An adaptive location estimator using tracking algorithms for indoor WLANs

Abstract

This paper presents adaptive algorithms for estimating the location of a mobile terminal (MT) based on radio propagation modeling (RPM), Kalman filtering (KF), and radio-frequency identification (RFID) assisting for indoor wireless local area networks (WLANs). The location of the MT of the extended KF positioning algorithm is extracted from the constant-speed trajectory and the radio propagation model. The observation information of the KF tracker is extracted from the empirical and RPM positioning methods. Specifically, a sensor-assisted method employs an RFID system to adapt the sequential selection cluster algorithm. As compared with the empirical method, not only can the RPM algorithm reduce the number of training data points and perform on-line calibration in the signal space, but the RPM and KF algorithms can alleviate the problem of aliasing. In addition, the KF tracker with the RFID-assisted scheme can calibrate the location estimation and improve the corner effect. Experimental results demonstrate that the proposed location-tracking algorithm using KF with the RFID-assisted scheme can achieve a high degree of location accuracy (i.e., more than 90% of the estimated positions have error distances of less than 2.1 m).

Appendices

Appendix A: radio propagation modeling equations (RPM equations)

A model of the simple form is used as the corresponding equation of the path loss. A polynomial of degree two is adopted to generate a fitting function for a given set of data. The response distance d is assumed as a random variable related to the input variable SNR. Therefore, based on the indoor environment in Fig. 7, the RPM equations around the hallway are described as follows.

$$ \begin{aligned} {\text{AP}}1:{\text{SNR}}^{o}_{[dB]} & = - 0.0801d^{2} - 0.7500d + 58.848 \\ {\text{AP2}}:{\text{SNR}}^{o}_{[dB]} & = 0.0127d^{2} + 1.5310d + 39.316 \\ {\text{AP3}}:{\text{SNR}}^{o}_{[dB]} & = 0.2149d^{2} - 1.4877d + 12.761 \\ {\text{AP4}}:{\text{SNR}}^{o}_{[dB]} & = 0.2319d^{2} - 4.1550d + 30.871 \\ \end{aligned} $$

(A.1)

$$ \begin{aligned} {\text{AP}}1:{\text{SNR}}^{o}_{[dB]} & = 0.0318d^{2} - 1.7976d + 38.585 \\ {\text{AP}}2:{\text{SNR}}^{o}_{[dB]} & = 0.0288d^{2} - 1.6510d + 58.249 \\ {\text{AP}}3:{\text{SNR}}^{o}_{[dB]} & = 0.0241d^{2} - 0.1193d + 31.043 \\ {\text{AP}}4:{\text{SNR}}^{o}_{[dB]} & = 0.0288d^{2} - 0.2067d + 15.022 \\ \end{aligned} $$

(A.2)

$$ \begin{aligned} {\text{AP}}1:{\text{SNR}}^{o}_{[dB]} & = 0.1695d^{2} - 1.1157d + 15.129 \\ {\text{AP2}}:{\text{SNR}}^{o}_{[dB]} & = 0.0818d^{2} - 2.0639d + 27.349 \\ {\text{AP3}}:{\text{SNR}}^{o}_{[dB]} & = - 0.0028d^{2} - 1.2242d + 56.262 \\ {\text{AP4}}:{\text{SNR}}^{o}_{[dB]} & = - 0.0469d^{2} + 1.6527d + 46.540 \\ \end{aligned} $$

(A.3)

$$ \begin{aligned} {\text{AP1}}:{\text{SNR}}^{o}_{{[{\text{dB}}]}} & = 0.0135d^{2} + 0.1659d + 29.976 \\ {\text{AP2}}:{\text{SNR}}^{o}_{{[{\text{dB}}]}} & = 0.0374d^{2} - 0.5423d + 15.317 \\ {\text{AP3}}:{\text{SNR}}^{o}_{{[{\text{dB}}]}} & = 0.0286d^{2} - 1.7279d + 35.293 \\ {\text{AP4}}:{\text{SNR}}^{o}_{{[{\text{dB}}]}} & = 0.0298d^{2} - 1.7640d + 55.220, \\ \end{aligned} $$

(A.4)

where Eqs. A.1, A.2, A.3, and A.4 are the predicted SNRs from the four APs as the MT moves along the left hallway from AP1 to AP2 (d is the distance between AP1 and the MT), the upper hallway from AP2 to AP3 (d is the distance between AP2 and the MT), the right hallway from AP3 to AP4 (d is the distance between AP3 and the MT), and the lower hallway from AP4 to AP1 (d is the distance between AP4 and the MT), respectively.

