Fuzzy Neural Network-Based Interacting Multiple

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Fuzzy Neural Network-Based Interacting Multiple Model for Multi-Node Target Tracking Algorithm Baoliang Sun *, Chunlan Jiang and Ming Li State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China; [email protected] (C.J.); iseagle@ bit.edu.cn (M.L.) * Correspondence: [email protected]; Tel.: +86-10-6891-4682 Academic Editors: Lyudmila Mihaylova, Byung-Gyu Kim and Debi Prosad Dogra Received: 28 June 2016; Accepted: 18 October 2016; Published: 1 November 2016

Abstract: An interacting multiple model for multi-node target tracking algorithm was proposed based on a fuzzy neural network (FNN) to solve the multi-node target tracking problem of wireless sensor networks (WSNs). Measured error variance was adaptively adjusted during the multiple model interacting output stage using the difference between the theoretical and estimated values of the measured error covariance matrix. The FNN fusion system was established during multi-node fusion to integrate with the target state estimated data from different nodes and consequently obtain network target state estimation. The feasibility of the algorithm was verified based on a network of nine detection nodes. Experimental results indicated that the proposed algorithm could trace the maneuvering target effectively under sensor failure and unknown system measurement errors. The proposed algorithm exhibited great practicability in the multi-node target tracking of WSNs. Keywords: wireless sensor network; multi-sensing data fusion; interacting multiple model; fuzzy neural network; target tracking

1. Introduction Wireless sensor networks (WSNs) are developing toward positive intelligence, a large scale, modularization, and integration with the advancement of sensors, electronic information, and integrated technology [1,2]. Given its technological superiority, including self-organization, fast deployment, high fault tolerance, and strong elusion, WSNs are extremely suitable for various applications, such as battlefield target location [3], intelligent transportation system [4], and ocean exploration [5]. The maneuvering target tracking technology is one of the current research hot spots in WSNs [6–9]. This technology uses multi-sensor data to conduct state estimations on targets. Numerous maneuvering target tracking algorithms are presently available; among those methods, the interacting multiple model (IMM) algorithm has been extensively applied, considering algorithm performance, complexity, storage, and engineering applied demands [10,11]. This method uses interacting integration among multiple maneuvering models to describe target motion state. The transition between models is controlled by a Markov chain [12,13]. The weakness of IMM is that it requires knowledge of the statistical property of noise signals, unknown in most situations. Several methods have been introduced recently to solve target tracking problems, and many scholars have introduced fuzzy technology into the IMM algorithm to estimate model parameters in real time. For example, a fuzzy logic-based adaptive Kalman filter was proposed in [14]; its main concept involves combining a fuzzy inference machine with a Kalman filter and realizing adaptive Kalman filtering by utilizing the excellent performance of fuzzy logical processing with inaccurate data and simple operations. In [15], a sensor fusion algorithm was proposed, which introduced a dynamic noise covariance matrix into the IMM. The proposed filter exhibits superior accuracy compared with a Kalman filter when a vehicle abruptly changes its trajectory. Several algorithms that combine fuzzy technology with the IMM algorithm have Sensors 2016, 16, 1823; doi:10.3390/s16111823

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been introduced in the literature [16–18], and such a combination has provided good state estimation results. A fuzzy-logic-based interacting multiple model (FIMM) algorithm for tracking a maneuvering target was proposed in [16]. A fuzzy system enhanced by a genetic algorithm is applied as a universal approximator to compute the variance of the overall process noise, which is time-varying because of the unknown target acceleration. The FIMM algorithm does not require prior information on the statistical properties of the target maneuver but the statistical characteristics of measurement errors. In [17], a fuzzy Kalman filter was proposed to combat the model-set adaptation problem and was found to be able to adaptively control the noise covariance such that the IMM algorithm can use an accurate model set to improve its performance. In [18], fuzzy logic was incorporated in a conventional IMM-PDA (probabilistic data association) method, which adopted the prediction error and change of the prediction error as fuzzy inputs. The feasibility of this proposed algorithm was examined by simulation. The Takagi–Sugeno fuzzy neural network (T–S FNN) model [19,20] is a fuzzy reasoning model proposed by Takagi and Sugeno that is widely applied in time-series prediction and parameter estimation [21,22]. The T–S FNN fuzzy inference machine can be realized using a combined neural network. An agile ENFS, termed PANFIS, was explored in [23]. PANFIS can commence its learning process from scratch with an empty rule base, and the fuzzy rules can be stitched up and expelled through the statistical contributions of fuzzy rules and injected datum afterward [23]. In [24], the author presented an architecture and learning procedure underlying an adaptive-network-based fuzzy inference system (ANFIS), which could construct input–output mapping based on both human knowledge and stipulated input–output data pairs. The proposed architecture of ANFIS can refine fuzzy if–then rules obtained from human experts. If human expertise is unavailable, then the proposed method can generate a set of fuzzy if–then rules by intuitively setting up reasonable initial membership functions and starting the learning process. Multi-node or multi-sensor data fusion is seldom introduced in these studies. A FNN-based IMM (FNN–IMM) filter is proposed in this work. Assuming each sensor node in a WSN to correspond to an FNN–IMM filter, several FNN–IMM filters were operated in parallel. The output results of all FNN–IMM filters were inputted into the fusion center during the multi-sensor data fusion stage to conduct data fusion. Finally, a network state estimation of the targets was obtained. The algorithm proposed in this study did not require the assumption of the statistical characteristics of measurement errors and exhibited good tracking performance in the network when some nodes outputted error data. This work is divided into five parts, including this section. The FNN–IMM algorithm of a single node is described in detail in Section 2. We discuss the multi-node target tracking and propose the multi-node tracking data fusion algorithm in Section 3. The capabilities of the proposed method are discussed and validated through simulation in Section 4. Finally, conclusions are drawn in Section 5. 2. The FNN–IMM Algorithm of Single Node A WSN can be divided into multiple structures based on networking modes: star configuration, symmetrical structure, dendritical structure, and mixed structure. The following hypotheses are proposed based on network power consumption and tactic utilization. 1. 2.

