Hybrid Unified Kalman Tracking Algorithms for
Heterogeneous Wireless Location Systems
Cheng-Tse Chiang, Po-Hsuan Tseng, Student Member, IEEE, and Kai-Ten Feng, Member, IEEE
Abstract—Location estimation and tracking for mobile stations
have attracted a significant amount of attention in recent years. Different types of signal sources are considered available to pro-vide measurement inputs for location estimation and tracking in heterogeneous wireless networks. Various techniques have been studied and combined for location tracking, e.g., the least square methods for location estimation associated with the Kalman filters for location tracking. In this paper, the hybrid unified Kalman tracking (HUKT) technique is proposed to provide an integrated algorithm for precise location tracking based on both time of ar-rival (TOA) and time difference of arar-rival (TDOA) measurements. A new variable is incorporated as an additional state within the Kalman filtering formulation to consider the nonlinear behavior in the measurement update process. The relationship between this new variable and the desired location estimate is applied in the state update process of the Kalman filter. Three different designs of hybrid factor are proposed to adaptively adjust the weighting value between the TOA and TDOA measurements. Moreover, similar concepts are also utilized in the design of unified Kalman tracking schemes for pure TOA and TDOA measurement inputs in this paper. Compared with existing schemes, numerical results illustrate that the proposed HUKT algorithm can achieve en-hanced accuracy for mobile location tracking, particularly under environments with an insufficient number of measurements in one of the signal paths.
Index Terms—Kalman filter, mobile location estimation and
tracking, time difference of arrival (TDOA), time of arrival (TOA).
I. INTRODUCTION
W
IRELESS location technologies, which are designed to estimate and track the position of a mobile station (MS), have drawn a lot of attention over the past few decades. Self-navigation and target tracking are the two main applications. With the acquisition of the MS’s location information,differ-Manuscript received May 29, 2011; revised October 10, 2011; accepted November 18, 2011. Date of publication December 21, 2011; date of current version February 21, 2012. This work was supported in part by the Aiming for the Top University and Elite Research Center Development Plan under Grant NSC 99-2628-E-009-005, Grant NSC 2221-E-009-065, and Grant NSC 98-2917-I-009-110, by the MediaTek Research Center at National Chiao Tung University, and by the Telecommunication Laboratories at Chunghwa Telecom Co. Ltd., Taiwan. This paper was presented in part at the IEEE 21st Interna-tional Symposium on Personal, Indoor, and Mobile Radio Communications, Istanbul, Turkey, September 2010. The review of this paper was coordinated by Prof. Dr. Y. Gao.
C.-T. Chiang and P.-H. Tseng were with the Department of Elec-trical Engineering and Control Engineering, National Chiao Tung Uni-versity, Hsinchu 30010, Taiwan (e-mail: henrychiang.cm97g@nctu.edu.tw; walker.cm90@nctu.edu.tw).
K.-T. Feng is with the Department of Electrical Engineering and Control Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: ktfeng@mail.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2011.2180939
ent types of location-based services (LBSs) can be explored, including enhanced 911 (E-911) subscriber safety services [2], location-based billing, navigation systems, and applications for intelligent transportation systems [3]. Due to the emergent interests in LBSs, it is required to provide enhanced precision in location estimation and tracking of the MS under different types of environments.
A variety of localization techniques have been investigated and proposed in wireless standards [4]. Network-based location estimation schemes are widely employed in wireless communi-cation systems. These schemes locate the position of an MS based on the measured radio signals from either its neighbor-hood base stations (BSs) in cellular networks or anchor nodes in wireless sensor networks (WSNs) [5], [6]. For convenience, these signal sources are represented as BSs in this paper. The location estimation algorithms can be categorized into range-free and range-based techniques. The range-range-free schemes [7]– [9] utilized the status of network connectivity between MS and BSs for localization, which possesses the benefits of simplicity and low cost. These schemes are primarily adopted in WSNs with the features of limited computation power and less require-ment on positioning accuracy. On the other hand, to provide precise location estimation, range-based schemes are consid-ered, which include received signal strength (RSS) [10], angle of arrival (AOA) [11], time of arrival (TOA) [12], and time difference of arrival (TDOA) [13]. The RSS schemes record the incoming signal strength from different wireless BSs for converting to distance measurement, and the AOA methods are in general implemented at the BSs to observe the signal bearing via the antenna array. The TOA schemes measure the arrival time of the radio signals coming from the BSs, whereas the TDOA algorithms measure the time difference between the radio signals.
One of the important issues for range-based positioning is its inherent nonlinear feature for location estimation, which results in complex computation and difficulties for analysis. Recursive Bayesian estimation [14]–[16] computes the poste-rior probability density function of the state variables based on both the incoming measurement inputs and the Markov state model recursively over time. With the estimated posterior probability density function, either the minimum mean square error (MMSE) or the maximum a posteriori estimation can be calculated. The Kalman filter is one of the simplest methods for Bayesian estimation and is proved to be an optimal realization of MMSE under the linear model perturbed by Gaussian noises [14]. Although the signal and noise are not jointly Gaussian, it is still considered an optimal linear MMSE (LMMSE) esti-mator. Moreover, the operation of the Kalman filter [16]–[18]
Fig. 1. (Left) HCLT scheme. (Middle) HKT scheme. (Right) Proposed HUKT scheme.
is linear, which offers efficient computation for real-time im-plementation. In [19] and [20], the Kalman filer has been extensively utilized to further enhance the precision for location estimation. It provides the estimation of internal states with dynamic weighting adjustment between the prediction and the observation input in recursion form. This feature alleviates the estimation outputs from severe signal variations and can finally converge to the true value. Several researches have adopted the Kalman filter to track the MS’s position with considerations of the nonline-of-sight (NLOS) interferences [21] and the mobility information of moving MS [22], [23]. Compared with the methods for stationary location estimation, these tracking schemes take advantage of the previous location and movement of the MS, which can result in a smoothed MS trajectory with better estimation accuracy.
On the other hand, owing to the feasibility of provid-ing synchronization between cellular BSs, the TDOA mea-surements have been extensively adopted for location esti-mation and tracking in existing telecommunication systems, e.g., the WiMax [24] standard. However, in urban canyons problem, it has been observed that the number of received Global Positioning System or cellular signals is insufficient for location estimation due to signal blockage in urban environ-ment. Moreover, the study in [25] suggests the adoption of TOA-based signal sources for dedicated short-range commu-nications (DSRC) to avoid the complex infrastructure required for TDOA measurements. To provide feasible precision for location estimation, it is sensible to combine these two types of signal sources under a variety of environments, e.g., to addi-tionally include TOA-based sensor anchors or roadside DSRC devices with TDOA-based cellular signal sources. Therefore, it will be beneficial to design a hybrid technique that can facilitate location estimation and tracking based on these two types of measurement inputs. Moreover, the performance of the location estimation schemes varies depending on the environmental conditions and the operational parameters. The Cramer–Rao lower bound [14] associated with the geometric dilution of precision (GDOP) [26] is utilized as the theoretical limitation on estimation variance to provide a benchmark for comparison between different estimators. Previous works [22], [27], [28]
have been dedicated to combining multiple location techniques for enhanced positioning precision with the theoretical lower bound derived in [27].
