Hybrid PDA/FIR Filtering for Indoor Localization Using Wireless Sensor Networks

Abstract

Indoor localization systems using wireless sensor networks (WSNs) are widely used to track the positions of workers, robots, and equipment. In indoor spaces, the occasional obstruction of radio propagation by physical objects such as furniture, appliances, and humans is referred to as the non-line-of-sight (NLOS) problem and has been a challenge for indoor localization. In this study, a new indoor localization algorithm to overcome the NLOS problem is proposed. We propose a new method to use redundant fixed nodes and nearest neighbor (NN) measurements, which increases the probability of avoiding NLOS-contaminated measurements. In addition, we propose a novel localization algorithm that can handle the contaminated measurements as clutters. The proposed algorithm is based on the hybrid filtering structure in which probabilistic data association (PDA) filter and a finite impulse response (FIR) filter are used as main and assisting filters, respectively. We adopt the extended minimum variance FIR (EMVF) filter as an assisting FIR filter, which recovers the main PDA filter from failures. Thus, the resulting filter is referred to as hybrid PDA/FIR filter (HPFF). Extensive simulations using an indoor localization scenario in a long corridor were performed for evaluation of the proposed localization algorithm. The EKF using NN measurements improves localization accuracy under temporary NLOS conditions, and the PDA filter further enhances the localization accuracy of EKF. However, EKF and PDA filter cannot completely overcome NLOS problem and exhibit significant increase in errors under certain conditions. The HPFF produced localization accuracy with the root time-averaged mean square (RTAMS) position error under 0.4 m and did not fail under NLOS conditions. The accurate and reliable localization performance of HPFF was demonstrated in comparison with the EKF and PDA filter through extensive WSN-based indoor localization simulations.

1. Introduction

In recent years, indoor localization systems based on wireless sensor network (WSN) technologies [1,2] have been widely used for various industrial applications, such as tracking workers and vehicles in factories, hospitals, and construction sites [3,4,5,6,7,8]. Indoor localization systems comprise sensors that communicate with one another through wireless signals. Some of these sensors are attached to the humans or vehicles that need to be tracked and called mobile nodes, while other sensors are installed at fixed positions in an indoor space and called fixed nodes. The wireless signal transmission between mobile and fixed nodes can be quantified by measurements such as the received signal strength (RSS) [9], time of arrival (TOA) [10], and time difference of arrival (TDOA) [11]. These measurements can then be processed using localization algorithms to obtain the position of mobile nodes.

Because wireless signal measurements are inevitably contaminated by noise, localization algorithms typically exploit stochastic (i.e., time-domain) filters that provide estimates of the state variables from noisy measurements [3,4,5,6,7,8]. The Kalman filter (KF) is one of the most well-known stochastic filters and is used for systems represented by linear state-space models. However, because indoor localization system measurement models are typically nonlinear, a nonlinear version of the KF called the extended KF (EKF) is widely used for indoor localization systems [3,4]. In recent years, stochastic finite impulse response (FIR) filters [12,13,14,15,16,17,18,19,20,21,22,23,24] have been studied because of their advantages over KFs. FIR filters do not require initialization and are robust against modeling and computational errors. However, if these errors are not significant, FIR filters are less accurate than KFs and require more computational time. Therefore, a hybrid filtering algorithm that combines the EKF and the FIR filter was proposed to compliment both filters [25].

In cluttered indoor spaces, wireless signals may be obstructed by physical objects such as furniture, appliances, and humans. The condition in which transmitter and receiver are not in direct visual line-of-sight contact is called the non-line-of-sight (NLOS) condition. Significant errors in wireless measurements may be present in NLOS conditions, especially if the wireless signal transmission is partially obstructed [7,26,27]. A measurement cannot be obtained if the transmission is completely obstructed. NLOS conditions occur frequently and cause severe problems in indoor localization systems [7,26,27]. It is therefore essential to make the localization algorithm robust against NLOS effects to achieve accurate and reliable localization.

In this study, a novel indoor localization algorithm is proposed to mitigate NLOS effects in WSNs. The proposed algorithm is primarily based on a probabilistic data association (PDA) filter. PDA filters have been widely used for target tracking in the presence of false measurements. The input to the PDA filter is a measurement set comprising three TOA measurements. A measurement set contaminated by the NLOS effect is considered as clutter (i.e., false measurements). The ability of the PDA filter to handle clutter mitigates NLOS effects to some extent. However, if the NLOS effect is severe, the PDA filter may fail to perform localization. An FIR filter is therefore applied to recover from PDA filter failures. The FIR filter is appropriate for assisting the recovery of the main filter because it does not require initialization and is robust against various types of errors. The proposed filter is referred to as the hybrid PDA/FIR filter (HPFF). The performance of HPFF is demonstrated by comparison with EKF and PDA filter through extensive indoor localization simulations under various NLOS conditions.

