velocity error vector
Figure GDA00035950750900000316
Wherein
Figure GDA00035950750900000317
Step C2, differentiating the above formula and further calculating to obtain
Figure GDA0003595075090000041
Wherein
Figure GDA0003595075090000042
Step C3, further calculating the available speed error equation according to the step C2
Figure GDA0003595075090000043
The carrier attitude matrix is decomposed into a carrier angular motion excitation term and a linear motion excitation term through coordinate system decomposition, wherein an error equation corresponding to the angular motion excitation term is an attitude error equation in the generalized PSI angular error model, and an error equation corresponding to the linear motion excitation term is a position error equation in the generalized PSI angular error model. On the premise of not introducing an approximate speed error term, the method has the advantage that an attitude angle error model is independent of speed and position errors, which is referred to in a 'broad sense'; meanwhile, all error equation deductions strictly follow a chain multiplication rule of an attitude matrix (quaternion), so that the deduction process is clear on one hand, and the deduced error equation is more directly corrected in the inertial navigation calculation result in the filtering application on the other hand; in addition, the velocity error equation takes into account
Figure GDA0003595075090000044
The error contained in the method improves the accuracy of the error model.
Drawings
Fig. 1 is an application flow of the PSI angle error model established in the present invention in the combined filtering.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments. The embodiments of the present invention have been presented for purposes of illustration and description, and are not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated.
The method for constructing the strapdown inertial navigation generalized PSI angle error model comprises the following steps of:
firstly, constructing a coordinate system
1. And a carrier coordinate system b, referred to as a b system for short, wherein the origin is the center of three axes of the inertial navigation system, the x axis points to the right of the carrier, and the y axis points to the front of the carrier, thereby forming a right-hand rectangular coordinate system Oxyz. And b is that the axial directions of the three shafts are respectively along the transverse rolling shaft, the pitching shaft and the yawing shaft of the carrier.
2. An inertial coordinate system i, i system for short, wherein an origin O is located at the earth centroid, coordinate axes are not rotated relative to an inertial space, an x axis and a y axis are located in the earth equatorial plane, the x axis points to the vernal point, and the z axis points to the earth rotation axis, so that a right-hand rectangular coordinate system Oxyz is formed.
3. And an earth coordinate system e, for short, an e system, wherein the origin is positioned at the center of the earth, the coordinate axis is fixedly connected with the earth, the z axis is along the direction of the rotation axis of the earth, and the x axis points to the intersection line of the Greenwich mean meridian plane and the equatorial plane of the earth, so that a right-hand rectangular coordinate system Oxyz is formed. The earth coordinate system rotates around the z-axis at an angular velocity omega relative to the inertial coordinate system ie And (4) rotating.
4. The navigation coordinate system N is called N system for short, and for a strapdown system, namely a local horizontal coordinate system, an original point is located at the center of mass of the carrier, and three coordinate axes respectively point to an east direction E, a north direction N and a vertical direction U (a vertical line of the local is in an upward direction). The angular rate of rotation of the navigational coordinate system relative to the terrestrial coordinate system depends on the motion of the center of mass of the vehicle relative to the earth.
5. Calculating a carrier coordinate system b ', which is called b' for short,
6. and calculating a navigation coordinate system n ', which is called n' system for short.
Since the inertial coordinate system i and the terrestrial coordinate system e are objectively present, they are assumed to contain no calculation error.
Definition of
Figure GDA0003595075090000051
The cosine matrix of the direction of rotation from a to b is usually used to represent the rotation relationship between two coordinate systems. The transformation from one coordinate system to another coordinate system can be realized by three successive rotations around different coordinate axes, the angle of the three successive rotations being called euler rotation angle. The coordinate transformation can also be expressed as the product of the independent transformations of the three rotations.