In the location tracking problem, the relationship between the measurement and the state variable is observed to be non-linear. Since the computation of recursive Bayesian estimation requires several integrals without analytic solutions, it is not possible to obtain an optimal solution to this problem in prac-tice. The particle filter [29] has been proposed as a nonlinear/ non-Gaussian method for this type of problem. However, the number of particles to establish the probability distribution function should be infinite to achieve the optimal solution. Therefore, in the location tracking problem, there are several methodologies for obtaining the suboptimal solution to deal with the nonlinear relationship between the distance measure-ment and the estimated position. As shown in the left plot of Fig. 1, the hybrid cascade location tracking (HCLT) scheme proposed in [22] utilizes the two-step LS method [12], [13] for initial location estimation of the MS. The two-stage architecture handles the nonlinear relation in the location estimator. There-fore, the Kalman filtering technique is adopted to smooth out the estimation error by tracking the positions and velocities of the MS. The fusion algorithm is utilized to combine the tracking results from two different sources to obtain the final location estimation of the MS. In the middle plot of Fig. 1, the hybrid Kalman tracking (HKT) scheme extends the Kalman tracking (KT) scheme in [23] by separating the linear components from the originally nonlinear equations for location tracking. The linear aspect is directly processed within the Kalman filtering formulation, whereas the nonlinear term is served as an external measurement input to the Kalman filter. However, both HCLT and HKT algorithms have the drawback of additional hardware cost due to their cascaded infrastructures. The Kalman filter is only utilized to deal with the linear behaviors of location tracking problem by adopting these two types of architectures. The nonlinear terms are considered outside of the Kalman filter by performing LS linearization technique, which can result in information loss and cause larger location tracking errors. This type of structure can result in information loss, which causes larger location tracking errors. Moreover, both algorithms
require sufficient numbers of signal sources from either the TOA or the TDOA path, which cannot resolve the signal insufficiency problem in urban canyons.
In this paper, a hybrid unified KT (HUKT) algorithm is proposed based on both TOA and TDOA signal inputs. Owing to the benefit of a linear relationship between the measurement and estimated states, the Kalman filter is adopted to provide computational efficiency for real-time implementation. As il-lustrated in the right plot of Fig. 1, the HUKT scheme integrates the nonlinear relation into the Kalman filtering formulation for location tracking based on both TOA and TDOA signal sources from heterogeneous BSs. The major design novelty of the HUKT scheme is that the nonlinear parameters within their respective TOA- and TDOA-based location estimators are mathematically combined into a single state variable, which is to be updated within the Kalman filter. This type of unified architecture for location tracking problem with heterogeneous signal inputs has not been proposed in previous works. In the measurement update of the Kalman filter, the nonlinear parameter is utilized to linearize the measurement equation by assigning all the nonlinear terms into an extra state variable. The constraint between this extra variable and the estimated position is further considered in the state update process of the Kalman filter. The proposed HUKT scheme is feasible to be adopted under environments with heterogeneous signal sources and is tolerant to an insufficient number of BSs from individ-ual signal paths. The determination of the hybrid factor that combines the TOA and TDOA signal sources is investigated based on different criterions. Furthermore, the proposed HUKT algorithm can directly be simplified into a unified KT (UKT) scheme for location tracking under the situation with only homogeneous signal sources, i.e., either the TOA or TDOA measurement input is available. Performance evaluation and comparison of the proposed HUKT and UKT schemes are conducted via simulations. Compared with existing schemes, simulation results show that the HUKT/UKT algorithm can achieve higher accuracy for location estimation and tracking.
The remainder of this paper is organized as follows. The mathematical modeling of signal sources and the existing location tracking techniques are summarized in Section II. Sections III and IV describe the proposed HUKT algorithm and the simplified UKT scheme, respectively. Performance evalua-tion and comparison of the proposed schemes are conducted in Section V via simulations. Section VI draws our conclusions.
II. SYSTEMMODELING ANDEXISTINGLOCATION
TRACKINGSCHEMES
A. Mathematical Modeling of Signal Inputs
In this section, the mathematical models for both TOA and TDOA measurements are presented. The 2-D coordinate of the MS is to be obtained in the proposed HUKT scheme. The TOA measured distance ri,k between the MS and the ith BS at the kth time step can be represented as
ri,k = c· ti,k= ζi,k+ ni,k+ ei,k i = 1, 2, . . . , N (1)
where ti,kdenotes the TOA measurement with respect to the ith
BS at the kth time step, and c is the speed of light. The measured
Fig. 2. Schematic diagram of Kalman filter.
distance ri,kis corrupted by both the measurement noises ni,k
and the NLOS error ei,k under urban and suburban areas. The
parameter N refers to the total number of TOA measurements. The noiseless distance ζi,kin (1) is
ζi,k=
(xk− xi,k)2+ (yk− yi,k)2
1/2
(2) where (xk, yk) represents the MS’s true position, and
(xi,k, yi,k) is the coordinate of the ith BS at time step k. Based
on the preceding TOA signal model, the TDOA measurement can be formulated as the subtraction of two TOA measure-ments, which conforms to the physical meaning of difference in propagation time. The relative distance ˜rjm,k1can be obtained
by computing the TDOA measurement ˜tjm,k, which is the time
difference between the MS with respect to the jth and mth BSs from (1) as
˜
rjm,k= c· ˜tjm,k= (˜ζj,k− ˜ζm,k) + (˜nj,k− ˜nm,k)
+ (˜ej,k− ˜em,k) m = 1; j = 2, . . . , ˜N . (3)
Note that the first BS of the TDOA system is in general denoted as the reference BS, e.g., the serving BS in cellular networks. The TDOA measurements are taken between the reference BS and the other neighbor BSs. The parameter ˜N is the number of BSs for the TDOA system, which comprises ( ˜N− 1) independent TDOA measurements.
B. Kalman Filter
The Kalman filter [18], [30], which is derived based on the Markov chain perturbed by Gaussian noises, is an efficient Bayesian estimator to solve the linear problem. Fig. 2 illustrates the concept of the Kalman filter that consists of measurement and state updates. With prior information coming from the state update and likelihood information from the measurement update, the Kalman filter will obtain a posterior estimate in the MMSE sense. Even with non-Gaussian noises, the LMMSE solution can still be acquired by adopting the Kalman filter. The
1In the paper, it is considered that the TDOA and the TOA measurements come from two different types of networks. For notational convenience, the variables with a tilde are denoted for the measurements from TDOA system, e.g., ˜rjm,k; while the variables without the tilde (e.g., ri,k) are utilized for
measurement and state equations for the Kalman filter can be represented as
yk = Mkxˆk+ mk (4)
ˆ
xk = Fkxkˆ −1+ uk−1+ pk−1 (5)
where ˆxk represents the estimated state/output, ykdenotes the
measurement input of the Kalman filter, and uk−1 indicates the control input for the state model. In the location tracking problem, the states that are of interest include the MS’s position, velocity, and acceleration. The matrices Mk and Fk refer to
the linear relations for the measurement and state models, respectively. The variables mk and pk−1 respectively denote
the measurement and the processing noises associated with the covariance matrices Rk and Qk within the Kalman filter
formulation. Based on the measurement and state equations as in (4) and (5), the Kalman filter estimates the states during the prediction phase in Fig. 2 as ˆx−k = Fkxk−1ˆ + uk−1 with
its estimate covariance C−k = FkCk−1FTk + Qk. On the other
hand, within the correction phase, the measurement input will be corrected via innovation process as ˜yk = yk− Mkxˆ−k with
innovation covariance ˜Ck= MkC−kMTk + Rk. The optimal
Kalman gain can be obtained as Kk = C−kMTk[ ˜Ck]−1.
