CN107561528A - The Joint Probabilistic Data Association algorithm that a kind of anti-flight path merges - Google Patents

The Joint Probabilistic Data Association algorithm that a kind of anti-flight path merges Download PDF

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CN107561528A
CN107561528A CN201710684778.2A CN201710684778A CN107561528A CN 107561528 A CN107561528 A CN 107561528A CN 201710684778 A CN201710684778 A CN 201710684778A CN 107561528 A CN107561528 A CN 107561528A
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井沛良
吴玉生
郭荣化
撒彦成
赵金红
赵鹏
姬强
马子龙
江山
宋平
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Abstract

The present invention relates to the Joint Probabilistic Data Association algorithm that a kind of anti-flight path merges, belong to data correlation, target following and statistic line loss rate field.Purpose is that solve classical JPDA (JPDA) algorithm existing flight path consolidation problem when target spatial domain is intensive.The essential reason that JPDA algorithms flight path merges phenomenon is analyzed first, is derived from key to the issue and is that adjacent target produces observation and the intersection of state each other is updated；Secondly each feasible relevance assumption of original JPDA algorithms is analyzed, assume the part for confirming state may be caused to intersect renewal, and the part is avoided to assume the cumulative effect to corresponding targetpath association probability, updated so as to effectively avoid intersecting, finally solve flight path consolidation problem.

