Target tracking based on improved cubature particle filter in UWSNs

IET Radar, Sonar & NavigationVolume 13, Issue 4 p. 638-645 Research ArticleFree Access Target tracking based on improved cubature particle filter in UWSNs Hailin Feng, Hailin Feng School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi, People's Republic of ChinaSearch for more papers by this authorZhiwei Cai, Corresponding Author Zhiwei Cai ZhiweiCai_XD@126.com School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi, People's Republic of ChinaSearch for more papers by this author Hailin Feng, Hailin Feng School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi, People's Republic of ChinaSearch for more papers by this authorZhiwei Cai, Corresponding Author Zhiwei Cai ZhiweiCai_XD@126.com School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi, People's Republic of ChinaSearch for more papers by this author First published: 04 February 2019 https://doi.org/10.1049/iet-rsn.2018.5489Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, an improved cubature particle filter based on the artificial bee colony (ABC) algorithm is proposed and applied to target tracking via underwater wireless sensor networks (UWSNs). In the proposed method, the square root cubature Kalman filter is used to generate the proposal distribution and the ABC algorithm is employed to optimise the particles before resampling, which makes the particles move toward the high likelihood region and maintain the diversity of the particles. Moreover, linear minimum variance criterion is utilised to fuse local estimates together in distributed fusion architectures of UWSNs. The simulation results show that the proposed method outperforms other classical algorithms in tracking accuracy. 1 Introduction Target tracking in underwater wireless sensor networks (UWSNs) is an indispensable part of modern underwater defence systems [1, 2]. The primary purpose of target tracking is estimating moving targets’ states using noisy measurements [3]. The extended Kalman filter (EKF) among suboptimal filters has been widely used to solve target tracking [4]. However, EKF only uses the first-order terms of the Taylor series expansion of non-linear system functions, which usually introduces large deviations in estimating the posterior distributions of the states and thus causes the target tracking diverging [5, 6]. To reduce the linearisation errors caused by EKF, the unscented KF (UKF) is proposed for target tracking [7]. The UKF utilises the determinate sigma points to capture the statistics of Gaussian random variable for non-linear systems [8]. However, when the dimension is more than three, the weights of UKF are negative, which will lead to the divergence of the filter [9]. Arasaratnam and Haykin [10] proposed the cubature KF (CKF) based on the spherical–radial cubature rule. It has been proofed that the CKF performs better than the UKF when the dimension of the system goes beyond three [9]. However, neither of them can be applied to general non-Gaussian distributions. To address this problem, the application of particle filter (PF) in target tracking is increasingly becoming popular and PF is considered to be more applicable than the traditional non-linear filters in recent years [11, 12]. There are numerous studies on target tracking based on PF. In [13], two tracking schemes based on distributed PF are proposed for the cluster-based UWSNs. One achieves higher accuracy than the EKF, while the other achieves reduction of communication cost. In [14], PF is applied to bear only tracking. It is enhanced by the variable neighbourhood search algorithm in resampling step and the simulation results show that the improved PF performs better than other estimators. In [15], extended PF (EPF) is used to track targets in stand-off jammer. The importance density function is generated by using EKF and simulation results show that it outperforms the EKF in tracking continuity and tracking accuracy. In [16], an improved unscented PF (UPF) is proposed. The target moves on a line through the networks and is tracked by using the UPF. Simulation results show that the improved UPF is capable of improving the utilisation efficiency of particles and has a stable tracking performance. In [3], cubature PF (CPF) with Markov chain Monte Carlo (MCMC) process is proposed. The CKF is used as proposal distribution and MCMC is employed to suppress the impoverish phenomenon after resampling. Simulation results show that it is superior to generic PF with MCMC. However, complex distributions may not be adequately represented by small number of sigma points in target tracking based on CPF. Moreover, the capacity of generic CPF will be restricted by the degeneracy. When a particle set estimating the states loses its diversity, the degeneracy happens. The main reason for a loss of diversity among the particles is the resampling process. It duplicates