Node Topology Effect on Target Tracking Based on ... - IEEE Xplore
Node Topology Effect on Target Tracking Based on Underwater Wireless Sensor Networks Qiang Zhang, Meiqin Liu, Senlin Zhang, and Huayan Chen College of Electrical Engineering, Zhejiang University, Hangzhou 310027, P. R. China Email: {zhangqiang, liumeiqin, slzhang, chenhuayan}@zju.edu.cn Abstract—Since underwater nodes provide measurements for target tracking based on underwater wireless sensor networks (UWSNs), node topology, which is made up of the underwater nodes, may affect the performance of target tracking. But all the existing target tracking schemes do not consider this effect. This paper studies the effect of node topology on the target tracking in UWSNs. Firstly, by using the knowledge of geometry, the effects of four typical topologies on target tracking based on UWSNs are analyzed qualitatively. The four typical topologies include four nodes form a square, four nodes are in line, four nodes are close to each other, and four nodes form a regular tetrahedron. Secondly, to evaluate the arbitrary topology, the relationship between the posterior Cramer-Rao lower bound (PCRLB) and node’s position is derived. Thirdly, our target tracking scheme consisting of the optimal topology selection scheme by minimizing PCRLB, the optimal fusion center selection scheme by minimizing energy consumption, and the multi-sensor particle filter (PF) is designed. Last, simulation results show the effectiveness of the proposed scheme.
I.
I NTRODUCTION
Nowadays, tracking targets in the underwater environment is an indispensable part in many military or civil fields, such as modern underwater defense systems, navigation and control, and traffic monitoring in intelligent transportation systems. For this purpose, a lot of sonar-array-based target tracking algorithms are designed [1], [2], [3]. However, since sonar arrays are mounted or towed by a ship or a submarine , sonar-arraybased algorithms may be impractical for on-demand tracking missions. Furthermore, once the platform which mounts or tows the sonar arrays breaks down, the entire systems will fail. In terms of underwater target tracking, the underwater wireless sensor networks (UWSNs) offer a promising approach. The main advantages of UWSNs include its low cost, rapid deployment, self-organization, and fault tolerance [4]. Some effort has been made for underwater target tracking based on UWSNs. Instead of designing a complete target tracking scheme, only location estimation is discussed in [5]. To detect underwater target size, a maximum likelihood estimation algorithm is proposed in [6], but a tracking scheme is also not designed. In [7], two tracking schemes based on the distributed particle filter are proposed for tracking targets in cluster-based underwater sensor networks. However, this tracking is only considered in two dimensions, which is a severe limitation for applications. In [8], combination of interacting multiple model with the particle filter for three dimensional target tracking scheme is designed to solve the nonlinear and maneuvering problems. However, this scheme does not consider the energy consumption problem, which is also a limitation for practical
applications. Isbitiren and Akan design a three dimensional underwater target tracking approach for UWSNs [9]. The range between the node and the target is determined based on the time of arrival (TOA) of the echoes from the target. Then trilateration is utilized to calculate target’s position. Node’s position and the calculated velocity are then used to track the underwater target. In [10], to save energy consumption, the waking-up sleep mechanism is utilized to select several nodes to take part in tracking underwater target at each time step. However, this node selection scheme does not consider the effect of the selected node topology on the tracking accuracy. As we know, node topology has great effect on underwater node/target localization in UWSNs [9], [11]. For example, all the node/target localization schemes try to avoid the case that four nodes are co-plane. Since localization for target provides measurements for target tracking, it can be inferred that the topology of selected nodes may also have effect on the tracking performance. But all the existing target tracking schemes based on UWSNs do not consider this effect. In this paper, we will study node topology effect on target tracking based on UWSNs. Firstly, we list four typical node topologies and analyze their effects on target tracking by using the knowledge of geometry. The four typical topologies include four nodes form a square, four nodes are in line, four nodes are close to each other, and four nodes form a regular tetrahedron, respectively. Then to evaluate the arbitrary topology, we derive the relationship between the posterior Cramer-Rao lower bound (PCRLB) and node’s position, which forms the basis for our optimal topology selection. Last, this novel topology selection approach is combined with the multi-sensor particle filter (PF) to track underwater target moving through UWSNs. The rest of the paper proceeds as follows. The target tracking problem is formulated in section II. In section III, the effects of four typical topologies on target tracking are analyzed qualitatively and the relationship between the PCRLB and the arbitrary topology is derived. Our target tracking scheme including the optimal topology selection scheme, the optimal fusion center selection scheme and the multi-sensor PF is presented in section IV. In section V, simulation results are presented and the performance evaluations of the proposed scheme are accomplished. Conclusions of this study are drawn in section VI. II.
P ROBLEM F ORMULATION
This section formulates the problem of single target tracking based on UWSNs. The issues to be covered include system description and the multi-sensor PF for target tracking based on UWSNs.
A. System Description Throughout this paper, the target is considered as a slowly manoeuvring point object moving in a three dimensional UWSNs. Thus this motion can be adequately described by the nearly constant-velocity (CV) model. Accordingly, the target motion equation can be given by: xk = Φk−1 xk−1 + Γk−1 wk−1 ,
(1)
where the target state at time k is given by xk = [x, vx , y, vy , z, vz ]. x, y, z are the target positions in x, y, and z coordinate; vx , vy , vz are the target velocities in x, y, and z coordinate, respectively; Φk−1 is the state transition matrix; Γk−1 is the process noise matrix; The process noise wk−1 is assumed to be Gaussian with zero mean and covariance matrix Qk−1 . The state transition matrix and the process noise matrix are given as follows: ⎤ ⎡ 1 T 0 0 0 0 ⎢0 1 0 0 0 0 ⎥ ⎥ ⎢ ⎢0 0 1 T 0 0 ⎥ (2) Φk−1 = ⎢ ⎥, ⎢0 0 0 1 0 0 ⎥ ⎣0 0 0 0 1 T ⎦ 0 0 0 0 0 1 ⎡ 2 ⎤ T /2 0 0 ⎢ T 0 0 ⎥ ⎢ ⎥ ⎢ 0 T 2 /2 0 ⎥ Γk−1 = ⎢ , (3) T 0 ⎥ ⎢ 0 ⎥ ⎣ 0 0 T 2 /2⎦ 0 0 T For the measurement model, TOA measurements are used. By multiplying the TOAs with the known propagation speed, the observed range at time k of the nth sensing node is: zkn =hnk (xk ) + vkn , (4) (xk − xn )2 + (yk − yn )2 + (zk − zn )2 , where hnk (xk ) = (xk , yk , zk ) and (xn , yn , zn ) are the positions of the target and the nth sensing node; vkn is the measurement noise of sensor n, and it is assumed to be independent with zero-mean, white, Gaussian: vkn ∼ N (0, Rkn ). (5) B. Multi-sensor PF for Target Tracking in UWSNs PF is an efficient way to solve nonlinear and non-Gaussian problems. Readers can refer to [12],[13] for the detailed PF. Here, we summarize the muti-sensor PF for target tracking based on UWSNs. It is assumed that Ns (in this paper Ns = 4) sensing nodes take part in tracking the underwater target and provide the fusion center with their measurements at each time step. Once receiving these measurements, the fusion center fuses the measurements into a single multi-sensor measurement likelihood. This leads to the measurement likelihood over all Ns sensing nodes: p(zk |xik ) =
Ns n
p(zkn |xik ),
(6)
Algorithm 1 multi-sensor PF for target tracking 1: if k = 0 then 2: 1. Particle Initialization: 3: for i = 1, 2, . . . , N do 4: draw particle xik from the prior of target state p(x0 ); 5: end for 6: end if 7: for k = 1, 2, . . . do 8: 2. Importance Sampling: 9: for i = 1, 2, . . . , N do 10: sample xik ∼ p(xk |xik−1 ); 11: end for 12: for i = 1, 2, . . . , N do 13: update the importance weights by Eqs. (6), (7), and (8); 14: end for 15: for i = 1, 2, . . . , N do 16: normalize the importance weights: w ˜ki =
N j i wk / j=1 wk ; 17: end for 18: 3. Resampling: 19: for i = 1, 2, . . . , N do 20: multiply (suppress) xik with high (low) weights w ˜ki to obtain N particles; 21: set wki = w ˜ki = N −1 ; 22: end for 23: 4. Estimate
N the State and Covariance:
N 24: x ˆk = i=1 wki xik , Covk = i=1 wki (xik − x ˆk )(xik − T xˆk ) ; 25: end for where exp[−0.5(zkn − hnk (xik ))T (Rkn )−1 (zkn − hnk (xik ))] 2πRkn (7) is probability density function (pdf) of the measurement likelihood regarding to the measurement acquired by the nth sensing node at time k. In Eq.s (6) and (7), zk = (zk1 , . . . , zkn , . . . , zkNs ) is the concatenated measurement over all Ns sensing nodes; xik is the ith particle at time k. Adopting the prior the transition prior p(xk |xk−1 ) as the proposal distribution [12], the importance weights of particles are calculated as: p(zkn |xik ) =
i wki = wk−1 p(zk |xik ).