There-fore, the corrected state estimate will be acquired as ˆxk= ˆx−k +
Kkxk˜ associated with the corrected estimate covariance Ck =
(I− KkMk)C−k. Based on the foregoing linear operations,
the Kalman filter can efficiently update the state estimates at different time instants.
C. HCLT Scheme
The left plot of Fig. 1 illustrates the architecture of the HCLT scheme [22]. The HCLT system consists of an LS location estimator, e.g., the two-step LS method, followed by a Kalman filtering technique at the next stage. Different versions of two-step LS methods have been proposed for distinct occasions, such as TOA [12], TDOA [13], and TDOA/AOA [31] mea-surement inputs. The concept of the two-step LS method is to acquire an intermediate location estimate in the first step with the definition of a new variable to represent the nonlinear term, which is mathematically related to the MS’s position. This assumption effectively transforms the nonlinear equations for location estimation into a set of linear equations, which can directly be solved by the LS method. The second step of the method primarily considers the fact that the newly defined vari-able is related to the MS position, which was originally assumed to be uncorrelated in the first step. An improved location esti-mate can be obtained after the adjustment from the second step. The MS’s estimated position from the output of the two-step LS estimator will be postprocessed by the Kalman filtering technique according to [17]. The Kalman filter smoothes out and tracks the estimation errors by adopting linear prediction from the previous estimation data while the MS is dynamically moving in the network. According to the Bayesian inference model [15], [32], the tracking results from the two disparate TOA and TDOA paths will be combined by the fusion mecha-nism based on their corresponding signal variations. The MS’s estimated position, i.e., (ˆxk, ˆyk), can therefore be acquired. The
detail algorithm of the HCLT scheme can be found in [22].
D. HKT Scheme
Since the equations associated with the network-based loca-tion estimaloca-tion are inherently nonlinear, different mechanisms, e.g., linearization, are utilized within the existing algorithms for location tracking. The KT scheme [23] considers the nonlinear term as an external measurement input to its Kalman filtering formulation. It distinguishes the linear part from the original nonlinear equations for location estimation and tracking. How-ever, the KT scheme does not specifically indicate the method for acquiring the nonlinear term. For comparison purpose, the KT scheme that was originally proposed based on the TDOA measurement inputs is reformulated and extended in this paper to consider both TOA and TDOA signal sources. The middle plot of Fig. 1 illustrates the architecture of the HKT scheme. The nonlinear terms can be obtained from external location estimators, e.g., by adopting the two-step LS method. With the formulation of the HKT scheme, a feasible accuracy can be acquired for location tracking, including position, velocity, and acceleration of the MS. However, the accuracy is significantly affected by the precision of the external location estimator. The detailed algorithm of the KT scheme can be found in [23].
III. PROPOSEDHYBRIDUNIFIEDKALMAN
TRACKINGSCHEME
The proposed HUKT scheme will be described in this sec-tion. The formulation of the HUKT algorithm is explained in Section III-A, and the determination of the hybrid factor βkat
time step k will be discussed in Section III-B. The variable βk
will be determined from three different approaches to address the various weighting factors between the TOA and TDOA measurements for the HUKT scheme.
A. Formulation of HUKT Algorithm
The right plot of Fig. 1 illustrates the architecture of the proposed HUKT scheme. Unlike the previous algorithms, e.g., the HCLT and HKT methods, the main design concept of the HUKT scheme is to provide a unified methodology for location estimation and tracking. The purpose of the HUKT algorithm is to obtain the updated state variables via the Kalman filtering technique directly from both TOA and TDOA measurements as the system inputs. The measurement update and the state update equations of the Kalman filter can be respectively acquired from (4) and (5), where ˆxk= [ˆxkyˆkˆkvˆx,kˆvy,kaˆx,kˆay,k]T is the
state vector that includes the MS’s estimated position (ˆxk, ˆyk),
the estimated velocity (ˆvx,k, ˆvy,k), the estimated acceleration
(ˆax,k, ˆay,k), and the estimated variable ˆk. Note that ˆk
rep-resents the estimated nonlinear term for the hybrid location estimation. The updating process of ˆkwill be addressed later.
To formulate the input/output relationship for Kalman filter based location tracking, error-free measurements will first be examined, i.e., ri,k= ζi,k. The following equation for TOA
measurement can be obtained by combining (1) and (2) as ri,k2 − Ki,k =−2xi,kxk− 2yi,kyk+ Rk (6)
where Ki,k= x2i,k+ y2i,k, and Rk= x2k+ yk2. Similarly, the
measurements (3) by substituting m = 1 as ˜
rj1,k2 − ( ˜Kj,k− ˜K1,k)
=−2(˜xj,k− ˜x1,k)xk− 2(˜yj,k− ˜y1,k)yk− 2˜r1,k˜rj1,k (7)
where ˜r1,k indicates the measured distance from the MS to
the reference BS via the TDOA system. To design a unified structure for location tracking, the purpose of the proposed HUKT scheme is to obtain an effective method to combine both TOA and TDOA measurements. More specifically, a new variable ˆk is introduced to combine the nonlinear terms Rk
in (6) and ˜r1,kin (7). Without loss of generality, the nonlinear
term ˜r1,kin (7) can be represented as
x2
k+ y2kby shifting the
entire coordinate, i.e., both TOA and TDOA systems, such that (˜x1,k, ˜y1,k) = (0, 0). With the definition of a hybrid factor βk,
the following relationship can be obtained by multiplying (7) with βk/˜rj1,kand adding to (6) as
r2i,k− Ki,k+ βkr˜j1,k− βk ˜ Kj,k− ˜K1,k ˜ rj1,k + βk2 =k− 2 xi,k+ βk ˜ xj,k− ˜x1,k ˜ rj1,k xk − 2 yi,k+ βk ˜ yj,k− ˜y1,k ˜ rj1,k yk (8) where k = ( x2
k+ y2k− βk)2 corresponds to the variable
that combines the effects from both TOA and TDOA measure-ments. It is included in the state vector ˆxk for state updating