some with larger-weights multiple times and roughly discards other particles, thus decreasing the diversity of particles. In other words, the information is always concentrated on a minority of higher-weight particles. In that case, losing the diversity makes them tend to underestimate their own uncertainty, which will lead to the inconsistency of the filter. Therefore, maintaining the diversity is extremely necessary to enhance the accuracy of tracking. To tackle this problem, in this paper, an improved CPF based on artificial bee colony (ABC) algorithm is proposed. In this approach, after the importance sampling step with the square root CKF (SCKF), the obtained particles are optimised by ABC algorithm before resampling step, which significantly increases the number of efficient particles and ensures that the diversity is maintained after the resampling step. Using UWSNs to track targets is drawing more and more attention. When tracking in UWSNs, wireless communication is the main consumption of energy of sensor node [17, 18]. In distributed architectures, sensor nodes just exchange information with their neighbours, which reduce the communication greatly and are more practical. At each time, in UWSNs, multi-sensors observe a same target and obtain different observations. The information coming from different sensors will be fused together to get an optimal result, which can improve the stability and accuracy of estimation. In this paper, fusion centre fuses the local estimates together via linear minimum variance criterion. The rest of this paper is organised as follows. Section 2 presents the problem formulation and the distributed fusion architectures, as well as the PF. In Section 3, the improved CPF based on ABC algorithm is proposed. Distributed information fusion is presented in Section 4. In Section 5, simulation results are compared and evaluating the performance of proposed algorithm is accomplished. Finally, conclusion of this paper is drawn in Section 6. 2 Problem formulation 2.1 System model Assume that there is a single target moving within a pre-set UWSNs and it can be modelled by the following discrete dynamic state equation: (1) where is the state vector of the target at time k, is the state transition matrix, is the process noise matrix and is the noise of the process. Considering that the underwater target moves in a three-dimensional (3D) physical environment, the state vector of the target in 3D Cartesian coordinate is defined as (2) where is the position of the target along the x-axis at time k. is the x-axis velocity of the target at time k. and are similarly defined. Assume that there are S wireless acoustic sensors randomly floating at different layers in seawater. The positions of sensors in 3D Cartesian coordinate are denoted by . Every sensor measures the distance to the target by transmitting acoustic pulses (ping) and computing the time of arrival of the pings and echoes. Therefore, the measurement of sensor at time k is defined as (3) where is the location of the target at time k and is the measurement noise. 2.2 Distributed fusion architectures At the same time, there are some different local sensors tracking the same target and thus acquiring different measurements. The information coming from different sensors should be fused together to obtain more accurate estimates of the states of the target. Given that distributed fusion architectures are superior to centralised fusion architectures in processing costs and lower communication, distributed fusion architectures are more applicable than centralised fusion architectures and preferential for UWSNs. Fig. 1 shows the structure of the distributed fusion system. Local sensors periodically obtain the measurements from the target. Then, they run filters out local estimates based on these new measurements and past information and transmit local estimates to the fusion centre. Finally, all local estimates are assembled and fused together by the fusion centre to obtain the fusion estimate . The fusion centre will send the fusion estimate back to local sensors to predict future states of the target. Fig. 1Open in figure viewerPowerPoint Distributed fusion architectures for target tracking 2.3 Particle filter In target tracking, the system model and measurement model are, in general, represented as follows: (4) (5) where are the state vector and measurement vector, respectively. Here, and are known as non-linear functions. are the process noise and the measurement noise, respectively, which are assumed to be mutually independent. For the purpose of recursively estimating from a sequence of measurements , it is required to estimate the posterior density of the state. When a new measurement arrives, a recursive update of posterior density can be given by Bayesian filter as follows: (6) (7) where is denoted as (8) Equation (6) is analytically solvable for linear Gaussian