CN107704432A - A kind of adaptive Interactive Multiple-Model method for tracking target of transition probability - Google Patents
A kind of adaptive Interactive Multiple-Model method for tracking target of transition probability Download PDFInfo
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Abstract
The invention discloses a kind of adaptive Interactive Multiple-Model method for tracking target of transition probability, the movement locus measured value of target is gathered by sensor and establishes the motion state Models Sets of target, the probability of initial model is set according to priori, Model transfer probability matrix, input interaction is carried out to state value, input value using interaction value as next step filtering, then filtered by the filter parallel under each submodel, obtain the filter value under different models, and then update the probability of each model, further according to the model rate of change after renewal, state-transition matrix is modified using hyperbolic sine inverse function, realize that transition probability matrix is adaptive, finally by the filter value weighted sum of submodel, realize target following.The present invention realizes the adaptive of interacting multiple model algorithm state-transition matrix；The tracking of motor-driven and nonmaneuvering target can be realized, obtains the real motion track of target, helps to lift the tracking performance of the method for tracking target based on Interactive Multiple-Model.
Description
Technical field
The invention belongs to target tracking technical field, and in particular to a kind of adaptive Interactive Multiple-Model target of transition probability
Tracking.
Background technology
Target following refers to by filtering to obtain target actual motion track to target observation track, its great practical value
Research direction, be widely applied and profound significance, especially at military aspect, it is army quickly and accurately to realize target following
One of thing technological core.When moving target generation is motor-driven, motion model changes, using Kalman filtering, particle filter
Deng the single filtering algorithm of model tracking performance can be caused to decline or even can not track.But in most cases, it is necessary to follow the trail of
Target there is mobility, therefore research aircraft object tracking is significant and with practical value.
Solving one of basic thought of maneuvering target tracing problem is：The Models Sets moved by establishing maneuvering target, make
Its motion state is described with multiple models, is redirected between model according to transfering state matrix, multiple filter parallel works
The tracking result under each model is obtained, then uses certain rule to be merged to filter result to obtain filter result.
The algorithm that early stage proposes according to this thought has：FIRST ORDER GENERALIZED DISTRIBUTED PARAMETER puppet bayes method and second order broad sense puppet bayes method.Interaction
Formula Multiple Models Algorithm (Interacting Multiple Model, IMM) is that Bar-Shalm and Blom are proposed on this basis
A kind of recursive algorithm redirected between model according to Markov Transition Probabilities matrix.The adaptive ability of MM algorithms is better than one
Rank broad sense puppet bayes method, under same filtering accuracy, its amount of calculation be only second order broad sense puppet bayesian algorithm point it
One (r is Models Sets neutron Number of Models), successfully avoids the full amount of calculation for assuming filtering algorithm and exponentially increase over time
The problem of long.It is just of great interest and be successfully applied to target tracking domain so once proposition.However, pass
The state-transition matrix of the IMM algorithms of system determines (being subjective determine) according to prior information and can not changed, and does not examine
Consider posterior information (i.e. the real-time probability of model changes the influence to transition probability), cause tracking inaccurate.
The content of the invention
It is an object of the invention to provide a kind of Interactive Multiple-Model target tracking algorism adaptive based on transition probability matrix,
The posterior information (model probability rate of change) obtained after being updated by model probability, the situation of change of transition probability matrix is analyzed,
Then it is corrected in real time using hyperbolic sine inverse function, solved in the prior art due to prior information mistake, and not
The problem of target tracking accuracy caused by energy real-time update is not high.
To solve the above problems, the present invention uses following technical scheme：
A kind of adaptive Interactive Multiple-Model method for tracking target of transition probability, comprises the following steps：
Step 1, the movement locus measured value of target is gathered by sensor and establishes the motion model collection of target；
Step 2, probability, Model transfer probability matrix according to priori setting initial model；
Step 3, according to state-transition matrix submodel is instructed to carry out input interaction；
Step 4, filter parallel work, are tracked to the target trajectory under each submodel；
Step 5, the probability according to each motion model of the likelihood function of submodel renewal；
Step 6, according to the model probability after renewal, state-transition matrix is modified using hyperbolic sine inverse function；
Step 7, according to motion model probability output interact.