The main contributions of this study can be summarized as follows:

• A novel method to increase the probability of avoiding measurements contaminated by NLOS errors is proposed. The proposed method is based on the use of redundant fixed nodes and nearest measurements.
• A novel filtering method, called the HPFF, that can mitigate NLOS effects is proposed. The main PDA filter in HPFF deals with the measurements contaminated by NLOS errors as false measurements (clutters). The assisting FIR filter recovers the main filter from failures.
• The proposed HPFF provides accurate and reliable localization under NLOS conditions, while conventional algorithms exhibit significant increases in localization errors.

The remainder of this paper is organized as follows. A brief review of related works about indoor localization, NLOS mitigation, and hybrid filtering is presented in Section 2. The WSN-based indoor localization scheme is described and the NLOS effect explained in Section 3. An indoor localization algorithm based on the EKF is introduced in Section 4, and an HPFF-based localization algorithm proposed in Section 5. Simulation results that demonstrate the superior performance of the proposed algorithm are presented in Section 6. Finally, Section 7 presents the conclusions.

2. Related Works

Indoor localization systems based on WSNs are typically classified according to wireless communication technologies or positioning methods. A huge number of localization algorithms based on various wireless measurements and positioning methods have been reported. In [28], a localization algorithm based on TOA measurements was proposed. The proposed algorithm used the neural networks to identify the undetected direct path (UDP). In [29], TOA and direction of arrival (DOA) measurements were used together. The localization algorithm proposed in [29] trying to solve the NLOS problem by converting it into an LOS problem using virtual stations. In [30], a modified probability hypothesis density (PHD) filter was proposed and applied to a TOA-based localization problem. The proposed algorithm was a joint tracking and mapping algorithm that was able to update an inaccurate floor plan map.

In addition to the TOA, RSS fingerprinting technologies has also been widely used in indoor localization. In [31], a RSS-based algorithm for localization of quadrotors in indoor environments. In [32], two approaches for self-localization based on RSS and wireless local area network (WLAN) were proposed. In [33], a clustering technique based on RSS was proposed for improving localization accuracy and for reducing computational time.

Three-dimensional (3D) indoor localization has also attracted attention of researchers. Optical wireless communications was used for 3D localization [34], RSS radio map was used for 3D localization in [35]. Recently, intelligence algorithms have been used for indoor localization. In [36], a recurrent neural network-based localization algorithm was proposed. In [37], a 3D localization system based on pattern recognition was proposed.

NLOS problem has been a challenge in the field of indoor localization, and a number of approaches have been proposed to solve the problem. In [38,39,40], NLOS identification algorithms based on neural networks were proposed. Such algorithms detects measurements contaminated by NLOS bias errors and avoids using it. In [41,42], stochastic filters that are robust against NLOS effects were proposed and applied to indoor localization. In [7], a distributed algorithm of hybrid filtering was proposed to mitigate NLOS effects, but the algorithm was heuristic and applicable to limited scenarios.

Hybrid filtering that combines infinite impulse response (IIR) and FIR filters was originally reported in [8], where a particle filter and an FIR filter were combined. In [25], an EKF and an FIR filter were combined in the same hybrid filtering structure, and the resulting filter was applied to mobile robot localization. In [43], a hybrid filter that combines PDA and FIR filters was proposed. The filter was designed to be appropriate for radar-based target tracking problems, such as the measurement/motion models.

In the previous works, the NLOS problem was overcome by identifying NLOS situations using intelligent or heuristic algorithms. However, our study aims to make the filtering algorithm deal with the measurements contaminated by NLOS effects. Thus, we use the PDA algorithm, which has been widely used to deal with false measurements. Moreover, we assist PDAF by using an FIR filter that is robust against error accumulation. The proposed filtering algorithm can be used with existing NLOS identification algorithms.

3. Indoor Localization Scheme and Problem Formulation

In this section, a typical WSN-based indoor localization scheme is explained and the NLOS problem that we solve in this study is formulated. Figure 1 shows the geometry of an indoor localization system in which fixed and mobile nodes are transmitters and receivers, respectively. The mobile and fixed nodes are assumed to be located in the same two-dimensional (2D) indoor space. The fixed nodes are installed at known positions and transmit wireless signals that contain information on their identifications and times of transmission. The mobile nodes receive the signals and compute the TOA, which is the difference between the transmission and reception times. Denoting the 2D coordinate of a mobile node as $(x,y)$, the TOA measurements can be expressed as

$$\begin{array}{cc}\hfill z_i& =\frac{1}{c}{d}_i+v,\hfill \\ \hfill d_i& =\sqrt{{(x-x_i)}^2+{(y-y_i)}^2},\hfill \end{array}$$

(1)

where $z_i$ is the TOA obtained from the i-th fixed node ($i=1,2,3,\cdots ,N$), c is the speed of light; $d_i$ is the distance between the mobile node and the i-th fixed node, and $(x_i,y_i)$ is the 2D coordinate of the i-th fixed node. v is a zero-mean white Gaussian measurement noise with the variance $r^2$.

