A real-time locating system for localization of high-speed mobile objects

Abstract

This article addresses a novel real-time locating system for localization of high-speed mobile objects in fading environments. The proposed locating system exploits time difference of arrival measurements based on ultra-wideband signals. However, the ultra-wideband signals cause a frequency-selective fading due to their short time duration, which induces severe inter-symbol interference. Moreover, high-speed objects cause fast fading due to large Doppler spread. Therefore, the fading cases considerably reduce the localization performance. The presented locating system relies on a new localization approach in order to overcome the fading issues, which utilizes a modification of extended Kalman filtering. Especially, the suggested locating method works well even in the zero time difference of arrival case, which occurs due to a very deep fading. Experiment results verify that the proposed real-time locating system gives excellent localization performance in severe fading environments. The results also exhibit that the presented locating system is superior to the conventional locating systems in the localization of high-speed mobile objects under fading environments.

Introduction

A lot of attention has been paid to real-time locating system (RTLS) for localization of high-speed mobile objects such as mobile robots or autonomous vehicles. Recently, the RTLS is considered a crucial component in the area of Internet-of-things (IoT) including machine-to-machine (M2M) and device-to-device (D2D) communications. ¹ Especially, in the cases of M2M/D2D under cooperative communications, ² the communication performance highly relies on the localization performance. If line-of-sight (LOS) signals are available in wireless channels, the RTLS usually exploits time difference of arrival (TDoA) from the LOS signals.^3–6 Otherwise, the RTLS can utilize received signal strength (RSS) or direction of arrival (DoA).^7,8 For accurate positioning, TDoA is considered a more desirable measurement in the RTLS. In wireless environments, the RTLS frequently utilizes the ultra-wideband (UWB) signals in order to considerably increase the resolution of TDoA measurements. Higher resolution leads to better localization performance in the RTLS. The UWB signals require very short time duration, which occupies very large bandwidth. ⁹ Especially, the IEEE802.15.4a specifies the standard of the UWB signals for the positioning. ¹⁰ When TDoA measurements are used, the RTLS relies on direct method^3,5,11 or extended Kalman filter (EKF)^4,12,13 The direct method exhibits the simplest calculation but renders poor locating performance if TDoA measurements are noisy. The EKF approach gives a fairly good performance even with noisy TDoA measurements. However, the EKF method is vulnerable to fading effects in UWB applications. Especially, the UWB signals exhibit frequency-selective fading due to the huge bandwidth. The fading causes inter-symbol interference (ISI), ¹⁴ which in turn reduces the accuracy of TDoA. In the case that mobile objects move with high speed, fast fading occurs due to large Doppler spread, ¹⁴ which also deteriorates the localization performance of the EKF.

This article presents a novel RTLS for localization of high-speed mobile objects in fading environments. The RTLS utilizes the UWB signals based on the IEEE802.15.4a in order to achieve TDoA measurements with high resolution. The RTLS relies on a proposed EKF approach in order to overcome the fading effects. The suggested EKF scheme is a modification of the conventional EKF technique. The modified EKF method includes a presented criterion. The criterion determines if the TDoA measurements are significantly distorted due to the ISI effects in the frequency-selective fading case. If the distortion is too severe, the corresponding TDoA measurements are excluded in the update process of the EKF. The exclusion significantly contributes to the robustness of the EKF to the frequency-selective fading effect. The modified EKF also includes a management technique for the zero-TDoA case. In the case of high-speed mobile objects, the fast fading sometimes causes a very deep fading, which leads to the zero-TDoA case. Any TDoA measurement is unavailable during the update process of the EKF in the zero-TDoA case. Sometimes, a tracking divergence occurs during the update process. The tracking divergence denotes a divergence of location estimates. In the case of tracking divergence, the location estimates are significantly deviated from the range of a track and never back to the range of the track. The suggested management method profoundly contributes to the divergence avoidance. Therefore, the proposed EKF version overcomes the weakness of the conventional EKF in fading environments.

Experiment results show that the novel RTLS gives excellent locating performance in severe fading environments. For the experiment of high-speed objects, the mobile objects move with the maximum speed of 60 km/h. The results exhibit that the modified EKF method outperforms the direct method and the conventional EKF approach in several tracks. They also confirm that the presented RTLS can accurately localize the high-speed mobile objects such as autonomous vehicles in fading environments.

In this article, the main motivations of our research are as follows:

• Recently, autonomous vehicles have emerged as main applications in the area of consumer electronics.

