Distributed Fusion Tracking Estimation under Range-Only Measurement

This paper concentrate on the distributed fusion estimation problem for range-only target tracking system with unknown but bounded noises, where the linear and nonlinear motion models are both considered. A kind of nonlinear transformation is used to convert the nonlinear distance measurement model into a linear one, which eliminates the corresponding linearization errors in the design of estimation error system. In spite of the transformed measurement noise becomes more complicated, while it is still bounded. Moreover, for the nonlinear target motion model, the state linearization error caused by the Taylor expansion is modeled by the state dependent matrix with uncertainty bounded matrix. In this case, based on the bounded recursive optimization algorithm, two kinds of convex optimization problem are established to determine the gains of the local/fusion estimators, and the stability of the designed estimators also can be guaranteed. Finally, two kinds of range-only target tracking systems are presented to show the effectiveness and advantages of the proposed methods.


I. INTRODUCTION
A S an essential application of multi-sensor fusion estimation, target location and tracking not only attracted significant interest in the military field initially, but also been a research focus in the civilian field during the past decades, such as emergency location [1], intelligent transport systems [2], cellular mobile station location [3]. Generally, the timeof-arrival (TOA), time-difference-of-arrival (TDOA), angleof-arrival (AOA) and received signal strength (RSS) are conventional measurement-based location and tracking methods [4]- [8], where AOA is based on the angle of the target, RSS is based on the signal strength, TOA and TDOA are based on the time information. Therefore, among these location methods, AOA requires the sensor to have the ability of angle measurement, TOA and TDOA require the clock synchronization between the sensors, and RSS requires sensors to identify the strength of the signal. However, the target tracking problem also can be viewed as a nonlinear state estimation problem for the range-only tracking systems. Then, different from above conventional methods, this paper will focus on distributed fusion estimation problem of the target tracking systems, which only requires to know the range-only measurement.
Notice that, since the target tracking problem is essen-tially a nonlinear state estimation problem, thus a variety of nonlinear filters have been developed in [9]- [13] to tracking the mobile target. Specifically, the extended Kalman filter (EKF) [14] usually linearized the nonlinear system process and measurement model by Taylor series expansion, then the classical Kalman filter (KF) can be used to estimate the target's position. However, the linearization errors were inevitably introduced by using the Taylor expansion, which might causing the nonlinear filtering was unstable. For example, a stability condition with respect to the linearization errors should be given to ensure the stability of the estimator in [15]. Though the second-order or higher-order EKF [16] can reduced the linearization errors, the Jacobian matrix or Hessian matrix were difficult to be calculated and thus the computation burden was increased as well. To better overcome this limitation, the unscented Kalman filter (UKF) [17] was developed to estimate the nonlinear system, where the sigma points were sampled to approximate the probability distribution of the nonlinear system, and the estimation precious can up to third-order Taylor expansion. However, the UKF performed a sensitive ability to nonpositive definite covariances, and the parameters α, β, κ also should be adjusted reasonably in practice. Then, a cubature Kalman filter (CKF) [18] was emerged to reduce the sampling points, thus the computation time can be reduced while keeping a fairly estimate performance and a better stability. Unfortunately, when the prediction error was larger, the UKF or CKF methods might be performed an unsatisfactory estimation performance that is introduced by the higher orders. Therefore, for the target tracking systems which employing range-only measurements, a kind of nonlinear transformation [19] is employed in this paper to transfer the nonlinear measurement into a linear form, which can avoids the linearization errors caused by above linearization approaches.
Moreover, whether the above single-sensor tracking estimation methods, or multi-sensor fusion tracking methods [20]- [23], the system uncertainty should be effectively handled. Particularly, for the most existing distributed estimation works, a saturated-constrained distributed filtering algorithm was developed in [24] under uncertain missing probabilities, and the boundedness and monotonicity were also discussed. In [25], a recursive and distributed Kalman filtering algorithm was developed for interconnected dynamic systems, where the each local estimator was designed by its own and neighbors' information. Additionally, a generalized nonlinear weighted measurement fusion UKF (WMF-UKF) algorithm was developed in [26], and the asymptotic global optimality has also been demonstrated. However, the above methods were all assumed that the noises have an prior information of covariance, while the process and measurement may contain various noises without statistical properties in practice [27]. In this case, a noise estimator was proposed in [28] for estimating the noises, and then eliminated the estimated noises to obtain a desirable performance. Although a nonlinear transformation developed in [19] can transformed the nonlinear distance measurement to a linear one [29], the converted noises were no longer Gaussian even if the original system noises assumed to be Gaussian. Then, by using a similar nonlinear transformation, the dimension of the augmented measurement in [30] was reduced by the Gauss-Markov measurement fusion, and an adaptive factor was introduced by the hypothesis test to deal with the non-Gaussian noises. Recently, a bounded recursive optimization algorithm was developed in [31] to solve the uncertainty caused by unknown bounded noises, while the stability of the designed nonlinear estimator was not discussed. Meanwhile, a distributed nonlinear estimation problem was considered in [32] based on the recursive optimization approach for rang-only target tracking system. Then, a decoupling strategy with sequentialstructure was proposed in [33] to solve the interconnected terms, where the designed algorithm can implemented fully distributed estimation. In spite of [34] presented the stability condition of the nonlinear estimator without noise statistical characteristics, the distance measurement function or the linearization errors of the range-only measurement were not easily judged as bounded, thus the stability condition was useless in the range-only tracking systems.
From the above analysis, the design problem of stable fusion tracking estimators for range-only target tracking sys-tems will be considered in this paper. The contribution of this paper can be summarized as follows: (1) A nonlinear transformation is employed to transform the distance measurement into a linear measurement model, which can avoid the linearization errors caused by the convention linearization approaches; (2) The linearization error of the nonlinear state model is modeled by the bounded uncertainty matrix with state-dependent matrix, and a bounded recursive optimization scheme is constructed to establish different convex optimization problems, which can deal with the transformed measurement noises without statistical characteristics. Then, the stable local estimator gains and fusion criterion can be determined by the standard software packages. Finally, two different target tracking examples are given to verify the effectiveness and advantages of the proposed methods.
Notations: R m×n denotes the set of m × n real matrices, the superscript "T" denotes the matrix transpose, and A < 0 denotes a negative definite matrix. O and I are zero and identity matrix with appropriate dimension, respectively, the symmetric terms of a symmetric matrix are denoted by " * ". Moreover, || · || 2 represents the 2-norm, and λ max (A) is the maximum eigenvalue of the A. col {b 1 , . . . , b n } denotes a column vector whose elements are b 1 , . . . , b n , while diag {·} denotes a block diagonal matrix.

