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CN111650577A - Maneuvering target tracking method containing Doppler measurement under polar coordinate system - Google Patents

Maneuvering target tracking method containing Doppler measurement under polar coordinate system Download PDF

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CN111650577A
CN111650577A CN202010533144.9A CN202010533144A CN111650577A CN 111650577 A CN111650577 A CN 111650577A CN 202010533144 A CN202010533144 A CN 202010533144A CN 111650577 A CN111650577 A CN 111650577A
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measurement
matrix
probability
moment
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CN111650577B (en
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程婷
李立夫
李茜
侯子林
檀倩倩
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University of Electronic Science and Technology of China
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/66—Radar-tracking systems; Analogous systems
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/50—Systems of measurement based on relative movement of target
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/50—Systems of measurement based on relative movement of target
    • G01S13/58—Velocity or trajectory determination systems; Sense-of-movement determination systems
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/418—Theoretical aspects

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Abstract

The invention belongs to the field of target tracking of Doppler radars, and particularly relates to a target tracking system and method utilizing Doppler measurement information. The invention introduces a conversion measurement Kalman filtering method with radial velocity into an interactive multi-model method, and provides a Doppler radar maneuvering target tracking method capable of accurately tracking maneuvering targets with Doppler measurement. When solving the correlation algorithm, firstly, calculating initial mixing probability and input state estimation interaction through a traditional interactive multi-model algorithm; then, obtaining an unbiased measurement value under a Cartesian coordinate system through unbiased measurement conversion, and obtaining a weighted result of one-step prediction of a target state based on model probability; then calculating the statistical characteristic of the measurement error based on the prediction information; and finally, performing information Kalman filtering and model probability updating on each model to obtain the final state estimation.

Description

Maneuvering target tracking method containing Doppler measurement under polar coordinate system
Technical Field
The invention belongs to the field of target tracking of Doppler radars, and particularly relates to a target tracking system and method utilizing Doppler measurement information.
Background
In radar target tracking, the state equation of the target is generally established in a rectangular coordinate system, and the measurement values are generally obtained in a polar coordinate system. Some radars can obtain doppler measurements in addition to position measurement information, and the position of the target and the doppler measurements are in a nonlinear relationship with the motion state. To address this non-linear relationship, various filtering methods have been proposed, including the following: a Measurement Conversion Kalman Filter method (SQ-MCKF) Based On Sequential filtering, a Measurement Conversion Kalman Filter Based On static Fusion (SF-MCKF), an Unbiased Measurement Conversion with radial velocity (Unbiased Measurement Kalman Filter with Range Rate, UCMKF-R), and the like. Wherein, the Sequential filtering (Z.Duan, C.Han, X.R.Li. "Sequential Nonlinear filtering with Range-rate Information in spectral Coordinates," in Proceedings of the 7th International Conference on Information Fusion,2004,50(1):300 Kalman 316) is to decorrelate the position measurement and the Doppler pseudo-measurement by Cholesky decomposition, first perform the standard Kalman filtering process on the position measurement, then perform the second-order expansion filtering on the Doppler pseudo-measurement by taking the filtering result as the input to obtain the final Sequential filtering result, but the Nonlinear filtering result of the Doppler pseudo-measurement is fed back to the next linear filtering, so that the Nonlinear filtering error is increased iteratively with the filtering; a static fusion filtering method (G.ZHou, Z.Guo, X.Chen, R.Xu and T.Kirubabajan, "statistical Fused Measurement Kalman Filter for phase-Array Radars," in IEEE Transactions on Aero space and electronic systems, vol.54, No.2, pp.554-568, April 2018.) is to separately perform standard Kalman filtering estimation on a target unknown state and a Doppler pseudo-Measurement state, and then perform static fusion on the position and the Doppler pseudo-state to obtain a final state estimation result; an unbiased measurement conversion method (h.liu, z.zhou, l.yu and c.lu, "two adjacent converted measurement Kalman filtering with range rate," in IET Radar, Sonar & Navigation, vol.12, No.11, pp.1217-1224,112018.) with radial velocity is to change the measurement matrix H so that the position of the target and the doppler measurement are in a linear relationship with the motion state, thereby performing completely linear Kalman filtering, but the theoretical value in the H matrix should be a real value, and only a measurement value can be actually used, so that the accuracy of the estimation result is reduced. The kalman filtering method for conversion measurement With radial velocity (s. bordonaro, p. willett and y. bar-shape, "coherent Linear Tracker With Converted Range, Bearing, and Range measurements," in IEEE Transactions on an Aerospace and Electronic Systems, vol.53, No.6, pp.3135-3149, dec.2017.) introduces an uninformative tangential velocity measurement on the basis of doppler measurement to convert the measurement of doppler measurement into a completely Linear form, so that a single Linear filter can be used to filter the measurement to obtain the final estimation result.