As the further scheme of this case invention, step 1 is specially：
Step (1.1), the measured value by sensor collection target trajectory.
Step (1.2), establish target movement model collection M=[M1,M2,…,Mr] (r is the probability of submodel in Models Sets),
Submodel needs the main motion model of coverage goal, but quantity can not be too much in order to avoid unnecessary competition between causing model, influences
Tracking accuracy.
J-th of model Mj(j=1,2 ... state equation r) is：
Xj(k+1)=Φj(k)Xj(k)+Gj(k)Wj(k)
Measurement equation is：
Z (k)=H (k) X (k)+V (k)
Wherein Xj(k+1) k+1 moment model Ms are representedj1 × n dimension state vectors of middle target, Z (k) represent k moment model Msj
1 × m observation vectors of middle target, Φj(k-1) it is model MjState-transition matrix, GjRepresent model MjNoise matrix, H is
Observing matrix, Wj(k) it is model MjState-noise sequence, V (k) are model MsjMeasuring value noise sequence, k represent the sampling time.
As the further scheme of this case invention, step 2 is specially：
Step (2.1), according to priori, subjectively in setting steps 1 each submodel probability ui(1) (i=
1,2,…r)；
Step (2.2), according to priori, the subjectively initial Markov state transfer in setting steps 1 between model
Matrix Π：
Wherein πijExpression is transferred to model j probability from model i,And the master couple of claimed condition transfer matrix
Diagonal element is dominant.
As the further scheme of this case invention, step 3 is specially：
Transition probability π in state-transition matrix Πij(k-1), model MiProbable value ui(k-1) (i=1,2 ...
R) calculation model MjPrediction probability：
Wherein πij(k-1) represent the k-1 moment from model MiIt is switched to model MjTransition probability, ui(k-1) when representing k-1
Carve model MiProbability；
The k-1 moment is calculated from model MiIt is switched to model MjMixing probability：
Calculate k-1 moment model MsjAdmixture estimation：
WhereinRepresent k-1 moment model MsiFiltering estimate；
Calculate k-1 moment model MsjMixing covariance estimation：
Wherein：Pi(k-1 | k-1) is the k-1 moment from model MiCovariance filtering estimate.
As the further scheme of this case invention, step 4 is specially：
According to the characteristic of noise in submodel and whether linear, corresponding filtering algorithm is selected, by what is obtained in step 3
Admixture is estimatedWith mixing covariance estimationCorresponding filtering is brought into as input
In algorithm, and filter parallel is worked, obtain model MjState filtering valueCovariance filter value Pj(k|k)、
New breath vjAnd observation covariance matrix S (k)j(k)。
As the further scheme of this case invention, step 5 is specially：
Step (5.1), according to the new breath v obtained in step (4)jAnd observation covariance matrix S (k)j(k) calculation model Mj
Likelihood function Λj(k)：
Step (5.2), according in step (5.1) obtain model MjLikelihood function Λj(k) model M is updatedjProbability：
As the further scheme of this case invention, step 6 is specially：
Step (6.1), according to the model probability u obtained in step (5.2)j(k), computation model probability rate of change and it is taken
Hyperbolic sine inverse function value：
Hyperbolic sine inverse function is increasing function, whenWhen,It is general in view of shifting
The nonnegativity of rate value, hyperbolic sine inverse function is translated up into a unit, obtains correction function：
L (k)=l (k)+1
Step (6.2), the correction function amendment state-transition matrix obtained according to step (6.1)：
πij' (k)=L (k) πij(k-1)
Step (6.3), obtained new state-transition matrix is normalized：
As the further scheme of this case invention, step 7 is specially：
The model M obtained using step 5jProbability uj(k) and step 4 obtains state filtering valueAnd association side
Poor filter value Pj(k | k), summation is weighted, obtains total filter valueAnd total covariance estimate P (k |
k)：
The beneficial effects of the invention are as follows：By considering posterior information (model probability rate of change), state transfer is analyzed in real time
The hyperbolic sine inverse function of the change of the generation of matrix, then computation model probability transformation rate, utilizes hyperbolic sine inverse function
Advantageous property on-line amending transfering state matrix, solving transfering state matrix in traditional interacting multiple algorithm can not in real time more
Newly, the problem of causing tracking result inaccuracy, the precision of target following is improved.