N detection nodes with the same type are present in a WSN, and the detection nodes only have one detection mode. Data measured by node detectors are 2D position coordinates of the invasion targets. When a single node is considered, the measuring model is shown as follows: Z( k ) = H j X( k ) + V j ( k )

j = 1, · · · , m

(1)

The target state transition equation is given as follows: X( k + 1) = F j X( k ) + G j W j ( k ),

j = 1, · · · , m

(2)

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X(k  1)  F j X(k )  G j Wj (k ),

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j  1, , m

(2)14 3 of

where m is the model number in the IMM algorithm, X(k) is the state vector, Fj is the system transition where m is the model number in the IMM algorithm, X(k) is the state vector, Fj is the system transition matrix, Gj is the noise disturbance matrix, and H is the measurement matrix. Wj and Vj are the white matrix, Gj is the noise disturbance matrix, and H is the measurement matrix. Wj and Vj are the noise sequences, and their means are 0; their covariance matrices are Qj and R, respectively. The white noise sequences, and their means are 0; their covariance matrices are Qj and R, respectively. transition between models is controlled using a Markov chain, and the transition matrix of the The transition between models is controlled using a Markov chain, and the transition matrix of the Markov chain is shown as follows: Markov chain is shown as follows:    11 π11   .. Π  Π =  . πm1 m1

 1m  · · · π1m  ..  .. . .    · · · πmm mm 

(3) (3)

Figure 11 illustrates illustrates the the principle principle of of the the IMM IMM algorithm. algorithm. Figure X1  k 1 k 1

X2  k 1 k 1

Xn  k 1 k 1



Input interaction

Xo1  k 1 k 1

Xo2 k 1 k 1

Xon k 1 k 1



Z(k ) (k 1 k 1)

 (k )

Filter K1

Filter K 2

Filter K n

X1  k k 

X2 k k 

Xn k k 

Update Model probability

 (k k )

Output

X k k 

Figure 1. Interacting multiple model (IMM) algorithm principle frame.

The process of the IMM algorithm can be described described using using Equations Equations (4)–(14). (4)–(14). a. Input interaction: interaction: a. Input Xˆ oj (ojk − 1/k − 1) =

m

∑m Xˆ iˆ(ik − 1/k − 1)µij (k − 1/k − 1),

j = 1, · · · , m ˆ (k  1 k  1) i= X ,m 1 X ( k  1 k  1) ij ( k  1 k  1), j  1,

(4) (4)

i 1

and and

m  Poj (k − 1/k − 1) = ∑ µij (k − 1/k − 1) Pi (k − 1/k − 1) i =m1





ˆi ˆ oi ˆi +[ ) − Xˆ oi (k − 1/k − 1)]T PXoj ((kk −  11/k k −1)1)− X (k (−k 1/k 1 − k 1)][ 1) XP(ik(− k 1/k 1 k−11) where

i 1

ij

m

ˆ i (k  1 kµ(1) ˆ oi (k  1 k  1)][ X ˆ i (k 1 k  1)  X ˆ oi (k  1 k  1)]T [ X X ij k − 1/k − 1) = πij µij ( k − 1) ∑ πij µij ( k − 1)

(5)



(5) (6)

i =1

where b. For the j-th mode: Prediction:

m

ij (k Xˆ1j (kk/k  1)   ij ij (ˆkoj 1)   ij ij (k  1) − 1) = F X (k − 1/k − 1) j

i 1

(6) (7)

Error covariance prediction: P j (k/k − 1) = F j Poj (k − 1/k − 1)FTj + G j Q j GTj

(8)

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Kalman gain: K j (k) = P j (k/k − 1)H j T [H j P j (k/k − 1)H j T + R j ]