II. PROBLEM FORMULATION
Consider a multi-sensor range only target tracking system (see Fig. 1), where the motion process of the target can be described by the following nonlinear state model: where the x(t) ∈ R n denotes the target's motion state in the planar coordinate, f (x(t)) is the nonlinear function that assume to be continuously differentiable. B(t) ∈ R n×r is time-varying matrix, w(t) ∈ R r×1 is the system noise satisfying: where σ w is a unknown upper bound.
Moreover, consider only the distance between the target and the sensors can be measured by the sensors, and there are q i ≥ 3 sensor nodes within each sensor group. Then, the measurement model of each sensor can be given by where i = 1, 2, . . . , m and m is the number of the sensor groups, (x p (t), y p (t)) is the target's position in the x − y planar, and (x i qi , y i qi ) is the sensor's position in the x − y coordinate, v i qi (t) is the sensor measurement noise satisfying: where σ i qi is also a unknown upper bound. Generally, the sink nodes collecting the measurements from each sensor of its corresponding sensor groups, then the measurement y i (t) ∈ R qi can be written as: where y i (t) col{d i 1 (t), d i 2 (t), . . . , d i qi (t)} is a nonlinear measurement. In order to reduce the influence of linearization errors in the nonlinear state estimation problem, a kind of nonlinear transformation, which has been developed in [19], with respect to range-only information of target tracking system is introduced to convert y i (t) as a linear model. Specifically, rearranging and squaring both sides of the equations in (3) to yield Then, taking the 1-th equation in (6) as a benchmark, and subtracting the 1-th equation from others, one has where Under this case, the measurements wherex p i (t) denotes one-step prediction, and K n i (t) is the nonlinear local gain to be determined in the next section.
Next, the LSEsx n i (t)(i = 1, 2, . . . , m) are used to design nonlinear distributed fusion estimate (DFE)x n (t) as follows: where the distributed weighting fusion matrices satisfy m i=1 W n i (t) = I. Notice that, when the nonlinear state model reduced to a linear state space: where A(t) is the state transition matrix with appropriate dimensions. Therefore, the linear LSEsx l i (t)(i = 1, 2, . . . , m) for linear systems (13) and (7) can be designed as: where K l i (t) is the linear local gain to be determined. Similarly, based on the LSEsx l i (t)(i = 1, 2, . . . , m), the linear DFEx l (t) is given by: where the weighting fusion matrices satisfy Consequently, the main objectives of this paper are to solve the following problems: • The first is to design local gains K n i (t) in (11) and K l i (t) in (14) such that the local linear/nonlinear estimators are stable, respectively, while the upper bound of square errors (SEs) of the corresponding LSEs are minimal at each time.
• The second is to design distributed weighting fusion matrices W n i (t) in (12) and W l i (t) in (15) such the fusion estimators are stable, while the upper bound of SEs of the corresponding DFEs are also minimal at each time. Remark 1: Notice that, the nonlinear raw measurement y i (t) in (5) are usually linearized by the Taylor series expansion and unscented transformation, while the linearization errors will inevitably be introduced. This is because the Taylor series usually expanding the nonlinear function in first or second order, while the higher-order terms are neglected on the designing of filtering, and the neglected higher-order terms also affect the accuracy of the estimation. At the same time, the calculate of Jacobian or Hessian matrices will increase the computation burden of the designed estimation algorithms. Though the unscented transformation by using the sigma points to approximate the probability distribution, which avoids solving the Jacobian or Hessian matrices and the estimation precision can up to the third order Taylor expansion, it still might be given a undesirable estimation error caused by the higher orders or prediction errors. In this case, a nonlinear transformation is introduced in this paper to convert the nonlinear measurement model (3) to a linear one (7), which can avoid the linearization errors caused by above linearization approaches and improve the stability of the designed estimators.
Remark 2: It can be seen from (9) that the measurement noise v i (t) in (5) after nonlinear transformation is no longer Gaussian and cross uncorrelated, even assuming that the measurement noises in (3) obey mutually independent Gaussian distribution, thus the various existing extension Kalman filtering methods cannot be directly employed to track the target. Though it increases the difficulty of estimator design, the developed estimator (11) in this paper are not require to have a prior knowledge of the noise statistical characteristics, which not only can performs an effective estimation performance, but also reduces the computation complexity of the noise covariances as compared with [30].