within the Kalman filtering formulation. The hybrid factor βkis
considered as a weighting between the TOA- and TDOA-based measurements, which can be determined according to the signal qualities of the two different paths. The detail design and selection for the value of βk will be addressed later in the
next section. Note that (8) is obtained as a linear combination from (6) and (7). Since the Kalman filter is well known for its linear operations, it is not required to apply scaling factors to achieve this combination from two different types of signal sources. As a result, the measurement data yk and the matrix
Mk associated with the measurement process in (4) can be
acquired as in (9), shown at the bottom of the page. Note that there are (N + ˜N− 2) linearly independent equations associated with both ykand Mk. There are N hybrid equations
formed by all the TOA measurements, i.e., from r1,k to rN,k,
and the first TDOA measurement ˜r21,k. The remaining ˜N− 2
hybrid equations are established by using the first TOA mea-surement, i.e., r1,k, and the remaining TDOA measurements,
i.e., from ˜r31,k to ˜rN 1,k˜ . Under the assumption of constant
acceleration model, the updating process of ˆxkand ˆykis
deter-mined as ˆ xk= ˆxk−1+ ˆvx,k−1Δt + 1 2ˆax,k−1Δt 2 (10) ˆ yk= ˆyk−1+ ˆvy,k−1Δt + 1 2ˆay,k−1Δt 2 (11) where Δt denotes the sampling time interval. To provide the updating process for the new variable k, similar to (8), the
relation among ˆk, ˆxk, and ˆyk can be acquired by summing all
yk = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ r2 1,k− K1,k+ βk˜r21,k− βk ˜ K2,k− ˜K1,k ˜ r21,k + β 2 k r2,k2− K2,k+ βkr˜21,k− βk ˜ K2,k− ˜K1,k ˜ r21,k + β 2 k .. . rN,k2− KN,k+ βkr˜21,k− βk ˜ K2,k− ˜K1,k ˜ r21,k + β 2 k r2 1,k− K1,k+ βk˜r31,k− βk ˜ K3,k− ˜K1,k ˜ r31,k + β 2 k r2 1,k− K1,k+ βk˜r41,k− βk ˜ K4,k− ˜K1,k ˜ r41,k + β 2 k .. . r2 1,k− K1,k+ βkr˜N 1,k˜ − βk ˜ KN ,k˜ − ˜K1,k ˜ rN 1,k˜ + β 2 k ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Mk = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −2 x1,k+ βkx˜2,kr˜21,k−˜x1,k −2 y1,k+ βky˜2,k˜r21,k−˜y1,k 1 01×4 −2 x2,k+ βkx˜2,kr˜21,k−˜x1,k −2 y2,k+ βky˜2,k˜r21,k−˜y1,k 1 01×4 .. . −2 xN,k+ βkx˜2,kr˜21,k−˜x1,k −2 yN,k+ βky˜2,k˜r21,k−˜y1,k 1 01×4 −2 x1,k+ βkx˜3,kr˜31,k−˜x1,k −2 y1,k+ βky˜3,k˜r31,k−˜y1,k 1 01×4 −2 x1,k+ βkx˜4,kr˜41,k−˜x1,k −2 y1,k+ βky˜4,k˜r41,k−˜y1,k 1 01×4 .. . −2 x1,k+ βk ˜ xN ,k˜ −˜x1,k ˜ rN 1,k˜ −2 y1,k+ βk ˜ yN ,k˜ −˜y1,k ˜ rN 1,k˜ 1 01×4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (9)
N TOA measurements of (6) and ˜N− 1 TDOA measurements of (7) as ˆ k= Wk+ 2XS,k· ˆxk+ 2YS,k· ˆyk (12) where Wk = βk2+ 1 N N i=1 ri,k2 − N i=1 Ki,k +˜1 N j=2r˜j1,k · ⎡ ⎣βk ˜ N j=2 ˜ rj1,k2 − βk ˜ N j=2 ˜ Kj,k+ βk( ˜N− 1) ˜K1,k ⎤ ⎦ XS,k = N i=1xi,k N + βk N˜ j=2(˜xj,k− ˜x1,k) N˜ j=2r˜j1,k YS,k = N i=1yi,k N + βk N˜ j=2(˜yj,k− ˜y1,k) N˜ j=2r˜j1,k .
Following the methodology as in (10) and (11), the updating process for the estimated variable ˆkbecomes
ˆ
k = ˆk−1+ 2(XS,k− XS,k−1)ˆxk−1
+ 2(YS,k− YS,k−1)ˆyk−1+ 2XS,kvˆx,k−1Δt
+ 2YS,kvˆy,k−1Δt + XS,kaˆx,k−1Δt2
+ YS,kˆay,k−1Δt2+ (Wk− Wk−1). (13)
Finally, the state matrix Fkassociated within the state equation
in (5) for the proposed HUKT scheme can be obtained as in (14), shown at the bottom of the page. The control input uk−1
can also be acquired as
uk−1= [ 0 0 (Wk− Wk−1) 0 0 0 0 ]T. (15)
To summarize, the proposed HUKT scheme integrates the measurement inputs from heterogeneous location estimation systems based on a unified Kalman filtering structure. The iterative operations of the Kalman filtering technique primarily consist of processes for state update as prediction and measure-ment update as correction. The equations for state update are represented as
ˆ
x−k = Fkxk−1ˆ + uk−1 (16)
C−k = FkCk−1FTk + Qk. (17)
The equations for measurement update become
Kk= C−kMTk MkC−kMTk + RhTOA,k+ RhTDOA,k −1 (18) ˆ xk= ˆx−k + Kk yk− Mkxˆ−k (19) Ck= C−kKkMkC−k (20)
where Kk represents the Kalman gain, and the matrix Ck is
denoted as the estimate error covariance. The covariance matri-ces associated with the TOA and TDOA measurement update processes for hybrid estimation are respectively represented as RhTOA,k= BRTOA,kBT and RhTDOA,k = ˜BRTDOA,kB˜T,
where the matrices B and ˜B are established to fulfill the
requirement for matrix Mkin (9), i.e.,
B = [I]N×N [C]( ˜N−2)×N = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 · · · 0 0 1 · · · 0 .. . ... . .. ... 0 0 · · · 1 1 0 · · · 0 .. . 1 0 · · · 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ˜ B = [D](N−1)× ˜N [E]( ˜N−1)× ˜N = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −1 1 0 0 · · · 0 .. . −1 1 0 0 · · · 0 −1 0 1 0 · · · 0 −1 0 0 1 · · · 0 .. . ... ... ... . .. ... −1 0 0 0 · · · 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
The corresponding covariance matrices RTOA,kand RTDOA,k
can respectively be acquired as
RTOA,k= LkJh,kLk (21)
RTDOA,k= ˜LkJ˜h,kL˜k (22)
where Lk= diag{ζ1,k, ζ2,k, . . . , ζN,k}, Jh,k= diag{σ1,k2 , σ2
2,k, . . . , σ2N,k}, ˜Lk= diag{˜ζ1,k, ˜ζ2,k, . . . , ˜ζN ,k˜ }, and ˜Jh,k=
diag{˜σ2
1,k, ˜σ2,k2 , . . . , ˜σ2N ,k˜ }.
B. Determination of Hybrid Factorβk
As shown in (8), the hybrid factor βk at time step k is
utilized to provide the weighting between the TOA and TDOA measurements to merge these two types of inputs for hybrid location tracking. Therefore, it is essential to develop feasible
Fk = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 Δt 0 12Δt2 0 0 1 0 0 Δt 0 12Δt2 2(XS,k− XS,k−1) 2(YS,k− YS,k−1) 1 2XS,kΔt 2YS,kΔt XS,kΔt2 YS,kΔt2 0 0 0 1 0 Δt 0 0 0 0 0 1 0 Δt 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (14)
mechanisms that can dynamically adjust the hybrid factor in accordance with the variations of estimation quality in the two signal paths. Note that the sign of the weighting value, i.e., the hybrid factor βk, will not be influential based on the design of
hybrid system in (8), whereas its magnitude is considered cru-cial to affect the performance of the hybrid location estimation. With larger absolute value of βk, more weighting is assigned
to the TDOA signal compared with TOA measurement input. In the following three sections, different types of design for the hybrid factor will be presented.