The location of the mobile node can be computed via triangulation using at least three TOA measurements. Under NLOS conditions, the TOA measurement includes large errors due to the multipath effect. The problem to be solved is how to obtain the coordinates of the mobile node from the TOA measurements contaminated by noise and NLOS errors. A typical solution is to use stochastic filters such as the EKF. To use stochastic filters, state-space models consisting of motion and measurement models are essential. Localization systems typically use a constant velocity (CV) motion model, which can be represented as

$$\begin{array}{cc}\hfill {\mathbf{x}}_k& =\mathbf{F}{\mathbf{x}}_{k-1}+\mathbf{G}{\mathbf{w}}_k,\hfill \\ \hfill \mathbf{F}& =\left[\begin{array}{cccc}1& T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T\\ 0& 0& 0& 1\end{array}\right],\mathbf{G}=\left[\begin{array}{cc}{T}^2/2& 0\\ T& 0\\ 0& T^2/2\\ 0& T\end{array}\right],\hfill \end{array}$$

(2)

where ${\mathbf{x}}_k$ is the state vector at time k and is defined as ${\mathbf{x}}_k={\left[x_k{\dot{x}}_k{y}_k{\dot{y}}_k\right]}^T$, and $({\dot{x}}_k,{\dot{y}}_k)$ are the 2D velocities. In (2), T is the sampling time and ${\mathbf{w}}_k$ is the process noise, which is modeled as a white Gaussian noise. The variance of ${\mathbf{w}}_k$ is $\mathbf{Q}=q^2{\mathbf{I}}_2$ where $q^2$ is the variance and ${\mathbf{I}}_2$ is the $2\times 2$ identity matrix.

The TOA measurement model is given by

$$\begin{array}{cc}\hfill {\mathbf{z}}_k& =\mathbf{h}\left({\mathbf{x}}_k\right)+{\mathbf{v}}_k,\hfill \end{array}$$

(3)

where ${\mathbf{z}}_k$ is the TOA measurement vector, which is defined as ${\mathbf{z}}_k={[z_{1,k}{z}_{2,k}\cdots z_{N,k}]}^T$; ${\mathbf{v}}_k$ is the measurement noise vector with covariance $\mathbf{R}=r^2{\mathbf{I}}_N$; and $\mathbf{h}(·)$ is the vector representation of the nonlinear function in (1).

Given the motion and measurement models, the state vector ${\mathbf{x}}_k$ can be estimated using stochastic filters. The EKF produces a state estimate by using the following equations:

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k^{-}& =\mathbf{F}{\widehat{\mathbf{x}}}_{k-1}^{+},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathbf{P}}_k^{-}& =\mathbf{F}{\mathbf{P}}_{k-1}^{+}{\mathbf{F}}^T+\mathbf{G}\mathbf{Q}{\mathbf{G}}^T,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathbf{K}}_k& ={\mathbf{P}}_k^{-}{\mathbf{H}}_k^T{({\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T+\mathbf{R})}^{-1},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k^{+}& ={\widehat{\mathbf{x}}}_k^{-}+{\mathbf{K}}_k({\mathbf{z}}_k-{\mathbf{H}}_k{\widehat{\mathbf{x}}}_k^{-}),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathbf{P}}_k^{+}& =({\mathbf{I}}_4-{\mathbf{K}}_k\mathbf{H}){\mathbf{P}}_k^{-},\hfill \end{array}$$

(8)

where ${\mathbf{P}}_k$ is the estimation error covariance matrix; ${\mathbf{K}}_k$ is Kalman gain; the superscripts − and + denote a priori and a posteriori, respectively; and ${\mathbf{H}}_k$ is the Jacobian matrix, which is defined as

$$\begin{array}{cc}\hfill {\mathbf{H}}_k& ={\left.\frac{\partial \mathbf{h}}{\partial \mathbf{x}}\right|}_{{\widehat{\mathbf{x}}}_k}^{-}.\hfill \end{array}$$

(9)

The conventional localization algorithm based on the EKF, i.e., (4)–(9), works successfully in LOS conditions. However, in NLOS conditions, the localization accuracy of the EKF is degraded or localization failures occur. This study focuses on the problem of localization failure/degradation owing to the NLOS effect. To solve this problem, we design a new localization algorithm that combines the PDA and FIR filters.