−1

(9)

Filter: ˆ j (k/k) = Xˆ j (k/k − 1)K j (k)[Z(k ) − H j Xˆ j (k/k − 1) X

(10)

Error covariance of filter: P j (k/k ) = [I − K j (k)]P j (k/k − 1)

(11)

Model probability updating: m

µ j (k) = µ j (k/k − 1)Λ j (k) / ∑ µi (k − 1)Λi (k)

(12)

i =1

where Λj (k) is the likelihood function of observational Z(k), and   1 T 1 −1 Λi (k) = N νi (k) ; 0, S j (k ) = 1/2 exp − νj (k) S j (k) νi (k ) , 2 (2π )1/2 S j (k)

(13)

υ j (k) = Z(k ) − H(k )Xˆ j (k/k − 1)

(14)

S j (k) = H(k)P j (k/k − 1)HT (k ) + R

(15)



where and c. Interacting output: Xˆ (k/k) =

m

∑ Xˆ j (k/k)µ j (k)

(16)

j =1 m

P(k/k) =

∑ µ j (k)

n

ˆ j (k/k) − X ˆ (k/k)] P j (k/k) + [Xˆ j (k/k ) −Xˆ (k/k )][X

T

o

(17)

j =1

The algorithm proposed in this study discusses system residual ε(k) of the models during the interacting output stage. ˆ (k/k) ε ( k ) = Z( k ) − H( k )X (18) According to [25,26], the theoretical value of the covariance matrix of the system residual is: T(k) = H(k )P(k/k )HT (k) + R

(19)

However, T(k) can be estimated but cannot be known in advance because R is unknown. We initialized R with a random value Rr . During the iteration progress, the algorithm adaptively adjusted Rr to the real value R. To achieve this self-adaptive progress, we set up a statistical value of the system residual as follows: E( k ) =

1 k ε(i )ε(i ) T , j = 1, 2, · · · , m W i∑ =i

(20)

0

where i0 = k − W + 1, and W is the time window length. The proposed algorithm applies the latest W datum to estimate the covariance matrix of the system residual. Given that the properties of target maneuvering are unknown, the performance of the IMM algorithm decreases when the target changes its trajectory abruptly, which makes ε(k), T(k), and E(k) difficult to estimate. A long time window covers much motion progress of the target, which leads to many maneuvering errors. By contrast, a short time window gains minimal statistical data, which cause significant error in E(k). Therefore,

and

E j (k ) =

1 W

k

 ε(i)ε(i)

T

, j = 1, 2, , m

(20)

i =i0

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where i0 = k – W + 1, and W is the time window length. The proposed algorithm applies the latest W datum to estimate the covariance matrix of the system residual. Given that the properties of target the value of time W isthe selected according theIMM sampling rate and maneuvering the maneuvering arewindow unknown, performance oftothe algorithm decreases whenstate the of target target. Section 3 presents the simulation part. The operational value of W after trial and error is the changes its trajectory abruptly, which makes ej(k), Tj(k), and Ej(k) difficult to estimate. A long time most idealcovers whenmuch W = 15. window motion progress of the target, which leads to many maneuvering errors. By Let contrast, a short time window gains minimal statistical data, which cause significant error in Ej(k). EoR = T(kaccording ) − E(k) to the sampling rate and maneuvering (21) Therefore, the value of time window W is selected stateEquation of the target. Section 3 presents the simulation part. The operational of W error after increases trial and (19) and (21) indicate that when the covariance matrix R of the value measuring error is the most ideal when W = 15. or decreases, the corresponding elements in the EoR matrix also increase or decrease. EoR detection Let can be used to adjust R and reduce the difference between T(k) and E(k). Only the elements on the principal diagonal of EoR can be detected because elements of matrix R are 0, except for those EoR = T( kthe )−E (k ) (21) elements on the principal diagonal that are not completely 0. Equation (15) indicates that when the are covariance matrix R of the measuring error increases or The principles of adaptive adjustment presented as follows: decreases, the corresponding elements in the EoR matrix also increase or decrease. EoR detection can 1. diag(EoR) is used to express the difference principal diagonal EoR.Only If diag(EoR) ≈ 0, then R be used to adjust R and reduce the between elements T(k) andofE(k). the elements on the remains unchanged. principal diagonal of EoR can be detected because the elements of matrix R are 0, except for those 2. If diag(EoR) > 0, thendiagonal the corresponding elements of R0.decrease. elements on the principal that are not completely The principles ofthen adaptive adjustment areelements presented asincrease. follows: 3. If diag(EoR) < 0, the corresponding of R Therefore, R can be adjusted as follows: Therefore, R can be adjusted as follows:

R(i, i) = R(i, i) +ΔR

(22) (22)

R(i, i ) = R(i, i ) + ∆Ri i

The membership membership functions functions of of EoR EoR and and ∆R ΔR are are established established as as shown shown in in Figure Figure2. 2. The EoR

Membership

0.8 0.6

EoR<0

△R 1

EoR>0

EoR≈0

0.4

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Decreasing ∆R

0.8

Membership

1

0.6 0.4

(a) 0 -a

In this figure,

-a/2Figure 2. 0

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Maintain ∆R

0.2

0.2

Increasing ∆R

(b) a/2 a Membership functions

0 of-0.3R (a) EoR

(a)

-0.1R 0 and (b) ΔR.

