CN115408652A - EKF (extended Kalman Filter) integrated navigation protection level determination method based on truncation error estimation - Google Patents

EKF (extended Kalman Filter) integrated navigation protection level determination method based on truncation error estimation Download PDF

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CN115408652A
CN115408652A CN202211341296.4A CN202211341296A CN115408652A CN 115408652 A CN115408652 A CN 115408652A CN 202211341296 A CN202211341296 A CN 202211341296A CN 115408652 A CN115408652 A CN 115408652A
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薛瑞
周里
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Beihang University
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Abstract

The invention relates to an EKF (extended Kalman filter) combined navigation protection level determination method based on truncation error estimation; establishing a GNSS/INS tight combination navigation system state space model considering truncation errors, and estimating truncation error values and covariance thereof; modifying the model with the estimated truncation error and its covariance; calculating the statistical characteristic of the corrected positioning error by considering the residual amount of the truncation error to obtain a protection level expression comprising the truncation error; acquiring actual pseudo-range, pseudo-range rate observed quantity samples and corresponding residual truncation error sample data; respectively enveloping pseudo range, pseudo range rate observation error and residual truncation error sample data based on an extreme value theory to obtain corresponding amplification factors; and substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tight combination navigation system. The invention combines more accurate protection level and more accurate error tail probability distribution, and ensures the reliability of the protection level.

Description

EKF (extended Kalman Filter) integrated navigation protection level determination method based on truncation error estimation
Technical Field
The invention belongs to the technical field of integrated navigation, and particularly relates to an EKF integrated navigation protection level determination method based on truncation error estimation.
Background
The single navigation system can not meet the requirement of accurate, credible and reliable flight of the unmanned aerial vehicle in the complex environment, and the GNSS/INS combined navigation has the advantages of all-weather long-term stable GNSS navigation and high INS short-time precision, and can be used as a navigation means for the unmanned aerial vehicle to fly in the complex environment. In order to ensure flight safety, the GNSS/INS integrated navigation has to meet given integrity requirements, the confidence upper limit of the positioning error under the designated risk probability, namely the protection level, can be calculated in real time, and when the protection level exceeds the alarm limit, the navigation system is unavailable at the moment, and the user needs to be alarmed in time.
When an EKF (extended kalman filter) is used to solve the nonlinear problem, taylor series is needed to linearize the nonlinear system, but the linearization needs to discard the high-order component in the taylor expansion. The core of the integrity monitoring of the GNSS/INS integrated navigation system is to estimate the statistical characteristics of positioning errors caused by various factors so as to calculate the protection level. When the EKF method is applied to GNSS/INS combined navigation, the calculation error of truncating the middle and high order terms of Taylor expansion is artificially generated and is called truncation error.
When the system nonlinearity is strong, the calculated positioning error covariance is inaccurate due to truncation error, so that the protection level calculation is not accurate enough and the integrity requirement cannot be met. The truncation error has to be taken into account during the filtering.
In addition, in the current correlation research of the combined navigation protection level, errors are assumed to follow a zero-mean gaussian distribution, but in practice, errors such as pseudorange and pseudorange rate are non-gaussian and non-zero-mean distribution, so that the calculated protection level cannot envelop the tail of the actual error, and a potential integrity risk may be caused.
Disclosure of Invention
In view of the above analysis, the present invention aims to disclose a method for determining an EKF integrated navigation protection level based on truncation error estimation, which is used to solve the problem that the truncation error generated in the EKF linearization process makes the protection level calculation not accurate enough.
The invention discloses an EKF (extended Kalman filter) combined navigation protection level calculation method based on truncation error estimation, which comprises the following steps of:
establishing a GNSS/INS tight combination navigation system state space model considering truncation errors, and estimating truncation error values and covariance thereof;
modifying the model with the estimated truncation error and its covariance; calculating the statistical characteristic of the corrected positioning error by considering the residual amount of the truncation error to obtain a protection level expression comprising the truncation error;
obtaining actual pseudo range, pseudo range rate observed quantity samples and corresponding residual truncation error sample data;
respectively enveloping pseudo range, pseudo range rate observation error and residual truncation error sample data based on an extreme value theory to obtain corresponding amplification factors;
and substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tight combination navigation system.
Further, the GNSS/INS tightly combined navigation system state space model considering truncation errors is:
Figure 849845DEST_PATH_IMAGE001
；
wherein,kis the time of day or the like,
Figure 100698DEST_PATH_IMAGE002
is thatnA state vector of dimensions;
Figure 66249DEST_PATH_IMAGE003
is thatmA measurement vector of dimensions;
Figure 402552DEST_PATH_IMAGE004
is composed ofnThe state of the stages is transferred to the matrix in one step,
Figure 721538DEST_PATH_IMAGE005
is composed ofm×nA measurement matrix of orders;
Figure 967843DEST_PATH_IMAGE006
to truncate the error vector;
Figure 714082DEST_PATH_IMAGE007
is composed ofnThe system noise vector of the dimension(s),
Figure 272102DEST_PATH_IMAGE008
is composed ofmThe measurement noise vector of the dimension.
