Noise‐aware manoeuvring target tracking algorithm in wireless sensor networks by a novel adaptive cubature Kalman filter

IET Radar, Sonar & NavigationVolume 14, Issue 11 p. 1795-1802 Research ArticleFree Access Noise-aware manoeuvring target tracking algorithm in wireless sensor networks by a novel adaptive cubature Kalman filter Xuming Fang, Corresponding Author Xuming Fang fangxm@smail.nju.edu.cn School of Software Engineering, Jinling Institute of Technology, Nanjing, People's Republic of ChinaSearch for more papers by this authorLijun Chen, Lijun Chen Department of Computer Science and Technology, Nanjing University, Nanjing, People's Republic of ChinaSearch for more papers by this author Xuming Fang, Corresponding Author Xuming Fang fangxm@smail.nju.edu.cn School of Software Engineering, Jinling Institute of Technology, Nanjing, People's Republic of ChinaSearch for more papers by this authorLijun Chen, Lijun Chen Department of Computer Science and Technology, Nanjing University, Nanjing, People's Republic of ChinaSearch for more papers by this author First published: 18 September 2020 https://doi.org/10.1049/iet-rsn.2020.0165Citations: 3AboutSectionsPDF 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 onFacebookTwitterLinked InRedditWechat Abstract Current target tracking algorithms for wireless sensor networks in noise environments have large positioning errors. Owing to the environmental noise, Kalman filters (KFs) are used to estimate the target position. To reduce the adverse effect of unknown or time-varying noise on KFs, adaptive KFs (AKFs) are developed. However, the present AKFs can only achieve second-order estimation accuracy. To improve the existing target tracking algorithm's positioning accuracy under unknown and time-varying noise environments, the authors propose a noise-aware algorithm based on a novel third-order adaptive cubature KF (ACKF) with higher estimation accuracy, which improves the accuracy of the existing algorithm by up to 63%. The innovative ACKF contains a new third-order noise statistic estimator and a traditional cubature KF without noise perception. A large number of numerical simulations and practical experiments show that the proposed noise-aware target tracking algorithm based on the novel ACKF is always more accurate than the target tracking algorithms based on the current KFs, no matter whether the moving target is manoeuvring or not, whether the strength of the noise is small or large, whether the number of anchor nodes is many or few, and whether the noise is time-varying or constant. 1 Introduction At present, high-precision target tracking algorithms based on wireless sensor networks (WSNs) in noisy environments have become a research hotspot [1-3]. In many industrial and commercial fields, the application requirements of mobile target tracking are needed, e.g. industrial automation, intelligent logistics, smart home, wisdom mall etc. [4-7]. In these applications, it has become a key technology to acquire the precise locations of mobile targets. If the spatiotemporal location of the moving target cannot be obtained, the data from the target will become meaningless. Therefore, it is very important to provide accurate target tracking services in indoor environments. The WSN is a self-organised network composed of many sensor nodes, where the nodes are equipped with various sensors, microprocessors, and wireless transceivers [8]. In the WSN, the nodes send and receive wireless signals to exchange information. The wireless signal's received signal strength indication (RSSI) measured by the transceiver is generally utilised to estimate the distance between nodes [9, 10]. However, because the measured RSSI will be interfered with by environmental noise, the Kalman filter (KF) is used to provide accurate position estimation [11, 12]. According to the state observation theory, the KF is an optimal state estimator for the dynamic system in noisy environments [13]. Therefore, the KF is usually employed to estimate the real-time locations of mobile targets in the WSN. In the KF, the measurement model for establishing the relationship of the node motion state and the RSSI measurement and the process model for expressing the target node's motion state change is required. However, the measurement noise of the measurement model and the process noise of the process model are time-varying and unknown in an actual target tracking scenario, which will greatly reduce the estimation accuracy of KFs [14]. Therefore, the adaptive KF (AKF) with noise sensing capability is established [15, 16]. At present, the existing AKFs have the noise statistic estimators (NSEs) that only can achieve the first- and second-order estimation accuracy when estimating the measurement noise and the process noise at the same time, which leads to their poor target tracking accuracy in time-varying and unknown noise environments [17, 18]. To improve the positioning accuracy of target tracking algorithms based on the AKF in the time-varying and unknown noise environment, in this paper, an innovative adaptive ACK (ACKF) is proposed. The proposed ACKF is composed of a traditional cubature KF (CKF) without noise perception ability and a newly-derived NSE with third-order estimation accuracy upon sensing the process noise and measurement noise simultaneously. The third-order noise statistics means that the calculation results of noise statistics can reach the third-order Taylor expansion accuracy in this paper. 