GDOP-Based Hybrid Factor (GHF): The main concept for the design of GHF βg,k is to consider the geometric
relation-ship between the TOA and TDOA signal inputs. The GDOP [26] describes the geometry influence on location estimation accuracy. For a set of spatially separated BSs or sensors, the set of relative distances from the MS to its respective BSs affects the estimation accuracy for the MS’s position. In general, when the MS locates around the center of the BSs, the GDOP value is lower than the case that the MS is situated around the geometric edge formed by the BSs. Therefore, the GDOP criterion that provides the relative distance information between the MS and BSs can be utilized to determine the hybrid factor βk that
represents the weighting between the TOA and TDOA mea-surements. Considering the MS located at xk= (xk, yk) with
the TOA range measurements ri,k for i = 1 to N associated
with Gaussian noise, the GDOP value Gxk,TOAfor xk at time
step k can be obtained as Gxk,TOA= trace HTG,kJ−1G,kHG,k −11/2 (23) where JG,kis the same as Jh,kin (21), and
HG,k= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ xk−x1,k r1,k yk−y1,k r1,k xk−x2,k r2,k yk−y2,k r2,k .. . ... xk−xN,k rN,k yk−yN,k rN,k ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (24)
On the other hand, considering the TDOA case with the range difference measurements ˜rj1,k for j = 2 to ˜N , the formulation
for the GDOP value can be obtained as Gxk,TDOA= trace H˜TG,k˜J−1G,kH˜G,k −11/2 (25) where ˜ JG,k= ⎡ ⎢ ⎢ ⎢ ⎣ ˜ σ2 2+ ˜σ21 σ˜21 · · · σ˜21 ˜ σ2 1 σ˜23+ ˜σ21 · · · σ˜21 .. . ... . .. σ˜2 1 ˜ σ2 1 σ˜21 ˜σ12 σ˜2N˜+ ˜σ 2 1 ⎤ ⎥ ⎥ ⎥ ⎦ (26) ˜ HG,k= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x−˜x2 ˜ r2 − x−˜x1 ˜ r1 y−˜y2 ˜ r2 − y−˜y1 ˜ r1 x−˜x3 ˜ r3 − x−˜x1 ˜ r1 y−˜y3 ˜ r3 − y−˜y1 ˜ r1 .. . ... x−˜xN˜ ˜ rN˜ − x−˜x1 ˜ r1 y−˜yN˜ ˜ rN˜ − y−˜y1 ˜ r1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (27)
Consequently, the GHF βg,k that is designed to be the ratio
between the TOA and TDOA estimation systems can be for-mulated as
βg,k =
Gxk,TOA
Gxk,TDOA
· ˜r1,k. (28)
Note that the original TDOA equation in (7) is divided by ˜ri1,k
to formulate the hybrid formulation as in (8). Therefore, the multiplication of ˜r1,k in (28) is to scale back to the original
magnitude order of the TDOA measurements in (7). For com-putational simplicity, the value of ˜r1,kis utilized instead of the
original ˜ri1,kvalue. Furthermore, it is noted that both Gxk,TOA
and Gxk,TDOAare nonzero values, which result in a countable
value of βg,k. The case with zero GDOP value denotes that
there is no signal variance that is unlikely to happen in realistic estimation problems. On the other hand, when the MS is located exactly on the same location as one of the BSs, singularity will occur in the above matrix operations, which leads to undefined behavior between MS and BSs. Both situations of zero signal variance and matrix singularity will not be considered in this paper.
Minimum Variance-Based Hybrid Factor (MVHF): The main purpose of this scheme is to obtain the hybrid factor MVHF βm,k to achieve minimum variance for the hybrid
estimation system. From the formulation of the HUKT scheme as shown in (8), the hybrid measurement update equation is composed by the TOA measurement from the ith BS and the TDOA measurement via the jth BS and the serving BS. To facilitate the design of MVHF βm,k, an equivalent set of BSs
is defined as (xeq,k(i, j), yeq,k(i, j)) = (xi,k+ βm,k(˜xj,k−
˜
x1,k/˜rj1,k), yi,k+ βm,k(˜yj,k− ˜y1,k/˜rj1,k)) for i = 1 to N if j = 2, and j = 3 to ˜N if i = 1. Note that there are a total of N + ˜N− 2 sets of equivalent BSs. Therefore, the original hybrid measurement update in (8) can be rewritten as
βm,kr˜j1,k− βm,k ˜ Kj,k− ˜K1,k ˜ rj1,k + βm,k2 − Ki,k + [rk(i, j)]2
=−2xeq,k(i, j)xk− 2yeq,k(i, j)yk+k (29)
where rk(i, j) = ri,k for i = 1 to N if j = 2, and rk(i, j) = r1,kfor j = 3 to ˜N if i = 1. Note that (29) can be considered
as an extended formulation of the TOA measurements in (6). Therefore, it is implicitly suggested by (29) that there exists a set of equivalent BSs (xeq,k(i, j), yeq,k(i, j)) for each entry
of the hybrid measurement equation, where the equivalent BS is a composition of both TOA and TDOA BSs with the ratio βm,k, i.e., (xeq,k(i, j), yeq,k(i, j)) = (xi,k+ βm,k(˜xj,k−
˜
x1,k/˜rj1,k), yi,k+ βm,k(˜yj,k− ˜y1,k/˜rj1,k)).
As a result, the target of MVHF is to acquire an optimal β∗m,ksuch that the variance of the hybrid system can be mini-mized as βm,k∗ = arg min ∀βm,k∈R trace HTM,kJ−1M,kHM,k −11/2 (30)
where HM,k= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ xk−xeq,k(1,2) rk(1,2) yk−yeq,k(1,2) rk(1,2) xk−xeq,k(2,2) rk(2,2) yk−yeq,k(2,2) rk(2,2) .. . ... xk−xeq,k(i,j) rk(i,j)
yk−yeq,k(i,j)
rk(i,j) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (31)
with its rank equal to (N + ˜N− 2). The matrix JM,k=
Rh
TOA,k+ RhTDOA,k can directly be acquired based on the
composition of (21) and (22). Note that the minimization problem in (30) can be interpreted as to search for the variance lower bound for the hybrid tracking system. Moreover, it is recognized that the complicate optimization process in (30) for obtaining βm,k will not be feasible for real-time
imple-mentation. An alternative method is to perform the numerical search for each specific network layout. For a predetermined BS topology that can be computationally divided by small grids in region, the optimal values of βm,kfor each grid can be acquired
to construct the offline table. Based on the inherent tracking information within the Kalman filter, the predicted a priori knowledge of the MS’s position will be provided to obtain βm,k
based on table lookup for real-time implementation.
Kalman Filter-Based Hybrid Factor (KHF): As stated in Section III-B1, the design concept of GHF is straightforward, which determines the hybrid factor based on the GDOP values acquired from TOA and TDOA measurements. However, the characteristics of the hybrid structure for location tracking have not been considered in the design of the GHF value. On the other hand, the MVHF designed in Section III-B2 considers the variances of the proposed HUKT system to explore the optimal solution for the hybrid factor. Nevertheless, an approximated solution is obtained due to the complexity of solving the opti-mization problem in real-time implementation. In this section, the KHF βf,k is designed based on the dynamic adjustment
of Kalman filtering formulation within the proposed HUKT scheme. It is closely related to the prediction and updating features of the Kalman filter-based location tracking system.