Brief description of the drawings
Fig. 1 is general flow chart of the present invention；
Fig. 2 is the adaptive interacting multiple algorithm flow chart of transition probability of the present invention；
Fig. 3 is the flow chart based on hyperbolic sine inverse function amendment transition probability matrix in the present invention.
Embodiment
The present invention is described in detail with reference to the accompanying drawings and detailed description.
In maneuvering target tracing problem, the filter tracks poor performance of single dynamic model is being comprised only, and IMM is calculated
Method is by establishing object movement Models Sets, by the system modeling that object is representative, adoption status transfer matrix
Carry out implementation model interaction, the input using interaction results as parallelism wave filter, track the motion state under different models respectively, so
The probability of each model is asked for respectively using maximum likelihood function afterwards, by filter result weighted sum.Transfering state as can be seen here
Matrix has a great impact to IMM algorithms, but transfering state matrix is set according to prior information in traditional IMM algorithms
, and can not change.This can cause tracking result to be influenceed the movement locus that can not truly reflect object by subjective factor, cause
Tracking accuracy declines.
As Figure 1-3, the adaptive Interactive Multiple-Model method for tracking target of a kind of transition probability of the present invention, specifically according to
Following steps are implemented：
Step 1, the movement locus measured value of target is gathered by sensor and establishes the motion state Models Sets of target, had
Body is：
Step (1.1), the measured value by sensor collection target trajectory.
Step (1.2), establish target movement model collection M=[M1,M2,…,Mr] (r is the probability of submodel in Models Sets),
Submodel needs the main motion model of coverage goal, but quantity can not be too much in order to avoid unnecessary competition between causing model, influences
Tracking accuracy.
J-th of model Mj(j=1,2 ... state equation r) is：
Xj(k+1)=Φj(k)Xj(k)+Gj(k)Wj(k)
Measurement equation is：
Z (k)=H (k) X (k)+V (k)
Wherein Xj(k+1) k+1 moment model Ms are representedj1 × n dimension state vectors of middle target, Z (k) represent k moment model Msj
1 × m observation vectors of middle target, Φj(k-1) it is model MjState-transition matrix, GjRepresent model MjNoise matrix, H is
Observing matrix, Wj(k) it is model MjState-noise sequence, V (k) are model MsjMeasuring value noise sequence, k represent the sampling time；
Step 2, probability, Model transfer probability matrix according to priori setting initial model, it is specially：
Step (2.1), according to priori, subjectively in setting steps 1 each submodel probability ui(1) (i=
1,2,…r)；
Step (2.2), according to priori, the subjectively initial Markov state transfer in setting steps 1 between model
Matrix Π：
Wherein πijExpression is transferred to model j probability from model i,And the master couple of claimed condition transfer matrix
Diagonal element is dominant.
