CN116660948A - Application method of improved volume Kalman in low-rail opportunistic signal positioning - Google Patents

Abstract

The invention relates to an application method of volume Kalman in low-orbit opportunistic signal positioning based on improvement, which comprises the steps of adding an adaptive filtering part into a volume Kalman filtering algorithm and correcting the statistical characteristics of process noise; SVD decomposition is used for replacing Cholesky decomposition, so that the positive qualitative requirement of a covariance matrix is avoided; an anti-difference factor is introduced, so that the influence of an abnormal observed value on positioning accuracy is reduced; before outputting the result, calculating a positioning error, if the positioning error exceeds a certain threshold value, introducing a fading factor, increasing the weight of the current epoch observation value, then continuously calculating an error covariance predicted value of the state quantity, and outputting the result; if the positioning error does not exceed the threshold, the result is directly output. The invention can reduce the error brought by the least square method in the linearization process.

Description

Application method of improved volume Kalman in low-rail opportunistic signal positioning

Technical Field

The invention relates to the field of opportunistic signal navigation and positioning, in particular to an application method of improved volume Kalman in low-orbit opportunistic signal positioning. The invention provides an application of Kalman filtering in the field of low-rail opportunistic signal positioning, and provides an improved concept of volume Kalman filtering for Doppler positioning calculation aiming at errors caused by a least square method in a linearization process.

Background

Under the condition that the global navigation satellite system (GlobalNavigationSatelliteSystem, GNSS) fails, a target can be navigated and positioned by utilizing opportunistic signals such as Bluetooth, ultra wideband and the like, but the coverage of the opportunistic signals is often insufficient for deserts, oceans and space regions, and the low-orbit (LowEarthOrbit, LEO) satellite signal is taken as one of the opportunistic signals, so that not only can the responsibility of navigation and positioning be born, but also a constellation formed by a plurality of low-orbit satellites has the capacity of covering the whole world.

The LEO satellite has high running speed and large Doppler frequency difference and high geometric position diversity of visible satellites, so that the positioning based on the LEO satellite is mainly based on Doppler frequency positioning, and the positioning of a target based on Doppler frequency shift characteristics of low-orbit opportunistic signals is a very effective means for assisting GNSS positioning.

The conventional resolving method is to establish a Doppler positioning equation set based on the idea of least square and combining with a Newton iteration method, and resolve by using an analytic method. However, in the process of resolving, taylor series expansion is generally adopted, and a linear part is reserved, which brings about a certain error. Meanwhile, the Doppler positioning expression is slightly complicated, so that the overall calculation complexity is improved when the Jacobian matrix is solved.

The volume kalman filter (CubatureKalmanFilter, CKF) is the approximation algorithm currently closest to bayesian filtering, and approximates the posterior mean and covariance of states by a third order spherical radial volume criterion to ensure that the posterior mean and variance of nonlinear gaussian states are approximated theoretically with a third order polynomial. Meanwhile, the solution of the jacobian matrix is replaced by a mode of calculating and transmitting the volume points, and the calculation complexity of an algorithm is reduced.

However, CKF has a certain drawback, and in practical positioning, due to complex and variable environment and problems of the measuring device, direct use of volume kalman filter solution brings about larger errors, so that some improvements are needed. Based on this, the present invention has been made.

Disclosure of Invention

In order to solve the problems in the prior art, the invention provides an application method of improved volume Kalman in low-rail opportunistic signal positioning so as to reduce errors caused by a least square method in a linearization process.

The technical scheme of the invention is as follows: an application method based on improved volume Kalman in low-orbit opportunistic signal positioning is provided, wherein an adaptive filtering part is added in a volume Kalman filtering algorithm, and the statistical characteristic of process noise is corrected; SVD decomposition is used for replacing Cholesky decomposition, so that the positive qualitative requirement of a covariance matrix is avoided; an anti-difference factor is introduced, so that the influence of an abnormal observed value on positioning accuracy is reduced; before outputting the result, calculating a positioning error, if the positioning error exceeds a certain threshold value, introducing a fading factor, increasing the weight of the current epoch observation value, then continuously calculating an error covariance predicted value of the state quantity, and outputting the result; if the positioning error does not exceed the threshold, the result is directly output.

Further, in the adaptive filtering, a windowing method is used to calculate the statistical characteristics of the process noise in combination with the idea of exponential decay.