Since the variable k consists of the hybrid factor and
is estimated along with other variables in the state vec-tor, the KHF βf,k can also be tracked to further enhance
the estimation performance under the presence of measure-ment errors. Considering the tracking process of the pro-posed HUKT scheme at the (k− 1)th time step, the posteriori estimation of the state vector can be acquired as ˆxk−1= [ˆxk−1yˆk−1ˆk−1ˆvx,k−1ˆvy,k−1ˆax,k−1aˆy,k−1]T. The KHF βf,k
at time step k can be predicted according to the definition of ˆ
k−1in (8) at the (k− 1)th time step as βf,k=
ˆ
x2k−1+ ˆyk2−11/2− (ˆk−1)1/2. (32) Note that the solution with minus sign is selected in (32) within its multiple solutions for computation simplicity since the sign of βf,k is not influential based on the original design
of the hybrid system in (8). The proposed KHF βf,k can be
implemented directly along with the real-time tracking process of the proposed HUKT scheme. In Section V, the performance of location tracking based on these three types of hybrid factor will be evaluated and compared via simulations.
IV. SIMPLIFIEDTIME OFARRIVAL-AND
TIMEDIFFERENCE OFARRIVAL-BASED
UNIFIEDKALMANTRACKINGSCHEMES
Considering environments with only a homogeneous type of signal inputs, the proposed HUKT algorithm can be simplified to the UKT scheme to support either TOA or TDOA measure-ments, i.e., the UKT-TOA and UKT-TDOA schemes. Note that the HUKT algorithm can be adopted under situations where there is insufficient number of measurements at one of the heterogeneous signal paths. With homogeneous signal sources, the MS and network operator that utilize either UKT-TOA or UKT-TDOA technique can have the flexibility to terminate the hybrid estimation mode to reduce computational complexity. In the next two sections, the formulations of both UKT-TOA and UKT-TDOA schemes will be described.
A. UKT-TOA Scheme
The formulation of the proposed HUKT algorithm can be reduced to the UKT-TOA scheme in the case that there only exists TOA measurements for location estimation and tracking. Based on the rearranged TOA measurements in (6), the same Kalman filter formulation as described in (4) and (5) can still be utilized for measurement and state updates, respectively, where the state vector becomes ˆxk= [ˆxkyˆkRˆkvˆx,kˆvy,kˆax,kaˆy,k]T.
Within the state vector, it can be observed that the original nonlinear term ˆk for the hybrid system is substituted into the
variable ˆRk = ˆx2k+ ˆy2k, which denotes the nonlinear variable
derived from pure TOA-based measurements. Therefore, the measurement data yk and the matrix Mk of the N TOA
measurements in the measurement update process become
yk = ⎡ ⎢ ⎢ ⎢ ⎣ r2 1,k− K1,k r22,k− K2,k .. . r2 N,k− KN,k ⎤ ⎥ ⎥ ⎥ ⎦ Mk = ⎡ ⎢ ⎢ ⎣ −2x1,k −2y1,k 1 01×4 −2x2,k −2y2,k 1 01×4 .. . −2xN,k −2yN,k 1 01×4 ⎤ ⎥ ⎥ ⎦ .
The covariance matrix RTOA,k associated with the
measure-ment equation in (4) is obtained from (21). Based on the same assumption of constant acceleration model, the state update process of ˆxk and ˆyk can still be acquired based on
(10) and (11). By summing up and rearranging all the N measurement equations, the following relationship can be ob-tained as ˆ Rk= WT ,k+ 2XT ,k· ˆxk+ 2YT ,k· ˆyk (33) where WT ,k = N i=1 r2i,k− N i=1 Ki,k XT ,k = N i=1 xi,k YT ,k= N i=1 yi,k. (34)
With (10), (11), and (33), the update process of the state variable ˆRkbecomes ˆ Rk = ˆRk−1+ 2(XT ,k− XT ,k−1)ˆxk−1 + 2(YT ,k− YT ,k−1)ˆyk−1+ 2XT ,kvˆx,k−1Δt + 2YT ,kˆvy,k−1Δt + XT ,kˆax,k−1Δt2 + YT ,kaˆy,k−1Δt2+ (WT ,k− WT ,k−1). (35)
Based on the derivations as above, all of the state variables can be obtained for the UKT-TOA scheme. The matrix Fk
associated with state equation (5) can be expressed by replacing XS,kand YS,kin (14) with XT ,kand YT ,kin (34), respectively,
for all k. The control input uk−1in (5) can also be acquired by
changing Wkin (15) to WT ,kfor all k.
B. UKT-TDOA Scheme
Under the network scenarios that there only exists TDOA measurement inputs, the UKT-TDOA scheme can be utilized to perform location estimation and tracking for the MS. The formulation of the UKT-TDOA scheme is similar to that of the UKT-TOA method, as stated in the previous section. The major difference is that the third nonlinear state variable in the state vector is replaced by ˆr1,k= [(ˆxk− ˜x1,k)2+ (ˆxk− ˜y1,k)2]1/2
instead of ˆRk for the UKT-TOA scheme, i.e., the state vector
becomes ˆxk = [ˆxk yˆk ˆr1,k vˆx,k ˆvy,k ˆax,k ˆay,k]T. Note that
the variable ˆr1,k represents the estimated distance from the
MS to the reference BS based on the TDOA system. With the available ˜N TDOA BSs, there will exist ˜N− 1 time difference measurements. Therefore, from (7), the measurement data yk and the matrix Mk in (4) can be acquired as shown at the
bottom of the page. The covariance matrix ˜RTDOA,kwithin the
Kalman filter measurement update is the same as (22). Similar to the methodology stated in the UKT-TOA scheme, the state variable ˆr1,kcan be represented as
ˆ r1,k= WD,k+ 2XD,k· ˆxk+ 2YD,k· ˆyk (36) where WD,k= 1 2Ni=2˜ ˜ri1,k × ⎡ ⎣ ˜ N i=2 ˜ Ki,k− ˜ N i=2 ˜ ri1,k2 − ( ˜N− 1) ˜K1,k ⎤ ⎦ (37) XD,k=− N˜ i=2(˜xi,k− ˜x1,k) N˜ i=2˜ri1,k YD,k=− N˜
i=2(˜yi,k− ˜y1,k)
N˜ i=2r˜i1,k
. (38)
Consequently, based on (10), (11), and (36), the update process of the variable ˆr1,kcan be obtained as
ˆ r1,k= ˆr1,k−1+ (XD,k− XD,k−1)ˆxk−1 + (YD,k− YD,k−1)ˆyk−1+ XD,kvˆx,k−1Δt + YD,kvˆy,k−1Δt +21XD,kˆax,k−1Δt2 +1 2YD,kaˆy,k−1Δt 2+ (W D,k− WD,k−1). (39)
Finally, the state matrix Fk of (5) can be obtained by
sub-stituting XS,k and YS,k in (14) with XD,k/2 and YD,k/2 in
(38), respectively. The control input uk−1in (5) is acquired by replacing Wkin (15) with WD,kin (37) for all k. Performance
evaluation of both UKT-TOA and UKT-TDOA schemes will be conducted in the next section.
V. PERFORMANCEEVALUATION
The performances of the proposed HUKT, UKT-TOA, and UKT-TDOA schemes are evaluated via MATLAB simulation platform. Realistic network simulations are performed to follow the models and parameters for practical systems, including TOA signals from the DSRC network and TDOA signals from the cellular network. Ranging schemes are utilized in these network systems to measure the signal arrival time between the MS and BS, which is adopted to align the time frame of re-ceived/transmitted packets and to measure the relative distances for positioning purpose. Since system bandwidth is reserved for the ranging schemes in both networks, perfect scheduling for the TOA and TDOA measurements can be assumed in network simulation. Based on the simulation procedure and parameters adopted in [33], the noise models that are utilized in the simulations are described in Section V-A. Performance comparisons of the proposed HUKT scheme under ideal and re-alistic network scenarios are conducted in Sections V-B and C, respectively. Section V-D describes the performance evaluation of UKT-TOA and UKT-TDOA schemes under homogeneous networks.
