CN115379560A - Target positioning and tracking method only under distance measurement information in wireless sensor network - Google Patents
Target positioning and tracking method only under distance measurement information in wireless sensor network Download PDFInfo
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Abstract
The invention relates to a target positioning and tracking method only under distance measurement information in a wireless sensor network, belonging to the technical field of information fusion. For target positioning under only distance measurement information, the invention deduces covariance information of an error item, reconstructs a 0-mean error item, obtains optimal multilateral positioning by using a generalized least square method, and then integrates the optimal multilateral positioning and Kalman filtering to realize target tracking under only distance measurement information. The optimal multilateral positioning precision is superior to that of the prototype multilateral positioning; and the optimal multilateral Kalman filtering method is also superior to other methods, and has excellent tracking consistency under different noise and motion models.
Description
Technical Field
The invention relates to a target positioning and tracking method only under distance measurement information in a wireless sensor network, belonging to the technical field of information fusion.
Background
Wireless sensor networks have been an active area of research. Various sensors, radars, sonars and the like are deployed to form a wireless sensor network, so that the wireless sensor network is widely applied. For example, vehicles carrying a sensor of a millimeter wave radar and other types of sensors realize automatic driving, autonomous underwater vehicles carry sensors to form a wireless sensing network for target tracking, unmanned aerial vehicle path planning and the like. An important application in wireless sensor networks is to realize the positioning and tracking of targets. For positioning and target tracking, due to the easy distributability of the sensor network, the sensors are arranged in areas without advanced infrastructure preparation, and tasks such as environment monitoring, safety monitoring, battlefield information acquisition and the like are injected. In addition, the sensor has the characteristics of small volume, light weight, mobility, convenience in deployment, strong real-time performance and the like, is suitable for being applied to various fields of military affairs, environmental monitoring, medical treatment and the like, and has wide application prospect.
In the wireless sensor network, each sensor can realize the positioning and tracking of the target by acquiring information such as distance measurement information, direction measurement information and the like, however, for some special application scenarios, only distance measurement information can be acquired, so that the problem of the target positioning and tracking of the wireless sensor network under the condition of only the distance measurement information is a research focus. In a radar wireless sensor network, the inverse synthetic aperture radar controls a range gate and an antenna direction by using the output of a distance measurement tracker only; due to the limitation of antenna aperture, high Frequency Ground Wave Radar (HFGWR) cannot provide antenna arrays of hundreds of meters or even thousands of meters to form narrow beams, so that the azimuth resolution is poor, and accurate azimuth information cannot be provided; the low frequency interference used in some passive radars, such as the "silent sentinel" system of rockschid martin, prevents them from providing accurate azimuth vignetting. Furthermore, range-only object location and tracking for multiple static or multiple single static sensors is considered to be a key technology for replacing large aperture antennas with low cost sensor networks in the future. In a wireless sensor network, ranging techniques such as time of arrival (DOA), time difference of arrival (TDOA), angle of arrival (AOA), and Received Signal Strength (RSS) are used to obtain range cropping information to achieve target location. In addition, the distance measurement information and the target state form a nonlinear mapping, namely Euclidean distance, so that only target tracking under the distance measurement information can be regarded as a nonlinear filtering problem. The nonlinear filtering has the problems of instability, complex linearization and the like, and the problems can influence the target tracking accuracy. Therefore, it is necessary to research the target positioning and tracking only under the information of the distance cultivator and improve the target tracking precision.
Disclosure of Invention
The invention provides a target positioning and tracking method only under distance measurement information in a wireless sensor network. For target positioning under the condition of only distance quantity information, the invention deduces covariance information of an error term, reconstructs a 0-mean error term, and then obtains optimal multilateral positioning by using a generalized least square method. By constructing pseudo-visual quantity, integrating optimal multilateral positioning and Kalman filtering, and using standard Kalman filtering to realize target tracking under only distance quantity information, instability of nonlinear filtering is avoided. The method can improve the target positioning precision under only distance measurement information and the target tracking precision left only by distance measurement, and solves the problems of instability and the like caused by nonlinear filtering.
