CN113432608A - Generalized high-order CKF algorithm based on maximum correlation entropy and suitable for INS/CNS integrated navigation system - Google Patents

Generalized high-order CKF algorithm based on maximum correlation entropy and suitable for INS/CNS integrated navigation system Download PDF

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CN113432608A
CN113432608A CN202110152328.5A CN202110152328A CN113432608A CN 113432608 A CN113432608 A CN 113432608A CN 202110152328 A CN202110152328 A CN 202110152328A CN 113432608 A CN113432608 A CN 113432608A
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navigation system
integrated navigation
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陈熙源
柳笛
刘晓
石春凤
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Southeast University
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    • 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
    • G01C21/20—Instruments for performing navigational calculations
    • 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
    • G01C21/165—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 combined with non-inertial navigation instruments

Abstract

The invention discloses a generalized high-order CKF algorithm based on maximum correlation entropy, which is suitable for an INS/CNS integrated navigation system, and the method comprises the following steps: (1) constructing an INS/CNS integrated navigation system filtering model; (2) carrying out time updating of a filtering algorithm according to the constructed filtering model; (3) and introducing a maximum correlation entropy criterion and a new judgment criterion, and performing measurement updating of the filtering algorithm. The method not only retains the advantages of the generalized high-order CKF algorithm, but also shows the robustness performance to non-Gaussian noise, thereby improving the navigation accuracy of the INS/CNS integrated navigation system.

Description

Generalized high-order CKF algorithm based on maximum correlation entropy and suitable for INS/CNS integrated navigation system
Technical Field
The invention relates to a filtering method, in particular to a generalized high-order CKF algorithm based on maximum correlation entropy and suitable for an INS/CNS integrated navigation system.
Background
In the integrated navigation system, a Kalman Filter (KF) is one of the most popular state estimation methods applied to a linear system, whereas the system equations of the INS/CNS integrated navigation system are nonlinear, and the filtering algorithms applied to the nonlinear integrated navigation system equations mainly include Extended Kalman (EKF), Unscented Kalman (UKF), volumetric kalman (CKF) and generalized high-order volumetric kalman filtering algorithms. The EKF simply carries out simple local linearization on a nonlinear system equation by using Taylor series expansion, and the linearized system has serious model description errors; the UKF utilizes a group of selected probability distribution of sigma point approximate states to overcome errors caused by local linearization of an EKF algorithm, however, the numerical instability of the UKF algorithm is caused because the weight value of the central point of unscented transformation is possibly negative; compared with UKF, CKF has better numerical stability, while the CKF algorithm is derived based on the third-order volume criterion, and can only ensure the third-order approximation precision, thus being not suitable for application scenes with higher precision requirements; the generalized high-order CKF algorithm constructed by using the complete symmetry formula has higher estimation precision compared with the CKF algorithm constructed based on the third-order volume rule. In practical applications, the generated measurement noise is usually non-gaussian noise, since the INS or CNS is affected by the external environment. However, the above filtering algorithm is only applicable to the case of gaussian white noise. In order to realize that the INS/CNS can still obtain high-precision navigation information under the condition that the measurement noise is non-Gaussian noise, the invention provides a method for introducing the maximum correlation entropy criterion into the measurement updating process of the generalized high-order CKF algorithm.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects of the prior art, the invention provides a generalized high-order CKF algorithm based on maximum correlation entropy, which is suitable for an INS/CNS combined navigation system.
The technical scheme is as follows: a generalized high-order CKF algorithm based on maximum correlation entropy suitable for an INS/CNS integrated navigation system comprises the following steps:
(1) constructing an INS/CNS integrated navigation system filtering model;
(2) carrying out time updating of a filtering algorithm according to the constructed filtering model;
(3) and introducing a maximum correlation entropy criterion and a new judgment criterion, and performing measurement updating of the filtering algorithm.
