CN111142358B - Deep space autonomous time keeping method based on natural X-ray source observation - Google Patents

Deep space autonomous time keeping method based on natural X-ray source observation Download PDF

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CN111142358B
CN111142358B CN202010063381.3A CN202010063381A CN111142358B CN 111142358 B CN111142358 B CN 111142358B CN 202010063381 A CN202010063381 A CN 202010063381A CN 111142358 B CN111142358 B CN 111142358B
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pulsar
timing
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CN111142358A (en
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宋诗斌
王海霞
卢晓
张晓玉
张治国
盛春阳
梁笑
聂君
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Shandong University of Science and Technology
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    • G04—HOROLOGY
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Abstract

The invention discloses a deep space autonomous time keeping method based on natural X-ray source observation, which corrects a satellite-borne clock error in an on-orbit manner by utilizing the periodic stability of an X-ray pulsar signal. The invention optimizes a pulsar timing observation model, models timing observation noise into the combination of observation error noise and timing noise, and provides a representation model of timing noise level; establishing an evolution model of a satellite-borne clock based on an improved GM model, and modeling constant parameters of the GM model into variables related to time; based on a timing observation model and a clock evolution model, the UKF is used for estimating the satellite-borne clock state, the purpose of correcting clock errors by using pulsar timing observation is achieved, and deep space autonomous time keeping based on natural X-ray source observation is achieved. The invention corrects the satellite-borne clock error in an on-orbit manner by using the signals radiated by the natural X-ray pulsar, gets rid of the dependence on a ground time system, has strong autonomy and also reduces the establishment and maintenance cost of the deep space time reference system.

Description

Deep space autonomous time keeping method based on natural X-ray source observation
Technical Field
The invention relates to a time keeping method in the field of aerospace application, in particular to a deep space autonomous time keeping method based on natural X-ray source observation.
Background
The time reference is critical information for deep space exploration. The deep space exploration range is gradually expanded, and the deep space exploration range is expanded from early moon exploration to Mars, asteroids and solar systems, and can be extended to the outside of the solar systems in the future. The extension of the detection range puts higher demands on the maintenance of the local time of the spacecraft. The existing deep space exploration relies on a ground time system, a satellite-borne clock is corrected through the ground time system, a large amount of manpower and material resources are consumed for maintaining the ground time system, and meanwhile, the ground time system is utilized for time distribution and correction of the deep space system, so that the defects of long delay time, high possibility of interference of a space complex electromagnetic environment and the like are overcome.
The X-ray pulsar is a naturally-existing star body, two poles of the X-ray pulsar can radiate X-ray signals in the rotating process, and the X-ray pulsar is a natural X-ray source. The autorotation period of the pulsar has high stability, and the period stability of part of pulsars is comparable to that of an atomic clock. Meanwhile, the pulsar is a naturally existing celestial body, is not interfered by other factors, has long service life, can continuously output stable signal pulses, has wide signal radiation range and can receive signals outside a solar system or even a solar system. These good characteristics of the pulsar signal enable the pulsar to serve as a good time reference in deep space.
Disclosure of Invention
Aiming at the problems that the existing deep space exploration depends on a ground time system, a large amount of manpower and material resources are consumed for maintenance, and the like, the invention provides a deep space autonomous time keeping method based on natural X-ray source observation, which is used for providing autonomous satellite-borne time keeping service for a spacecraft or a detector for deep space exploration.
The invention adopts the following technical scheme:
a deep space autonomous time keeping method based on natural X-ray source observation comprises the following steps:
step 1: establishing a timing observation model of the X-ray source, modeling observation noise in the timing observation process into the composition of observation error noise and timing noise, and providing a characterization method of the timing noise level;
step 2: the method comprises the steps that a satellite-borne clock evolution model is expressed based on an improved GM model, a gray acting quantity in a traditional GM model is modeled into a parameter related to time, an initial error in the traditional GM model is replaced by a clock error value at the previous moment from a constant, and the clock error value at the previous moment changes along with the time, so that the dependence of the clock evolution model based on the traditional GM model on the initial error is weakened;
and step 3: and correcting the satellite-borne clock by using unscented Kalman filtering based on a timing observation model and a satellite-borne clock evolution model to realize satellite-borne self-timing.