• Autonomous vehicle systems usually require real-time locating information for their proper operation.

• This indicates that a locating system is required for localization of high-speed objects such as autonomous vehicles.

• However, most locating systems focus on indoor localization or outdoor localization of low-speed objects.

• Therefore, we propose a RTLS for localization of high-speed objects such as autonomous vehicles.

In addition, the main contributions of our research are as follows:

• The proposed RTLS can considerably mitigate the frequency-selective fading and the deep fading effects, which leads to accurate localization of high-speed objects.

• The proposed RTLS is also very robust to long-time zero-TDoA case due to a very deep fading.

• Therefore, the proposed RTLS is very suitable for localization of high-speed objects such as autonomous vehicles.

Related works

The related works include two conventional location-estimation methods. The location estimations are based on direct method and EKF. To our best knowledge, the current positioning works are classified into the two conventional methods. The positioning works^3,5,11 belong to one category: Direct method. In addition, the positioning techniques^4,12,13 belong to the other category: EKF.

Many localization systems rely on TDoA-based UWB approaches for mobile objects. Note that the UWB techniques are also classified into the two conventional methods. However, most UWB methods focus on indoor localization.^10,15 The UWB methods are just applicable to low-speed objects. Recently, a TDoA algorithm was analyzed for rapid moving target with UWB tag. ¹⁶ However, the algorithm is just an iterative version of the direct method. Moreover, this method is just applicable to the target moving on a straight line. This indicates that the UWB scheme ¹⁶ is not suitable for localization of high-speed objects such as autonomous vehicles. Unlike the UWB approaches,^10,15,16 the proposed RTLS focuses on outdoor localization and is very suitable for localization of high-speed objects (including autonomous vehicles) moving on various long-distance tracks.

The conventional methods utilize TDoA measurements in order to locate targets. Figure 1 illustrates the service model for TDoA acquisition in generic RTLS. In the figure, the target is denoted as $R$ and the target position $(x,y)$ needs to be estimated. As indicated in Figure 1, the target sends its signal (such as UWB signal) to $N+1$ stations. In Figure 1, each station is denoted as $P_i(i=0,1,2,\dots ,N)$ . When each station receives the target signal, the station measures time-of-arrival (ToA) from the received signal. In Figure 1, the ToA measurement for station $P_i$ is denoted as $T_i(i=0,1,2,\dots ,N)$ . From the $N+1$ ToA measurements, $N$ TDoA values are calculated as follows

$${\tau }_i=T_i-T_0$$

(1)

where ${\tau }_i$ denotes the TDoA value for station $P_i(i=0,1,2,\dots ,N)$ . In equation (1), $T_0$ indicates the reference ToA. In Figure 1, $P_0$ denotes the reference station for the reference ToA.

Figure 1.

The service model for TDoA acquisition in generic RTLS.

The direct method directly finds the target location $(x,y)$ using the TDoA measurements and the station positions. From equation (1), the following equation can be set up

$$c{\tau }_i=c{T}_i-c{T}_0=R_i-R_0$$

(2)

where $c$ denotes the light speed and $i=0,1,2,\dots ,N$ . In equation (2), $R_i$ and $R_0$ denote the distances from station i and reference station to the target, respectively. If both sides are squared in equation (2), the following equation can be achieved

$$c^2{{\tau }_i}^2+2c{\tau }_i{R}_0+{R_0}^2-{R_i}^2=0$$

(3)

If equation (3) is divided by $c{\tau }_i$ , the equation can be changed as follows

$$c{\tau }_i+2R_0+\frac{{R_0}^2-{R_i}^2}{c{\tau }_i}=0$$

(4)

If i is set to 1 in equation (4), the following equation can be found

$$c{\tau }_1+2R_0+\frac{{R_0}^2-{R_1}^2}{c{\tau }_1}=0$$

(5)

If equation (4) is subtracted from equation (5), the following equation can be achieved

$$c{\tau }_i-c{\tau }_1+\frac{{R_0}^2-{R_i}^2}{c{\tau }_i}-\frac{{R_0}^2-{R_1}^2}{c{\tau }_1}=0$$

(6)

where $i=2,3,\dots ,N$ . In Figure 1, the squared distance from each station to the target can be expressed as follows

$${R_i}^2={(x-x_i)}^2+{(y-y_i)}^2$$

(7)