Preferably, in the step (1), the specific process of constructing the INS/CNS integrated navigation system filtering model is as follows:
(11) setting a state vector of the INS/CNS integrated navigation system as x ═ phi, delta v, delta r, epsilon and delta, wherein [ phi, delta v, delta r, epsilon and delta ] respectively represent INS attitude error, speed error, position error, gyro constant drift and accelerometer constant offset;
further, [ phi, delta v, delta r, epsilon, delta ] respectively represents three-dimensional INS attitude error, three-dimensional speed error, three-dimensional position error, three-dimensional gyro constant drift and three-dimensional accelerometer constant bias;
(12) establishing a state equation of the system according to a state vector x of the INS/CNS integrated navigation system:
Figure BDA0002931770010000021
where F represents the state transition matrix, G represents the process noise input matrix, and W represents the process noise. Discretizing the equation to obtain a discretized system state equation:
xk＝f(xk-1,Wk-1)
where f (-) is a known non-linear function, xk-1And xkRespectively representing the state vectors at time k-1 and at time k, Wk-1Representing the process noise at time k-1 and having an average value of 0, Wk-1The covariance of (a) can be expressed as:
Figure BDA0002931770010000022
taking the mathematical platform error angle equation of the navigation system as the measurement equation of the system,
zk＝Hkxk+Vk
wherein z iskA measurement vector representing time k, HkA measurement matrix representing the time k, VkThe measured noise at time k satisfies a mean of 0 and its covariance can be expressed as:
Figure BDA0002931770010000023
and VkAnd Wk-1The satisfaction is not related to each other.
Preferably, in step (2), the time updating process performed according to the constructed filtering model is as follows:
(21) selecting a proper kernel width lambda and a smaller positive threshold c, and setting the initial state vector estimation value as
Figure BDA0002931770010000024
Initial error covariance of P0|0K is 1, for P0|0Cholescent decomposition is carried out to obtain the characteristic square root initial value S of the error covariance matrix0|0；
(22) According to the formula
Figure BDA0002931770010000025
Calculate the volume point Xi,k-1|k-1(i＝1,...,2n2+1), where n represents the dimension of the state vector,
Figure BDA0002931770010000026
[·]ia set of representations [ ·]Column i, e.g.
Figure BDA0002931770010000027
When, there is [1]＝{[1 0]T,[0 1]T,[-1 0]T,[0 -1]T}；
(23) Calculating propagation volume points
Figure BDA0002931770010000028
(24) Predicting a state vector at a current time
Figure BDA0002931770010000029
Error covariance matrix Pk|k-1And calculating Pk|k-1Characteristic square root of Sk|k-1，
Figure BDA0002931770010000031
Figure BDA0002931770010000032
Wherein
Figure BDA0002931770010000033
Preferably, in the step (3), the measurement updating process of the filter algorithm by introducing the maximum correlation entropy criterion includes:
(31) based on predicted state vectors
Figure BDA0002931770010000034
And Pk|k-1Characteristic square root of Sk|k-1Generating a new volume point Xi,k|k-1And propagation volume point Zi,k|k-1，
Figure BDA0002931770010000035
Zi,k|k-1＝HkXi,k|k-1Wherein i is 12+1；
(32) Predicting time k measurements
Figure BDA0002931770010000036
(33) Based on the measurement equations of the INS/CNS integrated navigation system and the state vector in step (24)
Figure BDA0002931770010000037
And Pk|k-1The following regression equation is constructed,
Figure BDA0002931770010000038
wherein, I represents a unit vector,
Figure BDA0002931770010000039
Figure BDA00029317700100000310
Mp,k|k-1、Mr,kand MkCan pass through
Figure BDA00029317700100000311
Is obtained by georges decomposition.
Multiplication on both sides of the regression equation
Figure BDA00029317700100000312
Obtaining: dk＝BkXk+ek. Wherein the content of the first and second substances,
Figure BDA00029317700100000313
Figure BDA00029317700100000314
(34) updating the measured noise covariance matrix
Figure BDA00029317700100000315
Figure BDA00029317700100000316
Wherein the content of the first and second substances,
Figure BDA00029317700100000317
diag (. circle.) denotes the diagonalization of the matrix, m denotes the dimension of the measurement vector, kernel function in the correlation entropy
Figure BDA00029317700100000318
di,kRepresents DkThe ith element of (b)i,kIs represented by BkRow i element of (1);
(35) calculating the covariance P of the measurement vectorzz,k|k-1，
Figure BDA00029317700100000319
(36) Computing
Figure BDA00029317700100000320
If | akIf | c (where c is a set threshold), then
Figure BDA00029317700100000321
Returning to the step (22) to continue executing the next filtering period if | akIf | is less than or equal to c, calculating
Figure BDA0002931770010000041
Figure BDA0002931770010000042
Returning to the step (22) to continue executing the next filtering period.
