Gaussian sum pseudolinear Kalman filter for bearings‐only tracking

IET Control Theory & ApplicationsVolume 14, Issue 3 p. 452-460 Research ArticleFree Access Gaussian sum pseudolinear Kalman filter for bearings-only tracking Haonan Jiang, Haonan Jiang orcid.org/0000-0002-5758-4372 School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this authorYuanli Cai, Corresponding Author Yuanli Cai ylicai@mail.xjtu.edu.cn School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this author Haonan Jiang, Haonan Jiang orcid.org/0000-0002-5758-4372 School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this authorYuanli Cai, Corresponding Author Yuanli Cai ylicai@mail.xjtu.edu.cn School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this author First published: 01 February 2020 https://doi.org/10.1049/iet-cta.2019.0597Citations: 2AboutSectionsPDF 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 The efficacy of a bearings-only tracking algorithm, to a great extent, depends on the target-sensor geometry and motion. Although the pseudolinear Kalman filter and its variants have demonstrated comparable performance with the elite non-linear filters, they still suffer from bias problems and the tracking performance is inevitably affected by the relative geometry and motion relationships. In this study, an observability metric based on classical control theory is first presented to characterise the relative relationships between the target and the sensor. Then an efficient bearings-only tracking algorithm called Gaussian sum pseudolinear Kalman filter is developed. It is based on the bias-compensated pseudolinear Kalman filter and is built within a Gaussian sum framework. In the novel algorithm, a splitting and merging procedure will be triggered when a low degree of observability is detected. Simulation results show the significant performance improvement of the proposed algorithm. 1 Introduction Bearings-only tracking (BOT) finds a wide range of important applications both in civilian and military missions, such as submarine tracking, aircraft surveillance, and vehicle navigation [1]. With the advantages of high performance, fine hiding property, and miniaturisation, passive sensors measuring bearing information have drawn much more attention. An example is their ever-increasing integration into unmanned aerial vehicles [2], which are normally in small size. The objective of BOT is to estimate the kinematic parameters of the target with noise-corrupted bearing measurements. However, the high non-linearity of this problem and its observability issue make this goal intractable to achieve. The last three decades have witnessed great achievements in tracking algorithms, which are dedicated to tackling the first challenge (high non-linearity) in BOT. Early research efforts are mainly concentrated on the extended Kalman filter (EKF) [3]. However, EKF adopts the Taylor series expansion to approximate the non-linear measurement equation, which often leads to unstable and even divergent performance. The similar situation applies to the improved algorithms based on EKF, including the modified polar coordinate EKF [4] and the range-parameterised EKF [5]. Based on the idea of selecting a list of deterministic points to approximate the probability density distribution (PDD), non-linear filtering algorithms such as the unscented Kalman filter [6], the cubature Kalman filter (CKF) [7], the quadrature Kalman filter [8] etc. demonstrate better stability and accuracy. Another kind of point-based filter, represented by the particle filter (PF) [9–11] and the ensemble Kalman filter (EnKF) [12] generates random points to approximate the PDD of the non-linear function. Owing to the heavy computational load, PF is not suitable for real-time applications. Even so, PF can still be utilised as a benchmark for BOT problem. Being first introduced in 1978, the pseudolinear Kalman filter (PLKF) [13] has gained new attention in recent years. Through transforming the non-linear measurement equation to the pseudolinear formulation, PLKF can follow the concise steps of the Kalman filter to handle the state estimation problem in BOT. Besides, it shows comparable performance with the mentioned non-linear filtering algorithms while requiring lower computational complexity. Aiming at mitigating the bias existing in the PLKF, bias-compensated PLKF (BCPLKF), instrumental variable-based