CN113792412A - Spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering - Google Patents

Spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering Download PDF

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CN113792412A
CN113792412A CN202110929287.6A CN202110929287A CN113792412A CN 113792412 A CN113792412 A CN 113792412A CN 202110929287 A CN202110929287 A CN 202110929287A CN 113792412 A CN113792412 A CN 113792412A
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spacecraft
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covariance
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CN113792412B (en
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曹璐
杨宝健
冉德超
肖冰
蒋臣
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National Defense Technology Innovation Institute PLA Academy of Military Science
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Abstract

The invention discloses a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering, which comprises the following steps: establishing a nonlinear system according to spacecraft measurement data and an attitude dynamics model; solving a volume sampling point by using a Cholesky decomposition method and a preset volume point according to the state and the state covariance of the spacecraft at the previous moment, carrying out state transmission, and obtaining a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment; solving a volume sampling point by using a Cholesky decomposition method and a preset volume point according to the one-step prediction state estimation value and the one-step prediction state covariance, transferring measurement information, and obtaining a one-step prediction value, a covariance and a cross covariance of the spacecraft measurement output quantity; and establishing a linear regression equation of the spacecraft state based on the central error entropy criterion, determining a cost function of filtering of the central error entropy criterion, and obtaining the spacecraft state and the state covariance at the current moment. The invention can improve the attitude estimation precision when processing non-Gaussian noise.

Description

Spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering
Technical Field
The invention relates to the technical field of spacecraft attitude estimation, in particular to a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering.
Background
The high-precision and high-reliability attitude determination is the basis of the spacecraft for the tasks such as space on-orbit service and the like. The existing attitude determination methods of the spacecraft can be divided into a deterministic method and a state estimation method according to different attitude calculation methods, wherein the state estimation method adopts a filtering method to estimate the state quantity of the spacecraft according to observation information, so that the uncertainty influence of a reference vector can be effectively overcome.
In the nonlinear attitude estimation process, an Extended Kalman Filter (EKF) algorithm is mainly used for attitude estimation. However, the extended kalman filter has low filtering accuracy under strong non-linear conditions due to its own limitations. In order to overcome the problems of using an extended Kalman filtering algorithm, a volume Kalman filtering (CKF) algorithm is proposed at present, and the CKF algorithm is based on Cubaure transformation, has higher precision in processing nonlinear problems and has good filtering effect under the condition of Gaussian noise compared with the EKF algorithm. However, in an actual spacecraft attitude determination system, due to conditions such as sensor faults and outlier interference, noise obeys thick-tail non-gaussian distribution, and at this time, the conventional CKF algorithm may have a phenomenon of reduced accuracy or even filter divergence, resulting in reduced attitude determination accuracy of a spacecraft.
To deal with non-gaussian noise, non-gaussian filters are mainly used at present, and include: particle Filters (PF), Huber volume filters (HCF), maximum correlation entropy volume filters (MCCKF) and minimum error entropy volume filters (MECKF). The particle filter adopts a sequential importance sampling method to approximately calculate the posterior density, and can process any non-Gaussian noise; the Huber volume filter is formed by combining Cubasic transformation and a Huber cost function and can process a nonlinear non-Gaussian system; the maximum correlation entropy volume filter and the minimum error entropy volume filter respectively take a maximum correlation entropy criterion and a minimum error entropy criterion as optimal criteria, and have better non-Gaussian noise processing effect compared with the traditional minimum mean square error criterion.
However, in the above-mentioned non-gaussian filter, the particle filter has a large amount of computation complexity, and there are problems of particle degradation and particle depletion that are difficult to handle; the Huber volume filter based on the Huber cost function has limited precision for dealing with non-Gaussian noise; the problem of low accuracy also occurs when the maximum correlation entropy volume filter faces more complex non-gaussian noise; the minimum error entropy volume filter has translation invariance, and cannot enable an error probability density function to tend to zero.
Disclosure of Invention
In order to solve part or all of the technical problems in the prior art, the invention provides a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering.
The technical scheme of the invention is as follows:
a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering is provided, and the method is used for estimating the spacecraft attitude and comprises the following steps:
establishing a nonlinear system determined by the attitude of the spacecraft according to the measurement data of the spacecraft and the attitude dynamics model of the spacecraft;
solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the state and the state covariance of the spacecraft at the previous moment, and carrying out state transmission through a state equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment;
solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment, and transmitting measurement information through a measurement equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction value, a covariance of the one-step prediction value and a cross covariance of the one-step prediction value and the one-step prediction state estimation value of the measured output quantity of the spacecraft;
establishing a linearized regression equation corresponding to the spacecraft state based on the central error entropy criterion, determining a cost function of central error entropy criterion filtering by using the linearized regression equation, and performing maximization processing on the cost function to obtain the state and state covariance of the spacecraft at the current moment.
