CN110417378B - Gravity value estimation method based on cold atom interference type gravity meter - Google Patents

Gravity value estimation method based on cold atom interference type gravity meter Download PDF

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CN110417378B
CN110417378B CN201910499629.8A CN201910499629A CN110417378B CN 110417378 B CN110417378 B CN 110417378B CN 201910499629 A CN201910499629 A CN 201910499629A CN 110417378 B CN110417378 B CN 110417378B
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张柳青
胡正珲
朱栋
吴彬
程冰
林强
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Abstract

The gravity value estimation method based on the cold atom interference type gravity meter comprises the following steps: the method comprises the steps of setting an initial value, calculating sigma points, estimating a one-step predicted value of a state vector and an error covariance square root matrix, estimating the statistical characteristics of measurement noise, estimating a one-step predicted value of the measurement vector, the error covariance square root matrix and a cross covariance matrix, and calculating a Kalman gain matrix to obtain a state vector estimated value and a corresponding estimated error variance matrix. The atomic gravimeter obtains a phase value by generating and fitting interference fringes, thereby obtaining gravitational acceleration. The cosine fitting principle is a least square method, and the statistical characteristics of observed data and noise are not considered, so that the gravity acceleration obtained directly by cosine fitting may have deviation. The self-adaptive square root unscented Kalman filtering method is introduced into the gravity value estimation, and the statistical characteristics of observed data and noise can be considered, so that more accurate gravity acceleration is estimated.

Description

Gravity value estimation method based on cold atom interference type gravity meter
Technical Field
The invention relates to a gravity value estimation method based on a cold atom interference type gravity meter.
Background
In general, one often regards the earth gravitational acceleration g as a constant value (9.8 m/s 2 ) In practice, the change in the gravitational acceleration of the earth is very complex and can vary with time and space. At present, a plurality of gravimeters with different principles can accurately measure the gravitational acceleration, and even the accuracy can reach 2uGal. Over the last twenty years, quantum sensors based on atomic interferometry have evolved rapidly, such as atomic gravimeters, which is oneThe sensitivity and precision of the novel absolute gravimeter are comparable with those of the traditional gravimeter.
The atomic gravimeter realizes interference of atomic substance wave through three Raman pulses, scans Raman light frequency to compensate Doppler frequency shift induced by gravity, changes scanning slope (chirp rate alpha) of the Raman light frequency to enable phase change of interference fringes, measures population number of atoms on two states to obtain atomic interference fringes, and then extracts phase through cosine fitting the atomic interference fringes to obtain gravitational acceleration. The cosine fitting principle is a least square method, and the statistical characteristics of observed data and noise are not considered, so that the gravity acceleration obtained directly by cosine fitting may have deviation.
Disclosure of Invention
The invention aims to overcome the defect of cosine fitting and find a gravity acceleration value estimation method which is more in line with the actual situation. Considering that the gravity value estimation function is a nonlinear function, the invention provides that an adaptive square root unscented Kalman filtering method is introduced into the gravity value estimation on the basis of cosine fitting so as to obtain more accurate gravity acceleration.
The invention discloses a gravity value estimation method based on a cold atom interference type gravity meter, which comprises the following steps:
step 1, establishing a state equation and a measurement equation:
Figure BDA0002089776070000021
wherein x is a system state vector, y is a measurement vector, f (x k ) As a nonlinear system state function, h (x k ) As a nonlinear measurement function, w k Is system process noise, is Gaussian white noise with mean value of 0 and covariance of Q, v k The measurement noise is Gaussian white noise with mean value R and covariance R.
Step 2, initializing a system state: order the
Figure BDA0002089776070000022
For->
Figure BDA0002089776070000023
Performing Cholesky decomposition>
Figure BDA0002089776070000024
chol (·) represents Cholesky decomposition; />
Figure BDA0002089776070000025
Figure BDA0002089776070000026
Is a matrix->
Figure BDA0002089776070000027
Is the square root of the lower triangular matrix; in the filtering process, use->
Figure BDA0002089776070000028
Replace->
Figure BDA0002089776070000029
And the non-negative qualitative and symmetry of the matrix can be ensured by performing operation.
Step 3, calculating sigma point at k-1 time,
Figure BDA00020897760700000210
where k represents the time in which,
Figure BDA00020897760700000211
n is the dimension of x, λ=α 2 (n+kappa) -n, alpha is a positive scale factor, 10 -4 Alpha is more than or equal to 1, kappa is a proportion parameter, and 0 or 3-n is taken; />
Figure BDA00020897760700000212
Representation matrix->
Figure BDA00020897760700000213
Is the j-th column of (2).