III. MAIN RESULTS
In this section, the distributed linear/nonlinear fusion estimation algorithms will be proposed. Firstly, an useful Lemma is introduced for deriving the main results as follows.
Lemma 1: [37] Let P T 1 = P 1 , P 2 and P 3 be real matrices of appropriate dimensions with F satisfying F T F ≤ I. Then if and only if there exists a positive scalar > 0 such that

LINEAR DISTRIBUTED FUSION ESTIMATION ALGORITHM DESIGN
For the linear target motion process (13), based on the designed local estimator (14) and fusion criterion (15), the local estimator gain K l i (t) and weighting fusion matrix W l i (t) will be given in Theorem 1. Define (16) Theorem 1: Each local estimator gain K l i (t) for linear systems (13) and (7) can be determined by solving the following convex optimization problem: Ci (t) and Γ K Ci (t) have been defined in (16). In this case, the local estimation error (LEE)x l i (t) will be bounded, that is lim where γ i > 0 is a scalar. Next, the distributed weighting fusion matrix W l i (t) can be solved by the following convex optimization problem: (16). Proof : Define the local estimation errorx l Then, from (13) and (14), one has where A K Ci (t) and Γ K Ci (t) are defined in (16). Next, in order to construct an upper bound of To ensure the right term of (21) can be viewed as the upper bound of [x l i (t)] Txl i (t), the condition J i (t) < 0 must be held. Therefore, by using the Schur complement lemma [37], the first inequality in (17) is equivalent to J i (t) < 0. Furthermore, when the second and third inequalities in (17) are satisfied, it can be obtained from the similar derivation in [31, Th.1] that the bounded condition (18) of local estimator holds. Let Then, it can be concluded that Similarly, the right term of (25) also can be taken as an upper bound. Thereby, the " min(Tr{Υ i (t)}+Tr{Ξ i (t)})" is selected as an optimization objective to minimize this upper bound, and then the optimization problem (17) is formulated to determine the K l i (t) based on the designed stable estimator (14).
Next, letx l (t) x(t)−x l (t) be the fusion estimation error (FEE), it follows from (13) and (15) that Then, based on the LEE (20) and FEE (26), an fusion error system is formulated to determine each distributed weighting fusion matrix W l i (t) as follows have been defined in (16).
Remark 3: By using the idea of bounded recursive optimization approach, the convex optimization problems (17) and (19) are established in terms of linear matrix inequalities (LMIs), which can be solved by using the function "mincx" of MATLAB LMI Toolbox [37]. Moreover, when considering the linear LSEsx l i (t)(i = 1, 2, . . . , m) determined by the stable estimators (14), then the designed fusion estimator (15) is stable, that is, the SE of the DFE for the linear systems (13) and (7) is bounded at each instant. VOLUME 4, 2016