Further, the state vector of the GNSS/INS tightly combined navigation system state space model is an error state vector of 17 dimensions:
Figure 394779DEST_PATH_IMAGE009
wherein,
Figure 980744DEST_PATH_IMAGE010
is the three-dimensional velocity error;
Figure 163463DEST_PATH_IMAGE011
is the three-dimensional attitude angle error;
Figure 474359DEST_PATH_IMAGE012
is the three-dimensional position error;
Figure 10514DEST_PATH_IMAGE013
is the three-dimensional accelerometer error;
Figure 90465DEST_PATH_IMAGE014
is the three-dimensional gyroscope error;
Figure 444086DEST_PATH_IMAGE015
is a one-dimensional receiver constant offset;
Figure 366911DEST_PATH_IMAGE016
is a one-dimensional receiver clock drift.
Further, the measurement of the GNSS/INS tightly combined navigation system state space model is the GNSS observation pseudo range
Figure 831391DEST_PATH_IMAGE017
Pseudorange rate
Figure 500270DEST_PATH_IMAGE018
Pseudoranges predicted from INS
Figure 165737DEST_PATH_IMAGE019
Pseudorange rates
Figure 185646DEST_PATH_IMAGE020
The difference between the two;
measuring noise vector
Figure 188237DEST_PATH_IMAGE021
Including GNSS observed pseudoranges and pseudorange rate noise vectors.
Further, estimating the truncation error value and the covariance thereof by using an NPF method specifically includes:
1) Obtaining the estimated residual error of the current-time observed quantity and the last-time observed quantity;
residual error
Figure 193098DEST_PATH_IMAGE022
；
Wherein,
Figure 154101DEST_PATH_IMAGE023
is composed ofkAt the first momentiThe pseudorange observations of the satellites,
Figure 864568DEST_PATH_IMAGE024
is composed ofkAt the first momentiPseudo range rate observations of the particles;
Figure 77374DEST_PATH_IMAGE025
and
Figure 455266DEST_PATH_IMAGE026
is composed ofk-1 time of dayiAn estimate of the observed quantity of particles, equal to a function
Figure 587170DEST_PATH_IMAGE027
In thatk-values of the high order terms omitted after taylor expansion at the estimated localization solution at time 1;i=1,…,p，pthe number of visible stars;
2) Estimating an estimate of truncation error using residual
Figure 971884DEST_PATH_IMAGE028
And its covariance
Figure 785119DEST_PATH_IMAGE029
：
Estimator
Figure 814255DEST_PATH_IMAGE030
；
Covariance
Figure 992427DEST_PATH_IMAGE031
；
Wherein,
Figure 739803DEST_PATH_IMAGE032
is a matrix of weights that is semi-positively determined,
Figure 356729DEST_PATH_IMAGE033
to measure the noise matrix.
Further, the GNSS/INS tightly combined navigation system state space model corrected by the estimated value of the truncation error and the covariance thereof is as follows:
Figure 132049DEST_PATH_IMAGE034
；
in the formula,
Figure 809018DEST_PATH_IMAGE035
in order to estimate the truncation error by the NPF method and truncate the residual error amount,
Figure 778111DEST_PATH_IMAGE036
。
further, the determination of the protection stage expression including the truncation error includes:
1) Determining a positioning error according to error propagation of Kalman filtering;
2) Performing integrity hypothesis distribution to obtain a vertical positioning error expression of the GNSS/INS tight combination navigation system under the H0 hypothesis;
3) Determining the distribution characteristic of the vertical positioning error according to the vertical positioning error expression;
4) And obtaining a protection level expression comprising the truncation error under the assumption of H0 according to the error distribution characteristics.
Further, H0 assumes that the vertical positioning error expression:
Figure 870832DEST_PATH_IMAGE037
；
wherein,
Figure 77823DEST_PATH_IMAGE038
；
Figure 722431DEST_PATH_IMAGE039
is the Kalman gain;vas error state vectorsXThe line number corresponding to the height information is obtained;
Figure 303454DEST_PATH_IMAGE040
is composed of
Figure 262182DEST_PATH_IMAGE041
The elements in the row of the matrix corresponding to the height information,
Figure 385996DEST_PATH_IMAGE042
is composed of
Figure 404768DEST_PATH_IMAGE043
The elements in the row of the matrix corresponding to the height information,
Figure 489398DEST_PATH_IMAGE044
is composed of
Figure 782976DEST_PATH_IMAGE045
Elements in a row of the matrix corresponding to the height information;
vertical positioning error
Figure 698980DEST_PATH_IMAGE046
Obey the characteristics of Gaussian distribution;
Figure 577068DEST_PATH_IMAGE047
；
under the assumption of H0, a guard level expression including truncation errors:
Figure 8050DEST_PATH_IMAGE048
；
wherein
Figure 308581DEST_PATH_IMAGE049
And the failure-free undetected coefficient is obtained.
Further, the method for obtaining the corresponding amplification factor based on the extreme value theory envelope pseudo range sample data comprises the following steps:
1) Arranging the obtained actual pseudo-range samples from small to large after taking absolute values, and calculating the standard deviation of the samples;
2) Selecting a pseudo range sample as a threshold value by using an average excess function graph method;
3) Re-sampling the pseudo-range samples for B times in a Bootstrap method to obtain a group B of samples;
3) According to an extreme value theory, performing parameter estimation on each group of samples to determine a parameter confidence limit value;
4) And substituting the parameter confidence limit value into the sample distribution, and determining a pseudo-range error amplification factor according to the integrity of the probability distribution of the combined navigation pseudo-range error.