4. Mitigation of NLOS Effects Using Nearest Neighbor Measurements

The TOA measurement for a NLOS transmitter includes a significant error, which leads to performance degradation or failure in indoor localization. A simple method for mitigating NLOS effects is to use redundant fixed nodes. Theoretically, at least three TOA measurements are required for triangulation-based positioning. However, if only three measurements are used, all the three measurements will need to be utilized, and the NLOS effect is unavoidable. Thus, many indoor localization systems utilize more than three measurements by installing redundant fixed nodes. The problem of how to select measurements from the redundant measurements then arises. A simple approach is to select the smallest TOA measurement based on the idea that NLOS errors arising from the multipath effect usually make the TOA measurement larger. Therefore, selecting the smallest measurements increases the probability of avoiding contamination. Because small TOA implies that the fixed node is close to the mobile node, and we call this method the nearest neighbor (NN) measurement. A localization method based on NN measurements can be summarized as follows:

• Use N fixed nodes, where $N>3$, and obtain N TOA measurements.
• Select the n smallest measurements (from the nearest fixed nodes) among the N measurements.

An EKF-based localization algorithm based on NN measurements is described in Algorithm 1. The EKF requires the initial values of the estimated state and estimation error covariance. The initial 2D position can be obtained by using a typical triangulation method, and initial velocity can be computed by the following relationship [44]:

$$\begin{array}{cc}\hfill {\dot{x}}_k& =(x_k-x_{k-1})/T,\hfill \\ \hfill {\dot{y}}_k& =(y_k-y_{k-1})/T.\hfill \end{array}$$

(10)

In addition, the initial estimation error covariance can be obtained as [44]

$$\begin{array}{cc}\hfill {\mathbf{P}}_0& =\left[\begin{array}{cccc}r& r/T& 0& 0\\ r/T& 2r/T^2& 0& 0\\ 0& 0& r& r/T\\ 0& 0& r/T& 2r/T^2\end{array}\right].\hfill \end{array}$$

(11)

Algorithm 1: EKF-based localization using nearest neighbor measurements.

Although the NN method can avoid NLOS measurement contamination to some extent, it is not complete. The PDA filter has been successfully applied for target tracking in the presence of false measurements. The robustness of the localization algorithm against NLOS errors can be improved when the PDA filter is used in place of the EKF. A key characteristic of the PDA filter is the use of all the obtained measurements by assigning an association probability for each measurement. The PDA filter is known to be more effective than methods based on selecting the most suitable measurement. The single measurement selected by the NN method is not suitable for the PDA filter because the PDA filter requires a mixture of true and false measurements as its input. To create the required mixture of true/false measurements from all the available measurements, we propose the following approach:

• Construct the set of NN measurements $S_k$ by selecting the n nearest TOA measurements among N measurements. The number of elements in $S_k$ is n.
• Construct a set of mixture measurements denoted by ${\overline{S}}_k$. The elements of ${\overline{S}}_k$ are the m-combinations of $S_k$. The number of m-combinations is $C_m^n$. $\overline{S}$ is represented as ${\overline{S}}_k=\{{\mathbf{z}}_k\left(1\right),{\mathbf{z}}_k\left(2\right),\cdots ,{\mathbf{z}}_k\left(C_m^n\right)\}$, where $\mathbf{z}$ is a vector consisting of m TOA measurements. m indicates the number of TOA measurements used for triangulation and should be greater than or equal to three.

We now describe the localization algorithm based on the PDA filter and the measurement set ${\overline{S}}_k$. The key feature of the PDA filter that distinguishes it from the EKF is its data association process. In the process, the association probability for each measurement, which quantifies the likelihood of the measurement, is computed. Subsequently, in the measurement update process, the state estimate is corrected using all the validated measurements and their association probabilities. The measurement validation process involves discarding the measurements that deviate excessively from the predicted measurement. If the statistical distance between the actual and predicted measurements exceeds a gate threshold, the measurement is discarded. The measurement validation can be described as

$$\begin{array}{cc}\hfill {\mathcal{V}}_k\left(i\right)& ={({\mathbf{z}}_k\left(i\right)-{\widehat{\mathbf{z}}}_k)}^T{\mathbf{S}}_k^{-1}({\mathbf{z}}_k\left(i\right)-{\widehat{\mathbf{z}}}_k)\le \gamma ,i=1,2,\cdots ,C_m^n,\hfill \\ \hfill {\mathbf{S}}_k& ={\mathbf{H}}_k{\mathbf{P}}_k^{-}{\mathbf{H}}_k^T+\mathbf{R},\hfill \end{array}$$

