CN101819267A - Target tracking method based on receipt signal energy indication measurement - Google Patents

Abstract

The invention discloses a target tracking method based on receipt signal energy indication measurement in the filed of wireless sensor network technology. The method comprises the steps of: establishing a network; acquiring a network address and geographic coordinates, establishing a state equation and a wireless channel together with a moving object node to be measured outside the network and receiving a data pocket; for a sensor, resolving the data pocket to obtain an energy loss value between the moving object node to be measured and a wireless sensor node, converting the energy loss value into a radial distance and sending the geographic coordinate of the wireless sensor node and the radial distance to a coordinator node; for the coordinator node, further obtaining state interactive covariance and measurement covariance according to sampling state detection and covariance detection by combining sampling point weight and carrying out state updating and detection by substituting Kalman filtering to the coordinate and speed values of the moving object to be measured. The invention has high tracking accuracy higher than an EKF (Extended Kalman Filter) method and a trilateral positioning method and has excellent application prospects in a real WSN (Wireless Sensor Network) system.

Description

Method for tracking target based on receipt signal energy indication measurement

Technical field

What the present invention relates to is a kind of method of networking technology area, specifically is a kind of based on the indication of received signal energy (ReceiveSignal Strength Indicator, RSSI) method for tracking target of Ce Lianging.

Background technology

Wireless sensor network is considered to one of most important emerging technology of 21st century.Along with the development of technology such as wireless communication technology, sensor technology, micro electro mechanical system (MEMS) technology, integrated circuit technique, distributed information processing, making a large amount of cheap, intelligent sensor devices form wireless sensor network by radio communication becomes possibility.From current wireless sensor network development trend, the location technology of moving target had using value more widely.As aspects such as military surveillance, air traffic control, road surface monitoring, industry manufacturings.The wireless sensor network The Location is mainly concentrated at present the Position Research of static target, and the target to be monitored in the practical application mostly is moving target, therefore how effectively to moving target, especially to carry out relative accurate localization still be a problem that remains to be furtherd investigate to maneuvering target.

Usually adopt ultrasonic locating method and GPS localization method to reach location in the wireless sensor network location at present to target.In recent years, the RSSI localization method begins to be applied in the wireless sensor network.There is following shortcoming in prior art.The RSSI technology is confined to the location to static wireless sensor node more, does not use as yet for the location of moving target.By the data that the RSSI commercial measurement arrives, how to handle with traditional Trilateration methods, this kind method precision is not high.

Through the retrieval of prior art is found, the meeting paper of delivering with Abdalkarim Awad in the tenth the miniature Design of Digital System meeting in Europe that in August, 2007, IEEE computer society held that is entitled as Adaptive Distance Estimation andLocalization in WSN using RSSI Measures (self-adaptation distance estimations and localization method in the wireless sensor network) based on receipt signal energy indication measurement.The document has been put down in writing the RSSI technology has been incorporated in the wireless sensor network location, but is only used for static node locating.And in position fixing process, for nonlinear positioning system, utilization be traditional Trilateration methods, so cause bearing accuracy not high.

Summary of the invention

The objective of the invention is to overcome deficiency of the prior art, a kind of method for tracking target based on receipt signal energy indication measurement is proposed, rssi measurement and UKF (Unscented Kalman Filter does not have the track Kalman filter) method are incorporated in the tracking of wireless sensor network to moving target, have obtained than more accurate localization tracking of three traditional limit location survey methods.

The present invention is achieved by the following technical solutions, the present invention includes following steps:

Step 1, first wireless sensor node begin to set up network as coordinator node, coordinator node determines that network indicates and wireless channel, all the other wireless sensor node scannings detect already present wireless channel and network indicates and send the adding network requests to coordinator node, and the adding of due-in arrival self-coordinating device node is answered the back and added network and set up radio channel with router form or terminal device form;

Step 2, all wireless sensor nodes are from the coordinator node acquisition network address and geographic coordinate separately, and arbitrary wireless sensor node active is set up state equation and radio channel and received packet with the moving target node to be measured outside the network;

Step 3, sensor node resolution data bag obtain the energy loss value between moving target node to be measured and this wireless sensor node, then the energy loss value are converted to radial distance and also the geographic coordinate and the radial distance of this wireless sensor node are sent to coordinator node;

Step 4, coordinator node receive that a plurality of wireless sensor nodes send include geographic coordinate and radial distance packet after, select the odd number sampled point and give described each sampled point weight, each sampled point carries out state-detection and covariance detects; Detect according to all state-detection and covariance and to combine the sampled point weight and further obtain the mutual covariance of state and measure covariance and do not have track Kalman filter formula by substitution and carry out state and upgrade and detect, obtain the coordinate figure and the velocity amplitude in this moment of moving target to be measured.