In some possible implementations, the nonlinear system for spacecraft attitude determination is established as:
Figure BDA0003210659420000021
wherein x iskRepresenting the n-dimensional state vector of the spacecraft at time k, f (-) representing the state equation of the system, xk-1Representing the n-dimensional state vector, omega, of the spacecraft at the time k-1k-1N-dimensional system noise sequence representing time k-1, zkM-dimensional measurement vector representing the k-time, h (-) represents the measurement equation of the system, vkRepresenting the m-dimensional measurement noise sequence at time k.
In some possible implementation modes, the previous moment is set as k-1 moment, the current moment is set as k moment, and the posterior probability density of the spacecraft state at the k-1 moment is set as
Figure BDA0003210659420000022
According to the state and the state covariance of the spacecraft at the previous moment, a Cholesky decomposition method and preset volume points are utilized to solve and obtain a plurality of volume sampling points, and the method comprises the following steps:
performing Cholesky decomposition on the state covariance of the spacecraft at the previous moment by using the following formula III and formula IV;
Figure BDA0003210659420000023
Figure BDA0003210659420000031
solving and obtaining 2n volume sampling points by using the following formula five according to a Cholesky decomposition result, the state of the spacecraft at the previous moment and a preset volume point;
Figure BDA0003210659420000032
wherein the content of the first and second substances,
Figure BDA0003210659420000033
representing the state of the spacecraft at time k-1, Pk-1Representing the state covariance matrix of the spacecraft at the time k-1, N (-) representing the Gaussian distribution, Sk-1Represents Pk-1The square root matrix of (a) is,
Figure BDA0003210659420000034
sample points, ξ, representing the state of the spacecraft at time k-1iRepresenting the ith pre-defined volume point.
In some possible implementations, the obtaining of the one-step predicted state estimation value and the one-step predicted state covariance of the spacecraft at the current time by performing state transfer through a state equation in a nonlinear system based on the obtained volume sampling points includes:
on the basis of the acquired volume sampling points, state transmission is carried out by utilizing the following formula eight, and state one-step prediction sampling points are acquired;
Figure BDA0003210659420000035
based on the state one-step prediction sampling point, performing state prediction estimation by using the following formula nine to obtain a one-step prediction state estimation value of the spacecraft at the current moment;
Figure BDA0003210659420000036
based on the one-step prediction state estimation value of the spacecraft at the current moment, acquiring one-step prediction state covariance of the spacecraft at the current moment by using the following formula;
Figure BDA0003210659420000037
wherein the content of the first and second substances,
Figure BDA0003210659420000038
represents a one-step prediction sampling point of the state after being transferred by a state equation in a nonlinear system,
Figure BDA0003210659420000039
one-step predicted state estimation value, P, representing spacecraft at time kkk-1One-step predicted state covariance matrix, Q, representing a spacecraft at time kk-1Representing system noise omegak-1The covariance matrix of (2).
In some possible implementation manners, solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current time, including:
performing Cholesky decomposition on the covariance of the one-step predicted state of the spacecraft at the current moment by using the following formula eleven;
Figure BDA0003210659420000041
according to the Cholesky decomposition result, the one-step prediction state estimation value of the spacecraft at the current moment and the preset volume point, solving by using a formula twelve to obtain 2n volume sampling points;
Figure BDA0003210659420000042
wherein S iskk-1Represents Pkk-1The square root matrix of (a) is,
Figure BDA0003210659420000043
and the sampling points represent the estimated value of the one-step prediction state of the spacecraft at the moment k.
In some possible implementations, based on the obtained volume sampling points, the measurement information is transmitted through a measurement equation in a nonlinear system, and a one-step predicted value, a covariance of the one-step predicted value, and a cross-covariance of the one-step predicted value and the one-step predicted state estimated value of the measured output quantity of the spacecraft are obtained, including:
based on the obtained volume sampling points, transmitting measurement information by using a formula thirteen below to obtain sampling points of one-step predicted values of the measured output quantity of the spacecraft;
Figure BDA0003210659420000044
based on sampling points of the one-step predicted value of the spacecraft measurement output quantity, carrying out measurement information prediction estimation by utilizing the following formula fourteen to obtain the one-step predicted value of the spacecraft measurement output quantity;
Figure BDA0003210659420000045
based on the one-step predicted value of the spacecraft measurement output quantity, acquiring the covariance of the one-step predicted value and the cross covariance of the one-step predicted value and the one-step predicted state estimated value by using the following formula fifteen;
Figure BDA0003210659420000046
wherein the content of the first and second substances,
Figure BDA0003210659420000047
sampling points representing one-step predicted values of spacecraft measurement outputs,
Figure BDA0003210659420000048
one-step predicted value, P, representing spacecraft measurement outputzz,kk-1Covariance matrix, P, representing one-step predicted values of spacecraft measured outputsxz,kk-1A cross-covariance matrix, R, representing a one-step predicted state estimate for the spacecraft and a one-step predicted value for a measured output of the spacecraftkRepresenting measurement noise vkThe covariance matrix of (2).
In some possible implementations, the one-step prediction error is set as:
Figure BDA0003210659420000049
setting the measurement slope matrix as:
Figure BDA0003210659420000051
will measure the vector zkThe approximation is:
Figure BDA0003210659420000052
establishing a linear regression equation corresponding to the spacecraft state as follows:
Figure BDA0003210659420000053
Figure BDA0003210659420000054
wherein the content of the first and second substances,
Figure BDA0003210659420000055
i denotes an identity matrix, rkRepresenting a high order error term.