Kalman filter (IVKF), and IVKF combined with the selective-angle-measurement (SAM) scheme (SAM-IVKF) [14–17] are put forward in succession. With no need to calculate the Jacobian matrix and no need to generate deterministic or random points, PLKF-based estimators are efficient alternatives for BOT problem. The second challenge in BOT is the observability issue [18]. Since the range information is not measurable, the solution of BOT to the target state might not be decided uniquely even if the obtained bearing measurements are without error. It is well known that the target-sensor geometry and motion play significant roles in determining the estimation performance. Since the target motion is usually unavailable, it is the sensor trajectory that we must take into consideration. A broad consensus is that the sensor must outmanoeuvre the target to maintain the observability. The manoeuvring strategies of the sensor have often been studied in the literature relative to BOT [19–23]. However, it seems that few efforts have been made to evaluate the sensor trajectory. Although sensor trajectories composed of a finite number of legs (rectilinear segments with constant speed and course) are efficient in most BOT cases, we still want to find out how well the sensor behaves. In this context, we derive an observability metric according to the classical control theory [24]. Given certain target-sensor geometry and motion relationships, this metric can quantitatively describe the observability of the tracking system. With the help of the proposed metric, we can have knowledge about the current observability degree of the tracking system, which can reflect the effectiveness of the sensor trajectory. Subsequently, in this study, we propose a novel efficient pseudolinear estimation algorithm for the BOT problem called Gaussian sum PLKF (GSPLKF). We first combine the bias compensation step with PLKF and then extend it within the Gaussian sum framework [25]. Different from the traditional Gaussian sum filters, GSPLKF is initialised by the range-parameterised strategy [5]. Besides, it consists of a Gaussian splitting [26] procedure when the observability degree of the tracking system is lower than a given threshold. Conventional algorithms could do nothing in this situation. However, the performance of the proposed algorithm could be improved by splitting one Gaussian component into several ones. Although the computational load becomes a little heavier, performance improvement is more significant. Overall, there are two main contributions presented in this study. First, a novel observability metric for BOT with a single moving sensor is derived. Second, a range-parameterised initialisation strategy and a Gaussian splitting strategy based on the observability degree of the BOT system are developed and incorporated into the Gaussian sum framework, with BCPLKF being utilised as the base algorithm. The performance of the novel algorithm is compared with the state-of-the-art algorithms and turns out to be the best. The paper is organised as follows. A novel observability metric for the BOT system is derived in Section 2. Section 3 derives the PLKF for the BOT problem and reviews some improvements on it. Section 4 gives a detailed description of the proposed GSPLKF algorithm, followed by simulation comparison and analysis presented in Section 5. Section 6 provides the concluding remarks. 2 Observability metric for BOT 2.1 System model We consider the BOT problem in the 2D plane. Using the Cartesian coordinates, the target state at time step k is defined as , where and denote the position and velocity of the target, respectively. Similarly, the sensor state at time step k is defined as . Then the relative state is given by . The target is assumed to move with a nearly constant velocity. The dynamic equation can be written as (1)where is the state transition matrix, represents the deterministic input vector and is the process noise, which is independent and follows zero-mean Gaussian distribution with covariance matrix . The matrices , and are, respectively, given by where T represents the sample interval, and denote the intensities of process noise in the x -coordinate and y -coordinate, respectively. The bearing measurement collected by the moving sensor at time step k is given by (2)where is the measurement noise, which is also independent and follows zero-mean Gaussian distribution with variance . The process noise and measurement noise are assumed to be independent of each other. 