CN110595503B - Self-alignment method of SINS strapdown inertial navigation system shaking base based on lie group optimal estimation - Google Patents

Self-alignment method of SINS strapdown inertial navigation system shaking base based on lie group optimal estimation Download PDF

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CN110595503B
CN110595503B CN201910718946.4A CN201910718946A CN110595503B CN 110595503 B CN110595503 B CN 110595503B CN 201910718946 A CN201910718946 A CN 201910718946A CN 110595503 B CN110595503 B CN 110595503B
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裴福俊
朱德森
杨肃
尹舒男
蒋宁
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Beijing University of Technology
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Abstract

The invention discloses a shaking base self-alignment method of an SINS strapdown inertial navigation system based on lie group optimal estimation, which adopts the lie group description to replace the traditional quaternion description to realize the calculation of SINS attitude transformation, utilizes the lie group differential equation to establish a linear initial alignment filtering model based on the lie group description, and designs the initial attitude matrix required by navigation determined by the lie group optimal estimation method. The initial attitude matrix is directly and optimally estimated by adopting a lie group optimal estimation algorithm, SO that the initial attitude estimation problem is converted into the optimal estimation problem of an SO (3) group, one-step direct self-alignment of SINS is realized, and the alignment time is greatly shortened; the problems of non-uniqueness and non-linearity caused by the fact that the initial attitude matrix is described by the traditional quaternion are solved, and the alignment precision is effectively improved; the problem that the orthogonality of the rotation matrix cannot be guaranteed due to additive errors and calculation errors in the lie group filtering method is solved. The invention has practical value in practical engineering.

Description

Self-alignment method of SINS strapdown inertial navigation system shaking base based on lie group optimal estimation
Technical Field
The invention discloses a shaking base self-alignment method of an SINS strapdown inertial navigation system based on lie group optimal estimation, and belongs to the technical field of navigation methods and application.
Background
Navigation is the process of properly guiding a carrier along a predetermined route to a destination with the required accuracy and within a specified time. The inertial navigation system calculates each navigation parameter of the carrier according to the output of the sensor of the inertial navigation system by taking Newton's second law as a theoretical basis. The autonomous navigation system is an autonomous navigation system, does not depend on external information when working, does not radiate any energy to the outside, has good concealment and strong interference resistance, and can provide complete motion information for a carrier all day long and all weather.
The early inertial navigation system is mainly based on platform inertial navigation, and with the maturity of inertial devices and the development of computer technology, a strapdown inertial navigation system with an inertial device and a carrier directly fixedly connected with each other begins to appear in the last 60 th century. Compared with platform inertial navigation, the strapdown inertial navigation system saves a complex entity stable platform and has the advantages of low cost, small volume, light weight, high reliability and the like. In recent years, a strapdown inertial navigation system is mature, the precision is gradually improved, and the application range is gradually expanded. The strapdown inertial navigation technology directly installs a gyroscope and an accelerometer on a carrier to obtain the acceleration and the angular velocity under a carrier system, and converts measured data into a navigation coordinate system through a navigation computer to complete navigation.
The research of the self-alignment process is of great significance in the strapdown inertial navigation system, and particularly the self-alignment method under the shaking base is the current research hotspot. Besides the lie group filtering estimation method, the self-alignment process can be completed by using methods such as singular value decomposition, quaternion Kalman and the like under the condition of shaking the base. These methods still have considerable drawbacks. Singular value decomposition is an optimization method based on matrix decomposition, and since the calculation process does not meet the operation rule of the lie group space, singular points are generated under the condition of a large misalignment angle, so that the alignment performance is influenced. Quaternion Kalman is a method for constructing a linear filtering model by using a pseudo measurement equation and completing an alignment task by using a Kalman filter, and although the method cannot generate singular points, the construction of the pseudo measurement equation also influences the convergence speed and the alignment precision. In the process of lie group filtering alignment, calculation errors are generated due to conversion between a lie algebra and a lie group space when a gain matrix and an error covariance matrix are constructed; the design principle of the filter is improved based on Kalman filtering, and the definition of innovation does not conform to the property of the lie group rotation matrix. And therefore has an effect on the alignment accuracy.
Aiming at the problems of the existing shaking base self-alignment method, the invention provides a novel optimal estimation algorithm for the plum blossom so as to further improve the performance of initial alignment by using the idea of optimal estimation. Due to the improvement of the lie group theory, the state quantity is ensured to meet the special orthogonal property at any time by improving the expression mode of innovation. The estimation model constructed based on the method avoids the problem of singular values in the traditional optimization method, and the iterative computation of each step conforms to the characteristics of the lie group rotation matrix. Compared with the method for self-aligning the lie group filtering, the method has the advantages that the advantage of the lie group filtering is kept, meanwhile, the aligning speed is improved, and the accuracy is improved. The feasibility of the algorithm is proved through simulation verification, and the method can be used as an upper-level substitute for the lie filtering to perform shaking base self-alignment.
Disclosure of Invention
Since the carrier is easily affected by various external interference factors during the initial alignment process, it is difficult to keep the carrier still during the alignment process. Therefore, the self-alignment algorithm under the shaking base has high research significance and application value. The invention aims to solve the problems of the existing shaking base self-alignment method: (1) according to the invention, the initial attitude matrix is described by replacing quaternion with the lie group, so that the problems of non-uniqueness and non-linearity of the traditional quaternion description method are avoided; (2) the linear self-alignment filtering model based on lie group description is established by utilizing the lie group differential equation, so that the one-step direct self-alignment process of the SINS is realized, and compared with the existing two-step alignment method, the alignment time can be greatly shortened and the alignment precision can be improved; (3) the SINS strapdown inertial navigation system shaking base self-alignment method based on lie group optimal estimation can effectively avoid the complex expression problem and a large amount of calculation errors generated by the conversion of quaternions to attitude matrixes in the existing quaternion Kalman filtering method, and can avoid the problem that the orthogonality of a rotation matrix cannot be ensured due to additive errors and calculation errors in the lie group filtering method.