Preferably, step 1 specifically comprises:
based on long-term observation of pulsar signals, a pulsar phase evolution model is established at the position of the solar system mass center:
Figure BDA0002375197870000021
where t represents the current time, t0In order to refer to the epoch, it is,
Figure BDA0002375197870000022
is t0The initial phase of the time, f is the rotation frequency of the pulsar,
Figure BDA0002375197870000023
is the first derivative of f and is,
Figure BDA0002375197870000024
representing the second derivative of f, wherein epsilon (t) represents a neglected high-order term, and the pulse arrival time at a certain moment can be predicted through a pulsar phase evolution model;
accurately predicting the pulsar pulse phase reaching the SSB by using a pulsar phase evolution model;
observing the arrival time t of pulsar signalsobsIs the arrival time t at the spacecraft that needs to be converted to the centroid of the solar systemSSBThe conversion process is expressed by equation (2):
Figure BDA0002375197870000025
where δ t is the satellite-borne clock error, μsDenotes the gravitational constant of the sun, c is the speed of light, D0Observing the distance of the pulsar for the sun centroid distance, n representing the unit vector of the observed pulsar radiation direction, v representing the timing observation noise, rSSBFor the vector of the solar system centroid pointing to the spacecraft position, | rSSBL is rSSBB represents a vector from the sun's centroid mass to the solar system's centroid, | b | is the length of b;
definition of
Figure BDA0002375197870000026
To predict the time of arrival of the resulting pulsar signal using equation (1), the timing residual can be described as:
Figure BDA0002375197870000027
the timing observation model of the X-ray source is then expressed as:
z＝ζ+v (4)
if m pulsar is observed at the same time, the timing observation model is as follows:
Figure BDA0002375197870000028
wherein H is an observation matrix, and the observed quantity is Z ═ Z1,…,zm]TV is observation noise, and is expressed as V ═ V1,…,vm]TV has a covariance matrix of RO，ROIs the observation noise matrix of the observation model;
the timing observation noise is modeled as the combination of observation error noise and timing noise as follows:
let n beO,iRepresenting observation error noise, nt,iRepresenting the timing noise, the timing observation noise is:
Figure BDA0002375197870000036
wherein n isviTiming observation noise, n, representing the ith pulsarO,iAnd nt,iGaussian noise with zero mean;
the observed error noise level is characterized by equations (7) and (8) with nO,iHas a standard deviation ofO,iThen, then
Figure BDA0002375197870000031
Figure BDA0002375197870000032
In the formula, SNR represents the signal-to-noise ratio of the X-ray pulsar signal, BxRepresenting background X-ray radiation flux intensity, F, in the observation environmentxFor the observed pulse signal radiation flux intensity, pfIs the pulse ratio of the pulsar signal, A is the effective area of the detector, tobsD is the pulse duty ratio of the pulsar signal, and d can be expressed as d ═ W/P, W is the pulsar pulse width, and P is the pulse period;
the timing noise level is characterized as:
Figure BDA0002375197870000033
in the formula, C, alpha, beta and gamma are parameters obtained by fitting data obtained by observing a large number of pulsar for a long time.
Preferably, step 2 specifically comprises:
the improved GM model was:
Figure BDA0002375197870000034
Figure BDA0002375197870000035
therein, ζk+1Indicates the clock error at time k +1, ζkThe clock error at the moment k is shown, a is the development coefficient of the GM model, and a is a positive number smaller than 1; u. ofk+1、ukIs the amount of gray contribution of the GM model, in the conventional GM model, ukIs a constant, is modeled as a variable that varies with time, the course of which is represented by the formula (11), b2Is constant and is obtained by data fitting;
make xi ═ ζ, u]TRepresenting state variables of the system, evolution of the satellite-borne clockThe model can be expressed as:
ξk+1＝Φkξk+Wp (12)
Φkbeing a state transition matrix, Wp＝[wζ,wu]TBeing process noise, wζAnd wuIs a zero-mean gaussian noise that is,
Figure BDA0002375197870000041
is the noise wζThe variance of (a) is determined,
Figure BDA0002375197870000042
denotes wuOf a covariance matrix of
Figure BDA0002375197870000043
diag () represents a diagonal matrix.
