CN107727102A - Astronomy test the speed combined with terrestrial radio Mars capture section air navigation aid - Google Patents

Astronomy test the speed combined with terrestrial radio Mars capture section air navigation aid Download PDF

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CN107727102A
CN107727102A CN201710987142.5A CN201710987142A CN107727102A CN 107727102 A CN107727102 A CN 107727102A CN 201710987142 A CN201710987142 A CN 201710987142A CN 107727102 A CN107727102 A CN 107727102A
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msub
mrow
mover
detector
represent
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张伟
尤伟
陈晓
张嵬
杜洋
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Shanghai Institute of Satellite Engineering
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Shanghai Institute of Satellite Engineering
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/24—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 specially adapted for cosmonautical navigation

Abstract

Test the speed the Mars capture section air navigation aid combined with terrestrial radio the invention provides a kind of astronomy, comprises the following steps：Earth station obtains distance between detector and ground by radio distance-measuring；Earth station obtains the radial velocity between detector and earth station by radio Doppler range rate measurement；Detector obtains the radial velocity between detector and some fixed star by the navigation sensor that independently tests the speed；By EKF, the Position And Velocity estimation of world integrated navigation is obtained.The inventive method, compared with the navigation for only relying on terrestrial radio, add the autonomous astronomy of detector and test the speed observed quantity, navigation accuracy can be effectively improved.

Description

Astronomy test the speed combined with terrestrial radio Mars capture section air navigation aid
Technical field
The present invention relates to deep space integrated navigation technology, in particular it relates to which a kind of astronomy tests the speed what is combined with terrestrial radio Mars captures section air navigation aid.
Background technology
Survey of deep space task proposes higher requirement to spacecraft homing capability.At present, terrestrial radio is still to realize The Main Means of deep space navigation task.Restricted by visible segmental arc and radio time delay, ground based radio navigation can not meet The continuous autonomous, navigation needs of real-time high-precision of survey of deep space.Therefore, domestic and foreign scholars propose deep space celestial autonomous navigation side Method, it is broadly divided into angle measurement, ranging and three classes that test the speed.Navigation of wherein testing the speed is a kind of autonomous navigation method emerging in recent years, main Will be by the radial velocity of the navigation sources such as measurement and fixed star, and then estimate the state of flight of detector.
The content of the invention
For in the prior art the defects of, it is an object of the invention to provide a kind of astronomy to test the speed what is combined with terrestrial radio Mars captures section air navigation aid.
Test the speed the Mars capture section air navigation aid combined with terrestrial radio according to astronomy provided by the invention, including as follows Step：
Step 1：N days before detector reaches Mars periareon, the navigation observation of continuous d days is come into effect, wherein d's takes Value is less than N, and distance, the radial velocity of earth station and detector are obtained by earth station；
Step 2：The radial velocity of specified fixed star and detector is independently obtained by detector；
Step 3：Joint observation equation is established, and combines position and speed that deep space kinetics equation obtains world integrated navigation Degree estimation.
Preferably, the value of the N in the step 1 includes：4 to 8, d value include：2~3.
Preferably, the step 2 includes：Vector correlation formula between the distance and the radial velocity of earth station's detector is remembered For G1(Xi,ti), abbreviation G1, calculation formula is as follows：
In formula：I represents observation frequency, tiAt the time of representing ith observation, XiRepresent tiThe detector's status at moment, detection Device state includes：Position r and speed v, ρ represent the distance between detector and earth station；Represent between detector and earth station The radial velocity；For 1 × 2 vector；Subscript T represents transposition.
Preferably, the step 3 includes：
Step 3.1：According to the distance of earth station and detector, the radial velocity, the radial velocity for specifying fixed star and detector Establish joint observation establishing equation joint equation；
Step 3.2：The position of world integrated navigation is obtained by expanded Kalman filtration algorithm with reference to deep space kinetics equation Put and velocity estimation.