1. Transformation between a carrier coordinate system b and a navigation coordinate system n
Let phi be the course angle of carrier motion, theta be the pitch angle, gamma be the roll angle, the navigation coordinate system is around z n Axially rotated in the negative direction by an angle phi, rotated about the x-axis by an angle theta, and then rotated about the y-axis b After the shaft rotates gamma, the shaft coincides with the carrier coordinate system, and then the conversion matrix is
Figure GDA0003595075090000052
Figure GDA0003595075090000053
2. Transformation between terrestrial coordinate system e and navigation coordinate system n
The longitude and latitude of a point P on the earth surface are lambda and L, then the navigation coordinate system at the point P can be obtained by rotating the earth coordinate system three times, and the rotation matrix between the points is
Figure GDA0003595075090000061
3. Transformation between earth coordinate system e and inertial coordinate system i
The coordinate conversion matrix of the earth coordinate system relative to the inertial coordinate system is
Figure GDA0003595075090000062
Wherein ω is ie The earth rotation angular velocity t is the time elapsed from the time (reference time 0) at which the inertial coordinate system coincides with the instantaneous spherical coordinate system.
Second, attitude coordinate system decomposition
According to the attitude matrix (quaternion) chain multiplication rule, the attitude decomposition form can be obtained
Figure GDA0003595075090000063
Wherein
Figure GDA0003595075090000064
Is an ideal attitude quaternion without any error.
In the strapdown inertial navigation attitude calculation, the introduction of various errors causes the calculation to have an attitude quaternion containing errors as
Figure GDA0003595075090000065
According to the attitude matrix chain multiplication rule, the attitude decomposition form can be obtained
Figure GDA0003595075090000066
Third, attitude error equation derivation
1. Defining attitude quaternion errors
Figure GDA0003595075090000067
2. The differential of the formula (3) can be obtained
Figure GDA0003595075090000068
Wherein
Figure GDA0003595075090000069
Figure GDA00035950750900000610
3. By substituting the formulae (5) and (6) into the formula (4)
Figure GDA00035950750900000611
4. The quaternion and three-dimensional vector multiplication satisfy the following relation
Figure GDA00035950750900000612
Figure GDA0003595075090000071
According to the formulas (8) and (9), the formula (7) can be further developed
Figure GDA0003595075090000072
Wherein the content of the first and second substances,
Figure GDA0003595075090000073
(11) in the formula, the included angle between the calculated carrier coordinate system b' and the real carrier coordinate system b is assumed to be a small angle, that is, the included angle is
Figure GDA0003595075090000074
Under the small angle assumption, the error quaternion can be approximated to be δ q 1 ＝[1 α T /2] T (13) The differential equation of α obtained by simultaneous equations (10), (11) and (13) is
Figure GDA0003595075090000075
(14) The formula is an attitude error equation in the generalized PSI angle error model, and it can be seen that the attitude error alpha is only caused by the error of the gyroscope, which is consistent with the traditional PSI angle error model.
Fourth, position error equation derivation
1. Defining error quaternion
Figure GDA0003595075090000076
2. Can be obtained by differentiating the formula (15)
Figure GDA0003595075090000077
Wherein the content of the first and second substances,
Figure GDA0003595075090000078
Figure GDA0003595075090000079
3. by substituting the formulae (17) and (18) into the formula (16)
Figure GDA00035950750900000710
4. Defining an error vector
Figure GDA00035950750900000711
Then the formula (19) can be further developed into
Figure GDA0003595075090000081
5. Under the small angle assumption, the error quaternion can be approximated as δ q 2 ＝[1 θ T /2] T (21) The differential equation of θ obtained by combining the expressions (20) and (21) is
Figure GDA0003595075090000082
(22) The formula is a position error equation in the generalized PSI angle error model and is given in the form of angle error and is related to the traditional longitude and latitude errorIs θ [ - δ L δ λ cosL δ λ sinL] T (23)。
Fifth, velocity error equation derivation
1. The ideal velocity differential equation is
Figure GDA0003595075090000083
The differential equation of velocity calculated is
Figure GDA0003595075090000084
Defining a velocity error vector
Figure GDA0003595075090000085
2. Differentiating the expression (26), and substituting the expressions (24) and (25) into one another
Figure GDA0003595075090000086
(27) In which is used
Figure GDA0003595075090000087
In the form of a matrix decomposition of which the detailed expansion is
Figure GDA0003595075090000088
Substitution of formula (28) into formula (27)
Figure GDA0003595075090000089
(29) The formula is a speed error equation in the generalized PSI angle error model, and can be seen from the derivation process, the formula is not ignored
Figure GDA00035950750900000810
The error contained in the method.
And sixthly, deducing an error model in a state transition matrix form, namely, sorting the error equation and writing the error equation into the state transition matrix form.
