CN104252178B - It is a kind of based on strong motor-driven method for tracking target - Google Patents
It is a kind of based on strong motor-driven method for tracking target Download PDFInfo
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Abstract
It is a kind of based on strong motor-driven method for tracking target, comprise the following steps, parameter initialization, mode input interact, judge that covariance matrix, parallel filtering, model probability are updated, model output is exchanged, smoothing lag is filtered, whether state renewal completes to judge；On the basis of IMM algorithms, the IMM algorithms for recalculating weight are used, i.e. RIMM, this method not only make use of model probability, be also fully utilized by filtering covariance matrix so that tracking precision is higher.In addition, using SRCKF methods in the filter forecasting stage, it utilizes spherical integration criterion and Radial Integrals criterion.Compared to wide UKF algorithms are used in nonlinear filtering, it optimizes sigma point samplings strategy and weight distribution in UKF.Meanwhile, QR is introduced in SRCKF and is decomposed, Matrix power operation is avoided, improves filter wave stability.On the basis of above-mentioned, the present invention introduces smoothing lag filtering again, so as to further increase the real-time and accuracy of target following.
Description
Technical field
The invention belongs to control Scientific Engineering technical field, it is related to the Model Matching and nonlinear filtering of maneuvering target tracking
Technology, for realizing the tracking of strong maneuvering target, and obtains the stability of tracking and the tracking robustness of response state mutation
Raising has been arrived, more particularly to it is a kind of based on strong motor-driven method for tracking target.
Background technology
Maneuvering target tracking is one of study hotspot and difficult point of state estimation, information fusion and target tracking domain.Mesh
Mark tracking problem is actually the tracking filter problem of dbjective state, i.e., the target metric data pair that oneself obtains according to sensor
Dbjective state is accurately estimated.The application of target following is very wide, in national defence, available for ballistic missile defense, empty day
Early warning, firepower control, interception guidance and low-level penetration etc..In civil area, then for traffic control, maritime surveillance, robot
Roading and obstacle avoidance, the tracking traveling of civilian driving car and electron medicine etc..Therefore, many scientists in decades
It is directed to the research of this problem always with engineer.
Target Tracking Problem can trace back to the World War II eve as the one side of scientific technological advance,
When i.e. there is first radar tracking station SCR-28 in the world in nineteen thirty-seven.Afterwards, various radars, infrared, sonar and laser
It is developed and is gradually improved in succession Deng Target Tracking System.
Motor-driven due to moving target can make the penalty of tracking system so that it studies relative difficulty.With the modern times
Aeronautical and space technology is developed rapidly, and the mobility of various aircraft is greatly improved, motor-driven form is no longer limited to acceleration
The forms such as step, Snake-wriggling maneuver, it is also possible to occur that the step of rate of acceleration change is motor-driven etc. more complicated motor-driven.The height of target
It is motor-driven cause tracking filter use target dynamics model and the actual kinetic model of maneuvering target may mismatch, cause with
Track filter divergence, tracking performance degradation.Therefore, improve and the tracking performance of highly maneuvering target is become to be more and more important
The problem of.
At present, although the research on maneuvering target tracking has achieved a series of achievements in the world, due to it is motor-driven go out
Existing complexity, randomness and diversity, no matter in theoretical or practice, high speed and high maneuvering target tracking has higher skill
The problem of art difficulty is still one challenging.
Up to the present, it has been proposed that the method for many analysis target trackings, but most of method also exist it is certain
Defect.The motion model and adaptive-filtering for setting up target are two key components of maneuvering target tracking.Early stage research mesh
The algorithm of tracking is marked, mainly including simple object module and linear Kalman filter method.