Description

The Joint Probabilistic Data Association algorithm that a kind of anti-flight path merges
Technical field
The present invention relates to the Joint Probabilistic Data Association algorithm that a kind of anti-flight path merges, belong to data correlation, target following And statistic line loss rate field.
Background technology
JPDA (JPDA) algorithm is the classical way of multiple target tracking under false-alarm, clutter environment.But When target spatial domain is intensive, often there is flight path and merge phenomenon in the target following result of JPDA algorithms, that is to say, that dbjective state Deviation be present in estimation.
To solve this problem, it has been proposed that accurate arest neighbors JPDA (ENNJPDA) algorithm, yardstick connection Close probabilistic data association (SJPDA) algorithm and JPDA* algorithms.Essentially, these methods are all using hypothesis shearing or false If the logic of probability scale scaling carrys out the soft interrelated decision of " hardening " JPDA algorithms, therefore these methods have run counter to JPDA algorithms Thought starting point (namely Bayes's total probability formula), the reasonability of algorithm lack tight derivation, and performance is difficult to be guaranteed.
The content of the invention
Present invention aim to address problems of the prior art：Analyze the essence that JPDA algorithms flight path merges phenomenon Reason, it is derived from key to the issue and is that adjacent target produces observation and the intersection of state each other is updated；There is provided one kind can keep away Fork renewal is exempted from, and then effectively avoids the JPDA algorithms of flight path merging phenomenon.
The principle of the present invention：By analyzing each feasible relevance assumption of original JPDA algorithms, confirming may State is caused to intersect the part of renewal it is assumed that and avoiding the part from assuming to corresponding target-track association probability Cumulative effect, updated so as to effectively avoid intersecting, and then avoid flight path from merging phenomenon.Theory analysis shows：The algorithm is compared to biography Algorithm logic of uniting is relatively sharp, tight.Emulation experiment shows：The algorithm performance is significantly better than traditional algorithm.
The present invention solves the above problems the technical scheme of use, evades mesh by shearing the relevance assumption of target and observation Intersect renewal between mark state, avoid flight path from merging phenomenon；The Joint Probabilistic Data Association algorithm step that anti-flight path merges is as follows：
S.1 door region is solved according to each targetpath state estimation, error covariance matrix and pre- gating probability；
Each moment k, target i door region are constrained to obtain by formula below：
Wherein, z represents observation space vector variable, and H represents observing matrix,Represent target i moment k's Predicted state estimation, Pi(k | k-1) error covariance matrix corresponding to expression, R expression observation error covariance matrixs, pgRepresent that door is general Rate, n represent the dimension of measurement vector, and chi2inv () is the test function of chi square distribution；
S.2 a region is S.1 solved according to step, calculates all target door regions and all the sensors observation (zj, 1≤j≤ Jk) inclusion relation, building may incidence matrix；
Possible incidence matrix is defined asWherein IkFor moment k targetpath total number, ΩI, jDefinition is such as Under
S.3 feasible relevance assumption set is built, calculates each feasible relevance assumption θlPosterior probability；
Wherein feasible relevance assumption θlPrior probability p (θl) be calculated as follows
In formula (4), alRepresent feasible relevance assumption θlMiddle false-alarm number, μ () represent false-alarm number distribution function, taken It is uniformly distributed or Poisson distribution, pdRepresent target detection probability, nlRepresent feasible relevance assumption θlThe number of middle target detection, seemingly Right function p (Zk|θl, Zk-1) be calculated as follows
In formula (5), V represents the volume of sensor observation space, γI, jRepresent θlMiddle local association assumes hI, j, seemingly Right function, flight path i are same targets with observation j, i.e.,
S.4 step S.2 and S.3 on the basis of, calculate each feasible relevance assumption to each flight path-measurement association probability Cumulative effect；
Defining flight path-measurement association probability cumulant matrix isWherein pI, jCalculation formula is
In formula (7), symbol " ∧ " represents logical "and" operation.
S.5 step S.4 on the basis of, the accumulation results of flight path-measurement association probability are normalized, and then to every One flight path builds weighted residual, so as to realize the renewal of Target state estimator and error covariance matrix；
It is as follows to the normalization operation of association probability cumulant matrix
State-updating formula is：
Wherein gain KiIt can be calculated by following formula
Ki=Pi(k|k-1)HT(HPi(k|k-1)HT+R)-1 (10)
Error covariance matrix more new formula is：
Wherein
Represent residual weighted and.
The real motion track of two targets is as shown in Figure 1 in scene.For the scene, two scenario parameters s are definedx And sy, such as following formula：
Algorithm is measured to the extent of deviation of Target state estimator using classical optimal subpattern distribution distance.With field Scape operation duration is 80 seconds, and interframe is divided into 1 second, sxFor 0.42, syExemplified by 0.009,100 Monte Carlos for obtaining 5 kinds of algorithms are put down Equal performance changes over time curve.The present invention is applied to the intensive multiple target status tracking algorithm under design false-alarm, clutter environment. Principle and step of the present invention in above-mentioned application are all identicals.