(8)
The above multi-sensor PF with N particle number is listed as Algorithm 1. From Eqs. (6), (7), and (8), it can be seen that measurements from sensing nodes have effects on the importance weights. The interesting problems are: •
Does the topology of sensing nodes affect the performance of target tracking based on the UWSNs?
•
What is the relationship between the topology and the performance of target tracking?
The answers to these questions will be described in the following sections. III.
N ODE T OPOLOGY E FFECT ON TARGET T RACKING
To answer the questions mentioned above, this section studies node topology effect on target tracking based on
UWSNs. A. The Analysis of Four Typical Topologies Effects on Target Tracking Assume that node A, node B, node C, and node D are the sensing nodes at k time step and their positions are (xa , ya , za ), (xb , yb , zb ), (xc , yc , zc ), (xd , yd , zd ), respectively; target’s position is (xk , yk , zk ). Based on TOA, we have the following equations: ⎧ ⎪ dat = (xa − xk )2 + (ya − yk )2 + (za − zk )2 , ⎪ k ⎪ ⎨dbt = (x − x )2 + (y − y )2 + (z − z )2 , k b k b k k b (9) ct 2 + (y − y )2 + (z − z )2 , ⎪ d = (x − x ) c k c k c k ⎪ k ⎪ dt ⎩ dk = (xd − xk )2 + (yd − yk )2 + (zd − zk )2 ,
Fig. 1. Four nodes form a square (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
bt ct dt where dat k , dk , dk , and dk are the ranges between the target and node A, B, C, and D, respectively. These equations form the basis for the analysis of four typical topologies effects on target tracking.
•
Topology 1: four nodes form a square (Fig. 1). For this topology, if the real target T1 satisfies the range constraint determined by Eq.s (9), the false target T2 , as the mirror of T1 , also satisfies this constraint. In other words, this topology cannot uniquely determine target’s position. Fortunately, there is only one false target, which means that the tracking performance may not be very bad with the help of particle prediction based on the CV model.
Fig. 2. Four nodes are in line (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
B. The Relationship Between the PCRLB and Node Topology
•
Topology 2: four nodes are in line (Fig. 2). For this topology, if the real target T1 satisfies the range constraint determined by Eq.s (9), any point on the dashed circle, which is orthogonal to line AD, also satisfies this constraint. Hence, there are infinite false targets when four sensing nodes are in line. So this topology is worse than the previous one.
•
Topology 3: four nodes are close to each other (Fig. 3). If four sensing nodes are very close to each other, the role of the four nodes is almost the same as one node. Hence, any point on the dashed sphere determined by Eq.s (9) can be the false target. So there are also infinite false targets when four sensing nodes are close to each other. Hence, this topology is not a good choice.
The relationship between the PCRLB and the error covariance matrix of the target estimation is:
Topology 4: four nodes form a regular tetrahedron (Fig. 4). This topology avoids the shortcomings of the previous three topologies. For this topology, target’s location can be uniquely decided by Eq.s (9). Hence, this topology may be a good choice.
where
•
According to above analysis, it can be seen that the topology of the sensing nodes really has effect on target tracking. However, this analysis is not only qualitative but also limited to some typical topologies. In the next subsection, we will derive the the relationship between the PCRLB and sensing node’s position to evaluate the arbitrary topology quantitatively.
To evaluate the arbitrary topology, this subsection calculates the relationship between the PCRLB and the sensing node’s position. The PCRLB is defined to be the inverse of the Fisher information matrix (FIM), which provides a lower bound on the mean square error (MSE) of target state estimation. There are several approaches to compute the PCRLB, readers can refer to [14], [15].
xk − xk ]T ≥ Jk−1 , E[ˆ xk − xk ][ˆ
(10)
where Jk is the FIM at the kth time step. Assuming that ∇α is the first-order partial derivation operator and Jk−1 has already been obtained, Jk can be computed as follows [14]: 22 21 11 12 − Dk−1 (Jk−1 + Dk−1 )−1 Dk−1 , Jk = Dk−1
(11)
11 Dk−1 = E{[−∇xk−1 ∇Txk−1 ln p(xk |xk−1 )]},
(12)
12 = E{[−∇xk ∇Txk−1 ln p(xk |xk−1 )]}, Dk−1
(13)
21 12 T = E{[−∇xk−1 ∇Txk ln p(xk |xk−1 )]} = [Dk−1 ] , (14) Dk−1 22 Dk−1 = E{[−∇xk ∇Txk ln p(xk |xk−1 )]}
+E{[−∇xk ∇Txk ln p(zk |xk )]}.
(15)
The recursion of Eq. (11) starts from an initial FIM J0 = E{−∇x0 ∇Tx0 ln p(x0 )}, where p(x0 ) is the priori pdf of
where dnp k =
Fig. 3. Four nodes are close to each other (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
(ˆ xk|k−1 − xnk )2 + (ˆ yk|k−1 − ykn )2 + (ˆ zk|k−1 − zkn )2 (23)
is the range between the nth sensing node and the predicted target’s position; (xnk , ykn , zkn ) is the position of nth sensing node at k time step; and (ˆ xk|k−1 , yˆk|k−1 , zˆk|k−1 ) is the predicted target’s position, which can be given by: ⎧
N ˆk|k−1 = N1 ( i=1 xik|k−1 ) i = 1, 2 · · · N ⎪ ⎨x
N i yˆk|k−1 = N1 ( i=1 yk|k−1 (24) ) i = 1, 2 · · · N ⎪
⎩ N 1 i zˆk|k−1 = N ( i=1 zk|k−1 ) i = 1, 2 · · · N i i where N is particle number and (xik|k−1 , yk|k−1 , zk|k−1 ) is the predicted position according to the ith particle. Substituting Eq.s (22), (23), and (24) into (21), Λk can be calculated as follows: ⎡ 11 ⎤ τk 0 τk13 0 τk15 0 ⎢ 0 0 0 0 0 0 ⎥ ⎢ 31 ⎥ ⎢ τ 0 τk33 0 τk35 0 ⎥ (25) Λk = ⎢ k ⎥, ⎢ 0 0 0 0 0 0 ⎥ ⎣ τ 51 0 τ 53 0 τ 55 0 ⎦ k k k 0 0 0 0 0 0
Fig. 4. Four nodes form a regular tetrahedron (the blue circle stands for the sensing node; the solid triangle stands for the real target; and there is no false target).
where τk13 = τk31 , τk15 = τk51 , τk35 = τk53 , τk11
=
N Ns [ N1 ( i=1 xik|k−1 ) − xnk ]2 n=1
target’s state. Eq.s (12), (13), (14) and the first part of Eq. (15) gives the dependency of PCRLB on the motion model. This paper adopts the CV model, the follows are obtained by ¯ k−1 = Γk−1 Qk−1 ΓT : defining Q k−1 11 ¯ −1 Φk−1 , Dk−1 = ΦTk−1 Q k−1
(17)
21 ¯ −1 ΦTk−1 , Dk−1 = −Q k−1
(18)
22 ¯ −1 + Λk , Dk−1 =Q k−1
(19)
Λk =
=
E{[−∇xk ∇Txk
ln p(zk |xk )]}.
(20)
Eq. (20) gives the dependency of PCRLB on the measurements, which may contain the information of sensing node topology. Similar to the PF tracking algorithm, we also adopt particles to compute the Λk . Assuming that there are Ns sensing nodes taking part in the tracking task, Λk can be approximated by [15]: Λk ≈
Ns
ˆ kn )T (Rkn )−1 (H ˆ kn ), (H
(21)
n=1
ˆ n is the Jacobian matrix for the nth sensing node at where H k time k, which can be calculated as: n n −xn (ˆ yk|k−1 −yk ) (ˆ zk|k−1 −zk ) k) ˆ kn = (ˆxk|k−1 H 0 0 0 np np dnp d d k k k (22)
=
N i Ns [ N1 ( i=1 yk|k−1 ) − ykn ]2 n=1
(16)
¯ −1 , −ΦTk−1 Q k−1
12 Dk−1
τk33
τk55
=
=
2 Rkn (dnp k )
N i Ns [ N1 ( i=1 zk|k−1 ) − zkn ]2 n=1
τk13
2 Rkn (dnp k )
2 Rkn (dnp k )
,
(26)
,
(27)
,
(28)
N
N i Ns [ N1 ( i=1 xik|k−1 ) − xnk ][ N1 ( i=1 yk|k−1 ) − ykn ] n=1
2 Rkn (dnp k )
,
(29)
N i 1 N Ns i n 1 [ N ( i=1 xk|k−1 ) − xk ][ N ( i=1 zk|k−1 ) − zkn ] , τk15 = 2 Rkn (dnp k ) n=1 (30)
N i
N i 1 Ns n 1 n [ ( y ) − y ][ ( z ) − z k N k] i=1 k|k−1 i=1 k|k−1 N τk35 = . 2 Rkn (dnp k ) n=1 (31) This completes the calculation of the PCRLB. Since the PCRLB is related to the positions of sensing nodes, the arbitrary topology can be evaluated by PCRLB. Specifically, the topology that minimizes the PCRLB is the best one and the topology that maximizes the PCRLB is the worst one. It is needed to point out that the calculation of PCRLB is only based on the prediction information, which means that PCRLB can be used to select the optimal topology for target tracking.