Preferably, the step 3.1 includes：The radial velocity of specified fixed star and detector is designated as G2(Xi,ti), referred to as G2, calculation formula is as follows：
In formula,The radial velocity of fixed star and detector is represented, i.e., the relative velocity vector of fixed star and detector is in fixed star Projection between detector on line direction；
According to the distance of earth station and detector, the radial velocity, the connection for specifying the radial velocity of fixed star and detector to establish Close observation equation and be designated as G (Xi,ti), calculation formula is as follows：
Then joint observation equation is as follows：
Yi=G (Xi,ti)+ei, i=1,2 ..., l；
In formula：I represents observation frequency, and l represents total observation frequency, XiRepresent tiThe detector's status at moment, YiRepresent ti The observation at moment, eiRepresent ith observation error.
Preferably, the deep space kinetics equation in the step 3.2 is as follows：
Consider detector when capturing section stressing conditions：
In formula：X (t) represents detector's status vector,Represent derivative of the detector's status vector on the time, F (X (t), t) represent and state equation function relevant X (t), a0Represent Mars disome gravitational acceleration；aiRepresent big i-th celestial body Caused point mass perturbation acceleration, N represent big celestial body quantity；Represent Mars gravitational field J2Perturbation acceleration caused by； aRRepresent perturbation acceleration caused by solar light pressure；R represents the position vector of detector；When representing r pairs of detector position vector Between derivative, i.e. velocity；Represent derivatives of the detector speed vector v to the time, i.e. acceleration.
Preferably, the expanded Kalman filtration algorithm comprises the following steps：
Step S1：Initialization, even initial i=1, ti-1=t0、Wherein, i represents observation Number, ti-1At the time of once observation before representing, t0Represent initial time, Pi-1The covariance square at moment is once observed before representing Battle array,Represent the conjecture value of initial time covariance matrix, X*(ti-1) represent before once observation the moment nominal state vector,Represent the nominal state vector of initial time；
Step S2：T is read in from navigation sensoriThe observed quantity Y at momenti, and weight matrix R corresponding to the observationi；
Step S3：With X*(ti-1) it is initial value, by state equationFrom ti-1Integral recursion is to ti, obtainRepresent the nominal state vector at current time；
Step S4：A (t) is asked for, A (t) calculation formula is as follows：
In formula：F (X, t) represents state equation function, and X is that X (t) represents state vector；
Step S5：Calculate Φ (ti,ti-1), Φ (ti,ti-1) represent previous moment to the state-transition matrix at current time；
Specifically, with Φ (ti-1,ti-1)=I is initial value, by the state-transition matrix differential equation From ti-1Integral recursion is to ti, obtain Φ (ti,ti-1)；Wherein, Φ (ti-1,ti-1) represent previous moment to the state of previous moment Transfer matrix, I represent unit matrix,Represent previous moment to the state-transition matrix of any time；
Step S6：Time updates, and calculation formula is as follows：
In formula：Represent the estimate of current time covariance matrix；
Step S7：Calculating observation amount error yi, calculation formula is as follows：
In formula：Represent observational equation function, abbreviation G；
Step S8：Ask forCalculation formula is as follows；
In formula：Represent partial derivative of the observation function to observation quantity, referred to as observing matrix；
Step S9：Ask for gain Ki, calculation formula is as follows：
In formula：KiRepresent the function gain at current time；
Step S10：Observed quantity updates, and calculation formula is as follows：
In formula：Represent the estimate of current time state deviation, PiRepresent current time covariance matrix；
Step S11：Estimated result exports, i.e.,：
In formula：Represent the estimate of current time state；
Step S12：Loop iteration updates, even ti-1=ti,
Step S13：Judge whether observed quantity had fully entered, terminated if fully entering；Otherwise, input next Individual observed quantity, return to step perform step S2.
Compared with prior art, the present invention has following beneficial effect：
The present invention can combine traditional ground ranging, test the speed and the astronomical advantage independently to test the speed, be filtered by spreading kalman Ripple carries out the estimation of joint navigational state, and step is clear, meets engineering demand, is a kind of solution survey of deep space high accuracy navigation problem Effective means.