Appendix B: Kalman filtering

The following formulas Eqs. A.5–A.16 are adapted from [19, 34, 42, 46–49]. The mathematical model of the MT motion (system dynamic model) and the measurement matrix of the MT (measurement model) are denoted by

$$ {\mathbf{x}}_{k} = {\varvec{\Upphi}}_{k - 1} {\mathbf{x}}_{k - 1} + {\mathbf{w}}_{k - 1} ,\quad {\mathbf{w}}_{k} \sim {\cal{N}}({\mathbf{0}},{\mathbf{Q}}_{k} ) $$

(A.5)

$$ E\left\{ {{\mathbf{w}}_{n} {\mathbf{w}}_{k}^{T} } \right\} = \left\{ {\begin{array}{*{20}c} {{\mathbf{Q}}_{k} } & {{\text{for }}n = k} \\ {\mathbf{0}} & {{\text{for }}n \ne k} \\ \end{array} } \right. = \delta (k - n){\mathbf{Q}}_{k} $$

(A.6)

$$ {\mathbf{y}}_{k} = {\mathbf{H}}_{k} {\mathbf{x}}_{k} + {\mathbf{v}}_{k} ,\quad {\mathbf{v}}_{k} \sim {\cal{N}}({\mathbf{0}},{\mathbf{R}}_{k} ) $$

(A.7)

$$ E\left\{ {{\mathbf{v}}_{n} {\mathbf{v}}_{k}^{T} } \right\} = \left\{ {\begin{array}{*{20}c} {{\mathbf{R}}_{k} } & {{\text{for }}n = k} \\ {\mathbf{0}} & {{\text{for }}n \ne k} \\ \end{array} } \right. = \delta (k - n){\mathbf{R}}_{k} , $$

(A.8)

where x _k , ${\varvec{\Upphi}}_{k}$, w _k , Q _k , y _k , H _k , v _k , and R _k are the state matrix, state transition matrix, MT model noise (process noise) matrix, MT model noise covariance matrix, actual measurement matrix, measurement transition matrix, measurement noise matrix, and measurement noise covariance matrix, respectively. When a new observation y _k occurs at time t _k , the estimation can be summarized as follows.

2.1 Prediction phase

From k − 1 to k, the state prediction and prediction error covariance are obtained as

$$ {\tilde{\mathbf{x}}}_{k} = {\varvec{\Upphi}}_{k - 1} {\hat{\mathbf{x}}}_{k - 1} $$

(A.9)

$$ {\tilde{\mathbf{P}}}_{k} = {\varvec{\Upphi}}_{k - 1} {\hat{\mathbf{P}}}_{k - 1} {\varvec{\Upphi}}_{k - 1}^{T} + {\mathbf{Q}}_{k - 1} $$

(A.10)

$$ {\mathbf{e}}_{k|j} = {\mathbf{x}}_{k|j} - {\mathbf{x}}_{k} ,\quad {\mathbf{e}}_{k|k} = {\hat{\mathbf{x}}}_{k} - {\mathbf{x}}_{k} = {\hat{\mathbf{e}}}_{k} ,\quad {\mathbf{e}}_{k|k - 1} = {\tilde{\mathbf{x}}}_{k} - {\mathbf{x}}_{k} = {\tilde{\mathbf{e}}}_{k} $$

(A.11)

$$ {\hat{\mathbf{P}}}_{k} = E\left\{ {{\hat{\mathbf{e}}}_{k} {\hat{\mathbf{e}}}_{k}^{T} } \right\},\quad {\tilde{\mathbf{P}}}_{k} = E\left\{ {{\tilde{\mathbf{e}}}_{k} {\tilde{\mathbf{e}}}_{k}^{T} } \right\}, $$