Further, substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tight combination navigation system
Figure 16774DEST_PATH_IMAGE050
(ii) a It is composed ofIn (1),
Figure 174086DEST_PATH_IMAGE051
；
wherein,
Figure 295626DEST_PATH_IMAGE052
is a firstiThe pseudorange error amplification factor for the visible satellites,
Figure 321219DEST_PATH_IMAGE053
is as followsiThe pseudo-range rate error magnification factor for the visible satellites,
Figure 211815DEST_PATH_IMAGE054
is as followsiThe pseudorange residue of the visible satellites truncates the error amplification factor,
Figure 805607DEST_PATH_IMAGE055
is a firstiThe pseudorange rate residual truncation error magnification factor for the visible satellites.
The invention can realize one of the following beneficial effects:
aiming at the problem that the truncation error generated in the EKF linearization process causes the calculation of the protection level to be inaccurate, the EKF integrated navigation protection level determining method based on the truncation error estimation reduces the uncertainty of the truncation error and corrects the positioning error and the distribution thereof through the estimated size of the truncation error and the covariance thereof, thereby obtaining a more accurate protection level and ensuring that the protection level meets the given integrity risk; and positioning more accurate error tail probability distribution by adopting the extreme value theory to envelop errors such as actually sampled pseudo range, pseudo range rate and the like and truncating residual error distribution. And the more accurate protection level and the more accurate error tail probability distribution are combined, so that the reliability of the protection level is ensured.
Drawings
The drawings are for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout the figures;
fig. 1 is a flowchart of a method for determining an EKF integrated navigation protection level based on truncation error estimation according to an embodiment of the present invention.
Detailed Description
The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form a part hereof, and which together with the embodiments of the invention serve to explain the principles of the invention.
An embodiment of the present invention discloses an EKF integrated navigation protection level determination method based on truncation error estimation, as shown in fig. 1, including the following steps:
s1, establishing a GNSS/INS tight integrated navigation system state space model considering truncation errors, and estimating truncation error values and covariance thereof;
s2, correcting the model by using the estimated truncation error and the covariance thereof; calculating the statistical characteristic of the corrected positioning error by considering the residual amount of the truncation error to obtain a protection level expression comprising the truncation error;
s3, obtaining actual pseudo range, pseudo range rate observed quantity samples and corresponding residual truncation error sample data;
the residual truncation error sample data comprises pseudo-range residual truncation error sample data and pseudo-range rate residual truncation error sample data;
s4, respectively enveloping pseudo range, pseudo range rate observation error and residual truncation error data based on an extreme value theory to obtain corresponding amplification factors;
and S5, substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tight combination navigation system.
In step S1, the state space model of the GNSS/INS tightly combined navigation system considering the truncation error is established as follows:
Figure 820968DEST_PATH_IMAGE056
；
wherein,kis the time of day or the like,
Figure 728881DEST_PATH_IMAGE057
is thatnA state vector of dimensions;
Figure 536300DEST_PATH_IMAGE003
is thatmA measurement vector of dimensions;
Figure 238677DEST_PATH_IMAGE058
and
Figure 492066DEST_PATH_IMAGE059
are known system configuration parameters, respectively callednA state one-step transition matrix of the order,m×nA measurement matrix of orders;
Figure 734828DEST_PATH_IMAGE060
to truncate the error vector;
Figure 600016DEST_PATH_IMAGE061
is thatnThe system noise vector of the dimension(s),
Figure 410977DEST_PATH_IMAGE062
is thatmThe dimensional measurement noise vectors are both zero-mean gaussian white noise vector sequences (following a normal distribution) and are uncorrelated with each other.
Namely:
Figure 525564DEST_PATH_IMAGE063
。
Figure 509700DEST_PATH_IMAGE064
is a system noise matrix;
Figure 416345DEST_PATH_IMAGE065
to measure the noise matrix.
Preferably, in this embodiment, the state vector of the GNSS/INS tightly combined navigation system state space model is an error state vector with 17 dimensions:
Figure 257262DEST_PATH_IMAGE066
wherein,
Figure 62407DEST_PATH_IMAGE067
is the three-dimensional velocity error (east, north, high) in the ENU navigation coordinate system;
Figure 787918DEST_PATH_IMAGE068
is the three-dimensional attitude angle error (pitch angle, roll angle, course angle);
Figure 362119DEST_PATH_IMAGE069
is the three-dimensional position error (latitude, longitude, altitude) in the geodetic coordinate system;
Figure 373937DEST_PATH_IMAGE070
is the three-dimensional accelerometer error (with zero bias);
Figure 89214DEST_PATH_IMAGE071
is the three-dimensional gyroscope error (including zero bias);
Figure 680733DEST_PATH_IMAGE072
always bias for the receiver;
Figure 171757DEST_PATH_IMAGE073
is the receiver clock drift.
Preferably, in this embodiment, the measurement of the GNSS/INS tight combination navigation system state space model is a GNSS observation pseudorange
Figure 557739DEST_PATH_IMAGE074
Pseudorange rate
Figure 9580DEST_PATH_IMAGE075
Pseudoranges to INS predictions
Figure 201527DEST_PATH_IMAGE076
Pseudorange rates
Figure 484741DEST_PATH_IMAGE077
The difference between them;
measuring noise vector
Figure 228575DEST_PATH_IMAGE021
Including GNSS observed pseudoranges and pseudorange rate noise vectors.