B. New Fusion Center Selection At each time step, the fusion center mainly takes responsibility for the optimal topology selection, measurement fusion, target state estimation, new fusion center selection, and passing the estimation results to the new fusion center. Since the fusion center plays an important role in target tracking based on UWSNs, a new fusion center is needed to be selected from the Ckopt . Due to the limited energy resources [4], the rule of the new fusion center selection is to minimize energy consumption. To quantify the energy consumption in data transmission, we adopt the energy dissipation model based on the underwater acoustic communication principle in [16]. To send a b-bit packet from one node to another over a range d, the energy consumption of the sender is: Fig. 5. Optimal topology selection (the blue circle stands for the qualified candidate; the black circle stands for the unqualified candidate; the red circle stands for the fusion center at time k − 1; the red solid triangle stands for target’s position at time k − 1; the red dashed triangle stands for the predicted position of the target at time k; R and r are node’s communication range and sensing range, respectively).
IV.
T HE D ETAILED TARGET T RACKING S CHEME WITH THE O PTIMAL T OPOLOGY
Based on the PCRLB, the optimal topology can be selected to take part in tracking the underwater target at each time step. This section presents the detailed target tracking with the optimal topology.
Es (b, d) = bP0 A(d),
and to receive this packet, the energy consumption of the receiver is: Er (b) = bPr , (36) where P0 is the power needed at the input to the receiver; Pr depends on the receiver; and A(d) is the attenuation given as: A(d) = dm ad ,
A. Optimal Node Topology Selection
nf dnp k ≤ r and dk−1 ≤ R.
(33)
(37)
where m is the energy spreading factor and a = 10α(f )/10 is a frequency-dependent term obtained by: α(f ) =
The basic idea of optimal node topology selection scheme is illustrated by Fig. 5. At time k − 1, the fusion center predicts target’s position at time k based on the CV model. Then according to the prediction information, dnp k which is the range between the nth node and the predicted position can be calculated by Eq. (23); and dnf k−1 which is the range between the nth node and the fusion center at time k − 1 can be computed by: f f dnf (xfk−1 − xnk )2 + (yk−1 − ykn )2 + (zk−1 − zkn )2 , k−1 = (32) f f , zk−1 ) is the position of the fusion center where (xfk−1 , yk−1 at time k − 1. The qualified candidates are the nodes that may have the chance to take part in tracking the target at k time step, which satisfies the following constraints:
(35)
0.11f 2 44f 2 + + 2.75 × 10−4f 2 + 0.003. (38) 2 1+f 4100 + f 2
Let F Ck−1 be the old fusion center at time k − 1, and F Ck be the new fusion center. Let sm , se , and sc be the sizes of data packets of the measurement, the estimation results, and the waking-up command, respectively. dnf k is the range between the new fusion center and the nth sensor node; dnf k−1 is the range between the old fusion center and the nth sensor node; and dfk f be the range between two fusion centers. Let Ns be the number of sensing nodes. Obviously, the energy consumption is a function of F Ck , which can be calculated as follows. If F Ck ≡ F Ck−1 ,
Ns −1 sm [P0 n=1 A(dnf ) + Pr (Ns − 1)]
Ns −1 knf Energy(F Ck ) = +sc [P0 n=1 A(dk−1 ) + Pr (Ns − 1)], (39) else,
Let Nq donate the number of the qualified candidates and Ck denote the combination of Ns (in this paper, Ns = 4.) candidates. If Nq ≥ Ns , our goal is to select an optimal topology Ckopt from the Nq candidates. Obviously, Ckopt can be determined by: Ckopt = arg min Jk−1 (1, 1) + Jk−1 (3, 3) + Jk−1 (5, 5) , (34)
⎧
Ns −1 nf ⎪ ⎨sm [P0 n=1 A(dk ) + Pr (Ns − 1)] nf s −1 Energy(F Ck ) = +sc [P0 N n=1 A(dk−1 ) + Pr (Ns − 1)] ⎪ ⎩+s [P A(df f ) + P ]. e 0 r k (40) Our goal is to select an optimal fusion center F Ckopt from the Ckopt . Obviously, F Ckopt can be determined by:
where Jk−1 is the inverse matrix of FIM at the kth time step.
F Ckopt = arg min Energy(F Ck ).
(41)
C. Target Tracking Scheme Based on the Optimal Topology By making use of the optimal topology selection, the optimal fusion center selection, the multi-sensor PF algorithm can be extended to tracking the underwater target based on UWSNs. The pseudo-code of our target tracking scheme is listed in Algorithm 2. Algorithm 2 Our target tracking scheme 1: 1.The Initial Fusion Center F C a Does the Following: 2: if k = 0 then 3: 1.1. Particle Initialization: 4: for i = 1, 2, . . . , N do 5: draw particle xik from the prior of target state p(x0 ); 6: end for 7: end if 8: for k = 1, 2, . . . do 9: 1.2 Importance Sampling: 10: for i = 1, 2, . . . , N do 11: sample xik ∼ p(xk |xik−1 ); 12: end for 13: select the optimal topology using Eq. (34); 14: select the optimal fusion center using Eq. (41); 15: if the new fusion center is the same as the old one then 16: get the measurements from the Ns sensing nodes; 17: for i = 1, 2, . . . , N do 18: update the importance weights by Eqs. (6), (7), and (8); 19: end for 20: for i = 1, 2, . . . , N do 21: normalize the importance weights: N w ˜ki = wki / j=1 wkj ; 22: end for 23: 1.3. Resampling: 24: for i = 1, 2, . . . , N do 25: multiply (suppress) xik with high (low) weights w ˜ki to obtain N particles; 26: set wki = w ˜ki = N −1 ; 27: end for 28: 1.4. Estimate the State and Covariance:
N 29: x ˆak = i=1 wki xik ,
N ˆk )T ; Covka = i=1 wki (xik − xˆk )(xik − x 30: else 31: forward x ˆak and Covka to the new fusion center F Cb ; 32: 2. The New Fusion Center F Cb Does the Following: 33: for i = 1, 2, . . . , N do 34: draw xik ∼ N (ˆ xak , Covka ); 35: end for 36: repeat the steps done by the previous fusion center; 37: end if 38: end for
V.
S IMULATIONS
In this section, we present simulations using Matlab to study node topology effect on target tracking based on UWSNs.
A. Simulation for Target Tracking Based on the Four Typical Topologies We simulate the four typical topologies mentioned in section III to study their effects on target tracking. Underwater nodes are deployed over a region of size 1000m × 1000m × 1000m. From 1s to 30s, four nodes form a square; from 31s to 70s, four nodes are in line; from 71s to 120s, four nodes form a regular tetrahedron; and from 121s to 160s, four nodes are close to each other. The actual initial state of the target is x0 = [0 5 0 5 0 5]T ; the initial estimation of the target is x ˆ0 = [5 5 5 5 5 5]T ; and covariance matrix M0 = I6×6 . The initial FIM J0 = M0−1 . The measurement process with variance R = 52 . The sampling time is 1s.The particle number N = 500 and the simulation runs M = 100. To indicate the accuracy of target tracking, we adopt root mean square error (RMSE), defined as follows, to measure the tracking performance: M [xk,i − xˆk,i ]2 + [yk,i − yˆk,i ]2 + [zk,i − zˆk,i ]2 Err(k) = , M i=1 (42) where (xk,i , yk,i , zk,i ) and (ˆ xk,i , yˆk,i , zˆk,i ) are the true and the estimated positions of the target at time k, respectively, in the ith simulation. Tracking results for the four typical topologies are shown in Fig. 6. From 1s to 30s, the tracking error is small. This is because there are only two false targets when four nodes form a square. When the initial estimation error is small, the tracking algorithm can perform well with the help of the particle prediction based on the CV model. From 31s to 70s, the tracking performance is very bad. This is because there are infinite false targets when four nodes are in line. In this case, the measurements provide little information about target’s state. From 71s to 120s, the tracking error is small. This is because target’s location can be uniquely determined by this topology. For this topology, the measurements can provide target’s location information. From 121s to 160s, the tracking performance turns bad again. The reason is the same as that of four nodes are in line. The tracking performance under different topologies can confirm our analysis in section III. Furthermore, the estimated PCRLB of the four typical topologies is also plotted in Fig. 6. It can be seen that the estimated PCRLB can well reflect the quality of node topology. Specially, the smaller PCRLB achieves, the better the topology is. This also demonstrates the effectiveness of the estimated PCRLB. In Fig. 6, the tracking error happens to be lower than the lower bound PCRLB at some time steps. This is mainly because the expectation in Λk = E{[−∇xk ∇Txk ln p(zk |xk )]} has no closed-form analytical solution and must be approximated. Furthermore, our approach for calculation of PCRLB only bases on the prediction information instead of the posterior pdf of target state, which leads to the degradation of the estimation accuracy. Fortunately, a rough estimation of PCRLB is enough to select a better node topology to take part in tracking target at each time step, which can be confirmed by the next simulation experiment.