2.2 Observability metric The target-sensor geometry and motion are of great importance in determining the tracking performance. Since the target trajectory is usually not available in advance, it is sometimes difficult to design an effective motion path for the sensor. If the scheduled sensor trajectory is not well designed, even the elite estimation algorithms may reach unexpected tracking performance. Hence, what we want is that the target-sensor geometry and motion could be monitored and evaluated at each time step. If so, some adjustments could be made once the scheduled sensor trajectory seems to make the target state difficult to estimate. Normally, the system observability could be used to evaluate the target-sensor geometry and motion. The most frequently-used metric for testing the observability of a system is the rank of the observability matrix. However, the rank can only give a qualitative result, which means a system is or is not observable. In this section, we derive a novel observability metric that can characterise the observability degree or observability strength of the BOT system. In a continuous-time framework, the system equations (1) and (2) can be replaced by (3) (4)where and represent the relative position and velocity, respectively, between the target and the moving sensor. According to the classical control theory, the local non-linear observability matrix of BOT systems (3) and (4) can be computed as [24] (5)where denotes the gradient operator and is the i -order Lie derivative of h along . For simplicity, we rewrite by only reserving the first two items as follows: (6)where (7) (8)where is the slant distance between the target and the sensor. Inspired by [27, 28], we define an observability metric as follows: (9)where represents the inverse of the condition number of matrix . and are the smallest and largest singular values of matrix , respectively, which are computed as (10) (11) and are the smallest and largest eigenvalues of matrix . By substituting (7) and (8) into (6), then substituting (6) into (10) and (11) and using the definition in (9), the observability metric for the BOT system is finally formulated as (12)Obviously, the derived observability metric is related to the target-sensor geometry (can be characterised by x and y) and motion (can be characterised by and ). Remark 1.In (4), the measurement without noise is considered. This is because we are primarily concerned with the theoretical aspects of observability. Remark 2.Based on the definition of the observability metric, it is evident that . The larger the value of , the stronger the observability of the BOT system. Remark 3.Equation (12) reveals that is a function of the current true relative state, which is unavailable to the sensor. To address this issue, the predicted relative state is used to calculate the current observability degree of the BOT system. Remark 4.There is no need to consider the existence of , which is proved in Appendix 1. 3 Derivation of PLKF In this section, we first derive the PLKF concerning the BOT systems (1) and (2). Subsequently, the bias compensation step is imbedded into the original PLKF to solve the inherent bias problem. Besides, some other improvements on PLKF are briefly reviewed. 3.1 Pseudolinear Kalman filter (PLKF) The measurement equation (2) can be equivalently rewritten as (13)Through some basic algebraic manipulations, (13) can be rewritten as follows: (14)where (15)Here is the true distance between the target and the sensor at time step k. The variance of the pseudolinear noise is given by . Letting and replacing by , (14) becomes (16)which is the pseudolinear formulation of the original measurement equation (2). Now the rebuilt pseudolinear system model can be summarised as follows: (17) (18) Applying the linear Kalman filter to the pseudolinear system model yields the PLKF. In the time update stage, the predicted state and covariance are given by (19) (20)In the measurement update stage, the filtered state and covariance are given by (21) (22)where (23)In (23), the variance is unknown because it is a function of , which is not available. To overcome this, we approximate by using the predicted state where and are the position components of . 