Brief description of the drawings
The detailed description made by reading with reference to the following drawings to non-limiting example, further feature of the invention, Objects and advantages will become more apparent upon：
Fig. 1 be astronomy provided by the invention test the speed combine with terrestrial radio Mars capture section air navigation aid schematic diagram；
Fig. 2 defines schematic diagram for B planes in the present invention with error ellipse；
Fig. 3 is deviation ellipse of the present invention (3 σ) with respect to Mars graph of a relation；
Fig. 4 is deviation ellipse of the present invention (3 σ) Local map.
Embodiment
With reference to specific embodiment, the present invention is described in detail.Following examples will be helpful to the technology of this area Personnel further understand the present invention, but the invention is not limited in any way.It should be pointed out that the ordinary skill to this area For personnel, without departing from the inventive concept of the premise, some changes and improvements can also be made.These belong to the present invention Protection domain.
Astronomy provided by the invention tests the speed the Mars capture section air navigation aid combined with terrestrial radio, including following step Suddenly：Earth station obtains distance (containing measurement noise) between detector and ground by radio distance-measuring；Earth station passes through radio Doppler range rate measurement, obtain the radial velocity (containing measurement noise) between detector and earth station；Detector is led by independently testing the speed Navigate sensor, obtains the radial velocity (containing measurement noise) between detector and some fixed star；By EKF, obtain The Position And Velocity estimation of world integrated navigation.Compared with the navigation for only relying on terrestrial radio, present invention adds detector Autonomous astronomy tests the speed observed quantity, can effectively improve navigation accuracy.
Establish the observational equation of distance between detector and earth station, including observation and observation noise.
Establish the observational equation of the radial velocity between detector and earth station, including observation and observation noise.
Establish the observational equation of the radial velocity between detector and specified fixed star, including observation and observation noise.
Joint observation equation is established with astronomical radial velocity observed quantity based on ground ranging, observed quantity of testing the speed.
Section dynamics environment is captured according to Mars, establishes dynamic differential equation, including Mars disome gravitation, Mars gravitation Field J2 perturbations, the perturbation of the body of the sun three, the body perturbation of major planet three, solar radiation pressure perturbation etc..
Based on expanded Kalman filtration algorithm, Position And Velocity estimation is carried out to above-mentioned navigation model.
Navigation accuracy is improved using B plane errors are oval.
Specifically, shown in Fig. 1, astronomy provided by the present invention test the speed combined with terrestrial radio Mars capture section navigation Method, N days before Mars periareon is reached, the navigation observation of continuous d days is come into effect, is stopped until reaching N-d days before Mars (wherein N is generally 4-8, and the general values of d are 2-3).T in Fig. 1pAt the time of fire braking near for detector.Detector edge is entered to enter the orbit Road moves closer to Mars.In the process, the distance for earth station and detector being measured by earth station is observed with the radial velocity Measure ρ,The radial velocity of specified fixed star and detector is independently obtained by detector, obtains observed quantityObserved quantity ρ is tieed up by 2,With 1 dimension observed quantitySimultaneous, it can obtain joint observation equation.
After establishing joint observation equation, with reference to deep space kinetics equation, navigation results are obtained using EKF Output.
Specifically include as follows：
Step 1：Establish earth station's ranging, the observational equation for observed quantity of testing the speed：
Step 2：Establish the observational equation of astronomical autonomous navigation observed quantity of testing the speed：
Step 3：Establish integrated navigation world joint observation equation：
Step 4：Mars capture section kinetics equation (state equation) is established, is expressed as：
Consider that detector is capturing section stressing conditions, have：
In formula：a0For Mars disome gravitational acceleration；aiIt is for point mass perturbation acceleration, N caused by big i-th celestial body Big celestial body quantity；For Mars gravitational field J2Perturbation acceleration caused by；aRFor perturbation acceleration caused by solar light pressure.