20
20
Tracking Error Estimated PCRLB
Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
18 16
15
Error/m
Error/m
14
10
12 10 8 6
5
4 2
0 0
20
40
60
80 100 Time step/s
120
140
0 0
160
Fig. 6. Tracking results for the four typical topologies (from 1s to 30s, four nodes form a square; from 31s to 70s, four nodes are in line; from 71s to 120s, four nodes form a regular tetrahedron; and from 121s to 160s, four nodes are close to each other).
The sizes of data packets are sm = 32 bits, se = 64 bits, and sc = 8 bits. For the sake of simplicity, let P0 = Pr = 10−9 J/bit, m = 1.5, and f = 15kHz. The node sensing range r = 200m and communication range R = 400m. All the other parameters are the same as those set in the previous simulation. The performance metrics include tracking error, energy consumption, and communication traffic. Tracking error is defined as Eq. (42), the energy consumption is defined as Eqs. (39) (40), and the communication traffic indicates the total amount of data transmission. Communication traffic can be defined as follows. (sm + sc )(Ns − 1) if F Ck ≡ F Ck−1 , Com(k) = (sm + sc )(Ns − 1) + se otherwise. (43) TABLE I.
T HE AVERAGED METRICS OF OUR SCHEME , THE SCHEME WITH MAXIMUM PCRLB, AND 4NN SCHEME
Algorithem Our scheme Maximum PCRLB 4N N
Error (m) 2.67 9.00 4.75
Energy (mJ) 0.48 0.51 0.29
Communication (bits) 149.65 125.25 124.66
The results of tracking error, energy consumption, and communication traffic are plotted in Fig. 7, Fig. 8 and Fig. 9, respectively. Illustrated from Fig. 7, our scheme with the minimum PCRLB achieves the highest tracking accuracy than that of the other two schemes. This is because our scheme can select the optimal topology at each time step by minimizing
40
60
80 100 Time step/s
120
140
160
Fig. 7. Tracking errors of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
B. Simulation for Our Target Tracking Scheme
Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
1.2 Energy consumption/mJ
In this subsection, we present simulation for our scheme with the minimum PCRLB. Besides our scheme, the scheme with the maximum PCRLB, and the four nearest neighbors(4NN) scheme [17] are also simulated for comparison. For the scheme with the maximum PCRLB, the worst topology that maximizes the PCRLB is selected and the way of the new fusion center selection is the same as that of our scheme. For the 4NN scheme, four nodes that are nearest to the predicted target’ position are selected and the node that is nearest to the predicted position is selected as the new fusion center.
20
1 0.8 0.6 0.4 0.2 0 0
20
40
60
80 100 Time step/s
120
140
160
Fig. 8. Energy consumption of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
the PCRLB, whose measurements contains the most useful information of the underwater target. Since the scheme with the maximum PCRLB selects the worst topology at each time step by maximizing the PCRLB, its tracking performance is the worst. For the 4NN scheme, neither the optimal topology nor the worst one is selected. Hence, its performance is between that of the other two schemes. Table I shows that our scheme achieves about 30% tracking error of the scheme with the maximum PCRLB and about 56% tracking error of the 4NN scheme. Illustrated from Fig. 8, The 4NN achieves the smallest energy consumption. This because the four nodes selected by the 4NN scheme are all close to the predicted target’s position, which means that they are also close to each other. From Eq.s (39) and (40), we can see that the smaller dnf k is, the less energy consumption can be achieved. Hence the 4NN scheme achieve less energy consumption than that of the other two schemes. Table I shows that 4NN achieves about 60% energy consumption of our scheme and about 57% energy consumption of the scheme with the maximum PCRLB. Illustrated from Fig.9, the communication traffic performance of our scheme is worse than that of the other two schemes. This is because, the topology selected by our scheme changes frequently to search for the best one at each time step. When
20120101110115, the Zhejiang Provincial Science and Technology Planning Projects of China under Grant 2012C21044, the Marine Interdisciplinary Research Guiding Funds for Zhejiang University under Grant 2012HY009B, the Fundamental Research Funds for the Central Universities under Grant 2014XZZX003-12, the Open Fund for the Aircraft Marine Measurement and Control Joint Laboratory under Grant FOM2014OF06, and the ASFC under Grant 20132076002.
Communication traffic/bits
200
150
100 Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
50
0 0
20
40
60
80 100 Time step/s
120
R EFERENCES
140
160
Fig. 9. Communication traffic of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
topology changes frequently, the fusion center also changes frequently. From Eq. (43), it can be seen that the more frequently the fusion center changes, the larger communication traffic can be achieved. Hence our scheme achieves a little worse communication traffic performance than that of the other two schemes. Table I shows that 4NN scheme achieves about 83% communication traffic of our scheme and about 99% communication traffic of the scheme with the maximum PCRLB. In summary, our scheme is optimal in the sense of tracking error. Compared to other two schemes, our scheme achieves the best performance in tracking error. For the energy consumption and communication traffic, the 4NN scheme is a better choice. Hence, our scheme is more suitable for the scene that requires high tracking accuracy. VI.
C ONCLUSIONS
This paper studies node topology effect on target tracking based on UWSNs. By using the knowledge of geometry, the effects of four typical topologies on target tracking are analyzed qualitatively. To evaluate the arbitrary topology quantitatively, the relationship between the PCRLB and node topology is also derived, which forms the basis for our optimal topology selection scheme. Simulation results show that the PCRLB reflects the quality of arbitrary topology well and our scheme with the minimum PCRLB achieves higher tracking accuracy than the existing scheme. However, our scheme is optimal in the sense of tracking error. Hence, it is suitable for the scene that requires high tracking accuracy. For the further research, we will improve our scheme in the aspects of energy consumption and communication traffic and extend our scheme to track multiple targets based on UWSNs. ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China under Grants 61374021, 61222310, and 61174142, the Zhejiang Provincial Natural Science Foundation of China under Grant LZ14F030003, the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP) under Grants 20130101110109 and
[1] M. Asif, M. Rizal, and A. Yahya, “An active contour for underwater target tracking and navigation,” in Pro. Int. Conf. Man-Mach. Syst., Langkawi Islands, Malaysia, Sep., 2006, pp. 1–6. [2] E. Dalberg, A. Lauberts, R. K. Lennartsson, M. J. Levoen, and L. Persson, “Underwater target tracking by means of acoustic and electromagnetic data fusion,” in Proc. 9th Int. Conf. Inf. Fusion, Florence, Italy,, Jul., 2006, pp. 1–7. [3] M. I. Pettersson, V. Zetterberg, and I. Claesson, “Detection and imaging of moving targets in wideband sas using fast time backprojection combined with space-time processing,” in Proc. MTS/IEEE Oceans, Washington DC, USA, Sep., 2005, pp. 2388–2393. [4] J.E. Faugstadmo, “Underwater wireless sensor networks,” in Sensor Technologies and Applications (SENSORCOMM), Venice, Jul. 2010, pp. 422–427. [5] S. Zhou and P. Willett, “Submarine location estimation via a network of detection-only sensors,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 3104–3115, Jun. 2007. [6] Q. Liang and X. Cheng, “Underwater acoustic sensor networks:target size detection and performance analysis,” Ad Hoc Netw., vol. 7, no. 4, pp. 803–808, Jun. 2009. [7] Y. Huang, W. Liang, H.-B. Yu, and Y. Xiao, “Target tracking based on a distributed particle filter,” Wireless Communications and Mobile Computing, vol. 8, no. 8, pp. 1023–1033, Oct. 2008. [8] X. Wang, M.-X. Xu, H.-B. Wang, Y. Wu, and H.-Y. Shi, “Combination of interacting multiple models with the particle filter for threedimensional target tracking in underwater wireless sensor networks,” Mathematical Problems in Engineering, pp. 1–16, Oct. 2012. [9] G. Isbitiren, and O. B. Akan, “Three-dimensional underwater target tracking with acoustic sensor networks,” IEEE Transaction on Vehicular Technology, vol. 60, no. 8, pp. 3897–3906, Oct. 2011. [10] C. Yu, K. Lee, J. Choi, and Y. Seo, “Distributed single target tracking in underwater wireless sensor network,” in Proc. SCIE Annu. Conf., Tokyo, Japan, Aug. 2008, pp. 1351–1356. [11] G.J. Han, J.F. Jiang, L. Shu, Y.J. Xu, and F. Wang, “Localization algorithms of underwater wireless sensor networks: a survey,” SENSORS, vol. 12, no. 2, pp. 2026–2061, Feb. 2012. [12] M.S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking,” IEEE Transaction on Signal Processing, vol. 50, no. 2, pp. 174–188, Feb. 2002. [13] N. Gordon, D. Salmond, and A.F.M. Smith, “Novel approach to nonlinear and non-gaussian bayesian state estiamtion,” Proceeding of IEE(F), vol. 140, no. 2, pp. 107–113, Feb. 1993. [14] P. Tichavsky, C.H. Muravchik, and A. Nehorai, “Posterior cramerrao bound for disctete-time nonlinear filtering,” IEEE Transactions on Signal Processing, vol. 46, no. 5, pp. 1386–1396, May 1998. [15] X. Zhang, P. Willett, and Y. Bar-Shalom, “Dynamic cramer-rao bound for target tracking in clutter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 41, no. 4, pp. 1154–1167, Oct. 2005. [16] E. Sozer, M. Stojanovic, and J. Proakis, “Underwater acoustic networks,” IEEE Journal of Oceanic Engineering, vol. 25, no. 1, pp. 72–83, Jan. 2000. [17] D. Guo and X. Wang, “Dynamic sensor collaboration via sequential monte carlo,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 1037–1047, Aug. 2004.