In step S1, the NPF (non-linear predictive filtering) method is sampled, and the magnitude of the truncation error and its covariance are estimated.
The method specifically comprises the following steps:
1) Obtaining a residual error of the current-time observed quantity and the estimation of the last-time observed quantity;
residual error
Figure 292346DEST_PATH_IMAGE078
；
Wherein,
Figure 960087DEST_PATH_IMAGE023
is composed ofkAt the first momentiThe pseudorange observations of the satellites,
Figure 301070DEST_PATH_IMAGE024
is composed ofkAt the first momentiPseudo range rate observations of the particles;
Figure 91171DEST_PATH_IMAGE025
and
Figure 314342DEST_PATH_IMAGE026
is composed ofk-1 time of dayiAn estimate of the observed quantity of particles, equal to a function
Figure 739770DEST_PATH_IMAGE079
In thatk-values of the high order terms omitted after taylor expansion at the positioning solution estimated at time 1;i=1,…,p，pto be at leastSee the number of stars.
2) Estimating an estimate of truncation error using residual
Figure 997576DEST_PATH_IMAGE080
And its covariance
Figure 958578DEST_PATH_IMAGE081
：
Estimator
Figure 934625DEST_PATH_IMAGE082
；
Covariance
Figure 881852DEST_PATH_IMAGE083
；
Wherein,
Figure 56482DEST_PATH_IMAGE084
for a weight matrix of positive semidefinite, the superscript "-1" represents the inverse of the matrix, the superscript "T"represents the transpose of the matrix.
In step S2, the process of correcting the original model with the estimated value of the truncation error and the covariance thereof includes:
1) Estimating the residual error of the current time to estimate the estimated quantity of the truncation error
Figure 391648DEST_PATH_IMAGE085
Substituting the value, and correcting a one-step prediction equation of the EKF;
Figure 510783DEST_PATH_IMAGE086
；
2) Obtaining a corrected one-step predicted covariance matrix;
Figure 651914DEST_PATH_IMAGE087
；
3) After the model is corrected, a certain residue still remains in the truncation error, and the residual truncation error is considered
Figure 884312DEST_PATH_IMAGE088
Suppose that
Figure 62484DEST_PATH_IMAGE089
Obeying a zero mean gaussian distribution
Figure 13122DEST_PATH_IMAGE090
，
Figure 426786DEST_PATH_IMAGE091
(ii) a The state space model of the GNSS/INS tightly combined navigation system which is obtained by rewriting the state equation and is corrected by the estimated value of the truncation error and the covariance thereof is as follows:
Figure 202106DEST_PATH_IMAGE092
。
in step S2, after the statistical characteristics of the positioning error are corrected in consideration of the residual truncation error amount, the process of determining the protection level expression including the truncation error includes:
1) Determining a positioning error according to error propagation of Kalman filtering;
wherein, the error propagation process of Kalman filtering comprises:
according to the Kalman filtering formula, there is a state posterior estimation value:
Figure 613496DEST_PATH_IMAGE093
；
substituting the measurement equation into the above equation:
Figure 848168DEST_PATH_IMAGE094
；
order to
Figure 940889DEST_PATH_IMAGE095
Subtracting from both sides simultaneously
Figure 882300DEST_PATH_IMAGE096
And substituting into the prediction equation
Figure 792488DEST_PATH_IMAGE097
Derivation is carried out:
Figure 452139DEST_PATH_IMAGE098
；
to obtainkError of state at moment
Figure 332239DEST_PATH_IMAGE099
Where the positioning error is the position component in the state error, the positioning error is simply written as
Figure 456053DEST_PATH_IMAGE099
。
Expressed by positioning error
Figure 474825DEST_PATH_IMAGE100
In a clear view of the above, it is known that,kthe time positioning error is expressed askState error at time-1, process noise at time-k-1, truncation error estimator at time-k,kThe truncation error residue at the time,kThe GNSS measurement at that time is a recursive function of five parts of noise.
2) Performing integrity hypothesis distribution to obtain a vertical positioning error expression of the GNSS/INS tightly-combined navigation system under the H0 hypothesis;
specifically, in two classes of assumptions:
h0 assumes that: no satellite fault occurs;
h1 hypothesis: a satellite failure condition.
Integrity is distributed among two classes of assumptions:
h0 assumes that system integrity is guaranteed by protection level integrity. And detecting the satellite fault by using a integrity fault monitoring algorithm under the assumption of H1, so as to realize the integrity of fault monitoring.
Wherein the protection level is an upper confidence limit of the positioning error,
Figure 293876DEST_PATH_IMAGE101
is the probability of no fault missing detection.
Specifically, under the assumption of H0, the vertical positioning error expression may be expressed as:
Figure 56296DEST_PATH_IMAGE102
；
wherein,
Figure 769037DEST_PATH_IMAGE103
，
Figure 647125DEST_PATH_IMAGE104
is the Kalman gain;
Figure 281369DEST_PATH_IMAGE105
is composed of
Figure 644217DEST_PATH_IMAGE106
The elements in the matrix in the rows corresponding to the height information in the geodetic coordinate system,
Figure 86831DEST_PATH_IMAGE107
is composed of
Figure 447405DEST_PATH_IMAGE108
The elements in the row corresponding to the height information in the geodetic coordinate system,
Figure 365682DEST_PATH_IMAGE044
is composed of
Figure 391276DEST_PATH_IMAGE045
Elements in a row corresponding to height information in a geodetic coordinate system;
since, in the present embodiment, the high-level error in the 17-dimensional error state vector is established to be the 9 th dimension; therefore, the temperature of the molten metal is controlled,
Figure 281872DEST_PATH_IMAGE109
is composed of
Figure 875664DEST_PATH_IMAGE110
Row 9 elements in the matrix;
Figure 891025DEST_PATH_IMAGE111
is composed of
Figure 533359DEST_PATH_IMAGE112
Line 9 elements;
Figure 340778DEST_PATH_IMAGE044
is composed of
Figure 454272DEST_PATH_IMAGE045
Line 9 elements.