A. Noise Models
Different noise models [33] for the TOA measurements are considered in the simulations. The measurement noise ni,k
in (1) is chosen as the zero-mean Gaussian distribution with standard deviation σ, i.e., ni,k ∼ N (0, σ2), where σ will be
selected in the following sections based on different network
yk= ⎡ ⎢ ⎢ ⎣ ˜ r2 21,k− ( ˜K2,k− ˜K1,k) ˜ r2 31,k− ( ˜K3,k− ˜K1,k) .. .˜r2N 1,k˜ − ( ˜KN ,k˜ − ˜K1,k) ⎤ ⎥ ⎥ ⎦ , Mk = ⎡ ⎢ ⎢ ⎣ −2(˜x2,k− ˜x1,k) −2(˜y2,k− ˜y1,k) −2˜r21,k 01×4 −2(˜x3,k− ˜x1,k) −2(˜y3,k− ˜y1,k) −2˜r31,k 01×4 .. . −2(˜xN ,k˜ − ˜x1,k) −2(˜yN ,k˜ − ˜y1,k) −2˜rN 1,k˜ 01×4 ⎤ ⎥ ⎥ ⎦
environments. On the other hand, the NLOS noise ei,k is
modeled by an exponential distribution pei,k as
pei,k(v) = 1 λiexp −v λi , v > 0 0, otherwise (40)
where λi= c· τi= c· τmζiρ. The parameter τi is the RMS
delay spread between the ith BS and MS, τm represents the
median of τi, and is the path loss exponent assumed to be 0.5.
The shadow fading factor ρ is a lognormal random variable with zero mean, and its standard deviation σρis set to be 4 dB in the
simulation. The value of τmwill be determined later according
to various circumstances.
For the TDOA measurements, since it is formed by the subtraction of two TOA signals, the same parameter set with the TOA noise model is utilized except for the standard deviation of Gaussian noise σ and the mean value of the RMS delay spread τm. In the hybrid scenario, both values of σ and τmfor
TDOA-based cellular signals are selected to be larger than that of the TOA-based sensor measurements. The reason for selecting larger values in the cellular network is mainly due to the larger communication ranges of the BSs, which will result in higher Gaussian and NLOS errors. Moreover, a constant acceleration model is assumed for the Kalman filter, and the sampling time interval Δt = 1 s is selected for the total simu-lation time of 300 s.
B. Performance Comparison of HUKT Scheme Under Ideal Network Scenarios
The effectiveness of the proposed HUKT scheme associated with the three hybrid factors is evaluated in this section. The simulation scenarios for validating the proposed HUKT algo-rithm are to consider ideal network environments with only Gaussian noises and sufficient signal sources. There are eight BSs deployed as a regular polygon in the network, which includes four TOA and four TDOA measurements, as illustrated in Fig. 3. Within the total 300-s simulation time, it is assumed that the signals from all the BSs can always be received by the MS such that the precision for location tracking will not be affected by different numbers of available BSs. The source of estimation error is restricted to only Gaussian noise for val-idation purposes. Zero-mean Gaussian distributions each with standard deviation of 60 mN (0, 3600) and 120 m N (0, 14 400) are chosen for TOA and TDOA measurements, respectively.
Fig. 4 shows the performance validation of the proposed HUKT scheme by observing the position errors in each time step associated with their corresponding hybrid factors, i.e., βg,k, βm,k, and βf,k, which are denoted as HUKT-GHF,
HUKT-MVHF, and HUKT-KHF schemes. Note that the aver-age position error is defined as ΔPk=
∀mˆxk− xk/m,
where xk is the MS’s true position at time k, and m = 10 is
the number of simulation rounds for each time step in the entire 300-s simulation time. It can be observed from Fig. 4 that the values of GHF βg,k vary in a relatively small range compared
with the other two hybrid factors since it is only determined by the geometric relationship between the MS and the associated BSs. The GHF βg,k cannot completely react to the operating
Fig. 3. BS layout and tracking route for the proposed GHF, HUKT-MVHF, and HUKT-KHF schemes (triangles: TDOA-based BSs; circles: TOA-based BSs).
Fig. 4. Position errors associated with the hybrid factors from the proposed HUKT-GHF, HUKT-MVHF, and HUKT-KHF schemes.
status of the proposed HUKT scheme, which results in a larger position error compared with that from the other two hybrid fac-tors βm,kand βf,k. It can be seen that the KHF βf,kcan quickly
respond to variations of position error, e.g., a larger value of βf,kis assigned to compensate for the excessive position error
at simulation time of around 200 s. Therefore, the proposed HUKT-KHF scheme can provide the smallest average posi-tion error of the MS compared with the other two methods.
Fig. 5 illustrates the performance comparison of average po-sition errors among the HCLT algorithm, the HKT method, and the proposed HUKT scheme based on the three determination methods for hybrid factors βg,k, βm,k, and βf,k. Note that
the two-step LS method is adopted as the location estimator for both HCLT and HKT schemes, as shown in Fig. 1. It can be seen that the proposed HUKT algorithms outperform the other two existing schemes, e.g., the HUKT-KHF scheme results in around 220 m less in position error compared with the HCLT scheme under 90% average position error. The estimation accuracy for both HCLT and HKT methods relies greatly on the performance of the location estimator. These two-stage location tracking schemes induce larger estimation error compared with the proposed single-stage HUKT algorithm. The nonlinear behavior is also predicted and updated within the
Fig. 5. Performance comparison among the HUKT-GHF, HUKT-MVHF, HUKT-KHF, HKT, and HCLT schemes.
HUKT formulation, which results in higher location estimation and tracking accuracy for the MS. Furthermore, similar to the observation in Fig. 4, the HUKT-KHF scheme results in the smallest position error in comparison with the HUKT-MVHF and HUKT-GHF methods. The main reason is that the HUKT-KHF algorithm closely follows the KT process for ad-justing the hybrid factor βf,k, which can effectively reduce the
tracking error for the MS.
C. Performance Comparison of HUKT Scheme Under Realistic Network Scenarios
In this section, performance comparisons among HUKT, HKT, and HCLT schemes are implemented under realistic net-work environments with NLOS noises and insufficient number of signal sources. The network scenario for the simulation is de-scribed as follows. As shown in Fig. 9, for MS’s location track-ing, the BSs deployed in a regular cellular layout are considered to perform TDOA measurements, whereas the randomly dis-tributed short-range sensors conduct TOA measurements. Note that the empty circles represent the locations of the cellular BSs, and the empty triangles indicate the sensor BSs. The noise distributions for the TOA and TDOA measurements are chosen as N (0, 3600) and N (0, 32400), i.e., with 60 and 180 m of standard deviation, respectively. The RMS delay spread τmfor
the NLOS noises is set to be 0.1 for TOA measurements and 0.3 for TDOA measurements. Fig. 6 illustrates the total number of available BSs for TOA and TDOA measurements, respectively, during the simulation time of 300 s. It is noticed that situations with insufficient signal sources are arranged in the simulations, i.e., the number of BSs is less than three and four for TOA and TDOA BSs, respectively.