systems. However, in practical, the systems of target tracking are almost non-linear/non-Gaussian. Fortunately, PF is not subject to these constraints. The PF approximates the density by a set of particles with associated weights (9) where is the Dirac's delta function. The value of the density is reflected by the position and weight of each particle. The particles recursively update their positions and weights with new measurements. It consists of three steps: importance sampling, weights updating and resampling. In the importance sampling step, particles are drawn according to the proposal distribution . Moreover, in weights updating step, the weights are updated as follows: (10) A known problem of PF is degeneracy, which means that after some iterations, significant weights are focused on few particles and thus much computational effort will be spent on the particles with negligible contributions to the approximation. The resampling is a common method to eliminate degeneracy. In the resampling step, all unequally weighted particles are resampled into a new set of particles with equal weights: It throws away the particles with negligible weights and simply copies the particles with high weights to fill up the vacancy. 3 Target tracking based on improved CPF The generic PF directly employs the transition prior as the proposal distribution. Obviously, it is easy to cause serious deficiencies in PF, especially when prior distribution is much broader than the likelihood. To overcome this drawback, many researchers introduced the current measurements into the proposal distribution and put forward many improved algorithms such as the EPF and the UPF. The EPF uses the EKF to generate the proposal distribution, and in the UPF; it is replaced by the UKF. However, these strategies will fail when facing high-dimensional state estimation and cannot overcome the weakness of divergence or the curse of dimensionality [19]. Afterwards, in order to overcome these drawbacks, the CPF is proposed based on the CKF. However, the CKF using the third-degree cubature rule also fails to perform adequately when the measurement noise is deviated from Gaussian probability distribution [20]. Moreover, small number of sigma points could not represent complicated distributions. In addition, resampling is performed in the CPF, which leads to a loss of diversity of particles. In this paper, to overcome these shortcomings of CPF in tracking problems, an improved CPF based on the ABC algorithm is presented. It consists of importance sampling, particles optimising, weights updating and resampling. 3.1 Importance sampling As mentioned, it is inefficient to use the transition prior as proposal distribution. A good proposal distribution must incorporate the current measurement. Since the SCKF is more accurate and stable than EKF, UKF and CKF [10], it is used to generate the proposal distribution for drawing samples. The importance sampling step is described as follows: (I) Initialisation: is set, sample N particles and represent them as from the prior distribution and obtain a square root of the covariance . (II) Importance sampling step: For assuming the particles and the square root of the covariance at time are . Update the particles using SCKF. (i) Evaluate cubature points and propagate the state equation (11) where the cubature points are denoted by and weights . (ii) Estimate the predicted particles and the square root of the predicted covariance (12) where and are denoted as a general triangularisation algorithm. The weighted, centred matrix is defined as (13) (iii) Evaluate the predicted cubature points and propagate the measurement equation (14) (iv) Estimate the predicted measurement, square root of the innovation covariance and cross-covariance (15) where and the weighted, centred matrixes and are defined as (16) (v) Evaluate the gain, estimated state and square root of estimated covariance (17) (vi) Sample and obtain particles . 3.2 Particles optimising by ABC algorithm 3.2.1 Introduction of ABC algorithm Karaboga in 2005 introduced a bee swarm algorithm called ABC algorithm for numerical optimisation problems [21]. In ABC algorithm, the colony of artificial bees consists of three groups of bees: employed bees, onlookers and scouts. The number of the employed bees is equal to the number of onlooker bees, as well as the number of food sources. Each employed bee produces a modification of the present location to determine a new food source, and then its nectar amount is tested. If the new nectar amount is higher than the previous one, then she memorises the new one and forgets the old one. After all employed bees complete their search process, they share their information about nectar amounts of food sources and their positions with the onlooker bees on the dance area. All onlooker bees successively choose a food source with a probability based on its nectar amounts. As in the case of employed bees, each onlooker bee determines a neighbour food source within the neighbourhood of the one that she has chosen and then evaluates the nectar amount of the candidate one. Providing that the nectar amount of the new one is higher than the previous one, she forgets the old one and remembers the new position. If a food source cannot be improved by a predetermined number of trials, the employed bee of this food source will become a scout bee and abandon this food source replacing its position with a randomly produced food position. 