I.
I NTRODUCTION
Nowadays, tracking targets in the underwater environment is an indispensable part in many military or civil fields, such as modern underwater defense systems, navigation and control, and traffic monitoring in intelligent transportation systems. For this purpose, a lot of sonar-array-based target tracking algorithms are designed [1], [2], [3]. However, since sonar arrays are mounted or towed by a ship or a submarine , sonar-arraybased algorithms may be impractical for on-demand tracking missions. Furthermore, once the platform which mounts or tows the sonar arrays breaks down, the entire systems will fail. In terms of underwater target tracking, the underwater wireless sensor networks (UWSNs) offer a promising approach. The main advantages of UWSNs include its low cost, rapid deployment, self-organization, and fault tolerance [4]. Some effort has been made for underwater target tracking based on UWSNs. Instead of designing a complete target tracking scheme, only location estimation is discussed in [5]. To detect underwater target size, a maximum likelihood estimation algorithm is proposed in [6], but a tracking scheme is also not designed. In [7], two tracking schemes based on the distributed particle filter are proposed for tracking targets in cluster-based underwater sensor networks. However, this tracking is only considered in two dimensions, which is a severe limitation for applications. In [8], combination of interacting multiple model with the particle filter for three dimensional target tracking scheme is designed to solve the nonlinear and maneuvering problems. However, this scheme does not consider the energy consumption problem, which is also a limitation for practical
applications. Isbitiren and Akan design a three dimensional underwater target tracking approach for UWSNs [9]. The range between the node and the target is determined based on the time of arrival (TOA) of the echoes from the target. Then trilateration is utilized to calculate target’s position. Node’s position and the calculated velocity are then used to track the underwater target. In [10], to save energy consumption, the waking-up sleep mechanism is utilized to select several nodes to take part in tracking underwater target at each time step. However, this node selection scheme does not consider the effect of the selected node topology on the tracking accuracy. As we know, node topology has great effect on underwater node/target localization in UWSNs [9], [11]. For example, all the node/target localization schemes try to avoid the case that four nodes are co-plane. Since localization for target provides measurements for target tracking, it can be inferred that the topology of selected nodes may also have effect on the tracking performance. But all the existing target tracking schemes based on UWSNs do not consider this effect. In this paper, we will study node topology effect on target tracking based on UWSNs. Firstly, we list four typical node topologies and analyze their effects on target tracking by using the knowledge of geometry. The four typical topologies include four nodes form a square, four nodes are in line, four nodes are close to each other, and four nodes form a regular tetrahedron, respectively. Then to evaluate the arbitrary topology, we derive the relationship between the posterior Cramer-Rao lower bound (PCRLB) and node’s position, which forms the basis for our optimal topology selection. Last, this novel topology selection approach is combined with the multi-sensor particle filter (PF) to track underwater target moving through UWSNs. The rest of the paper proceeds as follows. The target tracking problem is formulated in section II. In section III, the effects of four typical topologies on target tracking are analyzed qualitatively and the relationship between the PCRLB and the arbitrary topology is derived. Our target tracking scheme including the optimal topology selection scheme, the optimal fusion center selection scheme and the multi-sensor PF is presented in section IV. In section V, simulation results are presented and the performance evaluations of the proposed scheme are accomplished. Conclusions of this study are drawn in section VI. II.
P ROBLEM F ORMULATION
This section formulates the problem of single target tracking based on UWSNs. The issues to be covered include system description and the multi-sensor PF for target tracking based on UWSNs.
A. System Description Throughout this paper, the target is considered as a slowly manoeuvring point object moving in a three dimensional UWSNs. Thus this motion can be adequately described by the nearly constant-velocity (CV) model. Accordingly, the target motion equation can be given by: xk = Φk−1 xk−1 + Γk−1 wk−1 ,
(1)
where the target state at time k is given by xk = [x, vx , y, vy , z, vz ]. x, y, z are the target positions in x, y, and z coordinate; vx , vy , vz are the target velocities in x, y, and z coordinate, respectively; Φk−1 is the state transition matrix; Γk−1 is the process noise matrix; The process noise wk−1 is assumed to be Gaussian with zero mean and covariance matrix Qk−1 . The state transition matrix and the process noise matrix are given as follows: ⎤ ⎡ 1 T 0 0 0 0 ⎢0 1 0 0 0 0 ⎥ ⎥ ⎢ ⎢0 0 1 T 0 0 ⎥ (2) Φk−1 = ⎢ ⎥, ⎢0 0 0 1 0 0 ⎥ ⎣0 0 0 0 1 T ⎦ 0 0 0 0 0 1 ⎡ 2 ⎤ T /2 0 0 ⎢ T 0 0 ⎥ ⎢ ⎥ ⎢ 0 T 2 /2 0 ⎥ Γk−1 = ⎢ , (3) T 0 ⎥ ⎢ 0 ⎥ ⎣ 0 0 T 2 /2⎦ 0 0 T For the measurement model, TOA measurements are used. By multiplying the TOAs with the known propagation speed, the observed range at time k of the nth sensing node is: zkn =hnk (xk ) + vkn , (4) (xk − xn )2 + (yk − yn )2 + (zk − zn )2 , where hnk (xk ) = (xk , yk , zk ) and (xn , yn , zn ) are the positions of the target and the nth sensing node; vkn is the measurement noise of sensor n, and it is assumed to be independent with zero-mean, white, Gaussian: vkn ∼ N (0, Rkn ). (5) B. Multi-sensor PF for Target Tracking in UWSNs PF is an efficient way to solve nonlinear and non-Gaussian problems. Readers can refer to [12],[13] for the detailed PF. Here, we summarize the muti-sensor PF for target tracking based on UWSNs. It is assumed that Ns (in this paper Ns = 4) sensing nodes take part in tracking the underwater target and provide the fusion center with their measurements at each time step. Once receiving these measurements, the fusion center fuses the measurements into a single multi-sensor measurement likelihood. This leads to the measurement likelihood over all Ns sensing nodes: p(zk |xik ) =
Ns n
p(zkn |xik ),
(6)
Algorithm 1 multi-sensor PF for target tracking 1: if k = 0 then 2: 1. Particle Initialization: 3: for i = 1, 2, . . . , N do 4: draw particle xik from the prior of target state p(x0 ); 5: end for 6: end if 7: for k = 1, 2, . . . do 8: 2. Importance Sampling: 9: for i = 1, 2, . . . , N do 10: sample xik ∼ p(xk |xik−1 ); 11: end for 12: for i = 1, 2, . . . , N do 13: update the importance weights by Eqs. (6), (7), and (8); 14: end for 15: for i = 1, 2, . . . , N do 16: normalize the importance weights: w ˜ki =
N j i wk / j=1 wk ; 17: end for 18: 3. Resampling: 19: for i = 1, 2, . . . , N do 20: multiply (suppress) xik with high (low) weights w ˜ki to obtain N particles; 21: set wki = w ˜ki = N −1 ; 22: end for 23: 4. Estimate
N the State and Covariance:
N 24: x ˆk = i=1 wki xik , Covk = i=1 wki (xik − x ˆk )(xik − T xˆk ) ; 25: end for where exp[−0.5(zkn − hnk (xik ))T (Rkn )−1 (zkn − hnk (xik ))] 2πRkn (7) is probability density function (pdf) of the measurement likelihood regarding to the measurement acquired by the nth sensing node at time k. In Eq.s (6) and (7), zk = (zk1 , . . . , zkn , . . . , zkNs ) is the concatenated measurement over all Ns sensing nodes; xik is the ith particle at time k. Adopting the prior the transition prior p(xk |xk−1 ) as the proposal distribution [12], the importance weights of particles are calculated as: p(zkn |xik ) =
i wki = wk−1 p(zk |xik ).
(8)
The above multi-sensor PF with N particle number is listed as Algorithm 1. From Eqs. (6), (7), and (8), it can be seen that measurements from sensing nodes have effects on the importance weights. The interesting problems are: •
Does the topology of sensing nodes affect the performance of target tracking based on the UWSNs?
•
What is the relationship between the topology and the performance of target tracking?
The answers to these questions will be described in the following sections. III.