(A.12)

where ${\hat{\mathbf{x}}}_{k}$, ${\tilde{\mathbf{x}}}_{k}$, ${\mathbf{e}}_{k|j}$, ${\hat{\mathbf{e}}}_{k}$, ${\tilde{\mathbf{e}}}_{k}$, ${\hat{\mathbf{P}}}_{k}$, and ${\tilde{\mathbf{P}}}_{k}$ are the state estimate matrix, state prediction matrix, state error matrix, estimation error matrix, one-step prediction error matrix, estimation error covariance matrix, and prediction error covariance matrix, respectively.

2.2 Innovation phase

The innovation that is based on the previous measurements is the difference between the actual measurement and the prediction. As a result, the innovation phase can be described by

$$ {\mathbf{z}}_{k} = {\mathbf{y}}_{k} - {\mathbf{H}}_{k} {\tilde{\mathbf{x}}}_{k} $$

(A.13)

$$ {\mathbf{K}}_{k} = {\tilde{\mathbf{P}}}_{k} {\mathbf{H}}_{k}^{T} \left[ {{\mathbf{H}}_{k} {\tilde{\mathbf{P}}}_{k} {\mathbf{H}}_{k}^{T} + {\mathbf{R}}_{k} } \right]^{ - 1} , $$

(A.14)

where z _k and K _k are the innovation matrix and the Kalman gain matrix, respectively.

2.3 Estimation phase

The state estimation and the estimation error covariance are updated as follows.

$$ {\hat{\mathbf{x}}}_{k} = {\tilde{\mathbf{x}}}_{k} + {\mathbf{K}}_{k} {\mathbf{z}}_{k} $$

(A.15)

$$ {\hat{\mathbf{P}}}_{k} = [{\mathbf{I}} - {\mathbf{K}}_{k} {\mathbf{H}}_{k} ]{\tilde{\mathbf{P}}}_{k} , $$

(A.16)

where I is the identity matrix. Equations A.5–A.16 of the KF algorithm fall into two group equations: time update and measurement update equations.

Appendix C: extended Kalman filtering

The following formulas A.17–A.28 for the first-order EKF algorithm of the nonlinear measurement relationship are adapted from [19, 47–49]. The motion matrix of the MT and the observation matrix of the MT are denoted by

$$ {\mathbf{x}}_{k} = {\mathbf{f}}({\mathbf{x}}_{k - 1} ,k - 1) + {\mathbf{w}}_{k - 1} ,\quad {\mathbf{w}}_{k} \sim {\cal{N}}({\mathbf{0}},{\mathbf{Q}}_{k} ) $$

(A.17)

$$ {\mathbf{y}}_{k} = {\mathbf{h}}({\mathbf{x}}_{k} ,k) + {\mathbf{v}}_{k} ,\quad {\mathbf{v}}_{k} \sim {\cal{N}}({\mathbf{0}},{\mathbf{R}}_{k} ), $$

(A.18)

where the functions ${\mathbf{f}}( \cdot )$ and ${\mathbf{h}}( \cdot )$ are the nonlinear state transition matrix and the nonlinear measurement transition matrix, respectively.

The Taylor approximation of function ${\mathbf{f}}( \cdot )$ that is done at the previous estimate is denoted by

$$ {\varvec{\Upphi}}({\hat{\mathbf{x}}},k) \approx \left. {{{\partial {\mathbf{f}}({\mathbf{x}},k)} \mathord{\left/ {\vphantom {{\partial {\mathbf{f}}({\mathbf{x}},k)} {\partial {\mathbf{x}}}}} \right. \kern-\nulldelimiterspace} {\partial {\mathbf{x}}}}} \right|_{{{\mathbf{x}} = {\hat{\mathbf{x}}}_{k} }} \equiv {\varvec{\Upphi}}^{[1]} ({\hat{\mathbf{x}}},k). $$

(A.19)

The Taylor approximation of function ${\mathbf{h}}( \cdot )$ that is done at the corresponding predicted position is denoted by

$$ {\mathbf{H}}({\hat{\mathbf{x}}},k) \approx \left. {{{\partial {\mathbf{h}}({\mathbf{x}},k)} \mathord{\left/ {\vphantom {{\partial {\mathbf{h}}({\mathbf{x}},k)} {\partial {\mathbf{x}}}}} \right. \kern-\nulldelimiterspace} {\partial {\mathbf{x}}}}} \right|_{{{\mathbf{x}} = {\tilde{\mathbf{x}}}_{k}}} \equiv {\mathbf{H}}^{[1]} ({\hat{\mathbf{x}}},k). $$

(A.20)

The models and implementing equations of the EKF algorithm are summarized as follows.