3.2.2 Particles optimising by ABC algorithm In this paper, we employ the ABC algorithm to optimise the particles drawn from the importance sampling step. The particles are taken as food sources. The work that the bees perform searching local optimal food source is the process of optimising particles. The optimisation encourages the particles move toward to high likelihood region and increase the number of efficient particles. The behaviour of neighbourhood searching of the bees preserves the diversity of particles. Particles optimising step is presented as follows. (i) Obtain particles from importance sampling step, and let be the fitness value of the particle , it represents nectar amount. Here, is defined as the following equation: (18) where is the acquired measurement at time and is the predicted measurement of . Obviously, the closer predicted measurement is to actual measurement , the greater the value of the fitness. Calculate all particles fitness values according to (18) obtaining . (ii) For , employed bee i produces a new food source by (19), and evaluate its fitness value according to (18) obtaining (19) where is a randomly produced index different from i, is a randomly chosen index, is the dimension of state vector and is a random number between . (iii) Apply the greedy selection mechanism to determine selecting the old one or the candidate one, and if the candidate one is selected, replace and by and , respectively. (iv) Calculate the probability values for all particles by the equation below: (20) (v) For , produce a new food source according to (19) for the onlooker bees from the particle selected depending on and evaluate them according to (18). (vi) Execute step (iii). (vii) Determine the abandon particle for the scout bee. If any particle did not improve by three times, replace it with a new randomly produced particle by the equation below: (21) where 3.3 Weights updating and resampling After the particles optimising step, we obtain a new set of particles. We update the importance weights of all particles and normalise them by (22) In the resampling step, particles with low weights are abandoned and replaced by high-weight particles. After this, the estimate state and corresponding covariance matrix can be obtained by (23) Fig. 2 shows the process of ABC optimisation. Fig. 2Open in figure viewerPowerPoint Process of ABC optimisation 4 Distributed information fusion When tracking in UWSNs, at each time, there are multiple active sensors near the target to track it. In distributed fusion architectures, these local sensors observe a same target and acquire different measurements. Each local sensor runs the improved CPF based on its new measurement and past information to obtain a local estimate. Then, they transmit local estimates to the fusion centre. In fusion centre, we fuse local estimates together via weighted linear combination and utilise linear minimum variance criterion to determine weighting factors. Assume that and are local estimates and corresponding covariance matrices from local sensors, respectively. is the number of local sensors waked up at time k, the ultimate fused result can be represented by (24) where are weighting factors and . To obtain weighting factors, a cost function is defined as (25) where is the error covariance matrix of the fusion estimate and is the error of the estimation. Therefore, the error of the fusion estimate is denoted as (26) On the basis of the unbiasedness assumption and different sensors are independent of each other in distributed fusion architectures, is uncorrelated to when . Thus we have (27) From (27), the cost function can be rewritten as (28) We determine weighting factors by minimising the cost function . To employ Lagrange multiplier method, an auxiliary function should be introduced as (29) Let and , and solving the equation group can get weighting factors, thus we have (30) where . Therefore, replace by , the fusion estimate and corresponding covariance matrix are denoted by (31) (32) To better understand the whole algorithm, a general block diagram is presented in Fig. 3. At each time, the same procedure is performed recursively. Fig. 3Open in figure viewerPowerPoint Flowchart of the target tracking 5 Simulation and results In this section, for the purpose of evaluating the performance of the proposed method, simulations are provided for target tracking in UWSNs. Two examples are presented. First, the proposed method is investigated on underwater non-manoeuvring target tracking problem. Then, its performance is evaluated on underwater manoeuvring target tracking. 