0.1R

0.3R

(b)

Figure 2. 2. Membership Membership functions functions of of (a) (a) EoR EoR and and (b) (b) ∆R. ΔR. Figure

1 r a   EoR(i, i) r i 1

In this figure, In this figure,

1 r a = ∑ |EoR(i, i )| where r is the dimension of the matrix EoR. r i=1 Ther maximum–minimum are utilized in the defuzzification progress: where is the dimension of theprinciples matrix EoR.

(23) (23)

The maximum–minimum principles are utilized in the defuzzification progress:

 f ( )   f ( ) R  i, i   µEoR1 Rf 1 (µEoR1 ) + µEoR 2 Rf2 (µEoR 2 ) EoR1 R1 EoR1   EoR2 R2 EoR2 ∆R (i, i ) =

EoR1

EoR 2

µ EoR1 + µ EoR2

(24) (24)

where f Ri = 1, 1, 2, 2, 3) 3) represents represents the theinverse inversefunction functionof of△R 4Runder underthe thecorresponding correspondingfuzzy fuzzyrules; rules; where Ri() (i = EoRi (i = 1, 2) represents the membership degree of EoR. µEoRi (i = 1, 2) represents the membership degree of EoR. In a real environment, R varies on a per-environment basis. In this study, we detect the change a real environment, R varies on a per-environment basis.during In thisiteration study, we detect the change of RInby examining EoR. The algorithm adjusts R adaptively progress to decrease of R by examining EoR. The algorithm adjusts R adaptively during iteration progress to decrease the tracking error of target. However, the relationship between EoR and ΔR is unclear because the measure environment is unknown. The FNN is introduced to learn this relationship and rebuild the membership functions of EoR and ΔR. During the iteration progress, we feed the EoR and ΔR obtained in each iteration into the FNN and use the output to adjust R online. The T–S FNN model is used in this study to construct the fuzzy inference machine between EOR

R  i, i  

EoR1 f R1 ( EoR1 )  EoR 2 f R 2 ( EoR 2 ) EoR1  EoR 2

(24)

f Ri () (i = 1, 2, 3) represents the inverse function of △R under the corresponding fuzzy rules; where Sensors 2016, 16, 1823

 EoRi (i = 1, 2) represents the membership degree of EoR.

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In a real environment, R varies on a per-environment basis. In this study, we detect the change

the tracking of target. However, the relationship between EoR and ∆R is to unclear because the of R byerror examining EoR. The algorithm adjusts R adaptively during iteration progress decrease the measure environment is unknown. The FNN is introduced to learn this relationship and rebuild tracking error of target. However, the relationship between EoR and ΔR is unclear because the the membership functionsisofunknown. EoR andThe ∆R. During the iteration we feed the EoR measure environment FNN is introduced to learn progress, this relationship and rebuild theand ∆R membership functions of EoR and ΔR. theoutput iteration feed the EoR and ΔR obtained in each iteration into the FNN andDuring use the to progress, adjust R we online. in each iteration intoin thethis FNN and use the output to R online. Theobtained T–S FNN model is used study to construct theadjust fuzzy inference machine between EOR The T–S FNN model is used in this study to construct the fuzzy inference machine between EOR and ∆R and adaptively deduce the relationship between learning EOR and R. The FNN inference and ΔR and adaptively deduce the relationship between learning EOR and R. The FNN inference machinemachine between EoR and R isRestablished MATLABaccording according to [24], and its structure between EoR and is establishedusing using MATLAB to [24], and its structure is type is type 3 of [24],3 of as[24], shown in Figure 3. 3. as shown in Figure Rule Input

Inputmf

Outputmf



 EoR1

 EoR1

N



f R1 EoR

EoR2

EoR2

N

Output



R

fR2



EoR3  EoR1

fR2

f R1

f R3 R 

 EoR1 f R1 (  EoR1 )   EoR 2 f R 2 (  EoR 2 )  EoR1   EoR 2

 EoR 2 3. Structure of the single-node fuzzy neural network (FNN) inference machine. FigureFigure 3. Structure of the single-node fuzzy neural network (FNN) inference machine.