1.1 Contributions The following are our main contributions. (i) An innovative noise statistic estimator for the ACKF is derived. It is the only third-order noise statistic estimator that can perceive both measurement noise and process noise at the same time by now. (ii) A novel ACKF based on the third-order noise statistic estimator is established. It is the only ACKF with third-order location estimation accuracy when both measurement noise and process noise are time-varying or unknown at present. (iii) A noise-aware manoeuvring target tracking algorithm for WSNs based on the new ACKF is proposed. The positioning accuracy of our proposed algorithm is always higher than that of the existing algorithms under different target manoeuvrability, noise strength, number of anchor nodes (ANs), and noise time-varying. 1.2 Organisation This paper's structure is organised as follows. Section 2 describes the relevant work. The problem formalisation of target tracking algorithms based on KFs is given in Section 3. Section 4 presents the proposed ACKF and the newly-derived third-order noise statistic estimator. Section 5 shows the results of actual experiments and numerical simulations. Finally, this paper is concluded in Section 6. 2 Related work According to whether it is necessary to obtain the physical distance between nodes to locate the mobile target in the WSN, the existing target tracking algorithms in the WSN are classified into range-based algorithms and range-free algorithms [19-22]. The distance and angle between nodes need not be derived to determine the locations of nodes for the range-free algorithm, but its positioning accuracy is worse than that of the range-based algorithm. At present, the commonly-used range-free location algorithms contain centroid, multidimensional scaling-map, convex position estimation, and distance vector-hop [23-26]. In the range-based algorithms, the main ranging technologies are RSSI, time of arrival (TOA), angle of arrival (AOA), and time difference of arrival (TDOA) [27-30]. Unlike TOA, TDOA, and AOA, the RSSI is a convenient and cheap location technology, because it does not need additional equipment and strict network synchronisation. However, the RSSI is susceptible to noise interference, so the KF is used to improve the positioning accuracy of moving targets [31]. Since the KF is only suitable for the estimation of linearly moving objects, the extended KF (EKF) for the estimation of curvilinear motion is designed [11]. Although the EKF can be employed to track the nodes of curvilinear motion, it needs to linearise the non-linear model, which seriously reduces the estimation accuracy of the target node position. This is caused by the linearisation that makes the EKF only reach the estimation accuracy of the first-order Taylor expansion. To avoid the adverse effect of linearisation on positioning accuracy, the unscented KF (UKF) and CKF without linearisation are created, whose estimation accuracy can reach the second-order Taylor expansion and the third-order Taylor expansion, respectively [32, 33]. Although traditional KF can be used to track moving targets in noisy environments, they need to obtain accurate noise statistics in advance. The inaccurate noise statistics will lead to a large moving target position deviation and even divergence estimated by the KF [14]. The NSE, which can estimate noise statistics in real-time, has been developed [16, 34]. Together with the KF, it forms the AKF. Like the EKF, the NSE for adaptive EKF (AEKF) needs to be linearised, so it can only achieve a first-order noise estimation accuracy, which reduces the AEKF's target tracking accuracy [35, 36]. Although the NSE for the adaptive UKF (AUKF) does not need to be linearised, its estimation accuracy can only reach the second-order Taylor expansion as the UKF does [14, 37]. To further improve the NSE's estimation accuracy, the ACKF's NSE is established, which has an accuracy of the third-order Taylor expansion [38, 39]. However, the estimation accuracy of ACKFs based on this NSE is not high or even divergent when updating the statistics of measurement noise and process noise at the same time [40, 41]. At present, the NSEs that can be used to estimate measurement noise and process noise simultaneously can only achieve the first- and second-order Taylor expansion estimation accuracy [42, 43]. We propose a novel third-order NSE with higher accuracy and more stable performance for ACKF, which can ensure that ACKF can obtain small positioning error and no divergence when tracking manoeuvring targets with unknown and time-varying process noise and measurement noise. 