In order to achieve the purpose, the invention provides the following technical scheme:
the SINS strapdown inertial navigation system shaking base self-alignment method based on lie group optimal estimation is characterized by comprising the following steps of:
step (1): the SINS strapdown inertial navigation system carries out system preheating preparation, starts the system, and obtains the longitude lambda and the latitude L of the position of the carrier and the projection g of the local gravity acceleration under the navigation systemnCollecting the projection of the rotation angular rate information of the carrier system output by the gyroscope in the inertial measurement unit IMU relative to the inertial system on the carrier system according to the basic information
Figure GDA0002248933830000021
And accelerationCarrier system acceleration information f output by the meterbEtc.;
step (2): preprocessing acquired data of the gyroscope and the accelerometer, and establishing a linear shaking base self-alignment system model based on the lie group description on the basis of a lie group differential equation:
the coordinate system in the detailed description of the method is defined as follows:
the earth coordinate system e is characterized in that the earth center is selected as an origin, the X axis is located in an equatorial plane and points to the original meridian from the earth center, the Z axis points to the geographic north pole from the earth center, and the X axis, the Y axis and the Z axis form a right-hand coordinate system and rotate along with the earth rotation;
the geocentric inertial coordinate system i is characterized in that the geocentric inertial coordinate system i is obtained by selecting the geocenter as an origin, an X axis is located in an equatorial plane and points to the spring equinox from the geocenter, a Z axis points to the geographical arctic from the geocenter, and the X axis, the Y axis and the Z axis form a right-hand coordinate system;
a navigation coordinate system N, which is a geographical coordinate system representing the position of the carrier, selects the gravity center of the carrier-based aircraft as an origin, points to the east E on the X-axis, points to the north N on the Y-axis, and points to the sky U on the Z-axis; in the method, a navigation coordinate system is selected as a geographic coordinate system;
a carrier coordinate system b system which represents a three-axis orthogonal coordinate system of the strapdown inertial navigation system, wherein the gravity center of the carrier-based aircraft is selected as an origin, and an X axis, a Y axis and a Z axis respectively point to the right along a transverse axis, point to the front along a longitudinal axis and point to the up along a vertical axis of the carrier-based aircraft body;
an initial navigation coordinate system n (0) system which represents a navigation coordinate system of the SINS at the startup running time and keeps static relative to an inertial space in the whole alignment process;
an initial carrier coordinate system b (0) system which represents a carrier coordinate system of the SINS at the starting-up running time and keeps static relative to an inertial space in the whole alignment process;
a navigation coordinate system n' system which represents an initial navigation coordinate system calculated by a lie group optimal estimation algorithm, wherein a rotation relation exists between the coordinate system and the real navigation coordinate system n system;
based on a lie group differential equation, establishing a linear self-alignment system model based on the lie group description:
according to the SINS strapdown inertial navigation system principle, the SINS shaking base self-alignment problem is converted into a posture estimation problem, the posture is converted into rotation transformation between two coordinate systems, and a navigation posture matrix is represented by a 3 multiplied by 3 orthogonal transformation matrix; the orthogonal transformation matrix conforms to the property of a special orthogonal group SO (n) of lie groups, and forms a three-dimensional rotation group SO (3):
Figure GDA0002248933830000022
wherein R ∈ SO (3) represents a specific navigation attitude matrix,
Figure GDA0002248933830000023
representing a 3 x 3 vector space, superscript T representing the transpose of the matrix, I representing a three-dimensional identity matrix, det (R) representing the determinant of matrix R;
the self-alignment attitude estimation problem of the shaking base is converted into a solving problem of an attitude matrix R based on the lie group description; navigating the attitude matrix according to the chain rule of the attitude matrix based on the lie group description
Figure GDA0002248933830000024
Decomposition is in the form of the product of three matrices:
Figure GDA0002248933830000025
wherein, t represents a time variable,
Figure GDA0002248933830000026
a pose matrix representing the current carrier frame relative to the current navigation frame,
Figure GDA0002248933830000027
representing an attitude matrix of an initial navigation system relative to a current navigation system, the initial attitude matrix
Figure GDA0002248933830000028
Indicating the initial carrier systemWith respect to the attitude matrix of the initial navigation frame,
Figure GDA0002248933830000029
representing a posture matrix of the current carrier system relative to the initial carrier system;
from the lie group differential equation, attitude matrix
Figure GDA00022489338300000210
And
Figure GDA00022489338300000211
the time-varying update process is:
Figure GDA00022489338300000212
Figure GDA00022489338300000213
wherein the content of the first and second substances,
Figure GDA00022489338300000214
a pose matrix representing the initial vehicle system relative to the current vehicle system,
Figure GDA00022489338300000215
representing the projection of the angular rate of rotation of the navigational system relative to the inertial system on the navigational system, which is equal to the angular rate of rotation of the earth under the condition of shaking the base
Figure GDA00022489338300000216
L represents the local latitude and the local latitude,
Figure GDA00022489338300000217
the angular rate of rotation of the carrier system representing the gyroscope output relative to the inertial system is projected onto the carrier system, the symbol (· ×) represents the operation of converting a three-dimensional vector into an antisymmetric matrix, and the operation rule is as follows:
Figure GDA0002248933830000031
as can be seen from the formulas (2) to (5),
Figure GDA0002248933830000032
and
Figure GDA0002248933830000033
calculated in real time from IMU sensor data, and
Figure GDA0002248933830000034
an attitude matrix representing an initial time, which does not change with time; thus, the attitude matrix during SINS self-alignment
Figure GDA0002248933830000035
Is converted into an initial attitude matrix based on the lie group description
Figure GDA0002248933830000036
Solving the problem of (1);
accelerometer measurement information f under base shaking conditionbExpressed as:
Figure GDA0002248933830000037
