CN107707220A - A kind of modified CKF methods applied in GNSS/INS - Google Patents

A kind of modified CKF methods applied in GNSS/INS Download PDF

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CN107707220A
CN107707220A CN201710770961.4A CN201710770961A CN107707220A CN 107707220 A CN107707220 A CN 107707220A CN 201710770961 A CN201710770961 A CN 201710770961A CN 107707220 A CN107707220 A CN 107707220A
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吴峻
王永杰
沈志乐
董国敏
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Southeast University
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
    • H03H21/00—Adaptive networks
    • H03H21/0012—Digital adaptive filters
    • H03H21/0025—Particular filtering methods
    • H03H21/0029—Particular filtering methods based on statistics
    • H03H21/003—KALMAN filters
    • 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
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01—Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/03—Cooperating elements; Interaction or communication between different cooperating elements or between cooperating elements and receivers
    • G01S19/07—Cooperating elements; Interaction or communication between different cooperating elements or between cooperating elements and receivers providing data for correcting measured positioning data, e.g. DGPS [differential GPS] or ionosphere corrections

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  • Engineering & Computer Science (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
  • Physics & Mathematics (AREA)
  • Probability & Statistics with Applications (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • General Physics & Mathematics (AREA)
  • Medicines Containing Antibodies Or Antigens For Use As Internal Diagnostic Agents (AREA)

Abstract

The invention discloses a kind of modified CKF methods applied in GNSS/INS, this method is in the High Order Nonlinear System of GNSS/INS tight integrations, due to the problem of local volumetric point sampling is inaccurate and system mode mutation causes filtering accuracy to reduce or even dissipate, it is proposed to improve nonlinear function mapping accuracy using new volume point sampling rule, and then improve filtering accuracy, and the fading factor of Strong tracking filter is introduced, robustness of the boosting algorithm in state mutation.It is final to construct conversion tracking volume Kalman filter.This method solve the problems, such as that volume point sampling is inaccurate and system mode mutation caused by filtering accuracy decline or even dissipate, have that precision is high, the optimization of strong robustness.

Description

A kind of modified CKF methods applied in GNSS/INS
Technical field
The present invention relates to air navigation aid field, more particularly to a kind of modified CKF methods applied in GNSS/INS.
Background technology
GNSS/INS integrated navigation systems are the popular research directions in Present navigation field, in large angle maneuver, or city The satellite-signals such as city, forest are easily blocked etc. under complex situations, use pseudorange and pseudorange rates as tight group of the GNSS/INS of observed quantity Close so that integrated navigation system can still keep integrated mode in the case where lacking star.
Complexity and polytropy due to environment, GNSS/INS tight integration models have non-linear and uncertain feature. Volume Kalman filtering (Cubature Kalman Filter, CKF) is to be proposed in recent years by Canadian scholar Arasaratnam Bayes's approximate nonlinear filtering algorithm, state variable nonlinear propagation rule is described by volume point sampling, be solution by no means The powerful tool of linear system state estimation.But traditional CKF volume point sampling rule is in high order system or strong nonlinearity feelings The problem of state estimation is inaccurate or even dissipates, in addition in the case where system mode is mutated, algorithm robustness occur under condition Reduce.CKF filtering accuracy, which reduces, even to be dissipated.In order to solve this problem, one kind is based on 5 rank sphere radial direction volume point samplings The CKF filtering algorithms of rule are suggested, and improve filtering accuracy.And due to based on Strong tracking filter (Strong Tracking Filter, STF) improvement CKF algorithms proposition, improve the robustness of wave filter correspondence system.
The content of the invention
In order to solve above-mentioned problem, the present invention provides a kind of modified CKF methods applied in GNSS/INS, It is in High Order Nonlinear System application, local volumetric point sampling is inaccurate and system mode mutation causes filtering accuracy to reduce The problem of even dissipating, this method are efficiently solved under filtering accuracy caused by volume point sampling inaccuracy is mutated with system mode The problem of drop even dissipates, there is precision height, the optimization of strong robustness, to provide one kind up to this purpose, the present invention and applying Modified CKF methods in GNSS/INS, comprise the following steps：
1) combined system nonlinear model is established：The foundation of state equation, observational equation comprising system, the environment of system The modeling of noise, and the solution of general volume Kalman filtering algorithm；
2) formulation of volume point sampling rule is converted：, can be because of volume for traditional CKF wave filters in the case of system higher-dimension Point mapping is improper, causes filtering accuracy to decline or even the situation of filtering divergence occur, proposes a kind of based on surface integral rule Orthogonal volume point sampling rule is converted, solves the problems, such as the decline of High Dimensional Systems filtering accuracy；
3) Strong tracking filter designs：It is difficult to quantify for noise model and system is undergone mutation, so as to causes wave filter to be sent out Dissipate, the problem of numerical value is unstable.It is proposed to introduce a kind of fading factor, output residual sequence is kept orthogonal, the Shandong of boosting algorithm Rod, improve the ability of tracking to state and parameter mutation.