The main task of object module research is to set up one to meet actual object module, so as to the filter for maneuvering target
Ripple tracking provides the maneuvering target model more accurately tallied with the actual situation.By the research of domestic and international decades, obtain
A series of classical target following models are arrived, being summed up mainly has：At the uniform velocity (Constant Velocity) model, it is even plus
Fast (Constant Acceleration) model, Singer models, Turn Models (Coordinate Turn), " current " statistics
Model (Current Statistical Model) and jerk models etc..CV, CA model are a priori to define the motion of target
For simply at the uniform velocity or uniformly accelerated motion, motor-driven to be counted as a kind of random input, its size is embodied in the association of process noise
In variance matrix.Target has good tracking effect without in the case of motor-driven.But in fact, target remains a constant speed or even acceleration
The situation of motion seldom occurs, and when the kinetic characteristic of target is complicated, the two models can bring about very big model error.CT
The movement characteristic of model is that the angular speed and velocity magnitude of target keep constant, and velocity attitude is changing constantly, therefore is one
Plant the special motion of automobile.The model is because needing the priori of target maneuver, so be seldom used alone in actual applications,
It is more to apply in Multiple Models Algorithm.1969, R.Singer proposed the zero-mean of maneuvering target, single order time correlation
Maneuver Acceleration model (i.e. Singer models), the Maneuver Acceleration of target is expressed as the knot of stochastic regime noise driving by he
Really, rather than statistical iteration white noise, and thus set up maneuvering target motion statistical model.Due to Singer models pair
It is usually not meet to obey equally distributed hypothesis in Maneuver Acceleration average for the probability density function of zero-sum Maneuver Acceleration
Actual, therefore, it is only applicable at the uniform velocity move with the target in even acceleration range.Nineteen eighty-three, professor Zhou Hongren proposes machine
Moving-target " present statistical model, it is believed that when target is motor-driven with a certain acceleration, the acceleration value of subsequent time is limited
And can only be in the neighborhood of " current " acceleration, the model is substantially a kind of time correlation model of Non-zero Mean, and it adds
The rayleigh distributed description of speed " current " probability density amendment, average is the predicted value of " current " acceleration, and random maneuver adds
Speed still conforms to single order time-duration-dependent process on a timeline.Compared with Singer models, " present statistical model is considered to work as
The preceding moment it is specific motor-driven, more truly reflect target maneuver scope and Strength Changes, be a kind of practical model, compare
The reality for being suitable for target is motor-driven.Kishore in 1997 proposes a kind of jerk models, in the state component of target maneuver model
In add three order derivatives of target location, i.e. jerk is expressed as a zero-mean by the rate of change or jerk, Kishore of acceleration
White-noise process, can obtain more accurately estimating acceleration with this, so as to reach the tracking to maneuvering target.Due to it
One-dimensional state component is added, therefore amount of calculation also increases therewith.In addition, on the basis of above-mentioned model, Blom and Shalom are carried
The interactive multi-model process (IMM) gone out is by setting up a group model collection come Parallel Tracking target, and the switching between model is assumed
Markov process is obeyed, the final estimation of target state is the weighting of all model filter outputs for participating in estimation
With.At present, IMM is widely used in maneuvering target tracking field.
Filtering algorithm is another important part in target following, and most classical is Kalman filtering algorithm, card
Kalman Filtering theory is to be proposed by R.E.Kalman in nineteen sixty at first, and this is one kind using unbiased minimum variance as optimal standard
Then, and using the linear filtering theory of recursive algorithm, new state estimation, amount of calculation and storage are provided with the time by recurrence equation
Amount is small, is easier to meet the requirement calculated in real time.Because it is that minimum mean square error criterion is utilized in the case of linear Gauss
The estimation of dbjective state is obtained, but in practice, the relation observed in many cases between data and target dynamic parameter is non-thread
Property.Due to maneuvering target track question in practical situations both non-linear so that the state estimation of nonlinear system no matter
It is in theory or all particularly significant in engineering.From 1970s, the State Estimation Theory of nonlinear system is obtained
Develop on a large scale very much, current most popular nonlinear filtering algorithm has：EKF (EKF), unscented kalman filter
(UKF), particle filter (PF), volume Kalman filtering (CKF) etc..
At present, a kind of more common nonlinear filtering algorithm is EKF, but when handling strongly non-linear system, EKF can cause
Very big linearized stability, causes filtering accuracy to reduce, or even diverging.UKF algorithms by using one group of determination weight sampling point
To approach the distribution function of stochastic variable, so as to avoid the error brought to nonlinear equation as approximate linearization and complexity is refined
Gram than matrix calculating.When this group of sampled point is propagated by nonlinear function, the statistical property of nonlinear function, precision are captured
Second order is can reach, but reasonably selection parameter can be only achieved preferable filter effect to UKF needs.CKF is a kind of new non-
Linear filtering algorithm, it is carried out approximately, i.e., using the weights such as one group using numerical integration principle to the posterior probability of dbjective state
Volume point calculate posterior probability density.Compared to EKF, UKF scheduling algorithm, CKF algorithms have preferably estimation performance, and
Its realization is more simple.But CKF is in recursive process, the shortcomings of there is computationally intensive and unstable numerical value is used for reference flat for this
The thought of root filtering, it is proposed that a kind of volume kalman filter method based on square root filtering, i.e. SRCKF.
The content of the invention
It is an object of the invention to overcome the shortcomings of above-mentioned prior art, propose a kind of based on the fixation for recalculating weight
Postpone square root volume filtering method, this method can preferably realize the tracking to maneuvering target, can not only press down well
Filtering divergence processed, and can reduce the complexity of calculating, and algorithm stability and the tracking robustness of state mutation
It is improved, there is certain meaning to the real time analysis of further maneuvering target tracking.