3.1 Equations for the EKF algorithm

$$ {\mathbf{x}}_{k} \approx {\tilde{\mathbf{x}}}_{k} + {\varvec{\Upphi}}_{k - 1} ({\mathbf{x}}_{k - 1} - {\hat{\mathbf{x}}}_{k - 1} ) + {\mathbf{w}}_{k - 1} $$

(A.21)

$$ {\mathbf{y}}_{k} \approx {\hat{\mathbf{y}}}_{k} + {\mathbf{H}}_{k} ({\mathbf{x}}_{k} - {\tilde{\mathbf{x}}}_{k} ) + {\mathbf{v}}_{k} , $$

(A.22)

where ${\hat{\mathbf{y}}}_{k}$ is the predicted measurement.

3.2 Prediction phase

$$ {\tilde{\mathbf{x}}}_{k} = {\mathbf{f}}_{k - 1} ({\hat{\mathbf{x}}}_{k - 1} ) $$

(A.23)

$$ {\tilde{\mathbf{P}}}_{k} = {\varvec{\Upphi}}_{k - 1} {\hat{\mathbf{P}}}_{k - 1} {\varvec{\Upphi}}_{k - 1}^{T} + {\mathbf{Q}}_{k - 1} $$

(A.24)

$$ {\hat{\mathbf{y}}}_{k} = {\mathbf{h}}_{k} ({\tilde{\mathbf{x}}}_{k} ). $$

(A.25)

3.3 Update phase

$$ {\mathbf{K}}_{k} = {\tilde{\mathbf{P}}}_{k} {\mathbf{H}}_{k}^{T} \left[ {{\mathbf{H}}_{k} {\tilde{\mathbf{P}}}_{k} {\mathbf{H}}_{k}^{T} + {\mathbf{R}}_{k} } \right]^{ - 1} $$

(A.26)

$$ {\hat{\mathbf{x}}}_{k} = {\tilde{\mathbf{x}}}_{k} + {\mathbf{K}}_{k} ({\mathbf{y}}_{k} - {\hat{\mathbf{y}}}_{k} ) $$

(A.27)

$$ {\hat{\mathbf{P}}}_{k} = \left[ {{\mathbf{I}} - {\mathbf{K}}_{k} {\mathbf{H}}_{k} } \right]{\tilde{\mathbf{P}}}_{k} . $$

(A.28)

The basic operation of the EKF algorithm is the same as the basic operation of the standard discrete KF algorithm.

Appendix D: performance evaluation and numerical simulations

4.1 SNR-based location estimation using fingerprinting (FP-based)

In the numerical simulation, according to the SS with a variance σ ² of 4.53 dB in an office room environment [14, 35, 36], the location accuracy of two different methods (SS-based and SNR-based) and the influence of the total number on APs are in Table 1. The location accuracy of the SNR-based method is better than that of the SS-based method when the total number of APs is equal to two or more [18]. Furthermore, a positioning technique using sample points to form a continuous sample space with an NN is introduced. This scheme can be based on the FP-based estimator feature. That is, in this approach, the location measurements based on the FP-based algorithm are used to train an NN. It is a good point to start further investigation in increasing accuracy. The algorithm described in Appendix E shows that the location estimator can mitigate the variations of measurement errors and reduce the corner effect of the indoor environment.