3 Problem formalisation The KF uses the measurement model and process model to track the position of the moving target in noisy environments. The expression for the KF is as follows [13]: (1) where represents a state function describing the motion state of a moving target, is the measurement function describing the relationship between a target's state and its RSSI, is a time-step-k estimation of the motion state, represents a time-step-k RSSI prediction, is an actual time-step-k RSSI measurement vector, and represents a Kalman gain matrix for correcting state. The process model for target tracking is expressed as follows: (2) where represents the moving target's time-step-k state vector and represents a vector composed of process noise random variables, which obeys a normal distribution of zero mean and covariance Q k. The measurement model's form is as follows, which is used to predict the target node's RSSI measured from the AN (3) where represents a predicted time-step-k measurement vector and represents a measurement noise vector, which is made up of random variables. These random variables obey the normal distribution of zero mean and R k covariance. As a process model, the coordinated turn (CT) model is usually used to describe the moving state of the manoeuvring target [44]. The target tracking scene based on the CT model is shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint CT target tracking scenario The CT model's state transition function is expressed as (4) where is the moving target's state vector; x and y represent the horizontal coordinate and the vertical coordinate, respectively; and are the horizontal and vertical velocities of the moving target, respectively; represents the time step's interval; is the moving target's turning rate. The covariance matrix Q of the process noise w k is as follows: (5) where , , and are the constants related to the process noise strength, respectively. In the WSN, the RSSI-based ranging models are utilised to express the relationship of the distance between nodes and the RSSI measurement [45]. In this paper, the measurement model uses the widely applied log-normal shadowing model to describe the measured RSSI between nodes, whose form is expressed as follows [46]: (6) where represents the target node's RSSI measurement at the distance of r m, represents a path attenuation exponent of RSSI propagating with the distance, is the average RSSI measured from reference distance r 0 (typically 1 m), and is the measurement noise of RSSI, which obeys the normal distribution of standard deviation and zero mean. The measurement model's measurement vector is composed of the RSSI values obtained from the ANs (7) where represents the RSSI value measured by the L -th AN, and L represents the total number of ANs participating in positioning. The measurement model's measurement function is defined as follows: (8) where represents the L th AN's coordinates, is the coordinates of the target node at time step k, indicates a symbol calculating the Euclidean distance between the static anchor and the mobile target. The covariance matrix R of the measurement noise v k is as follows: (9) where represents the L th AN measurement noise's standard deviation and represents the diagonal matrix. Since the RSSI measurements of different ANs are independent of each other, the non-diagonal elements of the measurement noise covariance matrix are uncorrelated, i.e. zero. 4 Proposed algorithm 4.1 Current AEKF The present AEKF is made up of a traditional EKF without noise perception ability and an NSE that can sense unknown and time-varying noise [42]. The AEKF estimates the position of the moving target by predicting state and correcting state. Step 1. Predict state (10) (11) (12) where is the time-step-k − 1 state estimation of the mobile target, represents the time-step-k state prediction, represents a Jacobian matrix computed based on , represents a time-step-k − 1 state estimation covariance expectation, and represents a time-step-k state estimation covariance prediction. Step 2. Correct state (13) (14) (15) (16) (17) (18) where represents the time-step-k − 1 RSSI measurement prediction, represents a Jacobian matrix calculated according to , is represents a time-step-k measurement covariance expectation, represents the matrix of Kalman gain, represents the time-step-k covariance matrix of state estimation, and represents the time-step-k state estimation correction. In the unknown or time-varying noise environment, the inaccurate Q and R will seriously reduce the estimation accuracy of EKF. At present, an NSE for the AEKF that can estimate Q and R simultaneously online is as follows [42]: (19) (20) (21) (22) where represents the difference between the measured RSSI and the predicted RSSI, represents a forgetting variable, and b represents an amnestic constant, whose range is between 0.95 and 0.995. In this paper, b is set to 0.96. 