wherein, gbShowing the projection of the local gravitational acceleration under the carrier system, δ fbWhich is indicative of other noise, is,
Figure GDA0002248933830000038
showing the projection of the centripetal acceleration caused by the shaking under the carrier system,
Figure GDA0002248933830000039
the IMU sensor data is obtained through real-time calculation, and the calculation process is as follows:
Figure GDA00022489338300000310
wherein r isbThe distance from the installation position of the inertial sensor to the axis of the shaking base is represented;
will acceleration of gravity gbMeasuring information f from an accelerometerbSeparating to obtain:
Figure GDA00022489338300000311
according to the SINS inertial navigation philosophy and lie chain law, gnAnd gbThere is the following relationship between:
Figure GDA00022489338300000312
and (3) performing item shifting and sorting operations on the formula (9) to obtain:
Figure GDA00022489338300000313
equation (10) is simplified as:
Figure GDA00022489338300000314
wherein the content of the first and second substances,
Figure GDA00022489338300000315
representing the projection of the gravitational acceleration under the initial navigation system,
Figure GDA00022489338300000316
representing a projection of gravitational acceleration under an initial carrier system;
integrating equation (11) over [0, t ] yields:
Figure GDA00022489338300000317
wherein the content of the first and second substances,
Figure GDA00022489338300000318
representing acceleration of gravity
Figure GDA00022489338300000319
The corresponding velocity vector is set to be,
Figure GDA00022489338300000320
representing acceleration of gravity
Figure GDA00022489338300000321
A corresponding velocity vector;
equation (12) can be simplified as:
Figure GDA00022489338300000322
wherein
Figure GDA00022489338300000323
Figure GDA00022489338300000324
Equation (12) relates to the initial attitude matrix
Figure GDA00022489338300000325
The mathematical equation of (a) is,
Figure GDA00022489338300000326
and
Figure GDA00022489338300000327
calculated from the sensor output; given in equation (12)
Figure GDA00022489338300000328
And
Figure GDA00022489338300000329
the method is in a continuous form, discretization is needed in the actual calculation process, and a linear measurement equation of the shaking base self-alignment system is established after discretization as follows:
Figure GDA00022489338300000330
since the attitude matrix will be solved
Figure GDA00022489338300000331
Is converted into solving the initial attitude matrix
Figure GDA00022489338300000332
And is not limited to
Figure GDA00022489338300000333
To conform to the constant matrix of lie group characteristics, the linear state equation for establishing the rocking base self-aligning system is as follows:
Rk＝Rk-1 (17)
according to the content, the solution problem of the attitude matrix is converted into the solution problem under the inertial coordinate system at the initial moment, and a shaking base self-alignment system equation with a plum cluster structure is established and expressed as follows:
Figure GDA00022489338300000334
and (3): according to the optimal estimation algorithm of the lie group, directly carrying out the initial attitude matrix based on the description of the lie group
Figure GDA00022489338300000335
And (3) carrying out optimal estimation:
the one-step prediction of the state matrix is represented as:
Figure GDA0002248933830000041
wherein the content of the first and second substances,
Figure GDA0002248933830000042
one-step prediction of an initial attitude matrix representing time k, Rk-1A posteriori estimation of an initial attitude matrix representing the time k-1;
the estimated value of the measurement vector is expressed as:
Figure GDA0002248933830000043
wherein the content of the first and second substances,
Figure GDA0002248933830000044
representing velocity vector at time k
Figure GDA0002248933830000045
Is determined by the estimated value of (c),
Figure GDA0002248933830000046
a one-step prediction of the initial attitude matrix representing time k,
Figure GDA0002248933830000047
calculating the data output by the sensor;
in the process of transforming the coordinate system, the existence of errors can cause the rotation relationship between the navigation coordinate system n' obtained by calculation and the real navigation coordinate system n; θ represents the rotation angle of the phase difference between the n' system and the n system, and using the vector product operation property, θ (k) at the k time can be expressed as:
Figure GDA0002248933830000048
wherein the content of the first and second substances,
Figure GDA0002248933830000049
representing velocity vector at time k
Figure GDA00022489338300000410
Is determined by the estimated value of (c),
Figure GDA00022489338300000411
and
Figure GDA00022489338300000412
calculating the data output by the sensor;
according to the right-hand rule, theta (k) is the projection of the rotation angle under the n' system, and theta (k) is projected under the b system to obtain the measurement innovation of the optimal estimation algorithm of the lie group
Figure GDA00022489338300000413
Comprises the following steps:
Figure GDA00022489338300000414
according to the principle of the SINS strapdown inertial navigation system based on lie group description, the state update equation of the lie group optimal estimation algorithm can be written as follows:
Figure GDA00022489338300000415
wherein R iskRepresenting a posterior estimate of the initial pose matrix at time k, exp () representing an exponential mapping from lie algebraic space to lie group space,
Figure GDA00022489338300000416
representing a pose update matrix based on the lie group structure,
Figure GDA00022489338300000417
representing vectors
Figure GDA00022489338300000418
The antisymmetric matrix form of (a);
the self-alignment algorithm of the shaking base of the SINS strapdown inertial navigation system based on the lie group optimal estimation is summarized as follows:
Figure GDA00022489338300000419
and (4): solving attitude matrices required by a navigation system
Figure GDA00022489338300000420
Thereby accomplish and rock base self-alignment process:
according to the attitude change matrix obtained by solving in the previous step
Figure GDA00022489338300000421
And
Figure GDA00022489338300000422
and (3) information, namely solving a navigation attitude matrix through a formula (2) to finish self-alignment of the SINS strapdown inertial navigation system shaking base.
Compared with the prior art, the invention has the following advantages and beneficial effects:
(1) the invention adopts the lie group to describe the initial attitude matrix
Figure GDA00022489338300000423
The initial attitude matrix described by the traditional quaternion can be effectively avoided
Figure GDA00022489338300000424
Non-uniqueness and non-linearity of acoustics.
(2) The method adopts the lie group method to establish the linear system model, realizes the one-step direct self-alignment process of the SINS, and compared with the traditional two-step alignment method, not only can greatly shorten the alignment time, but also can improve the alignment precision.