Step 4, updating time: the sigma point is transmitted through a nonlinear state function f (& gt) to obtain a state vector in one stepPrediction
Figure BDA00020897760700000214
Error covariance square root matrix->
Figure BDA00020897760700000215
Figure BDA0002089776070000031
Wherein the method comprises the steps of
Figure BDA0002089776070000032
W in the formula m ，W c The weights when the mean and variance are calculated, respectively; for gaussian distribution, β=2 is optimal; q is the process noise covariance. QR (·) represents QR decomposition of the matrix, cholupdate (S, u, + -1) represents Cholesky update of the lower triangular matrix, corresponding to calculation of chol (SS) T ±uu T )。
And 5, measuring and updating: by means of
Figure BDA0002089776070000033
And->
Figure BDA0002089776070000034
Again the sigma point is calculated and,
Figure BDA0002089776070000035
propagating sigma points through a nonlinear measurement function h (), and obtaining one-step prediction of a measurement vector
Figure BDA0002089776070000036
Error covariance square root matrix->
Figure BDA0002089776070000037
And cross covariance matrix->
Figure BDA0002089776070000038
Figure BDA0002089776070000039
Estimating the statistical characteristics of the measured noise R:
Figure BDA00020897760700000310
wherein the method comprises the steps of
Figure BDA0002089776070000041
Is a forgetting factor.
When (when)
Figure BDA0002089776070000042
When the square matrix is not the semi-positive square matrix, the variance matrix updating strategy of the measurement noise is as follows:
Figure BDA0002089776070000043
wherein l k Is an adjustment factor and is related to a state error variance matrix; p=1, taking p=p+1 loop execution as needed
Figure BDA0002089776070000044
Is semi-positive array.
Step 6, in obtaining a new measurement y k Then, filtering and updating are carried out to obtain the estimated value of the state vector
Figure BDA0002089776070000045
And corresponding estimation error variance matrix>
Figure BDA0002089776070000046
Figure BDA0002089776070000047
Wherein ε is k For residual sequences, K k Is a filter gain matrix.
The atomic gravimeter obtains a phase value by generating and fitting interference fringes, thereby obtaining gravitational acceleration. The cosine fitting principle is a least square method, and the statistical characteristics of observed data and noise are not considered, so that the gravity acceleration obtained directly by cosine fitting may have deviation. The invention introduces the self-adaptive square root unscented Kalman filtering method into the gravity value estimation, and can consider the statistical characteristics of observed data and noise, thereby estimating more accurate gravity acceleration. The method comprises the steps of setting an initial value, calculating sigma points, estimating a one-step predicted value of a state vector and an error covariance square root matrix, estimating statistical characteristics of measured noise, estimating the one-step predicted value of the measured vector, the error covariance square root matrix and the cross covariance matrix, and calculating a Kalman gain matrix to obtain the state vector estimated value and a corresponding estimated error variance matrix.
The beneficial effects of the invention are as follows:
the statistical characteristics of observed data and noise can be considered more through the self-adaptive square root unscented Kalman filtering, the minimum mean square error is taken as the optimal criterion of estimation, the optimal estimated value of the previous moment and the observed value of the current moment are utilized to update the estimation of the state variable, and the optimal estimated value of the current moment is obtained. And the self-adaptive square root unscented Kalman filtering can estimate the mean value and variance of the noise when the statistical characteristics of the measured noise are unknown, and can also overcome the problem of calculation divergence of the unscented Kalman filtering. By using the method in the gravity value estimation, more accurate gravity acceleration can be obtained.
Drawings
FIG. 1 is a flow chart of adaptive square root unscented Kalman filtering.
Fig. 2 is a cosine fit atomic interference fringe.
Fig. 3 shows k P (3) values after filtering, and P (3) obtained by mean and cosine fitting.
Fig. 4 shows the filtered y values and measured values and fitting values.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings.
The invention discloses a gravity value estimation method based on a cold atom interference type gravity meter, which comprises the following steps:
the following calculations are performed for i=1, 3,5, (when the chirp rate is negative) or i=2, 4,6, (when the chirp rate is positive) interference fringes:
step 1, establishing a state equation and a measurement equation:
Figure BDA0002089776070000061
the state variable is x= [ P (1) P (2) P (3) P (4)] T The equation of state is x k ＝x k-1 +w k-1 . The measurement equation is y k ＝P(4)+P(1)*[1-cos(2π(-α x -P(3)-α 0 )*P(2) 2 )]+v k Or y k ＝P(4)+P(1)*[1-cos(2π(α x -P(3)+α 0 )*P(2) 2 )]+v k And respectively represent the measurement equation when the chirp rate is positive and negative. y is k Is the atomic population, alpha 0 Alpha is the position of the approximate chirp rate alpha corresponding to absolute gravity x For the chirp rate changed during scanning, the gravity value can be obtained by finding the offset coordinate P (3).