4) structure of TSCKF wave filters：A kind of TSCKF wave filters are built based on orthogonal transformation principle and Strong tracking filter, And carry out time renewal for GNSS/INS systems, measure renewal.
Further improvement of the present invention, the specific steps of combined system nonlinear model are established in the step 1) to be included：
(1.1) discretization combined system state equation isChoosing quantity of state isThe respectively misalignment in three directions, velocity error, site error, acceleration The biasing of degree meter constant value, gyroscope constant value drift；
(1.2) GNSS/SINS tight integrations navigation system measurement equation includes the system pseudo range measurement of pseudo range observed quantity composition The system pseudorange rates measurement equation of equation and pseudorange rates observed quantity composition, the pseudo range measurement equation of combined system areρiFor relative to the pseudorange of i-th satellite,For pseudorange rates, V (t) is measurement Noise；
(1.3) solution of state-transition matrix f in (1.1) is converted to by integrated form based on CKF principleWherein ωi=1/m, i.e. sampled point weight, i=1,2 ..., m=2n；E is that n ties up unit square formation, [1]iRepresent i-th row of [1].
Further improvement of the present invention, the specific steps of the formulation of step 2) the conversion volume point sampling rule include：
(2.1) integrated form in (1.3) is changedWherein, x =Crz+ μ,P=CCT, | | Z | |=1,UnUnit sphere space is represented,Wherein, βi It is to project to the volume point that spherical coordinate is fastened, and | | β | |=1, k are volume numbers, and ω is weight corresponding to volume point；
(2.2) according to sphere orthogonal transformation rule, βiIt is as follows with ω selection rule：
Wherein, i=1,2 ..., 2n and r=1,2 ..., [n/2], and when n is odd number, ri,n=(- 1)i-1, take λ=r2/ 2, Obtain：
(2.3) equation in Gauss-Lagrangian Integration Solving (2.2) is utilized, can be obtained：
Wherein, m is chooses orthogonal points number, therefore obtains converting orthogonal volume point and weight is asked for：
λiFor Gauss-Lagrangian orthogonal points, AiFor corresponding weight, obtained by following formula：
Further improvement of the present invention, the specific steps of step 3) the Strong tracking filter design include：
(3.1) system residual error output sequence is：ηk=yk-yk|k-1, therefore strong tracking filfer should meet following condition：
The covariance matrix of system environments noise is defined as：
Wherein, ρ is forgetting factor, 0.95≤ρ≤0.98；
(3.2) calculating matrix：
Mk=Pzz,k|k-1-Vk+Nk；
Wherein, Pk|k-1For varivance matrix；Pxz,k|k-1For Cross-covariance；Pzz,k|k-1For auto-covariance matrix.Cause This fading factor represents as follows：
Wherein, the mark of tr [] representing matrix.
Further improvement of the present invention, the specific steps of the structure of step 4) the TSCKF wave filters include：
(4.1) volume point is calculated according to improved volume point sampling rule in 3：
(4.2) time updates：The SVD of matrix decomposes to be represented with svd (), then can be asked：
Sk-1|k-1=svd (Pk-1|k-1)；
Calculate cubature points：
Cubature points are propagated by state equation：
ξi,k-1|k-1=f (ξi,k-1|k-1), i=1 ..., 2n；
Calculate status predication value and error covariance matrix：
Calculate fading factor λk, pass through SVD decomposed Psk|k-1：
Sk|k-1=svd (Pk|k-1)；
Calculate Cubature points：
(4.3) renewal is measured：Cubature points are propagated by measurement equation：
Zi,k|k-1=h (ξi,k|k-1)；
Calculate measurement predictorAuto-covariance battle array Pzz,k|k-1With cross covariance battle array Pxz,k|k-1：
Calculated according to formula and introduce fading factor λk,
WillAnd Pk|k-1It is iterated, solves new volume point, and introduce fading factor λkAnd Pk|k-1Recalculate Pzz,k|k-1And Pxz,k|k-1.Calculate filtering gain, state estimation and error covariance matrix：
Pk=Pk|k-1-Kk。
The present invention provides a kind of modified CKF methods applied in GNSS/INS, for high order system local volumetric point The influence that sampling is inaccurate and system mode mutation is to system filter precision, on the basis of CKF, propose the improved orthogonal appearance of conversion Plot point sampling rule, on this basis with reference to the basic theories framework of STF algorithms, propose that a kind of modified CKF (TSCKF) is calculated Method, filtering accuracy caused by this method efficiently solves the inaccurate mutation with system mode of volume point sampling decline even diverging Problem, there is precision height, the optimization of strong robustness.