If i is set to 0 and 1 in equation (7), the corresponding equations can be expressed as follows

$${R_0}^2={(x-x_0)}^2+{(y-y_0)}^2$$

(8)

$${R_1}^2={(x-x_1)}^2+{(y-y_1)}^2$$

(9)

If equation (8) is subtracted from equation (9), the following equation can be achieved

$$R_0^2-R_1^2=x_0^2+y_0^2-x_1^2-y_1^2-2x_{0}x-2y_{0}y+2x_{1}x+2y_{1}y$$

(10)

If equation (8) is also subtracted from equation (7), the following equation can be achieved

$$R_0^2-R_i^2=x_0^2+y_0^2-x_i^2-y_i^2-2x_{0}x-2y_{0}y+2x_{i}x+2y_{i}y$$

(11)

where $i=2,3,\dots ,N$ . If equations (10) and (11) are substituted into equation (6), the simultaneous equations can be achieved as follows

$$A_{i}x+B_{i}y=C_i$$

(12)

where $i=2,3,\dots ,N$ . In equation (12), $A_i$ , $B_i$ , and $C_i$ are calculated using equations (13)–(15), respectively

$$A_i=\frac{2}{c{\tau }_i}(x_i-x_0)-\frac{2}{c{\tau }_1}(x_1-x_0)$$

(13)

$$B_i=\frac{2}{c{\tau }_i}(y_i-y_0)-\frac{2}{c{\tau }_1}(y_1-y_0)$$

(14)

$$C_i=\frac{1}{c{\tau }_1}(x_0^2+y_0^2-x_1^2-y_1^2)-\frac{1}{c{\tau }_i}(x_0^2+y_0^2-x_i^2-y_i^2)+c{\tau }_1-c{\tau }_i$$

(15)

In equations (13)–(15), $i=2,3,\dots ,N$ . In the simultaneous equations of equation (12), the target location $(x,y)$ can be determined if $N\ge 3$ in equations (12)–(15). Therefore, at least four stations are required for estimation of the location $(x,y)$ in Figure 1. As indicated in equation (12), the parameters ( $A_i$ , $B_i$ , and $C_i$ ) should have exact values for accurate determination of location $(x,y)$ . The exactness of the parameters heavily depends on the TDoA measurements as shown in equation (13)–(15). Therefore, noisy TDoA values significantly degrade the performance of the location estimation.

The EKF is frequently used for location estimation with noisy TDoA measurements. The EKF consists of prediction and update steps. ⁴ In the prediction step, a priori state vector ${\mathbf{X}}_k^{-}$ is generated as follows

$${\mathbf{X}}_k^{-}=\mathbf{F}{\mathbf{X}}_{k-1}+{\mathbf{W}}_{k-1}$$

(16)

where $\mathbf{F}$ and ${\mathbf{W}}_{k-1}$ denote state transition matrix and process noise vector, respectively. In equation (16), the previous state vector ${\mathbf{X}}_{k-1}$ is given as follows

$${\mathbf{X}}_{k-1}={\left[p_{x,k-1}{p}_{y,k-1}{v}_{x,k-1}{v}_{y,k-1}{a}_{x,k-1}{a}_{y,k-1}\right]}^T$$

(17)

where $p_{x,k-1}$ and $p_{y,k-1}$ denote the x-position and the y-position, respectively of target at the (k– 1)th iteration (previous iteration). In equation (17), $v_{x,k-1}$ and $v_{y,k-1}$ denote the x-direction velocity and the y-direction velocity, respectively, of target at the (k– 1)th iteration (previous iteration). In addition, $a_{x,k-1}$ and $a_{y,k-1}$ denote the x-direction acceleration and the y-direction acceleration, respectively, of target at the (k– 1)th iteration (previous iteration) in equation (17). The process noise vector of equation (16) includes the following elements

$${\mathbf{W}}_{k-1}={\left[\frac{{\eta }_{x,k-1}\mathrm{\Delta }{T}^2}{2}\frac{{\eta }_{y,k-1}\mathrm{\Delta }{T}^2}{2}{\eta }_{x,k-1}\mathrm{\Delta }T{\eta }_{y,k-1}\mathrm{\Delta }T{\eta }_{x,k-1}{\eta }_{y,k-1}\right]}^T$$