4.2 SNR-based location estimation using radio propagation modeling (RPM-based)

From the RPM-based equations in Eq. A.2 [cf. Appendix A], for example, there are 8 roots of distance d. The numerical values of the model parameters from a sample point of the four APs are shown in Table 2. If the two closest distances, 16.87 and 16.77 m, are picked to yield an average of 16.82 m, as compared with the actual distance d = 17.02 m, there is only 0.18 m deviation. In addition, to take changes into account in the indoor environment, we can use a calibrating propagation modeling method to overcome the changes in current environmental conditions. Namely, we can effectively adopt the RPM-based equations as an on-line calibrating mechanism to characterize the real dynamic indoor environments.

4.3 Performance comparison between the FP-based and RPM-based positioning methods

With data points fitted to a polynomial of degree two, a comparison of simulation results between the FP-based and the RPM-based positioning methods is shown in Table 3. In the situations with higher variation in measurements and higher CDF, the results show that the location accuracy of the RPM-based algorithm is better than that of the FP-based algorithm. That is, the RPM-based algorithm can mitigate the aliasing phenomenon and make a better decision for the location estimate.

Table 3 Comparison among NNSNRS, RPM, and EKF positioning methods

4.4 Location estimation using combined radio propagation modeling and adaptive Kalman filtering (EKF-based, SNR Information tracker)

Based on the corresponding equation of the simple-form path loss model [cf. Appendix A] and the tracking SNR information measurements, the comparisons between KF and EKF algorithms and the performances of the proposed EKF-based positioning method include the follows.

4.4.1 Performance comparison between the KF-based and EKF-based positioning methods

In the non-linear measurement system, based on the simulation results in [19], the results show that the KF algorithm to track the SNR Information based on a larger value of σ ^w can track closely the true SNR between the corresponding AP and MT, where the $w_{k} \sim {\cal{N}}(0,(\sigma_{k}^{w} )^{2} )$ is the process noise which depends on the acceleration of the MT, and σ ^w is the standard deviation of the process noise. However, in the situation of higher measurement noise (variation) of the non-linear system, the performance of the SNR estimation will not be good enough. That is, the estimation error of the SNR is caused by the SNR (or SS) measurement system which is non-linear. Furthermore, the results based on the proposed EKF-based positioning method in [19] demonstrate that the proposed EKF-based positioning method can mitigate efficiently the measurement error in tracking SNR information and perform much better than the KF-based positioning method.

4.4.2 Effect of the sampling time and the variance of the EKF-based positioning method

A comparison of the EKF-based positioning method with the influence of different variances and different sampling times in terms of the CDF of the error distance are summarized in Table 3. From the results in Table 3 and Fig. 10, increasing the value of variances has a considerable effect on performance. There are diminishing positioning accuracies when the value of variances becomes high. Furthermore, the sampling time is decreased from 1 to 0.001 s, and the response of the estimated location accuracy is increased.

4.5 Location tracking using Kalman filtering (KF-based, position tracker)

The location estimator of the KF-based tracking algorithm can extract the observed location information of an MT from the FP-based and RPM-based positioning methods. For convenience, through the simulation, four APs are located at coordinates AP1 (0, 0), AP2 (0, 20), AP3 (40, 20), and AP4 (40, 0; in meters). Using AP2 as the starting point, an MT moves in the clockwise direction with a steady speed of 0.5 m/sec and arrives at positions AP3, AP4, AP1, and AP2. In this simulation of the KF-based tracker, the parameters R _k and Q _k are chosen to be 4.5 and 0.01, respectively. In Fig. 12, the simulation results show that the KF-based tracker can track trajectory and speed of an MT effectively. Furthermore, Fig. 14 shows the simulation results of the KF-based tracker with a different sampling time as the MT moved along different paths. When the sampling time is decreased from 1 to 0.1 s, the response of the estimated location accuracy is increased, and a smoother trajectory can be obtained using a larger sampling time. The simulation results investigate the validity of the algorithm as the tracking method.

