Cascade Location Tracking (CLT) Algorithm

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(2.38)

assuming that the measurements obtained are uncorrelated σ_{T DOA}² is a diagonal matrix of dimension equal to the number of avalaible measurements N.

2.4.3 Cascade Location Tracking (CLT) Algorithm

The Cascade Location Tracking (CLT) scheme as proposed in [21] utilizes the two-step Least Square (LS) method [12] [13] for initial location estimation of the MS. The two-step LS method computes the solution z_kwhich is also the measurement input of Kalman filter. After the Kalman filter, we can get the final solution ˆs_k of MS. The Kalman filtering technique is employed to smooth out and to trace the position of the MS based on its previously estimated data. The details of the two-step LS method and Kalman filter are illustrated in front sections.

Chapter 3

Architectural Overview of the Proposed PLT and GPLT

Algorithms

The objective of the proposed Predictive Location Tracking (PLT) and the Geometric-assisted Predictive Location Tracking (GPLT) algorithms is to utilize the predictive information ac-quired from the Kalman filter to serve as the assisted measurement inputs while the environ-ments are deficient with signal sources. Fig. 3.1, Fig. 3.2 and Fig. 3.3 illustrate the system architectures of the KT [18], the CLT [21] and the proposed PLT/GPLT schemes. The TOA signals (r_k as in (2.1)) associated with the corresponding location set of the BSs (P_BS,k) are obtained as the signal inputs to each of the system, which result in the estimated state vector of the MS, i.e. ˆs_k= [ˆx_kˆv_kˆa_k]^T where ˆx_k = [ˆx_k yˆ_k] represents the MS’s estimated position, ˆv_k

= [ˆv_x,k vˆ_y,k] is the estimated velocity, and ˆa_k = [ˆa_x,k ˆa_y,k] denotes the estimated acceleration.

Since the equations (i.e.(2.1) and (2.2)) associated with the network-based location es-timation are intrinsically nonlinear, different mechanisms are considered within the existing algorithms for location tracking. The KT scheme [18] (as shown in Fig. 3.1) explores the linear aspect of location estimation within the Kalman filtering formulation; while the non-linear term (i.e. ˆβ_k= ˆx²_k+ ˆy_k²) is treated as an additional measurement input to the Kalman

Kalman Tracking

Location Estimator (2-Step LS) r_k

TOA Signals

Nk < 3

k

s_k= [x_k v_k a_k]^T PBS,k

-Figure 3.1: The Architecture Diagrams of the Kalman Tracking (KT) Scheme

Location Estimator (2-Step LS)

Kalman Filter Nk < 3 z_k

s_k = [x_k v_k a_k]^T r_k TOA Signals

PBS,k

-Figure 3.2: The Architecture Diagrams of the Cascade Location Tracking (CLT) Scheme

Location Estimator

Figure 3.3: The Architecture Diagrams of the Proposed Predictive Location Tracking (PLT) and Geometric-assisted Predictive Location Tracking (GPLT) Scheme

filter. It is stated within the KT scheme that the value of the nonlinear term can be obtained from an external location estimator, e.g. via the two-step LS method. Consequently, the estimation accuracy of the KT algorithm greatly depends on the precision of the additional location estimator. On the other hand, the CLT scheme [21] (as illustrated in Fig. 3.2) adopts the two-step LS method to acquire the preliminary location estimate of the MS. The Kalman Filter is utilized to smooth out the estimation error by tracing the estimated state vector ˆs_k of the MS.

The architecture of the proposed PLT and GPLT schemes is illustrated in Fig. 3.3. It is noticed that the GPLT algorithm involves additional transformation via the GDOP calculation comparing with the PLT scheme. It can be seen that the PLT/GPLT algorithms will be the same as the CLT scheme while N_k ≥ 3, i.e. the number of available BSs is greater than or equal to three. On the other hand, the effectiveness of the PLT/GPLT schemes is revealed as 1 ≤ N_k < 3, i.e. with deficient measurement inputs. The predictive state information obtained from the Kalman filter is utilized for acquiring the assisted information, which will be fed back into the location estimator. The extended sets for the locations of the BSs (i.e.

P^e_BS,k = {P_BS,k, P_BSv,k}) and the measured relative distances (i.e. r^e_k = {r_k, r_v,k}) will be

utilized as the inputs to the location estimator. The sets of the virtual BS’s locations P_BSv,k and the virtual measurements r_v,k are defined as follows.

Definition 1 (Virtual Base Stations) Within the PLT/GPLT formulation, the virtual Base Stations are considered as the designed locations for assisting the location tracking of the MS under the environments with deficient signal sources. The set of virtual BSs P_BSv,k is defined under two different numbers of N_k as

P_BSv,k=

Definition 2 (Virtual Measurements) Within the PLT/GPLT formulation, the virtual measurements are utilized to provide assisted measurement inputs while the signal sources are insufficient. Associating with the designed set of virtual BSs P_BSv,k, the corresponding set of virtual measurements r_v,k is defined as

r_v,k=

It is noticed that the major tasks of both the PLT and GPLT schemes are to design and to acquire the values of P_BSv,k and r_v,k for the two cases (i.e. N_k = 1 and 2) with inadequate signal sources. In both the KT and the CLT schemes, the estimated state vector ˆ

s_k can only be updated by the internal prediction mechanism of the Kalman filter while there are insufficient numbers of BSs (i.e. N_k < 3 as shown in Fig. 3.1 and 3.2 with the dashed lines). The location estimator (i.e. the two-step LS method) is consequently disabled owing to the inadequate number of the signal sources. The tracking capabilities of both schemes significantly depend on the correctness of the Kalman filter’s prediction mechanism.