Preferably, step 3 specifically comprises:
step 3.1: first, the sigma point is calculated:
Figure BDA0002375197870000044
in the formula, the first step is that,
Figure BDA0002375197870000045
represents the optimal estimate of time k-1, ()iRepresenting the ith row of the matrix, n is twice the number of state variables, k represents the scale parameter of the model, Pξ,k-1Indicating the time of k-1
Figure BDA0002375197870000046
The covariance matrix of (a);
step 3.2: calculating the one-step state transition of sigma point in formula (13)
Figure BDA0002375197870000047
Figure BDA0002375197870000048
Weighted mean value after one-step state transition of sigma point
Figure BDA0002375197870000049
Sum covariance matrix
Figure BDA00023751978700000410
The calculation is as follows:
Figure BDA00023751978700000411
Figure BDA00023751978700000412
wherein, Wi m、Wi cRepresenting a weighting coefficient, which is calculated by the following equation:
Figure BDA00023751978700000413
wherein alpha is a positive number smaller than 1, lambda represents a proportionality coefficient, 0 or 3-n is taken, n is the number of state variables, beta is a state distribution parameter and is set as 0;
step 3.3: weighted observation result of sigma point after one-step transfer
Figure BDA00023751978700000414
Expressed as:
Figure BDA00023751978700000415
in the formula, the first step is that,
Figure BDA00023751978700000416
representing the sigma observation result after the one-step transfer, wherein H is an observation matrix;
step 3.4:
Figure BDA00023751978700000417
for estimating the effect of (2) using a covariance matrix Pz,kRepresents:
Figure BDA00023751978700000418
ROto observe the noise matrix;
in addition, the cross-covariance matrix is defined as:
Figure BDA0002375197870000051
the filter gain of the UKF is then calculated as:
Figure BDA0002375197870000052
step 3.5: optimal estimation of clock states
Figure BDA0002375197870000053
The calculation is as follows:
Figure BDA0002375197870000054
zksystem observation results obtained based on observation model for time k
At the same time, update
Figure BDA0002375197870000055
Covariance matrix of (2):
Figure BDA0002375197870000056
the invention has the beneficial effects that:
the method has the advantages that the satellite-borne clock error is corrected on track by utilizing the stability of the natural X-ray source-X-ray pulse signal period and through the timing observation of the natural X-ray source, the dependence on a foundation time system is eliminated, the implementation and maintenance cost is low, the autonomy is high, and the method which is low in implementation difficulty and maintenance cost and high in autonomy is provided for the long-distance and long-time deep space task to be in time.
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FIG. 1 is a schematic block diagram of a deep space autonomous time keeping method based on natural X-ray source observation.
Detailed Description
The following description of the embodiments of the present invention will be made with reference to the accompanying drawings:
with reference to fig. 1, a deep space autonomous time keeping method based on natural X-ray source observation includes the following steps:
step 1: establishing a timing observation model of the X-ray source, modeling timing observation noise into the combination of observation error noise and timing noise, and providing a characterization method of the timing noise level.
Based on the long-term observation of pulsar signals, a pulsar phase evolution model is established at the position of the solar system mass center:
Figure BDA0002375197870000057
where t represents the current time, t0In order to refer to the epoch, it is,
Figure BDA0002375197870000058
is t0The initial phase of the time, f is the rotation frequency of the pulsar,
Figure BDA0002375197870000059
is the first derivative of f and is,
Figure BDA00023751978700000510
the second derivative of f is shown, epsilon (t) shows a neglected high-order term, and the pulse arrival time at a certain moment can be predicted through a pulsar phase evolution model.
And accurately predicting the pulsar pulse phase reaching the SSB by using a pulsar phase evolution model.
Accurately predicting the pulsar pulse phase reaching the SSB by using a pulsar phase evolution model;
observing the arrival time t of pulsar signalsobsIs the arrival time t at the spacecraft that needs to be converted to the centroid of the solar systemSSBThe conversion process is expressed by equation (2):
Figure BDA0002375197870000061
where δ t is the satellite-borne clock error, μsDenotes the gravitational constant of the sun, c is the speed of light, D0Observing the distance of the pulsar for the sun centroid distance, n representing the unit vector of the observed pulsar radiation direction, v representing the timing observation noise, rSSBFor the vector of the solar system centroid pointing to the spacecraft position, | rSSBL is rSSBB represents a vector from the sun's centroid mass to the solar system's centroid, | b | is the length of b.