The adaptive IMM algorithms particular flow sheet of transition probability matrix as shown in Figure 2 (including step 3,4,5,6,7), has
Body is：
Step 3, according to state-transition matrix motion model is instructed to carry out input interaction, according in state-transition matrix Π
Transition probability πij(k-1), model MiProbable value ui(k-1) (i=1,2 ... r) calculation model MjPrediction probability：
Wherein πij(k-1) represent the k-1 moment from model MiIt is switched to model MjTransition probability, ui(k-1) when representing k-1
Carve model MiProbability；
The k-1 moment is calculated from model MiIt is switched to model MjMixing probability：
Calculate k-1 moment model MsjAdmixture estimation：
WhereinRepresent k-1 moment model MsiFiltering estimate；
Calculate k-1 moment model MsjMixing covariance estimation：
Wherein：Pi(k-1 | k-1) is the k-1 moment from model MiCovariance filtering estimate；
Step 4, filter parallel work, are tracked to the target trajectory under each submodel, are specially：
According to the characteristic of noise in submodel and whether linear, corresponding filtering algorithm is selected, by what is obtained in step 3
Admixture is estimatedWith mixing covariance estimationCorresponding filtering is brought into as input
In algorithm, and filter parallel is worked, obtain model MjState filtering valueCovariance filter value Pj(k|k)、
New breath vjAnd observation covariance matrix S (k)j(k)。
The probability of step 5, each motion model of renewal, it is specially：
Step (5.1), according to the new breath v obtained in step (4)jAnd observation covariance matrix S (k)j(k) calculation model Mj
Likelihood function Λj(k)：
Step (5.2), according in step (5.1) obtain model MjLikelihood function Λj(k) model M is updatedjProbability：
Step 6, according to the model probability after renewal, state-transition matrix is modified using hyperbolic sine inverse function,
Flow chart is as shown in figure 3, be specially：
Step (6.1), according to the model probability u obtained in step (5.2)j(k), computation model probability rate of change and it is taken
Hyperbolic sine inverse function value：
Hyperbolic sine inverse function is increasing function, whenWhen,It is general in view of shifting
The nonnegativity of rate value, hyperbolic sine inverse function is translated up into a unit, obtains correction function：
L (k)=l (k)+1
Step (6.2), the correction function amendment state-transition matrix obtained according to step (6.1)：
πij' (k)=L (k) πij(k-1)
Step (6.3), obtained new state-transition matrix is normalized：
Step 7, according to motion model probability output interact, real-time tracking maneuvering target, be specially：Obtained using step 5
Model MjProbability uj(k) and step 4 obtains state filtering valueAnd covariance filter value Pj(k | k), added
Power summation, obtains total filter valueAnd total covariance estimate P (k | k)：
The situation of change of posterior information (model interconversion rate) analysis state-transition matrix is obtained after being updated by model probability,
Then state-transition matrix is corrected using the hyperbolic sine inverse function of model probability rate of change.And then realize and believed by posteriority
Breath corrects state-transition matrix in real time, reduces the interference that wrong prior information interacts to mode input, improves based on interaction
The precision of the target tracking algorism of multi-model.
Described above is present pre-ferred embodiments, for the ordinary skill in the art, according to the present invention's
Teaching, in the case where not departing from the principle of the present invention and spirit, the changes, modifications, replacement and the change that are carried out to embodiment
Type is still fallen within protection scope of the present invention.
Claims (8)
1. the adaptive Interactive Multiple-Model method for tracking target of a kind of transition probability, it is characterised in that comprise the following steps：
Step 1, the movement locus measured value of target is gathered by sensor and establishes the motion model collection of target；
Step 2, probability, Model transfer probability matrix according to priori setting initial model；
Step 3, according to state-transition matrix submodel is instructed to carry out input interaction；
Step 4, filter parallel work, are tracked to the target trajectory under each submodel；
Step 5, the probability according to each motion model of the likelihood function of submodel renewal；
Step 6, according to the model probability after renewal, state-transition matrix is modified using hyperbolic sine inverse function；
Step 7, according to motion model probability output interact.
A kind of 2. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 1 is specially：
Step (1.1), the measured value by sensor collection target trajectory；
Step (1.2), establish target movement model collection M=[M1,M2,…,Mr] (r is the probability of submodel in Models Sets), submodule
Type needs the main motion model of coverage goal, but quantity can not be too much in order to avoid unnecessary competition between causing model, influences to track
Precision；
J-th of model Mj(j=1,2 ... state equation r) is：
Xj(k+1)=Φj(k)Xj(k)+Gj(k)Wj(k)
Measurement equation is：
Z (k)=H (k) X (k)+V (k)
Wherein Xj(k+1) k+1 moment model Ms are representedj1 × n dimension state vectors of middle target, Z (k) represent k moment model MsjMiddle mesh
Target 1 × m observation vectors, Φj(k-1) it is model MjState-transition matrix, GjRepresent model MjNoise matrix, H for observation
Matrix, Wj(k) it is model MjState-noise sequence, V (k) are model MsjMeasuring value noise sequence, k represent the sampling time.
