- Open Access
Maneuvering target tracking using fuzzy logic-based recursive least squares filter
© Fan et al.; licensee Springer. 2014
- Received: 20 December 2013
- Accepted: 31 March 2014
- Published: 16 April 2014
In this paper, a fuzzy logic-based recursive least squares filter (FLRLSF) is presented for maneuvering target tracking (MTT) in situations of observations with unknown random characteristics. In the proposed filter, fuzzy logic is applied in the standard recursive least squares filter (RLSF) by the design of a set of fuzzy if-then rules. Given the observation residual and the heading change in the current prediction, these rules are used to determine the magnitude of the fading factor of RLSF. The proposed filter has an advantage in which the restrictive assumptions of statistical models for process noise, measurement noise, and motion models are relaxed. Moreover, it does not need a maneuver detector when tracking a maneuvering target. The performance of FLRLSF is evaluated by using a simulation and real test experiment, and it is found to be better than those of the traditional RLSF, the fuzzy adaptive α-β filter (FAα-βF), and the hybrid Kalman filter in tracking accuracy.
- Maneuvering target tracking
- Adaptive filter
- Recursive least squares filter
- Fading factor
- Fuzzy logic
- Heading change
Maneuvering target tracking (MTT) is always a critical problem in target tracking area[1–5]. In the literature on MTT in sensor network, its survey primarily consists of target dynamic models, observation models and techniques, decision-based methods, multiple-model methods, and nonlinear filtering methods. The adaptive filters are popular to apply in MTT as a nonlinear-type filter; they are mainly classified as follows: structured adaptive filters, such as the interacting multiple model (IMM) and the variable structured model; parametric adaptive filters, such as the Singer model, the current statistical model, and the adaptive acceleration model. Structured adaptive filters, usually with more computation, require sufficient prior knowledge such as all possible motion models of a moving target. So, they are less suitable for real situations with limited prior knowledge. Parametric adaptive filters, generally with less computation, describe maneuver characteristics as unknown random parameters with certain probability distribution functions (pdf) and estimate maneuver models jointly by both the parameters and the target states. Recently, particle filters are broadly adopted in various applications, such as location estimation[6–8]. However, their tracking performances are combined in proportion to the number of the corresponding particles in target tracking. Moreover, the computation of these filters is generally complicated. Hence, they are unsuitable to apply in real-time tracking[9–11]. In real sensor networks, observations or state estimates are generally transmitted without their covariances. They are processed in sensor nodes or fusion centers. To relax transmission burdens and save communication bandwidths, the filters designed need to possess good performances in both computational and time complexity. For the above reasons, this paper is only concerned with parametric adaptive filters.
Maneuver modeling which is a usual prerequisite for MTT has a direct influence on tracking results. Several techniques have been introduced to overcome different maneuver modeling. MTT methods based on statistical models often need to establish motion models exactly and obtain the target position timely. Unfortunately, these requirements are difficult to satisfy because of unknown maneuver characteristics in real situations. To solve the problem, one of the most popular techniques has been applied in MTT. Fuzzy logic with intelligent adaptation capabilities has been widely used to relax or avoid the restrictive assumptions of motion models[12–16]. Due to the limitation of the standard IMM algorithm in real applications, the adaptive fuzzy IMM filter (AFIMMF) proposed in defines several basis sub-models and time-varying mode transition probabilities to reduce its computational complexity. Nevertheless, it suffers from a deficiency originated in expensive computation of the time-varying mode transition probabilities. Moreover, its performance depends on the assumptions on the basis sub-models. Two different modified Kalman filters proposed in and tend to extend the standard Kalman filter for MTT. Nonetheless, their computation is still expensive, and the proposed hybrid Kalman filter (HKF) in is mainly applied to track an accelerating target. To track a target making sharp turns or accelerating at nonuniform rates in stressful environments, the fuzzy gain α-β filter (FGα-βF) is proposed in. It can avoid any assumptions on statistical models by utilizing fuzzy rules to determine the magnitudes of α and β. However, its tracking result in the case where sharp turns appear is not improved markedly. For perfect tracking accuracy, the fuzzy adaptive α-β filter (FAα-βF) proposed in can detect maneuvers. However, its tracking result is still undesirable.