Brief description of the drawings
Fig. 1 is the inventive method flow chart；
Fig. 2 is GNSS/INS tight integrations system diagram in the present invention；
Fig. 3 is that volume point samples rule schema in the present invention；
Embodiment
The present invention is described in further detail with embodiment below in conjunction with the accompanying drawings：
The present invention provides a kind of modified CKF methods applied in GNSS/INS, and it should for High Order Nonlinear System In, local volumetric point sampling is inaccurate and system mode mutation causes filtering accuracy to reduce the problem of even dissipating, this method Solve the problems, such as that filtering accuracy caused by the inaccurate mutation with system mode of volume point sampling declines or even dissipated, have Precision is high, the optimization of strong robustness.
As Figure 1-3, kind of the present invention applies the modified CKF methods in GNSS/INS, and this method includes Following steps：
Step 1：Combined system nonlinear model is established, the foundation of state equation, observational equation comprising system, system The modeling of ambient noise, and the solution of general volume Kalman filtering algorithm；
Step 2：The formulation of volume point sampling rule is converted, can be because for traditional CKF wave filters in the case of system higher-dimension The mapping of volume point is improper, causes filtering accuracy to decline or even occur the situation of filtering divergence, proposes that one kind is based on sphere area divider The orthogonal volume point sampling rule of conversion then, solves the problems, such as the decline of High Dimensional Systems filtering accuracy；
Step 3：Strong tracking filter designs, and is difficult to quantify for noise model and system is undergone mutation, so as to cause filtering The problem of device dissipates, numerical value is unstable.It is proposed to introduce a kind of fading factor, output residual sequence is kept orthogonal, boosting algorithm Robustness, improve to state and parameter mutation ability of tracking.
Step 4：The structure of TSCKF wave filters：A kind of TSCKF filtering is built based on orthogonal transformation principle and Strong tracking filter Device, and carry out time renewal, measurement renewal for GNSS/INS systems.
Wherein, step 1：Combined system nonlinear model is established, the foundation of state equation, observational equation comprising system, The modeling of the ambient noise of system, and the solution of general volume Kalman filtering algorithm；It is specific as follows：
1) discretization combined system state equation isChoosing quantity of state isThe respectively misalignment in three directions, velocity error, site error, acceleration The biasing of degree meter constant value, gyroscope constant value drift.
2) GNSS/SINS tight integrations navigation system measurement equation includes the system pseudo range measurement equation of pseudo range observed quantity composition With the system pseudorange rates measurement equation of pseudorange rates observed quantity composition.The pseudo range measurement equation of combined system isρiFor relative to the pseudorange of i-th satellite,For pseudorange rates, V (t) is measurement Noise.
3) solution of state-transition matrix f in (1.1) is converted to by integrated form based on CKF principleWherein ωi=1/m, i.e. sampled point weight, i=1,2 ..., m=2n；E is that n ties up unit square formation, [1]iRepresent i-th row of [1].
Step 2：Convert the formulation of volume point sampling rule：, can be because for traditional CKF wave filters in the case of system higher-dimension The mapping of volume point is improper, causes filtering accuracy to decline or even occur the situation of filtering divergence, proposes that one kind is based on sphere area divider The orthogonal volume point sampling rule of conversion then, solves the problems, such as the decline of High Dimensional Systems filtering accuracy；It is specific as follows：
1) by step 1 3) in integrated form changeWherein, X=Crz+ μ, P=CCT, | | Z | |=1, UnRepresent unit sphere space.Wherein, βiIt is to project to the volume point that spherical coordinate is fastened, and | | β | |=1, k are volume numbers, and ω is weight corresponding to volume point.
