Since the equations associated with the network-based location estimation are inherently nonlinear, different mechanisms, e.g., linearization, are considered within the existing algo-rithms for location tracking. The Kalman tracking (KT) scheme [25] considers the nonlinear term as an external measurement input to its Kalman filtering formulation. It distinguishes the linear part from the originally nonlinear equations for location estimation and tracking.
However the KT scheme does not specifically indicate the method for acquiring the value of the nonlinear term.
For comparison purpose, the KT scheme that was originally proposed based on the TDOA measurement inputs is reformulated and extended to consider both the TOA and TDOA signal sources. The middle plot of Fig. 1.1 illustrates the architecture of the hybrid KT (HKT) scheme. The nonlinear terms can be obtained from the external location estimators, i.e., by adopting the two-step LS method. With the formulation of the HKT scheme, feasible accuracy for location tracking (including position, velocity, and acceleration) can be acquired. However, the accuracy is significantly affected by the precision of the external location estimator. The detail algorithm of the KT scheme can be found in [25].
Chapter 3
The Proposed Hybrid Unified
Kalman Tracking (HUKT) Scheme
The proposed HUKT scheme will be addressed in this chapter. The formulation of HUKT algorithm will be explained in section 3.1, and the determination of the hybrid variable β will be discussed in section 3.2. The variable β will be determined from three different approaches in order to allocate the weighting factors between the TOA and TDOA measurements for the HUKT scheme.
3.1 Formulation of HUKT Algorithm
The right plot of Fig. 1.1 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 HUKT algorithm is to obtain the updated state variables via the Kalman filtering technique directly from both the TOA and TDOA measurements as the system inputs. The measurement update and the state update equations of the Kalman filter can be
represented as
yk = Mˆxk+ mk (3.1)
ˆ
xk = Hˆxk−1+ uk−1+ pk−1 (3.2)
where ˆxk= [ˆxk ˆyk <ˆk ˆvx,k ˆvy,k ˆax,k ˆay,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 ˆ<k. It is noted that ˆ<k represents the estimated nonlinear term for the hybrid-based location estimation. The updating process of ˆ<k will be addressed later.
The variables mk and pk−1 denote the measurement and the processing noises respectively.
With the assumption that ri,k2 ≥ ζi,k2 due to the existence of NLOS errors ei,k, the following inequality can be obtained by rearranging the TOA measurements (2.1) and (2.2) as
ri,k2 − Ki,k ≥ −2xi,kxk− 2yi,kyk+ Rk (3.3)
where Ki,k = x2i,k+y2i,kand Rk= x2k+y2k. Similarly, the following relation can also be acquired from the TDOA measurements (2.3) by substituting j = 1 as:
˜
ri1,k2 − ( ˜Ki,k− ˜K1,k) ≥ −2(˜xi,k− ˜x1,k)xk− 2(˜yi,k− ˜y1,k)yk− 2˜r1,k˜ri1,k (3.4)
where ˜r1,k indicates the measured distance from the MS to the reference BS via the TDOA system. In order to design a unified structure for location tracking, the purpose of proposed HUKT scheme is to obtain an effective method to combine both the TOA and TDOA mea-surements. More specifically, a new variable ˆ<kis introduced to combine the nonlinear terms Rk in (3.3) and ˜r1,k in (3.4). Without loss of generality, the nonlinear term ˜r1,k in (3.4) can be represented as
q
x2k+ y2kby shifting the entire coordinate (i.e., both TOA and TDOA sys-tems) such that (˜x1,k, ˜y1,k) = (0, 0). Let the parameter βk be defined as a hybrid factor. By
multiplying (3.4) with βk/˜ri1,k and adding to (3.3), the following equation can be obtained:
ri,k2− Ki,k+ βkr˜j1,k− βkK˜j,k− ˜K1,k
˜
rj1,k + β2k=
− 2(xi,k+ βkx˜j,k− ˜x1,k
˜
rj1,k )xk− 2(yi,k + βky˜j,k− ˜y1,k
˜
rj1,k )yk+ <k (3.5)
where <k = ( q
x2k+ y2k− βk)2 corresponds to the variable that combines the effects from both the TOA and TDOA measurements. It is included within the state vector ˆxk for state updating within the Kalman filtering formulation. Therefore, the measurement data yk and the matrix M associated with the measurement process (as in (3.1)) can be acquired in (3.6).
It is noted that there are (N + ˜N − 2) linearly independent equations associated with both yk and M. 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 measurement (i.e., r1,k) and the remaining TDOA measurements (i.e., from ˜r31,k to ˜rN 1,k˜ ). The parameter hybrid factor βk is utilized to merge the TOA and TDOA based measurements, which can be determined according to the signal qualities of the two different paths. The detail of choosing appropriate value for βk will be addressed later in the next section.
Under the assumption of constant acceleration model, the updating process of ˆxk and ˆyk are determined as
ˆ
xk= ˆxk−1+ ˆvx,k−1∆t +1
2ˆax,k−1∆t2 (3.8)
ˆ
yk= ˆyk−1+ ˆvy,k−1∆t + 1
2ˆay,k−1∆t2 (3.9)
where ∆t denotes the sampling time interval. In order to provide the updating process for the new variable <k, similar to (3.5), the relation between <k, xk, and yk can be acquired by summing all N TOA measurements of (3.3) and ˜N − 1 TDOA measurements of (3.4) as
<k= Wk+ 2XS,k· xk+ 2YS,k· yk (3.10)
yk=
Following the methodology as in (3.8) and (3.9), the updating process for the estimated variable ˆ<k becomes
<k=<k−1+ 2(XS,k− XS,k−1)xk−1+ 2(YS,k− YS,k−1)yk−1 + 2 · XS,k· vx,k−1· ∆t + 2 · YS,k· vy,k−1· ∆t
+ XS,k· ax,k−1· ∆t2+ YS,k· ay,k−1· ∆t2 (3.12)
Finally, the state matrix H associated within the state equation in (3.2) for the proposed HUKT scheme can be obtained in (3.7). The control input uk−1 can also be acquired as
uk−1=
·
0 0 (Wk− Wk−1) 0 0 0 0
¸T
(3.13)
To summarize, the proposed HUKT scheme integrates the measurement inputs from het-erogeneous location estimation systems based on a unified Kalman filtering structure. The iterative operations of the Kalman filtering technique primarily consist of the processes for state update (i.e., prediction) and measurement update (i.e., correction). The equations for state update is represented as
ˆ
x−k = Hˆxk−1+ uk−1 (3.14)
C−k = HCk−1HT + Qk (3.15)
The equations for measurement update becomes
Kk = C−kMT(MC−kMT + RT OA,k+ RT DOA,k)−1 (3.16)
ˆ
xk= ˆx−k + Kk(yk− Mˆx−k) (3.17)
Ck= C−k − KkMC−k (3.18)
where Kk represents the Kalman gain and the matrix Ckis denoted as the estimate error co-variance. The covariance matrices associated with the TOA and TDOA measurement update
processes are respectively represented as
are arranged according to the TOA and TDOA measurements pairs as in (3.6). The matrices Lk= diag{ζ1,k, ζ2,k, . . . , ζN,k}, ˜Lk= diag{˜ζ1,k, ˜ζ2,k, . . . , ˜ζN,k} and the covariances of TOA and TDOA measurements are respectively represented as
Jh,k = diag{σ1,k2 , σ22,k, . . . , σN,k2 } (3.21) J˜h,k = diag{˜σ1,k2 , ˜σ22,k, . . . , ˜σN,k2 } (3.22)