The technical scheme is that using proposed by the present invention based on recalculating weight, with fixed delay square
Root volume Kalman Algorithm is nonlinear filtering algorithm, and the interactive multi-model constituted using CV, CA, CT is object module, to machine
Moving-target is tracked.Parameter first to model is initialized, then with the interactive multi-model for recalculating weight
Algorithm (RIMM), RIMM algorithms and IMM algorithms are essentially the same, and topmost is not both to input interaction and output interaction stage,
Input and output of the IMM algorithms using model probability as weight to several parallel wave filters are handled, and RIMM algorithms
Model probability and filtering covariance matrix are used simultaneously as weight.On this basis, with square root volume Kalman's method
(SRCKF) it is filtering algorithm, finally after RIMM fusion output state estimations, using smoothing technique, i.e. fixed delay method,
Further improve the tracking performance of maneuvering target.
To achieve the above object, the present invention uses following technical scheme, a kind of based on strong motor-driven method for tracking target, bag
Include following steps：
(1) parameter initialization：Motion conditions for target choose corresponding model, then to the corresponding model of selection
State-transition matrix, measurement matrix, process noise covariance matrix, measure noise covariance matrix, state-transition matrix and
Forecast model probability matrix is initialized；
(2) mode input is interacted：The model j Jing Guo initialization process is chosen, and inputs the original state at k moment, is passed through
The state of interaction k-1 moment each model filter obtains mixing covariance matrix value；Calculation formula is as follows：
ζ (k-1 | k-1)=μij(k-1|k-1)M0j(k-1)Mij -1(k-1)Fj
In formula, μi|j(k-1)=P { mi(k-1)|mj(k),zk-1}
μij(k-1 | k-1) it is model probability matrix, FjIt is state-transition matrix, P is Model transfer probability matrix, ζ (k-1 |
K-1) it is weighting weight of the model probability matrix with filtering covariance matrix, zk-1For observing matrix, M0j(k-1) it is Model transfer
The hybrid weight of probability and covariance matrix, Mij -1(k-1) it is to recalculate the inverse matrix of weight matrix, cjFor normalized parameter,
T represents transposed matrix；
(3) covariance matrix is judged：According to the mixing covariance matrix value obtained in step (2), i.e.,
When the matrix is that real symmetric tridiagonal matrices, the i.e. matrix meet orthotropicity, continue next step, when covariance matrix is unsatisfactory for positive definite
Property, then return in Parameter Initialization procedure, from new iteration；
(4) parallel filtering：According to the k-1 moment model primary condition obtained in step (2), using SRCKF filtering methods,
Calculate the state estimation of each model of k momentWith covariance matrix Pj(k|k)；
(5) model probability updates：Utilize the forecast model probability μ at the k-1 moment obtained in step (2)ij(k-1), and lead to
Cross following formula and obtain model modification probability μj(k)；
With following formula computation model update probability μj(k)
Wherein c is normaliztion constant,
In formula, For k moment each model
Likelihood function, pljFor Model transfer probability matrix；
(6) model output interaction：Using combining the final estimation that each wave filter obtains the dbjective state of k moment wave filter groups
And its covariance matrix, it is specific as follows：
(7) smoothing lag is filtered：On the basis of step (6), delay d is drawn with smoothing lag method
The smooth valuation of delay of sampling instant dbjective state and corresponding error co-variance matrix, it is specific such as following formula：
Therefore, being augmented to dbjective state, make：
In formula：
Introducing enhanced situation variableAfterwards, the state equation and observational equation of system are rewritten as：
In formula：
The process noise covariance matrix of system is modified toFor extension
The control matrix of dimension, wk-1It is interval for acceleration,For the observing matrix after extension dimension, vkFor observation noise matrix, zk
For observation, E is represented | the end value obtained under primary condition below；
(8) whether state updates completes to judge：Terminated according to the run time of setting track come whether decision state updates,
If run time does not terminate, return to step (4) continues through parallel filtering to obtain the predicted value of state, then by measurement equation
Obtained measuring value, is modified to status predication value, obtains state estimation, while covariance matrix is have updated, until
Run time terminates；If end of run, final filter forecasting track is exported；
Wherein described in step (2) is based on the admixture estimate and covariance value for recalculating weight：Introduce
Calculation formula is as follows：
In formula,For normalized parameter.M0j(k-1) it is Model transfer probability and covariance matrix
Hybrid weight, Mij -1(k-1) it is to recalculate the inverse matrix of weight matrix, FjFor state-transition matrix, QjFor process noise square
Battle array, μijFor model probability, PiFor Model transfer probability matrix；
The measurement in model modification new probability formula wherein described in step (5) newly ceases γj(k) with covariance matrix Ωj(k) it is：
Wherein variance matrix square root Szz(k | k-1) determined as the square root volume filtering method described in step (4), z (k)
For observation,For observation predicted value.