In order to achieve the purpose, the invention is realized by the following technical scheme, which comprises the following specific steps:
where i =1,2,3.. M, denotes the i-th sensor, and m denotes the number of sensors in the wireless sensor network;
a distance nimble representing the view of the i-th sensor station at time k;
is the corresponding measurement noise, which is obeyed to a zero mean with a variance of
(ii) a gaussian distribution of; p x,k And P y,k Respectively representing the position of the target in the x direction and the y direction at the k time; s. the x，i And S y，i Respectively, the position of the ith sensor observation station in the x-direction and the y-direction.
for the distance measurement information:
in the formula
It is indicated that during the movement of the object, at the time k, the range vignetting measured by the ith sensor,
distance measurement information representing the i-th sensor without noise interference.
Is shown as
P x，k And P y,k Respectively representing the position of the target in the x direction and the y direction at the k time; s x,i And S y,j Respectively representing the position of the ith sensor observation station in the x-direction and the y-direction,
is the corresponding measurement noise, which is obeyed to a zero mean with a variance of
A gaussian distribution of (a). It is also expressed by a standard normal distribution as:
wherein w 0 Obeying a standard normal distribution, k i Are coefficients to summarize the distribution of measurement noise at different variances.
From the above analysis, if the true distance information is known
Distance measurement information
Becomes a random variable whose distribution is gaussian centered around the true value:
in practice, only the true distance information can be estimated from the distance observation information. For the function expressed by the expression (10),
is a known parameter, and
is the only unknown variable, thus obtaining
The likelihood function of (a) is as follows:
from the above formula, it is found that
The likelihood function and the Gaussian probability density function have the same mathematical form, namely, the method obtains
Likelihood distribution of (2). Thus, for
Is provided with
The positions of the unknown target and the sensor satisfy the following equation (without accounting for observation noise):
wherein
Distance information of the i-th sensor observation station indicating no noise interference at time k.
Subtracting the ith equation in the above equation from the remaining equations yields:
the above formula is expanded and simplified to be rewritten into a matrix form:
the above formula is abbreviated as:
S＝Mp k +U k
wherein:
p k ＝[p x,k p y,k ] T
in the above formula, S, M is the information known for determination, however U is
Since it contains the true distance
In practice
Is agnostic and thus
Information unknown to the user. In the case of sufficiently considering noise interference, U is a problem causing an assumption that the gaussian-markov theorem is not satisfied. In reality
Are not available. Can only pass through
And distance observation information containing noise interference
To obtain the relation between
Thereby estimating the position of the target.
Thus, by the step (2) for
And with
The following relationships exist:
the relevant information of the distance truth value is removed from the distance observation information containing noise interference, so that the state information of the target is more accurate.
Therefore, will solve for U
First order moment statistical information and second order moment statistical information of
I.e. to solve for
And
in the formula E [. C]D[·]And Cov [. C]Respectively, mathematical expectation, variance and covariance. And further constructing a 0 mean error term e to meet the assumption of the Gaussian-Markov theorem.
So for the first moment of U:
the second moment of U includes a second-order central moment and a second-order mixed moment. For the second central moment of U, there are:
for the second order mixing moments of U:
in summary, U is at r 1：i ，n 1：i Lower second moment
Has an analytic formula R · As follows.
The unified analytic formula (c) is as follows, and on this basis, the scenario of multiple sensors can be extended:
for 1 ≦ i, j ≦ m-1, i = j:
for I is less than or equal to I, j is less than or equal to m-1,i is not equal to j, the following:
i, j respectively represent covariance matrices
The row of (c), m is the total number of sensors,
and
respectively represent covariance matrices
Diagonal elements and non-diagonal elements which together form a covariance matrix
Covariance of U
Is the key to causing the hypothesis that the gaussian-markov theorem is not satisfied. Construction of 0 mean error term to S = Mp k +U k The formula satisfies the assumption of the gaussian-markov theorem. Constructing a random vector e as an error term, which is a mean of 0,
gaussian distribution of (a). The analysis was as follows: since E is 0 mean, there is E (U) + E = U, resulting from 0 mean gaussian distribution symmetry:
for the same specification of sensor, the distribution of the halo noise is the same, i.e. when all k are equal i When the phase difference is equal to each other,
the sum of c is recorded as s:
the corresponding specific S is as follows:
in summary, the optimal polygon positioning formula after reconstruction is:
S＝Mp k +e
wherein
optimal multilateration S = Mp after reconstruction k In + e, its error term e is the 0 mean, which satisfies the assumption of the gaussian-markov theorem. Therefore, by the generalized least squares method, the least variance unbiased estimate of the target state can be obtained:
And 4, constructing pseudo measurement of the target position, and designing a corresponding Kalman filter to realize target tracking.