Has the advantages that: compared with the prior art, the invention has the following advantages:
1. the invention predicts the prior state vector and the covariance matrix of the combined navigation system by using the time updating process of the generalized high-order CKF filtering algorithm, and then introduces the maximum correlation entropy criterion and the judgment criterion in the measurement updating process of the filtering algorithm to realize the estimation of the posterior state vector and the covariance, thereby ensuring that the INS/CNS combined navigation system has better robustness in the non-Gaussian noise environment.
2. The invention replaces the original criterion judged in an iterative mode with the new judgment criterion, and can execute the next step only by comparing and judging with the set threshold value c, thereby improving the operation efficiency.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
Example 1
As shown in fig. 1, the invention discloses a generalized high-order CKF algorithm based on maximum correlation entropy suitable for an INS/CNS integrated navigation system, comprising the following steps:
(1) and setting the state vector of the INS/CNS integrated navigation system as x ═ phi, delta v, delta r, epsilon and delta, wherein the [ phi, delta v, delta r, epsilon and delta ] respectively represent three-dimensional INS attitude error, three-dimensional speed error, three-dimensional position error, three-dimensional gyro constant drift and accelerometer constant offset.
Establishing a state equation of the system according to a state vector x of the INS/CNS integrated navigation system:
Figure BDA0002931770010000043
where F represents the state transition matrix, G represents the process noise input matrix, and W represents the process noise. Discretizing the equation to obtain a discretized system state equation:
xk＝f(xk-1,Wk-1) (2)
where f (-) is a known non-linear function, xk-1And xkRespectively representing the state vectors at time k-1 and at time k, Wk-1Representing the process noise at time k-1 and having an average value of 0, Wk-1Can be expressed as
Figure BDA0002931770010000044
Taking a mathematical platform error angle equation of a navigation system as a measurement equation of the system:
zk＝Hkxk+Vk (3)
wherein z iskA measurement vector representing time k, HkA measurement matrix representing the time k, VkThe measured noise at time k satisfies a mean of 0 and its covariance can be expressed as:
Figure BDA0002931770010000051
and VkAnd Wk-1The satisfaction is not related to each other.
(2) Selecting a proper kernel width lambda and a smaller positive threshold c, and setting the initial state vector estimation value as
Figure BDA0002931770010000052
Initial error covariance of P0|0K is 1, for P0|0Cholescent decomposition is carried out to obtain the characteristic square root initial value S of the error covariance matrix0|0And calculating the capacity according to the formula (4)Integral point Xi,k-1|k-1(i＝1,...,2n2+1)，
Figure BDA0002931770010000053
Where n represents the dimension of the state vector,
Figure BDA0002931770010000054
[·]ia set of representations [ ·]Column i, e.g.
Figure BDA0002931770010000055
When, there is [1]＝{[1 0]T,[0 1]T,[-1 0]T,[0 -1]T}。
Calculating the propagation volume point by the formula (5)
Figure BDA0002931770010000056
Figure BDA0002931770010000057
Predicting a state vector at a current time
Figure BDA0002931770010000058
Error covariance matrix Pk|k-1And calculating Pk|k-1Characteristic square root of Sk|k-1，
Figure BDA0002931770010000059
Figure BDA00029317700100000510
Figure BDA00029317700100000511
Wherein
Figure BDA00029317700100000512
(3) Based on predicted state vectors
Figure BDA00029317700100000513
And Pk|k-1Characteristic square root of Sk|k-1Generating a new volume point Xi,k|k-1And propagation volume point Zi,k|k-1；
Figure BDA00029317700100000514
Zi,k|k-1＝HkXi,k|k-1 (10)
Then predicting the k time measurement
Figure BDA00029317700100000517
Figure BDA00029317700100000516
According to the measurement equation and the state vector of the INS/CNS integrated navigation system
Figure BDA0002931770010000061
And Pk|k-1The following regression equation is constructed,
Figure BDA0002931770010000062
wherein, I represents a unit vector,
Figure BDA0002931770010000063
Figure BDA0002931770010000064
Mp,k|k-1、Mr,kand MkCan pass through
Figure BDA0002931770010000065
Is obtained by georges decomposition.