2) according to sphere orthogonal transformation rule, βiIt is as follows with ω selection rule：
Wherein, i=1,2 ..., 2n and r=1,2 ..., [n/2], and when n is odd number, ri,n=(- 1)i-1, take λ=r2/ 2, Obtain：
3) equation in Gauss-Lagrangian Integration Solving (2.2) is utilized, can be obtained：
Wherein, m is chooses orthogonal points number, therefore obtains converting orthogonal volume point and weight is asked for：
λiFor Gauss-Lagrangian orthogonal points, AiFor corresponding weight, obtained by following formula：
Step 3：Strong tracking filter designs, and is difficult to quantify for noise model and system is undergone mutation, so as to cause filtering The problem of device dissipates, numerical value is unstable.It is proposed to introduce a kind of fading factor, output residual sequence is kept orthogonal, boosting algorithm Robustness, improve to state and parameter mutation ability of tracking；It is specific as follows：
1) system residual error output sequence is：ηk=yk-yk|k-1, therefore strong tracking filfer should meet following condition：
The covariance matrix of system environments noise is defined as：
Wherein, ρ is forgetting factor, 0.95≤ρ≤0.98.
2) calculating matrix：
Mk=Pzz,k|k-1-Vk+Nk；
Wherein, Pk|k-1For varivance matrix；Pxz,k|k-1For Cross-covariance；Pzz,k|k-1For auto-covariance matrix.Cause This fading factor represents as follows：
Wherein, the mark of tr [] representing matrix.
Step 4：The structure of TSCKF wave filters：A kind of TSCKF filtering is built based on orthogonal transformation principle and Strong tracking filter Device, and carry out time renewal, measurement renewal for GNSS/INS systems；It is specific as follows：
1) volume point is calculated according to improved volume point sampling rule in step 2.
2) time updates：The SVD of matrix decomposes to be represented with svd (), then can be asked：
Sk-1|k-1=svd (Pk-1|k-1)；
Calculate cubature points：
Cubature points are propagated by state equation：
ξi,k-1|k-1=f (ξi,k-1|k-1), i=1 ..., 2n；
Calculate status predication value and error covariance matrix：
Calculate fading factor λk, pass through SVD decomposed Psk|k-1：
Sk|k-1=svd (Pk|k-1)；
Calculate Cubature points：
3) renewal is measured：Cubature points are propagated by measurement equation：
Zi,k|k-1=h (ξi,k|k-1)；
Calculate measurement predictorAuto-covariance battle array Pzz,k|k-1With cross covariance battle array Pxz,k|k-1：
Calculated according to formula and introduce fading factor λk,
WillAnd Pk|k-1It is iterated, solves new volume point, and introduce fading factor λkAnd Pk|k-1Recalculate Pzz,k|k-1And Pxz,k|k-1.Calculate filtering gain, state estimation and error covariance matrix：
Pk=Pk|k-1-Kk；
The above described is only a preferred embodiment of the present invention, it is not the limit for making any other form to the present invention System, and any modification made according to technical spirit of the invention or equivalent variations, still fall within present invention model claimed Enclose.

Claims (5)

  1. A kind of 1. modified CKF methods applied in GNSS/INS, it is characterised in that：Comprise the following steps：
    1) combined system nonlinear model is established：The foundation of state equation, observational equation comprising system, the ambient noise of system Modeling, and the solution of general volume Kalman filtering algorithm；
    2) formulation of volume point sampling rule is converted：For traditional CKF wave filters in the case of system higher-dimension, it can be reflected because of volume point It is improper to penetrate, and causes filtering accuracy to decline or even occur the situation of filtering divergence, proposes a kind of conversion based on surface integral rule Orthogonal volume point sampling rule, solves the problems, such as the decline of High Dimensional Systems filtering accuracy；
    3) Strong tracking filter designs：It is difficult to quantify for noise model and system is undergone mutation, so as to causes filter divergence, number It is worth the problem of unstable.It is proposed to introduce a kind of fading factor, output residual sequence is kept orthogonal, the robustness of boosting algorithm, Improve the ability of tracking to state and parameter mutation.
    4) structure of TSCKF wave filters：A kind of TSCKF wave filters, and pin are built based on orthogonal transformation principle and Strong tracking filter Time renewal is carried out to GNSS/INS systems, measures renewal.