3) Determining the distribution characteristic of the vertical positioning error according to the vertical positioning error expression;
Figure 81562DEST_PATH_IMAGE113
、
Figure 527587DEST_PATH_IMAGE114
、
Figure 330458DEST_PATH_IMAGE115
、
Figure 203736DEST_PATH_IMAGE116
、
Figure 52743DEST_PATH_IMAGE117
in which the components are independent of each other, vertical positioning error
Figure 489410DEST_PATH_IMAGE118
Obeying the gaussian distribution property:
Figure 209104DEST_PATH_IMAGE119
；
4) According to the error distribution characteristics, obtaining a protection level expression including a truncation error under the assumption of H0:
Figure 315601DEST_PATH_IMAGE120
；
wherein,
Figure 527270DEST_PATH_IMAGE121
the failure-free undetected coefficient.
Figure 315098DEST_PATH_IMAGE122
The function is defined as:
Figure 951615DEST_PATH_IMAGE123
；
the protection level considers the influence of truncation errors generated in the linearization process of the nonlinear system on positioning errors, so that the protection level is more rigorous and accurate.
In step S3, the pseudorange, pseudorange rate observation samples, and residual truncation error sample data are obtained from the observation data of the actual receiver.
In order to ensure the independence of sample data, the time span of data acquisition is as large as possible, and the receiver positions are scattered, for example, the same constellation is observed in Beijing, shanghai and Lhasa 3 for 1 month, all the pseudo-range and pseudo-range rate error samples are sampled at time intervals of 30s, and about 40000 pseudo-range and pseudo-range rate error samples of each satellite are collected.
And calculating and obtaining the residual truncation error sample according to an actual positioning result. And substituting the combined positioning solution and the inertial navigation positioning solution of the GNSS/INS combined navigation at each moment into an EKF observation equation, and subtracting a first-order term reserved by Taylor expansion from the original observed quantity to obtain the residual truncation error at the moment. At least 40000 samples were collected.
In step S4, a pseudorange error amplification factor, a pseudorange rate error amplification factor, and a residual truncation error amplification factor are obtained by respectively enveloping the pseudorange error, the pseudorange rate error, and the residual truncation error based on an extremum theory by using the same process.
Wherein the residual truncation error amplification factor comprises a pseudorange residual truncation error amplification factor and a pseudorange rate residual truncation error amplification factor.
Specifically, the process of obtaining the pseudorange error amplification factor based on the extreme value theory envelope pseudorange error includes:
1) Arranging the obtained actual pseudo-range samples from small to large after taking absolute values, and calculating the standard deviation of the samples
Figure 166696DEST_PATH_IMAGE124
；
Arranging the obtained actual pseudo-range samples from small to large after taking absolute values to form a sample sequence
Figure 147553DEST_PATH_IMAGE125
；NThe total number of samples taken. And calculating the standard deviation of the sample
Figure 270229DEST_PATH_IMAGE124
。
2) Selecting a pseudo-range sample as a threshold value by using an average excess function graph method;
specifically, the average excess function is:
Figure 964516DEST_PATH_IMAGE126
i.e. for more than a threshold value in a sampleTAveraging the excess of (A) to obtain a threshold valueTPoint of time
Figure 288181DEST_PATH_IMAGE127
Continuously taking the data from small to large with small steps at equal intervalsTThe threshold value can be selected as one of the parts of the curve which is close to linearity and has positive slopeTThe value is obtained.
3) Re-sampling the pseudo range samples B times by using a Bootstrap method in a returning way to obtain a group B of samples
Figure 599077DEST_PATH_IMAGE128
；
Figure 728707DEST_PATH_IMAGE129
，
Figure 464450DEST_PATH_IMAGE130
；
3) According to an extreme value theory, performing parameter estimation on each group of samples to determine a parameter confidence limit value;
the excess is greater than the threshold value in each group of samplesTOf a sample andTa difference of (i) that
Figure 818071DEST_PATH_IMAGE131
，
Figure 819525DEST_PATH_IMAGE132
，
Figure 690529DEST_PATH_IMAGE133
，
Figure 93829DEST_PATH_IMAGE134
Greater than a threshold for each set of samplesTThe number of the samples of (a) is,
Figure 883930DEST_PATH_IMAGE135
distribution of (2)
Figure 795517DEST_PATH_IMAGE136
Distribution of available samples
Figure 532529DEST_PATH_IMAGE137
Represents:
Figure 852652DEST_PATH_IMAGE138
；
thus, the sample distribution can be obtained
Figure 751337DEST_PATH_IMAGE139
。
Let the probability of greater than the threshold sample in each set of samples be
Figure 399488DEST_PATH_IMAGE140
，
Figure 736928DEST_PATH_IMAGE141
。
The theory of the extreme values states that,
Figure 849240DEST_PATH_IMAGE142
the maximum value distribution of the maximum value follows Gumbel distribution, the excess amount
Figure 371358DEST_PATH_IMAGE143
Obeying a type I generalized Pareto distribution:
Figure 569121DEST_PATH_IMAGE144
；
to obtain the sample distribution, two parameters need to be estimated
Figure 444673DEST_PATH_IMAGE145
And
Figure 349175DEST_PATH_IMAGE146
。
probability of a sample in each resample sample exceeding a threshold
Figure 855243DEST_PATH_IMAGE147
Similarly, the probability of the sample in the population exceeding the threshold may be obtained
Figure 602619DEST_PATH_IMAGE148
. Statistics of memory aid
Figure 642381DEST_PATH_IMAGE149
。
Figure 729286DEST_PATH_IMAGE146
Maximum likelihood estimation can be performed.