Described substitution does not have track Kalman filter formula and carries out state and upgrade with detection and be meant: the average X of known state amount and covariance P, select one group of sampling point set to merge and give each sampled point weight, by to sampling point set with carry out the probability density distribution that nonlinear transformation is similar to nonlinear function.

The selection rule of described sampled point is as follows:

$\left\{\begin{array}{ccc}{A}_0=\stackrel{\&OverBar;}{X}& {\mathrm{<figure-callout\; id="0"\; label="W"\; filenames="HDA0000020351360000013.png,HDA0000020351360000014.png"\; state="\{\{state\}\}">W</figure-callout>}}_0=\frac{\mathrm{\&lambda;}}{(n+\mathrm{\&lambda;})}& i=0\\ A_i=\stackrel{\&OverBar;}{X}+{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=1,...,n\\ A_i=\stackrel{\&OverBar;}{X}-{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=n+1,...,2n\end{array}\right.$

Wherein: X is the average X of quantity of state (coordinate figure and velocity amplitude), and P is the covariance of quantity of state, and the number of sampled point is 2n+1 in the sampled point set, and n is the state vector dimension, and sampled point is A _i, corresponding weights are W _i, λ is a scaling function.Usually get n+ λ=3.

Above-mentioned steps two is carried out to increase progressively circulation in time to step 4.

Compared with prior art, the present invention has following beneficial effect: the UKF method is higher than EKF method and three limit localization methods to the position and the tracking accuracy on the speed of target, and tracking accuracy is the highest; For actual WSN system, RSSI is a kind of range finding means commonly used.Therefore UKF and rssi measurement technology are combined, in actual WSN system, have good practical prospect; Therefore, this patent has wide practical use in the wireless sensor network target in fields such as military affairs, environmental monitoring, health care, industrial automation, public safety is followed the tracks of.

Description of drawings

Fig. 1 is the schematic diagram of the rssi measurement technology among the present invention;

Fig. 2 is the tracking effect figure of embodiment to the uniform motion target.

Fig. 3 be embodiment to the uniform motion target directions X apart from filtering error figure.

Fig. 4 be embodiment to the uniform motion target the Y direction apart from filtering error figure.

Fig. 5 is that embodiment is to the pie slice Error Graph of uniform motion target at directions X.

Fig. 6 is that embodiment is to the pie slice Error Graph of uniform motion target in the Y direction.

Fig. 7 is the tracking effect figure of embodiment to the uniformly accelerated motion target.

Fig. 8 be embodiment to the uniformly accelerated motion target directions X apart from filtering error figure.

Fig. 9 be embodiment to the uniformly accelerated motion target the Y direction apart from filtering error figure.

Figure 10 is that embodiment is to the pie slice Error Graph of uniformly accelerated motion target at directions X.

Figure 11 is that embodiment is to the pie slice Error Graph of uniformly accelerated motion target in the Y direction.

Embodiment

Below embodiments of the invention are elaborated, present embodiment is being to implement under the prerequisite with the technical solution of the present invention, provided detailed embodiment and concrete operating process, but protection scope of the present invention is not limited to following embodiment.

The application of present embodiment comprises: the network distributed areas are that radius is the border circular areas of 50m; 100 sensors are evenly distributed in the border circular areas that radius is 50m at random; An external target is moved in this zone;

Target setting moves in two dimensional surface, and search coverage is to be the center of circle with (0,0), and 50m is the border circular areas of radius.Process noise and measurement noise are white Gaussian noise.Wherein the average of process noise is 0, and variance is 0.3m; The average of measurement noise is 0, and variance is 1.5m.Sampling time is 1s.

Initial value is: uniform motion X ₀=[30 ,-30,1,1] ^TUniformly accelerated motion X ₀=[30,0,1,1] ^T

Step 100, first wireless sensor node begin to set up network as coordinator node, coordinator node determines that network indicates and wireless channel, all the other wireless sensor node scannings detect already present wireless channel and network indicates and send the adding network requests to coordinator node, and the adding of due-in arrival self-coordinating device node is answered the back and added network and set up radio channel with router form or terminal device form;

Step 200, all wireless sensor nodes are from the coordinator node acquisition network address and geographic coordinate separately, and arbitrary wireless sensor node active is set up state equation and radio channel and received packet with the moving target node to be measured outside the network;