In practice, the standard recursive least squares filter (RLSF)[17, 18] is quite well received in linear models with the unknown random characteristics of observations. In this case, it has less computational complexity than the standard Kalman filter. Unfortunately, its performance degrades seriously during maneuvering. For perfect output results in nonlinear models, the traditional fuzzy adaptive filters make the relationship between the input and output variables maps into a set of fuzzy rules, and these rules determine the control parameters of the filters. So, their performances depend on whether the rules can describe maneuver characteristics exactly and completely. Due to the unknown motion models during real tracking procedures, a complex nonlinear relationship exists between observations and state estimates. For this reason, the fuzzy rules are usually difficult to establish by directly utilizing the relationship between the observations and the state estimates. Therefore, the traditional fuzzy adaptive filters are restricted in the applications of MTT.
Applying fading factors and fuzzy logic into traditional filters are popular because of some practical concerns including easy implementation and effectiveness. In this respect, it provides a direction to design a filter for MTT. Considering these facts, a novel fuzzy logic-based recursive least squares filter (FLRLSF) is proposed to deal with the problem mentioned above. First, the proposed filter utilizes observation residuals and heading changes to describe maneuver characteristics of a maneuvering target and employs the fading factor of RLSF to express the magnitude of the target maneuver. A set of fuzzy rules is designed according to the relationship of the fading factor with observation residuals and heading changes over time. Next, given the observation residual and the heading change in the current prediction, these rules are applied to determine the magnitude of the fading factor. Then, the fuzzy system designed based on the rules is utilized to adjust the fading factor in response to the changes in speed and direction without maneuver detectors. Finally, FLRLSF is applied to estimate the target state. Its performance is evaluated by using a simulation and real test experiment, compared with the existing filters mentioned above.
where is δ kt the Kronecker delta function.
Though targets often move at constant velocity, they are easy to maneuver suddenly. In this respect, maneuver modeling is a key problem of MTT. To solve this problem, many methods have been proposed on the restrictive assumptions of the motion models[17, 20]. Considering the aforementioned facts, these assumptions are often inconsistent with real environments. Then, two strategies of motion modeling exist with unknown maneuver information: describe the target trajectory as several typical motion models with known parameters or their combination or incorporate control variables in the target motion equation as the random variables with the certain pdf. Unfortunately, it is difficult to obtain the prior information in real situations, and various uncertainties generally exist in maneuver motions during real tracking procedures. So, the methods based on the statistic framework are complicated and difficult due to establishing the accurate motion models or obtaining the exact pdf by using the probability and statistics theory. Nowadays, fuzzy systems with the universal approximation capabilities have been widely applied in nonlinear complicated system[21, 22]. Incorporation of fuzzy logic in fuzzy systems is easy and flexible to describe various uncertainties or random variables by the use of linguistic terms. From this point of view, it is a good idea that fuzzy systems are designed to adjust the filter parameter for MTT.
where x k ,, and are the components of z k ,, and in the x-axis direction respectively, while y k ,, and are their corresponding components in the y-axis direction. To simplify both computation and discussion, ∆z k and ∆θ k are necessary in normalization processing with the following forms: ∆z k ' = ∆z k /∆zmax and ∆θ k ' = ∆θ k /∆θmax instead of ∆z k and ∆θ k . Here, ∆zmax and ∆θmax are their corresponding maximum values, and they are usually related with the target's type and the sensor's performance.
where z k = (z1, z2, …, z k )T, H k = (H1, H2, …, H k )T, and w k = (w1, w2, …, w k )T.
From Equations 14 and 15, is the sum of the state prediction and the deviation between the observation and the predicted position at time k on the one hand, and the magnitude of depends on how large the deviation is modified by P k on the other hand. Meanwhile, P k can vary with λ k . From this point of view, can be indirectly modified by adjusting λ k . While the modification of becomes weaker with the increase of λ k , it becomes greater with the decrease of λ k . In this respect, if one can obtain the right λ k according to maneuver information, it implies that can be exactly expressed as the modified value by adjusting λ k at each discrete time. Therefore, RLSF can be applied to estimate the target state by employing λ k to describe maneuver characteristics.
4.1 Design of the fuzzy system
As shown in Figure 1, the design procedure of the fuzzy system, which consists of four blocks, includes the following: first, utilize a singleton fuzzifier as block 2; next, design the fuzzy rules of block 3 according to the relationship between the input and output variable of block 4; then, determine the reference engine of block 4, which maps the spaces of all variables into the fuzzy spaces and gives their membership functions; finally, select a defuzzifier of block 5. Each step is elaborated in detail as follows.