Utilization SRCKF filtering methods in the step (4), are comprised the following steps that：
(4a) time updates, and first seeking volume point, (i=1,2,3 ... m)
In formula, m=2n, parameterIt is given by：
Volume point is after propagation
Status predication estimate is
The square root of predicated error variance matrix is S (k | k-1)=qr ([X*(k|k-1) SQ(k)])
(4b), which is measured, to be updated, and first seeks volume point
Calculate the volume point z after propagatingi(k | k-1)=h (xi(k|k-1))
(4c) measurement prediction
Calculate the square root S of variance matrixzz(k | k-1)=qr ([z (k | k-1) SR(k)])
Seek cross covariance battle array Pxz(k | k-1)=x (k | k-1) zT(k|k-1)
(4d) gain battle array be
(4e) more new state
Update square root S (k | k)=qr ([x (k | k-1)-K (k) z (k | k-1) K (k) S of variance matrixR(k)])。
State Estimation is augmented in the step (7)And corresponding error co-variance matrixIt is to utilize SRCKF algorithms
It is rightIt is filtered, in filtering implementation process, is substituted using the state initial value after being augmented and covariance initial value former
The initial value come；In addition, with the state transition function after being augmented, observing matrix function, noise matrix replaces original corresponding letter
Number；
Specific formula is as follows：
Smoothing lag valuation and corresponding error co-variance matrix can be obtained, i.e.,：
E is represented | base below
The end value obtained under the conditions of this.
The present invention uses above-mentioned technical proposal, with advantages below：On the basis of IMM algorithms, use and recalculated
The IMM algorithms of weight, i.e. RIMM, this method not only make use of model probability, be also fully utilized by filtering covariance matrix so that
Tracking precision is higher.In addition, using SRCKF methods in the filter forecasting stage, it utilizes spherical integration criterion and Radial Integrals
Criterion.Compared to wide UKF algorithms are used in nonlinear filtering, it optimizes sigma point samplings strategy and weight in UKF
Distribution.Meanwhile, QR is introduced in SRCKF and is decomposed, Matrix power operation is avoided, improves filter wave stability.In above-mentioned base
On plinth, the present invention introduces smoothing lag filtering again, so as to further increase the real-time and accuracy of target following.
The present invention is described in further details below with reference to accompanying drawing.
Brief description of the drawings
Fig. 1 is the FB(flow block) of the present invention；
Fig. 2 is target following real trace schematic diagram in the embodiment of the present invention；
Fig. 3 is target following filtering track result figure in the embodiment of the present invention；
Fig. 4 is the position root-mean-square error result figure of filtering track in the embodiment of the present invention；
Fig. 5 is the speed root-mean-square error result figure of filtering track in the embodiment of the present invention.
Embodiment
Further carried out specifically based on strong motor-driven method for tracking target to a kind of with reference to the accompanying drawings and examples
It is bright.
Similar with IMM algorithms, RIMM algorithms are also classified into four steps.That is input interaction, filtering, model probability is updated, and output is handed over
Mutually.If mi(k) represent at k moment, model miWith system model matching, μi(k) be i-th model (i=1,2,3 ..., N) when
Between interval (k-1, k) in the probability that works.Therefore have
In formula, zkThe set of all observations before the expression k moment, it is assumed that the transfer between each model meets Markov
Rule, then Model transfer probability can be expressed as：
Pij=P { mj(k)|mi(k-1)}
The state estimation that is obtained with j-th of model filtering of note k moment and corresponding covariance matrix are respectivelyPj
(k|k).RIMM-SRCKF-FLS methods are mainly made up of following 9 steps.