Figure 468572DEST_PATH_IMAGE150
The probability density function of (a) is:
Figure 640927DEST_PATH_IMAGE151
；
Figure 733648DEST_PATH_IMAGE152
；
the log-likelihood function of a type I generalized Pareto distribution is:
Figure 737376DEST_PATH_IMAGE153
；
order to
Figure 585246DEST_PATH_IMAGE154
To obtain the following solution:
Figure 431849DEST_PATH_IMAGE155
；
i.e. of the b-th group of resampled samples
Figure 390577DEST_PATH_IMAGE156
The maximum likelihood estimate is
Figure 514391DEST_PATH_IMAGE157
. The parameter estimation of the overall sample can be obtained by the same method
Figure 939687DEST_PATH_IMAGE158
. Statistics of memory aid
Figure 86635DEST_PATH_IMAGE159
。
Thus, the parameter values of B groups of resample sample estimation can be obtained
Figure 911371DEST_PATH_IMAGE160
And
Figure 250211DEST_PATH_IMAGE161
。
using Bootstrap method, according to the defined confidence
Figure 439884DEST_PATH_IMAGE162
Determining a parameter confidence limit
Figure 870865DEST_PATH_IMAGE163
And
Figure 171397DEST_PATH_IMAGE164
。
4) And substituting the parameter confidence limit value into the sample distribution, and determining a pseudo-range error amplification factor according to the integrity of the probability distribution of the combined navigation pseudo-range error.
Will be provided with
Figure 879590DEST_PATH_IMAGE165
And
Figure 302481DEST_PATH_IMAGE166
value substitution sample distribution:
Figure 158441DEST_PATH_IMAGE167
；
obtaining the probability distribution of the GNSS/INS combined navigation pseudo range error:
Figure 449614DEST_PATH_IMAGE168
；
and order
Figure 340210DEST_PATH_IMAGE169
，
Figure 934002DEST_PATH_IMAGE170
Is a specified risk of integrity. Can be solved to obtain
Figure 949363DEST_PATH_IMAGE171
Quantile of
Figure 591697DEST_PATH_IMAGE172
：
Figure 399116DEST_PATH_IMAGE173
；
Determining an amplification factor
Figure 524329DEST_PATH_IMAGE174
：
Figure 354882DEST_PATH_IMAGE175
。
Respectively carrying out error envelope on the pseudo-range error, the pseudo-range rate error and the residual truncation error to obtain a pseudo-range error amplification factor
Figure 863223DEST_PATH_IMAGE176
Pseudorange rate amplification factor
Figure 462832DEST_PATH_IMAGE177
Pseudorange residual truncation error amplification factor
Figure 8214DEST_PATH_IMAGE178
Pseudorange rate residual truncation error amplification factor
Figure 388380DEST_PATH_IMAGE179
。
In step S5, in calculating the protection level,
measuring noise by Kalman filtering, measuring noise vector
Figure 638095DEST_PATH_IMAGE180
The variance of each component in the set ofkTime of day measurement noise matrix
Figure 279161DEST_PATH_IMAGE181
Diagonal values, whereiniPseudorange measurement noise variance of visible satellites equal tokTime of day measurement noise matrix
Figure 323341DEST_PATH_IMAGE182
2 nd (2)iLine 1, line 2iA column value of-1, i.e
Figure 925223DEST_PATH_IMAGE183
The false of the ith visible starRange rate measurement noise variance equal tokTime of day measurement noise matrix 2 ndiLine 2iColumn values, i.e.
Figure 650734DEST_PATH_IMAGE184
. First, theiResidual truncation error variance of visible star pseudoranges equal tokTruncation error matrix for time of day
Figure 224935DEST_PATH_IMAGE185
No. 2iLine 1, line 2iA column value of-1, i.e
Figure 502332DEST_PATH_IMAGE186
(ii) a First, theiResidual truncation error variance of visible star pseudorange rate equal tokTruncation error matrix for time of day
Figure 217609DEST_PATH_IMAGE185
2 nd (2)iLine 2iColumn values, i.e.