3.2 Bias compensation The bias in the PLKF is mainly caused by two components. One is the correlation between the pseudolinear measurement matrix and the process noise , the other is the correlation between and the pseudolinear noise . Under the assumption of process noise with small covariance, we can just care about the bias caused by the correlation between and . One idea of bias compensation for the PLKF is directly subtracting the produced bias from the filtered state, yielding the BCPLKF with the bias compensation step [16] as follows: (24)Here is approximated by using the filtered state and Another idea to mitigate the bias is eliminating the correlation between and , the examples of which are IVKF and SAM-IVKF. Refer to [15–17] for more details about this kind of algorithm. 4 Gaussian sum PLKF (GSPLKF) In order to further improve the efficacy, we extend the BCPLKF within the Gaussian sum framework. The proposed GSPLKF algorithm consists of a bank of BCPLKFs. In addition, with a range-parameterised initialisation strategy and a Gaussian splitting mechanism, the novel algorithm is able to achieve the performance with significant enhancement. 4.1 Range-parameterised initialisation Inspired by the range-parameterised idea [5], we can divide the uncertain range region constructed by the target and the sensor into a number of intervals. In each interval, the mean and standard deviation are defined as (25) (26)where , is the total number of parallel BCPLKFs and the ratio is given by (27)The initial weight of each BCPLKF is set equally, namely . An illustration of the range-parameterised initialisation strategy with is shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Range-parameterised initialisation diagram For the BCPLKF, the initial estimate and covariance are computed as (28) (29)where , , and . They represent the initial bearing measurement, the initial estimates of target speed and target course, respectively. , , , and denote the true values of the initial bearing, the velocity of the target, the velocity of the sensor and the initial course of the sensor, respectively. The items in are calculated by The detailed derivation of the initialisation is given in Appendix 2. 4.2 Time update stage The time update stage of GSPLKF is constructed by the predictions of all the BCPLKFs. Under Gaussian assumptions, the predicted state and corresponding covariance achieved by the n th BCPLKF are given by (30) (31) 4.3 Measurement update stage With the predicted state achieved by the n th BCPLKF, the observability of the BOT system is first evaluated. If the current observability degree is lower than a threshold , the Gaussian component alone is deemed not qualified to propagate a satisfactory estimate. Therefore, when , the predicted density will be split and approximated by Gaussian components as follows: (32)where and are the mean and covariance of the m th Gaussian component split from the predicted density of the n th BCPLKF. is the corresponding weight and . While when , the Gaussian splitting procedure will not be triggered and . If , based on the idea of matching the moments of the Gaussian mixture density (the right-hand side of (32)) to that of the single Gaussian density (the left-hand side of (32)) [26], we split the Gaussian density into three components with respective means and weights given by (33)and the same covariance given by (34)where is the displacement parameter, which is usually chosen as . , and are the largest eigenvalue of and its associated eigenvector, respectively. denotes the Euclidean norm. The usage of Gaussian splitting is reasonable. On the one hand, since the observability degree is low, it will be difficult to achieve a satisfactory estimate of the target state just by using one Gaussian component. On the other hand, the density approximated using Gaussian mixtures is much more accurate than the approximation with only one Gaussian component. The covariances of the splitting Gaussian components will have smaller eigenvalues than that of the original single Gaussian distribution. The error ellipse thereby (or the confidence region containing target position) can be narrowed at the same time (illustrations will be given in Section 5). When the splitting procedure is done, all the Gaussian components in each BCPLKF will be propagated to get the filtered states. Given the bearing measurement at time step k, the filtered state and corresponding covariance of the m