Further, the general formula of the SVD decomposition is:

wherein A is any M x N matrix, U is a left singular matrix of A,the singular value matrix of A, V is the right singular matrix of A, and the superscript T represents the matrix transposition.

Further, the robust factor is used for adjusting the measurement matrix, reducing the influence of abnormal values on the positioning accuracy, and the expression is that

，

wherein , and />Represents a constant, in general +.>The value range of (2) is 1.5-2.0%>The value range of (2) is 3.0-8.5; />Representing normalized robust differences.

Further, the fading factor is a parameter not less than 1.

Further, the application method of the improved volume Kalman-based low-rail opportunistic signal positioning comprises the following steps:

step1: initializing a state;

step2: SVD decomposition is carried out on the covariance matrix;

step3: calculating a volume point;

step4: propagating the volume points and calculating predicted values of the state quantities;

step5: adaptively estimating statistical characteristics of process noise;

step6: calculating an error covariance predicted value of the state quantity;

step7: SVD decomposition is carried out on the covariance matrix;

step8: calculating a volume point;

step9: propagating the volume points and calculating a predicted value of the observed quantity;

step10: adding an anti-difference factor, and calculating a measurement error covariance and a predicted value of a cross covariance;

step11: and calculating the Kalman gain, and obtaining the final estimated value of the state quantity and the corresponding error covariance.

Further, step12: calculating a positioning error, if the error is greater than a threshold value, introducing an fading factor, returning to Step6, recalculating, and outputting a new state quantity estimated value; if the error is less than or equal to the threshold value, the state quantity estimation value in Step11 is output.

The invention has the advantages that: 1. aiming at complex and changeable environments and problems of measuring equipment, correcting the statistical characteristics of process noise by adding an adaptive filtering part into CKF; 2. SVD decomposition is used for replacing Cholesky decomposition, so that the positive qualitative requirement of a covariance matrix is avoided; 3. adding an anti-difference factor, and reducing the influence of an abnormal observed value on positioning accuracy; 4. the fading factor is introduced when the positioning error exceeds a certain threshold value to obtain a state quantity estimation value with the error smaller than the threshold value.

Drawings

Fig. 1 is a flow chart of the improved CKF algorithm of the present invention.

Figure 2 is a schematic diagram of doppler positioning.

Detailed Description

The invention is further described below with reference to fig. 1 and 2. It will be apparent that the described embodiments are only some of the embodiments of the present invention and are not exhaustive of all embodiments.

1. Illustrating the principle of Doppler location based on low-rail signals of opportunity

The low orbit satellite has low orbit height and high running speed, so the Doppler phenomenon is obvious. The Doppler shift of a user receiver can be expressed as

wherein ,indicating the Doppler shift received by the receiver, < >>Representing the carrier frequency of the satellite,crepresenting the speed of signal propagation, +.>Indicating the position of the receiver->Indicating the position of the satellite(s),indicating the relative speed between satellite and receiver, < >>Represents the receiver frequency drift (derivative of receiver clock difference with respect to time),/and the like>Other errors are indicated. Doppler positioning is the position of a receiver given knowledge of the receiver frequency, satellite carrier frequency, and the position and velocity information of the satellites. It is often necessary to obtain a plurality of doppler frequency values to construct a system of equations to solve for a plurality of unknowns.

The parameters are as defined above, subscripts 1,2, …,mthe sequence number of the formula/parameter is expressed.

For example, a single star Doppler is shown in FIG. 2, which is a schematic diagram of the location of a stationary user on the ground. The user is att ₁ Andt ₂ two moments are respectively connectedTwo times of Doppler information from a satellite is received, so that two equal-frequency conical surfaces can be obtained; based on the information that the user is located on the earth surface, the positions of the two intersection points 1 and 2 in fig. 2 can be obtained; finally, according to priori knowledge or capturing Doppler information at a moment, the position of the user can be finally determined.

2. Introduction to the volume Kalman Filter algorithm step

The Kalman filtering (KalmanFilter, KF) theory is based on the minimum mean square error criterion and is widely applied to the fields of target tracking, dynamic system control, navigation and the like. The state quantity is estimated by repeatedly recursively using a set of recursion equations while the system is running, and a theoretical method for filtering system measured values through input and output values and intelligently estimating linear dynamic system states is provided.