The above methods all consider non-motorized target tracking, and a motorized target exists in an object tracked by an actual radar, so that the maneuverability of the target and the measured nonlinearity need to be considered at the same time, and an interactive multi-model filtering algorithm (e.mazor, a.avenbuch, y.bar-shape and j.dayan, "interactive multiple model methods in targeting: a following," in IEEE Transactions on aeronautics and Electronic Systems, vol.34, No.1, pp.103-123, jan.1998.) is considered as the most effective motorized target tracking method. Document (s.m. ally, r.e. food and h.braka, "Extended Kalman filtering and Interacting Multiple Model for tracking and evaluating targets in sensor networks," 2009 sensing work kshop on interactive sources in Embedded Systems, Ancona,2009, pp.149-156.) combining traditional Extended Kalman filtering method (Extended Kalman Filter) with interactive Multiple Model (Interacting Multiple Model) results in a mobile target tracking algorithm that can handle non-linearity measurements, whose performance is far superior to that of the non-linear filtering algorithm of single Model, but has poor tracking accuracy or tracking divergence problem when the non-linearity is high; the literature (d.xiaolong and z.pingfang, "a New Interacting multiple model Algorithm Based on the unknown Particle Filter,"2009 fine international Conference Information assessment and Security, Xi' an,2009, pp.419-422.) combines the Unscented Particle filtering method with the interactive multiple model Algorithm to achieve non-linear metrology-Based maneuvering target tracking, but the filtering estimation result is still not ideal because the Unscented transformation cannot completely eliminate the transformation bias. Aiming at the problems, the invention introduces a conversion measurement Kalman filtering method with radial velocity into an interactive multi-model method, and provides a Doppler radar maneuvering target tracking method capable of accurately tracking maneuvering targets with Doppler measurement.
Disclosure of Invention
Assume that the state of the model q at time k is estimated as
Figure BDA0002536075710000021
And an estimated error covariance of
Figure BDA0002536075710000022
The total number of models is M. Measurement information Z obtained at the moment of k +1k+1Including distance measurement
Figure BDA0002536075710000023
Azimuth angle measurement
Figure BDA0002536075710000024
And Doppler measurements
Figure 1
Measurement noise for range, azimuth and doppler measurements
Figure BDA0002536075710000026
And
Figure BDA0002536075710000027
is zero mean additive white Gaussian noise, and the measured variances are respectivelyAnd
Figure BDA0002536075710000029
the filtering steps from the k moment to the k +1 moment of the maneuvering target tracking method containing Doppler measurement under a polar coordinate system are as follows:
step 1: an initial mixing probability and input state estimate interaction are calculated.
Assuming that the model matched at time k is q and the model matched at time k +1 is r, the metrology data is Zk+1Under the condition (1), the initial mixing probability is:
Figure BDA00025360757100000210
Figure BDA0002536075710000031
wherein,
Figure BDA0002536075710000032
to normalize constant, piqrThe transition probability of the Markov model is expressed,
Figure BDA0002536075710000033
is the probability of the occurrence of the model q at time k.
And obtaining the input of the model r at the moment k +1 after interaction:
Figure BDA0002536075710000034
Figure BDA0002536075710000035
step 2: an unbiased metrology transformation at time k +1 is calculated.
Compensation factors for introducing multiplicative deviations
Figure BDA0002536075710000036
Obtaining unbiased measurement conversion at the moment k + 1:
Figure BDA0002536075710000037
wherein D is a direction cosine matrix:
Figure BDA0002536075710000038
and step 3: and obtaining a weighted result of one-step prediction of the target state by using the model probability.
One-step prediction of model r is:
Figure BDA0002536075710000039
Figure BDA00025360757100000310
wherein
Figure BDA00025360757100000311
Is the state transition matrix of model r.