Definition of
Figure BDA0002375197870000062
To predict the time of arrival of the resulting pulsar signal using equation (1), the timing residual can be described as:
Figure BDA0002375197870000063
the timing observation model of the X-ray source is then expressed as:
z＝ζ+v (4)
if m pulsar is observed at the same time, the timing observation model is as follows:
Figure BDA0002375197870000064
wherein H is an observation matrix, and the observed quantity is Z ═ Z1,…,zm]TV is observation noise, and is expressed as V ═ V1,…,vm]TV has a covariance matrix of RO，ROIs the observation noise matrix of the observation model.
The modeling of timing observation noise as a composite of observation error noise and timing noise can be described as:
let n beO,iRepresenting observation error noise, nt,iRepresenting timing noise, then the timing observation noise can be represented as:
Figure BDA0002375197870000065
wherein the content of the first and second substances,
Figure BDA0002375197870000066
timing observation noise, n, representing the ith pulsarO,iAnd nt,iGaussian noise with zero mean.
The observed error noise level can be characterized by equations (7) and (8) with nO,iHas a standard deviation ofO,iThen, then
Figure BDA0002375197870000071
Figure BDA0002375197870000072
In the formula, SNR represents the signal-to-noise ratio of the X-ray pulsar signal, BxRepresenting background X-ray radiation flux intensity, F, in the observation environmentxFor the observed pulse signal radiation flux intensity, pfIs the pulse ratio of the pulsar signal, A is the effective area of the detector, tobsFor the signal observation time, d is the pulse duty ratio of the pulsar signal, and d can be expressed as d ═ W/P, where W is the pulsar pulse width and P is the pulse period.
The timing noise level is characterized as:
Figure BDA0002375197870000073
in the formula, C, alpha, beta and gamma are parameters obtained by fitting data obtained by observing a large number of pulsar for a long time.
Step 2: representing the space-borne clock evolution model based on an improved GM model, wherein the improved GM model can be represented as follows:
Figure BDA0002375197870000074
uk+1＝uk+b2k (11)
therein, ζk+1Indicates the clock error at time k +1, ζkThe clock error at the moment k is shown, a is the development coefficient of the GM model, and a is a positive number smaller than 1; u. ofk+1、ukIs the amount of gray contribution of the GM model, in the conventional GM model, ukIs a constant, is modeled as a variable that varies with time, the course of which is represented by the formula (11), b2Is constant and is obtained by data fitting. By improving, overcome the model and rely on the initial error and the defect that the error is cumulative over time.
Make xi ═ ζ, u]TRepresenting the state variables of the system, the satellite-borne clock evolution model can be expressed as:
ξk+1＝Φkξk+Wp (12)
Φkbeing a state transition matrix, Wp＝[wζ,wu]TBeing process noise, wζAnd wuIs a zero-mean gaussian noise that is,
Figure BDA0002375197870000075
is the noise wζThe variance of (a) is determined,
Figure BDA0002375197870000076
denotes wuOf a covariance matrix of
Figure BDA0002375197870000077
diag () represents a diagonal matrix.
And step 3: based on a timing observation model and a satellite-borne clock evolution model, an Unscented Kalman Filtering (UKF) is used for correcting a satellite-borne clock, so that satellite-borne self-timing is realized.