As shown in figure 1, the present invention's comprises the following steps that：
(1) parameter initialization：Selection Model, and the state-transition matrix of the model to selection, measurement matrix, process noise
Covariance matrix, measures noise covariance matrix, state-transition matrix and forecast model probability matrix are initialized；Choose
For example strong maneuvering target of target can choose Turn Models, current statistical model etc., weak maneuvering target can choose at the uniform velocity model
With even acceleration model, the present embodiment is mainly for a snakelike motor-driven track, from uniform motion model (CV), uniformly accelerated motion
Model (CA), Turn Models (CT) constitute interactive multi-model structure.Initialize each motion model parameters；
(2) mode input is interacted：The model j Jing Guo initialization process is chosen, and inputs the original state at k moment, is passed through
The state of interaction k-1 moment each model filter obtains mixing covariance matrix value；Calculation formula is as follows：
ζ (k-1 | k-1)=μij(k-1|k-1)M0j(k-1)Mij -1(k-1)Fj
In formula, μij(k-1)=P { mi(k-1)|mj(k),zk-1}
In formula,
μij(k-1 | k-1) it is model probability matrix, FjIt is state-transition matrix, P is Model transfer probability matrix, ζ (k-1 |
K-1) it is weighting weight of the model probability matrix with filtering covariance matrix, zk-1For observing matrix, M0j(k-1) it is Model transfer
The hybrid weight of probability and covariance matrix, Mij -1(k-1) it is to recalculate the inverse matrix of weight matrix, cjFor normalized parameter,
T represents transposed matrix；
In the input interaction stage, using the method for recalculating weight, i.e., the stage is interactively entered in model, using obtaining again
The weight factor obtained obtains the mixing Initial state estimation value and mixing covariance initial value of target, while it is general to obtain model prediction
Rate and hybrid weight.Due to next delay smothing filtering is fixed to status predication value and covariance matrix, thus together
When to model state variable carry out be augmented processing, the target movement model state variable after being augmented；
(3) covariance matrix is judged：According to the mixing covariance matrix value obtained in step (2), i.e.,
Orthotropicity is met if the matrix is real symmetric tridiagonal matrices, the i.e. matrix, the decision condition of real symmetric matrix is its all feature
Value have to be larger than zero.Otherwise covariance matrix is unsatisfactory for orthotropicity, then in Parameter Initialization procedure, readjusts covariance square
The initial value of battle array, can be adjusted by increasing matrix value in proportion.Herein, judge covariance matrix whether positive definite it is main
Purpose is in order to prevent the filtering divergence in step (4), so as to cause BREAK TRACK.
(4) parallel filtering：According to the k-1 moment model primary condition obtained in step (2), using SRCKF filtering methods,
Calculate the state estimation of each model of k momentWith covariance matrix Pj(k|k)；Specifically follow the steps below
's：
(a) time updates, and first seeking volume point, (i=1,2,3 ... m)
In formula, m=2n, parameterIt is given by：
Volume point is after propagation
Status predication estimate is
The square root of predicated error variance matrix is S (k | k-1)=qr ([X*(k|k-1) SQ(k)])
(b) measure and update, first seek volume point
Calculate the volume point z after propagatingi(k | k-1)=h (xi(k|k-1))
(c) measurement prediction
Calculate the square root S of variance matrixzz(k | k-1)=qr ([z (k | k-1) SR(k)])
Seek cross covariance battle array Pxz(k | k-1)=x (k | k-1) zT(k|k-1)
(d) gain battle array is
(e) more new state
Update square root S (k | k)=qr ([x (k | k-1)-K (k) z (k | k-1) K (k) S of variance matrixR(k)])；
Mainly on the basis of above-mentioned steps, square root volume kalman filter method is used in the filter forecasting stage.
State dimension in the present embodiment is 6 dimensions, then volume points elect 12 as, and weights are the inverse that volume is counted.Pass through filter forecasting
Afterwards, the square root of predicated error variance matrix is obtained, so that it is determined that fading factor, is then decided whether to trace back to by fading factor
Previous step, i.e., adaptively adjust predicting covariance by fading factor.Then by obtained filtering gain value, shape is finally obtained
State valuation and covariance updated value, and predicated error variance matrix square root updated value.
(5) model probability updates：Utilize the forecast model probability μ at the k-1 moment obtained in step (2)ij(k-1), and lead to
Cross following formula and obtain model modification probability μj(k)；
The new breath γ of measurement is calculated firstj(k) with covariance matrix Ωj(k)
Then following formula computation model update probability μ is usedj(k)
Wherein c is normaliztion constant,
In formula, For k moment each model
Likelihood function, pliFor model probability transfer matrix.