Step 5：Simultaneous state equation and integrated navigation observational equation, navigation shape is carried out using expanded Kalman filtration algorithm State is estimated；Expanded Kalman filtration algorithm comprises the following steps that：
1), initialize：I=1, ti-1=t0、
2) t, is read iniThe observed quantity Y at momentiAnd its weight matrix Ri；
3), with X*(ti-1) it is initial value, by state equationFrom ti-1Integral recursion is to ti, obtain
4), ask for
5), with Φ (ti-1,ti-1)=I is initial value, by the state-transition matrix differential equationFrom ti-1Integral recursion is to ti, obtain Φ (ti,ti-1)；
6), the time updates：
7), calculating observation residual error：
8), ask for
9) gain, is asked for
10) renewal, is measured：
11), estimated result exports：
12), loop iteration updates：ti-1=ti,
Finished 13), if data have been read, algorithm terminates；Otherwise next observed quantity, return to step 2 are read).
Step 6：It is oval using B plane errors, lift navigation accuracy.
As shown in Figure 2 to 4, Fig. 2 gives detector B plane errors oval definition.V in figure∞,inFly for detector Row velocity, along hyperbola close to Mars.B is that the vector for entering hyperbola asymptote and the intersection point of B planes is pointed to from the fiery heart； Unit vector S through overdo the heart and with enter asymptote it is parallel；Unit vector T is parallel to ecliptic plane and vertical with S；Unit vector R Equal to S × T.The physical meaning of error ellipse is：Because evaluated error, therefore vector B end be present in the Position And Velocity of detector There is dimensional probability distribution in the position at end, i.e. vector B end will be fallen within error ellipse with certain probability.Error ellipse face Product is smaller, then it represents that detector position, the estimated accuracy of speed are higher, i.e., navigation accuracy is also higher.
In Fig. 3, the definition of reference axis BT, BR is projection of the vector B on T and R directions.Right side semicircle is martian surface, Left side is error ellipse.Fig. 4 is the partial enlargement of Fig. 3 error ellipses.Four error ellipses in Fig. 4 correspond to different imitate respectively True operating mode.Emulation operating mode illustrates that table is as follows：
One skilled in the art will appreciate that except realizing system provided by the invention in a manner of pure computer readable program code And its beyond each device, completely can by by method and step carry out programming in logic come system provided by the invention and its Each device is in the form of gate, switch, application specific integrated circuit, programmable logic controller (PLC) and embedded microcontroller etc. To realize identical function.So system provided by the invention and its every device are considered a kind of hardware component, and it is right What is included in it is used to realize that the device of various functions can also to be considered as the structure in hardware component；It will can also be used to realize respectively The device of kind of function, which is considered as, not only can be the software module of implementation method but also can be the structure in hardware component.
The specific embodiment of the present invention is described above.It is to be appreciated that the invention is not limited in above-mentioned Particular implementation, those skilled in the art can make a variety of changes or change within the scope of the claims, this not shadow Ring the substantive content of the present invention.In the case where not conflicting, the feature in embodiments herein and embodiment can any phase Mutually combination.

Claims (7)

  1. The Mars capture section air navigation aid combined with terrestrial radio 1. a kind of astronomy tests the speed, it is characterised in that including following step Suddenly：
    Step 1：N days before detector reaches Mars periareon, the navigation observation of continuous d days is come into effect, wherein d value is small In N, pass through acquisition earth station of earth station and distance, the radial velocity of detector；
    Step 2：The radial velocity of specified fixed star and detector is independently obtained by detector；
    Step 3：Joint observation equation is established, and the Position And Velocity for combining the acquisition world integrated navigation of deep space kinetics equation is estimated Meter.
  2. The Mars capture section air navigation aid combined with terrestrial radio 2. astronomy according to claim 1 tests the speed, its feature It is, the value of the N in the step 1 includes：4 to 8, d value include：2~3.