The specific process comprises the following steps:
step 3.1: first, the sigma point is calculated:
Figure BDA0002375197870000081
in the formula, the first step is that,
Figure BDA0002375197870000082
represents the optimal estimate of time k-1, ()iRepresenting the ith row of the matrix, n is twice the number of state variables, k represents the scale parameter of the model, Pξ,k-1Indicating the time of k-1
Figure BDA0002375197870000083
The covariance matrix of (a);
step 3.2: calculating the one-step state transition of sigma point in formula (13)
Figure BDA0002375197870000084
Figure BDA0002375197870000085
Weighted mean value after one-step state transition of sigma point
Figure BDA0002375197870000086
Sum covariance matrix
Figure BDA0002375197870000087
The calculation is as follows:
Figure BDA0002375197870000088
Figure BDA0002375197870000089
wherein, Wi m、Wi cRepresenting a weighting coefficient, which is calculated by the following equation:
Figure BDA00023751978700000810
wherein alpha is a positive number smaller than 1, lambda represents a proportionality coefficient, 0 or 3-n is taken, n is the number of state variables, beta is a state distribution parameter and is set as 0;
step 3.3: weighted observation result of sigma point after one-step transfer
Figure BDA00023751978700000811
Expressed as:
Figure BDA00023751978700000812
in the formula, the first step is that,
Figure BDA00023751978700000813
representing the sigma observation result after the one-step transfer, wherein H is an observation matrix;
step 3.4:
Figure BDA00023751978700000814
for estimating the effect of (2) using a covariance matrix Pz,kRepresents:
Figure BDA00023751978700000815
ROto observe the noise matrix;
in addition, the cross-covariance matrix is defined as:
Figure BDA0002375197870000091
the filter gain of the UKF is then calculated as:
Figure BDA0002375197870000092
step 3.5: optimal estimation of clock states
Figure BDA0002375197870000093
The calculation is as follows:
Figure BDA0002375197870000094
zksystem observation results obtained based on observation model for time k
At the same time, update
Figure BDA0002375197870000095
Covariance matrix of (2):
Figure BDA0002375197870000096
the invention corrects the satellite-borne clock by timing observation of the X-ray pulsar signal, thereby realizing on-orbit timekeeping of the satellite-borne clock. The deep space autonomous time keeping based on the natural X-ray source does not depend on a ground system, time keeping cost is low, autonomy is high, and the method is suitable for special requirements of deep space exploration.
It is to be understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make modifications, alterations, additions or substitutions within the spirit and scope of the present invention.

Claims (1)

1. A deep space autonomous time keeping method based on natural X-ray source observation is characterized by comprising the following steps:
step 1: establishing a timing observation model of the X-ray source, modeling observation noise in the timing observation process into the composition of observation error noise and timing noise, and providing a characterization method of the timing noise level;
the step 1 specifically comprises the following steps:
based on long-term observation of pulsar signals, a pulsar phase evolution model is established at the position of the solar system mass center:
Figure FDA0003034319310000011
where t represents the current time, t0In order to refer to the epoch, it is,
Figure FDA0003034319310000012
is t0The initial phase of the time, f is the rotation frequency of the pulsar,
Figure FDA0003034319310000013
is the first derivative of f and is,
Figure FDA0003034319310000014
representing the second derivative of f, wherein epsilon (t) represents a neglected high-order term, and the pulse arrival time at a certain moment can be predicted through a pulsar phase evolution model;
accurately predicting the pulsar pulse phase reaching the SSB by using a pulsar phase evolution model;
observing the arrival time t of pulsar signalsobsIs the arrival time t at the spacecraft that needs to be converted to the centroid of the solar systemSSBThe conversion process is expressed by equation (2):
Figure FDA0003034319310000015
wherein, deltat is the satellite-borne clock error, musDenotes the gravitational constant of the sun, c is the speed of light, D0Observing the distance of the pulsar for the sun centroid distance, n representing the unit vector of the observed pulsar radiation direction, v representing the timing observation noise, rSSBFor the vector of the solar system centroid pointing to the spacecraft position, | rSSBL is rSSBB represents a vector from the sun's centroid mass to the solar system's centroid, | b | is the length of b;
definition of
Figure FDA0003034319310000016
To predict the time of arrival of the resulting pulsar signal using equation (1), the timing residual can be described as:
Figure FDA0003034319310000017
the timing observation model of the X-ray source is then expressed as:
z＝ζ+v (4)
if m pulsar is observed at the same time, the timing observation model is as follows:
Figure FDA0003034319310000021
wherein H is an observation matrix, and the observed quantity is Z ═ Z1,…,zm]TV is observation noise, and is expressed as V ═ V1,…,vm]TV has a covariance matrix of RO，ROIs the observation noise matrix of the observation model;