The second-order statistical information of the error term e provides a basis for realizing target tracking, and a target tracking estimator under the distance quantity is designed.
In the designed target tracking estimator under only distance halo measurement, the estimation of the target state is generated through at least three times of distance measurement and then pseudo measurement is constructed and combined with a Kalman filter. The method mainly comprises the following steps:
obtaining target minimum variance unbiased estimation
Meanwhile, the noise covariance of the target state estimation is:
to this end, the model of the target can be represented as: x k ＝F k-1 X k-1 +G k-1 W k-1 . On the basis of optimal multilateral positioning, pseudo-cut Z of a target is constructed k The measurement equation can expressComprises the following steps: z k ＝HX k +N k . Where Z is k The pseudo quantity nimble of the target position at time k obtained by the optimal multilateral positioning is shown. x is the number of k Is the target state, G k-1 Coefficient matrix of the corresponding dimension, W k-1 As covariance of Q k-1 Process noise of, N k As covariance of R k And W is k-1 Is irrelevant and H is the measurement matrix. R is k And H is as follows:
in order to achieve target tracking, the last step is to use the pseudo-sensing and corresponding noise information in combination with kalman filtering to update the state of the target. Therefore, the expression of the optimal multilateral positioning combined with kalman filtering to realize target tracking is as follows:
1) And (3) state prediction:
2) Prediction of covariance matrix:
3) Calculating a gain matrix:
4) And (3) updating the state:
5) And (3) updating the covariance matrix:
P k/k ＝(I 4×4 -K k H)P k/k-1
the obvious progress and the creative technical characteristics of the invention are as follows: through analysis and derivation, an optimal multilateral positioning method and a method for realizing target tracking by combining Kalman filtering are provided. Covariance information derived from the error term in optimal multilateration is exactly what is needed for noise covariance in kalman filtering. The optimal multilateral positioning is combined with Kalman filtering to realize real-time updating of the measured noise covariance along with the target motion in target tracking, and the method is different from the conventional filtering technology and improves the target tracking precision. A series of simulation researches show that the precision of the proposed optimal multilateral positioning is superior to that of the prototype multilateral positioning. The tracking algorithm combining optimal multilateral positioning and Kalman filtering is also superior to other methods in the aspect of tracking precision, and has excellent tracking consistency under different noise and motion models. The method can realize the positioning and tracking of the target in the wireless sensor network under the condition of only distance measurement, and has high precision and high consistency under different noise and motion models; the technical purpose is realized as follows: only the distance is stripped, and high-precision target positioning and tracking can be realized; the beneficial effects are that: according to the invention, only distance reclamation and measurement information is available, and target positioning and target tracking can be realized; the method solves the problems of target positioning under only distance measurement, instability caused by nonlinear filtering and the like.
Drawings
Fig. 1 is a general flow chart of the present invention.
FIG. 2 is a graph comparing the root mean square error of the positioning method of the present invention with the original multilateral positioning and the Cramer-Lo boundary.
FIG. 3 is a comparison graph of contour lines of the positioning method of the present invention and the original multilateral positioning method.
FIG. 4 is a comparison graph of the near-uniform linear motion trajectory of the target tracking method and different algorithms in the present invention.
FIG. 5 is a graph showing the result of the consistency of the target tracking method in the present invention under the condition of near-uniform linear motion.
FIG. 6 is a diagram showing the comparison of the position root mean square error of the target tracking method with different algorithms and the posterior Clalmelo boundary under the nearly uniform linear motion.
FIG. 7 is a graph comparing the root mean square error of the target tracking method with different algorithms and posterior Clalmelo boundaries under near uniform linear motion.
FIG. 8 is a comparison graph of the constant rate turning motion trajectory of the target tracking method and different algorithms in the present invention.