First, the derivation of the error vectors involved therein is expanded.
Figure GDA0003595075090000091
Figure GDA0003595075090000092
Wherein the content of the first and second substances,
Figure GDA0003595075090000093
Figure GDA0003595075090000094
Figure GDA0003595075090000095
wherein R is M 、R N Respectively representing the radius of curvature of the earth meridian and the radius of curvature of the prime unit circle of the place where the carrier is located;
second, for terrestrial and marine applications, the altitude error is ignored, and the state vector is defined
Figure GDA0003595075090000096
Then its corresponding state space model is
Figure GDA0003595075090000097
Wherein the content of the first and second substances,
Figure GDA0003595075090000101
Figure GDA0003595075090000102
Figure GDA0003595075090000103
Figure GDA0003595075090000104
M θv ＝M 3
Figure GDA0003595075090000105
fig. 1 is an application flow of the PSI angle error model established in the invention in combined filtering, and it can be known by referring to the above error equations that the generalized PSI error equation established in the invention contains fewer terms, has a simpler form, is convenient for kalman filtering to be implemented, and enables the derived error equation to more intuitively perform inertial navigation solution result correction in filtering application.
It is to be understood that the described embodiments are merely a few embodiments of the invention, and not all embodiments. All other embodiments, which can be derived by one of ordinary skill in the art and related arts based on the embodiments of the present invention without any creative effort, shall fall within the protection scope of the present invention.

Claims (3)

1. A strapdown inertial navigation generalized PSI angle error model construction method is characterized in that an attitude error equation of an error model is
Figure FDA0003595075080000011
The position error equation is
Figure FDA0003595075080000012
Wherein θ [ - δ L δ λ cosL δ λ sinL] T (ii) a The velocity error equation is
Figure FDA0003595075080000013
Wherein
Figure FDA0003595075080000014
Figure FDA0003595075080000015
Defining state vectors
Figure FDA0003595075080000016
Then its corresponding state space model is
Figure FDA0003595075080000017
Wherein the content of the first and second substances,
Figure FDA0003595075080000018
Figure FDA0003595075080000019
M θv ＝M 3 ，
Figure FDA00035950750800000110
the attitude error equation derivation includes the steps of:
step A1, defining attitude quaternion error
Figure FDA00035950750800000111
Step A2, differentiating the above formula to obtain
Figure FDA00035950750800000112
Wherein
Figure FDA00035950750800000113
Figure FDA00035950750800000114
Step A3, obtainable according to step A2
Figure FDA00035950750800000115
Step A4, combining the relation of quaternion and three-dimensional vector multiplication to obtain
Figure FDA0003595075080000021
Wherein the content of the first and second substances,
Figure FDA0003595075080000022
in step A5, it is assumed that the angle between the calculated carrier coordinate system b' and the real carrier coordinate system b is a small angle, that is, the included angle is
Figure FDA0003595075080000023
The error quaternion may be approximated by δ q 1 ＝[1 α T /2] T Deducing the available attitude error equation
Figure FDA0003595075080000024
2. The strapdown inertial navigation generalized PSI angle error model building method of claim 1, wherein the position error equation derivation comprises the steps of:
step B1, defining error quaternion
Figure FDA0003595075080000025
Step B2, differentiating the above formula to obtain
Figure FDA0003595075080000026
Wherein the content of the first and second substances,
Figure FDA0003595075080000027
Figure FDA0003595075080000028
step B3, obtained according to step A2
Figure FDA0003595075080000029
Step B4, defining an error vector
Figure FDA00035950750800000210
Further obtain
Figure FDA00035950750800000211
Step B5, under the assumption of small angle, the error quaternion can be approximated to be δ q 2 ＝[1θ T /2] T Derived position error equation of
Figure FDA00035950750800000212
3. The strapdown inertial navigation generalized PSI angle error model building method of claim 1, wherein the velocity error equation derivation comprises the steps of:
step C1, defining a velocity error vector
Figure FDA00035950750800000213
Wherein
Figure FDA00035950750800000214
Figure FDA00035950750800000215
Step C2, differentiating the above formula and further calculating to obtain
Figure FDA00035950750800000216
Wherein
Figure FDA00035950750800000217
Step C3, further calculating the available speed error equation according to the step C2
Figure FDA0003595075080000031
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