5.1 Simulation setup The proposed method is simulated using MATLAB and is compared with EPF and CPF. To make it closer to the real situation, the target is assumed to move in a 3D underwater environment. The considered region is a space with 50 identical sensors deployed randomly. The general underwater target motion model is as follows: (33) where T is a sampling interval, which is assumed to be 1 s. For non-manoeuvring target, constant velocity (CV) model and coordinated turn (CT) model are utilised. In the CV model, the transition matrix is denoted as (34) In this case, the target makes rectilinear motion and moves at CV. In the CT model, the transition matrix is denoted as (35) where corresponds to turn rate, when , the target makes clockwise turn manoeuvre; when , it makes anticlockwise turn. Moreover, we consider that it only moves as a CT model on xoy-plane and moves at CV in the z-axis direction. Moreover, for manoeuvring target, the CT model is the most common model [22, 23]. The process noise is assumed to be zero-mean Gaussian with covariance . All identical sensors are assumed to have same measurement noises, which is zero-mean Gaussian with covariance . The tracking position error and velocity error are and respectively, where is the estimated location of the target at time k; denote the estimated velocity components at time k. 5.2 Underwater non-manoeuvring target tracking 5.2.1 CV model In this scenario, the proposed method and other algorithms are used to track an underwater non-manoeuvring target. Assuming that , and the target moves rectilinear with CV. The simulation results are obtained over 50 MC runs with 50 particles. Fig. 4a depicts the real trajectory and estimated trajectories obtained by EPF, CPF and the proposed method. The root mean square error (RMSE) is adopted as the performance index. The position RMSE and velocity RMSE of algorithms are shown in Figs. 4b and c, respectively. As can be seen, the proposed method has better tracking performance than EPF and CPF. From Fig. 4, it can be seen that EPF often loses the target and cannot perform stably. This is because tracking targets in UWSNs is a non-linear and non-Gaussian problem. The strong non-linearity involved in the measurement model makes its performance degraded. In addition, though CPF keeps the tracking, the estimation error is large and its performance may still be unsatisfactory. However, the proposed method follows the target with small estimation errors throughout the whole simulation. To further compare the performance of three algorithms with quantified indicators, Fig. 5 shows the mean and variance of RMSE based on each algorithm. It is obvious that the mean and variance of position RMSE and velocity RMSE based on the proposed method are much smaller than those of EPF and CPF, which means that the proposed method has superior performance for underwater non-manoeuvring target tracking. Fig. 4Open in figure viewerPowerPoint Tracking results (a) Real and estimated trajectories, (b) Position RMSEs of three filters, (c) Velocity RMSEs of three filters Fig. 5Open in figure viewerPowerPoint RMSE of algorithms (a) Position RMSEs, (b) Velocity RMSEs Table 1 (see the Appendix for table of results) shows the RMSE of position and RMSE of velocity by three algorithms with different numbers of particles. It is evident that the proposed method outperforms EPF and CPF even when the numbers of particles is small. Moreover, it is not hard to see that computational efficiency of the proposed method is much higher than EPF and CPF. The main reason for this is that the proposed method employs the ABC algorithm to optimise the particles before resampling step. It encourages the particles to move toward high likelihood region and makes them better approximate the true state of the target. Meantime, it figures out the problem of particle impoverishment. Therefore, the proposed method has an advantage that using fewer particles can realise the same estimation accuracy as that of EPF and CPF. As Table 1 shows, the estimation error of the proposed method using ten particles is smaller than that of EPF and CPF with 50 particles. It means that the proposed method obtains more accuracy with lower computational cost. Table 1. Performance of algorithms with different numbers of particles Number of particles Algorithms Position RMSE Velocity RMSE Processing time, s 50 EPF 14.77 3.14 9.63 CPF 7.53 1.42 17.54 proposed algorithm 2.62 0.14 18.74 30 EPF 17.55 3.64 5.89 CPF 14.85 3.14 10.61 proposed algorithm 5.68 0.34 11.73 10 EPF 29.45 6.59 2.07 CPF 20.37 3.53 3.67 proposed algorithm 6.58 0.49 3.70 5.2.2 CT model In this scenario, the initial state