In some possible implementations, the cost function of the central error entropy criterion filtering is:
Figure BDA0003210659420000056
wherein, λ represents a weight coefficient,
Figure BDA0003210659420000057
represents a kernel width of σ1Gaussian kernel function of ei,kRepresenting an error variable ekThe state of the (i) th dimension of (c),
Figure BDA0003210659420000058
represents a kernel width of σ2Gaussian kernel function of ej,kRepresenting an error variable ekDimension j of (d).
In some possible implementations, the state of the spacecraft at the current time is determined using the following equation twenty-four;
Figure BDA0003210659420000059
wherein the content of the first and second substances,
Figure BDA00032106594200000510
represents an optimal estimate of the state of the spacecraft at time k,
Figure BDA00032106594200000511
in some possible implementation manners, the maximization processing is performed on the cost function to obtain the state and the state covariance of the spacecraft at the current moment, and the method comprises the following steps:
calculating the gradient of the cost function, expressing a gradient calculation formula into a matrix form, and enabling the gradient to be equal to 0;
and obtaining the state and the state covariance of the spacecraft at the current moment by adopting a fixed point iterative algorithm based on a gradient calculation formula in a matrix form.
The technical scheme of the invention has the following main advantages:
according to the spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering, the Cabauure transformation is utilized to obtain the one-step prediction state estimation value and the one-step prediction state covariance, then the linearized regression equation is constructed, the central error entropy criterion is utilized to solve the posterior state of the spacecraft, the non-Gaussian noise occurring in the nonlinear system determined by the spacecraft attitude can be effectively dealt with, and the spacecraft attitude estimation precision and robustness during processing of the non-Gaussian noise are improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering in accordance with an embodiment of the present invention;
FIG. 2 is a schematic diagram of a comparison of root mean square errors of a roll angle of a spacecraft, which is obtained by using a conventional volume Kalman filtering algorithm, a maximum correlation entropy volume Kalman filtering algorithm, a minimum error entropy volume Kalman filtering algorithm and a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a comparison of root mean square errors of a pitch angle of a spacecraft, which is obtained by using a conventional volume Kalman filtering algorithm, a maximum correlation entropy volume Kalman filtering algorithm, a minimum error entropy volume Kalman filtering algorithm and a spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering of an embodiment of the invention;
fig. 4 is a schematic diagram of a comparison of root mean square errors of the yaw angles of the spacecraft, which is obtained by using a conventional volume kalman filtering algorithm, a maximum correlation entropy volume kalman filtering algorithm, a minimum error entropy volume kalman filtering algorithm, and a spacecraft attitude determination method based on the central error entropy criterion volume kalman filtering according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the specific embodiments of the present invention and the accompanying drawings. 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 a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
The technical scheme provided by the embodiment of the invention is described in detail below with reference to the accompanying drawings.
Referring to fig. 1, an embodiment of the present invention provides a spacecraft attitude determination method based on a central error entropy criterion volume kalman filter, the method is used for estimating a spacecraft attitude, and includes the following steps:
s1, establishing a nonlinear system determined by the attitude of the spacecraft according to the measurement data of the spacecraft and the attitude dynamics model of the spacecraft;
s2, solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the state and state covariance of the spacecraft at the previous moment, and performing state transmission through a state equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment;
s3, solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the one-step prediction state estimation value and the one-step prediction state covariance of the spacecraft at the current moment, and transmitting measurement information through a measurement equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction value, the covariance of the one-step prediction value and the cross covariance of the one-step prediction value and the one-step prediction state estimation value of the measured output quantity of the spacecraft;
s4, establishing a linearized regression equation corresponding to the spacecraft state based on the central error entropy criterion, determining a cost function of central error entropy criterion filtering by using the linearized regression equation, and performing maximization processing on the cost function to obtain the state and state covariance of the spacecraft at the current moment.
The following specifically describes steps and principles of the spacecraft attitude determination method based on the central error entropy criterion volume kalman filtering, provided by an embodiment of the present invention, with the former time being k-1 and the current time being k.
And step S1, establishing a nonlinear system determined by the attitude of the spacecraft according to the measurement data of the spacecraft and the attitude dynamics model of the spacecraft.
Specifically, according to the measurement data of the spacecraft and a spacecraft attitude dynamics model, a nonlinear system for determining the spacecraft attitude is established as follows:
Figure BDA0003210659420000071
wherein x iskRepresenting the n-dimensional state vector of the spacecraft at time k, f (-) representing the state equation of the system, xk-1Representing the n-dimensional state vector, omega, of the spacecraft at the time k-1k-1N-dimensional system noise sequence representing time k-1, zkM-dimensional measurement vector representing the k-time, h (-) represents the measurement equation of the system, vkM-dimensional measurement noise sequence representing k time, and ωkV and vkAre not related to each other.