  3. The Mars capture section air navigation aid combined with terrestrial radio 3. astronomy according to claim 1 tests the speed, its feature It is, the step 2 includes：Vector correlation formula between the distance and the radial velocity of earth station's detector is designated as G1(Xi, ti), abbreviation G1, calculation formula is as follows：
    <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mi>&amp;rho;</mi> <mo>,</mo> <mover> <mi>&amp;rho;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>;</mo> </mrow>
    In formula：I represents observation frequency, tiAt the time of representing ith observation, XiRepresent tiThe detector's status at moment, detector shape State includes：Position r and speed v, ρ represent the distance between detector and earth station；Represent the line of vision between detector and earth station Speed；For 1 × 2 vector；Subscript T represents transposition.
  4. The Mars capture section air navigation aid combined with terrestrial radio 4. astronomy according to claim 3 tests the speed, its feature It is, the step 3 includes：
    Step 3.1：According to the distance of earth station and detector, the radial velocity, the radial velocity of fixed star and detector is specified to establish Joint observation establishing equation combines equation；
    Step 3.2：With reference to deep space kinetics equation by expanded Kalman filtration algorithm obtain world integrated navigation position with Velocity estimation.
  5. The Mars capture section air navigation aid combined with terrestrial radio 5. astronomy according to claim 4 tests the speed, its feature It is, the step 3.1 includes：The radial velocity of specified fixed star and detector is designated as G2(Xi,ti), abbreviation G2, calculation formula It is as follows：
    <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mover> <mi>&amp;rho;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>S</mi> <mi>u</mi> <mi>n</mi> </mrow> </msub> </mrow>
    In formula,The radial velocity of fixed star and detector is represented, i.e., the relative velocity vector of fixed star and detector is in fixed star and spy Projection between survey device on line direction；
    Sight is combined with what the radial velocity of detector was established with the distance, the radial velocity, specified fixed star of detector according to earth station Survey equation and be designated as G (Xi,ti), calculation formula is as follows：
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>G</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>G</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>&amp;rho;</mi> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&amp;rho;</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&amp;rho;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>S</mi> <mi>u</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    Then joint observation equation is as follows：
    Yi=G (Xi,ti)+ei, i=1,2 ..., l；
    In formula：I represents observation frequency, and l represents total observation frequency, XiRepresent tiThe detector's status at moment, YiRepresent tiMoment Observation, eiRepresent ith observation error.
  6. The Mars capture section air navigation aid combined with terrestrial radio 6. astronomy according to claim 5 tests the speed, its feature It is, the deep space kinetics equation in the step 3.2 is as follows：
    <mrow> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    Consider detector when capturing section stressing conditions：
    <mrow> <mover> <mi>r</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>a</mi> <msub> <mi>J</mi> <mn>2</mn> </msub> </msub> <mo>+</mo> <msub> <mi>a</mi> <mi>R</mi> </msub> <mo>;</mo> </mrow>
    In formula：X (t) represents detector's status vector,Derivative of the expression detector's status vector on the time, F (X (t), T) represent and state equation function relevant X (t), a0Represent Mars disome gravitational acceleration；aiRepresent that big i-th celestial body causes Point mass perturbation acceleration, N represents big celestial body quantity；Represent Mars gravitational field J2Perturbation acceleration caused by；aRTable Show perturbation acceleration caused by solar light pressure；R represents the position vector of detector；Represent detector position vector r to the time Derivative, i.e. velocity；Represent derivatives of the detector speed vector v to the time, i.e. acceleration.