The weighted result of the target one-step prediction is:
Figure BDA00025360757100000312
Figure BDA00025360757100000313
and 4, step 4: computing an inverse matrix of prediction information based metrology error covariance
Figure BDA00025360757100000314
And (3) estimating parameters under an LOS coordinate system by rotating the predicted target state and the covariance:
Figure BDA00025360757100000315
Figure BDA0002536075710000041
wherein the orientation is predicted
Figure BDA0002536075710000042
Is composed of
Figure BDA0002536075710000043
Obtaining a measurement error covariance inverse matrix based on the prediction information
Figure BDA0002536075710000044
Figure BDA0002536075710000045
Wherein
Figure BDA0002536075710000046
Comprises the following steps:
Figure BDA0002536075710000047
RRare respectively as follows:
Figure BDA0002536075710000048
Figure BDA0002536075710000049
Figure BDA00025360757100000410
Figure BDA00025360757100000411
Figure BDA00025360757100000412
Figure BDA00025360757100000413
wherein
Figure BDA00025360757100000414
To predict the azimuthal variance, it can be obtained by linearization
Figure BDA00025360757100000415
Figure BDA00025360757100000416
To represent
Figure BDA00025360757100000417
The (n) th element of (a),
Figure BDA00025360757100000418
representation matrix PR,k+1|kRow i and column j.
And 5: information kalman filtering (taking model r as an example) is performed on each model.
Obtaining the covariance of the input state estimation error of the model r at the moment k +1 after interaction in the step 1
Figure BDA0002536075710000051
One-step prediction error covariance matrix of available model r
Figure BDA0002536075710000052
Comprises the following steps:
Figure BDA0002536075710000053
wherein
Figure BDA0002536075710000054
And
Figure BDA0002536075710000055
a state transition matrix and a process noise driving matrix of the k +1 moment model r respectivelyAnd a process noise matrix.
Filter gain of model r
Figure BDA0002536075710000056
Comprises the following steps:
Figure BDA0002536075710000057
the state update information matrix is:
Figure BDA0002536075710000058
Figure BDA0002536075710000059
step 6: model probability update and final state estimation.
The information filtering likelihood function of the model r is:
Figure BDA00025360757100000510
where det (-) is the determinant of the matrix,
Figure BDA00025360757100000511
an innovation error matrix of the model r is shown, and an inverse matrix expression of the innovation error matrix is as follows:
Figure BDA00025360757100000512
wherein,
Figure BDA00025360757100000513
and is
Figure BDA00025360757100000514
The probability of model r is updated as:
Figure BDA00025360757100000515
wherein c is a normalization constant:
Figure BDA00025360757100000516
the final state estimation and covariance obtained by model probability weighting are respectively:
Figure BDA00025360757100000517
Figure BDA0002536075710000061
principle of the invention
Interactive multi-model algorithm based on information Kalman filtering
And aiming at the maneuvering moving target, modeling and estimating a plurality of possible motion modes of the target by adopting an interactive multi-model method. The model of the assumed k time consists of M models to form a model set
Figure BDA0002536075710000062
In the interactive multi-model algorithm, each sensor usually employs kalman filtering, and weights the result and obtains the final state estimation result. And the covariance of the measurement error in the conventional Kalman filter is RCBut if only the inverse of the covariance of the measurement transformation error is known
Figure BDA0002536075710000063
And is
Figure BDA0002536075710000064
When the filter is irreversible, an information Kalman filter is required to be used as a sub-filter. The information kalman filter is updated by the inverse matrix of the state estimation error covariance and the measurement error covariance, as shown in equations (25) and (14), and the state estimation thereof can be updated according to the conventional kalman filtering method, as shown in equation (24).
Based on the state estimation result of each sub-filter, the IMM algorithm needs to calculate the model probability when performing state weighting, the model probability is updated based on the likelihood function, the information filtering likelihood function of the model r is calculated as shown in formula (26), wherein the form of the innovation covariance needs to be based on the inverse matrix of the measured conversion error covariance
Figure RE-GDA0002566021740000065
And (6) reckoning.
The traditional innovation error calculation method comprises the following steps:
Figure BDA0002536075710000067
then derived by matrix inversion lemma
Figure BDA0002536075710000068
The inverse matrix of (a), is of the form:
Figure BDA0002536075710000071
thus, the result shown in the formula (27) was obtained.