B. NONLINEAR DISTRIBUTED FUSION ESTIMATION ALGORITHM DESIGN
When considering the nonlinear target dynamic process (1), according to the proposed nonlinear local estimator (11) and fusion criterion (12), the nonlinear local estimator gain K n i (t) and weighting fusion matrix W n i (t) are given in Theorem 2. Define Theorem 2: Each nonlinear local estimator gain K n i (t) for nonlinear systems (1) and (7) can be determined by solving the following convex optimization problem: Tr{Ψ i (t)} + Tr{Φ i (t)} s.t. : where O I i (t) and ∆ i (t) have been defined in (30). In this case, the nonlinear LEEx n i (t) will be bounded, that is where α i > 0 is a scalar. Then, the distributed weighting fusion matrix W n i (t) can be solved by the following convex optimization problem: min Π(t)>0,Ψ(t)>0,Φ(t)>0, (t) Π 1 (t), Π 2 (t), Ψ 1 (t), W(t) Tr{Π(t)} + Tr{Ψ(t)} + Tr{Φ(t)} s.t. : where O I (t), ∆(t), E N W (t) are defined by:  (1) and (11) Then, by using the Taylor expansion to linearize the nonlinear represents the high order-terms of Taylor series expansion.
In order to further analyze the stability of the designed estimators, the linearization errors O f ([x n i (t − 1)] 2 ) in (36) can be modeled by state-dependent scaling matrix with uncertain matrix [36], one obtains where E ni (t) is the state-dependent scaling matrix, and F ni (t) is a unknown matrix with norm constraint, i.e.
Then, substituting (36) and (37) into (35) yields Ei (t) and B N Ki (t) have been defined in (30). Since (38) has a similar form of (20), thus to ensure [x n i (t)] Txn i (t) has an upper bound, the following inequality is held from the similar derivation in Theorem 1:  Then, it can be derived from Lemma 1 that the first inequality in (31) holds. Moreover, the optimization problems (31) and (33) can be formulated by using the similar proof in [35, Th.1,Th.2], the detailed derivation is omitted here. Based on Theorem 2, the computation procedures for the nonlinear LSEsx n i (t)(i = 1, · · · , m) and the DFEx n (t) can be summarized in Algorithm 2.

Algorithm 2: Distributed Fusion Estimation Based on
Range-only Measurement for Nonlinear Target Dynamic.

Remark 4:
Though the range-only measurements can using a nonlinear transformation to avoid the linearization errors, it cannot be used for the nonlinear state model (1). In this case, the Taylor expansion has been used in [31] to linearize nonlinear function f (x), while the higher-order terms were modeled by bounded noises, thus the linearization errors were still be ignored and the stability of the estimator was not discussed as well. In fact, the estimation algorithm performance will be affected by the linearization errors, thus the (36) is modeled by a state-dependent matrix with unknown bounded matrix, which can further to analyze the characteristic of the linearization errors. Then, the stability of the designed nonlinear local/fusion estimators can be guaranteed, and the estimation performance also can be further improved.
Remark 5: Notice that the computational complexity of Algorithm 1 and Algorithm 2 depends on solving different convex optimization problems, while these optimization problems on the design of the local estimators and distributed fusion criteria are established in terms of linear matrix inequalities (LMIs). In this case, the calculation amount of the proposed algorithms are mainly determined by following aspects: one is the dimension of the LMIs, and the other is the number of introduced parameters for these optimization problems. For example, the dimension of the first LMI in (17) is D l = 2n+r+q i and the number of the introduced unknown parameters is S l = n 2 + r 2 + q 2 i , then the computational complexity of optimization problem (17) can be assumed to be O(D 2 l S l ). Consequently, dwindling the dimension of the LMIs and decreasing the number of the introduced unknown parameters are feasible schemes to reduce the search space for these optimization problems, then the solution time can be shortened. Moreover, the computational time will also have a significant impact on the online running of Algorithm 1 and Algorithm 2, while the calculating speed can be improved with the rapid development of computer software and hardware technology.