  7. The Mars capture section air navigation aid combined with terrestrial radio 7. astronomy according to claim 6 tests the speed, its feature It is, the expanded Kalman filtration algorithm comprises the following steps：
    Step S1：Initialization, even initial i=1, ti-1=t0、Wherein, i represents observation frequency, ti-1At the time of once observation before representing, t0Represent initial time, Pi-1The covariance matrix at moment is once observed before representing,Table Show the conjecture value of initial time covariance matrix, X*(ti-1) represent before once observation the moment nominal state vector,Represent just The nominal state vector at moment beginning；
    Step S2：T is read in from navigation sensoriThe observed quantity Y at momenti, and weight matrix R corresponding to the observationi；
    Step S3：With X*(ti-1) it is initial value, by state equationFrom ti-1Integral recursion is to ti, obtain Xi *, Xi * Represent the nominal state vector at current time；
    Step S4：A (t) is asked for, A (t) calculation formula is as follows：
    <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;part;</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <mo>&amp;part;</mo> <mi>X</mi> <msub> <mo>|</mo> <mrow> <mi>X</mi> <mo>=</mo> <msup> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>*</mo> </msup> </mrow> </msub> <mo>;</mo> </mrow>
    In formula：F (X, t) represents state equation function, and X is that X (t) represents state vector；
    Step S5：Calculate Φ (ti,ti-1), Φ (ti,ti-1) represent previous moment to the state-transition matrix at current time；
    Specifically, with Φ (ti-1,ti-1)=I is initial value, by the state-transition matrix differential equationFrom ti-1Integral recursion is to ti, obtain Φ (ti,ti-1)；Wherein, Φ (ti-1,ti-1) represent that the state of previous moment to previous moment turns Matrix is moved, I represents unit matrix,Represent previous moment to the state-transition matrix of any time；
    Step S6：Time updates, and calculation formula is as follows：
    <mrow> <msub> <mover> <mi>P</mi> <mo>&amp;OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <mi>&amp;Phi;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>&amp;Phi;</mi> <mi>T</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    In formula：Represent the estimate of current time covariance matrix；
    Step S7：Calculating observation amount error yi, calculation formula is as follows：
    <mrow> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo>-</mo> <mi>G</mi> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>,</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    In formula：Represent observational equation function, abbreviation G；
    Step S8：Ask forCalculation formula is as follows；
    <mrow> <msub> <mover> <mi>H</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <mo>&amp;part;</mo> <mi>G</mi> <mo>/</mo> <mo>&amp;part;</mo> <mi>X</mi> <msub> <mo>|</mo> <mrow> <mi>X</mi> <mo>=</mo> <msubsup> <mi>X</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </msub> <mo>;</mo> </mrow>
    In formula：Represent partial derivative of the observation function to observation quantity, referred to as observing matrix；
    Step S9：Ask for gain Ki, calculation formula is as follows：
    <mrow> <msub> <mi>K</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mover> <mi>P</mi> <mo>&amp;OverBar;</mo> </mover> <mi>i</mi> </msub> <msubsup> <mover> <mi>H</mi> <mo>~</mo> </mover> <mi>i</mi> <mi>T</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>H</mi> <mo>~</mo> </mover> <mi>i</mi> </msub> <msub> <mover> <mi>P</mi> <mo>&amp;OverBar;</mo> </mover> <mi>i</mi> </msub> <msubsup> <mover> <mi>H</mi> <mo>~</mo> </mover> <mi>i</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>;</mo> </mrow>
    In formula：KiRepresent the function gain at current time；
    Step S10：Observed quantity updates, and calculation formula is as follows：
    <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>K</mi> <mi>i</mi> </msub> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <msub> <mi>K</mi> <mi>i</mi> </msub> <msubsup> <mover> <mi>H</mi> <mo>~</mo> </mover> <mi>i</mi> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>&amp;OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>;</mo> </mrow>
    In formula：Represent the estimate of current time state deviation, PiRepresent current time covariance matrix；
    Step S11：Estimated result exports, i.e.,：
    <mrow> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msubsup> <mi>X</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>+</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>;</mo> </mrow>
    <mrow> <msubsup> <mi>X</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>=</mo> <msub> <mover> <mi>X</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>;</mo> </mrow>
    In formula：Represent the estimate of current time state；
    Step S12：Loop iteration updates, even ti-1=ti,
    Step S13：Judge whether observed quantity had fully entered, terminated if fully entering；Otherwise, next sight is inputted Measurement, return to step perform step S2.
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