(3) The SINS strapdown inertial navigation system shaking base self-alignment method based on the lie group optimal estimation can effectively avoid the complex expression problem and a large amount of calculation errors generated by the conversion of quaternions to attitude matrixes in the traditional quaternion Kalman filtering method, and can avoid the problem that the orthogonality of the rotation matrixes cannot be ensured due to additive errors and calculation errors in the lie group filtering method.
Drawings
FIG. 1 is a schematic diagram of a strapdown inertial navigation system
FIG. 2 flow chart of strapdown inertial navigation system
FIG. 3 is a schematic diagram of a rotation relationship between a navigation coordinate system and a body coordinate system
FIG. 4 is a schematic diagram of the rotation angle between the calculated coordinate system and the real coordinate system
FIG. 5 is a flow chart of an optimal estimation method for lie groups
FIG. 6 is a diagram of a self-aligned simulation result of a wobble base
FIG. 7 is a graph of the results of a self-alignment comparison experiment of a rocking base
FIG. 8 is a schematic view of a page of the upper computer collecting real navigation data
Detailed Description
The invention relates to a self-alignment method design of a shaking base of an SINS strapdown inertial navigation system based on lie swarm optimal estimation, and the specific implementation steps of the invention are described in detail by combining the system flow chart of the invention:
the invention provides a shaking base self-alignment method of an SINS strapdown inertial navigation system based on lie group optimal estimation, which comprises the steps of firstly, acquiring real-time data of a sensor; processing the acquired data, and establishing a linear self-alignment system model based on the lie group description based on the lie group differential equation; estimating to obtain an initial attitude matrix based on the lie group description by using an optimal lie group estimation algorithm
Figure GDA0002248933830000051
And solving the attitude matrix
Figure GDA0002248933830000052
During the self-alignment period, the accurate initial attitude matrix is finally obtained through multiple estimation and calculation
Figure GDA0002248933830000053
And attitude matrix
Figure GDA0002248933830000054
The self-alignment process is completed.
Step 1: starting and initializing an SINS inertial navigation system, and obtaining longitude lambda and latitude L and local gravity acceleration g of the position of a carriernCollecting angular rate information output by a gyroscope in an inertial measurement unit IMU (inertial measurement Unit) according to basic information
Figure GDA0002248933830000055
And acceleration information f output by the accelerometerb。
Step 2: processing the acquired data of the gyroscope and the accelerometer, establishing a linear shaking base self-alignment system model based on the lie group description based on the lie group differential equation,
step (2.1): by passing
Figure GDA0002248933830000056
Updating computations
Figure GDA0002248933830000057
Attitude matrix chain rule and attitude matrix based on lie group description
Figure GDA0002248933830000058
The decomposition is as follows:
Figure GDA0002248933830000059
due to the fact that
Figure GDA00022489338300000510
Represents the angular velocity of rotation of the navigation system relative to the inertial system, and
Figure GDA00022489338300000511
the change is usually very slow, and the solving process is updated according to the attitude matrix based on the lie group description, tk-1Time tkAttitude matrix of time of day
Figure GDA00022489338300000512
The approximation is:
Figure GDA00022489338300000513
wherein the content of the first and second substances,
Figure GDA00022489338300000514
according to equations (25) - (27), the attitude matrix
Figure GDA00022489338300000515
The iterative process is approximated as:
Figure GDA00022489338300000516
step (2.2): angular rate information output by a gyroscope
Figure GDA00022489338300000517
Updating computations
Figure GDA00022489338300000518
tk-1Time tkAttitude matrix of time of day
Figure GDA00022489338300000519
The approximation is:
Figure GDA00022489338300000520
wherein, according to a double-subsample rotation vector method, the following can be obtained:
Figure GDA00022489338300000521
wherein, Delta theta1And Δ θ2Respectively representing two adjacent half-sample periods
Figure GDA00022489338300000522
Outputting the calculated angle increment by the gyroscope;
the attitude matrix according to equations (29) - (30)
Figure GDA0002248933830000061
The iterative process is approximated as:
Figure GDA0002248933830000062
step (2.3): establishing initial rotation matrix based on lie group description
Figure GDA0002248933830000063
The self-aligning system model mode is as follows:
accelerometer measurement information f under base shaking conditionbExpressed as:
Figure GDA0002248933830000064
will acceleration of gravity gbMeasuring information f from an accelerometerbSeparating to obtain:
Figure GDA0002248933830000065
according to the SINS inertial navigation philosophy and lie chain law, gnAnd gbThere is the following relationship between:
Figure GDA0002248933830000066
integrating equation (34) over [0, t ], and obtaining after sorting:
Figure GDA0002248933830000067
substituting the formula (33) into the formula (35) to obtain:
Figure GDA0002248933830000068
equation (36) reduces to:
Figure GDA0002248933830000069
wherein
Figure GDA00022489338300000610
Figure GDA00022489338300000611
Equation (37) relates to the initial attitude matrix
Figure GDA00022489338300000612
The mathematical equation of (a) is,
Figure GDA00022489338300000613
and
Figure GDA00022489338300000614
all in a continuous form, solved by an integral iterative algorithm
Figure GDA00022489338300000615
And
Figure GDA00022489338300000616
discrete value at time k
Figure GDA00022489338300000617
And
Figure GDA00022489338300000618
time k
Figure GDA00022489338300000619
The approximation is:
Figure GDA00022489338300000620
wherein, T is the sampling period,
Figure GDA00022489338300000621
the unit matrix is represented by a matrix of units,
Figure GDA00022489338300000622
the calculation is iterated by the formula (28);
due to δ fbMuch smaller than the acceleration of gravity gbIn a discrete process, δ fbConsidering the minimum error amount, equation (39) is therefore approximated as:
Figure GDA00022489338300000623
wherein the content of the first and second substances,
Figure GDA00022489338300000624
time k
Figure GDA00022489338300000625
The approximation is:
Figure GDA00022489338300000626
wherein the content of the first and second substances,
Figure GDA00022489338300000627
the calculation is iterated by the formula (31);
according to the formula, a linear measurement equation of the shaking base self-alignment system is established as follows:
Figure GDA00022489338300000628
due to the matrix of the posture
Figure GDA00022489338300000629
Is converted into an initial attitude matrix
Figure GDA00022489338300000630
Solve the problem, and
Figure GDA00022489338300000631
to conform to the constant matrix of lie group characteristics, the linear state equation for establishing the rocking base self-aligning system is:
Rk＝Rk-1 (44)
according to the content, the solution problem of the attitude matrix is converted into the solution problem under the inertial coordinate system at the initial moment, and a shaking base self-alignment system equation with a plum cluster structure is established and expressed as follows:
Figure GDA00022489338300000632
and (3): direct estimation using lie group optimal estimation algorithm
Figure GDA00022489338300000633
The plum cluster optimal estimation algorithm estimates the shaking base self-alignment model, and the whole process is as follows:
Figure GDA0002248933830000071
wherein the content of the first and second substances,
Figure GDA0002248933830000072
for one-step prediction of the initial attitude matrix at time k, Rk-1For a posteriori estimation of the initial attitude matrix at time k-1,
Figure GDA0002248933830000073
is an estimated value of the measurement vector at the time k, theta (k) is a projection of the rotation angle between the system n ' and the system n ' under the system n ',
Figure GDA0002248933830000074
for the b is the measured innovation of the optimal estimation algorithm of the next lie group,
Figure GDA0002248933830000075
updating the matrix, R, for lie group structure based poseskA posteriori estimation of the initial attitude matrix at time k, i.e. as calculated
Figure GDA0002248933830000076
And (4): solving an attitude matrix
Figure GDA0002248933830000077
And the attitude information is resolved,
in step (2), the SINS is self-aligned
Figure GDA0002248933830000078
Is converted into a pair
Figure GDA0002248933830000079
And will solve the problem
Figure GDA00022489338300000710
Decomposition is in the form of three matrix products:
Figure GDA00022489338300000711
solved according to formula (28), formula (31) and formula (46)
Figure GDA00022489338300000712
And
Figure GDA00022489338300000713
attitude matrix
Figure GDA00022489338300000714
The solving method is as follows:
Figure GDA00022489338300000715
and obtaining an attitude matrix according to the solution
Figure GDA00022489338300000716
And resolving attitude information.
The invention has the following beneficial effects:
(1) the method was subjected to simulation experiments under the following simulation conditions:
in the step (1), the simulation carrier is influenced by wind waves under the condition of shaking the base, the course angle psi, the pitch angle theta and the roll angle gamma of the simulation carrier periodically change, and the posture change conditions are as follows:
Figure GDA00022489338300000717
Figure GDA00022489338300000718
Figure GDA00022489338300000719
in step (1), the initial geographic position: east longitude 118 degrees, north latitude 40 degrees;
in the step (1), the output frequency of the sensor is 100 Hz;
in the step (1), the gyroscope drifts: the gyro constant drift on the three directional axes is 0.02 degree/h, and the random drift is 0.005 degree/h;
in the step (1), zero offset of the accelerometer: the constant bias of the accelerometer on three direction axes is 2 multiplied by 10-4g, randomly biasing to
Figure GDA00022489338300000720
In step (2), the rotation angular rate of the earth 7.2921158e-5rad/s；
In the step (2), the time interval T is 0.02 s;
in step (3), the initial value of the optimal estimation algorithm of the plum group
Figure GDA00022489338300000721
The simulation result of the method is as follows:
300s simulation is carried out, the estimation error of the attitude angle is taken as a measurement index, and the simulation result is shown in figure (6). As can be seen from the figure, the pitch attitude completes alignment around 120s, converging to 0.42'; the transverse rolling posture is aligned in about 120s and converged to 0.48'; the heading attitude completes alignment around 150s, converging to 1.2'. According to the simulation result, the method can quickly and effectively complete the self-alignment task under the shaking base.
(2) The shaking base self-alignment method of the SINS strapdown inertial navigation system based on the optimal estimation of the lie group provided by the invention is verified through a real experiment. In a real experiment, no external auxiliary information is provided, and the experiment lasts for 300 s. The upper navigation computer controls a navigation system, actual three-axis attitude information with course accuracy of 0.1 degree and attitude accuracy of 0.05 degree is collected at a data updating rate of 100HZ and a baud rate of 115200bps, and the carrier attitude information obtained by resolving is compared with the high-accuracy real carrier attitude information obtained in the step, so that the feasibility and the effectiveness of the method and the system are proved.
The system experiment results are as follows:
a real experiment was performed for 300s, and the estimation error of the attitude angle was used as an index for measuring the alignment accuracy, and the result is shown in fig. 7. As can be seen in the figure, the pitch attitude completes alignment at about 65s, converging to-6.82'; the rolling gesture is aligned in about 50s and converged to 4.37'; the heading pose is aligned around 120s, converging to 23.58'. The method can quickly and effectively complete the self-alignment task under the shaking base, and compared with a quaternion Kalman filtering method, a singular value decomposition method and a lie group filtering method, the method is small in overshoot, high in convergence speed and good in filtering precision.