Setting: initial state x of spacecraft0And omegakV and vkIndependent of each other, omegakAnd vkIndependently of one another,. omegakV and vkThe statistical properties of (a) are as follows:
Figure BDA0003210659420000072
wherein E (-) represents the mathematical expectation, QkRepresenting system noise omegakOf the covariance matrix, ωkN-dimensional system noise sequence representing time k, RkRepresenting measurement noise vkThe covariance matrix of (2).
And step S2, solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the state and the state covariance of the spacecraft at the previous moment, and performing state transmission through a state equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment.
Specifically, assume a posteriori probability density of spacecraft states at time k-1 of
Figure BDA0003210659420000081
According to the state and the state covariance of the spacecraft at the previous moment, a Cholesky decomposition method and a preset volume point are utilized to solve and obtain a plurality of volume sampling points, which can include:
performing Cholesky decomposition on the state covariance of the spacecraft at the previous moment by using the following formula III and formula IV;
Figure BDA0003210659420000082
Figure BDA0003210659420000083
solving and obtaining 2n volume sampling points by using the following formula five according to a Cholesky decomposition result, the state of the spacecraft at the previous moment and a preset volume point;
Figure BDA0003210659420000084
wherein the content of the first and second substances,
Figure BDA0003210659420000085
representing the state of the spacecraft at time k-1, Pk-1Representing the state covariance matrix of the spacecraft at the time k-1, N (-) representing the Gaussian distribution, Sk-1Represents Pk-1The square root matrix of (a) is,
Figure BDA0003210659420000086
representing the state of the spacecraft at time k-1Sample point xiiRepresenting the ith preset volume point and n representing the system state dimension.
In an embodiment of the present invention, the preset volume point may be determined as follows:
determining the ith preset volume point xi by using the following formulai；
Figure BDA0003210659420000087
Determining the corresponding weight of the ith preset volume point by using the following formula
Figure BDA0003210659420000088
Figure BDA0003210659420000089
Wherein [1 ]]iThe ith column, i ═ 1,2, …,2n, represents the set of sample points.
Further, based on the acquired volume sampling points, performing state transfer through a state equation in a nonlinear system, and acquiring a one-step predicted state estimation value and a one-step predicted state covariance of the spacecraft at the current time may include:
on the basis of the acquired volume sampling points, state transmission is carried out by utilizing the following formula eight, and state one-step prediction sampling points are acquired;
Figure BDA00032106594200000810
based on the state one-step prediction sampling point, performing state prediction estimation by using the following formula nine to obtain a one-step prediction state estimation value of the spacecraft at the current moment;
Figure BDA0003210659420000091
based on the one-step prediction state estimation value of the spacecraft at the current moment, acquiring one-step prediction state covariance of the spacecraft at the current moment by using the following formula;
Figure BDA0003210659420000092
wherein the content of the first and second substances,
Figure BDA0003210659420000093
represents a one-step prediction sampling point of the state after being transferred by a state equation in a nonlinear system,
Figure BDA0003210659420000094
one-step predicted state estimation value, P, representing spacecraft at time kkk-1One-step predicted state covariance matrix, Q, representing a spacecraft at time kk-1Representing system noise omegak-1The covariance matrix of (2).
S3, solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the one-step prediction state estimation value and the one-step prediction state covariance of the spacecraft at the current moment, and transmitting measurement information through a measurement equation in a nonlinear system based on the obtained volume sampling points to obtain the one-step prediction value, the covariance of the one-step prediction value and the cross covariance of the one-step prediction value and the one-step prediction state estimation value of the measured output quantity of the spacecraft.
Specifically, solving and obtaining a plurality of volume sampling points according to a one-step predicted state estimation value and a one-step predicted state covariance of the spacecraft at the current moment by using a Cholesky decomposition method and preset volume points may include:
performing Cholesky decomposition on the covariance of the one-step predicted state of the spacecraft at the current moment by using the following formula eleven;
Figure BDA0003210659420000095
according to the Cholesky decomposition result, the one-step prediction state estimation value of the spacecraft at the current moment and the preset volume point, solving by using a formula twelve to obtain 2n volume sampling points;
Figure BDA0003210659420000096
wherein S iskk-1Represents Pkk-1The square root matrix of (a) is,
Figure BDA0003210659420000097
and the sampling points represent the estimated value of the one-step prediction state of the spacecraft at the moment k.
Further, based on the obtained volume sampling points, measurement information transmission is performed through a measurement equation in a nonlinear system, and a one-step predicted value, a covariance of the one-step predicted value, and a cross-covariance of the one-step predicted value and the one-step predicted state estimated value of the spacecraft measured output quantity are obtained, which may include:
based on the obtained volume sampling points, transmitting measurement information by using a formula thirteen below to obtain sampling points of one-step predicted values of the measured output quantity of the spacecraft;
Figure BDA0003210659420000101
based on sampling points of the one-step predicted value of the spacecraft measurement output quantity, carrying out measurement information prediction estimation by utilizing the following formula fourteen to obtain the one-step predicted value of the spacecraft measurement output quantity;
Figure BDA0003210659420000102
based on the one-step predicted value of the spacecraft measurement output quantity, acquiring the covariance of the one-step predicted value and the cross covariance of the one-step predicted value and the one-step predicted state estimated value by using the following formula fifteen;
Figure BDA0003210659420000103
wherein the content of the first and second substances,
Figure BDA0003210659420000104
sampling points representing one-step predicted values of spacecraft measurement outputs,
Figure BDA0003210659420000105
one-step predicted value, P, representing spacecraft measurement outputzz,kk-1Covariance matrix, P, representing one-step predicted values of spacecraft measured outputsxz,kk-1And a cross covariance matrix representing the estimated value of the one-step prediction state of the spacecraft and the one-step predicted value of the measured output quantity of the spacecraft.