Fuzzy rules on ∆ θ k ′, ∆ z k ′, and
∆θ k ′
∆z k ′
where,, and denote the membership functions of ∆z k ', ∆θ k ', and, respectively; at each discrete time is equivalent to the corresponding value of when the membership function of is maximized at each fuzzy set defined on.
4.2 FLRLSF method
4.3 Computational complexity analysis
Computational complexity for an algorithm is an important index whether the algorithm provides processing capability in real time, and it is crucial for real-time systems. Hence, many filters with good performances in accuracy are limited in target tracking area due to their big computation. Here, computational complexity mainly denotes the times in terms of multiplications, divisions, additions, and subtraction for each run of an algorithm under no consideration of the process of generation, move, and measure for a target.
Computational complexity for each filter
14n3+ (4 m - 1)n2 + (12 m2 + 2 m - 1)n + 2 m3 - m2 + 4
14n3+ (4 m - 1)n2 + (12 m2 + 2 m - 1)n + 2 m3 - m2 + 3 M + 3
2n2 + (2 m - 1)n + 3 M - 1
4n3 + (12 m + 6)n2 + (12 m2 + 2)n + 4 m3 + 2 m2 - 4 m + 3 M
A stimulation experiment and a real test experiment have been carried out to evaluate the performance of the FLRLSF method in comparison with the other three existing methods, the traditional RLSF, FAα-βF, and HKF for MTT.
5.1 Simulation experiment
5.2 Real test experiment
In this paper, considering the properties and drawbacks of the traditional adaptive filters, FLRLSF is proposed for MTT in the situation of observations with unknown random characteristics. The paper employs the fading factor of RLSF to describe maneuver characteristics of the motion model, analyzes the relationship of the fading factor with observation residuals and heading changes, and maps the relationship into a set of fuzzy rules. By applying fuzzy logic in the standard RLSF and designing the fuzzy system, the fading factor is allowed to be adjusted adaptively based on the fuzzy rules. These rules determine the fading factor dynamically according to the magnitudes of the observation residual and heading change in the current prediction. Therefore, FLRLSF is able to detect maneuvers timely and estimate their magnitude accurately. The effectiveness of the proposed filter is evaluated by using the simulation and real test experiment. Its performance in terms of tracking accuracy and the average run time is compared against those of RLSF, FAα-βF, and HKF. The results of the simulation experiment show that the proposed filter can track a maneuvering target adaptively, and it achieves better performance in tracking accuracy than the other three methods. In addition, the real test experiment validates its feasibility in real environment.
In future work, we will extend the proposed filter in tracking multiple targets in cluster in distributed multisensor system. Furthermore, we will also explore FLRLSF for the state estimates of global tracks after local track-to-local track fusion in the fusion center of the system.
EF received his Bachelor's Degree in Electronic Information Science and Technology from Hubei Engineering University, Xiaogan in 2002 and Master's Degree in Signal and Information Processing from Nanchang Hangkong University, Nanchang in 2006. He is currently pursuing his PhD in Signal and Information Processing at Xidian University, Xi'an, China. His fields of interests include intelligent information processing and multisensor data fusion. WX graduated from Xidian University, Xi'an and remained at the university as a faculty member in 1965. From 1981 to 1983, he was a visiting scholar at the University of Pennsylvania, USA. In 1989, he was invited to the same university as a visiting professor. He is currently working in the School of Information Engineering, Shenzhen University, Shenzhen, China. His fields of interests include intelligent information processing, fuzzy information processing, image processing, pattern recognition, etc. ZL received his bachelor's degree and master's degree from Tianjin University, Tianjin in 1985 and 1990, respectively, and PhD degree from Xidian University, Xi'an in 2005. He is currently working in the School of Information Engineering, Shenzhen University, Shenzhen, China. His fields of interests include intelligent information processing, fuzzy information processing, and multisensor data fusion.
This work was supported by the National Natural Science Foundation of China (Grant: 61271107, 61301074), Key Project in the National Science & Technology Pillar Program (2011BAH24B12), Specialized Research Fund for the Doctoral Program of Higher Education (20104408120001), and Natural Science Foundation of Ministry of Education (S2012010009417).