(12)

where ${\mathcal{V}}_k\left(i\right)$ is the innovation for ${\mathbf{z}}_k\left(i\right)$; ${\mathbf{z}}_k\left(i\right)$ is the i-th measurement vector in ${\overline{S}}_k$; $\gamma$ is the gate threshold, which can be obtained from the chi-square table; ${\widehat{\mathbf{z}}}_k=\mathbf{h}\left({\widehat{\mathbf{x}}}_k^{-}\right)$ is the predicted measurement; and ${\mathbf{S}}_k\left(i\right)$ is the innovation covariance [44]. The overall localization algorithm based on the PDA filter is summarized in Algorithm 2.   

Algorithm 2: PDAF-based localization the true/false mixture measurements.

5. Hybrid PDA/FIR Filter for Indoor Localization

The NLOS effects can be mitigated by using the NN method and a PDA filter. However, the PDA filter has a problem in that localization may fail if there is no surviving measurement after the validation process (i.e., $n_v\left(k\right)=0$) [43]. In this section, we propose an indoor localization algorithm based on HPFF with NN measurements.

HPFF is a hybrid filter that complements the PDA and FIR filters. The PDA filter is a recursive Bayesian filter with an infinite impulse response (IIR) structure. Therefore, all the past measurements are used by the PDA filter to estimate the current state. If some errors (e.g., modeling/computation errors) occur in the state estimation, the errors may accumulate over time and lead to filter divergence. Because the FIR filter performs batch processing using finite recent measurements, it is robust against modeling and computational errors. However, under ideal conditions, the FIR filter is less accurate than the PDA filter. Therefore, combining PDA and FIR filters has a complementary effect on both filters. The key ideas in HPFF can be summarized as follows:

• HPFF consists of main and assisting filters.
• The PDA filter is the main filter and is used to produce state estimates in normal situations.
• The FIR filter is the assisting filter and operates only if the main filter fails.
• When failure occurs, the main filter is reset using the output of the assisting filter.

A localization algorithm based on HPFF is designed. The main filter that produces the estimated position is the PDA filter described in Algorithm 2. We adopted an extend minimum variance FIR (EMVF) filter [21] as the assisting filter. Various types of FIR filters have been developed, but the EMVF filter has a special advantage that is suitable for the assisting filter. The EMVF filter is the only nonlinear FIR filter that can provide the estimation error covariance. To reset the PDA filter, estimated state and estimation error covariance are essential. Estimated state can be obtained from any FIR filter, but the estimated error covariance cannot. Most FIR filters perform batch-processing without initialization and do not produce estimation error covariance. However, the minimum variance FIR (MVF) filter can provide the estimation error covariance. The EMVF filter is a nonlinear version of the MVF filter, and hence, it is applicable to the nonlinear indoor localization measurement models. The reset of PDA filter can be done by re-initializing the PDA filter using the estimated state and the estimation error covariance obtained from the EMVF filter. The EMVF filter does not require initialization and performs batch-processing using several recent measurements. In FIR filtering, the length of the time horizon of the measurements to be used is called the horizon (memory) size. The horizon size of the EMVF filter is denoted by M. An estimate of the current state at time k is produced using measurements on the horizon $[k-M,k-1]$ through the EMVF filter equations shown in Algorithm 3.  

Algorithm 3: EMVF filter.

Because the PDA filter uses multiple measurements (i.e., a mixture of true/false measurements) at each time step, whereas the EMVF filter does not, thus, the most suitable measurement should be chosen for the EMVF filter. We exploit the association probabilities computed at each time step by the PDA filter for the EMVF filter. The measurement with the maximal association probability is selected as the most suitable measurement and saved in the memory.   

Algorithm 4: Localization based on HPFF.

We then design a failure detection method. A localization failure is indicated by the following signs:

• There is no surviving measurement after the validation process, and the PDA filter performs only a time update.
• The predicted measurement is too far away from the actual measurement in terms of a certain statistical distance (e.g., the Mahalanobis distance).