th component in the n th BCPLKF can be computed as (35) (36)where is a identity matrix and (37) (38)Note that and are computed using and , respectively, due to the unavailability of the true range information. The corresponding weight is updated by (39)where (40)is the likelihood of the measurement for the m th component in the n th BCPLKF. is the predicted bearing measurement of the m th component in the n th BCPLKF. is given by (41)Here is the linearised measurement matrix for the m th component in the n th BCPLKF, which is given by (42)where . has a similar definition. Finally, through the fusion of all the Gaussian components in each BCPLKF, the filtered state and covariance of GSPLKF are given by (43) (44) In order to keep the number of parallel BCPLKFs fixed, a merging procedure [25] is needed once Gaussian splitting is stimulated. The components generated from the same predicted density will be merged into one. Hence, the state mean and covariance of the n th BCPLKF are updated as (45) (46)where is the normalised weight. The block diagram of the proposed GSPLKF algorithm from time step to k is shown in Fig. 2. Remark 5.In the measurement update stage, the predicted density can be split into more than three Gaussian components when a low degree of observability is detected. It is a trade-off between tracking performance and computational load. Correspondingly, the mean and covariance of each component need to be adjusted according to [26]. Fig. 2Open in figure viewerPowerPoint Block diagram of the proposed GSPLKF algorithm 5 Simulation results In this section, the performance of the proposed GSPLKF is compared with several classical algorithms in a typical BOT scenario [1, 25]. The algorithms used for comparison include EKF, CKF, PLKF, BCPLKF, SAM-IVKF, and PF. The posterior Cramér–Rao lower bound (CRLB) [29], which represents the best possible performance one filter can achieve, is also used for comparison. The target-sensor geometry with zero process noise is shown in Fig. 3. The trajectory of the sensor is composed of two legs, in both of which the sensor travels at the same constant speed. The first leg is from point to point and the other one ends by the point . The target is assumed to move with a constant velocity during the whole simulation. Fig. 3Open in figure viewerPowerPoint Trajectories of the target and sensor Each algorithm can be initialised through (28) and (29). Table 1 lists the parameter values used in the simulations. For the GSPLKF, a bank of BCPLKFs is initialised with their own range mean and standard deviation, which can be calculated by using (25) and (26). The number of parallel BCPLKFs in the GSPLKF is . The predicted density of a BCPLKF is split into three Gaussian components when the observability degree of the BOT system is lower than the threshold . Table 1. Parameter setup Parameter Physical meaning Value minimum range in GSPLKF maximum range in GSPLKF velocity of the target velocity of the sensor course of the target initial course of the sensor standard deviation of the initial range standard deviation of the measurement standard deviation of the target velocity standard deviation of the target course For other algorithms, the initialisation process is completed by replacing and with and , where . denotes the true initial relative distance. The PF uses 5000 particles in our simulations. Besides, the sample interval is set to and the noise intensities are set to . Two different performance indexes, including root-mean-square error (RMSE) and bias norm, are utilised in the simulations to evaluate the performance of different algorithms. The definitions of RMSE of position, RMSE of velocity and bias norm are given as follows: (47) (48) (49)where M is the number of Monte Carlo simulation runs. and are, respectively, the filtered state and the true state at time step k in the i th run. , , and are the items of . Similarly, , , and are the corresponding items of . Our simulation results are based on 500 Monte Carlo runs, each of which lasts 32 min. The comparisons of the proposed GSPLKF with other algorithms in terms of RMSE of position and RMSE of velocity are shown in Figs. 4 and 5. Owing to the high non-linearity of the original measurement equation, the linearisation operation in the EKF leads to great performance degradation. The PLKF suffers from severe bias problems and it performs only better than EKF. Since