S4, establishing a linearized regression equation corresponding to the spacecraft state based on the central error entropy criterion, determining a cost function of central error entropy criterion filtering by using the linearized regression equation, and performing maximization processing on the cost function to obtain the state and state covariance of the spacecraft at the current moment.
Specifically, establishing a linearized regression equation corresponding to the spacecraft state based on the central error entropy criterion includes:
defining a one-step prediction error as:
Figure BDA0003210659420000106
defining the measurement slope matrix as:
Figure BDA0003210659420000107
will measure the vector zkThe approximation is:
Figure BDA0003210659420000108
establishing a linear regression equation corresponding to the spacecraft state as follows:
Figure BDA0003210659420000111
wherein r iskWhich represents a high-order error term that is,
Figure BDA0003210659420000112
i denotes an identity matrix.
Further, setting:
Figure BDA0003210659420000113
wherein S isk、Sp,kk-1And Sr,kRespectively represent matrices
Figure BDA0003210659420000114
Pk|k-1And RkCholesky decomposition of (1).
The equation of the linearized regression equation expressed by the formula nineteen above is multiplied by the equation on both sides
Figure BDA0003210659420000115
To transform the linearized regression equation into:
dk＝Wkxk+ekformula twenty-one
Wherein the content of the first and second substances,
Figure BDA0003210659420000116
further, setting: e.g. of the typek＝[e1,k,e2,k,…,eL,k]T，dk＝[d1,k,d2,k,…,dL,k]T，Wk＝[w1,k,w2,k,…,wL,k]T，ei,k＝di,k-wi,kxk(i＝1,…,L)，L＝m+n，ei,kDenotes ekThe ith element of (1), di,kDenotes dkThe ith element of (1), wi,kRepresents WkThe ith row vector of (1);
the cost function of the central error entropy criterion filtering (CEEKF) is then:
Figure BDA0003210659420000117
wherein, λ represents a weight coefficient,
Figure BDA0003210659420000118
represents a kernel width of σ1The gaussian kernel function of (a) is,
Figure BDA0003210659420000119
represents a kernel width of σ2Gaussian kernel function of (1).
Setting:
Figure BDA00032106594200001110
the formula twenty-two can be expressed as:
Figure BDA00032106594200001111
under the criterion of Central Error Entropy (CEE), the optimal estimation value of the state of the spacecraft at the current time can be obtained by maximizing the cost function, and the optimal estimation value is the estimated state of the spacecraft at the current time.
Specifically, taking the current moment as the moment k as an example, the state of the spacecraft at the moment k can be determined by the following formula twenty-four;
Figure BDA0003210659420000121
wherein the content of the first and second substances,
Figure BDA0003210659420000122
represents the optimal estimated value of the state of the spacecraft at the moment k, namely the state of the spacecraft at the moment k,
Figure BDA0003210659420000123
denotes JL(xk) X corresponding to maximum valuekThe value is obtained.
Further, in an embodiment of the present invention, the maximizing the cost function to obtain the state and the state covariance of the spacecraft at the current time may include the following steps:
calculating the gradient of the cost function, expressing a gradient calculation formula into a matrix form, and enabling the gradient to be equal to 0;
and obtaining the state and the state covariance of the spacecraft at the current moment by adopting a fixed point iterative algorithm based on a gradient calculation formula in a matrix form.
Specifically, taking the current time as the time k as an example, the gradient may be calculated by using the following formula twenty-five pairs of cost functions, and the gradient is made equal to 0;
Figure BDA0003210659420000124
wherein the content of the first and second substances,
Figure BDA0003210659420000125
Figure BDA0003210659420000126
expressing a cost function gradient calculation formula shown in the above formula twenty-five into a matrix form shown in the following formula twenty-six;
Figure BDA0003210659420000131
wherein the content of the first and second substances,
Figure BDA0003210659420000132
Figure BDA0003210659420000133
(Ωk)ijrepresents omegakThe ith row and the jth column of elements,
Figure BDA0003210659420000134
based on a cost function gradient calculation formula in a matrix form, a fixed point iterative algorithm is adopted to obtain the state of the spacecraft at the moment k as follows:
Figure BDA0003210659420000135
wherein the content of the first and second substances,
Figure BDA0003210659420000136
representing the state of the spacecraft at time k for the t +1 th iteration.