And finally, carrying out probability updating at the moment of k +1 and final state estimation of the interactive multi-model.
Unbiased measurement conversion method with Doppler measurement information
The traditional measurement conversion method only considers the measurement of the target position, and the target tracking precision can be further improved after the radial velocity measurement information is introduced. Meanwhile, in order to improve the measurement conversion mode, the tangential velocity measurement without information is considered to be introduced.
The measured value at the time k +1 is
Figure BDA0002536075710000072
Constructing a coordinate transformation matrix
Figure BDA0002536075710000073
So that the measurement conversion result is:
Figure BDA0002536075710000074
wherein,
Figure BDA0002536075710000075
the specific form of (A) is as follows:
Figure BDA0002536075710000076
since the tangential velocity cannot be exactly measured by the radar, a parameter
Figure BDA0002536075710000077
Are not informative. However, a priori information about the desired distribution of tangential velocities (which may be obtained based on the target velocity being known) may be used to calculate the metrology conversion error covariance.
Equation (34) is desirably:
Figure BDA0002536075710000078
it can be seen that the measurement transform of equation (34) is biased, thereby introducing a compensation factor for multiplicative bias
Figure BDA0002536075710000079
And obtaining the unbiased measurement conversion result of the formula (5).
The measurement transformation error can be expressed as:
Figure BDA0002536075710000081
since the true value is not available in practice, a one-step prediction of the radar is used here for the approximation:
Figure BDA0002536075710000082
substituting equation (38) into equation (37) yields the measured conversion error as:
Figure BDA0002536075710000083
to ensure unbiased and consistent post-raw measurement conversion, the mean and covariance of the measurement conversion error are calculated here based on the predicted values:
Figure BDA0002536075710000091
covariance matrix R for easy calculation of unbiased metrology conversion errorCBy solving for R in line of sight (LOS) coordinatesRTo obtain a measurement error covariance RC：
Figure BDA0002536075710000092
When in use
Figure BDA0002536075710000093
The method comprises the following steps:
Figure BDA0002536075710000094
measurement error covariance R based on prediction informationRThe derivation of each element in (1) is as follows:
Figure BDA0002536075710000095
Figure BDA0002536075710000096
Figure BDA0002536075710000101
Figure BDA0002536075710000102
Figure BDA0002536075710000103
Figure BDA0002536075710000104
wherein,
Figure BDA0002536075710000105
Figure BDA0002536075710000106
to predict the azimuthal variance, it can be obtained by linearization
Figure BDA0002536075710000107
Figure BDA0002536075710000108
To represent
Figure BDA0002536075710000109
The (n) th element of (a),
Figure BDA00025360757100001010
representation matrix PR,k+1|kRow i and column j.
Due to tangential velocity measurement
Figure BDA00025360757100001011
Is non-informative, with standard deviation of measurement error
Figure BDA00025360757100001012
The value of (c) should be infinite. But may be valued based on an a priori estimate of the target tangential velocity standard deviation. Thus, the residual in the covariance of the measurement transformation error in the LOS coordinate system (e.g.
Figure BDA00025360757100001013
) Set to infinity, R is inverted using a matrix inversion methodRRedefining the formula (15) can be obtained.
Taking the orthogonality of the directional cosine matrix (the transpose of the direction cosine matrix is equal to the inverse matrix), the measurement conversion error covariance matrix is inversely transformed, and finally the measurement error covariance inverse matrix of the formula (14) in the step 4 is obtained.
Drawings
FIG. 1 is a schematic diagram of an interactive multi-model approach;
FIG. 2 is a diagram of a true trajectory of a target and a state estimation trajectory of the present invention in accordance with an embodiment of the present invention;
FIG. 3 is a diagram showing a target true track, an estimated trajectory of the algorithm (CMKFRR-IMM) of the present invention, and a trajectory of a comparison algorithm (single CV model, single CT model, interactive multi-model method CMKF-IMM without Doppler measurement) in accordance with an embodiment of the present invention;
FIG. 4 is a graph of the position RMSE of the CMKFRR-IMM algorithm and the CMKF-IMM algorithm in accordance with an embodiment of the present invention;
FIG. 5 is a velocity RMSE curve of the CMKFRR-IMM algorithm and the CMKF-IMM algorithm in the embodiment of the present invention;
FIG. 6 shows the model probability of the CMKFRR-IMM algorithm in an embodiment of the present invention.