the intensity of process noise is small, the effect of the correlation between and can be negligible. Hence, the bias in the PLKF is almost totally created by the correlation between and . With the bias compensation step, BCPLKF demonstrates an obvious advancement against the PLKF. The RMSE curves of BCPLKF are close to those of SAM-IVKF, which means BCPLKF has a comparable efficacy with SAM-IVKF. In reality, the parameter in the SAM-IVKF used for judging whether instrumental variable (IV) estimation is adopted instead of BCPLKF must be carefully chosen. PF exhibits obvious superiority against the EKF, PLKF, and CKF but is inferior to the improved PLKF algorithms. Although PF will yield better results if one were to increase the number of particles, the computational burden will be much heavier, which is not desirable. The results of the bias norm curves of different algorithms are similar to those of RMSE curves, as shown in Fig. 6. Fig. 4Open in figure viewerPowerPoint RMSE of position comparison after sensor manoeuvre Fig. 5Open in figure viewerPowerPoint RMSE of velocity comparison after sensor manoeuvre Fig. 6Open in figure viewerPowerPoint Bias norm comparison after sensor manoeuvre According to the simulation results, it is evident that the proposed algorithm outperforms all the other methods. As shown in Figs. 4 and 5, the position and velocity errors of GSPLKF are very close to the CRLB towards the end of the simulation time. Especially in the last 10 min, GSPLKF exhibits a significant improvement on the time-averaged RMSE, as shown in Table 2. Under the condition of large measurement noise variance, GSPLKF can still reach position accuracy of around and velocity accuracy of around . Table 2. Time-averaged RMSE for all the algorithms in the last 10 min Time-averaged Time-averaged Algorithm RMSE of position, km RMSE of velocity, m/s EKF 4.5363 4.1217 CKF 2.0575 1.9200 PLKF 2.5558 4.5225 BCPLKF 1.8140 1.5964 SAM-IVKF 1.5896 1.5596 GSPLKF 1.3750 1.2361 PF 1.9343 1.6932 Next, the effect of the parameter in the developed algorithm is investigated. The time-averaged RMSEs using the GSPLKF with five different values are shown in Table 3. The conclusion is that GSPLKF with larger has better tracking performance. When is small, which means the filter has a strong tolerance for low system observability, the splitting mechanism will not be easily triggered. On the contrary, GSPLKF with large has little tolerance for low system observability, which leads to driving more splitting components from the original Gaussian density. The simulation results, in the meanwhile, validate the practicability of the derived metric. Table 3. Time-averaged RMSE for GSPLKF with different values in the last10 min Time-averaged Time-averaged Value of RMSE of position, km RMSE of velocity, m/s 0.5 1.3750 1.2361 0.1 1.3765 1.2402 1.4099 1.3658 1.4113 1.4029 1.5163 1.4474 In Figs. 7-9–10, the confidence ellipse is used to further demonstrate the superiority of the GSPLKF. The ellipse of each algorithm at time step k is constructed using the equation as follows: (50)where , is the covariance related to , is the parameter determining the size of the ellipse. Specifically, corresponds to the confidence ellipse and corresponds to the confidence ellipse. Based on the results of Figs. 7-9–10, the confidence ellipses of the EKF, CKF, and PLKF cannot incorporate the true target position. In the 27th minute, the confidence ellipses of the BCPLKF, SAM-IVKF, and PF, as shown in Fig. 8, are able to incorporate the true target position. However, as shown in Fig. 7, the target is nearly out of their confidence ellipses. The confidence ellipse of the GSPLKF has a higher probability of containing the target position with the smallest radius no matter which confidence level is considered. Fig. 7Open in figure viewerPowerPoint confidence ellipses of different algorithms in the 27th minute of one Monte Carlo run Fig. 8Open in figure viewerPowerPoint confidence ellipses of different algorithms in the 27th minute of one Monte Carlo run Fig. 9Open in figure viewerPowerPoint confidence ellipses of different algorithms in the 29th minute of one Monte Carlo run Fig. 10Open in figure viewerPowerPoint confidence ellipses of different algorithms in the 29th minute of one Monte Carlo run Finally, the computational complexities of different algorithms are investigated. The computation time for