N ODE T OPOLOGY E FFECT ON TARGET T RACKING
To answer the questions mentioned above, this section studies node topology effect on target tracking based on
UWSNs. A. The Analysis of Four Typical Topologies Effects on Target Tracking Assume that node A, node B, node C, and node D are the sensing nodes at k time step and their positions are (xa , ya , za ), (xb , yb , zb ), (xc , yc , zc ), (xd , yd , zd ), respectively; target’s position is (xk , yk , zk ). Based on TOA, we have the following equations: ⎧ ⎪ dat = (xa − xk )2 + (ya − yk )2 + (za − zk )2 , ⎪ k ⎪ ⎨dbt = (x − x )2 + (y − y )2 + (z − z )2 , k b k b k k b (9) ct 2 + (y − y )2 + (z − z )2 , ⎪ d = (x − x ) c k c k c k ⎪ k ⎪ dt ⎩ dk = (xd − xk )2 + (yd − yk )2 + (zd − zk )2 ,
Fig. 1. Four nodes form a square (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
bt ct dt where dat k , dk , dk , and dk are the ranges between the target and node A, B, C, and D, respectively. These equations form the basis for the analysis of four typical topologies effects on target tracking.
•
Topology 1: four nodes form a square (Fig. 1). For this topology, if the real target T1 satisfies the range constraint determined by Eq.s (9), the false target T2 , as the mirror of T1 , also satisfies this constraint. In other words, this topology cannot uniquely determine target’s position. Fortunately, there is only one false target, which means that the tracking performance may not be very bad with the help of particle prediction based on the CV model.
Fig. 2. Four nodes are in line (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
B. The Relationship Between the PCRLB and Node Topology
•
Topology 2: four nodes are in line (Fig. 2). For this topology, if the real target T1 satisfies the range constraint determined by Eq.s (9), any point on the dashed circle, which is orthogonal to line AD, also satisfies this constraint. Hence, there are infinite false targets when four sensing nodes are in line. So this topology is worse than the previous one.
•
Topology 3: four nodes are close to each other (Fig. 3). If four sensing nodes are very close to each other, the role of the four nodes is almost the same as one node. Hence, any point on the dashed sphere determined by Eq.s (9) can be the false target. So there are also infinite false targets when four sensing nodes are close to each other. Hence, this topology is not a good choice.
The relationship between the PCRLB and the error covariance matrix of the target estimation is:
Topology 4: four nodes form a regular tetrahedron (Fig. 4). This topology avoids the shortcomings of the previous three topologies. For this topology, target’s location can be uniquely decided by Eq.s (9). Hence, this topology may be a good choice.
where
•
According to above analysis, it can be seen that the topology of the sensing nodes really has effect on target tracking. However, this analysis is not only qualitative but also limited to some typical topologies. In the next subsection, we will derive the the relationship between the PCRLB and sensing node’s position to evaluate the arbitrary topology quantitatively.
To evaluate the arbitrary topology, this subsection calculates the relationship between the PCRLB and the sensing node’s position. The PCRLB is defined to be the inverse of the Fisher information matrix (FIM), which provides a lower bound on the mean square error (MSE) of target state estimation. There are several approaches to compute the PCRLB, readers can refer to [14], [15].
xk − xk ]T ≥ Jk−1 , E[ˆ xk − xk ][ˆ
(10)
where Jk is the FIM at the kth time step. Assuming that ∇α is the first-order partial derivation operator and Jk−1 has already been obtained, Jk can be computed as follows [14]: 22 21 11 12 − Dk−1 (Jk−1 + Dk−1 )−1 Dk−1 , Jk = Dk−1
(11)
11 Dk−1 = E{[−∇xk−1 ∇Txk−1 ln p(xk |xk−1 )]},
(12)
12 = E{[−∇xk ∇Txk−1 ln p(xk |xk−1 )]}, Dk−1
(13)
21 12 T = E{[−∇xk−1 ∇Txk ln p(xk |xk−1 )]} = [Dk−1 ] , (14) Dk−1 22 Dk−1 = E{[−∇xk ∇Txk ln p(xk |xk−1 )]}
+E{[−∇xk ∇Txk ln p(zk |xk )]}.
(15)
The recursion of Eq. (11) starts from an initial FIM J0 = E{−∇x0 ∇Tx0 ln p(x0 )}, where p(x0 ) is the priori pdf of
where dnp k =
Fig. 3. Four nodes are close to each other (the blue circle stands for the sensing node; the solid triangle stands for the real target; and the dashed triangle stands for the false target).
(ˆ xk|k−1 − xnk )2 + (ˆ yk|k−1 − ykn )2 + (ˆ zk|k−1 − zkn )2 (23)
is the range between the nth sensing node and the predicted target’s position; (xnk , ykn , zkn ) is the position of nth sensing node at k time step; and (ˆ xk|k−1 , yˆk|k−1 , zˆk|k−1 ) is the predicted target’s position, which can be given by: ⎧
N ˆk|k−1 = N1 ( i=1 xik|k−1 ) i = 1, 2 · · · N ⎪ ⎨x
N i yˆk|k−1 = N1 ( i=1 yk|k−1 (24) ) i = 1, 2 · · · N ⎪
⎩ N 1 i zˆk|k−1 = N ( i=1 zk|k−1 ) i = 1, 2 · · · N i i where N is particle number and (xik|k−1 , yk|k−1 , zk|k−1 ) is the predicted position according to the ith particle. Substituting Eq.s (22), (23), and (24) into (21), Λk can be calculated as follows: ⎡ 11 ⎤ τk 0 τk13 0 τk15 0 ⎢ 0 0 0 0 0 0 ⎥ ⎢ 31 ⎥ ⎢ τ 0 τk33 0 τk35 0 ⎥ (25) Λk = ⎢ k ⎥, ⎢ 0 0 0 0 0 0 ⎥ ⎣ τ 51 0 τ 53 0 τ 55 0 ⎦ k k k 0 0 0 0 0 0
Fig. 4. Four nodes form a regular tetrahedron (the blue circle stands for the sensing node; the solid triangle stands for the real target; and there is no false target).
where τk13 = τk31 , τk15 = τk51 , τk35 = τk53 , τk11
=
N Ns [ N1 ( i=1 xik|k−1 ) − xnk ]2 n=1
target’s state. Eq.s (12), (13), (14) and the first part of Eq. (15) gives the dependency of PCRLB on the motion model. This paper adopts the CV model, the follows are obtained by ¯ k−1 = Γk−1 Qk−1 ΓT : defining Q k−1 11 ¯ −1 Φk−1 , Dk−1 = ΦTk−1 Q k−1
(17)
21 ¯ −1 ΦTk−1 , Dk−1 = −Q k−1
(18)
22 ¯ −1 + Λk , Dk−1 =Q k−1
(19)
Λk =
=
E{[−∇xk ∇Txk
ln p(zk |xk )]}.
(20)
Eq. (20) gives the dependency of PCRLB on the measurements, which may contain the information of sensing node topology. Similar to the PF tracking algorithm, we also adopt particles to compute the Λk . Assuming that there are Ns sensing nodes taking part in the tracking task, Λk can be approximated by [15]: Λk ≈
Ns
ˆ kn )T (Rkn )−1 (H ˆ kn ), (H
(21)
n=1
ˆ n is the Jacobian matrix for the nth sensing node at where H k time k, which can be calculated as: n n −xn (ˆ yk|k−1 −yk ) (ˆ zk|k−1 −zk ) k) ˆ kn = (ˆxk|k−1 H 0 0 0 np np dnp d d k k k (22)
=
N i Ns [ N1 ( i=1 yk|k−1 ) − ykn ]2 n=1
(16)
¯ −1 , −ΦTk−1 Q k−1
12 Dk−1
τk33
τk55
=
=
2 Rkn (dnp k )
N i Ns [ N1 ( i=1 zk|k−1 ) − zkn ]2 n=1
τk13
2 Rkn (dnp k )
2 Rkn (dnp k )
,
(26)
,
(27)
,
(28)
N
N i Ns [ N1 ( i=1 xik|k−1 ) − xnk ][ N1 ( i=1 yk|k−1 ) − ykn ] n=1
2 Rkn (dnp k )
,
(29)
N i 1 N Ns i n 1 [ N ( i=1 xk|k−1 ) − xk ][ N ( i=1 zk|k−1 ) − zkn ] , τk15 = 2 Rkn (dnp k ) n=1 (30)
N i
N i 1 Ns n 1 n [ ( y ) − y ][ ( z ) − z k N k] i=1 k|k−1 i=1 k|k−1 N τk35 = . 2 Rkn (dnp k ) n=1 (31) This completes the calculation of the PCRLB. Since the PCRLB is related to the positions of sensing nodes, the arbitrary topology can be evaluated by PCRLB. Specifically, the topology that minimizes the PCRLB is the best one and the topology that maximizes the PCRLB is the worst one. It is needed to point out that the calculation of PCRLB is only based on the prediction information, which means that PCRLB can be used to select the optimal topology for target tracking.
B. New Fusion Center Selection At each time step, the fusion center mainly takes responsibility for the optimal topology selection, measurement fusion, target state estimation, new fusion center selection, and passing the estimation results to the new fusion center. Since the fusion center plays an important role in target tracking based on UWSNs, a new fusion center is needed to be selected from the Ckopt . Due to the limited energy resources [4], the rule of the new fusion center selection is to minimize energy consumption. To quantify the energy consumption in data transmission, we adopt the energy dissipation model based on the underwater acoustic communication principle in [16]. To send a b-bit packet from one node to another over a range d, the energy consumption of the sender is: Fig. 5. Optimal topology selection (the blue circle stands for the qualified candidate; the black circle stands for the unqualified candidate; the red circle stands for the fusion center at time k − 1; the red solid triangle stands for target’s position at time k − 1; the red dashed triangle stands for the predicted position of the target at time k; R and r are node’s communication range and sensing range, respectively).