FIG. 9 is a graph showing the results of the consistency of the target tracking method in the present invention under a constant rate turning motion.
FIG. 10 is a graph comparing the position root mean square error of the target tracking method with different algorithms and the posterior Clarithrome bound under the constant velocity turning motion in the present invention.
FIG. 11 is a graph comparing the root mean square error of the velocity of the target tracking method with different algorithms and the posterior Clalmelo bound under constant rate turning motion.
FIG. 12 is a graph showing the result of the mean normalized estimation error squared under different noise variances in uniform linear motion according to the present invention.
FIG. 13 is a graph of the result of the mean normalized estimation error squared for different noise variances in constant rate cornering maneuvers for the target tracking method of the present invention.
Detailed Description
The invention is further described with reference to the following figures and specific examples.
step 1-1, acquiring distance observation of moving target by using m sensors which are deployed in space and only acquire distance information
The deployment of the sensors is not on the same line, and the deployed sensor distance observation station acquires the distance observation information of the moving target
The distance measurement model of the sensor viewing station is as follows:
the motion state of the object is selected from the motion of a two-dimensional planar structure, where x k ＝[p x,k ，v x,k ，p y,k ，v y,k ] T Each representing the abscissa, the lateral velocity, the ordinate and the longitudinal velocity relative to a fixed coordinate system, the motion model of the object of the invention moving as follows:
x k ＝F k-1 x k-1 +G k-1 w k-1
wherein F k-1 Is a state transition matrix, x k Is composed of p x,k 、p y,k 、v x,k And v y,k Target state of equal correlation quantity, G k-1 A matrix of coefficients of corresponding dimensions, w k-1 Obey a zero mean with a covariance of Q k-1 And process noise of
Are not relevant.
Step 1-2, calculating a second order statistical information of error items in the distance measurement equation set
The 0-mean error term e is reconstructed.
Firstly, the distance measurement model is measured
Analysis is carried out to obtain information about
Likelihood distribution of (2); then, the distance is the second order statistical information of the error term in the equation set
The 0 mean error term e is reconstructed.
Step 1-3, obtaining the minimum variance unbiased estimation of the position by using a generalized least square method;
optimal multilateration S = Mp after reconstruction k + e, its errorThe difference term e is a 0-mean, which satisfies the assumption of the gaussian-markov theorem. Therefore, by the generalized least squares method, the least variance unbiased estimate of the target state can be obtained:
the least-squares unbiased estimate of the target location is shown.
Step 1-4, constructing a pseudo-measurement Z of the target position k And designing a corresponding Kalman filter to realize target tracking.
The second-order statistical information of the error term e provides a basis for realizing target tracking, and a target tracking estimator under only distance measurement is designed. In the designed target tracking estimator under only distance measurement, the estimation of the target state is generated by at least three times of distance measurement and then constructing a pseudo measurement combined with a Kalman filter. In the case of deriving the amount of statistical information for the error term e, the noise covariance of the target state estimate is:
to this end, the model of the target can be represented as: x is the number of k ＝F k-1 x k-1 +G k-1 w k-1 . The metrology model for converting a plurality of non-linear distance measurements to linear measurements with respect to the target location may be represented as: z is a radical of k ＝Hx k +N k . Where Z is k Representing a new measurement, x, at time k with respect to the target position transformed from a plurality of non-linear distance measurements k ＝[p x,k ，v x,k ,p y,k ，v y,k ] T Is the target state, G k-1 Coefficient matrix of the corresponding dimension, w k-1 Is covariance Q k-1 Process noise of, N k As covariance of R k Is equal to W k-1 Is irrelevant and H is the measurement matrix.
Example 2 the calculation was carried out according to the method in example 1, wherein 4 sensor stations are considered, which take only the distance reclamation as distinct, and their positions are (30,40) m, respectively; (0,150) m; (100, 150) m; (100, O) m. The target motion model is uniform linear motion (NCV) and constant velocity of a two-dimensional planeTurning motion (CT), initial state of [60m,1.5m/s,60m,2.5m/s]The running time step is 100 steps, gaussian white noise is added to the state in each time step, the noise is 2-dimensional, the obedient mean value is zero, and the variance is 1m 2 A gaussian distribution of (a).