each of the algorithms, normalised by the EKF runtime, is provided in Table 4. With the simplest linear filtering process, the PLKF is faster than other algorithms. BCPLKF and SAM-IVKF are slightly slower due to the bias compensation step and the judgement for IV estimation. The additional computational burden of CKF results from generating the cubature points. The proposed GSPLKF has more extra computations caused by the modifications we make on it, but it still shows a major computational advantage over the PF. It is just because of these modifications that the GSPLKF outperforms other algorithms. Table 4. Relative computation time of different algorithms Algorithm Relative computation time EKF 1 CKF 1.9 PLKF 0.6 BCPLKF 0.7 SAM-IVKF 0.8 GSPLKF 12 PF 1462 6 Conclusions In this study, we first derive an observability metric to characterise the relative geometry and motion relationships between the target and the sensor in BOT. Based on the definition, a larger value of this metric corresponds to better target-sensor geometry and motion. In reality, this also means that the sensor trajectory is more effective. After that, we develop a novel BOT algorithm based on the PLKF, which has the advantages of simple structure and low complexity. We combine the original PLKF with a bias compensation step and then extend it within the Gaussian sum framework. In order to increase the flexibility of the algorithm, a Gaussian splitting and merging procedure will be carried out when a low degree of system observability is detected. Simulation results show that the proposed algorithm outperforms the existing algorithms in terms of several performance indexes. Although only non-manoeuvring target tracking is considered, the observability metric is also suitable for tracking manoeuvring target with bearings-only measurements. In future work, we will focus more on BOT algorithms for manoeuvring target with better efficacy, efficiency, and flexibility. 7 Acknowledgments This work was supported by the China Scholarship Council (CSC) under grant no. 201806280305 and the National Key R&D Program of China under grant no. 2018YFB1700100. 9 Appendix 9.1 Appendix 1 By substituting into , we have (51)Therefore, . This result assures that we do not have to worry about the existence of the proposed observability metric for BOT with a single moving sensor. 9.2 Appendix 2 The position components of the state can be initialised based on the bearing measurement at time and the prior knowledge of the initial range. The initial range estimate and measured bearing are defined as (52)where the errors in range and bearing are independent with zero mean and variances and , respectively. Then the estimated x and y components of the initial state vector can be computed by (53) By substituting (52) into (53), we can obtain (54)where is the true initial position, and are the errors in the x -coordinate and y -coordinate, respectively. Then, we can expand (54) by using trigonometric identities to obtain (55)Further by using the first-order Taylor series expansion, the errors in (55) can be approximated as (56)Note that and , which explains the rationality of using as the initial estimate. The elements of the initial covariance with respect to the position components are calculated as (57) The derivation of initialisation with respect to the velocity components is similar. We omit it for brevity. Since the true range and bearing are unknown, we modify the formula for the initial covariance by replacing and with and . For the n th BCPLKF in the GSPLKF algorithm, we use and as the initial range estimate and its variance. 8 References 1Ristic B. Arulampalam S., and Gordon N.: ' Beyond the Kalman filter: particle filters for tracking applications' ( Artech House, Norwood, MA, USA, 2004) 2Xu S. Doğançay K., and Hmam H.: 'Distributed pseudolinear estimation and UAV path optimization for 3D AOA target tracking', Signal Process., 2017, 133, pp. 64– 78 3Inoue R.S. Terra M.H., and Cerri J.P.: 'Extended robust Kalman filter for attitude estimation', IET Control Theory Appl., 2016, 10, (2), pp. 162– 172 4Aidala V.J., and Hammel S.E.: 'Utilization of modified polar coordinates for bearings-only tracking', IEEE Trans. Autom. Control, 1983, 28, (3), pp. 283– 294 5Peach N.: 'Bearings-only tracking using a set of range-parameterised extended Kalman filters', IEE Proc. Control Theory Appl., 1995, 142, (1), pp. 