3. Multi-Node Target Tracking Data Fusion Algorithm

3. Multi-Node Target Tracking Data Fusion Algorithm

During the multi-node data fusion stage, a large amount of sensor data is sent to the infusion

center the for infusion in WSNs; wrong data ainevitably appear. of The data infusion mustinfusion During multi-node datathus, fusion stage, large amount sensor data isalgorithm sent to the possess high fault tolerance and efficiency. The FNN fusion system (FNNFS) proposed in this study center for infusion in WSNs; thus, wrong data inevitably appear. The data infusion algorithm must possess Sensors high 2016, fault16,tolerance and efficiency. The FNN fusion system (FNNFS) proposed in this 1823 7 of 14 study estimates the working status of nodes with EoR and R. EoR is the difference between the theoretical the working of nodes with EoR andof R.the EoRsensor. is the difference between the theoretical value Testimates and actual value Estatus of the system residual R is the covariance matrix of the value T and actual value E of the system residual of the sensor. R is the covariance matrix of the current measurement error. This algorithm then classifies the working status of nodes into ideal, current measurement error. This algorithm then classifies the working status of nodes into ideal, good, and poor. this fuzzy classification, output data ofnodes the nodes are endowed good, and After poor. After this fuzzy classification,the the interaction interaction output data of the are endowed with thewith corresponding confidence weight vector w (k). Finally, w (k) is used to complete the data the corresponding confidence weight vector wjj(k). Finally, wj(k)j is used to complete the data defuzzification and derive the target status estimationof of the the network time k ask the output. FigureFigure 4 defuzzification and derive the target status estimation networkatat time as the output. 4 illustrates the principle of FNNFS. illustrates the principle frameframe graphgraph of FNNFS. Node 1

Node 2

Z1(k)

Z2(k)

. . . Node M

FNN-IMM_1

X1(k/k)

R EoR Fuzzy Inference Machine

W1(k)

FNN-IMM_2

X2(k/k)

EoR R Fuzzy Inference Machine

W2(k)

FNN-IMM_M

Xm(k/k)

...

Zm(k)

EoR

×

× ... ×

Defuzzification Data Fusion System

Xˆ(k k )

R

Fuzzy Inference Machine

Wm(k)

Figure 4. Principle framegraph graph of of FNN system (FNNFS). Figure 4. Principle frame FNNfusion fusion system (FNNFS).

3.1. Calculation of Confidence Weight Vector The working status of node detectors in the network can be estimated using the measurement error. If the measurement error of some nodes is large or possesses a singular value (i.e., the value significantly deviates from the average value), then the data output of the node output for the current time is unreliable. A relatively low confidence weight is obtained, and the working status of the nodes

Machine

Figure 4. Principle frame graph of FNN fusion system (FNNFS). Sensors 2016, 16, 1823

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3.1. Calculation of Confidence Weight Vector

Fuzzy decision coefficient of R(MS)

Fuzzy decision coefficient of EoR (MSe)

The 3.1. working status of nodeWeight detectors Calculation of Confidence Vectorin the network can be estimated using the measurement error. If the measurement error of some nodes is large or possesses a singular value (i.e., the value The working status of node detectors in the network can be estimated using the measurement significantly deviates from the average value), then the data output aofsingular the node output forvalue the current error. If the measurement error of some nodes is large or possesses value (i.e., the time is unreliable. Adeviates relatively low weight is data obtained, working status of the nodes significantly from the confidence average value), then the output and of thethe node output for the current time isas unreliable. A relatively low confidence obtained, and the main working status of of theEoR nodesand R is is determined poor. When the absolute valueweight of theiselements in the diagonal is determined as poor. When the absolute value of the elements in the main diagonal of EoR and close to 0, the working status of the node is ideal. Otherwise, the detection error of theRnode is is close to 0, the working status of the node is ideal. Otherwise, the detection error of the node is considerable, and the working status is poor. The reliability of the node data is decided. Figure 5 considerable, and the working status is poor. The reliability of the node data is decided. Figure 5 shows theshows critical of EoR andand R. R. the functions critical functions of EoR 2

EoR(i,i) =L

1.5

EoR(i,i) =S

EoR(i,i) =ZE 1

2

R(i,i) = L

1.5

R(i,i) = S

R(i,i) = ZE 1

(a)

(b)

Figure 5. Decision functions (a)EoR EoRand and Figure 5. Decision functions of of (a) (b)(b) R. R. In the figure, In the figure,  MSei = EoR(i, i ) mean {diag(|EoR|)}

(25)

 MSi = R(i, i ) mean(diag(|R|))

(26)

and The decision relationship of wj (k) with EoRj (k) and Rj (k) is shown in Table 1. Table 1. Decision table of wj (k). Rj (i,i) EoRj (i,i) ZE S L

ZE

S

L

wi j (k) = 1 wi j (k) = 1 wi j (k) = 0.5

wi j (k) = 1 wi j (k) = 0.5 wi j (k) = 0

wi j (k) = 0.5 wi j (k) = 0 wi j (k) = 0

wi j (k) represents the i-th element of the j-th weight vector at time k, and element i is the confidence weight of the i-th element in the corresponding target status vector X(k). The multi-node FNN structure is shown in Figure 6.