(6) model output is exchanged：Using combining the final estimation that each wave filter obtains the dbjective state of k moment wave filter groups
And its covariance matrix, it is specific as follows：
Before the model interaction output stage, by the weight factor for recalculating weight used, melting for each wave filter is obtained
Conjunction state estimate and covariance estimate；
(7) smoothing lag is filtered：On the basis of step (6), delay d is drawn with smoothing lag method
The smooth valuation of delay of sampling instant dbjective state and corresponding error co-variance matrix, it is specific such as following formula：
Therefore, being augmented to dbjective state, make：
In formula：
Introducing enhanced situation variableAfterwards, the state equation and observational equation of system are rewritten as：
In formula： The process noise covariance matrix of system is modified toTo extend the control of dimension
Matrix processed, wk-1It is interval for acceleration,For the observing matrix after extension dimension, vkFor observation noise matrix, zkFor observation
Value, E is represented | the end value obtained under primary condition below；
The SRCKF algorithms pair of applying step (4)It is filtered, in filtering implementation process, after being augmented
State initial value and covariance initial value substitute original initial value；In addition, with the state transition function after being augmented, observing matrix
Function, noise matrix replaces original respective function；
State Estimation is augmented using what SRCKF filtering methods can obtain targetAnd corresponding error co-variance matrix
I.e.
Smoothing lag valuation and corresponding error co-variance matrix can be obtained, i.e.,：
E is represented | the end value obtained under primary condition below；
(8) whether state updates completes to judge：Terminated according to the run time of setting track come whether decision state updates,
If run time does not terminate, return to step (4) continues through parallel filtering to obtain the predicted value of state, then by measurement equation
Obtained measuring value, is modified to status predication value, obtains state estimation, while covariance matrix is have updated, until
Run time terminates；If end of run, final filter forecasting track is exported.
The effect of the present invention can be further illustrated by following experiment：
1. simulated conditions：
It is core22.4GHZ, emulated using Matlab in internal memory 2G, WINDOWS XP systems in CPU.
2. emulation content：
A weaving target track is chosen, the run duration of target trajectory is 50 seconds, Monte Carlo simulation times
For 100, the sampling interval is T=1 seconds.In emulation, we are set to the initial state of target：Target motion detailed process be：x0
=[x, vx,ax,y,vy,ay]T, i.e., corresponding occurrence x0==[29.8km, -250m/s, 0,35km, -215m/s, 0]T, mould
Type initialization probability μ0=[0.60.20.2] state-transition matrixes areMeasuring noise square difference battle array is R=
150I, process-noise variance battle array is Q=25I.As shown in Fig. 2 the time of day and speed and acceleration change feelings of target trajectory
Condition.Table 1 show target maneuver process, and table 2 show the relative operation time of each method.
In experiment, the state dimension that we choose is 6 dimensions, then volume points are m=26=12, and weights are ω=112,
Choose d=3 sampling instant.Experimental result is realized motor-driven as shown in figure 3, target prediction track and real trace are coincide
The tracking of target.
From experimental result Fig. 4 and Fig. 5, using the method for tracking target proposed in the present invention, conventional method is overcome
Deficiency, not only can real-time and accurately track target trajectory, can also be improved the tracking robustness of state mutation, together
When enhance the convergence rate and numerical stability of filtering.As shown in Table 2, using the method for tracking target that is filtered based on EKF with
The method for adding smoothing lag is compared, and the latter's complexity is higher.And the method for tracking target based on fixed delay and this hair
The method of bright proposition is compared, because the present invention uses SRCKF filtering methods so that complexity is substantially reduced.Pass through three methods
Entirety, which compares, to be understood, smoothing lag method adds complexity, but SRCKF causes complexity to reduce.Therefore, using this hair
The method of bright proposition, though being added in complexity, overall tracking accuracy is significantly improved.
The target maneuver state of table 1
| Time (s) | Motion state |
| 0-9 | Target is uniform motion |
| 10-22 | Target is anticlockwise campaign |
| 23-25 | Target is uniform motion |
| 26-38 | Target is right rotation campaign |
| 29-50 | Target is uniform motion |
The each method relative operation time of table 2
| RIMM | RIMM-FL | RIMM-SRCKF-FL |
| 1.1070 | 2.5438 | 1.3627 |
Above-mentioned embodiment is only an example of the present invention, does not constitute any limitation of the invention, for example, is sent out with this
Bright method can also be tracked to the track with different motor-driven situations that computer is generated, and to target track in practice
Mark is tracked.
It is exemplified as above be only to the present invention for example, do not constitute the limitation to protection scope of the present invention, it is all
It is to be belonged to the same or analogous design of the present invention within protection scope of the present invention.The portion that the present embodiment is not described in detail
Part and structure belong to the well-known components and common structure or conventional means of the industry, do not describe one by one here.