The Root Mean Square Error (RMSE) is calculated as:
in the above formula, M represents the number of Monte Carlo experiments, N is the time step,
represents the estimated target state, x, at time k of the i-th trial k The real target state of the signal is represented, and | is | · | | | | represents the Frobenius norm of the matrix.
The measure of consistency analysis is the mean Normalized Estimation Error Squared (ANEES), which represents the stability of the filter and is calculated as:
where M represents the number of monte carlo experiments, the ANEES is close to 1 if the estimation error matches the estimation covariance, i.e. the filter is stable.
Assuming that the target starts to move in an initial state of 60m,1.5m/s,60m,2.5m/s, the method is compared with the conventional Unscented Kalman Filter (UKF) method, particle Filter (Particle Filter PF) method and post-test Claritol Bound (PCRLB) respectively, and 100-step and 100-time Monte Carlo experiments are carried out to calculate corresponding RMSE respectively.
For the simulation setup of target positioning, assume that the position deployment of the range sensors is as follows: sensor1 (30m, 100m), sensor2 (40m, 0m), sensor3 (50m, 120m) and sensor4 (0m, 30m). For the simulation results, root Mean Square Error (RMSE), position estimation Bias (Bias), and Cramer-Rao Lower Bound (CRLB) were used as performance evaluation indexes. The calculation formula of the root mean square error and the position estimation deviation is as follows:
here, the
The estimate for the target position obtained in the i-th experiment is shown, and M represents the number of monte carlo simulations. z denotes the actual position of the unknown object.
Fig. 2 shows a comparison of the root mean square error of the positioning of the unknown target [100,80] by the Optimal multilateral positioning (OM) of the positioning method of the present invention and other methods under 10000 monte carlo simulations under different observation noise conditions. As can be seen from fig. 2, under the different observation of nimble noise, the OM has a smaller root mean square error (PM) than the original multilateration (PM) and is closer to the lower boundary of cramer. Furthermore, as the measurement noise increases, the root mean square error of the OM also gets closer to the CRLB.
Fig. 3 shows a plot of the mean value of the target locations and their distribution contour obtained by locating an unknown target with OM and prototype multilateral locating PM under 10000 monte carlo simulations. Wherein, + and o represent OM and the positioning mean value of prototype multilateral positioning respectively, and Δ represents the actual position of the unknown target. The solid line and the dotted line respectively represent the contour lines of the respective distribution of OM and prototype multilateral positioning to the unknown target after positioning. As shown in fig. one, after OM and prototype multilateral positioning are subjected to monte carlo simulation, the OM estimates the target unit are closer to the actual unit, and the deviation of the prototype multilateral positioning from the target position is significantly larger than OM. The contour range of OM is smaller than that of the prototype multilateral localization, so the variance of OM estimation is smaller than that of the prototype multilateral localization estimation as a whole. In the central region, the contour of OM is observed to be contained by the contour of the prototype multilateral location, which further illustrates that the variance of the OM estimate is smaller than the variance of the prototype multilateral location estimate.
FIG. 4 is an optimal multilateration Kalman Filter (optimal-multilateration Kalman Filter) of the target tracking method of the present invention
OMKF) under uniform linear motion is compared with the track maps of different algorithms, and the OMKF can realize the tracking of the target.
Fig. 5 is a result diagram of tracking consistency of the target tracking method OMKF under uniform linear motion. The mean normalized estimation error square of the OMKF is 1, which shows that the designed filter is stable and effective.
The results of the mean square error tests of the position and the speed under the uniform linear motion are shown in fig. 6 and 7, respectively, wherein: the abscissa represents the step size and the ordinate represents the mean square error.
As seen in figure 6, the bit units Root Mean Square Error (RMSE) for all three methods decreased with increasing time in 100 monte carlo experiments, but the RMSE of the inventive method OMKF converged faster, significantly less than the other two algorithms and closer to the posterior cralmelo boundary.
In the speed root mean square error comparison of FIG. 7, the RMSE of the OMKF of the present invention is significantly less than that of the other methods and is closer to the posterior Cramer-Lo boundary, demonstrating the effectiveness of the method.
Fig. 8 is a comparison between the optimal multilateral kalman filtering OMKF of the target tracking method of the present invention and the trajectory diagrams of different algorithms under the constant velocity turning motion.