Figure 543549DEST_PATH_IMAGE187
。
According to the pseudo range error amplification factor, the pseudo range rate amplification factor and the residual truncation error amplification factor; after the pseudo-range error variance, the pseudo-range rate error variance and the residual truncation error variance are amplified, the distribution of pseudo-range measurement noise, the pseudo-range rate measurement noise and the residual truncation error is as follows:
Figure 34573DEST_PATH_IMAGE188
；
Figure 358238DEST_PATH_IMAGE189
；
Figure 872396DEST_PATH_IMAGE190
；
Figure 64343DEST_PATH_IMAGE191
；
Figure 534507DEST_PATH_IMAGE192
is as followsiThe pseudorange error amplification factors for the visible satellites,
Figure 91390DEST_PATH_IMAGE053
is as followsiThe pseudo-range rate error magnification factor for the visible satellites,
Figure 155161DEST_PATH_IMAGE054
is a firstiThe pseudorange residue of the visible satellites truncates the error amplification factor,
Figure 88482DEST_PATH_IMAGE055
is as followsiThe pseudo-range rate residue of the visible satellites truncates the error amplification factor.
Substituting the difference into the calculation to obtain the final protection level:
Figure 163886DEST_PATH_IMAGE193
；
wherein,
Figure 953987DEST_PATH_IMAGE051
。
in the embodiment, the high-degree error in the established 17-dimensional error state vector is 9 th dimension; corresponding final protection level
Figure 442737DEST_PATH_IMAGE194
(ii) a Namely thatv=9。
In summary, according to the method for determining the EKF integrated navigation protection level based on the truncation error estimation in the embodiment, aiming at the problem that the truncation error generated in the EKF linearization process makes the protection level calculation inaccurate, the uncertainty of the truncation error is reduced through the estimated size of the truncation error and the covariance thereof, and the positioning error and the distribution thereof are corrected, so that a more accurate protection level is obtained and meets the given integrity risk; and the error such as pseudo range, pseudo range rate and the like of actual sampling is enveloped by adopting an extreme value theory, and residual error distribution is truncated, so that more accurate error tail probability distribution is positioned. And the more accurate protection level and the more accurate error tail probability distribution are combined, so that the reliability of the protection level is ensured. And the integrity of the nonlinear GNSS/INS integrated navigation system is further guaranteed.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention.

Claims (10)

1. An EKF integrated navigation protection level calculation method based on truncation error estimation is characterized by comprising the following steps:
establishing a GNSS/INS tight combination navigation system state space model considering truncation errors, and estimating truncation error values and covariance thereof;
modifying the model with the estimated truncation error and its covariance; calculating the statistical characteristic of the corrected positioning error by considering the residual amount of the truncation error to obtain a protection level expression comprising the truncation error;
acquiring actual pseudo-range, pseudo-range rate observed quantity samples and corresponding residual truncation error sample data;
respectively enveloping pseudo range, pseudo range rate observation error and residual truncation error sample data based on an extreme value theory to obtain corresponding amplification factors;
and substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tight combination navigation system.
2. The EKF integrated navigation protection level calculation method based on truncation error estimation according to claim 1,
the GNSS/INS tightly combined navigation system state space model considering truncation errors comprises the following steps:
Figure 297192DEST_PATH_IMAGE001
；
wherein,kis the time of day or the like,
Figure 620857DEST_PATH_IMAGE002
is thatnA state vector of dimensions;
Figure 666173DEST_PATH_IMAGE003
is thatmA measurement vector of dimensions;
Figure 326962DEST_PATH_IMAGE004
is composed ofnThe state of the stage is transferred to the matrix in one step,
Figure 298591DEST_PATH_IMAGE005
is composed ofm×nA measurement matrix of orders;
Figure 386633DEST_PATH_IMAGE006
to truncate the error vector;
Figure 919245DEST_PATH_IMAGE007
is composed ofnThe system noise vector of the dimension(s),
Figure 524670DEST_PATH_IMAGE008
is composed ofmThe measurement noise vector of the dimension.
3. The method of claim 2, wherein the state vector of the GNSS/INS tightly-integrated navigation system state space model is a 17-dimensional error state vector:
Figure 459128DEST_PATH_IMAGE009
wherein,
Figure 718071DEST_PATH_IMAGE010
is the three-dimensional velocity error;
Figure 737980DEST_PATH_IMAGE011
is the three-dimensional attitude angle error;
Figure 130784DEST_PATH_IMAGE012
is the three-dimensional position error;
Figure 919748DEST_PATH_IMAGE013
is the three-dimensional accelerometer error;
Figure 615172DEST_PATH_IMAGE014
is the three-dimensional gyroscope error;
Figure 263322DEST_PATH_IMAGE015
is a one-dimensional receiver constant offset;
Figure 804025DEST_PATH_IMAGE016
is a one-dimensional receiver clock drift.
4. The EKF combined navigation protection level calculation method based on truncation error estimation as claimed in claim 3, wherein,
measurement of GNSS/INS tightly combined navigation system state space model quantity as GNSS observation pseudo range
Figure 713075DEST_PATH_IMAGE017
Pseudorange rates
Figure 939919DEST_PATH_IMAGE018
Pseudoranges to INS predictions
Figure 934420DEST_PATH_IMAGE019
Pseudorange rate
Figure 544393DEST_PATH_IMAGE020
The difference between them;
measuring noise vectors
Figure 183316DEST_PATH_IMAGE021
Including GNSS observed pseudoranges and pseudorange rate noise vectors.