of the target is assumed to be and the associated covariance is assumed to be . The initial state estimate is chosen randomly from [24]. The target makes a CT with turn rate 0.05 rad/s. The simulation results are shown in Fig. 6. The real trajectory and estimated trajectories can be seen in Fig. 6a. The RMSE of position and RMSE of velocity are shown in Figs. 6b and c, respectively. Similar to the previous example, the proposed method follows the target with small estimation errors throughout the whole simulation. It can be found that the proposed method has better tracking accuracy than that of EPF and CPF. It owes much to ABC algorithm optimisation process. Fig. 6Open in figure viewerPowerPoint Tracking results (a) Real and estimated trajectories, (b) Position RMSEs of three filters, (c) Velocity RMSEs of three filters 5.3 Underwater manoeuvring target tracking In this section, the three algorithms are used to track an underwater manoeuvring target. The initial state of the target is assumed to be . and the associated covariance is assumed to be . The initial state estimate is chosen randomly from [24]. For 50 s, the target starts to move at CT with turn rate of about 0.05 rad/s. From 51 to 100 s, it moves with negative turn rate of about −0.05 rad/s. From 101 to 150 s, it moves with positive turn rate of about 0.05 rad/s again. The real trajectory and the estimated trajectories from algorithms are shown in Fig. 7. It can be confirmed that the proposed method can successfully track the target with better tracking accuracy than that of EPF and CPF. The EPF performs unstably and the tracking error often subsequently becomes too large. Moreover, the CPF falls into the similar situation. However, the proposed algorithm can follow the target with small estimation errors throughout the whole simulation. Figs. 7b and c show the RMSE of position and RMSE of velocity, respectively. Clearly, the proposed method can well estimate the state of the target and converge more quickly than EPF and CPF. EPF and CPF are unbearable to the abrupt state change, which makes the performance of tracking reduced. However, the proposed method does not suffer it. Fig. 8 shows that the mean and variance of position RMSE and velocity RMSE based on the proposed method are much smaller than those of EPF and CPF. It means that the proposed method outperforms EPF and CPF and is more robust than them. The main reason is that the particles optimising step increases the number of efficient particles and maintains the diversity of particles. Fig. 7Open in figure viewerPowerPoint Tracking results (a) Real and estimated trajectories, (b) Position RMSEs of three filters, (c) Velocity RMSEs of three filters Fig. 8Open in figure viewerPowerPoint RMSE of algorithms (a) Position RMSEs, (b) Velocity RMSEs Table 2 (see the Appendix for table of results) shows the RMSE of position and RMSE of velocity by three algorithms with different numbers of particles. As can be seen, the proposed method has better performance than the others in spite of that number of particles is small. Table 2. Performance of algorithms with different numbers of particles Number of particles Algorithms Position RMSE Velocity RMSE Processing time, s 50 EPF 28.77 3.88 8.65 CPF 11.36 1.44 16.67 proposed algorithm 3.10 0.21 17.75 30 EPF 44.78 6.27 5.31 CPF 23.08 2.85 10.85 proposed algorithm 4.75 0.45 11.73 10 EPF 60.63 10.57 1.86 CPF 36.45 4.32 3.47 proposed algorithm 5.62 0.68 3.72 6 Conclusion In this paper, an improved CPF based on the ABC algorithm is proposed and applied to target tracking using UWSNs. In the proposed method, the SCKF is utilised to generate the proposal distribution, which is more stable and accurate than CKF, and the ABC algorithm is employed to optimise particles before resampling step. Every particle is regarded as a food source; the work that the bees perform searching local optimal food source is the process of optimising particles. It makes the particles move toward high likelihood region and meantime preserves the diversity of particles. Moreover, linear minimum variance criterion is used to fuse local estimates together for distributed fusion architectures. By minimising the cost function, the weighting factors are obtained, and further the fusion estimate is obtained. For the purpose of evaluating the proposed method, we investigate it on the problem of non-manoeuvring and manoeuvring target tracking via UWSNs. The results demonstrate that the proposed method has better tracking performance than EPF and CPF. It has the advantage that using few particles can obtain the same estimation accuracy as that of EPF and CPF. Therefore, the proposed method is an effective algorithm for applications. 7 References 1Ma, H., Ng, W.H.: ‘ Collaborative data and information processing for target tracking in wireless sensor networks’. IEEE Int. Conf. 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