The method converts the shaking base self-alignment problem into the optimal estimation problem of the initial rotation matrix, adopts the method based on the lie group to describe the initial attitude matrix, and can effectively avoid the problems of non-uniqueness and non-linearity caused by the description of the initial attitude matrix based on the quaternion method in the prior art; the method utilizes the lie group differential equation to establish the one-step alignment linear system model, and compared with the traditional two-step alignment method, the method can greatly shorten the self-alignment time and improve the alignment precision; the invention adopts the lie group optimal estimation method, can avoid the complex expression problem and a large amount of calculation errors generated by the conversion of quaternion to attitude matrix in the quaternion Kalman filtering process, and can avoid the problem that the orthogonality of the rotation matrix can not be ensured due to additive errors and calculation errors in the lie group Kalman filtering process.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. It should be noted that, for a person skilled in the art, several modifications and variations can be made without departing from the principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (4)

1. The SINS strapdown inertial navigation system shaking base self-alignment method based on lie group optimal estimation is characterized by comprising the following steps of:
step (1): the SINS strapdown inertial navigation system carries out system preheating preparation, starts the system, and obtains the longitude lambda and the latitude L of the position of the carrier and the projection g of the local gravity acceleration under the navigation systemnAcquiring basic information, and acquiring rotation angle of a carrier system output by a gyroscope in an inertial measurement unit IMU relative to an inertial systemProjection of rate information on a carrier
Figure FDA00027786723200000120
And carrier system acceleration information f output by accelerometerb；
Step (2): preprocessing acquired data of the gyroscope and the accelerometer, and establishing a linear shaking base self-alignment system model based on the lie group description on the basis of a lie group differential equation:
the coordinate system in the detailed description of the method is defined as follows:
the earth coordinate system e is characterized in that the earth center is selected as an origin, the X axis is located in an equatorial plane and points to the original meridian from the earth center, the Z axis points to the geographic north pole from the earth center, and the X axis, the Y axis and the Z axis form a right-hand coordinate system and rotate along with the earth rotation;
the geocentric inertial coordinate system i is characterized in that the geocentric inertial coordinate system i is obtained by selecting the geocenter as an origin, an X axis is located in an equatorial plane and points to the spring equinox from the geocenter, a Z axis points to the geographical arctic from the geocenter, and the X axis, the Y axis and the Z axis form a right-hand coordinate system;
a navigation coordinate system N, which is a geographical coordinate system representing the position of the carrier, selects the gravity center of the carrier-based aircraft as an origin, points to the east E on the X-axis, points to the north N on the Y-axis, and points to the sky U on the Z-axis; in the method, a navigation coordinate system is selected as a geographic coordinate system;
a carrier coordinate system b system which represents a three-axis orthogonal coordinate system of the strapdown inertial navigation system, wherein the gravity center of the carrier-based aircraft is selected as an origin, and an X axis, a Y axis and a Z axis respectively point to the right along a transverse axis, point to the front along a longitudinal axis and point to the up along a vertical axis of the carrier-based aircraft body;
an initial navigation coordinate system n (0) system which represents a navigation coordinate system of the SINS at the startup running time and keeps static relative to an inertial space in the whole alignment process;
an initial carrier coordinate system b (0) system which represents a carrier coordinate system of the SINS at the starting-up running time and keeps static relative to an inertial space in the whole alignment process;
a navigation coordinate system n' system which represents an initial navigation coordinate system calculated by a lie group optimal estimation algorithm, wherein a rotation relation exists between the coordinate system and the real navigation coordinate system n system;
based on a lie group differential equation, establishing a linear self-alignment system model based on the lie group description:
according to the SINS strapdown inertial navigation system principle, the SINS shaking base self-alignment problem is converted into a posture estimation problem, the posture is converted into rotation transformation between two coordinate systems, and a navigation posture matrix is represented by a 3 multiplied by 3 orthogonal transformation matrix; the orthogonal transformation matrix conforms to the property of a special orthogonal group SO (n) of lie groups, and forms a three-dimensional rotation group SO (3):
Figure FDA00027786723200000121
wherein R ∈ SO (3) represents a specific navigation attitude matrix,
Figure FDA00027786723200000122
representing a 3 x 3 vector space, superscript T representing the transpose of the matrix, I representing a three-dimensional identity matrix, det (R) representing the determinant of matrix R;
the self-alignment attitude estimation problem of the shaking base is converted into a solving problem of an attitude matrix R based on the lie group description; navigating the attitude matrix according to the chain rule of the attitude matrix based on the lie group description
Figure FDA0002778672320000011
Decomposition is in the form of the product of three matrices:
Figure FDA00027786723200000123
wherein, t represents a time variable,
Figure FDA0002778672320000012
a pose matrix representing the current carrier frame relative to the current navigation frame,
Figure FDA0002778672320000013
representing an attitude matrix of an initial navigation system relative to a current navigation system, the initial attitude matrix
Figure FDA00027786723200000117
A pose matrix representing the initial carrier system relative to the initial navigation system,
Figure FDA00027786723200000118
representing a posture matrix of the current carrier system relative to the initial carrier system;
from the lie group differential equation, attitude matrix
Figure FDA00027786723200000124
And
Figure FDA0002778672320000017
the time-varying update process is:
Figure FDA00027786723200000119
Figure FDA0002778672320000018
wherein the content of the first and second substances,
Figure FDA0002778672320000019
a pose matrix representing the initial vehicle system relative to the current vehicle system,
Figure FDA00027786723200000110
representing the projection of the angular rate of rotation of the navigational system relative to the inertial system on the navigational system, which is equal to the angular rate of rotation of the earth under the condition of shaking the base
Figure FDA00027786723200000111
L represents the local latitude and the local latitude,
Figure FDA00027786723200000112