L

Wij(k) = 0

0.5

Wij(k) = 0

Wij(k) represents the i-th element of the j-th weight vector at time k, and element i is the confidence

weight of the i-th element in the corresponding target status vector X(k). The multi-node FNN 8 of 14 in Figure 6.

Sensors 2016,is16, 1823 structure shown

Input

Inputmf

Rule

Outputmf

Output

EOR

W

R

Figure 6. 6. Structure Structure of of the the multi-node multi-node FNN. FNN. Figure

The following two conditions are considered to to avoid avoid possible possible error: error: 1.

2.

Allthe thedetection detectionnodes nodesin inthe thenetwork networkare are judged judged by by the the system system as as having having poor poor status at a certain All time.The Thefusion fusionalgorithm algorithmdirectly directlyadopts adoptsthe thenode node data data with with the the lowest lowest average average of MSe and MS time. Equations(25) (25)and and(26). (26). ininEquations The Thenode nodedata dataare areabandoned abandonedififthe theMSe MSe of of some some nodes nodes is is higher higher than than 2. 2. If If the the MSe MSe values of all the nodes are higher than 2, the algorithm operation stops, and an error is reported. the nodes are higher than 2, the algorithm operation stops, and an error is reported.

3.2. Defuzzification Data Infusion The larger MSe andand MS values are, theare, poorer statusthe of the nodeof willthe be. node The defuzzification largerthe the MSe MS values the the poorer status will be. The progress is implemented asimplemented follows: defuzzification progress is as follows: wi j ( k ) = wi j (

1 1 + ) MSi MSei

(27)

At time k, the target status output in the network is estimated to be    i i ˆ ˆ X (k) = argmax(w j (k)); X(k) =   j 

Xˆ 1 (k) Xˆ 2 (k) .. . M ˆ X (k)

   ,  

(28)

where argmax(wi j (k )) represents the largest i-th element of all return target status estimation vectors j

at time k. Given that the maximum value may not be only 1 in an actual situation, the average value of the corresponding elements of all maximum values comprises the fusion estimation data. 4. Simulation Experiment 4.1. Experiment Description In a WSN, multi-node cooperative detection is an effective means to improve network detection accuracy and system detection robustness. Networked cooperative detection consumes more resources than single-node detection. The number of detection nodes is directly proportional to the consumption of network resources. The number of detection nodes distributed in the network should be controlled within a reasonable value when tracking a single maneuvering target to achieve the best target tracking

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Sensors 2016, 16, 1823 be controlled within

9 ofthe 14 a reasonable value when tracking a single maneuvering target to achieve best target tracking performance. This study performs calculation 100 times using the Monte Carlo method and verifies the performance of the proposed algorithm with a network of nine detection performance. This study performs calculation 100 times using the Monte Carlo method and verifies nodes. the performance of the proposed algorithm with a network of nine detection nodes. The status estimation vector of the target is The status estimation vector of the target is

X =h[ xx xx. yy y. y x..  x y.. iyT]. . T

(29) (29)

X=

TheFNN–IMM FNN–IMMalgorithm algorithmisiscomposed composedof ofthree threemotion motionmodels: models: constant constant velocity, velocity,acceleration, acceleration, The and turn. The simulation experiment is completed in a MATLAB platform. The variances ofthe the and turn. The simulation experiment is completed in a MATLAB platform. The variances of 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 measurementerrors errorsofof nodes and 3 are supposed d1 = 200 m , d2 = 200 m , and d3 = 400 m22, measurement nodes 1, 1, 2, 2, and 3 are supposed as das 1 = 200 m , d2 = 200 m , and d3 = 400 m , 2 2 2 mm 2 2(i(i==4,4,5,5,. .…, respectively,and andthose thoseofofthe theother other nodes 100 10). Node Node 33 simulates simulatesaafault fault respectively, nodes areare di 2di= =100 . , 10). node, which cannot measure the coordinate of the target normally. The real motion and observation node, which cannot measure the coordinate of the target normally. The real motion and observation curvesof ofthe thetarget targettrajectory trajectoryare arepresented presentedininFigure Figure7.7. curves Node 1

4

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Figure Detected target trajectoryofofthe thenet netnodes. nodes.(a) (a)Detected trajectory of Figure 7. 7. Detected target trajectory Detected trajectory of node node 1; 1; (b) (b)Detected Detected trajectory node 2; (c) Detected trajectory of (d) node 3; (d) Detected trajectory of node 4. trajectory of node of 2; (c) Detected trajectory of node 3; Detected trajectory of node 4.