Further, the setting is made as in the following formulas twenty-eight to thirty-seven:
Figure BDA0003210659420000137
Figure BDA0003210659420000138
Ck＝(Cx,k Cy,k) Formula thirty
Figure BDA0003210659420000141
Figure BDA0003210659420000142
Figure BDA0003210659420000143
Figure BDA0003210659420000144
Figure BDA0003210659420000145
Figure BDA0003210659420000146
Figure BDA0003210659420000147
Wherein, Λx,kRepresenting an n x n dimensional matrix, Λxy,kIs an m × n dimensional matrix, Λyx,kIs an n x m dimensional matrix, Λy,kIs an m x m dimensional matrix, Λi,j；kIs represented bykA matrix formed by the ith row and the jth column of the matrix, xii,j；kDenotes xikThe ith row and the jth column of the matrix form a matrix, Ωi,j；kRepresents omegakThe ith row and the jth column of the matrix.
According to the formula twenty-one and the set formulas twenty-eight to thirty-seven, the state of the spacecraft at the time k can be expressed as:
Figure BDA0003210659420000148
wherein the content of the first and second substances,
Figure BDA0003210659420000149
Figure BDA00032106594200001410
Figure BDA00032106594200001411
obtained by using a formula of thirty-eight
Figure BDA00032106594200001412
For the spacecraft state x at time kkCan be used as the estimated state of the spacecraft at the current k momentState.
Further, based on the setting, the covariance matrix P of the posterior state of the spacecraft at the current k momentkCan be updated as:
Figure BDA0003210659420000151
wherein I represents an identity matrix.
According to the spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering, provided by the embodiment of the invention, the Cabauure transformation is utilized to obtain the one-step prediction state estimation value and the one-step prediction state covariance, then the linearized regression equation is constructed, and the central error entropy criterion is utilized to solve the posterior state of the spacecraft, so that the non-Gaussian noise occurring in a non-linear system determined by the spacecraft attitude can be effectively dealt with, and the spacecraft attitude estimation precision and robustness in processing the non-Gaussian noise are improved.
Based on the above defined steps and contents, in an embodiment of the present invention, the iterative solution process of the filtering step of the spacecraft attitude determination method based on the central error entropy criterion volume kalman filtering may include the following steps:
s201, selecting kernel width sigma1And σ2Selecting a weight coefficient lambda, setting an iteration stop condition, selecting a numerical value of a positive number epsilon, and giving an initial value of a spacecraft state and a state covariance
Figure BDA0003210659420000152
And P0；
S202, calculating to obtain a one-step prediction state estimation value of the spacecraft by utilizing the formula nine
Figure BDA0003210659420000153
Calculating to obtain the covariance P of the one-step predicted state of the spacecraft by using the formulak|k-1Obtaining the parameter S through Choleskey decompositionp,k|k-1And Sr,kObtaining the parameter d by using the above formula twenty onekAnd Wk；
S203, setting: t is equal to 1, and t is equal to 1,
Figure BDA0003210659420000154
wherein the content of the first and second substances,
Figure BDA0003210659420000155
representing the state estimation value of the spacecraft at the k moment in the t iteration;
s204, using the measured value { z1,z2,…,znUpdating the state, and calculating the state of the spacecraft at the moment k by using the formula thirty-eight
Figure BDA0003210659420000156
S205, the following determination conditions are compared:
Figure BDA0003210659420000157
if the above-mentioned determination condition is satisfied, setting
Figure BDA0003210659420000158
Step S206 is executed, and if the determination condition is not satisfied, t is set to t +1, and the process returns to step S204;
s206, updating the posterior covariance matrix P of the spacecraft at the moment k by utilizing the formula to achieve thirty-ninekK is set to k +1, and the process returns to step S202.
The following describes beneficial effects of the spacecraft attitude determination method based on the central error entropy criterion volume kalman filtering according to an embodiment of the present invention with reference to specific examples.
The system state equation and the measurement equation of a certain spacecraft attitude determination system are shown in formula forty:
Figure BDA0003210659420000161
wherein q isboAttitude of spacecraft expressed by quaternion, b gyroscopeIs also estimated as a state variable, ωgAs gyroscope measurements, omegagIs input, ηgAnd ηbIs zero mean system noise, ηgAnd ηbHas a covariance of QgAnd Qb，ΩdTo transfer the matrix, it can be expressed as:
Figure BDA0003210659420000162
Figure BDA0003210659420000163
qoptfor the observed output of the attitude sensor of the spacecraft, qNFor measuring noise quaternion, omegaoi＝[0 -ω0 0]Representing the angular velocity of the orbit under the inertial system, EboThe coordinate transformation matrix representing the transformation from orbital to star system can be expressed as:
Figure BDA0003210659420000164
the parameters for spacecraft attitude determination are set as follows:
constant drift of the gyro: b ═ 303030]T(°)/h, constant drift white noise mean square error σb0.5(°)/h, gyro measurement noise σg0.5(°)/h, the initial value of filtering is selected as: q. q.sbo(0)＝[0,0,0,1]T，b(0)＝[30 30 30]T(°)/h，ωbo＝10-4×[cos(10ω0t) cos(8ω0t) cos(5.7ω0t)]Angular velocity of orbit omega00.0012rad/s, initial covariance P0＝diag(I3×3 0.04I3×3) The measured noise of the star sensor is mixed Gaussian noise,
Figure BDA0003210659420000171
σv＝8”。
figures 2 to 2 of the drawingsAnd 4, a comparison schematic diagram of different attitude estimation results obtained by using a traditional volume Kalman filtering algorithm (CKF), a maximum correlation entropy volume Kalman filtering algorithm (MCCKF), a minimum error entropy volume Kalman filtering algorithm (MEECKF) and a spacecraft attitude determination method (CEECKF) based on central error entropy criterion volume Kalman filtering of the embodiment of the invention is given. In the drawing, RMSE of
Figure BDA0003210659420000172
Root mean square error of roll angle, RMSE of θ: root mean square error of pitch angle, RMSE of ψ: yaw angle root mean square error, time: time.