IV. SIMULATION EXAMPLES
In this section, two kinds of target tracking systems are presented to show the effectiveness and advantages of the proposed fusion estimation algorithms.

A. LINEAR TARGET MOTION SYSTEM
Consider a single target tracking example in X-Y planar, where only the distance information can be collected by each sensor. Generally, the position and velocity of the target are taken as the state, i.e. x(t) col{x p (t),ẋ p (t), y p (t),ẏ p (t)}, and the motion process of the target can be modeled by (13), where the specific parameters are given as follows [39] where angular velocity ω is taken as π/15 rad, and sampling period T 0 is given by 0.5. It is assumed that 12 sensors are installed to obtain the measurements, which can be di-  (42) where κ (t)(∈ [0, 1])(κ = 1, 2) and ij (t)(∈ [0, 1])(i = 1, 2, 3, j = 1, 2, 3, 4) are random variables. The initial state is assumed to be known, which is taken as x(0) =x l i (0) = [10, 0.5, 10, 0.5] T .
By implementing the range-only measurement fusion estimation method in Algorithm 1, the tracking effect of the local estimators and fusion estimator are drawn in Fig. 2. It is seen from the subfigure (a-c) of Fig. 2 that each local estimator can tracking the target's motion trajectory well, and subfigure (d) shows the distributed fusion estimator also tracking the target's trajectory well, which indicate that the proposed fusion estimation Algorithm 1 performing a effective tracking performance, that is the designed local estimator (14) and fusion estimator (15) in this paper are useful. Moreover, the mean square error (MSE) is employed to further assess the tracking capabilities, which is defined by wherex j (t) denotes the corresponding local or fusion estimates at time t, and N is the number of Monte Carlo simulations. Then, the MSEs over an average of 100 runs of Monte Carlo method [9] for local estimators (14) and fusion estimator (15) are drawn in Fig. 3 (a), respectively. It can be found that the MSE of the linear fusion estimator is smaller than each local estimator, which shows that the tracking capabilities can be further improved by the designed fusion estimation algorithm. This is because for the distributed fusion estimator (15), a optimization problem (19) is established with respect to each local estimation, and then a better tracking performance can be obtained based on the local estimates. Moreover, the stability condition of the designed estimators has been addressed in the theoretical analysis section, then it can be concluded from Theorem 1 that (18) holds when the optimization problem (17) can be solvable at each time. In this case, the following condition must be held: which has been verified by Fig. 3 (b), and then the stability of the local estimators can be guaranteed. Since the distributed fusion estimator is a linear combination of local estimators, thus the designed fusion estimator (15) is also stable.   To further show the advantages of the Algorithm 1 for tracking the motion target, the tracking performance of proposed local estimation algorithm, EKF [14], UKF [17] and CKF [18] methods are presented in Fig. 4. It is seen that the MSE of local estimator (14) is smaller than EKF, UKF and CKF methods, which implies that the developed local estimators in this paper have an better tracking capabilities. The main reason for this situation is that the statistical characteristics of noises are difficult to be accurately obtained in the target motion process, while the normal filtering methods are required a prior information of the noises. Thus, for the unknown bounded noises (42), the designed tracking algorithm with independent of the noise statistical information has a better tracking performance, and it also shows the advantages of the proposed method in this paper.   [14], UKF [17] and CKF [18].