Therefore, the performance for location tracking can be severely degraded due to the changing behavior of the MS, i.e. with the variations from the MS’s acceleration.

On the other hand, the proposed PLT/GPLT algorithms can still provide satisfactory

these circumstances, the location estimator is still effective with the additional virtual BSs P_BSv,k and the virtual measurements r_v,k, which are imposed from the predictive output of the Kalman filter (as shown in Fig. 3.3). It is also noted that the PLT/GPLT schemes will perform the same as the CLT method under the case with no signal input, i.e. under N_k = 0. Furthermore, the GPLT algorithm enhances the precision and the robustness of the location estimation from the PLT scheme by considering the GDOP effect, i.e. the geographic relationship between the locations of the BSs and the MS. By adopting the GPLT scheme, the locations of the virtual BSs P^{P LT}_BSv,k obtained from the PLT method are adjusted into P^{GP LT}_BSv,k in order to make the predicted MS possess with a minimal GDOP value. Consequently, smaller estimation errors can be acquired by exploiting the GPLT algorithm comparing with the PLT scheme. The virtual BS’s location set P^{P LT}_BSv,k and the virtual measurements r^{P LT}_v,k by exploiting the PLT formulation is presented in the next section; while the adjusted location set of the virtual BSs P^{GP LT}_BSv,k adopting from the GPLT algorithm will be derived in chapter 5.

Chapter 4

Formulation of the PLT Algorithm

The proposed Predictive Location Tracking (PLT) scheme will be explained in this section.

As shown in Fig. 3.3, the measurement and state equations for the Kalman filter can be represented as

z_k = Mˆs_k+ m_k (4.1)

ˆ

s_k = Fˆs_k−1+ p_k (4.2)

where ˆs_k = [ˆx_k ˆv_k aˆ_k]^T. The variables m_k and p_k denote the measurement and the process noises associated with the covariance matrices R and Q within the Kalman filtering formula-tion. The measurement vector z_k = [ˆx_ls,k yˆ_ls,k]^T represents the measurement input which is obtained from the output of the two-step LS estimator at the k^thtime step (as in Fig. ??.(c)).

rv , k

Figure 4.1: The Schematic Diagram of the Two-BSs Case for the proposed PLT and GPLT Schemes

The matrix M and the state transition matrix F can be obtained as

M =

where ∆t denotes the sample time interval. The main concept of the PLT scheme is to provide additional virtual measurements (i.e. r_v,k as in (3.2)) to the two-step LS estimator while the signal sources are insufficient. Two cases (i.e. the two-BSs case and the single-BS case) are considered as follows:

4.1 The Two-BSs Case

As shown in Fig. 4.1, it is assumed that only two BSs (i.e. BS₁ and BS₂) associated with two TOA measurements are available at the time step k in consideration. The main target is to introduce an additional virtual BS along with its virtual measurement (i.e. P^{P LT}_BSv,k = {x^{P LT}_v1,k } and r^{P LT}_v,k = {r_v^{P LT}_1,k }) by acquiring the predictive output information from the Kalman filter. Knowing that there are predicting and correcting phases within the Kalman filtering formulation, the predictive state can therefore be utilized to compute the supplementary virtual measurement r^{P LT}_v1,k as

r_v^{P LT}_1,k = kˆx_k|k−1− ˆx_k−1|k−1k

= kM F ˆs_k−1|k−1− ˆx_k−1|k−1k (4.5)

where ˆx_k|k−1 denotes the predicted MS’s position at time step k; while ˆx_k−1|k−1 is the cor-rected MS’s position obtained at the (k − 1)^th time step. It is noticed that both values are available at the (k − 1)^th time step. The virtual measurement r^{P LT}_v1,k is defined as the distance between the previous location estimate (ˆx_k−1|k−1) as the position of the virtual BS (i.e. BS_v,1: x^{P LT}_v1,k , ˆx_k−1|k−1) and the predicted MS’s position (ˆx_k|k−1) as the possible position of the MS (as shown in Fig. 4.1). It is also noted that the corrected state vector ˆs_k−1|k−1 is available at the current time step k; while ˆs_k|k is unobtainable at the k^th time step. By adopting r^{P LT}_v1,k (in (4.5)) as the additional signal input, the measurement vector z_k can be acquired after the three measurement inputs r^e_k = {r_1,k, r_2,k, r^{P LT}_v1,k } and the locations of the BSs P^e_BS,k = {x_1,k, x_2,k, x^{P LT}_v1,k } have been imposed into the two-step LS estimator. Therefore, the state vector ˆs_k|k can be obtained with the implementation of the correcting phase of the Kalman filter at the time step k as

ˆ

s_k|k = ˆs_k|k−1+ P_k|k−1M^T[MP_k|k−1M^T + R]⁻¹(z_k− Mˆs_k|k−1) (4.6)



Figure 4.2: The Schematic Diagram of the Single-BS Case for the proposed PLT and GPLT Schemes

where

P_k|k−1 = FP_k−1|k−1F^T + Q (4.7)

P_k−1|k−1 = [ I − P_k−1|k−2M^T (MP_k−1|k−2M^T + R)⁻¹M ] P_k−1|k−2 (4.8)

It is noted that P_k|k−1 and P_k−1|k−1represent the predicted and the corrected estimation covariances within the Kalman filter. I in (4.8) is denoted as an identity matrix. As can been observed from Fig. 4.1, the virtual measurement r_v^{P LT}_1,k associating with the other two existing measurements r_1,k and r_2,k provide a confined region for the estimation of the MS’s location at the time step k, i.e. ˆx_k|k.