Thus, we determine that a localization failure has occurred if there is no surviving measurement and the Mahalanobis distance between the predicted and actual measurements exceeds a threshold. The Mahalanobis distance can be computed as

$$\begin{array}{cc}\hfill {\mathcal{D}}_k& =({\mathbf{z}}_k^{*}-{\widehat{\mathbf{z}}}_k){\mathbf{R}}_k^{-1}({\mathbf{z}}_k^{*}-{\widehat{\mathbf{z}}}_k),\hfill \\ \hfill {\mathbf{z}}_k^{*}& ={[{\overline{z}}_1{\overline{z}}_2\cdots {\overline{z}}_n]}^T,\hfill \\ \hfill {\widehat{\mathbf{z}}}_k& ={\mathbf{h}}_k\left({\widehat{\mathbf{x}}}_k^{-}\right),\hfill \end{array}$$

(13)

where ${\mathbf{z}}_k^{*}$ is the measurement vector comprising the n measurements. The threshold is obtained from the ${\chi }_m^2$ values in the chi-square table. For instance, if the confidence interval is $99\%$ and the dimension of the measurement vector is five, the value of ${\chi }_m^2$ is $15.086$. When a failure is detected, the EMVF filter operates and produces an estimated location. In addition, the failed PDA filter is reset using the output of the EMVF filter. The overall algorithm for HPFF-based localization is summarized in Algorithm 4.

6. Simulation

The results of an indoor localization simulation to demonstrate the performance of the proposed HPFF-based localization algorithm are presented in this section. The localization algorithms were evaluated under both LOS and NLOS conditions in the simulations. The ability of EKF-based localization algorithms to mitigate the NLOS effect using NN measurements is first evaluated. The PDA filter and HPFF are then compared to demonstrate the reliability of HPFF. Finally, we compare HPFF and EKF using NN measurements under various NLOS conditions.

Figure 2 shows the simulation scenario in which the transmission of wireless signals along a long corridor is frequently disturbed by people and obstacles. The corridor is 30 m long and 4 m wide. Eight transmitters (fixed nodes) are installed along the two sides of the corridor. A person with a receiver (mobile node) travels along the corridor for a minute (60 s). The 2D coordinates of the person are estimated using the localization algorithms under LOS and NLOS conditions. We consider the following two NLOS conditions:

• A transmitter is temporarily NLOS for five seconds.
• A transmitter is constantly NLOS for the entire simulated duration.

The measurement and process noise variances were set as $r=0.5$ and $q=0.5$, respectively. The sampling interval T was set to 0.5 s. The design parameters for the PDA filter were set as follows: The detection probability was set to $P_D=1$, the gate probability to $P_G=0.99$, and the gate threshold to $\gamma =11.345$. The horizon size of EMVF filter was set to $M=4$. The numbers of measurements used for HPFF were set as $n=5$ and $m=3$.

The TOA measurements under NLOS conditions were generated using the following noise model:

$$\begin{array}{cc}\hfill \overline{v}& =v+b,\hfill \end{array}$$

(14)

where b is the NLOS error, and we assume that it is a Gaussian noise with the mean and variance, ${\mu }_b$ and ${\sigma }_b^2$, respectively. ${\mu }_b$ is not zero and that the NLOS error is generally much larger than the zero-mean measurement noise v. We set the NLOS error parameters to ${\mu }_b=5$ and ${\sigma }_b=5$.

We performed 100 Monte-Carlo (MC) simulations for each scenario. The root mean square (RMS) position error at time k was computed as

$$\begin{array}{c}\hfill \mathrm{RMS}=\sqrt{\frac{1}{100-N_d}\sum _{i=1}^{100-N_f}[{(x_k^i-{\widehat{x}}_k^i)}^2+{(y_k^i-{\widehat{y}}_k^i)}^2]},\end{array}$$

(15)

where $(x_k,y_k)$ are the true 2D positions and $({\widehat{x}}_k,{\widehat{y}}_k)$ are the estimated positions. The root time-averaged mean square (RTAMS) position error was computed as

$$\begin{array}{c}\hfill \mathrm{RTAMS}=\sqrt{\frac{1}{(100-N_d)(t_{\max }-t_{\mathrm{init}}+1)}\sum _{k=t_{\mathrm{init}}}^{t_{\max }}\sum _{i=1}^{100-N_d}[{(x_k^i-{\widehat{x}}_k^i)}^2+{(y_k^i-{\widehat{y}}_k^i)}^2]},\end{array}$$

(16)

where the superscript i indicates that the data is obtained in the i-th MC simulation, $t_{\max }$ is the final time step of the simulation, and $t_{\mathrm{init}}$ is a time index after which the filters are initialized. In our simulation, $t_{\max }=120$ and $t_{\mathrm{init}}=3$. During the computation of the RMS/RTAMS position errors, the data for divergent localizations with diverging errors were discarded. This is because the very large errors in divergent localizations led to abnormal increases in the RMS/RTAMS position errors. We counted the number of divergent localizations in the 100 MC simulations and denoted is as $N_d$. A localization was judged to be divergent if the RTAMS position error in the localization simulation exceeded 5 m.