A kind of 3. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 2 is specially：
Step (2.1), according to priori, subjectively in setting steps 1 each submodel probability ui(1) (i=1,
2,…r)；
Step (2.2), according to priori, the subjectively initial Markov state transfer matrix in setting steps 1 between model
Π：
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A kind of 4. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 3 is specially：
Transition probability π in state-transition matrix Πij(k-1), model MiProbable value ui(k-1) (i=1,2 ... r) count
Calculate model MjPrediction probability：
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Wherein：Pi(k-1 | k-1) is the k-1 moment from model MiCovariance filtering estimate.
A kind of 5. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 4 is specially：
According to the characteristic of noise in submodel and whether linear, corresponding filtering algorithm is selected, the mixing that will be obtained in step 3
State estimationWith mixing covariance estimationCorresponding filtering algorithm is brought into as input
In, and filter parallel is worked, obtain model MjState filtering valueCovariance filter value Pj(k | k), new breath
vjAnd observation covariance matrix S (k)j(k)。
A kind of 6. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 5 is specially：
Step (5.1), according to the new breath v obtained in step (4)jAnd observation covariance matrix S (k)j(k) calculation model MjSeemingly
Right function Λj(k)：
<mrow>
<msub>
<mi>&Lambda;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mi>&pi;</mi>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msup>
<mo>|</mo>
<msub>
<mi>S</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<msup>
<mo>|</mo>
<mrow>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
<mi>exp</mi>
<mo>{</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msup>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>T</mi>
</msup>
<msubsup>
<mi>S</mi>
<mi>j</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mo>}</mo>
</mrow>
Step (5.2), according in step (5.1) obtain model MjLikelihood function Λj(k) model M is updatedjProbability：
<mrow>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>&Lambda;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>c</mi>
<mo>&OverBar;</mo>
</mover>
<mi>j</mi>
</msub>
<mo>/</mo>
<mi>c</mi>
</mrow>
<mrow>
<mi>c</mi>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>r</mi>
</munderover>
<msub>
<mi>&Lambda;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>c</mi>
<mo>&OverBar;</mo>
</mover>
<mi>j</mi>
</msub>
<mo>.</mo>
</mrow>
A kind of 7. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 6 is specially：
Step (6.1), according to the model probability u obtained in step (5.2)j(k), computation model probability rate of change and its hyperbolic is taken
Sinusoidal inverse function value：
<mrow>
<msub>
<mover>
<mi>u</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>l</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>ln</mi>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>j</mi>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>+</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>j</mi>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msqrt>
<mo>)</mo>
</mrow>
</mrow>
Hyperbolic sine inverse function is increasing function, whenWhen,In view of transition probability value
Nonnegativity, hyperbolic sine inverse function is translated up into a unit, obtains correction function：
L (k)=l (k)+1
Step (6.2), the correction function amendment state-transition matrix obtained according to step (6.1)：
πij' (k)=L (k) πij(k-1)
Step (6.3), obtained new state-transition matrix is normalized：
<mrow>
<msub>
<mi>&pi;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<msub>
<mi>&pi;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>r</mi>
</munderover>
<msup>
<msub>
<mi>&pi;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>.</mo>
</mrow>
A kind of 8. adaptive Interactive Multiple-Model method for tracking target of transition probability as claimed in claim 1, it is characterised in that
Step 7 is specially：
The model M obtained using step 5jProbability uj(k) and step 4 obtains state filtering valueAnd covariance filter
Wave number Pj(k | k), summation is weighted, obtains total filter valueAnd total covariance estimate P (k | k)：
<mrow>
<mover>
<mi>X</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>r</mi>
</munderover>
<msub>
<mover>
<mi>X</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>*</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