Step 300, sensor node resolution data bag obtain the energy loss value between moving target node to be measured and this wireless sensor node, then the energy loss value are converted to radial distance and also the geographic coordinate and the radial distance of this wireless sensor node are sent to coordinator node;

For an external movement target that enters in the wireless senser zone, its state equation is:

X (k+1)=φ X (k)+L ω _k(formula one)

Because adopt three limit localization methods, sensor node to the distance measuring equation of moving target is:

$Z_1=\sqrt{{(X_1-X)}^2+{(Y_1-Y)}^2}+v$

${\mathrm{<figure-callout\; id="2"\; label="Z"\; filenames="HDA0000020351360000013.png,HDA0000020351360000014.png"\; state="\{\{state\}\}">Z</figure-callout>}}_2=\sqrt{{(X_2-X)}^2+{(Y_2-Y)}^2}+v$

$Z_3=\sqrt{{(X_3-X)}^2+{(Y_3-Y)}^2}+v$ (formula two)

1, move with uniform velocity when target, its parameter respectively:

$X={[x,\stackrel{\&CenterDot;}{x},y,\stackrel{\&CenterDot;}{y}]}^T,$ $\mathrm{\&phi;}=\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right],$ $L=\left[\begin{array}{cc}0.5& 0\\ 1& 0\\ 0& 0.5\\ 0& 1\end{array}\right],$ ω _k＝[ω _x，ω _y] ^T；

2, do uniformly accelerated motion when target, its parameter is distinguished:

$X={[x,\stackrel{\&CenterDot;}{x},\stackrel{\&CenterDot;\&CenterDot;}{x}]}^T{[y,\stackrel{\&CenterDot;}{y},\stackrel{\&CenterDot;\&CenterDot;}{y}]}^T,$ $\mathrm{\&phi;}=\left[\begin{array}{ccc}1& 1& 0.5\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],$ L＝[001] ^T，ω _k＝ω _xω _y；

Wherein X is the quantity of state of moving target.X among the X, y are the coordinates of targets value.

For target respectively at x, the velocity amplitude of y direction.

For target respectively at x, the accekeration of y direction.Z _nMeasure the measurement distance of self distance objective for sensor node.ω _kBe respectively process noise and measurement noise with v, be white Gaussian noise.

Step 401, known state vector dimension n=4 are selected 2n+1=9 si gma sampled point and are carried out the conversion of trackless mark.

Calculate 9 sigma sampled point A _iWith corresponding weights W _i:

$\left\{\begin{array}{ccc}{A}_0=\stackrel{\&OverBar;}{X}& {\mathrm{<figure-callout\; id="0"\; label="W"\; filenames="HDA0000020351360000013.png,HDA0000020351360000014.png"\; state="\{\{state\}\}">W</figure-callout>}}_0=\frac{\mathrm{\&lambda;}}{(n+\mathrm{\&lambda;})}& i=0\\ A_i=\stackrel{\&OverBar;}{X}+{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=1,...,n\\ A_i=\stackrel{\&OverBar;}{X}-{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=n+1,...,2n\end{array}\right.$

(formula three)

Wherein, X is the average X of quantity of state (coordinate figure and velocity amplitude), and P is the covariance of quantity of state.The sampled point number is 2n+1, n=4, and, sampled point is A _i, corresponding weights are W _i, λ is a scaling function.Usually get n+ λ=3.

Step 402 is carried out UKF filtering by the sigma sampled point and the corresponding weight that obtain.

To all sigma sampled point A _iA step detect and to be:

ξ _i(k+1|k)=φ A _i(formula four)

One step of state is detected and is:

$\hat{X}(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i{\mathrm{\&xi;}}_i(k+1|k)$ (formula five)

One step of covariance is detected and is:

$P(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i[{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)]{[{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T+Q\left(k\right)$ (formula six)

Measure sigma sampling one step detection:

${\mathrm{\&zeta;}}_i(k+1|k)=h(k+1,A_i(k+1|k))$ (formula seven)

Wherein h is a measurement equation.