  2. A kind of 2. modified CKF methods applied in GNSS/INS according to claim 1, it is characterised in that：
    The specific steps of combined system nonlinear model are established in the step 1) to be included：
    (1.1) discretization combined system state equation isChoosing quantity of state isThe respectively misalignment in three directions, velocity error, site error, acceleration The biasing of degree meter constant value, gyroscope constant value drift；
    (1.2) GNSS/SINS tight integrations navigation system measurement equation includes the system pseudo range measurement equation of pseudo range observed quantity composition With the system pseudorange rates measurement equation of pseudorange rates observed quantity composition, the pseudo range measurement equation of combined system is.I=1,2,3,4, ρiFor relative to the pseudorange of i-th satellite,For pseudorange rates, V (t) is amount Survey noise；
    (1.3) solution of state-transition matrix f in (1.1) is converted to by integrated form based on CKF principleWherein ωi=1/m, i.e. sampled point weight, i=1,2 ..., m=2n；[1]=[E ,-E], E are that n ties up unit square formation, [1]iRepresent i-th row of [1].
  3. A kind of 3. modified CKF methods applied in GNSS/INS according to claim 1, it is characterised in that：
    The specific steps of the formulation of step 2) the conversion volume point sampling rule include：
    (2.1) integrated form in (1.3) is changedWherein, x=Crz + μ, P=CCT, | | Z | |=1, UnUnit sphere space is represented,Wherein, βiIt is to throw The volume point that shadow is fastened to spherical coordinate, and | | β | |=1, k are volume numbers, and ω is weight corresponding to volume point；
    (2.2) according to sphere orthogonal transformation rule, βiIt is as follows with ω selection rule：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> <mi>r</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <mn>2</mn> <mo>/</mo> <mi>n</mi> </mrow> </msqrt> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>i</mi> <mi>&amp;pi;</mi> <mo>/</mo> <mi>n</mi> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;beta;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <mn>2</mn> <mo>/</mo> <mi>n</mi> </mrow> </msqrt> <mi>sin</mi> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>r</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>i</mi> <mi>&amp;pi;</mi> <mo>/</mo> <mi>n</mi> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    <mrow> <mi>&amp;omega;</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <msup> <mi>&amp;pi;</mi> <mi>n</mi> </msup> </msqrt> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mi>&amp;Gamma;</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>;</mo> </mrow>
    Wherein, i=1,2 ..., 2n and r=1,2 ..., [n/2], and when n is odd number, rI, n=(- 1)i-1, take λ=r2/ 2, obtain：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mi>&amp;Gamma;</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mi>&amp;lambda;</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&amp;infin;</mi> </msubsup> <msup> <mi>f</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>&amp;lambda;</mi> <mo>)</mo> </mrow> <msup> <mi>&amp;lambda;</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;lambda;</mi> </mrow> </msup> <mi>d</mi> <mi>&amp;lambda;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>f</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>&amp;lambda;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>2</mi> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <mi>C</mi> <msqrt> <mrow> <mn>2</mn> <mi>&amp;lambda;</mi> </mrow> </msqrt> <msub> <mi>&amp;beta;</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>&amp;mu;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    (2.3) equation in Gauss-Lagrangian Integration Solving (2.2) is utilized, can be obtained：
    <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>C&amp;xi;</mi> <mi>i</mi> </msub> <mo>+</mo> <mi>&amp;mu;</mi> <mo>)</mo> </mrow> <msub> <mi>W</mi> <mi>j</mi> </msub> <mo>;</mo> </mrow>
    Wherein, m is chooses orthogonal points number, therefore obtains converting orthogonal volume point and weight is asked for：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;xi;</mi> <mi>i</mi> </msub> <mo>=</mo> <msqrt> <mrow> <mn>2</mn> <msub> <mi>&amp;lambda;</mi> <mi>j</mi> </msub> </mrow> </msqrt> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mn>2</mn> <mi>n</mi> <mi>&amp;Gamma;</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    λiFor Gauss-Lagrangian orthogonal points, AiFor corresponding weight, obtained by following formula：
    <mrow> <msub> <mi>L</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;lambda;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>m</mi> </msup> <msup> <mi>&amp;lambda;</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mi>&amp;lambda;</mi> </msup> <mfrac> <msup> <mi>d</mi> <mi>m</mi> </msup> <mrow> <msup> <mi>d&amp;lambda;</mi> <mi>m</mi> </msup> </mrow> </mfrac> <msup> <mi>&amp;lambda;</mi> <mrow> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;lambda;</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>m</mi> <mo>!</mo> <mi>&amp;Gamma;</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> <mo>+</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>&amp;lambda;</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>L</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;lambda;</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>.</mo> </mrow>