4.2 Novel ACKF Based on the AEKF, we derive an innovative ACKF, whose steps are as follows: Step 1. State prediction (i) Select the cubature points (23) where is the weight of the cubature point (24) where e is an n -dimensional unit vector and n represents a state vector's dimension. (ii) Propagate the cubature points (25) (iii) Calculate the state prediction (26) (iv) Calculate the state covariance prediction (27) Step 2. State correction (i) Select the cubature points (28) (ii) Propagate the cubature points (29) (iii) Calculate the measurement prediction (30) (iv) Calculate the measurement covariance expectation (31) (v) Calculate the cross-covariance of state and measurement (32) (vi) Calculate the Kalman gain feedback matrix (33) (vii) Calculate the state correction (34) (viii) Calculate the state covariance correction (35) (ix) Noise statistic estimation (36) (37) where represents a point set, which is propagated by the cubature point recalculated based on the correction of state estimation covariance, and represents the RSSI prediction estimated by (38) (39) (40) Since ACKF is based on the third-order cubature rule, its estimation accuracy can reach the third-order Taylor expansion [47]. The detailed derivation of the novel NSE for the third-order ACKF is shown in the Appendix (Section 9.1). 4.3 ACKF-based algorithm The detailed steps of the proposed noise-aware manoeuvring target tracking algorithm based on the new ACKF are exhibited in Algorithm 1 (see Fig. 2). Fig. 2Open in figure viewerPowerPoint Algorithm 1: the ACKF-based algorithm 5 Performance evaluation In this section, we carry out a large number of numerical simulations and actual experiments under different target manoeuvrability, different noise strength, different numbers of ANs and different noise time-varying to verify the proposed algorithm's efficiency and effectiveness. The results show that our proposed noise-aware target tracking algorithm for WSNs improves the positioning accuracy by up to 63% compared with the existing algorithm. 5.1 Setup of numerical simulation We conduct the numerical simulations in a square area of 10 m×10 m to evaluate the proposed algorithm's tracking performance. Four static ANs with known positions are placed at four corners of this area, and the mobile target node of the estimated position makes a random curvilinear motion in this region. The target node's position is estimated through the RSSIs measured from the ANs. Assuming that the measurement noise of RSSI obeys the normal distribution and the standard deviation, i.e. the noise strength is 3 dB. The tracking performance of all evaluated algorithms is tested under 200 simulations. Since the target tracking range is small and the number of ANs is not large, it is assumed that the moving target node can always be detected by ANs in this paper [48, 49]. The moving target makes a curvilinear motion with coordinates (4, 6) as the starting point. The movement speed is 0.2 m/s, and the total duration of each move is 100 s. In the periods of 0–15, 20–35, 40–65, 70–85, and 90–100 s, the mobile target makes a uniform motion. In the periods of 15–20, 35–40, 65–70, and 85–90 s, the mobile target makes a random turning manoeuvre, whose turn rate is 0.35 rad/s. The values of initial covariances of state estimation, process noise, and measurement noise of all the evaluated KFs are assigned 10−1 × I 5×5, 10−2 × I 5×5, and 10−2 × I 4×4, in the simulation, respectively, where I is the unit matrix. According to the parameters obtained from the actual experiment, the signal attenuation exponent and the average RSSI measurement of the reference distance are assigned 2.62 and 45.55 dBm, respectively [42]. Since each simulation moving target moves for 100 s, the number of time steps for the KFs is set to 100. In this paper, the target node position estimation's root mean square error (RMSE) is used to measure the location accuracy of all target tracking algorithms (41) where represents the actual target location at time step k, represents the estimated one at time step k, and N represents the total number of time steps in a simulation (42) where represents the average value of RMSE, represents the RMSE obtained in the i th simulation, and M is the number of simulations. 