Multiplication on both sides of the regression equation
Figure BDA0002931770010000066
Obtaining:
Dk＝BkXk+ek (13)
wherein the content of the first and second substances,
Figure BDA0002931770010000067
the measured noise covariance matrix is updated by the formula (14)
Figure BDA0002931770010000068
And calculating the covariance P of the measurement vectorzz,k|k-1，
Figure BDA0002931770010000069
Figure BDA00029317700100000610
Wherein the content of the first and second substances,
Figure BDA00029317700100000611
diag (. circle.) denotes the diagonalization of the matrix, m denotes the dimension of the measurement vector, kernel function in the correlation entropy
Figure BDA00029317700100000612
di,kRepresents DkThe ith element of (b)i,kIs represented by BkRow i element of (1);
eta is calculated by the equations (16) and (17), respectivelykAnd ak，
Figure BDA00029317700100000613
Figure BDA00029317700100000614
If | akIf | is > c, the estimated value of the state vector is calculated by equation (18)
Figure BDA00029317700100000615
The covariance matrix is updated by equation (19), and k is made k +1, and the next filtering cycle is continued.
Figure BDA00029317700100000616
Pk|k＝Pk|k-1 (19)
If | akC is less than or equal to l, the estimated value of the state vector is calculated by the formula (20)
Figure BDA00029317700100000617
Updating the covariance matrix through the formula (21), making k equal to k +1, continuing to execute the next filtering period,
Figure BDA0002931770010000071
Figure BDA0002931770010000072
wherein the content of the first and second substances,
Figure BDA0002931770010000073
(4) in a non-gaussian noise environment, the Method (MCGHCKF), Extended Kalman (EKF), Unscented Kalman (UKF), volumetric kalman (CKF), generalized high-order volumetric kalman (GHCKF), and maximum entropy unscented kalman (MCUKF) algorithms of the present invention are respectively applied to the INS/CNS integrated navigation system, and the root mean square errors of the attitude angles (pitch angle, course angle, and roll angle) estimated by these methods are compared, and the comparison results are shown in table 1. As can be seen from Table 1, the root mean square error of the attitude angle estimated by the method of the present invention is the smallest.
Table 1 shows the root mean square error comparison between the method of the present invention (MCGHCKF) and the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), volumetric kalman filter (CKF), generalized high-order volumetric kalman filter (GHCKF), maximum entropy unscented kalman filter (MCUKF) algorithms, which are applied to the attitude angles (pitch angle, course angle, and roll angle) estimated in the INS/CNS integrated navigation system, in the non-gaussian noise environment.
TABLE 1 RMS error comparison of attitude angles (pitch, course, and roll angles) under non-Gaussian noise
Figure BDA0002931770010000074

Claims (4)

1. A generalized high-order CKF algorithm based on maximum correlation entropy suitable for an INS/CNS integrated navigation system is characterized by comprising the following steps:
(1) constructing an INS/CNS integrated navigation system filtering model;
(2) carrying out time updating of a filtering algorithm according to the constructed filtering model;
(3) and introducing a maximum correlation entropy criterion and a judgment criterion, and carrying out measurement updating of the filtering algorithm.
2. The generalized high-order CKF algorithm based on maximum correlation entropy for an INS/CNS integrated navigation system as claimed in claim 1, wherein the constructing the INS/CNS integrated navigation system filter model in step (1) comprises the following steps:
(11) setting a state vector of the INS/CNS integrated navigation system as x ═ phi, delta v, delta r, epsilon and delta, wherein [ phi, delta v, delta r, epsilon and delta ] respectively represent INS attitude error, speed error, position error, gyro constant drift and accelerometer constant offset;
(12) establishing a state equation of the system according to a state vector x of the INS/CNS integrated navigation system:
Figure FDA0002931770000000011
wherein F represents a state transition matrix, G represents a process noise input matrix, and W represents process noise; discretizing the equation to obtain a discretized system state equation:
xk＝f(xk-1,Wk-1)
where f (-) is a known non-linear function, xk-1And xkRespectively representing the state vectors at time k-1 and at time k, Wk-1Representing the process noise at time k-1 and having an average value of 0, Wk-1The covariance of (a) can be expressed as:
Figure FDA0002931770000000012
taking the mathematical platform error angle equation of the navigation system as the measurement equation of the system,
zk＝Hkxk+Vk
wherein z iskA measurement vector representing time k, HkA measurement matrix representing the time k, VkThe measured noise at time k satisfies a mean of 0 and its covariance is:
Figure FDA0002931770000000013
and VkAnd Wk-1The satisfaction is not related to each other.