Detailed Description
In consideration of tracking simulation of a maneuvering target with two motion states of constant speed (CV) and Constant Turning (CT) under a two-dimensional condition, the algorithm is a maneuvering target tracking method (CMKFRR-IMM) with Doppler measurement under a polar coordinate system, and is respectively compared with a single CV model, a single CT model and an interactive multi-model method (CMKF-IMM) without Doppler measurement. The specific kinetic time profiles of the experiments are shown in table 1. The initial position coordinates of the target are: (1000m 800m), initial velocity:
Figure BDA0002536075710000111
the fixed turning angle of the constant turning model is 8 degrees left turning. The radar sampling period is 1s, the measurement of the target comprises the measurement of the distance, the azimuth angle and the radial velocity of the target, and the correlation coefficient of the distance and the radial velocity is 0.9. Assuming that each measured noise is white Gaussian zero mean noise, the standard deviation of the noise is defined as shown in Table 2Shown in the figure. The number of monte carlo cycles for the entire simulation was 100.
TABLE 1 temporal distribution of motion states
Time of exercise 1s-60s 61s-75s 76s-125s 126s-140s 141s-200s
State of motion Uniform motion Constant turning Uniform motion Constant turning Uniform motion
TABLE 2 measurement of standard deviation of noise
Measure standard deviation Standard deviation of distance Standard deviation of azimuth Radial velocity standard deviation Standard deviation of tangential velocity
Numerical value 50m 0.2° 0.1m/s 10m/s
Fig. 2 shows the real track of the target and the state estimation track of the present invention, from which it can be seen that the real track of the target is more closely attached to the estimation track, and the deviation is not large when the target motion state is switched. FIG. 3 shows the true target track, the estimated track of the algorithm (CMKFRR-IMM) of the present invention and the track of the comparison algorithm (single CV model, single CT model, and interactive multi-model method CMKF-IMM without Doppler measurement), from which it can be seen that the single model target tracking algorithm cannot track the target well when the target motion state is not matched or changes, and therefore the position and speed root mean square error comparison of the single model is not considered hereinafter; a tracking algorithm (CMKF-IMM) which does not include Doppler measurement can basically track the maneuvering target, but is inferior to the CMKFRR-IMM algorithm in performance. FIGS. 4 and 5 show the RMSE curves for the position and velocity of the CMKFRR-IMM algorithm and the CMKF-IMM algorithm, respectively, from which it can be seen that the position and velocity root mean square errors of both algorithms can converge, but the CMKFRR-IMM converges to a smaller value; when the motion state is switched, the variation amplitude of the root mean square error of the CMKFRR-IMM algorithm is smaller, which shows that the method has better real-time performance on the maneuvering target tracking and is obviously superior to the traditional maneuvering target tracking method without Doppler measurement. The model probability of the CMKFRR-IMM algorithm is shown in FIG. 6, and it can be seen that the algorithm of the invention can accurately judge the motion state of the target, the resolution among different models is very clear, and when the motion state is switched, the algorithm can quickly distribute the model weight to the correct model, thereby ensuring the correctness of the algorithm.
And (4) carrying out result analysis: the maneuvering target tracking method (CMKFRR-IMM) containing Doppler measurement in a polar coordinate system can timely adjust model probability when the motion state is switched, has good tracking precision within the time that a target keeps a certain motion, and can better realize maneuvering target tracking compared with a single model and a maneuvering target tracking algorithm without Doppler measurement.
In conclusion, the CMKFRR-IMM algorithm provided by the invention is an effective maneuvering target tracking method comprising Doppler measurement in a polar coordinate system.