Claims (1)
1. it is a kind of based on strong motor-driven method for tracking target, it is characterised in that to comprise the following steps：
(1) parameter initialization：Motion conditions for target choose corresponding model, then to the state of the corresponding model of selection
Transfer matrix, measurement matrix, process noise covariance matrix measures noise covariance matrix, state-transition matrix and prediction
Model probability matrix is initialized；
(2) mode input is interacted：The model j Jing Guo initialization process is chosen, and inputs the original state at k moment, passes through interaction
The state of k-1 moment each model filter obtains admixture initial valueWith mixing covariance matrix valueCalculation formula is as follows：
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<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<msub>
<mi>&zeta;</mi>
<mrow>
<mi>i</mi>
<mo>|</mo>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
ζ (k-1 | k-1)=μij(k-1|k-1)M0j(k-1)Mij -1(k-1)Fj
In formula, μij(k-1)=P { mi(k-1)|mj(k),zk-1}
<mrow>
<msubsup>
<mi>P</mi>
<mi>j</mi>
<mn>0</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mtable>
<mtr>
<mtd>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mi>&zeta;</mi>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>{</mo>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>&lsqb;</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msubsup>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
<mn>0</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>&times;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msubsup>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
<mn>0</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>}</mo>
<mo>;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
μij(k-1 | k-1) it is model probability matrix, mi(k) represent at k moment, model miWith system model matching, zkWhen representing k
The set of all observations before carving, FjIt is state-transition matrix, P is Model transfer probability matrix, and ζ (k-1 | k-1) it is model
The weighting weight of probability matrix and filtering covariance matrix, zk-1For observing matrix, M0j(k-1) it is Model transfer probability and association side
The hybrid weight of poor matrix, Mij -1(k-1) it is to recalculate the inverse matrix of weight matrix, T represents transposed matrix；
(3) covariance matrix is judged：According to the mixing covariance matrix value obtained in step (2), i.e.,When this
Matrix is real symmetric tridiagonal matrices, i.e. the matrix meets orthotropicity, continues next step, when covariance matrix is unsatisfactory for orthotropicity,
Then return in Parameter Initialization procedure, from new iteration；
(4) parallel filtering：According to the k-1 moment model primary condition obtained in step (2), using SRCKF filtering methods, calculate
Go out the state estimation of each model of k momentWith covariance matrix Pj(k|k)；
(5) model probability updates：Utilize the forecast model probability μ at the k-1 moment obtained in step (2)ij(k-1), and by with
Lower formula obtains model modification probability μj(k)；
With following formula computation model update probability μj(k)
<mrow>
<msub>
<mi>&mu;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>&Lambda;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
<mi>c</mi>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>l</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>p</mi>
<mrow>
<mi>l</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>&mu;</mi>
<mi>l</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein c is normaliztion constant,
In formula,Λj(k) it is the likelihood of k moment each model
Function, pljTo be transition probability that model l goes to model j；
(6) model output is exchanged：Using combine each wave filter obtain k moment wave filter groups dbjective state final estimation and its
Covariance matrix, it is specific as follows：
<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>N</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>
<msub>
<mi>&zeta;</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>
<mo>|</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mover>
<mo>&Sigma;</mo>
<mi>N</mi>
</mover>
<msub>
<mi>&zeta;</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>{</mo>
<msub>
<mi>P</mi>
<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>
<msup>
<mrow>
<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>
</mrow>
<mi>T</mi>
</msup>
<mo>}</mo>
<mo>;</mo>
</mrow>
(7) smoothing lag is filtered：On the basis of step (6), d sampling of delay is drawn with smoothing lag method
The smooth valuation of delay of moment dbjective state and corresponding error co-variance matrix, it is specific such as following formula：
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
<mo>|</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>E</mi>
<mo>{</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
</mrow>
</msub>
<mo>|</mo>
<msub>
<mi>Z</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
</mrow>
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
<mo>|</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>E</mi>
<mo>{</mo>
<mo>&lsqb;</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
<mo>|</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>&rsqb;</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mi>d</mi>
<mo>|</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>|</mo>
<msub>
<mi>Z</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
</mrow>
Therefore, being augmented to dbjective state, make：
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>T</mi>
</msup>
</mrow>
In formula：
Introducing enhanced situation variableAfterwards, the state equation and observational equation of system are rewritten as：
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msub>
<mo>=</mo>
<mover>
<mi>f</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mover>
<mi>G</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msub>
<msub>
<mi>w</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>T</mi>
</msup>
<mo>+</mo>
<msub>
<mover>
<mi>G</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msub>
<msub>