The figure shows that the target motion status consists of three uniform rectilinear motions and The figure shows that the target motion status consists of three uniform rectilinear motions and . x = 0 two accelerated motions. After the uniform rectilinear motion during the initial stage where two accelerated motions. After the uniform rectilinear motion during the initial stage where x = 0 m/s 2 .. m/s .and y = −15 m/s, the accelerated velocity after the first rotation is x = 0.075 m/s.. and y = −0.075 and y = −15 m/s, the accelerated velocity after the first rotation is x = 0.075 m/s2 and y = −0.075 m/s2 , .. .. 2 2 . 2The  =0.3 m/s2, whereas thatthe after the second rotation −0.3 m/s . Theinitial initialvalues valuesare areset set x = −0.3 whereas that after second rotation is x = is −0.3 m/s2m/s andand y = y− m/s asfollows: follows: as The Markov chain of the node:

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The Markov chain of the node: 

 0.05 0.05  0.9 0.05  0.05 0.9

0.9  Π1 = Π2 = Π3 =  0.05 0.05

(30)

System process noise: 

0  Q1 = Q2 = Q3 =  0 0

 0  0  0.01114

0 0.001 0

(31)

The sampling rate is f =

1 = 0.5 Hz T

(32)

where T represents the sampling period, and T = 2 s. The angular velocity of the CT model is ω = 0.001 rad/s

(33)

The performance of the algorithm is evaluated using the location information error in the target status estimation vector, and the evaluation function is as follows: J=

1 mont mont i∑ =1

q

ˆ i (:, 1) X(:, 1) − X

2

+ X(:, 3) − Xˆ i (:, 3)

2

(34)

where mont is the number of times the Monte Carlo calculation was performed, and Xˆ i represents the estimated output vector of the network target status in the i-th experiment. Sensors 2016, 16, 1823 4.2. Result Analysis Figure 8 illustrates the algorithm operation results. 600

Node1 Node2 Node3 Node4 Node5 Node6 Node7 Node8 Node9 Network

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Figure Figure 8. Computational results of the FNN–IMM. (a) Position error of each node. (b) Estimated 8. Computational results of the FNN–IMM. (a) Position error of each node. (b) Estimated trajectory of Nodes 1, 2, 1,and 3 and the output. trajectory of Nodes 2, and 3 and thenetwork network output.

Figure Figure 8 shows that that the the target position afterinfusion infusion is smaller the position 8 shows target positionerror error after is smaller than than the position error of error of any node. The algorithm reduces the influence of nodes with poor working status on target tracking any node. The algorithm reduces the influence nodes with poor working status on target tracking effects and improves the accuracy target tracking. tracking. effects and improves the accuracy ofoftarget 4.2.1. Performance Contrast Experiment of the Single-Node FNN–IMM Algorithm

4.2.1. Performance Contrast Experiment of the Single-Node FNN–IMM Algorithm

The performance of the single-node FNN–IMM algorithm is verified by comparing it with

The performance of the algorithm is verified comparing with IMMIMM-EKF, IMM-UKF, andsingle-node VB-IMM [27].FNN–IMM We perform the calculation 100 times by using the Monte it Carlo method and verify the performance of the proposed algorithm with Node 3 described above. The other EKF, IMM-UKF, and VB-IMM [27]. We perform the calculation 100 times using the Monte Carlo constant. methodinitial and values verifyare the performance of the proposed algorithm with Node 3 described above. The Random white noise with an average value of 0 and a variance of 100 is added based on the other initial values are constant. target observation data before the beginning of each time iteration of the algorithms. In this manner, Random white noise withtoan average value of 0noise and ofa the variance of proposed 100 is added the self-adaptive capability adjust the measurement algorithm in this based study on the target observation dataThe before the beginning of each time iteration of the algorithms. In this manner, can be examined. operation results are presented in Figure 9.

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the self-adaptive self-adaptive capability capability to to adjust adjust the the measurement measurement noise noise of of the the algorithm algorithm proposed proposed in in this this study study the Sensors 16, 1823 The can be be2016, examined. The operation operation results results are are presented presented in in Figure Figure 9. 9. can examined.

Estimation of of Target Target Trajectory Trajectory Estimation 4 IMM-UKF FNN-IMM 104 IMM-UKF 10 FNN-IMM xx10 xx10 44

00 2000 2000

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FNN-IMM FNN-IMM IMM-UKF IMM-UKF IMM-EKF IMM-EKF VB-IMM VB-IMM

600 600

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yy // m m

yy // m m

m xx // m IMM-EKF 44IMM-EKF x 10

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(a) (a)

(b) (b)

Figure 9. Computational Computational results results of of FNN–IMM, FNN–IMM, IMM-UKF, IMM-UKF, IMM-EKF, IMM-EKF, and and VB-IMM. VB-IMM. ((aa)) Mean Mean Figure Figure 9.9.Computational results of FNN–IMM, IMM-UKF, IMM-EKF, and VB-IMM. (a) Mean trajectory trajectory after 100 Monte Carlo trials. (b) Estimation errors of target position by FNN-IMM, IMMtrajectory after 100 Monte Carlo trials. (b) Estimation errors of target position by FNN-IMM, IMMafter 100 Monte Carlo trials. (b) Estimation errors of target position by FNN-IMM, IMM-UKF, IMM-EKF, UKF, IMM-EKF, and and VB-IMM VB-IMM UKF, IMM-EKF, and VB-IMM.