FIG. 9 is a graph of the tracking consistency results of the target tracking method OMKF of the present invention under constant rate cornering maneuvers. Under the constant-speed turning motion, the square of the average normalized estimation error of the OMKF is 1, which shows that the designed filter is stable and effective.
The results of the mean square error test of the position and the speed under the constant-speed turning motion are respectively shown in fig. 10 and fig. 11, wherein: the abscissa represents the step size and the ordinate represents the mean square error.
From fig. 10, it can be seen that the RMSE of the OMKF of the present invention method is smaller than the other two algorithms in the Root Mean Square Error (RMSE) of the three methods in 100 monte carlo experiments, and converges faster and closer to the posterior cralmelo boundary.
In the speed root mean square error comparison of FIG. 11, the RMSE of the OMKF of the present invention is significantly less than that of the other methods and is closer to the posterior Cramer-Lo boundary, demonstrating the effectiveness of the method.
Fig. 12 and 13 are graphs showing the results of the mean normalized estimation error squared of the target tracking method OMKF under different noise variances in different motion models in the present invention.
As seen from FIG. 12, the mean normalized estimation error square of the target tracking method OMKF in the invention is around 1 under different noise variances in uniform linear motion, which shows that the method of the invention is stable and effective under different noise conditions.
Similarly, as seen from fig. 13, the mean normalized estimation error square of the target tracking method OMKF in the constant velocity turning motion under different noise variances is also around 1, which shows that the method of the present invention is stable and effective under different noise conditions in different motion models.
In conclusion, the method can realize target tracking under the condition of only distance measurement and segmentation information, and has the advantages of small estimation error, and good stability and consistency.
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the spirit of the present invention.
Claims (4)
1. A target positioning and tracking method only under distance measurement information in a wireless sensor network is characterized in that: the method comprises the following specific steps:
step 1: obtaining distance measurement information between a moving target and each sensor by using m sensors which are deployed in space and only obtain distance information
The deployment of the sensors is not in-line;
step 2: for the distance measurement information
Analyzing and calculating the first moment statistical information and the second moment statistical information of the error items in the distance measurement equation set
Reconstructing a 0 mean error term e;
and step 3: obtaining a minimum variance unbiased estimate of a position using generalized least squares
And 4, step 4: constructing pseudo-measurements Z of target locations k And designing a corresponding Kalman filter to realize target tracking.
2. The method of claim 1, wherein the target is located and tracked in a wireless sensor network with only distance measurement information, and wherein the method comprises: the information about the distance measurement in step 2 is
Get about
The method of likelihood distribution of (1) is as follows:
for the distance measurement information:
in the formula
It shows the distance measurement measured by the ith sensor at the time k in the process of moving the target,
distance measurement information representing an ith sensor without noise interference;
is shown as
P x，k And P y，k Respectively representing the position of the target in the x direction and the y direction at the k time; s. the x，i And S y，i Respectively representing the position of the ith sensor observation station in the x-direction and the y-direction,
is the corresponding measurement noise, which is obeyed to a zero mean with a variance of
(ii) a gaussian distribution of; it is expressed by a standard normal distribution as:
wherein w 0 Obeying a standard normal distribution, k i Is a coefficient to generally represent the distribution of measurement noise under different variances;
from the above analysis, if the true distance information is known
Distance measurement information
Becomes a random variable whose distribution is gaussian centered around the true value:
in practice, only the actual distance information can be estimated according to the distance observation information; for the function expressed by the expression (10),
is a known parameter, and
is the only unknown variable, thus obtaining
The likelihood function of (a) is as follows:
by the formula as above, the compound has the advantages of high purity,
the likelihood function and the Gaussian probability density function have the same mathematical form, and obtain
Likelihood distribution of (2); therefore, for
Is provided with
The positions of the unknown target and the sensor satisfy the following equation:
wherein
Distance information of the ith sensor which shows no noise interference at the k moment;