Fig. 7 shows the position errors along with the corre-sponding hybrid factors from the proposed HUKT-GHF, HUKT-MVHF, and HUKT-KHF schemes. It can still be ob-served that the proposed HUKT-KHF scheme outperforms the other two methods under the existence of NLOS noises. Fig. 8 illustrates the performance comparison on the average position errors among HKT, HCLT, and the three proposed HUKT
Fig. 6. Number of available BSs from TOA and TDOA measurements.
Fig. 7. Position errors associated with the hybrid factors from the proposed HUKT-GHF, HUKT-MVHF, and HUKT-KHF schemes.
Fig. 8. Performance comparison among the HUKT-GHF, HUKT-MVHF, HUKT-KHF, HKT, and HCLT schemes.
schemes. The proposed HUKT-KHF algorithm can provide better performance compared with all the other schemes, e.g., around 100 m less in position error compared with the HKT and HCLT schemes under 67% average position error. The HUKT
Fig. 9. Trajectory tracking of the MS using the HCLT, HKT, and HUKT-KHF schemes (solid lines: true trajectories; dotted lines: estimated trajectories; triangles: TDOA BSs; circles: TOA BSs).
Fig. 10. Velocity tracking of the MS using the HCLT, HKT, and HUKT-KHF schemes (solid lines: true velocities; dotted lines: estimated velocities).
formulation tracks the nonlinear behavior as the information feedback to enhance the measurement update, which results in higher location estimation and tracking accuracy for the MS. Furthermore, the signal insufficiency problem from the individual measurement path can also be alleviated by adopting the proposed HUKT algorithm. Figs. 9–11 show the trajectory tracking for the MS’s position, velocity, and acceleration by adopting the HUKT-KHF, HCLT, and HKT schemes. It can be seen that the proposed HUKT-KHF algorithm can provide better tracking capability compared with the other two schemes. Both tracking results obtained from HCLT and HKT schemes severely deviate from their true trajectories as the acceleration has been altered. Furthermore, at the tail of route, the insuffi-ciency of signal sources made both HCLT and HKT algorithms unable to maintain accurate location tracking for the MS. The proposed HUKT-KHF algorithm can still provide consistent performance, including position, velocity, and acceleration, under the variations of MS’s mobility.
Fig. 11. Acceleration tracking of the MS using the HCLT, HKT, and HUKT-KHF schemes (solid lines: true accelerations; dotted lines: estimated accelerations).
Fig. 12. Performance comparison among the UKT-TOA, KT, and CLT schemes for TOA measurements.
D. Performance Comparison of UKT-TOA and UKT-TDOA Schemes
In this section, the performances of the proposed UKT scheme for pure TOA and TDOA measurement inputs are evaluated. The BSs are designed to locate in the regular cellular layout for both situations. The noise model for both types of signal inputs are Gaussian measurement noises with 60 m standard deviation, i.e., ni,k∼ N (0, 3600), and exponential
NLOS noises as (40) with RMS delay spread τm= 0.3. Figs. 12
and 13 respectively show the performance evaluation for the UKT-TOA and UKT-TDOA schemes compared with both KT and cascade location tracking (CLT) algorithms. Note that the KT and CLT schemes are also implemented with pure TOA and TDOA measurement inputs for the purpose of performance comparison.
Similar to the results obtained from the HUKT algorithm, the simplified versions, i.e., the UKT-TOA and UKT-TDOA schemes, can still outperform both KT and CLT algorithms with
Fig. 13. Performance comparison among the UKT-TDOA, KT, and CLT schemes for TDOA measurements.
homogeneous measurement inputs. For example, as shown in Figs. 12 and 13, under 67% average position error, the proposed UKT-TOA scheme can provide around 60 m less position error compared with the other two methods, whereas the UKT-TDOA algorithm results in 120 m less of position error compared with the CLT scheme. The proposed UKT schemes can additionally track the variation of the nonlinear variable to provide better location tracking accuracy. The effectiveness of the proposed single-stage architecture can be revealed by directly extracting the observation results from original measurement inputs to mitigate the error propagation phenomenon in multiple-stage systems. This benefit of adopting the unified structure for achieving higher precision on location estimation and tracking can therefore be observed.
VI. CONCLUSION
In this paper, an HUKT technique has been proposed for lo-cation estimation and tracking. Based on heterogeneous signal inputs, the HUKT scheme integrates the location estimation and tracking problems within a unified Kalman filtering for-mulation. Different hybrid factors are designed for the HUKT algorithm to enhance the location tracking accuracy. Compared with other existing wireless location techniques, simulation results show that the proposed HUKT scheme can both provide higher precision for mobile location tracking and adapt to environments with insufficient signal sources.
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Cheng-Tse Chiang received the B.S. degree in
com-munication engineering and the M.S. degree from the National Chiao Tung University, Hsinchu, Taiwan, in 2008 and 2010, respectively.
He is currently with Realtek Semiconductor Corp., Hsinchu. His research interests include network pro-tocol design for mobile ad hoc networks, wireless sensor networks, and wireless location estimation and tracking technologies.
Po-Hsuan Tseng (S’08) received the B.S. and Ph.D.
degrees in communication engineering from the Na-tional Chiao Tung University, Hsinchu, Taiwan, in 2005 and 2011, respectively.
He is currently in the military service, Taiwan. From January 2010 to October 2010, he was a Vis-iting Researcher with the University of California at Davis. His research interests are in the areas of signal processing for networking and communications, in-cluding location estimation and tracking, cooperative localization, and mobile broadband wireless access system design.
Kai-Ten Feng (M’03) received the B.S. degree from
the National Taiwan University, Taipei, Taiwan, in 1992, the M.S. degree from the University of Michi-gan, Ann Arbor, in 1996, and the Ph.D. degree from the University of California, Berkeley, in 2000.
Between 2000 and 2003, he was an In-Vehicle De-velopment Manager/Senior Technologist with On-Star Corporation, a subsidiary of General Motors Corporation, where he worked on the design of fu-ture Telematics platforms and in-vehicle networks. Since August 2007, he has been an Associate Pro-fessor with the Department of Electrical Engineering, National Chiao Tung University (NCTU), Hsinchu, Taiwan, where he was an Assistant Professor from August 2007 to July 2011 and from February 2003 and July 2007, respectively. From July 2009 to March 2010, he was a Visiting Scholar with the Department of Electrical and Computer Engineering, University of California at Davis. He has also been the Convener of the NCTU Leadership Development Program since August 2011. Since August 2011, he has been a full Professor with the Department of Electrical Engineering, National Chiao Tung University (NCTU), Hsinchu, Taiwan, where he was an Associate Professor and Assistant Professor from August 2007 to July 2011 and from February 2003 to July 2007, respectively. Since October 2011, he has been serving as the Director of the Digital Content Production Center at the same university. His current research interests include broadband wireless networks, cooperative and cog-nitive networks, smart phone and embedded system designs, wireless location technologies, and intelligent transportation systems.
Dr. Feng received the Best Paper Award from the Spring 2006 IEEE Vehicular Technology Conference, which ranked his paper first among the 615 accepted papers. He also received the Outstanding Youth Electrical Engineer Award in 2007 from the Chinese Institute of Electrical Engineering and the Distinguished Researcher Award from NCTU in 2008, 2010, and 2011. He has served on the technical program committees of the Vehicular Technology, International Communications, and Wireless Communications and Networking Conferences.