3. The generalized high-order CKF algorithm based on maximum correlation entropy for INS/CNS integrated navigation system as claimed in claim 1, wherein the step (2) comprises the following steps
(21) Setting the values of kernel width lambda and threshold value c according to the filtering precision requirement of INS/CNS integrated navigation system, and setting the initial state vectorThe estimated value is
Figure FDA0002931770000000021
Initial error covariance of P0|0K is 1, for P0|0Cholescent decomposition is carried out to obtain the characteristic square root initial value S of the error covariance matrix0|0；
(22) According to the formula
Figure FDA0002931770000000022
Calculate the volume point Xi,k-1|k-1(i＝1,...,2n2+1), where n represents the dimension of the state vector,
Figure FDA0002931770000000023
[·]ia set of representations [ ·]Column i, e.g.
Figure FDA0002931770000000024
When, there is [1]＝{[1 0]T,[0 1]T,[-1 0]T,[0 -1]T}；
(23) Calculating propagation volume points
Figure FDA0002931770000000025
Figure FDA0002931770000000026
(24) Predicting a state vector at a current time
Figure FDA0002931770000000027
Error covariance matrix Pk|k-1And calculating Pk|k-1Characteristic square root of Sk|k-1，
Figure FDA0002931770000000028
Figure FDA0002931770000000029
Wherein
Figure FDA00029317700000000210
4. The generalized higher-order CKF algorithm based on maximum correlation entropy as claimed in claim 3, wherein: introducing the maximum correlation entropy criterion and the judgment criterion, performing measurement update of the filter algorithm,
(31) based on predicted state vectors
Figure FDA00029317700000000211
And Pk|k-1Characteristic square root of Sk|k-1Generating a new volume point Xi,k|k-1And propagation volume point Zi,k|k-1，
Figure FDA00029317700000000212
Zi,k|k-1＝HkXi,k|k-1Wherein i is 12+1；
(32) Predicting time k measurements
Figure FDA00029317700000000213
Figure FDA00029317700000000214
(33) Based on the measurement equations of the INS/CNS integrated navigation system and the state vector in step (24)
Figure FDA00029317700000000215
And Pk|k-1The following regression equation is constructed,
Figure FDA00029317700000000216
wherein, I represents a unit vector,
Figure FDA0002931770000000031
Figure FDA0002931770000000032
Mp,k|k-1、Mr,kand MkBy passing
Figure FDA0002931770000000033
Is obtained by georges decomposition;
multiplication on both sides of the regression equation
Figure FDA0002931770000000034
Obtaining: dk＝BkXk+ek(ii) a Wherein the content of the first and second substances,
Figure FDA0002931770000000035
Figure FDA0002931770000000036
(34) updating the measured noise covariance matrix
Figure FDA0002931770000000037
Figure FDA0002931770000000038
Wherein the content of the first and second substances,
Figure FDA0002931770000000039
diag (. circle.) denotes the diagonalization of the matrix, m denotes the dimension of the measurement vector, kernel function in the correlation entropy
Figure FDA00029317700000000310
di,kRepresents DkThe ith element of (b)i,kIs represented by BkRow i element of (1);
(35) calculating the covariance P of the measurement vectorzz,k|k-1，
Figure FDA00029317700000000311
(36) Computing
Figure FDA00029317700000000312
If | akIf | is greater than c, then
Figure FDA00029317700000000313
Pk|k＝Pk|k-1K is k +1, return to step (22) and continue to execute the next filtering cycle, if | akIf | is less than or equal to c, calculating
Figure FDA00029317700000000314
Figure FDA00029317700000000315
And k is k +1, and the step (22) is returned to continue to execute the next filtering cycle.
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