B. NONLINEAR INTELLIGENT VEHICLE TRACKING SYSTEM
In this example, consider a nonlinear intelligent vehicle tracking system, where also only the distance information from the vehicle to sensors can be obtained. When taking the vehicle's state as x(t) col {x p (t), y p (t), θ(t)}, where x p (t) and y p (t) are vehicle's position in the X and Y axis of plane coordinates, respectively, and θ(t) is the angular orientation. Then the motion trajectory of the vehicle can be modeled by (1), and the parameters can be set as [38]: (45) where t 0 = 0.5 is the sampling period, w(t) is the process noise, c t = 0.2 and c r = 0.8 can the control translational and rotational velocity, respectively. Similarly, there are 12 sensors that are divided into 3 groups to obtain the measurements, and the position of sensors are taken as (x 11 , y 11 ) = (5,20), (x 12 , y 12 ) = (9, 23), (x 13 , y 13 where ρ κ (t)(∈ [0, 1])(κ = 1, 2) are random variables, and v i,qi are given in (42). Meanwhile, the initial state is given by x(0) =x n i (0) = [0.1, 0.05, 0.08] T . In addition, the matrix A ni (t − 1) can be calculated by Then, implementing the fusion estimation Algorithm 2, the tracking effect of the local estimators and fusion estimator are drawn in Fig. 5. As shown in subfigure (a-d) that all nonlinear local estimators and distributed nonlinear fusion estimator can tracking the target's motion trajectory well, it is verified that the proposed nonlinear estimation Algorithm 2 addressing a good tracking ability, and the designed nonlinear estimators (11) and (12) are also effective. Then, the MSEs of local estimators (11) and fusion estimator (12) with 100 Monte Carlo runs are performed in Fig. 6 (a), respectively. It shows that the tracking capability of the nonlinear DFE is better than that of each LSE, thus the tracking performance of the target tracking system can be further improved by the fusion estimation Algorithm 2. Particularly, Fig. 6 (b) shows the condition A N Ei (t) 2 < 1 holds, when the optimization problem (31) is solvable. Then, it can be concluded from Theorem 2 that the stability of local/fusion nonlinear estimators is held. Notice that, the linearization errors of the f (x(t)) in (45) are not easily judged as bounded errors, thus the stability condition (C.2) of [34, Th.2] is useless for this example. However, this paper modeling the linearization errors by the state-dependent scaling matrices with unknown bounded matrices, which can modified the stability conditions of [  Similarly, when considering the unknown bounded noises (42) and (46), the MSEs of local estimation algorithm proposed in Algorithm 2, EKF, UKF and CKF methods are plotted in Fig. 7, respectively. From Fig. 7, it shows that the VOLUME 4, 2016 proposed local nonlinear estimator (11) in this article has a better tracking performance than that of EKF, UKF and CKF as well. This is because these nonlinear tracking methods usually require to know the covariances of Gaussian noises, but the statistical properties of noises in (42) and (46) are still without knowledge. Therefore, it follows from the analysis of previous example that the developed nonlinear tracking Algorithm 2 is more suitable for practical tracking system. (a-c) The comparison of tracking performance among the nonlinear estimation Algorithm 2, EKF [14], UKF [17] and CKF [18].

V. CONCLUSION
In this article, a distributed fusion tracking problem with range-only measurement has been investigated. For the nonlinear distance measurement model, which is converted to a linear model in regard to the position of the target by using a nonlinear transformation. When considering the nonlinear motion dynamic of the target, a uncertainty bounded matrix combined with sate-dependent matrix were introduced to describe the state linearization error caused by the Taylor expansion. Though the measurement noises are more complicated and the uncertainty parameters are introduced by the corresponding nonlinear transformation, a robust designed approach based on the bounded recursive optimization scheme was developed in this paper, which can avoided the instability of nonlinear estimator. Then, the stable local estimator gains and fusion criterion have been determined by the constructed convex optimization problems. Finally, two range-only target tracking examples were addressed to verify the effectiveness of the proposed tracking algorithms. Furthermore, the target tracking and location problems over the WSNs are also give a significant research direction. Particularly, the time-delay, packet-dropouts, resource constraints and cyber-attacks problems in the networked structure, which are widely existed. In this case, based on the developed dimensionality reduction approach, quantization approach and event-triggered scheme, how to design the stable estimator for WSNs-based target tracking systems will be our future works.