IV.
T HE D ETAILED TARGET T RACKING S CHEME WITH THE O PTIMAL T OPOLOGY
Based on the PCRLB, the optimal topology can be selected to take part in tracking the underwater target at each time step. This section presents the detailed target tracking with the optimal topology.
Es (b, d) = bP0 A(d),
and to receive this packet, the energy consumption of the receiver is: Er (b) = bPr , (36) where P0 is the power needed at the input to the receiver; Pr depends on the receiver; and A(d) is the attenuation given as: A(d) = dm ad ,
A. Optimal Node Topology Selection
nf dnp k ≤ r and dk−1 ≤ R.
(33)
(37)
where m is the energy spreading factor and a = 10α(f )/10 is a frequency-dependent term obtained by: α(f ) =
The basic idea of optimal node topology selection scheme is illustrated by Fig. 5. At time k − 1, the fusion center predicts target’s position at time k based on the CV model. Then according to the prediction information, dnp k which is the range between the nth node and the predicted position can be calculated by Eq. (23); and dnf k−1 which is the range between the nth node and the fusion center at time k − 1 can be computed by: f f dnf (xfk−1 − xnk )2 + (yk−1 − ykn )2 + (zk−1 − zkn )2 , k−1 = (32) f f , zk−1 ) is the position of the fusion center where (xfk−1 , yk−1 at time k − 1. The qualified candidates are the nodes that may have the chance to take part in tracking the target at k time step, which satisfies the following constraints:
(35)
0.11f 2 44f 2 + + 2.75 × 10−4f 2 + 0.003. (38) 2 1+f 4100 + f 2
Let F Ck−1 be the old fusion center at time k − 1, and F Ck be the new fusion center. Let sm , se , and sc be the sizes of data packets of the measurement, the estimation results, and the waking-up command, respectively. dnf k is the range between the new fusion center and the nth sensor node; dnf k−1 is the range between the old fusion center and the nth sensor node; and dfk f be the range between two fusion centers. Let Ns be the number of sensing nodes. Obviously, the energy consumption is a function of F Ck , which can be calculated as follows. If F Ck ≡ F Ck−1 ,
Ns −1 sm [P0 n=1 A(dnf ) + Pr (Ns − 1)]
Ns −1 knf Energy(F Ck ) = +sc [P0 n=1 A(dk−1 ) + Pr (Ns − 1)], (39) else,
Let Nq donate the number of the qualified candidates and Ck denote the combination of Ns (in this paper, Ns = 4.) candidates. If Nq ≥ Ns , our goal is to select an optimal topology Ckopt from the Nq candidates. Obviously, Ckopt can be determined by: Ckopt = arg min Jk−1 (1, 1) + Jk−1 (3, 3) + Jk−1 (5, 5) , (34)
⎧
Ns −1 nf ⎪ ⎨sm [P0 n=1 A(dk ) + Pr (Ns − 1)] nf s −1 Energy(F Ck ) = +sc [P0 N n=1 A(dk−1 ) + Pr (Ns − 1)] ⎪ ⎩+s [P A(df f ) + P ]. e 0 r k (40) Our goal is to select an optimal fusion center F Ckopt from the Ckopt . Obviously, F Ckopt can be determined by:
where Jk−1 is the inverse matrix of FIM at the kth time step.
F Ckopt = arg min Energy(F Ck ).
(41)
C. Target Tracking Scheme Based on the Optimal Topology By making use of the optimal topology selection, the optimal fusion center selection, the multi-sensor PF algorithm can be extended to tracking the underwater target based on UWSNs. The pseudo-code of our target tracking scheme is listed in Algorithm 2. Algorithm 2 Our target tracking scheme 1: 1.The Initial Fusion Center F C a Does the Following: 2: if k = 0 then 3: 1.1. Particle Initialization: 4: for i = 1, 2, . . . , N do 5: draw particle xik from the prior of target state p(x0 ); 6: end for 7: end if 8: for k = 1, 2, . . . do 9: 1.2 Importance Sampling: 10: for i = 1, 2, . . . , N do 11: sample xik ∼ p(xk |xik−1 ); 12: end for 13: select the optimal topology using Eq. (34); 14: select the optimal fusion center using Eq. (41); 15: if the new fusion center is the same as the old one then 16: get the measurements from the Ns sensing nodes; 17: for i = 1, 2, . . . , N do 18: update the importance weights by Eqs. (6), (7), and (8); 19: end for 20: for i = 1, 2, . . . , N do 21: normalize the importance weights: N w ˜ki = wki / j=1 wkj ; 22: end for 23: 1.3. Resampling: 24: for i = 1, 2, . . . , N do 25: multiply (suppress) xik with high (low) weights w ˜ki to obtain N particles; 26: set wki = w ˜ki = N −1 ; 27: end for 28: 1.4. Estimate the State and Covariance:
N 29: x ˆak = i=1 wki xik ,
N ˆk )T ; Covka = i=1 wki (xik − xˆk )(xik − x 30: else 31: forward x ˆak and Covka to the new fusion center F Cb ; 32: 2. The New Fusion Center F Cb Does the Following: 33: for i = 1, 2, . . . , N do 34: draw xik ∼ N (ˆ xak , Covka ); 35: end for 36: repeat the steps done by the previous fusion center; 37: end if 38: end for
V.
S IMULATIONS
In this section, we present simulations using Matlab to study node topology effect on target tracking based on UWSNs.
A. Simulation for Target Tracking Based on the Four Typical Topologies We simulate the four typical topologies mentioned in section III to study their effects on target tracking. Underwater nodes are deployed over a region of size 1000m × 1000m × 1000m. From 1s to 30s, four nodes form a square; from 31s to 70s, four nodes are in line; from 71s to 120s, four nodes form a regular tetrahedron; and from 121s to 160s, four nodes are close to each other. The actual initial state of the target is x0 = [0 5 0 5 0 5]T ; the initial estimation of the target is x ˆ0 = [5 5 5 5 5 5]T ; and covariance matrix M0 = I6×6 . The initial FIM J0 = M0−1 . The measurement process with variance R = 52 . The sampling time is 1s.The particle number N = 500 and the simulation runs M = 100. To indicate the accuracy of target tracking, we adopt root mean square error (RMSE), defined as follows, to measure the tracking performance: M [xk,i − xˆk,i ]2 + [yk,i − yˆk,i ]2 + [zk,i − zˆk,i ]2 Err(k) = , M i=1 (42) where (xk,i , yk,i , zk,i ) and (ˆ xk,i , yˆk,i , zˆk,i ) are the true and the estimated positions of the target at time k, respectively, in the ith simulation. Tracking results for the four typical topologies are shown in Fig. 6. From 1s to 30s, the tracking error is small. This is because there are only two false targets when four nodes form a square. When the initial estimation error is small, the tracking algorithm can perform well with the help of the particle prediction based on the CV model. From 31s to 70s, the tracking performance is very bad. This is because there are infinite false targets when four nodes are in line. In this case, the measurements provide little information about target’s state. From 71s to 120s, the tracking error is small. This is because target’s location can be uniquely determined by this topology. For this topology, the measurements can provide target’s location information. From 121s to 160s, the tracking performance turns bad again. The reason is the same as that of four nodes are in line. The tracking performance under different topologies can confirm our analysis in section III. Furthermore, the estimated PCRLB of the four typical topologies is also plotted in Fig. 6. It can be seen that the estimated PCRLB can well reflect the quality of node topology. Specially, the smaller PCRLB achieves, the better the topology is. This also demonstrates the effectiveness of the estimated PCRLB. In Fig. 6, the tracking error happens to be lower than the lower bound PCRLB at some time steps. This is mainly because the expectation in Λk = E{[−∇xk ∇Txk ln p(zk |xk )]} has no closed-form analytical solution and must be approximated. Furthermore, our approach for calculation of PCRLB only bases on the prediction information instead of the posterior pdf of target state, which leads to the degradation of the estimation accuracy. Fortunately, a rough estimation of PCRLB is enough to select a better node topology to take part in tracking target at each time step, which can be confirmed by the next simulation experiment.
20
20
Tracking Error Estimated PCRLB
Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
18 16
15
Error/m
Error/m
14
10
12 10 8 6
5
4 2
0 0
20
40
60
80 100 Time step/s
120
140
0 0
160
Fig. 6. Tracking results for the four typical topologies (from 1s to 30s, four nodes form a square; from 31s to 70s, four nodes are in line; from 71s to 120s, four nodes form a regular tetrahedron; and from 121s to 160s, four nodes are close to each other).