the timing observation noise is modeled as the combination of observation error noise and timing noise as follows:
let n beO,iRepresenting observation error noise, nt,iRepresenting the timing noise, the timing observation noise is:
nvi＝nt,i+nO,i (6)
wherein n isviTiming view of the ith pulsarMeasuring noise, nO,iAnd nt,iGaussian noise with zero mean;
the observed error noise level is characterized by equations (7) and (8) with nO,iHas a standard deviation ofO,iThen, then
Figure FDA0003034319310000022
Figure FDA0003034319310000023
In the formula, SNR represents the signal-to-noise ratio of the X-ray pulsar signal, BxRepresenting background X-ray radiation flux intensity, F, in the observation environmentxFor the observed pulse signal radiation flux intensity, pfIs the pulse ratio of the pulsar signal, A is the effective area of the detector, tobsD is the pulse duty ratio of the pulsar signal, and d can be expressed as d ═ W/P, W is the pulsar pulse width, and P is the pulse period;
the timing noise level is characterized as:
Figure FDA0003034319310000024
in the formula, C, alpha, beta and gamma are parameters obtained by fitting data obtained by observing a large number of pulsar for a long time;
step 2: the method comprises the steps that a satellite-borne clock evolution model is expressed based on an improved GM model, a gray acting quantity in a traditional GM model is modeled into a parameter related to time, an initial error in the traditional GM model is replaced by a clock error value at the previous moment from a constant, and the clock error value at the previous moment changes along with the time, so that the dependence of the clock evolution model based on the traditional GM model on the initial error is weakened;
the step 2 specifically comprises the following steps:
the improved GM model was:
Figure FDA0003034319310000031
uk+1＝uk+b2k (11)
therein, ζk+1Indicates the clock error at time k +1, ζkThe clock error at the moment k is shown, a is the development coefficient of the GM model, and a is a positive number smaller than 1; u. ofk+1、ukIs the amount of gray contribution of the GM model, in the conventional GM model, ukIs a constant, is modeled as a variable that varies with time, the course of which is represented by the formula (11), b2Is constant and is obtained by data fitting;
make xi ═ ζ, u]TRepresenting the state variables of the system, the satellite-borne clock evolution model can be expressed as:
ξk+1＝Φkξk+Wp (12)
Φkbeing a state transition matrix, Wp＝[wζ,wu]TBeing process noise, wζAnd wuIs a zero-mean gaussian noise that is,
Figure FDA0003034319310000032
is the noise wζThe variance of (a) is determined,
Figure FDA0003034319310000033
denotes wuOf a covariance matrix of
Figure FDA0003034319310000034
diag () represents a diagonal matrix;
and step 3: correcting the satellite-borne clock by using unscented Kalman filtering based on a timing observation model and a satellite-borne clock evolution model to realize satellite-borne autonomous time keeping;
the step 3 specifically comprises the following steps:
step 3.1: first, the sigma point is calculated:
Figure FDA0003034319310000035
in the formula, the first step is that,
Figure FDA0003034319310000036
represents the optimal estimate of time k-1, ()iRepresenting the ith row of the matrix, n is twice the number of state variables, k represents the scale parameter of the model, Pξ,k-1Indicating the time of k-1
Figure FDA0003034319310000037
The covariance matrix of (a);
step 3.2: calculating the one-step state transition of sigma point in formula (13)
Figure FDA0003034319310000038
Figure FDA0003034319310000039
Weighted mean value after one-step state transition of sigma point
Figure FDA00030343193100000310
Sum covariance matrix
Figure FDA00030343193100000311
The calculation is as follows:
Figure FDA00030343193100000312
wherein the content of the first and second substances,
Figure FDA00030343193100000313
representing a weighting coefficient, which is calculated by the following equation:
Figure FDA0003034319310000041
wherein alpha is a positive number smaller than 1, lambda represents a proportionality coefficient, 0 or 3-n is taken, n is the number of state variables, beta is a state distribution parameter and is set as 0;
step 3.3: weighted observation result of sigma point after one-step transfer
Figure FDA0003034319310000042
Expressed as:
Figure FDA0003034319310000043
in the formula, the first step is that,
Figure FDA0003034319310000044
representing the sigma observation result after the one-step transfer, wherein H is an observation matrix;
step 3.4:
Figure FDA0003034319310000045
for estimating the effect of (2) using a covariance matrix Pz,kRepresents:
Figure FDA0003034319310000046
ROto observe the noise matrix;
in addition, the cross-covariance matrix is defined as:
Figure FDA0003034319310000047
the filter gain of the UKF is then calculated as:
Figure FDA0003034319310000048
step 3.5: optimal estimation of clock states
Figure FDA0003034319310000049
The calculation is as follows:
Figure FDA00030343193100000410
zksystem observation results obtained based on observation model for time k
At the same time, update
Figure FDA00030343193100000411
Covariance matrix of (2):
Figure FDA00030343193100000412
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