Beneficial effects of the present invention：Effectively realize and the intensive multiple target state under false-alarm, clutter environment is tracked； To the deviation of Target state estimator significantly less than conventional method.
Brief description of the drawings
Fig. 1 illustrates the scene objects flight path of a case study on implementation
In figure：Linear uniform motion target real trace curve is represented, represents track initial point, and zero represents rail Mark terminating point；Ordinate is Y-axis, and abscissa is X-axis.
Fig. 2 illustrates each algorithm OSPA performances under scene special parameter and changes over time curve
In figure：This paper algorithm performance curves are represented,Correct data association algorithm performance curve is represented,JPDA algorithm performance curves are represented,ENNJPDA algorithm performance curves are represented,Represent JPDA* algorithms Can curve；Ordinate is that logarithmic mean subpattern distributes distance, and abscissa is the time.
Fig. 3 illustrates each algorithm OSPA performances with scenario parameters change curve
In figure：This paper algorithm performance curves are represented,Correct data association algorithm performance curve is represented,JPDA algorithm performance curves are represented,ENNJPDA algorithm performance curves are represented,Represent JPDA* algorithms Can curve；Ordinate is that logarithmic mean subpattern distributes distance, and abscissa is scenery control parameter sy。
Embodiment
Below in conjunction with the accompanying drawings, embodiment is made to the present invention and illustrated.
Example 1
The present invention comprises the following steps：
S.1 door region is solved according to each targetpath state estimation, error covariance matrix and pre- gating probability；
Each moment k, target i door region can be constrained to obtain by formula below：
Wherein, z represents observation space vector variable, and H represents observing matrix,Represent target i moment k's Predicted state estimation, Pi(k | k-1) error covariance matrix corresponding to expression, R expression observation error covariance matrixs, pgRepresent that door is general Rate, n represent the dimension of measurement vector, and chi2inv () is the test function of chi square distribution.
S.2 a region is S.1 solved according to step, calculates all target door regions and all the sensors observation (zj, 1≤j≤ Jk) inclusion relation, and then build may incidence matrix；
Possible incidence matrix is defined asWherein IkFor moment k targetpath total number, ΩI, jDefinition is such as Under
S.3 feasible relevance assumption set is built, calculates each feasible relevance assumption θlPosterior probability；
Wherein feasible relevance assumption θlPrior probability p (θl) can be calculated as follows
In formula (4), alRepresent feasible relevance assumption θlMiddle false-alarm number, μ () represent false-alarm number distribution function (one As take be uniformly distributed or Poisson distribution), pdRepresent target detection probability, nlRepresent feasible relevance assumption θlOf middle target detection Number, likelihood function p (Zk|θl, Zk-1) can be calculated as follows
In formula (5), V represents the volume of sensor observation space, γI, jRepresent θlMiddle local association assumes hI, j(flight path i With observation j be same target) likelihood function, i.e.,
S.4 step S.2 and S.3 on the basis of, calculate each feasible relevance assumption to each flight path-measurement association probability Cumulative effect；
Defining flight path-measurement association probability cumulant matrix isWherein pI, jCalculation formula is
In formula (7), symbol " ∧ " represents logical "and" operation.
S.5 step S.4 on the basis of, the accumulation results of flight path-measurement association probability are normalized, and then to every One flight path builds weighted residual, so as to realize the renewal of Target state estimator and error covariance matrix；
It is as follows to the normalization operation of association probability cumulant matrix
State-updating formula is：
Wherein gain KiIt can be calculated by following formula
Ki=Pi(k|k-1)HT(HPi(k|k-1)HT+R)-1 (24)
Error covariance matrix more new formula is：
Wherein
Below by the use of the simple scenario of a Bi-objective cross flying as embodiment, above-mentioned algorithm performance is illustrated And analysis.The real motion track of two targets is as shown in Figure 1 in scene.For the scene, two scenario parameters s are definedx And sy, such as following formula：
Algorithm is measured to the extent of deviation of Target state estimator using classical OSPA distances.When scene set is run Shi Changwei 80 seconds, interframe is divided into 1 second, sx=0.42, syWhen=0.009,100 Monte Carlo average behaviors of 5 kinds of algorithms can be obtained It is as shown in Figure 2 to change over time curve.As can be seen from Figure 2：
For 1.JPDA algorithms due to flight path consolidation problem be present, there is relatively large deviation in its state estimation so that its average OSPA Metric is noticeably greater than other algorithms；
2.ENNJPDA algorithms and JPDA* algorithm performances relatively, but compared with the performance boost limitation of JPDA algorithms；
3. set forth herein algorithm performance to be significantly better than JPDA algorithms, ENNJPDA algorithms and JPDA* algorithms, in all algorithms Performance of the middle performance closest to hypothetic algorithm (namely correct data association algorithm).
Fixed scene parameter sx=0.42, running parameter sy, all algorithm performances can be obtained with scenario parameters syThe performance of change Curve, as shown in Figure 3.From Fig. 3 it can be found that this paper algorithms are in parameter sySelection range in still show and be significantly better than The superperformance of other algorithms, and the algorithm quality order disclosed in Fig. 3 is consistent substantially with Fig. 2.