When optimally estimating a nonlinear gaussian system, linear kalman filtering (kalman filter) is no longer applicable, whereas the positioning equation based on the doppler shift of the low-rail signal-of-opportunity is exactly nonlinear. The extended Kalman filtering (ExtendedKalmanFilter, EKF) can bring larger calculation errors and has poorer positioning accuracy because only a first-order term is reserved in the calculation process. In 2009, the volume kalman filter (CubatureKalmanFilter, CKF) proposed by araararatnam I et al approximates the posterior mean and covariance of states by a third order spherical radial volume criterion, with accuracy reaching above the third order.

(1) The general expression for a nonlinear system is:

wherein ,represent the firstkA state vector of time; />Representing a state transfer function; />Represent the firstk-a state vector at time 1; />Represent the firstk-system input at time 1; />Represent the firstk-process noise at time-1; />Represent the firstkA measurement vector of time; />Representing a measurement function; />Represent the firstkSystem input quantity at moment; />Represent the firstkMeasurement noise at time.

(2) The volume kalman filter algorithm steps are as follows:

(1) initializing: initializing state vectors（kTime of day), error covariance matrix->（kTime of day), covariance matrix of process noise +.>And covariance matrix of measurement noise +.>。

(2) Calculating volume points:

in the above-mentioned method, the step of,representation ofkError covariance of time,/>Representing Cholesky decomposition of the error covariance matrix,/A->Representation pair->The matrix obtained by Cholesky decomposition, the superscript T denotes the matrix transpose,representing a set of volume points>Representing the dimension of a state vector

(3) Propagation volume point:

(4) calculating a state quantity predicted value and a predicted value of an error covariance:

(5) again calculate the volume point, the propagation volume point:

(6) calculating a measurement predicted value, a covariance of a measurement error and a cross covariance:

(7) calculating the Kalman gain and updating the state quantity and the corresponding error covariance:

TABLE 1 symbol illustrations

In actual positioning, because of complex and changeable environment and possible problems of measuring equipment, the direct use of the volume Kalman filter solution brings about larger errors, so the invention improves the method, and provides a Doppler positioning solution method based on improved volume Kalman filter. The improvement thought is as follows:

(1) adding an adaptive filtering part, and correcting the statistical characteristics of process noise;

(2) SVD decomposition is used for replacing Cholesky decomposition, so that the positive qualitative requirement of a covariance matrix is avoided;

(3) adding an anti-difference factor, and reducing the influence of an abnormal observed value on positioning accuracy;

(4) and calculating a positioning error, and if the positioning error exceeds a certain threshold value, introducing an fading factor and increasing the weight of the current epoch observation value.

The improvement thought is specifically described as follows:

(1) adaptive filtering

In the actual positioning process, some instability which is difficult to predict occurs in equipment and the like in the system, meanwhile, noise is often not regular due to the existence of interference of external factors, and the noise cannot be replaced by simple Gaussian white noise. On the premise that the system obtains the optimal estimation, an accurate noise statistical model is known before filtering, so that self-adaptive estimation needs to be introduced into CKF, and the state is updated through measurement data while the statistical characteristics of noise are corrected, so that the purpose of improving the accuracy is achieved.

According to the prior art, the simultaneous estimation of the covariance of the unknown process noise and the covariance of the measured noise cannot be realized, and only the correction of the process noise is considered based on the application scene of the invention. The measurement noise is mainly influenced by the performance of system equipment (such as a receiver), and the statistical characteristics are generally stable; the process noise is caused by various discretized errors, so that the randomness is stronger, and the invention establishes a time-varying noise correction model to describe the statistical characteristics of the process noise more accurately.

Let the mean value of the process noise beCovariance matrix +.>Both time-varying (where subscriptskRepresent the firstkTime of day). The traditional Sage-Husa self-adaptive filtering is mainly applied to a discrete linear system, and the invention refers toBy utilizing the thought of measuring the statistical characteristics of the innovation correction process noise and combining a sliding window method, the self-adaptive filtering of Sage-Husa is improved, and a nonlinear self-adaptive volume Kalman (NL-ACKF) algorithm is designed. From the state equation of the nonlinear system, the equation of state is formulated to be:

in the formula ,representation ofkProcess noise of time of day->Representation ofkReal state vector at +1. Since the true state quantity is unknown, use the firstk+1 time state estimation value ∈1>Instead of the actual value. Combining the nonlinear system model with the specific steps of CKF described above, one can derive:

wherein ,representation ofkReal state quantity at +1 moment, +.>Representation->The state estimation error of the moment in time,representing new information->Representation->The other parameters are as defined in table 1.