subtracting the ith equation in the above equation from the remaining equations yields:
the above formula is simplified and rewritten into a matrix form:
the above formula is abbreviated as:
S＝Mp k +U
wherein:
p k ＝[p x，k p y，k ] T 
in the above formula, S, M is the information known for certain, however U is
Since it contains the true distance
In practice
Is agnostic and thus
Is unknown information; in the case of noise interference, U is an assumption that does not satisfy the gaussian-markov theorem; in reality
Is not available; can only pass through
And distance observation information containing noise interference
To obtain the relation between
Thereby estimating the position of the target;
for the
And
the following relationships exist:
the relevant information of the distance truth value is counted in the distance observation information containing the noise interference, so that the state information of the target is more accurate;
solve for U is
Of a second moment, i.e. solving
And
in the formula E [. C]、D[·]And Cov [. C]Respectively representing mathematical expectation, variance and covariance; further constructing a 0 mean error term e to meet the assumption of Gauss-Markov theorem;
so for the first moment of U:
the second moment of U has a second-order central moment and a second-order mixed moment. For the second central moment of U, there are:
for the second order mixing moments of U:
in summary, U is at r 1∶i ，n 1∶i Lower second moment
The analytic formula (2) R DEG is as follows;
the unified analytical formula (2) is shown below, and hereThe scenario is scalable to a maximum number of sensors on a basic basis:
for 1 ≦ i, j ≦ m-1, i = j:
for i is more than or equal to 1, j is more than or equal to m-1,i is not equal to j, the following components are provided:
i, j respectively represent covariance matrices
M is the total number of sensors,
and
respectively represent covariance matrices
Diagonal elements and non-diagonal elements which together form a covariance matrix
Covariance of U
Is the key to causing the hypothesis that the gaussian-markov theorem is not satisfied; construction of 0 mean error term to S = Mp k +U k The formula satisfies the assumption of the Gauss-Markov theorem; constructing a random vector e as an error term, which is a mean of ●,
(ii) a gaussian distribution of;
since E is 0 mean, there is E (U) + E = U, resulting from 0 mean gaussian distribution symmetry:
for sensors of the same specification, the distribution of the measurement noise is the same, i.e. when all k are equal i When the phase difference is equal to each other,
and the sum of c is recorded as S:
the corresponding specific S is as follows:
in summary, the optimal polygon positioning formula after reconstruction is:
S＝Mp k +e
wherein
3. The method of claim 1, wherein the target is located and tracked in a wireless sensor network with only distance measurement information, and wherein the method comprises: the specific process of the step 3 is as follows:
optimal multilateration S = Mp after reconstruction k In + e, the error term e is the mean value of 0, which meets the assumption of Gauss-Markov theorem; obtaining the minimum variance unbiased estimation of the target state by a generalized least square method:
4. The method of claim 1, wherein the target is located and tracked in a wireless sensor network with only distance measurement information, and wherein the method comprises: the specific process of the step 4 is as follows:
second moment statistics of error term e
Providing a basis for realizing target tracking, and designing a target tracking estimator only under distance measurement; in a designed target tracking estimator under only distance measurement, the estimation of a target state is generated by at least three times of distance measurement and then constructing a pseudo measurement combined with a Kalman filter; the method mainly comprises the following steps:
obtaining target minimum variance unbiased estimation
Meanwhile, the noise covariance of the target state estimation is:
to this end, the model of the target may be expressed as: x is the number of k ＝F k-1 x k-1 +G k-1 W k-1 (ii) a On the basis of optimal multilateral positioning, pseudo-measurement Z of a target is constructed k The measurement equation is expressed as: z k ＝Hx k +N k (ii) a Where Z is k A pseudo-metric representing the target location at time k from the optimal multilateration; x is a radical of a fluorine atom k Is the target state, G k-1 Coefficient matrix of the corresponding dimension, W k-1 As covariance of Q k-1 Process noise of, N k As covariance of R k And W is k-1 Is irrelevant, H is the measurement matrix; r is k And H is as follows:
in order to realize target tracking, the last step is to use pseudo measurement and corresponding noise information to update the state of the target in combination with Kalman filtering; therefore, the expression of the optimal multilateral positioning combined with kalman filtering to realize target tracking is as follows:
1) And (3) state prediction:
2) Prediction of covariance matrix:
3) Calculating a gain matrix:
4) And (3) updating the state:
5) Covariance matrix update:
P k/k ＝(I 4×4 -K k H)P k/k-1 。
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