Step 2, initializing a system state:
Figure BDA0002089776070000062
Figure BDA0002089776070000063
representing the estimated value of the ith stripe obtained by cosine fitting as the initial state value of the filtering,/>
Figure BDA0002089776070000064
The average of the estimates of all fringes obtained by cosine fitting at each measurement is shown. For->
Figure BDA0002089776070000065
The decomposition of Cholesky was performed and,
Figure BDA0002089776070000066
fig. 2 is a cosine fit interference fringe when the chirp rate is negative.
Step 3, calculating sigma point at k-1 time,
Figure BDA0002089776070000067
where k represents the time in which,
Figure BDA0002089776070000068
λ＝α 2 (n+κ)-n，n＝4，κ＝0，α＝1。
step 4, updating time: the sigma points are propagated through a state function. But because the state function is a linear function in this example, x k ＝x k-1 Therefore, step 3 can be omitted, and the formula (4) is simplified to obtain one-step prediction of the state vector
Figure BDA0002089776070000069
Error covariance square root matrix->
Figure BDA00020897760700000610
Equation (4) can be reduced to
Figure BDA00020897760700000611
Q is the process noise covariance, taken as q=diag ([ 7e-5,1e-2,5e-12,1e-4 ]).
And 5, measuring and updating: by means of
Figure BDA0002089776070000071
And->
Figure BDA0002089776070000072
Again the sigma point is calculated and,
Figure BDA0002089776070000073
propagating sigma points through a nonlinear measurement function h (), and obtaining one-step prediction of a measurement vector
Figure BDA0002089776070000074
Error covariance square root matrix->
Figure BDA0002089776070000075
And cross covariance matrix->
Figure BDA0002089776070000076
Figure BDA0002089776070000077
Wherein the method comprises the steps of
Figure BDA0002089776070000078
W in the formula m ，W c The weights when the mean and variance are calculated, respectively; beta=2.
Estimating the statistical characteristics of the measured noise R:
Figure BDA0002089776070000079
wherein the method comprises the steps of
Figure BDA00020897760700000710
As forgetting factor, in this example, noise is measured>
Figure BDA00020897760700000711
Taking the variance of the cosine fit residual error of each stripe,>
Figure BDA00020897760700000712
taking the residual error level of each stripe after cosine fittingAnd (5) an average value.
When (when)
Figure BDA00020897760700000713
When the square matrix is not the semi-positive square matrix, the variance matrix updating strategy of the measurement noise is as follows:
Figure BDA0002089776070000081
wherein l k Is an adjustment factor and is related to a state error variance matrix; p=1, taking p=p+1 loop execution as needed
Figure BDA0002089776070000082
Is semi-positive array.
Step 6, in obtaining a new measurement y k Then, filtering and updating are carried out to obtain the estimated value of the state vector
Figure BDA0002089776070000083
And corresponding estimation error variance matrix>
Figure BDA0002089776070000084
Figure BDA0002089776070000085
Wherein ε is k For residual sequences, K k Is a filter gain matrix.
And 7, k P (3) can be obtained by utilizing the steps, wherein the comparison of the filtered y value, the measured value and the fitting value is shown in fig. 4.k pieces of P (3) are averaged as P (3) of the ith interference fringe. P (3) obtained by combining one-time positive and negative scan estimation + And P (3) - The systematic errors can be eliminated to obtain the gravitational acceleration,
Figure BDA0002089776070000086
compared with the gravity acceleration obtained by direct calculation of cosine fitting, the self-adaptive square root unscented KalrThe gravity acceleration after the Mannich filter processing is closer to the reference gravity acceleration, which proves that the method has a certain effect.