5. The EKF integrated navigation protection level calculation method based on truncation error estimation according to claim 4,
estimating a truncation error value and a covariance thereof by using an NPF method, specifically comprising:
1) Obtaining a residual error of the current-time observed quantity and the estimation of the last-time observed quantity;
residual error
Figure 220542DEST_PATH_IMAGE022
；
Wherein,
Figure 436760DEST_PATH_IMAGE023
is composed ofkAt the first momentiThe pseudorange observations of the satellites,
Figure 975057DEST_PATH_IMAGE024
is composed ofkAt the first momentiPseudo range rate observed quantity of the particle;
Figure 593120DEST_PATH_IMAGE025
and
Figure 801248DEST_PATH_IMAGE026
is composed ofk-1 time of dayiEstimated value of observed quantity of particles, equal to function
Figure 645707DEST_PATH_IMAGE027
In thatk-values of the high order terms omitted after taylor expansion at the estimated localization solution at time 1;i=1,…,p，pnumber of visible stars;
2) Estimating truncation error using residual errorIs estimated by
Figure 331903DEST_PATH_IMAGE028
And its covariance
Figure 70052DEST_PATH_IMAGE029
：
Estimation quantity
Figure 329040DEST_PATH_IMAGE030
；
Covariance
Figure 519850DEST_PATH_IMAGE031
；
Wherein,
Figure 275316DEST_PATH_IMAGE032
is a matrix of weights that is semi-positively determined,
Figure 477759DEST_PATH_IMAGE033
to measure the noise matrix.
6. The EKF integrated navigation protection level calculation method based on truncation error estimation according to claim 5,
the GNSS/INS tightly combined navigation system state space model corrected by the estimated value of the truncation error and the covariance thereof is as follows:
Figure 27689DEST_PATH_IMAGE034
；
in the formula,
Figure 971374DEST_PATH_IMAGE035
in order to estimate the truncation error residual quantity by an NPF method,
Figure 389586DEST_PATH_IMAGE036
。
7. the EKF integrated navigation protection level calculation method based on truncation error estimation according to claim 6, wherein,
the determination of the protection level expression including the truncation error comprises:
1) Determining a positioning error according to error propagation of Kalman filtering;
2) Performing integrity hypothesis distribution to obtain a vertical positioning error expression of the GNSS/INS tight combination navigation system under the H0 hypothesis;
3) Determining the distribution characteristic of the vertical positioning error according to the vertical positioning error expression;
4) And obtaining a protection level expression comprising the truncation error under the assumption of H0 according to the error distribution characteristics.
8. The EKF integrated navigation protection level calculation method based on truncation error estimation according to claim 7,
under the assumption of H0, the vertical positioning error expression is as follows:
Figure 836748DEST_PATH_IMAGE037
；
wherein,
Figure 557579DEST_PATH_IMAGE038
；
Figure 598347DEST_PATH_IMAGE039
is the Kalman gain;vas error state vectorsXThe line number corresponding to the height information is obtained;
Figure 695616DEST_PATH_IMAGE040
is composed of
Figure 262864DEST_PATH_IMAGE041
Corresponding to height information in a matrixThe elements in the row of (a) and (b),
Figure 780695DEST_PATH_IMAGE042
is composed of
Figure 433393DEST_PATH_IMAGE043
The elements in the row of the matrix corresponding to the height information,
Figure 68774DEST_PATH_IMAGE044
is composed of
Figure 100315DEST_PATH_IMAGE045
Elements in rows of the matrix corresponding to the height information;
vertical positioning error
Figure 428528DEST_PATH_IMAGE046
Obeying the characteristics of Gaussian distribution;
Figure 302943DEST_PATH_IMAGE047
；
h0 assumes that the guard level expression including truncation error:
Figure 866649DEST_PATH_IMAGE048
；
wherein
Figure 408488DEST_PATH_IMAGE049
The failure-free undetected coefficient.
9. The EKF combined navigation protection level calculation method based on truncation error estimation as claimed in claim 8,
the method for obtaining the corresponding amplification factor based on the extreme value theory envelope pseudo range sample data comprises the following steps:
1) Arranging the obtained actual pseudo-range samples from small to large after taking absolute values, and calculating the standard deviation of the samples;
2) Selecting a pseudo range sample as a threshold value by using an average excess function graph method;
3) Re-sampling the pseudo-range samples for B times in a Bootstrap method to obtain a group B of samples;
3) According to an extreme value theory, performing parameter estimation on each group of samples to determine a parameter confidence limit value;
4) And substituting the parameter confidence limit value into the sample distribution, and determining a pseudo-range error amplification factor according to the integrity of the probability distribution of the combined navigation pseudo-range error.
10. The EKF combined navigation protection level calculation method based on truncation error estimation as claimed in claim 9, wherein,
substituting the amplification factor into the protection level expression to obtain the protection level of the GNSS/INS tightly combined navigation system
Figure 642024DEST_PATH_IMAGE050
(ii) a Wherein,
Figure 879101DEST_PATH_IMAGE051
；
wherein,
Figure 856284DEST_PATH_IMAGE052
is as followsiThe pseudorange error amplification factor for the visible satellites,
Figure 987051DEST_PATH_IMAGE053
is as followsiThe pseudo-range rate error magnification factor for the visible satellites,
Figure 283166DEST_PATH_IMAGE054
is as followsiThe pseudorange residue of the visible satellites truncates the error amplification factor,
Figure 132173DEST_PATH_IMAGE055
is as followsiThe pseudorange rate residual truncation error magnification factor for the visible satellites.
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