Appendix E: location estimation using neural networks (NN-based)

An NN can be termed as a parallel and distributed process, which includes multiple processing element. In this method, the EKF algorithm is used to train the multi-layer perception (MLP) NNs [50]. For convenience, we assumed that the location estimator based on the NN-based algorithm can extract the training and observed location information of an MT from the positioning methods in the simulation. Consequently, the training of the weights in an NN can be treated as an estimation problem. Here, the output of the neuron is given by

$$ y_{j} = \phi (v_{j} ) = \phi \left( {\sum\limits_{i = 1}^{n} {w_{ji} x_{i} + b} } \right),\quad \phi (v) = \tanh (v) $$

(A.29)

where y, $\phi ( \cdot )$, x _i, w _ji , and b are the output variable, activation function, input variable, connective weight (link between the jth neuron and the ith input variable), and bias, respectively. As a result, the equation of the MLP-NN is denoted by

$$ {\mathbf{y}} \, = \, {\mathbf{w}}_{{\mathbf{2}}} \cdot \tanh ({\mathbf{w}}_{1} \cdot {\mathbf{x}} + {\mathbf{b}}_{1} ) + {\mathbf{b}}_{2} , $$

(A.30)

where y, w ₁, w ₂, b ₁, and b ₂ are the output matrix of the variable, corresponding weight matrix of the tanh hidden neuron, corresponding weight matrix of the linear output neuron, bias matrix of the hidden neuron, and bias matrix of the output neuron, respectively. The MLP-NN algorithm is to search the optimal weighted parameter. Therefore, the weights of neurons and bias values, w = [w ₁ b ₁ w ₂ b ₂]^T, should be adjusted by a training algorithm, and the output of the NN will match the desired results for specific inputs. In brief, after training the network, it can estimate the position for a set of the test data.

In this indoor environment, the path is set a 90-degree corner of a hallway. While the parameters are trained to generalize the network, the simulation is set to run for 500 iterations with a location variance of 5 m and with the hidden layer of 32 neurons. In Fig. 17, the results show that the NN-based estimator can overcome the variations of measurement errors. Specifically, the NN-based estimator can mitigate the most corner effect of the hallway except in the close vicinity of the corner. To sum up, according to the phenomenon of the simulation, if the NN extracts a little information (under learning situations, the network doesn’t achieve an acceptable performance), one may try to add more neurons to the hidden layer(s) or add an extra hidden layer. On the contrary, if the NN extracts too much information from the individual cases (over learning situations, the network achieves an over-fitting phenomenon), it may forget the relevant information of the general cases [51]. One may try to remove hidden units or possible layers. That is, the under learning or over learning situations may lead to lower location accuracy and to increase the corner effect.

Fig. 17

Simulation results for estimating the location of an MT, which is based on the NN algorithm as the MT moves along the 90-degree corner of the hallway with a location variance of 5 m: a original trajectory, observed trajectory, and estimated trajectory; b measured (observed) error and estimated error

Appendix F: a layout planning scheme of WLAN APs and RFID tags

Our experimental results demonstrate that the Friis’ transmission formula of a propagation model is not suitable in indoor environments, and the standard deviation of the received SS varies according to the signal level. As a result, one must avoid coverage gaps to improve the network services for users and must separate the APs as far apart as possible to minimize cost in selecting AP locations. A design procedure of a layout planning scheme of APs based on the signal estimation algorithm was reported in [44]. For convenience, the APs are placed in the corners of a rectangular region in this paper.

After the selection of AP locations is complete, the concept of the reference tag can be applied to improve the location accuracy. The sensor-assisted scheme can alleviate the aliasing problem and the corner effect by using the clustering scheme. Therefore, in an indoor environment, a design procedure of a layout planning scheme of RFID tags based on the SNR information can be summarized as follows.

• 1. Create the radio-map information of APs in the building.

• 2. Assign RFID tags at the corners of the paths.

• 3. Search the locations of larger variations of the SNR information (the threshold value of the variation can be modified for different environments).

• 4. Assign RFID tags at the locations of larger SNR variations.

• 5. Search the locations of the aliasing in signal space.

• 6. Assign RFID tags at the locations with the aliasing phenomenon.

Appendix G: performance of various indoor location estimation methods

See Tables 1, 2, 3, 4, 5 and 6.

Table 4 Experimental results (FP, RPM, EKF, RM&KF, RPM&KF)

Table 5 State reduction and assignment

Table 6 Computational complexity comparisons of the positioning and tracking algorithms