  4. A kind of 4. modified CKF methods applied in GNSS/INS according to claim 1, it is characterised in that：
    The specific steps of step 3) the Strong tracking filter design include：
    (3.1) system residual error output sequence is：ηk=yk-yk|k-1, therefore strong tracking filfer should meet following condition：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>E</mi> <mo>&amp;lsqb;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;rsqb;</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>=</mo> <mi>M</mi> <mi>I</mi> <mi>N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>E</mi> <mo>&amp;lsqb;</mo> <msubsup> <mi>&amp;eta;</mi> <mi>k</mi> <mi>T</mi> </msubsup> <msub> <mi>&amp;eta;</mi> <mrow> <mi>k</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    The covariance matrix of system environments noise is defined as：
    <mrow> <msub> <mi>V</mi> <mi>k</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;eta;</mi> <mn>1</mn> </msub> <msubsup> <mi>&amp;eta;</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>&amp;rho;V</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;eta;</mi> <mi>k</mi> </msub> <msubsup> <mi>&amp;eta;</mi> <mi>k</mi> <mi>T</mi> </msubsup> <mo>)</mo> <mo>/</mo> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>&amp;rho;</mi> <mo>)</mo> <mo>,</mo> <mi>k</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    Wherein, ρ is forgetting factor, 0.95≤ρ≤0.98；
    (3.2) calculating matrix：
    Mk=Pzz,k|k-1-Vk+Nk；
    Wherein, Pk|k-1For varivance matrix；Pxz,k|k-1For Cross-covariance；Pzz,k|k-1For auto-covariance matrix.Therefore gradually The factor representation that disappears is as follows：
    <mrow> <msub> <mi>&amp;lambda;</mi> <mi>k</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>&amp;lambda;</mi> <mn>0</mn> </msub> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&amp;lambda;</mi> <mn>0</mn> </msub> <mo>&amp;GreaterEqual;</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&amp;lambda;</mi> <mn>0</mn> </msub> <mo>&lt;</mo> <mn>1</mn> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>&amp;lambda;</mi> <mn>0</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>t</mi> <mi>r</mi> <mo>&amp;lsqb;</mo> <msub> <mi>N</mi> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mi>t</mi> <mi>r</mi> <mo>&amp;lsqb;</mo> <msub> <mi>M</mi> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> </mfrac> <mo>;</mo> </mrow>
    Wherein, the mark of tr [] representing matrix.
  5. A kind of 5. modified CKF methods applied in GNSS/INS according to claim 1, it is characterised in that：
    The specific steps of the structure of step 4) the TSCKF wave filters include：
    (4.1) volume point is calculated according to improved volume point sampling rule in 3：
    (4.2) time updates：The SVD of matrix decomposes to be represented with svd (), then can be asked：
    Sk-1|k-1=svd (Pk-1|k-1)；
    Calculate cubature points：
    <mrow> <msub> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo>;</mo> </mrow>
    Cubature points are propagated by state equation：
    ξi,k-1|k-1=f (ξi,k-1|k-1), i=1 ..., 2n；
    Calculate status predication value and error covariance matrix：
    <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>Q</mi> <mi>k</mi> </msub> <mo>;</mo> </mrow>
    Calculate fading factor λk, pass through SVD decomposed Psk|k-1：
    Sk|k-1=svd (Pk|k-1)；
    Calculate Cubature points：
    <mrow> <msub> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>;</mo> </mrow>
    (4.3) renewal is measured：Cubature points are propagated by measurement equation：
    Zi,k|k-1=h (ξi,k|k-1)；
    Calculate measurement predictorAuto-covariance battle array Pzz,k|k-1With cross covariance battle array Pxz,k|k-1：
    <mrow> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>P</mi> <mrow> <mi>z</mi> <mi>z</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mi>R</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>;</mo> </mrow>
    <mrow> <msub> <mi>P</mi> <mrow> <mi>x</mi> <mi>z</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>Z</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>;</mo> </mrow>
    Calculated according to formula and introduce fading factor λk,
    <mrow> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>&amp;lambda;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msub> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <msubsup> <mi>&amp;xi;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Q</mi> <mi>k</mi> </msub> <mo>;</mo> </mrow>
    WillAnd Pk|k-1It is iterated, solves new volume point, and introduce fading factor λkAnd Pk|k-1Recalculate Pzz,k|k-1 And PXz, k | k-1.Calculate filtering gain, state estimation and error covariance matrix：
    <mrow> <msub> <mi>K</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>x</mi> <mi>z</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>P</mi> <mrow> <mi>z</mi> <mi>z</mi> <mo>,</mo> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>;</mo> </mrow>
    <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Pk=Pk|k-1-Kk。
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Application publication date: 20180216