Detect and measure:

$\hat{Z}(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i\mathrm{\&phi;}{\mathrm{\&zeta;}}_i(k+1|k)$ (formula eight)

Measure and the mutual covariance of state:

$P_{\mathrm{xz}}=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i[{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)]{[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T$ (formula nine)

Measure covariance:

$P_{\mathrm{zz}}=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]{[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T$ (formula ten)

New breath covariance is:

S (k+1)=P _Zz+ R (k+1) (formula 11)

Filter gain is:

K (k+1)=P _XzS ^-1(k+1) (formula 12)

New breath is:

$v(k+1)=Z(k+1)-\hat{Z}(k+1|k)$ (formula 13)

The state renewal equation:

$X(k+1|k+1)=\hat{X}(k+1|k)+K(k+1)v(k+1)$ (formula 14)

Covariance is upgraded variance:

P (k+1|k+1)=P (k+1|k)-K (k+1) S (k+1) K ^T(k+1) (formula 15)

So far, obtain X (k+1|k+1), i.e. the coordinate figure and the velocity amplitude of K+1 moment moving target.Along with moment k increases, above-mentioned steps 401 and 402 constantly circulates, and constantly obtains coordinate figure and the velocity amplitude of moving target at moment k

Present embodiment based on the UKF filtering of rssi measurement technology as shown in Figure 2 to the tracking results of uniform motion target, Fig. 3 Fig. 4 is respectively X, the Y direction apart from filtering error figure.Fig. 5 Fig. 6 is respectively X, the pie slice Error Graph of Y direction.To the tracking results of uniformly accelerated motion target as shown in Figure 7, Fig. 8 Fig. 9 is respectively X, the Y direction apart from filtering error figure.Figure 10 Figure 11 is respectively X, the pie slice Error Graph of Y direction.

From Fig. 2,3,4,5,6 as can be seen, to the tracking results of uniform motion target,, all locate and the EKF method less than three traditional limits at distance and the filtering error on the speed based on the UKF method of rssi measurement technology.From Fig. 7,8,9,10,11 as can be seen, to the tracking results of uniformly accelerated motion target,, all locate and the EKF method less than three traditional limits at distance and the filtering error on the speed based on the UKF method of rssi measurement technology.Has higher filtering accuracy.

The accuracy of detection of present embodiment is as shown in the table:

The target uniform motion	Three limit localization methods (variance)	EKF (variance)	UKF (variance)
				The filtering error of directions X	??0.5117	?0.2318	??0.0239
The filtering error of Y direction	??0.6126	?0.2680	??0.0255
				??V xFiltering error		?0.0446	??0.000018884
??V yFiltering error		?0.0512	??0.000012020
				The target shift speed motion	Three limit localization methods (average)	EKF (average)	UKF (average)
The filtering error of directions X	??0.0264	?0.0139	??0.0290
				The filtering error of Y direction	??-0.0350	?-0.0581	??-0.0563
??V xFiltering error		?0.0185	??0.0012

The target uniform motion	Three limit localization methods (variance)	EKF (variance)	UKF (variance)
				??V yFiltering error		?-0.0457	??-0.0026

Claims (5)

1. the method for tracking target based on receipt signal energy indication measurement is characterized in that, comprises the steps:

Step 1, first wireless sensor node begin to set up network as coordinator node, coordinator node determines that network indicates and wireless channel, all the other wireless sensor node scannings detect already present wireless channel and network indicates and send the adding network requests to coordinator node, and the adding of due-in arrival self-coordinating device node is answered the back and added network and set up radio channel with router form or terminal device form;

Step 2, all wireless sensor nodes are from the coordinator node acquisition network address and geographic coordinate separately, and arbitrary wireless sensor node active is set up state equation and radio channel and received packet with the moving target node to be measured outside the network;

Step 3, sensor node resolution data bag obtain the energy loss value between moving target node to be measured and this wireless sensor node, then the energy loss value are converted to radial distance and also the geographic coordinate and the radial distance of this wireless sensor node are sent to coordinator node;

Step 4, coordinator node receive that a plurality of wireless sensor nodes send include geographic coordinate and radial distance packet after, select the odd number sampled point and give described each sampled point weight, each sampled point carries out state-detection and covariance detects; Detect according to all state-detection and covariance and to combine the sampled point weight and further obtain the mutual covariance of state and measure covariance and do not have track Kalman filter formula by substitution and carry out state and upgrade and detect, obtain the coordinate figure and the velocity amplitude in this moment of moving target to be measured.