In the application scene of the invention, the statistical characteristics of the process noise are calculated by adopting a windowing method and combining the idea of exponential decay in consideration of the fact that the process noise is time-varying and only needs to pay attention to recent historical data. Firstly, selecting the length of a memory window asN（k≥N) From the slavekSampling forward at the moment; then, the data at the adjacent time is given a larger weight value and a weight coefficient by the idea of exponentially weighted attenuationThe method meets the following conditions:

，/>

in the formula ,indicating forgetting factor, the range of values is (0.95,0.99) in the normal case,bthe larger the description, the more reliable the recent historical data, and the more pronounced the noise time-varying characteristics.

Mean and covariance of original noise from the firstk-NAnd multiplying the time +1 by the corresponding weight to obtain new statistical characteristics. The recursive formulas of the mean and variance of the process noise after the self-adaptive correction are respectively as follows:

wherein ,representation->Posterior covariance after transfer through nonlinear state transfer functions.

(2) SVD decomposition replaces Cholesky decomposition

In CKF there are two Cholesky decompositions, which requires that the decomposed matrix be a positive definite matrix, and in the calculation process, it often happens that the state covariance matrix is not positive. The usual matrix decomposition includes eigenvalue decomposition, singular value decomposition and orthogonal triangular decomposition, but both eigenvalue decomposition and orthogonal triangular decomposition require the decomposed matrix to be a square matrix, so the present invention replaces Cholesky decomposition with singular value decomposition (SingularValueDecomposition, SVD) decomposition.

Any mxn matrix can be subjected to SVD decomposition:

(3) introduction of an anti-aliasing factor

Some abnormal values (also called outliers) are unavoidable in a real measurement environment, and an anti-difference factor is introduced to adjust a measurement matrix, so that the influence of the abnormal values on positioning accuracy is reduced. The basic principle is as follows: when the accuracy of the measurement information is high, the weight of the measurement value in the state estimation is improved; otherwise, the weight is reduced.

Robust factorSimilar to the expression of IGGIII equivalent function

wherein , and />Represents a constant, in general +.>The value range of (2) is 1.5-2.0%>The value range of (2) is 3.0-8.5; />Representing normalized robust differences. The corrected measurement noise error covariance matrix is

(4) Introduction of an evanescent factor

The fading filter is commonly used in a dynamic system, when the dynamic model error is large, the contribution of the dynamic model can be reduced through the fading factor, and the fading is effectively inhibited by more depending on the measurement information of the current epoch. The basic idea of the evanescent filtering is to increase the effect of the newly acquired data on the result while reducing the effect of the previous data.

The invention refers to paper 'comprehensive comparison analysis of two kinds of fading filtering and self-adaptive anti-difference filtering', takes the measurement information of the current epoch as a single random quantity, and adds an error covariance matrix after fading factors for standard Kalman filteringThe formula of (c) is as follows:

wherein （/>1) represents an evanescent factor; />Representation according tokInformation calculated at timekCovariance matrix of state estimation value at +1 moment;tr[·]representing the trace of the matrix; />、/>Jacobian matrices representing state transfer functions and measurement functions, respectively; />Covariance representing process noise, ++>Representing the covariance of the measurement noise.

It should be noted that if the state transition matrix fits the user's position change very well, i.e. the dynamics model accuracy is high, the prior information has a high reference value, and it is not appropriate to discard it. Therefore, the algorithm designed by the invention adds a threshold judgment link, namely, only when the threshold is exceeded (namely, the positioning error is larger), the fading factor is added.

4. Fusion of improved volumetric Kalman filtering and Doppler location algorithm

For low-rail signal-of-opportunity based doppler positioning systems:

(1) state variablesSix parameters are used for representing three-dimensional coordinates and speeds of users respectively;

(2) state transitionfThe (-) function is represented according to a priori information. For example for a user at rest,f(. Cndot.) can be described as a state transition matrixThe method comprises the steps of carrying out a first treatment on the surface of the For a user to move straight at a uniform speed,f(. Cndot.) can be described as a state transition matrix +.>, wherein tThe presentation time … … adjusts the state transfer function for users of different motion states.