the angular rate of rotation of the carrier system representing the gyroscope output relative to the inertial system is projected onto the carrier system, the symbol (· ×) represents the operation of converting a three-dimensional vector into an antisymmetric matrix, and the operation rule is as follows:
Figure FDA00027786723200000113
as can be seen from the formulas (2) to (5),
Figure FDA00027786723200000114
and
Figure FDA00027786723200000115
calculated in real time from IMU sensor data, and
Figure FDA00027786723200000116
an attitude matrix representing an initial time, which does not change with time; thus, the attitude matrix during SINS self-alignment
Figure FDA0002778672320000021
Is converted into an initial attitude matrix based on the lie group description
Figure FDA0002778672320000022
Solving the problem of (1);
accelerometer measurement information f under base shaking conditionbExpressed as:
Figure FDA0002778672320000023
wherein, gbShowing the projection of the local gravitational acceleration under the carrier system, δ fbWhich is indicative of other noise, is,
Figure FDA0002778672320000024
showing the projection of the centripetal acceleration caused by the shaking under the carrier system,
Figure FDA0002778672320000025
the IMU sensor data is obtained through real-time calculation, and the calculation process is as follows:
Figure FDA0002778672320000026
wherein r isbThe distance from the installation position of the inertial sensor to the axis of the shaking base is represented;
will acceleration of gravity gbMeasuring information f from an accelerometerbSeparating to obtain:
Figure FDA0002778672320000027
according to the SINS inertial navigation philosophy and lie chain law, gnAnd gbThere is the following relationship between:
Figure FDA0002778672320000028
and (3) performing item shifting and sorting operations on the formula (9) to obtain:
Figure FDA0002778672320000029
equation (10) is simplified as:
Figure FDA00027786723200000210
wherein the content of the first and second substances,
Figure FDA00027786723200000211
representing the projection of the gravitational acceleration under the initial navigation system,
Figure FDA00027786723200000212
representing a projection of gravitational acceleration under an initial carrier system;
integrating equation (11) over [0, t ] yields:
Figure FDA00027786723200000213
wherein the content of the first and second substances,
Figure FDA00027786723200000214
representing acceleration of gravity
Figure FDA00027786723200000215
The corresponding velocity vector is set to be,
Figure FDA00027786723200000216
representing acceleration of gravity
Figure FDA00027786723200000217
A corresponding velocity vector;
equation (12) can be simplified as:
Figure FDA00027786723200000218
wherein
Figure FDA00027786723200000219
Figure FDA00027786723200000220
Equation (12) relates to the initial attitude matrix
Figure FDA00027786723200000221
The mathematical equation of (a) is,
Figure FDA00027786723200000222
and
Figure FDA00027786723200000223
calculated from the sensor output; given in equation (12)
Figure FDA00027786723200000224
And
Figure FDA00027786723200000225
the method is in a continuous form, discretization is needed in the actual calculation process, and a linear measurement equation of the shaking base self-alignment system is established after discretization as follows:
Figure FDA00027786723200000226
since the attitude matrix will be solved
Figure FDA00027786723200000227
Is converted into solving the initial attitude matrix
Figure FDA00027786723200000228
And is not limited to
Figure FDA00027786723200000229
To conform to the constant matrix of lie group characteristics, the linear state equation for establishing the rocking base self-aligning system is as follows:
Rk＝Rk-1 (17)
according to the content, the solution problem of the attitude matrix is converted into the solution problem under the inertial coordinate system at the initial moment, and a shaking base self-alignment system equation with a plum cluster structure is established and expressed as follows:
Figure FDA00027786723200000230
and (3): according to the optimal estimation algorithm of the lie group, directly carrying out the initial attitude matrix based on the description of the lie group
Figure FDA00027786723200000231
And (3) carrying out optimal estimation:
the one-step prediction of the state matrix is represented as:
Figure FDA00027786723200000232
wherein the content of the first and second substances,
Figure FDA00027786723200000233
one-step prediction of an initial attitude matrix representing time k, Rk-1A posteriori estimation of an initial attitude matrix representing the time k-1;
the estimated value of the measurement vector is expressed as:
Figure FDA00027786723200000234
wherein the content of the first and second substances,
Figure FDA0002778672320000031
representing velocity vector at time k
Figure FDA0002778672320000032
Is determined by the estimated value of (c),
Figure FDA0002778672320000033
one of the initial attitude matrix representing time kThe step of predicting is carried out by predicting,
Figure FDA0002778672320000034
calculating the data output by the sensor;
in the process of transforming the coordinate system, the existence of errors can cause the rotation relationship between the navigation coordinate system n' obtained by calculation and the real navigation coordinate system n; θ represents the rotation angle of the phase difference between the n' system and the n system, and using the vector product operation property, θ (k) at the k time can be expressed as:
Figure FDA0002778672320000035
wherein the content of the first and second substances,
Figure FDA0002778672320000036
representing velocity vector at time k
Figure FDA0002778672320000037
Is determined by the estimated value of (c),
Figure FDA0002778672320000038
and
Figure FDA0002778672320000039
calculating the data output by the sensor;
according to the right-hand rule, theta (k) is the projection of the rotation angle under the n' system, and theta (k) is projected under the b system to obtain the measurement innovation of the optimal estimation algorithm of the lie group
Figure FDA00027786723200000310
Comprises the following steps:
Figure FDA00027786723200000311
according to the principle of the SINS strapdown inertial navigation system based on lie group description, the state update equation of the lie group optimal estimation algorithm can be written as follows:
Figure FDA00027786723200000312
wherein R iskRepresenting a posterior estimate of the initial pose matrix at time k, exp () representing an exponential mapping from lie algebraic space to lie group space,
Figure FDA00027786723200000313
representing a pose update matrix based on the lie group structure,
Figure FDA00027786723200000314
representing vectors
Figure FDA00027786723200000315
The antisymmetric matrix form of (a);
the self-alignment algorithm of the shaking base of the SINS strapdown inertial navigation system based on the lie group optimal estimation is summarized as follows:
Figure FDA00027786723200000316
and (4): solving attitude matrices required by a navigation system
Figure FDA00027786723200000317
Thereby accomplish and rock base self-alignment process:
according to the attitude change matrix obtained by solving in the previous step
Figure FDA00027786723200000318
And
Figure FDA00027786723200000319
and (3) information, namely solving a navigation attitude matrix through a formula (2) to finish self-alignment of the SINS strapdown inertial navigation system shaking base.