<mi>w</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>v</mi>
<mi>k</mi>
</msub>
</mrow>
In formula：
The process noise covariance matrix of system is modified to For extension dimension
Control matrix, wk-1It is interval for acceleration,For the observing matrix after extension dimension, vkFor observation noise matrix, zkFor observation
Value, d represents the sampling instant of delay, and E is represented | the end value obtained under primary condition below,It is procedure function
Matrix, procedure function represents the evolution function that dbjective state changes with discrete time,Represent d row therein；N tables
Show wkThe dimension of matrix, variable Qc,kRepresent wkCovariance matrix, variableRepresent process noise transfer matrix；
(8) whether state updates completes to judge：Terminated according to the run time of setting track come whether decision state updates, if fortune
The row time does not terminate, then return to step (4) continues through parallel filtering to obtain the predicted value of state, then is obtained by measurement equation
Measuring value, status predication value is modified, state estimation is obtained, while have updated covariance matrix, until operation
Time terminates；If end of run, final filter forecasting track is exported；
Wherein described in step (2) is based on the admixture estimate and covariance value for recalculating weight：The calculating of introducing
Formula is as follows：
<mrow>
<msup>
<msub>
<mi>M</mi>
<mrow>
<mn>0</mn>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>&mu;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<msubsup>
<mi>M</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>M</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>/</mo>
<msub>
<mi>c</mi>
<mi>j</mi>
</msub>
<msub>
<mi>F</mi>
<mi>j</mi>
</msub>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<msubsup>
<mi>F</mi>
<mi>j</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msub>
<mi>Q</mi>
<mi>j</mi>
</msub>
</mrow>
In formula,For normalized parameter, QjFor process noise matrix, μijFor model probability, Pi(k-1|k-
1) covariance matrixes of the table model i at the k-1 moment；
The measurement in model modification new probability formula in described step (5) newly ceases γj(k) with covariance matrix Ωj(k) it is：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&gamma;</mi>
<mi>j</mi>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>=</mo>
<mi>z</mi>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>-</mo>
<msub>
<mover>
<mi>z</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&Omega;</mi>
<mi>j</mi>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>=</mo>
<msub>
<mi>S</mi>
<mrow>
<mi>z</mi>
<mi>z</mi>
</mrow>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
<msubsup>
<mi>S</mi>
<mrow>
<mi>z</mi>
<mi>z</mi>
</mrow>
<mi>T</mi>
</msubsup>
<mo>(</mo>
<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein variance matrix square root Szz(k | k-1) determined as the square root volume filtering method described in step (4), z (k) is sight
Measured value,For observation predicted value；
State Estimation is augmented in the step (7)And corresponding error co-variance matrixIt is to utilize SRCKF algorithms pairzkIt is filtered, in filtering implementation process, is substituted using the state initial value after being augmented and covariance initial value original
Initial value；In addition, with the state transition function after being augmented, observing matrix function, noise matrix replaces original respective function；
Specific formula is as follows：
<mrow>
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<mo>&OverBar;</mo>
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<mo>^</mo>
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<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>E</mi>
<mo>{</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mi>k</mi>
</msub>
<mo>|</mo>
<msub>
<mi>Z</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
</mrow>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
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<mi>k</mi>
<mo>|</mo>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>E</mi>
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<mrow>
<mo>(</mo>
<msub>
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<mi>x</mi>
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</mover>
<mi>k</mi>
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<mo>-</mo>
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</msup>
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<msub>
<mi>Z</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
</mrow>
Smoothing lag valuation and corresponding error co-variance matrix can be obtained, i.e.,：
<mrow>
<msub>
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<mi>x</mi>
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</mover>
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<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mi>k</mi>
<mi>l</mi>
</msubsup>
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<msub>
<mi>Z</mi>
<mi>k</mi>
</msub>
<mo>}</mo>
</mrow>
E is represented | basic bar below
The end value obtained under part；
Utilization SRCKF filtering methods in the step (4), are comprised the following steps that：
(4a) time updates, and first seeking volume point, (i=1,2,3 ... m)
In formula, m=2n, parameterIt is given by：
Volume point is after propagation
Status predication estimate is
The square root of predicated error variance matrix is S (k | k-1)=qr ([X*(k|k-1)SQ(k)])
Wherein, SQ(k) root mean square of process noise covariance, S are representedR(k) root mean square of observation noise covariance is represented；
(4b), which is measured, to be updated, and first seeks volume point
Calculate the volume point z after propagatingi(k | k-1)=h (xi(k|k-1))
(4c) measurement prediction
Calculate the square root S of variance matrixzz(k | k-1)=qr ([z (k | k-1) SR(k)])
Seek cross covariance battle array Pxz(k | k-1)=x (k | k-1) zT(k|k-1)
(4d) gain battle array be
(4e) more new state
Update square root S (k | k)=qr ([x (k | k-1)-K (k) z (k | k-1) K (k) S of variance matrixR(k)])。
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