6.1. Effect of the Use of Nearest Neighbor Measurements

We simulated EKF-based localization based on Algorithm 1 with three different numbers of NN measurements $n=3$, 5, and 8. Figure 3 shows the localization simulation results for the long corridor scenario under the LOS condition. In this figure, EKF3 and EKF5 denote EKFs based on three and five NN measurements, respectively. EKF8 denotes an EKF that uses all eight measurements and is hence not the proposed NN measurement-based algorithm but a conventional algorithm. In Figure 3a, the trajectories from all the three algorithms are indistinguishable because their differences are insignificant. The RMS position errors in this simulation are shown in Figure 3b. We can see that the error decreased when a greater number of measurements was used. However, the differences are insignificant (≤3 cm).

We next evaluated the three EKF-based algorithms under NLOS conditions. One of the eight transmitters was set to a temporary NLOS condition for five seconds ($50\le k<60$). The simulation results are shown in Figure 4. The position error of the EKF using all eight measurements (i.e., EKF8) exhibited a sharp increase in all eight cases while that of the EKF using five NN measurements (EKF5) exhibited a significant increase when FN3, FN4, FN5, and FN6 were NLOS. NLOS effects were avoided in EKF5 in four of the eight cases. The EKF using three NN measurements (EKF3) exhibited a significant increase in error in only two cases. Moreover, EKF3 exhibited the fastest decreases in error after the significant increases, as shown in Figure 4c,d. The sharp increases in errors occurred when EKFs cannot avoid using the TOA measurements from the FNs in NLOS situations. The temporary NLOS situation occurred when $50\le k<60$. At that time, the mobile node was passing by halfway of the corridor, and FNs 3 and 4 were nearest to the mobile node. The NN methods uses nearest (i.e., smallest TOA) measurements. Thus, all three EKF algorithms could not help using the measurements from FNs 3 and 4. When FNs 3 and 4 were in NLOS situation for $50\le k<60$, all three EKFs used TOA measurements from them and exhibited sharp error increases. In conclusion, EKF3 provided the most reliable localization under NLOS conditions, and there was little difference between the EKFs under the LOS condition.

6.2. Comparison of PDAF and HPFF

We now evaluate PDAF- and HPFF-based localization in the long corridor scenario by simulating Algorithms 2 and 4 under LOS and NLOS conditions. Figure 5 shows the localization results for the LOS scenario. The HPFF results are identical to the PDAF results. This is because the main filter in the HPFF did not fail and the assisting filter did not operate. Thus, HPFF performed similarly to pure PDAF.

Figure 6 shows the localization results for temporary NLOS conditions in which one of the eight FNs was NLOS for five seconds ($50\le k<60$). PDAF did not exhibit a drastic error increase or failure under temporary NLOS conditions. This is because PDAF has the ability to deal with false measurements. PDAF performs a measurement update process by merging true and false measurements according to their association probabilities. Thus, PDAF is not easily affected by false measurements. In localization problems, PDAF’s ability can be used to deal with measurements contaminated by NLOS errors. Simulations in Figure 6 demonstrates that the PDAF’s ability. If PDAF is successful, HPFF performs identically to PDAF. Under the temporary NLOS conditions, both PDAF and HPFF successfully performed localization, whereas the the EKFs exhibited degraded localization accuracy.

We tested PDAF and HPFF under more severe conditions. We assumed that one of the FNs was constantly in the NLOS condition over the entire simulated time. Figure 7 shows the localization results obtained under this scenario. PDAF exhibited degraded performance and larger errors compared with those of HPFF in six cases (Figure 7a–f). When FNs 7 and 8 were in NLOS (Figure 7g,h), the PDAF used the measurements from FNs 1–6, which are closer to the mobile node than FNs 7 and 8. Thus, PDAF could successfully work for most of simulation time. In the constant NLOS conditions, PDAF with NN measurements cannot avoid using the NLOS-contaminated measurements. PDAF unequally used the measurements by imposing association probabilities on them. Thus, PDAF exhibited much better accuracy than the EKFs. However, if there was no surviving measurement after validation process in PDAF algorithm, the PDAF skipped the measurement update process. This situation occurred due to the constant NLOS conditions and led to degradation of PDAF’s estimation accuracy. On the contrary, HPFF can use the assisting EMVF filter in the same situation. EMVF filter is not very accurate but is robust against various types of errors and can provide state estimates under such severe conditions. When the main PDAF algorithm failed due to lack of surviving measurements, the HPFF algorithm operated the assisting EMVF filter. State estimates produced by the EMVF filter become the output of HPFF and were used to reset the main PDAF. In Figure 7a–d, although the initial errors of HPFF were larger than those of PDAF, the HPFF error decreased rapidly. In the cases in which FN7 and FN8 were in a constant NLOS condition, localization using PDAF was successful and HPFF performed identically to pure PDAF.