The embodiments described in the present specification are merely examples of implementation forms of the inventive concept, and the scope of protection of the present invention should not be construed as being limited to the specific forms set forth in the embodiments, and the scope of protection of the present invention and equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims (1)

1. A gravity value estimation method based on a cold atom interference type gravity meter comprises the following steps:
the following calculations are performed for i interference fringes, where i=1, 3,5 when the chirp rate is negative; chirp rate is positive, i=2, 4,6,;
step 1, establishing a state equation and a measurement equation:
Figure FDA0004209843030000011
wherein x is a system state vector, x= [ P (1) P (2) P (3) P (4)] T The equation of state is x k ＝x k-1 +w k-1 The method comprises the steps of carrying out a first treatment on the surface of the y is the measurement vector, and the measurement equation is y k ＝P(4)+P(1)*[1-cos(2π(-α x -P(3)-α 0 )*P(2) 2 )]+v k Or y k ＝P(4)+P(1)*[1-cos(2π(α x -P(3)+α 0 )*P(2) 2 )]+v k Respectively representing measurement equations when the chirp rate is positive and negative; y is k Is the atomic population, alpha 0 Is the chirp rate alpha position corresponding to absolute gravity, alpha x For the chirp rate changed in the scanning process, finding an offset coordinate P (3) to obtain a gravity value; f (x) k ) As a nonlinear system state function, h (x k ) As a nonlinear measurement function, w k Is system process noise, is Gaussian white noise with mean value of 0 and covariance of Q, v k The Gaussian white noise with the mean value of R and the covariance of R is used for measuring the noise;
step 2, initializing a system state: order the
Figure FDA0004209843030000012
I.e.
Figure FDA0004209843030000013
For->
Figure FDA0004209843030000014
Performing Cholesky decomposition>
Figure FDA0004209843030000015
chol (·) represents Cholesky decomposition; />
Figure FDA0004209843030000016
Figure FDA0004209843030000017
Is a matrix->
Figure FDA0004209843030000018
Is the square root of the lower triangular matrix; in the filtering process, use->
Figure FDA0004209843030000019
Replace->
Figure FDA00042098430300000110
The non-negative qualitative and symmetry of the matrix can be ensured by performing operation; />
Figure FDA00042098430300000111
Representing the estimated value of the ith stripe obtained by cosine fitting as the initial state value of the filtering,/>
Figure FDA00042098430300000112
Representing the average value of the estimated values obtained by cosine fitting all stripes at each measurement;
step 3, calculating sigma points at the time of k-1:
Figure FDA0004209843030000021
where k represents the time in which,
Figure FDA0004209843030000022
n is the dimension of x, λ=α 2 (n+kappa) -n, alpha is a positive scale factor, 10 -4 Alpha is more than or equal to 1, kappa is a proportion parameter, and 0 or 3-n is taken; />
Figure FDA0004209843030000023
Representation matrix->
Figure FDA0004209843030000024
Is the j-th column of (2);
step 4, updating time: propagating sigma points through a nonlinear state function f (& gt) to obtain a state vector one-step prediction
Figure FDA0004209843030000025
Error covariance square root matrix->
Figure FDA0004209843030000026
Figure FDA0004209843030000027
Wherein the method comprises the steps of
Figure FDA0004209843030000028
W in the formula m ，W c The weights when the mean and variance are calculated, respectively; for gaussian distribution, β=2 is optimal; q is the process noise covariance; QR (·) represents QR decomposition of the matrix, cholupdate (S, u, + -1) represents Cholesky update of the lower triangular matrix, corresponding to calculation of chol (SS) T ±uu T )；
And 5, measuring and updating: by means of
Figure FDA0004209843030000029
And->
Figure FDA00042098430300000210
Again the sigma point is calculated and,
Figure FDA00042098430300000211
propagating sigma points through a nonlinear measurement function h (), and obtaining one-step prediction of a measurement vector
Figure FDA00042098430300000212
Error covariance square root matrix->
Figure FDA00042098430300000213
And cross covariance matrix->
Figure FDA00042098430300000214
Figure FDA0004209843030000031
Figure FDA0004209843030000032
Figure FDA0004209843030000033
Figure FDA0004209843030000034
Figure FDA0004209843030000035
Estimating the statistical characteristics of the measured noise R:
Figure FDA0004209843030000036
wherein the method comprises the steps of
Figure FDA0004209843030000037
Is a forgetting factor;
when (when)
Figure FDA0004209843030000038
When the square matrix is not the semi-positive square matrix, the variance matrix updating strategy of the measurement noise is as follows:
Figure FDA0004209843030000039
wherein l k Is an adjustment factor and is related to a state error variance matrix; p=1, taking p=p+1 loop execution as needed
Figure FDA00042098430300000310
Is a semi-positive array;
step 6, in obtaining a new measurement y k Then, filtering and updating are carried out to obtain the estimated value of the state vector
Figure FDA00042098430300000311
And corresponding estimation error variance matrix>
Figure FDA00042098430300000312
Figure FDA0004209843030000041
Wherein ε is k For residual sequences, K k Is a filter gain matrix;
step 7, obtaining k P (3) by utilizing the steps, and comparing the y value obtained after filtering with the measured value and the fitting value; k P (3) are averaged to be P (3) of the ith interference fringe; p (3) obtained by combining one-time positive and negative scan estimation + And P (3) - The system error is eliminated to obtain the gravitational acceleration,
Figure FDA0004209843030000042
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