Figure 99 indicates indicates that that the the IMM-EKF IMM-EKF algorithm algorithm cannot cannot self-adaptively self-adaptively adjust adjust the the measurement measurement Figure Figure 9 indicates that the IMM-EKF algorithm cannot self-adaptively adjust the measurement error. The The algorithm algorithm becomes becomes ineffective ineffective and and stops stops operating operating when when the the statistical statistical characteristics characteristics of of error. error. The algorithm becomes ineffective and stops operating when the statistical characteristics the measurement error change. By contrast, the proposed FNN–IMM algorithm can effectively and the measurement error change. By contrast, the proposed FNN–IMM algorithm can effectively and of the measurement error change. By contrast, the proposed FNN–IMM algorithm can effectively self-adaptively adjust adjust the the measurement measurement error, error, and and its its applicability applicability is is higher higher than than that that of of the the IMMIMMself-adaptively and self-adaptively adjust the measurement error, and its applicability is higher than that of the UKF algorithm. UKF algorithm. IMM-UKF algorithm. 4.2.2. Performance Verification Verification Experiment Experiment of of Multi-Node Multi-Node Infusion Infusion Algorithm Algorithm 4.2.2. 4.2.2. Performance Performance Verification Experiment of Multi-Node Infusion Algorithm The fault fault tolerance tolerance of of the the data data infusion infusion algorithm algorithm proposed proposed in in this this study study is is verified. verified. At At the The The fault tolerance of the data infusion algorithm proposed in this study is verified. At the the beginning of of each each time iteration iteration of of the the algorithm, random random white white noise noise with with an an average value value of of 00 and and beginning beginning of each time time iteration of the algorithm, algorithm, random white noise with an average average value of 0 and variance of 100 100 is added added to the the observation data data of Node Node 3, and and the other other initial values values are constant. constant. aaa variance variance of of 100 is is added to to the observation observation data of of Node 3, 3, and the the other initial initial values are are constant. In this this manner, manner, the the invalidation invalidation of of the the node node detection detection data data is is simulated. simulated. The The operation results results are In In this manner, the invalidation of the node detection data is simulated. The operation operation results are are presented in in Figure Figure 10. 10. presented presented in Figure 10. 500 500

Node3 output output Node3 Network output output Network

Position Position Error Error // m m

400 400 300 300 200 200 100 100 000 0

100 100

200 200 Times // ss Times

300 300

400 400

Figure 10. Target Target Target positional positional errors errors of of the the network networkoutputs outputsand andNode Node3.3. Figure

Figure10 10and andTable Table demonstrate that the proposed proposed multi-node FNN–IMM algorithm can Figure 10 and Table 22 demonstrate the multi-node FNN–IMM algorithm can Figure 2 demonstrate thatthat the proposed multi-node FNN–IMM algorithm can reduce reduce the influence of erroneous data on output results at the time of multi-node data infusion. reduce the influence of erroneous data on output results at the time of multi-node data infusion. the influence of erroneous data on output results at the time of multi-node data infusion. Consequently, the fault tolerance and accuracy of the maneuvering target tracking system of the WSN are improved.

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Table 2. Position accuracy of the multi-node FNN–IMM and the single-node FNN–IMM.

Average Position Error ±∆ ( x, y) Average Error of x axis ±∆x Average Error of y axis ±∆y

Multi-Node Output of FNN–IMM

Single Node Output of FNN–IMM

46.9680 34.6952 34.7728

162.6039 119.7772 120.1503

5. Conclusions An IMM multi-node target tracking algorithm based on FNN is herein proposed to solve the maneuvering target tracking problem in WSNs. This algorithm can self-adaptively adjust system measurement errors without assuming the statistical characteristics of the system measurement errors. Consequently, the proposed algorithm can effectively overcome the shortcoming of the IMM algorithm of requiring a priori knowledge of the system measurement errors. The information infusion system of the FNN algorithm reduces the influence of poor nodes on target status estimation during the multi-node infusion stage. The fault tolerance mechanism is proposed according to the possible faults in data infusion, thus improving the fault tolerance of the algorithm. The simulation experiment indicates that the multi-node FNN–IMM algorithm exhibits high practicability. Acknowledgments: This research was supported by State Key laboratory of Explosion Science and Technology (4040xxx09). Author Contributions: Baoliang Sun and Chunlan Jiang designed the overall method for tracing the maneuvering target based on the FNN–IMM. Baoliang Sun wrote and revised the paper. Ming Li implemented the algorithm and helped with the experiments. Conflicts of Interest: The authors declare no conflicts of interest.

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