5.2 Impact of turning manoeuvre Figs. 3-6 show the estimated positions and errors of EKF, UKF, CKF, AEKF, AUKF, and ACKF at different time steps of the turn manoeuvre simulation, respectively. As shown in the figure, the location errors of EKF, UKF, and CKF are larger than those of AEKF, AUKF, and ACKF when the mobile node turns, respectively. The main reason is that AEKF, AUKF, and ACKF have the ability of noise perception, which makes the influence of turning manoeuvre on position estimation deviation smaller. In particular, ACKF has a more accurate noise estimator, so its positioning error is smaller than those of AEKF and AUKF upon turning manoeuvring. Fig. 3Open in figure viewerPowerPoint Estimated positions of EKF, UKF, and CKF at different time steps in the simulation of turn manoeuvre Fig. 4Open in figure viewerPowerPoint Location errors of EKF, UKF, and CKF at different time steps in the simulation of turn manoeuvre Fig. 5Open in figure viewerPowerPoint Estimated positions of AEKF, AUKF, and ACKF at different time steps in the simulation of turn manoeuvre Fig. 6Open in figure viewerPowerPoint Location errors of AEKF, AUKF, and ACKF at different time steps in the simulation of turn manoeuvre In Fig. 7, the average RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF in 200 manoeuvring-turn simulations are exhibited, whose values are 2.29, 2.15, 2.55, 1.64, 1.61 and 1.39 m, respectively. The positioning accuracy of ACKF is 39, 35, 45, 14, and 13% higher than that of EKF, UKF, CKF, AEKF, and AUKF, respectively. It can be observed from the figure that although CKF has higher third-order theoretical estimation accuracy, its positioning error is higher than EKF and UKF when tracking the manoeuvring target. This is because CKF cannot perceive noise, which leads to the accumulation of tracking error in the case of inaccurate noise parameters. Since the proposed ACKF has a more accurate third-order noise estimator, the average tracking error of ACKF is smaller than those of AEKF and AUKF even when the initial noise parameters are not accurate. Fig. 7Open in figure viewerPowerPoint Average RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF in the simulations 5.3 Impact of noise strength In Fig. 8, the change of average positioning errors of EKF, UKF, CKF, AEKF, AUKF, and ACKF is described under different noise strength in 600 simulations. To evaluate the influence of noise strength on the positioning accuracy of the tracking algorithm, the noise strength is assigned 1, 2, and 3 dB, respectively. As shown in the figure, when the noise strength increases, the positioning errors of all evaluated algorithms also increase. However, the error growth rate of ACKF is obviously slower than that of other algorithms. The main reason is that compared with other algorithms, ACKF has a more accurate noise estimator. This makes ACKF always obtain a relatively smaller positioning error no matter whether the strength of the noise is small or large. Fig. 8Open in figure viewerPowerPoint Average RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF under different noise strengths in the simulations 5.4 Impact of the number of ANs The mean location errors of EKF, UKF, CKF, AEKF, AUKF, and ACKF are revealed under a different number of ANs through 600 times of turning curve motion simulation in Fig. 9. Since the positioning based on KF is a multi-lateration using the least square method, the number of ANs is at least three. However, due to the existence of noise in the actual environment, to ensure the applicable positioning accuracy, the number of ANs is usually set to at least four. To observe the change of positioning accuracy of the target tracking algorithm with the increase of the number of ANs, we set the number of ANs to 4, 6, and 8, respectively. All additional ANs are symmetrically and evenly placed around the positioning area, and the target node still makes a random turning manoeuvre. As can be seen from the figure, with the increase of the number of ANs participating in positioning, all the evaluated algorithms have a downward trend of location error. However, no matter how the number of ANs changes, our proposed ACKF always has the minimum positioning error. Fig. 9Open in figure viewerPowerPoint Average RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF under different number of ANs in the simulations 5.5 Setup of practical experiment We use a real WSN testbed to verify the proposed algorithm's efficiency and validity in the time-varying noise environment. The testbed is based on the off-the-shelf Micaz motes platform, which consists of four ANs and one target node. Four ANs are symmetrically deployed at four corners of a 6 m × 6 m area, respectively. The target node makes the manoeuvring-turning curve movement in the region, which sends the signal to the AN every 0.1 s. The values used for all KF parameters are the same as those set in the simulation. 