- Li XR, Jilkov VP: A survey of maneuvering target tracking-part VIc: approximate nonlinear density filtering in discrete time. In Proceedings of the SPIE Conference on Signal and Data Processing of Small Targets (CSDPST'12), vol. 8393. SPIE; 2012:1-12.Google Scholar
- Pan Q, Liang Y, Yang F, Cheng YM: Modern Target Tracking and Information Fusion. Beijing: National Defense Industry Press; 2009:5-61.Google Scholar
- Li XR, Jilkov VP: Survey of maneuvering target tracking. Part V: multiple-model methods. IEEE Trans. Aero. Elec. Sys 2005, 41: 1255-1321. 10.1109/TAES.2005.1561886View ArticleGoogle Scholar
- Nadarajah N, Tharmarasa R, Mcdonald M, Kirubarajan T: IMM forward filtering and backward smoothing for maneuvering target tracking. IEEE Trans. Aero. Elec. Sys. 2012, 48(3):2673-2678.View ArticleGoogle Scholar
- Li XR, Jilkov VP: Survey of maneuvering target tracking. Part I: dynamic models. IEEE Trans. Aero. Elec. Sys. 2003, 39: 1333-1364. 10.1109/TAES.2003.1261132View ArticleGoogle Scholar
- Chiu WY, Chen BS: Mobile location estimation in urban areas using mixed Manhattan/Eulidean norm and convex optimization. IEEE Trans. Wir. Commun. 2009, 8: 414-423.View ArticleGoogle Scholar
- Chiu WY, Chen BS, Yang CY: Robust relative location estimation in wireless sensor networks with inexact position problems. IEEE Trans. Mobile Comput. 2012, 11: 935-946.View ArticleGoogle Scholar
- Chiu WY, Chen BS: A mixed-norm approach using simulated annealing with changeable neighborhood for mobile location estimation. IEEE Trans. Mobile Comput. 2010, 9: 633-642.View ArticleGoogle Scholar
- Sutharsan S, Kirubarajan T, Lang T, McDonald M: An optimization based parallel particle filter for multitarget tracking. IEEE Trans. Aero. Elec. Sys. 2012, 48(2):1601-1618.View ArticleGoogle Scholar
- Li LQ, Xie WX: Intuitionistic fuzzy joint probabilistic data association filter and its application to multitarget target tracking. Signal Process. 2014, 96: 433-444.View ArticleGoogle Scholar
- Mihaylova L, Hegyi A, Gning A, Boel RK: Parallelized particle and Gaussian sum particle filters for large-scale freeway traffic systems. IEEE Trans. Intell. Transp. Sys. 2012, 13(1):36-48.View ArticleGoogle Scholar
- Kim H, Kim I: Design of adaptive fuzzy IMM algorithm for tracking the maneuvering target with time-varying measurement noise. Int. J. Control Autom. 2007, 5(3):307-316.Google Scholar
- Jwo D, Wang S: Adaptive fuzzy strong tracking extended Kalman filtering for GPS Navigation. IEEE Sens. J. 2007, 7(5):778-789.View ArticleGoogle Scholar
- Barhari MH, Karsaz A, Pariz N: High maneuvering target tracking using a novel hybrid Kalman filter-fuzzy logic architecture. Int. J. Innov. Comput. 2011, 7: 501-510.Google Scholar
- Chan KCC, Lee V, Leung H: Radar tracking for air surveillance in a stressful environment using a fuzzy-gain filter. IEEE Trans. Fuzzy Syst. 1997, 5: 80-89. 10.1109/91.554452View ArticleGoogle Scholar
- Li PF, Yu JP, Li LQ: Maneuvering target tracking based on fuzzy adaptive α-β filter. J. Syst. Eng. Electron. 2008, 30(11):2138-2141.Google Scholar
- Zhu YZ: Efficient recursive state estimator for dynamic systems without knowledge of noise covariances. IEEE Trans. Aero. Elec. Sys. 1999, 35: 102-114. 10.1109/7.745684View ArticleGoogle Scholar
- Peng DL, Lin WC, Xue AK: Multisensor Multisource Information Fusion Theory with Application. Beijing: Science Press; 2010:68-71.Google Scholar
- Bahari MH, Sistani MBN, Pariz N: Intelligent fading memory for high maneuvering target tracking. Int. J. Physic. Sci. 2009, 4: 548-554.Google Scholar
- Liang Y, Zhou DH, Zhang L, Pan Q: Adaptive filtering for stochastic systems with generalized disturbance inputs. IEEE Signal Proc. Let. 2008, 15: 645-648.View ArticleGoogle Scholar
- Wang LX: Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Beijing: National Defense Industry Press; 1995:210-232.Google Scholar
- Su XJ, Shi P, Song YD: A novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay. IEEE Trans. Fuzzy Syst. 2012, 20: 1114-1129.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.