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<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>r</mi>
</munderover>
<mo>{</mo>
<msub>
<mover>
<mi>P</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>&lsqb;</mo>
<msub>
<mover>
<mi>X</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mover>
<mi>X</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>&CenterDot;</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mo>&CenterDot;</mo>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>}</mo>
<mo>*</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
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| CN109188443A (en) * | 2018-06-29 | 2019-01-11 | 中国船舶重工集团公司第七〇五研究所 | A kind of passive target tracking method based on Interactive Multiple-Model |
| CN109687844A (en) * | 2018-08-17 | 2019-04-26 | 西安理工大学 | A kind of intelligent maneuver method for tracking target |
| CN109684771A (en) * | 2019-01-11 | 2019-04-26 | 西安电子科技大学 | Maneuvering target state prediction optimization method based on interactive multi-model |
| CN110426671A (en) * | 2019-07-04 | 2019-11-08 | 重庆邮电大学 | Model probability modified IMM method for tracking target and device in real time are based in WSN |
| CN110927727A (en) * | 2019-11-25 | 2020-03-27 | 深圳市智慧海洋科技有限公司 | Target positioning and tracking method and device |
| CN111142100A (en) * | 2020-01-06 | 2020-05-12 | 南京鹰目电子科技有限公司 | Radar target tracking method |
| CN111157983A (en) * | 2020-01-06 | 2020-05-15 | 南京鹰目电子科技有限公司 | Radar target tracking method |
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| CN111837048A (en) * | 2018-05-17 | 2020-10-27 | 罗伯特·博世有限公司 | Method and apparatus for filtering positioning data |
| CN112041699A (en) * | 2018-02-23 | 2020-12-04 | 卡帕奇欧特私人有限公司 | Reconstruction system and method |
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| CN113190960A (en) * | 2021-03-26 | 2021-07-30 | 大连海事大学 | Parallel IMM maneuvering target tracking method based on non-equal-dimension state hybrid estimation |
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| CN114415157A (en) * | 2021-12-30 | 2022-04-29 | 西北工业大学 | Underwater target multi-model tracking method based on underwater acoustic sensor network |
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| CN111837048B (en) * | 2018-05-17 | 2024-05-31 | 罗伯特·博世有限公司 | Method and apparatus for filtering positioning data |
| CN111837048A (en) * | 2018-05-17 | 2020-10-27 | 罗伯特·博世有限公司 | Method and apparatus for filtering positioning data |
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| CN109188443A (en) * | 2018-06-29 | 2019-01-11 | 中国船舶重工集团公司第七〇五研究所 | A kind of passive target tracking method based on Interactive Multiple-Model |
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| CN109684771A (en) * | 2019-01-11 | 2019-04-26 | 西安电子科技大学 | Maneuvering target state prediction optimization method based on interactive multi-model |
| CN109684771B (en) * | 2019-01-11 | 2021-09-03 | 西安电子科技大学 | Maneuvering target state prediction optimization method based on interactive multi-model |
| CN110426671A (en) * | 2019-07-04 | 2019-11-08 | 重庆邮电大学 | Model probability modified IMM method for tracking target and device in real time are based in WSN |
| CN110927727A (en) * | 2019-11-25 | 2020-03-27 | 深圳市智慧海洋科技有限公司 | Target positioning and tracking method and device |
| CN110927727B (en) * | 2019-11-25 | 2021-09-03 | 深圳市智慧海洋科技有限公司 | Target positioning and tracking method and device |
| CN111157983A (en) * | 2020-01-06 | 2020-05-15 | 南京鹰目电子科技有限公司 | Radar target tracking method |
| CN111157983B (en) * | 2020-01-06 | 2022-02-15 | 南京鹰目电子科技有限公司 | Radar target tracking method |
| CN111142100A (en) * | 2020-01-06 | 2020-05-12 | 南京鹰目电子科技有限公司 | Radar target tracking method |
| CN111208506A (en) * | 2020-01-08 | 2020-05-29 | 中国船舶重工集团公司第七二四研究所 | Simplified interactive multi-model tracking method |
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| CN112083377B (en) * | 2020-09-17 | 2023-03-17 | 哈尔滨工程大学 | Position estimation method and device for underwater robot |
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| CN112784506B (en) * | 2021-01-29 | 2023-04-07 | 中国人民解放军空军工程大学 | Reentry maneuvering trajectory target tracking algorithm based on variable structure multi-model |
| CN113190960A (en) * | 2021-03-26 | 2021-07-30 | 大连海事大学 | Parallel IMM maneuvering target tracking method based on non-equal-dimension state hybrid estimation |
| CN113190960B (en) * | 2021-03-26 | 2024-03-12 | 大连海事大学 | Parallel IMM maneuvering target tracking method based on non-equal-dimensional state hybrid estimation |
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Application publication date: 20180216 |