Claims (1)

  1. A kind of 1. Joint Probabilistic Data Association algorithm that anti-flight path merges, it is characterized in that associating vacation with what is observed by shearing target If intersecting renewal between carrying out evading target state, flight path is avoided to merge phenomenon；The Joint Probabilistic Data Association algorithm that anti-flight path merges Step is as follows：
    S.1 door region is solved according to each targetpath state estimation, error covariance matrix and pre- gating probability；
    Each moment k, target i door region are constrained to obtain by formula below：
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>HP</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>H</mi> <mi>T</mi> </msup> <mo>+</mo> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;le;</mo> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mn>2</mn> <mi>i</mi> <mi>n</mi> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mi>g</mi> </msub> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, z represents observation space vector variable, and H represents observing matrix,Represent prediction shapes of the target i in moment k State estimation, Pi(k | k-1) error covariance matrix corresponding to expression, R expression observation error covariance matrixs, pgRepresent door probability, n tables Show the dimension of measurement vector, chi2inv () is the test function of chi square distribution；
    S.2 a region is S.1 solved according to step, calculates all target door regions and all the sensors observation (zj, 1≤j≤Jk) Inclusion relation, building may incidence matrix；
    Possible incidence matrix is defined asWherein IkFor moment k targetpath total number, ΩI, jIt is defined as follows
    <mrow> <msub> <mi>&amp;Omega;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mi>c</mi> <mi>h</mi> <mi>i</mi> <mn>2</mn> <mi>i</mi> <mi>n</mi> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mi>g</mi> </msub> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>e</mi> <mi>l</mi> <mi>s</mi> <mi>e</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    S.3 feasible relevance assumption set is built, calculates each feasible relevance assumption θlPosterior probability；
    <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>|</mo> <msup> <mi>Z</mi> <mi>k</mi> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>,</mo> <msup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <munder> <mi>&amp;Sigma;</mi> <mi>l</mi> </munder> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>,</mo> <msup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Wherein feasible relevance assumption θlPrior probability p (θl) be calculated as follows
    <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>a</mi> <mi>l</mi> </msub> <mo>!</mo> </mrow> <mrow> <msub> <mi>J</mi> <mi>k</mi> </msub> <mo>!</mo> </mrow> </mfrac> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>d</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>l</mi> </msub> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>p</mi> <mi>d</mi> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <msub> <mi>I</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mi>n</mi> <mi>l</mi> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    In formula (4), alRepresent feasible relevance assumption θlMiddle false-alarm number, μ () represent false-alarm number distribution function, take uniformly Distribution or Poisson distribution, pdRepresent target detection probability, nlRepresent feasible relevance assumption θlThe number of middle target detection, likelihood letter Number p (Zk|θl, Zk-1) be calculated as follows
    <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>Z</mi> <mi>k</mi> </msub> <mo>|</mo> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> <mo>,</mo> <msup> <mi>Z</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>V</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>l</mi> </msub> </mrow> </msup> <munder> <mi>&amp;Pi;</mi> <msub> <mi>&amp;theta;</mi> <mi>l</mi> </msub> </munder> <mi>&amp;gamma;</mi> <mrow> <mo>(</mo> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    In formula (5), V represents the volume of sensor observation space, γI, jRepresent θlMiddle local association assumes hI, j, i.e. flight path i with Observation j is the likelihood function of same target
    <mrow> <msub> <mi>&amp;gamma;</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>HP</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>H</mi> <mi>T</mi> </msup> <mo>+</mo> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <mi>det</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>HP</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>H</mi> <mi>T</mi> </msup> <mo>+</mo> <mi>R</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msqrt> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    S.4 step S.2 and S.3 on the basis of, calculate each feasible relevance assumption and each flight path-measurement association probability tired out Product effect；
    Defining flight path-measurement association probability cumulant matrix isWherein pI, jCalculation formula is
    In formula (7), symbol " ∧ " represents logical "and" operation；
    S.5 step S.4 on the basis of, the accumulation results of flight path-measurement association probability are normalized, and then to each boat Mark builds weighted residual, so as to realize the renewal of Target state estimator and error covariance matrix；
    It is as follows to the normalization operation of association probability cumulant matrix
    <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>J</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>I</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    State-updating formula is：
    <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>i</mi> </msub> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>J</mi> <mi>k</mi> </msub> </munderover> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    Wherein gain KiCalculated by following formula
    Ki=Pi(k|k-1)HT(HPi(k|k-1)HT+R)-1 (10)
    Error covariance matrix more new formula is：
    <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>P</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>J</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>K</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>HP</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msup> <mi>H</mi> <mi>T</mi> </msup> <mo>+</mo> <mi>R</mi> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>K</mi> <mi>i</mi> <mi>T</mi> </msubsup> <mo>+</mo> <msub> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
    Wherein
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>P</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>J</mi> <mi>k</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>J</mi> <mi>k</mi> </msub> </munderover> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <msub> <mover> <mi>z</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <msubsup> <mover> <mi>z</mi> <mo>^</mo> </mover> <mi>i</mi> <mi>T</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>K</mi> <mi>i</mi> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
    Error covariance matrix amendment component is represented,
    <mrow> <msub> <mover> <mi>z</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>J</mi> <mi>k</mi> </msub> </munderover> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>-</mo> <mi>H</mi> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>|</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
    Represent residual weighted and.
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