2. the method for tracking target based on receipt signal energy indication measurement according to claim 1 is characterized in that, the state equation described in the step 2 is:

X(k+1)＝φX(k)+Lω _k，

Sensor node to the distance measuring equation of moving target is:

$Z_1=\sqrt{{(X_1-X)}^2+{(Y_1-Y)}^2}+v$

$Z_2=\sqrt{{(X_2-X)}^2+{(Y_2-Y)}^2}+v$

$Z_3=\sqrt{{(X_3-X)}^2+{(Y_3-Y)}^2}+v$

When target moves with uniform velocity, its parameter is respectively:

$X={[x,\stackrel{\&CenterDot;}{x},y,\stackrel{\&CenterDot;}{y}]}^T,$ $\mathrm{\&phi;}=\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right],$ $L=\left[\begin{array}{cc}0.5& 0\\ 1& 0\\ 0& 0.5\\ 0& 1\end{array}\right],$ ω _k＝[ω _x，ω _v] ^T；

When target is done uniformly accelerated motion, its parameter is respectively:

$X={[x,\stackrel{\&CenterDot;}{x},\stackrel{\&CenterDot;\&CenterDot;}{x}]}^T{[y,\stackrel{\&CenterDot;}{y},\stackrel{\&CenterDot;\&CenterDot;}{y}]}^T,$ $\mathrm{\&phi;}=\left[\begin{array}{ccc}1& 1& 0.5\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],$ L＝[001] ^T，ω _k＝ω _xω _y；

Wherein: X is the quantity of state of moving target, and x among the X, y are the coordinates of targets value,

For target respectively at x, the velocity amplitude of y direction, For target respectively at x, the accekeration of y direction, Z _nFor sensor node measures the measurement distance of self distance objective, ω _kBe respectively process noise and measurement noise with v, be white Gaussian noise.

3. the method for tracking target based on receipt signal energy indication measurement according to claim 1, it is characterized in that, substitution described in the step 4 does not have track Kalman filter formula and carries out state and upgrade with detection and be meant: the average X of known state amount and covariance P, select one group of sampling point set to merge and give each sampled point weight, by to sampling point set with carry out the probability density distribution that nonlinear transformation is similar to nonlinear function.

4. the method for tracking target based on receipt signal energy indication measurement according to claim 3 is characterized in that, the selection rule of described sampled point is as follows:

$\left\{\begin{array}{ccc}{A}_0=\stackrel{\&OverBar;}{X}& W_0=\frac{\mathrm{\&lambda;}}{(n+\mathrm{\&lambda;})}& i=0\\ A_i=\stackrel{\&OverBar;}{X}+{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=1,...,n\\ A_i=\stackrel{\&OverBar;}{X}-{\left(\sqrt{(n+\mathrm{\&lambda;})P}\right)}_i& W_i=\frac{\mathrm{\&lambda;}}{2(n+\mathrm{\&lambda;})}& i=n+1,...,2n\end{array}\right.$

Wherein: X is the average X of quantity of state (coordinate figure and velocity amplitude), and P is the covariance of quantity of state, and the number of sampled point is 2n+1 in the sampled point set, and n is the state vector dimension, and sampled point is A _i, corresponding weights are W _i, λ is a scaling function.

5. the method for tracking target based on receipt signal energy indication measurement according to claim 3 is characterized in that, described substitution does not have track Kalman filter formula and carries out state and upgrade with detection and be meant:

To all sigma sampled point A _iA step detect and to be:

ξ _i(k+1|k)＝φA _i

One step of state is detected and is:

$\hat{X}(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i{\mathrm{\&xi;}}_i(k+1|k)$

One step of covariance is detected and is:

$P(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i{[{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)][{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T+Q\left(k\right)$

A step that measures the sigma sampled point is detected:

ζ _i(k+1|k)＝h(k+1，A _i(k+1|k))

Wherein h is a measurement equation;

Detect and measure:

$\hat{Z}(k+1|k)=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i\mathrm{\&phi;}{\mathrm{\&zeta;}}_i(k+1|k)$

Measure and the mutual covariance of state:

$P_{\mathrm{xz}}=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i[{\mathrm{\&xi;}}_i(k+1|k)-\hat{X}(k+1|k)]{[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T$

Measure covariance:

$P_{\mathrm{zz}}=\underset{i=0}{\overset{9}{\mathrm{\&Sigma;}}}{W}_i[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]{[{\mathrm{\&zeta;}}_i(k+1|k)-\hat{X}(k+1|k)]}^T$

New breath covariance is:

S(k+1)＝P _zz+R(k+1)

Filter gain is:

K(k+1)＝P _xzS ^-1(k+1)

New breath is:

$v(k+1)=Z(k+1)-\hat{Z}(k+1|k)$

The state renewal equation:

$X(k+1|k+1)=\hat{X}(k+1|k)+K(k+1)v(k+1)$

Covariance is upgraded variance:

P(k+1|k+1)＝P(k+1|k)-K(k+1)S(k+1)K ^T(k+1)

So far, the above-mentioned steps that increases progressively along with the time constantly circulates to finish no track Kalman filtering.

Images