Claims (1)

1. Assume that the state of the model q at time k is estimated as
Figure FDA0002536075700000011
And an estimated error covariance of
Figure FDA0002536075700000012
The total number of the models is M; measurement information Z obtained at the moment of k +1k+1Including distance measurement
Figure FDA0002536075700000013
Azimuth angle measurement
Figure FDA0002536075700000014
And Doppler measurements
Figure FDA0002536075700000015
Measurement noise for range, azimuth and doppler measurements
Figure FDA0002536075700000016
And
Figure FDA0002536075700000017
is zero mean additive white Gaussian noise, and the measured variances are respectively
Figure FDA0002536075700000018
And
Figure FDA0002536075700000019
the filtering steps from the k moment to the k +1 moment of the maneuvering target tracking method containing Doppler measurement under a polar coordinate system are as follows:
1. calculating initial mixing probability and input state estimation interaction;
assuming that the model matched at time k is q and the model matched at time k +1 is r, the metrology data is Zk+1Under the condition (1), the initial mixing probability is:
Figure FDA00025360757000000110
Figure FDA00025360757000000111
wherein,
Figure FDA00025360757000000112
to normalize constant, piqrThe transition probability of the Markov model is expressed,
Figure FDA00025360757000000113
is the probability of the occurrence of the model q at time k;
and obtaining the input of the model r at the moment k +1 after interaction:
Figure FDA00025360757000000114
Figure FDA00025360757000000115
。
2. calculating unbiased measurement conversion at the moment of k + 1;
compensation factors for introducing multiplicative deviations
Figure FDA00025360757000000116
Obtaining unbiased measurement conversion at the moment k + 1:
Figure FDA00025360757000000117
wherein D is a direction cosine matrix:
Figure FDA00025360757000000118
。
3. obtaining a weighted result of one-step prediction of the target state by using the model probability;
one-step prediction of model r is:
Figure FDA00025360757000000119
Figure FDA0002536075700000021
wherein
Figure FDA0002536075700000022
A state transition matrix for model r;
the weighted result of the target one-step prediction is:
Figure FDA0002536075700000023
Figure FDA0002536075700000024
。
4. computing an inverse matrix of prediction information based metrology error covariance
Figure FDA0002536075700000025
And (3) estimating parameters under an LOS coordinate system by rotating the predicted target state and the covariance:
Figure FDA0002536075700000026
Figure FDA0002536075700000027
wherein the orientation is predicted
Figure FDA0002536075700000028
Is composed of
Figure FDA0002536075700000029
Obtaining a measurement error covariance inverse matrix based on the prediction information
Figure FDA00025360757000000210
Figure FDA00025360757000000211
Wherein
Figure FDA00025360757000000212
Comprises the following steps:
Figure FDA00025360757000000213
RRare respectively as follows:
Figure FDA00025360757000000214
Figure FDA00025360757000000215
Figure FDA00025360757000000216
Figure FDA0002536075700000031
Figure FDA0002536075700000032
Figure FDA0002536075700000033
wherein
Figure FDA0002536075700000034
To predict the azimuthal variance, it can be obtained by linearization
Figure FDA0002536075700000035
Figure FDA0002536075700000036
To represent
Figure FDA0002536075700000037
The (n) th element of (a),
Figure FDA0002536075700000038
representation matrix PR,k+1kRow i and column j.
5. Performing information Kalman filtering (taking the model r as an example) on each model;
obtaining the covariance of the input state estimation error of the model r at the moment k +1 after interaction in the step 1
Figure FDA0002536075700000039
One-step prediction error covariance matrix of available model r
Figure FDA00025360757000000310
Comprises the following steps:
Figure FDA00025360757000000311
wherein
Figure FDA00025360757000000312
And
Figure FDA00025360757000000313
a state transition matrix, a process noise driving matrix and a process noise matrix of the k +1 moment model r respectively;
filter gain of model r
Figure FDA00025360757000000314
Comprises the following steps:
Figure FDA00025360757000000315
the state update information matrix is:
Figure FDA00025360757000000316
Figure FDA00025360757000000317
。
6. updating model probability and estimating final state;
the information filtering likelihood function of the model r is:
Figure FDA00025360757000000318
where det (-) is the determinant of the matrix,
Figure FDA00025360757000000319
an innovation error matrix of the model r is shown, and an inverse matrix expression of the innovation error matrix is as follows:
Figure FDA0002536075700000041
wherein,
Figure FDA0002536075700000042
and is
Figure FDA0002536075700000043
The probability of model r is updated as:
Figure FDA0002536075700000044
wherein c is a normalization constant:
Figure FDA0002536075700000045
the final state estimation and covariance obtained by model probability weighting are respectively:
Figure FDA0002536075700000046
Figure FDA0002536075700000047
。
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