2. The method for self-aligning the shaking base of the SINS strapdown inertial navigation system based on lie optimal estimation according to claim 1, wherein in the step (2), a linear shaking base self-alignment system model based on the lie description is established based on the lie differential equation;
step (2.1): by passing
Figure FDA00027786723200000320
Updating computations
Figure FDA00027786723200000321
Attitude matrix chain rule and attitude matrix based on lie group description
Figure FDA00027786723200000322
The decomposition is as follows:
Figure FDA00027786723200000323
due to the fact that
Figure FDA00027786723200000324
Represents the angular velocity of rotation of the navigation system relative to the inertial system, and
Figure FDA00027786723200000325
the change is usually very slow, and the solving process is updated according to the attitude matrix based on the lie group description, tk-1Time tkAttitude matrix of time of day
Figure FDA00027786723200000326
The approximation is:
Figure FDA00027786723200000330
wherein the content of the first and second substances,
Figure FDA00027786723200000327
according to equations (25) - (27), the attitude matrix
Figure FDA00027786723200000328
The iterative process is approximated as:
Figure FDA00027786723200000329
step (2.2): angular rate information output by a gyroscope
Figure FDA0002778672320000041
Updating computations
Figure FDA0002778672320000042
tk-1Time tkAttitude matrix of time of day
Figure FDA0002778672320000043
The approximation is:
Figure FDA0002778672320000044
wherein, according to a double-subsample rotation vector method, the following can be obtained:
Figure FDA0002778672320000045
wherein, Delta theta1And Δ θ2Respectively representing two adjacent half-sample periods
Figure FDA0002778672320000046
Outputting the calculated angle increment by the gyroscope;
the attitude matrix according to equations (29) - (30)
Figure FDA0002778672320000047
The iterative process is approximated as:
Figure FDA0002778672320000048
step (2.3): establishing initial rotation matrix based on lie group description
Figure FDA0002778672320000049
The self-aligning system model mode is as follows:
accelerometer measurement information f under base shaking conditionbExpressed as:
Figure FDA00027786723200000410
will acceleration of gravity gbMeasuring information f from an accelerometerbSeparating to obtain:
Figure FDA00027786723200000411
according to the SINS inertial navigation philosophy and lie chain law, gnAnd gbThere is the following relationship between:
Figure FDA00027786723200000412
integrating equation (34) over [0, t ], and obtaining after sorting:
Figure FDA00027786723200000413
substituting the formula (33) into the formula (35) to obtain:
Figure FDA00027786723200000414
equation (36) reduces to:
Figure FDA00027786723200000415
wherein
Figure FDA00027786723200000416
Figure FDA00027786723200000417
Equation (37) relates to the initial attitude matrix
Figure FDA00027786723200000418
The mathematical equation of (a) is,
Figure FDA00027786723200000419
and
Figure FDA00027786723200000420
all in a continuous form, solved by an integral iterative algorithm
Figure FDA00027786723200000421
And
Figure FDA00027786723200000422
discrete value at time k
Figure FDA00027786723200000423
And
Figure FDA00027786723200000424
time k
Figure FDA00027786723200000425
The approximation is:
Figure FDA00027786723200000426
wherein, T is the sampling period,
Figure FDA00027786723200000427
the unit matrix is represented by a matrix of units,
Figure FDA00027786723200000428
the calculation is iterated by the formula (28);
due to δ fbMuch smaller than the acceleration of gravity gbIn a discrete process, δ fbConsidering the minimum error amount, equation (39) is therefore approximated as:
Figure FDA00027786723200000429
wherein the content of the first and second substances,
Figure FDA00027786723200000430
time k
Figure FDA00027786723200000431
The approximation is:
Figure FDA00027786723200000432
wherein the content of the first and second substances,
Figure FDA00027786723200000433
the calculation is iterated by the formula (31);
according to the formula, a linear measurement equation of the shaking base self-alignment system is established as follows:
Figure FDA00027786723200000434
due to the matrix of the posture
Figure FDA00027786723200000435
Is converted into an initial attitude matrix
Figure FDA00027786723200000436
Solve the problem, and
Figure FDA00027786723200000437
to conform to the constant matrix of lie group characteristics, the linear state equation for establishing the rocking base self-aligning system is:
Rk＝Rk-1 (44)
according to the content, the solution problem of the attitude matrix is converted into the solution problem under the inertial coordinate system at the initial moment, and a shaking base self-alignment system equation with a plum cluster structure is established and expressed as follows:
Figure FDA0002778672320000051
3. the method for self-aligning the swaying base of SINS strapdown inertial navigation system based on lie group optimal estimation according to claim 1, wherein the step (3) uses the lie group optimal estimation algorithm to directly estimate
Figure FDA0002778672320000052
The plum cluster optimal estimation algorithm estimates the shaking base self-alignment model, and the whole process is as follows:
Figure FDA0002778672320000053
wherein the content of the first and second substances,
Figure FDA0002778672320000054
for one-step prediction of the initial attitude matrix at time k, Rk-1For a posteriori estimation of the initial attitude matrix at time k-1,
Figure FDA0002778672320000055
is an estimated value of the measurement vector at the time k, theta (k) is a projection of the rotation angle between the system n ' and the system n ' under the system n ',
Figure FDA0002778672320000056
for the b is the measured innovation of the optimal estimation algorithm of the next lie group,
Figure FDA0002778672320000057
updating the matrix, R, for lie group structure based poseskA posteriori estimation of the initial attitude matrix at time k, i.e. as calculated
Figure FDA0002778672320000058
4. The method for self-aligning the swaying base of SINS strapdown inertial navigation system based on lie group optimal estimation according to claim 1, wherein the attitude matrix is solved in the step (4)
Figure FDA00027786723200000518
And resolving attitude information;
in step (2), the SINS is self-aligned
Figure FDA00027786723200000510
Is converted into a pair
Figure FDA00027786723200000511
And will solve the problem
Figure FDA00027786723200000519
Decomposition is in the form of three matrix products:
Figure FDA00027786723200000512
solved according to formula (28), formula (31) and formula (47)
Figure FDA00027786723200000513
And
Figure FDA00027786723200000514
attitude matrix
Figure FDA00027786723200000515
The solving method is as follows:
Figure FDA00027786723200000516
and obtaining an attitude matrix according to the solution
Figure FDA00027786723200000517
And resolving attitude information.
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