6.3. Comparison of HPFF and EKF3

Finally, we compared the proposed HPFF with EKF3 by performing the indoor localization simulation under temporary/constant NLOS conditions. The simulation results are shown in Figure 8 and Figure 9. Under the temporary NLOS conditions, HPFF exhibited smaller errors than EKF3. HPFF did not exhibit sharp increases in errors while EKF3 did in Figure 8c,d. For the time interval of LOS condition, HPFF was more accurate than EKF3. Under the constant NLOS conditions, EKF3 produced significant error peaks larger than $1\mathrm{m}$. The EKF’s error peaks occurred when the FN in NLOS was near the mobile node. In Figure 9a,b, the error peak occurs in the early stage because the FNs 1 and 2 are near the mobile node at the beginning of simulation. In Figure 9g,h, the error peak occurs in the late stage because the FNs 7 and 8 are near the mobile node at the end of simulation. HPFF did not exhibit a drastic increase in error except in the early stage. Although the error increased in the early stage, the error gradually decreased because the assisting EMVF filter recovered the main filter. Overall, HPFF consistently provided reliable localization without significant increases in error.

The RTAMS position errors for the NLOS conditions are listed in

Table 1. RTAMS position errors (unit: meter) in temporary NLOS conditions.

Localization Algorithm	FN in NLOS Condition
	FN1	FN2	FN3	FN4	FN5	FN6	FN7	FN8
EKF3	0.18	0.17	0.24	0.24	0.18	0.18	0.18	0.17
PDAF	0.15	0.15	0.15	0.15	0.15	0.15	0.15	0.15
HPFF	0.15	0.15	0.15	0.15	0.15	0.15	0.15	0.15

and

Table 2. RTAMS position errors (unit: meter) in constant NLOS conditions.

Localization Algorithm	FN in NLOS Condition
	FN1	FN2	FN3	FN4	FN5	FN6	FN7	FN8
EKF3	0.36	0.37	0.59	0.59	0.59	0.60	0.39	0.39
PDAF	0.38	0.32	0.53	0.46	0.19	0.19	0.16	0.16
HPFF	0.26	0.22	0.38	0.36	0.17	0.17	0.16	0.16

. The three algorithms in the tables are proposed algorithms that use NN measurements, and EKF8 is the only conventional algorithm. EKF8 was excluded in the comparison, because it has been shown that EKF8 exhibits poor performance under NLOS conditions. Except for the case of FN1 being under a constant NLOS condition, in which the RTAMS error of PDAF larger, the RTAMS errors of PDAF are much smaller than those of EKF3. PDAF outperformed the EKF3 when it operated normally. However, PDAF sometimes diverged and produced large errors under severe NLOS conditions.

Table 3. Percentage of divergent localizations in constant NLOS conditions.

Localization Algorithm	FN in NLOS Condition
	FN1	FN2	FN3	FN4	FN5	FN6	FN7	FN8
EKF3	0	0	0	0	0	0	0	0
PDAF	3	6	24	25	1	0	0	0
HPFF	0	0	0	0	0	0	0	0

lists the percentage of divergent localizations for the three algorithms. Both HPFF and EKF3 did not diverge all, and only the PDAF diverged. This is because the measurement update process was skipped in the PDAF algorithm if there were no surviving measurements after the measurement validation process. Thus, PDAF sometimes failed to track the target and the error gradually increased over time. In contrast, the divergence problem due to error accumulation was overcome in HPFF because HPFF could recover from failures by resetting the main filter using data obtained from the assisting filter.

7. Conclusions

In this study, we proposed a new indoor localization algorithm based on HPFF to overcome the problems that result from NLOS conditions. We demonstrated using NN measurements can mitigate NLOS effects. Indoor localization simulations were performed using the scenario in which a person with a mobile node travels along a long corridor where eight fixed nodes were installed. EKF3, which used three NN measurements, was more reliable in NLOS conditions compared with EKFs that used more measurements. We applied PDAF to the indoor localization problem and found that its RMS position errors were generally smaller than those of EKF3. However, PDAF sometimes exhibited significantly large errors or even diverged when there is no surviving measurement after the measurement validation process. Thus, we proposed HPFF, which can recover the PDAF algorithm from its failures. HPFF did not diverge at all and achieved superior localization accuracy compared with both EKF3 and pure PDAF under temporary/constant NLOS conditions. The RTAMS position errors of HPFF were under 0.4 m. Therefore, the proposed HPFF can provide accurate and reliable indoor localization. In the future, we will develop a different version of HPFF for other applications, such as three-dimensional (3D) localization of unmanned aerial vehicles (UAVs).