73– 80 6Chen X. Sun R., and Wang F. et al.: 'Two-stage unscented Kalman filter algorithm for fault estimation in spacecraft attitude control system', IET Control Theory Appl., 2018, 12, (13), pp. 1781– 1791 7Zhang Y. Huang Y., and Li N. et al.: 'Interpolatory cubature Kalman filters', IET Control Theory Appl., 2015, 9, (11), pp. 1731– 1739 8Qin W. Wang X., and Cui N.: 'Maximum correntropy sparse Gauss–Hermite quadrature filter and its application in tracking ballistic missile', IET Radar Sonar Navig., 2017, 11, (9), pp. 1388– 1396 9Arulampalam M.S. Maskell S., and Gordon N. et al.: 'A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking', IEEE Trans. Signal Process., 2002, 50, (2), pp. 174– 188 10Merlinge N. Dahia K., and Piet-Lahanier H. et al.: 'A box regularized particle filter for state estimation with severely ambiguous and non-linear measurements', Automatica, 2019, 104, pp. 102– 110 11Bréhard T., and Le Cadre J.P.: 'Hierarchical particle filter for bearings-only tracking', IEEE Trans. Aerosp. Electron. Syst., 2007, 43, (4), pp. 1567– 1585 12Evensen G.: 'The ensemble Kalman filter: theoretical formulation and practical implementation', Ocean Dyn., 2003, 53, (4), pp. 343– 367 13Lindgren A.G., and Gong K.F.: 'Position and velocity estimation via bearing observations', IEEE Trans. Aerosp. Electron. Syst., 1978, AES-14, (4), pp. 564– 577 14Doğançay K.: 'Bias compensation for the bearings-only pseudolinear target track estimator', IEEE Trans. Signal Process., 2006, 54, (1), pp. 59– 68 15Doğançay K.: '3D pseudolinear target motion analysis from angle measurements', IEEE Trans. Signal Process., 2015, 63, (6), pp. 1570– 1580 16Nguyen N.H., and Doğançay K.: 'Improved pseudolinear Kalman filter algorithms for bearings-only target tracking', IEEE Trans. Signal Process., 2017, 65, (23), pp. 6119– 6134 17Nguyen N.H., and Doğançay K.: 'Instrumental variable based Kalman filter algorithm for three-dimensional AOA target tracking', IEEE Signal Process. Lett., 2018, 25, (10), pp. 1605– 1609 18Le Cadre J.P., and Jauffret C.: 'Discrete-time observability and estimability analysis for bearings-only target motion analysis', IEEE Trans. Aerosp. Electron. Syst., 1997, 33, (1), pp. 178– 201 19Passerieux J.M., and Van Cappel D.: 'Optimal observer maneuver for bearings-only tracking', IEEE Trans. Aerosp. Electron. Syst., 1998, 34, (3), pp. 777– 788 20Kronhamn T.R.: 'Bearings-only target motion analysis based on a multihypothesis Kalman filter and adaptive ownship motion control', IEE Proc. Radar Sonar Navig., 1998, 145, (4), pp. 247– 252 21Trémois O., and Le Cadre J.P.: 'Optimal observer trajectory in bearings-only tracking for manoeuvring sources', IEE Proc. Radar Sonar Navig., 1999, 146, (1), pp. 31– 39 22Zhang H. Dufour F., and Anselmi J. et al.: ' Piecewise optimal trajectories of observer for bearings-only tracking by quantization'. Proc. 20th Int. Conf. on Information Fusion (FUSION), Xi'an, China, July 2017, pp. 1458– 1464 23Gilitschenski I. Kurz G., and Hanebeck U.D.: ' Bearings-only sensor scheduling using circular statistics'. Proc. 16th Int. Conf. on Information Fusion (FUSION), Istanbul, Turkey, July 2013, pp. 515– 521 24Hermann R., and Krener A.J.: 'Nonlinear controllability and observability', IEEE Trans. Autom. Control, 1977, 22, (5), pp. 728– 740 25Leong P.H. Arulampalam S., and Lamahewa T.A. et al.: 'Gaussian-sum cubature Kalman filter with improved robustness for bearings-only tracking', IEEE Signal Process. Lett., 2014, 21, (5), pp. 513– 517 26Leutnant V. Krueger A., and Haeb-Umbach R.: ' A versatile Gaussian splitting approach to non-linear state estimation and its application to noise-robust ASR'. Proc. 12th Annual Conf. of the Int. Speech Communication Association (INTERSPEECH), Florence, Italy, August 2011, pp. 1641– 1644 27Jiang H., and Cai Y.: 'Adaptive fifth-degree cubature information filter for multi-sensor bearings-only tracking', Sensors, 2018, 18, (10), p. 3241 28Williams R.K., and Sukhatme G.S.: ' Observability in topology-constrained multi-robot target tracking'. Proc. 2015 IEEE Int. Conf. on Robotics and Automation (ICRA), Seattle, USA, May 2015, pp. 1795– 1801 29Tichavský P. Muravchik C.H., and Nehorai A.: 'Posterior Cramér–Rao bounds for discrete-time nonlinear filtering', IEEE Trans. Signal Process., 1998, 46, (5), pp. 1386– 1396 Citing Literature Volume14, Issue3February 2020Pages 452-460 FiguresReferencesRelatedInformation