5.6 Impact of time-varying noise Fig. 10 shows the time-varying measurement noise of RSSI from AN 1 and the noise statistics estimated by AEKF, AUKF, and ACKF in the experiment. The standard deviation of time-varying measurement noise is calculated offline through 50 sampling sliding windows. Since AEKF needs to be linearised, its estimated noise statistics are not as accurate as ACKF. Although AUKF does not need to be linearised, its noise estimation accuracy is not as high as that of ACKF. The estimation results of other noise parameters are still that ACKF is better than AEKF and AUKF. Since the measurement noise is time-varying, there is always a bias in the estimated noise statistics. Although there are errors in measurement noise estimation, KF combines process noise adaptation and state feedback correction to jointly reduce the positioning error. The reason why adaptation makes the algorithm more efficient compared with the estimation of noise statistics is that the state feedback correction result of the algorithm is jointly optimised by the correlation results of measurement noise adaptation and process noise adaptation. In the environment of slowly-varying measurement noise, extending the length of the sliding window does not greatly improve the results. Fig. 10Open in figure viewerPowerPoint Time-varying measurement noise from AN 1 and noise statistics estimated by AEKF, AUKF, and ACKF in the experiment The mean RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF are exhibited under the time-varying noise environment in 100 manoeuvring turn experiments in Fig. 11, i.e. 1.90, 1.80, 2.54, 1.09, 1.05, and 0.91 m. Based on these data, the positioning errors of ACKF are 51, 49, 63, 16, and 13% less than those of EKF, UKF, CKF, AEKF, and AUKF, respectively. It follows that even when the process noise and measurement noise are time-varying, the positioning accuracy of our proposed algorithm is still higher than that of the existing algorithms. Fig. 11Open in figure viewerPowerPoint Average RMSEs of EKF, UKF, CKF, AEKF, AUKF, and ACKF in the experiments 5.7 Time complexity analysis Based on MATLAB installed on a computer with a 2.2-GHz processor, the time complexity of all the evaluated algorithms is measured. The mean time step execution time of EKF, UKF, CKF, AEKF, AUKF, and ACKF is calculated according to all simulation and experimental data, which are 2.1, 7.8, 7.1, 3.7, 11.3, and 10.9 ms, respectively. The AKF's time complexity is slightly increased compared with the traditional KF. A main reason is that the noise perception calculation is increased in each time step. Although the calculation time of ACKF is a little longer than that of CKF, it is still far less than the period of the signal sent by the target node, i.e. 100 ms. Thus, it can be seen that the proposed ACKF can meet the application requirement of WSNs for real-time tracking mobile targets. The time complexity of EKF, UKF, CKF, AEKF, AUKF, and ACKF are O(n 2), O(n 3), O(n 3), O(n 2), O(n 3), and O(n 3), respectively [50, 51]. 6 Conclusion Compared with the AEKF with the first-order estimation accuracy and the AUKF with the second-order estimation accuracy, the proposed ACKF has higher position estimation accuracy when the measurement noise and process noise are time-varying or unknown. A large number of numerical simulations and practical experiments show that the positioning accuracy of our proposed target tracking algorithm using this new ACKF is always higher than that of the target tracking algorithms based on the existing KFs, no matter whether the moving target is manoeuvring or not, whether the strength of the noise is small or large, whether the number of ANs is many or few, and whether the noise is time-varying or constant. The proposed algorithm improves the positioning accuracy by up to 63% compared with the existing algorithm. The current work only focuses on tracking one target in the indoor environment. The future work is to study tracking multiple targets in the indoor environment, such as monitoring in the airport [52, 53]. 7 Acknowledgments This research was financially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 19KJB520034), the Doctoral Scientific Research Startup Foundation of Jinling Institute of Technology (no. jit-b-201723), the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (no. PPZY2015B140), and the National Natural Science Foundation of China (no. 61272418). 9 Appendix