Wherein, the selection of the kernel width parameter of each algorithm is shown in table 1;
TABLE 1
Figure BDA0003210659420000173
The obtained Average Root Mean Square Error (ARMSE) of the three-axis attitude angles under different algorithms is shown in Table 2;
TABLE 2
Algorithms CKF MEECKF MCCKF CEECKF
Row(deg) 16.342865 35.746283 7.951591 3.451482
Pitch(deg) 5.867528 86.384785 9.125961 3.204264
Yaw(deg) 50.765787 77.421710 5.560464 2.939755
Therefore, the spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering provided by the embodiment of the invention has the highest filtering precision under the non-Gaussian noise condition, and has the lowest estimation error covariance after filtering convergence, namely the best filtering stability, so that the spacecraft attitude determination method can better cope with the non-Gaussian noise.
It is noted that, in this document, relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. In addition, "front", "rear", "left", "right", "upper" and "lower" in this document are referred to the placement states shown in the drawings.
Finally, it should be noted that: the above examples are only for illustrating the technical solutions of the present invention, and not for limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (10)

1. A spacecraft attitude determination method based on central error entropy criterion volume Kalman filtering is characterized in that the method is used for estimating the spacecraft attitude and comprises the following steps:
establishing a nonlinear system determined by the attitude of the spacecraft according to the measurement data of the spacecraft and the attitude dynamics model of the spacecraft;
solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to the state and the state covariance of the spacecraft at the previous moment, and carrying out state transmission through a state equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment;
solving and obtaining a plurality of volume sampling points by using a Cholesky decomposition method and preset volume points according to a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment, and transmitting measurement information through a measurement equation in a nonlinear system based on the obtained volume sampling points to obtain a one-step prediction value, a covariance of the one-step prediction value and a cross covariance of the one-step prediction value and the one-step prediction state estimation value of the measured output quantity of the spacecraft;
establishing a linearized regression equation corresponding to the spacecraft state based on the central error entropy criterion, determining a cost function of central error entropy criterion filtering by using the linearized regression equation, and performing maximization processing on the cost function to obtain the state and state covariance of the spacecraft at the current moment.
2. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 1, characterized in that the nonlinear system for spacecraft attitude determination is established as follows:
Figure FDA0003210659410000011
wherein x iskRepresenting the n-dimensional state vector of the spacecraft at time k, f (-) representing the state equation of the system, xk-1Representing the n-dimensional state vector, omega, of the spacecraft at the time k-1k-1N-dimensional system noise sequence representing time k-1, zkM-dimensional measurement vector representing the k-time, h (-) represents the measurement equation of the system, vkRepresenting the m-dimensional measurement noise sequence at time k.
3. The method for spacecraft attitude determination based on central error entropy criterion volume Kalman filtering according to claim 2, characterized in that the previous time is set as time k-1, the current time is time k, and the posterior probability density of the spacecraft state at time k-1 is set as
Figure FDA0003210659410000012
According to the state and the state covariance of the spacecraft at the previous moment, a Cholesky decomposition method and preset volume points are utilized to solve and obtain a plurality of volume sampling points, and the method comprises the following steps:
performing Cholesky decomposition on the state covariance of the spacecraft at the previous moment by using the following formula III and formula IV;
Figure FDA0003210659410000013
Figure FDA0003210659410000021
solving and obtaining 2n volume sampling points by using the following formula five according to a Cholesky decomposition result, the state of the spacecraft at the previous moment and a preset volume point;
Figure FDA0003210659410000022
wherein the content of the first and second substances,
Figure FDA0003210659410000023
representing the state of the spacecraft at time k-1, Pk-1Representing the state covariance matrix of the spacecraft at the time k-1, N (-) representing the Gaussian distribution, Sk-1Represents Pk-1The square root matrix of (a) is,
Figure FDA0003210659410000024
sample points, ξ, representing the state of the spacecraft at time k-1iRepresenting the ith pre-defined volume point.
4. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 3, characterized in that, based on the obtained volume sampling points, state transfer is performed through a state equation in a nonlinear system, and a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment are obtained, and the method comprises the following steps:
on the basis of the acquired volume sampling points, state transmission is carried out by utilizing the following formula eight, and state one-step prediction sampling points are acquired;
Figure FDA0003210659410000025
based on the state one-step prediction sampling point, performing state prediction estimation by using the following formula nine to obtain a one-step prediction state estimation value of the spacecraft at the current moment;
Figure FDA0003210659410000026
based on the one-step prediction state estimation value of the spacecraft at the current moment, acquiring one-step prediction state covariance of the spacecraft at the current moment by using the following formula;
Figure FDA0003210659410000027
wherein the content of the first and second substances,
Figure FDA0003210659410000028
represents a one-step prediction sampling point of the state after being transferred by a state equation in a nonlinear system,
Figure FDA0003210659410000029
one-step predicted state estimation value, P, representing spacecraft at time kk|k-1One-step predicted state covariance matrix, Q, representing a spacecraft at time kk-1Representing system noise omegak-1The covariance matrix of (2).
5. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 4, characterized in that a Cholesky decomposition method and preset volume points are utilized to solve and obtain a plurality of volume sampling points according to a one-step prediction state estimation value and a one-step prediction state covariance of the spacecraft at the current moment, and the method comprises the following steps:
performing Cholesky decomposition on the covariance of the one-step predicted state of the spacecraft at the current moment by using the following formula eleven;
Figure FDA0003210659410000031
according to the Cholesky decomposition result, the one-step prediction state estimation value of the spacecraft at the current moment and the preset volume point, solving by using a formula twelve to obtain 2n volume sampling points;
Figure FDA0003210659410000032
wherein S isk/k-1Represents Pk/k-1The square root matrix of (a) is,
Figure FDA0003210659410000033
and the sampling points represent the estimated value of the one-step prediction state of the spacecraft at the moment k.
6. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 5, characterized in that, based on the obtained volume sampling points, measurement information transmission is performed through a measurement equation in a nonlinear system, and a one-step predicted value, a covariance of the one-step predicted value and a cross covariance of the one-step predicted value and the one-step predicted state estimated value of a spacecraft measurement output quantity are obtained, and the method comprises the following steps:
based on the obtained volume sampling points, transmitting measurement information by using a formula thirteen below to obtain sampling points of one-step predicted values of the measured output quantity of the spacecraft;
Figure FDA0003210659410000034
based on sampling points of the one-step predicted value of the spacecraft measurement output quantity, carrying out measurement information prediction estimation by utilizing the following formula fourteen to obtain the one-step predicted value of the spacecraft measurement output quantity;
Figure FDA0003210659410000035
based on the one-step predicted value of the spacecraft measurement output quantity, acquiring the covariance of the one-step predicted value and the cross covariance of the one-step predicted value and the one-step predicted state estimated value by using the following formula fifteen;
Figure FDA0003210659410000036
wherein the content of the first and second substances,
Figure FDA0003210659410000037
sampling points representing one-step predicted values of spacecraft measurement outputs,
Figure FDA0003210659410000038
one-step predicted value, P, representing spacecraft measurement outputzz,k/k-1Covariance matrix, P, representing one-step predicted values of spacecraft measured outputsxz,k/k-1A cross-covariance matrix, R, representing a one-step predicted state estimate for the spacecraft and a one-step predicted value for a measured output of the spacecraftkRepresenting measurement noise vkThe covariance matrix of (2).
7. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 6, characterized by setting one-step prediction error as:
Figure FDA0003210659410000041
formula sixteen
Setting the measurement slope matrix as:
Figure FDA0003210659410000042
seventeen formula
Will measure the vector zkThe approximation is:
Figure FDA0003210659410000043
eighteen formulas
Establishing a linear regression equation corresponding to the spacecraft state as follows:
Figure FDA0003210659410000044
formula nineteen
Wherein the content of the first and second substances,
Figure FDA0003210659410000045
i denotes an identity matrix, rkRepresenting a high order error term.
8. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering of claim 7, characterized in that the cost function of the central error entropy criterion filtering is:
Figure FDA0003210659410000046
wherein, λ represents a weight coefficient,
Figure FDA0003210659410000047
represents a kernel width of σ1Gaussian kernel function of ei,kRepresenting an error variable ekThe state of the (i) th dimension of (c),
Figure FDA0003210659410000048
represents a kernel width of σ2Gaussian kernel function of ej,kRepresenting an error variable ekDimension j of (d).
9. The spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 8, characterized by determining the state of the spacecraft at the current moment by using twenty-four of the following formula;
Figure FDA0003210659410000049
wherein the content of the first and second substances,
Figure FDA00032106594100000410
represents an optimal estimate of the state of the spacecraft at time k,
Figure FDA00032106594100000411
10. the spacecraft attitude determination method based on the central error entropy criterion volume Kalman filtering according to claim 9, characterized in that the maximization processing is performed on the cost function to obtain the state and the state covariance of the spacecraft at the current moment, and the method comprises the following steps:
calculating the gradient of the cost function, expressing a gradient calculation formula into a matrix form, and enabling the gradient to be equal to 0;
and obtaining the state and the state covariance of the spacecraft at the current moment by adopting a fixed point iterative algorithm based on a gradient calculation formula in a matrix form.
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