The sizes of data packets are sm = 32 bits, se = 64 bits, and sc = 8 bits. For the sake of simplicity, let P0 = Pr = 10−9 J/bit, m = 1.5, and f = 15kHz. The node sensing range r = 200m and communication range R = 400m. All the other parameters are the same as those set in the previous simulation. The performance metrics include tracking error, energy consumption, and communication traffic. Tracking error is defined as Eq. (42), the energy consumption is defined as Eqs. (39) (40), and the communication traffic indicates the total amount of data transmission. Communication traffic can be defined as follows. (sm + sc )(Ns − 1) if F Ck ≡ F Ck−1 , Com(k) = (sm + sc )(Ns − 1) + se otherwise. (43) TABLE I.
T HE AVERAGED METRICS OF OUR SCHEME , THE SCHEME WITH MAXIMUM PCRLB, AND 4NN SCHEME
Algorithem Our scheme Maximum PCRLB 4N N
Error (m) 2.67 9.00 4.75
Energy (mJ) 0.48 0.51 0.29
Communication (bits) 149.65 125.25 124.66
The results of tracking error, energy consumption, and communication traffic are plotted in Fig. 7, Fig. 8 and Fig. 9, respectively. Illustrated from Fig. 7, our scheme with the minimum PCRLB achieves the highest tracking accuracy than that of the other two schemes. This is because our scheme can select the optimal topology at each time step by minimizing
40
60
80 100 Time step/s
120
140
160
Fig. 7. Tracking errors of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
B. Simulation for Our Target Tracking Scheme
Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
1.2 Energy consumption/mJ
In this subsection, we present simulation for our scheme with the minimum PCRLB. Besides our scheme, the scheme with the maximum PCRLB, and the four nearest neighbors(4NN) scheme [17] are also simulated for comparison. For the scheme with the maximum PCRLB, the worst topology that maximizes the PCRLB is selected and the way of the new fusion center selection is the same as that of our scheme. For the 4NN scheme, four nodes that are nearest to the predicted target’ position are selected and the node that is nearest to the predicted position is selected as the new fusion center.
20
1 0.8 0.6 0.4 0.2 0 0
20
40
60
80 100 Time step/s
120
140
160
Fig. 8. Energy consumption of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
the PCRLB, whose measurements contains the most useful information of the underwater target. Since the scheme with the maximum PCRLB selects the worst topology at each time step by maximizing the PCRLB, its tracking performance is the worst. For the 4NN scheme, neither the optimal topology nor the worst one is selected. Hence, its performance is between that of the other two schemes. Table I shows that our scheme achieves about 30% tracking error of the scheme with the maximum PCRLB and about 56% tracking error of the 4NN scheme. Illustrated from Fig. 8, The 4NN achieves the smallest energy consumption. This because the four nodes selected by the 4NN scheme are all close to the predicted target’s position, which means that they are also close to each other. From Eq.s (39) and (40), we can see that the smaller dnf k is, the less energy consumption can be achieved. Hence the 4NN scheme achieve less energy consumption than that of the other two schemes. Table I shows that 4NN achieves about 60% energy consumption of our scheme and about 57% energy consumption of the scheme with the maximum PCRLB. Illustrated from Fig.9, the communication traffic performance of our scheme is worse than that of the other two schemes. This is because, the topology selected by our scheme changes frequently to search for the best one at each time step. When
20120101110115, the Zhejiang Provincial Science and Technology Planning Projects of China under Grant 2012C21044, the Marine Interdisciplinary Research Guiding Funds for Zhejiang University under Grant 2012HY009B, the Fundamental Research Funds for the Central Universities under Grant 2014XZZX003-12, the Open Fund for the Aircraft Marine Measurement and Control Joint Laboratory under Grant FOM2014OF06, and the ASFC under Grant 20132076002.
Communication traffic/bits
200
150
100 Our scheme Mean of our scheme Maximum PCRLB Mean of maximum PCRLB 4NN Mean of 4NN
50
0 0
20
40
60
80 100 Time step/s
120
R EFERENCES
140
160
Fig. 9. Communication traffic of our scheme, the scheme with maximum PCRLB, and the 4NN scheme.
topology changes frequently, the fusion center also changes frequently. From Eq. (43), it can be seen that the more frequently the fusion center changes, the larger communication traffic can be achieved. Hence our scheme achieves a little worse communication traffic performance than that of the other two schemes. Table I shows that 4NN scheme achieves about 83% communication traffic of our scheme and about 99% communication traffic of the scheme with the maximum PCRLB. In summary, our scheme is optimal in the sense of tracking error. Compared to other two schemes, our scheme achieves the best performance in tracking error. For the energy consumption and communication traffic, the 4NN scheme is a better choice. Hence, our scheme is more suitable for the scene that requires high tracking accuracy. VI.
C ONCLUSIONS
This paper studies node topology effect on target tracking based on UWSNs. By using the knowledge of geometry, the effects of four typical topologies on target tracking are analyzed qualitatively. To evaluate the arbitrary topology quantitatively, the relationship between the PCRLB and node topology is also derived, which forms the basis for our optimal topology selection scheme. Simulation results show that the PCRLB reflects the quality of arbitrary topology well and our scheme with the minimum PCRLB achieves higher tracking accuracy than the existing scheme. However, our scheme is optimal in the sense of tracking error. Hence, it is suitable for the scene that requires high tracking accuracy. For the further research, we will improve our scheme in the aspects of energy consumption and communication traffic and extend our scheme to track multiple targets based on UWSNs. ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China under Grants 61374021, 61222310, and 61174142, the Zhejiang Provincial Natural Science Foundation of China under Grant LZ14F030003, the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP) under Grants 20130101110109 and
[1] M. Asif, M. Rizal, and A. Yahya, “An active contour for underwater target tracking and navigation,” in Pro. Int. Conf. Man-Mach. Syst., Langkawi Islands, Malaysia, Sep., 2006, pp. 1–6. [2] E. Dalberg, A. Lauberts, R. K. Lennartsson, M. J. Levoen, and L. Persson, “Underwater target tracking by means of acoustic and electromagnetic data fusion,” in Proc. 9th Int. Conf. Inf. Fusion, Florence, Italy,, Jul., 2006, pp. 1–7. [3] M. I. Pettersson, V. Zetterberg, and I. Claesson, “Detection and imaging of moving targets in wideband sas using fast time backprojection combined with space-time processing,” in Proc. MTS/IEEE Oceans, Washington DC, USA, Sep., 2005, pp. 2388–2393. [4] J.E. Faugstadmo, “Underwater wireless sensor networks,” in Sensor Technologies and Applications (SENSORCOMM), Venice, Jul. 2010, pp. 422–427. [5] S. Zhou and P. Willett, “Submarine location estimation via a network of detection-only sensors,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 3104–3115, Jun. 2007. [6] Q. Liang and X. Cheng, “Underwater acoustic sensor networks:target size detection and performance analysis,” Ad Hoc Netw., vol. 7, no. 4, pp. 803–808, Jun. 2009. [7] Y. Huang, W. Liang, H.-B. Yu, and Y. Xiao, “Target tracking based on a distributed particle filter,” Wireless Communications and Mobile Computing, vol. 8, no. 8, pp. 1023–1033, Oct. 2008. [8] X. Wang, M.-X. Xu, H.-B. Wang, Y. Wu, and H.-Y. Shi, “Combination of interacting multiple models with the particle filter for threedimensional target tracking in underwater wireless sensor networks,” Mathematical Problems in Engineering, pp. 1–16, Oct. 2012. [9] G. Isbitiren, and O. B. Akan, “Three-dimensional underwater target tracking with acoustic sensor networks,” IEEE Transaction on Vehicular Technology, vol. 60, no. 8, pp. 3897–3906, Oct. 2011. [10] C. Yu, K. Lee, J. Choi, and Y. Seo, “Distributed single target tracking in underwater wireless sensor network,” in Proc. SCIE Annu. Conf., Tokyo, Japan, Aug. 2008, pp. 1351–1356. [11] G.J. Han, J.F. Jiang, L. Shu, Y.J. Xu, and F. Wang, “Localization algorithms of underwater wireless sensor networks: a survey,” SENSORS, vol. 12, no. 2, pp. 2026–2061, Feb. 2012. [12] M.S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking,” IEEE Transaction on Signal Processing, vol. 50, no. 2, pp. 174–188, Feb. 2002. [13] N. Gordon, D. Salmond, and A.F.M. Smith, “Novel approach to nonlinear and non-gaussian bayesian state estiamtion,” Proceeding of IEE(F), vol. 140, no. 2, pp. 107–113, Feb. 1993. [14] P. Tichavsky, C.H. Muravchik, and A. Nehorai, “Posterior cramerrao bound for disctete-time nonlinear filtering,” IEEE Transactions on Signal Processing, vol. 46, no. 5, pp. 1386–1396, May 1998. [15] X. Zhang, P. Willett, and Y. Bar-Shalom, “Dynamic cramer-rao bound for target tracking in clutter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 41, no. 4, pp. 1154–1167, Oct. 2005. [16] E. Sozer, M. Stojanovic, and J. Proakis, “Underwater acoustic networks,” IEEE Journal of Oceanic Engineering, vol. 25, no. 1, pp. 72–83, Jan. 2000. [17] D. Guo and X. Wang, “Dynamic sensor collaboration via sequential monte carlo,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 1037–1047, Aug. 2004.