(3) Measurement functiong(. Cndot.) is a Doppler shift location equation, and the equations and principles have been described in the first section;

(4) system inputProcess noise->And measuring noise->Setting according to actual conditions. The system input and measurement noise is related to the performance of the device; process noise is affected by the environment in which the system is located.

The improved volume Kalman filtering algorithm comprises the following specific steps:

step1: state initialization

Step2: for covariance matrixSVD decomposition->

Step3: calculating and propagating volume points

，/>

Step4: adaptive estimation of statistical properties of process noise

Step5: calculating a predicted value of a state quantity

Step6: calculating an error covariance prediction value of a state quantity

Step7: for covariance matrixSVD decomposition is performed

Step8: calculating volume points

，/>

Step9: propagating volume points and calculating predictions of observed quantities

Step10: adding an anti-difference factor, and calculating a measurement error covariance and a prediction value of the cross covariance

/>

Step11: calculating Kalman gain and obtaining final estimation value of state quantity and corresponding error covariance

Step12: calculating positioning errors

if error > threshold

Introducing an evanescent factor;

returning to Step6, and recalculating;

outputting a new state quantity estimation value;

break；

else

outputting a state quantity estimated value in Step 11;

end。

it should be understood that the foregoing examples of the present invention are provided merely for clearly illustrating the present invention and are not intended to limit the embodiments of the present invention, and that various other changes and modifications may be made therein by one skilled in the art without departing from the spirit and scope of the present invention as defined by the appended claims.

Claims (7)

1. An improved application method of volume Kalman in low-rail opportunistic signal positioning is characterized in that an adaptive filtering part is added in a volume Kalman filtering algorithm, and the statistical characteristics of process noise are corrected; SVD decomposition is used for replacing Cholesky decomposition, so that the positive qualitative requirement of a covariance matrix is avoided; an anti-difference factor is introduced, so that the influence of an abnormal observed value on positioning accuracy is reduced; before outputting the result, calculating a positioning error, if the positioning error exceeds a certain threshold value, introducing a fading factor, increasing the weight of the current epoch observation value, then continuously calculating an error covariance predicted value of the state quantity, and outputting the result; if the positioning error does not exceed the threshold, the result is directly output.

2. The method of claim 1, wherein the adaptive filtering uses a windowing method in combination with an exponential decay concept to calculate the statistical properties of the process noise.

3. The method of claim 1, wherein the SVD decomposition is of the general formula:

wherein A is any M x N matrix, U is a left singular matrix of A,the matrix is a singular value matrix of A, V is a right singular matrix of A, and T represents matrix transposition.

4. The method of using improved volume kalman in low rail signal-of-opportunity localization of claim 1, wherein the robust factor is used to adjust the measurement matrix to reduce the effect of outliers on localization accuracy by the expression of

，

wherein , and />Represents a constant, in general +.>The value range of (2) is 1.5-2.0%>The value range of (2) is 3.0-8.5; />Representing normalized robust differences.

5. A method of using improved volume kalman in low rail signal-of-opportunity location as recited in claim 1, whereinThe fading factorIs a parameter not less than 1.

6. A method of using improved volumetric kalman in low rail signal-of-opportunity location as defined in claim 1, comprising the steps of:

step1: initializing a state;

step2: SVD decomposition is carried out on the covariance matrix;

step3: calculating a volume point;

step4: propagating the volume points and calculating predicted values of the state quantities;

step5: adaptively estimating statistical characteristics of process noise;

step6: calculating an error covariance predicted value of the state quantity;

step7: SVD decomposition is carried out on the covariance matrix;

step8: calculating a volume point;

step9: propagating the volume points and calculating a predicted value of the observed quantity;

step10: adding an anti-difference factor, and calculating a measurement error covariance and a predicted value of a cross covariance;

step11: and calculating the Kalman gain, and obtaining the final estimated value of the state quantity and the corresponding error covariance.

7. The method of using improved volumetric kalman in low rail signal-of-opportunity localization of claim 6, further comprising Step12: calculating a positioning error, if the error is greater than a threshold value, introducing an fading factor, returning to Step6, recalculating, and outputting a new state quantity estimated value; if the error is less than or equal to the threshold value, the state quantity estimation value in Step11 is output.