(18)

where ${\eta }_{x,k-1}$ and ${\eta }_{y,k-1}$ denote the zero-mean white Gaussian noise (WGN) components for $a_{x,k-1}$ and $a_{y,k-1}$ , respectively in equation (17). ¹⁷ In equation (18), $\mathrm{\Delta }T$ denotes the time interval between current and previous iterations. From equation (16), a priori error covariance matrix ${\mathbf{P}}_k^{-}$ is defined as follows

$${\mathbf{P}}_k^{-}=E\left[({\mathbf{X}}_k^{-}-E\left[{\mathbf{X}}_k^{-}\right]){({\mathbf{X}}_k^{-}-E\left[{\mathbf{X}}_k^{-}\right])}^T\right]$$

(19)

where E[ ] denotes the operator of ensemble average. From equation (19), the a priori covariance matrix ${\mathbf{P}}_k^{-}$ can iteratively be calculated as follows

$${\mathbf{P}}_k^{-}=\mathbf{F}{\mathbf{P}}_{k-1}{\mathbf{F}}^T+{\mathbf{Q}}_{k-1}$$

(20)

where ${\mathbf{Q}}_{k-1}$ denotes the covariance matrix of ${\mathbf{W}}_{k-1}$ and is expressed as

$${\mathbf{Q}}_{k-1}=E\left[{\mathbf{W}}_{k-1}{\mathbf{W}}_{k-1}^T\right]$$

(21)

In the update state, the a priori state vector ${\mathbf{X}}_k^{-}$ and the a priori covariance matrix ${\mathbf{P}}_k^{-}$ are updated into the a posteriori state vector ${\mathbf{X}}_k$ and the a posteriori covariance matrix ${\mathbf{P}}_k$ , respectively. For the updates, the observation vector at the kth iteration (current iteration) ${\mathbf{Y}}_k$ is generated using TDoA measurements as follows

$${\mathbf{Y}}_k={[c{\tau }_{1}c{\tau }_{2}c{\tau }_3\cdots c{\tau }_N]}^T$$

(22)

where ${\tau }_i$ denotes the TDoA measurement defined in equation (1). Using the a priori state vector ${\mathbf{X}}_k^{-}$ of equation (16), the observation vector ${\mathbf{Y}}_k$ can also be expressed as follows

$${\mathbf{Y}}_k=\mathbf{h}\left({\mathbf{X}}_k^{-}\right)+{\mathbf{V}}_k$$

(23)

where $\mathbf{h}$ and ${\mathbf{V}}_k$ denote nonlinear measurement matrix and measurement noise vector, respectively. The measurement matrix $\mathbf{h}$ includes the nonlinear formulas for ${\mathbf{Y}}_k$ of equation (22). Using the a priori covariance matrix ${\mathbf{P}}_k^{-}$ of equation (20), the Kalman gain matrix $\mathbf{K}$ is achieved as follows

$$\mathbf{K}={\mathbf{P}}_k^{-}{\mathbf{H}}^T{({\mathbf{HP}}_k^{-}{\mathbf{H}}^T+{\mathbf{R}}_k)}^{-1}$$

(24)

where ${\mathbf{R}}_k$ denotes the covariance matrix of ${\mathbf{V}}_k$ and is expressed as

$${\mathbf{R}}_k=E\left[{\mathbf{V}}_k{\mathbf{V}}_k^T\right]$$

(25)

In equation (24), $\mathbf{H}$ is defined as follows

$$\mathbf{H}=\frac{\partial \mathbf{h}}{\partial {\mathbf{X}}_k^{-}}$$

(26)

Using equations (22)–(24), the a priori state vector ${\mathbf{X}}_k^{-}$ can be updated into the a posteriori state vector ${\mathbf{X}}_k$ as follows

$${\mathbf{X}}_k={\mathbf{X}}_k^{-}+\mathbf{K}[{\mathbf{Y}}_k-\mathbf{h}({\mathbf{X}}_k^{-})]$$

(27)

Then, the a priori covariance matrix ${\mathbf{P}}_k^{-}$ can be updated into the a posteriori covariance matrix ${\mathbf{P}}_k$ using equations (24) and (26) as follows

$${\mathbf{P}}_k=(\mathbf{I}-\mathbf{KH}){\mathbf{P}}_k^{-}$$

(28)

where $\mathbf{I}$ denotes the identity matrix.

At the next iteration, the a posteriori state vector ${\mathbf{X}}_k$ replaces the previous state vector ${\mathbf{X}}_{k-1}$ of equation (16). In addition, the a posteriori covariance matrix ${\mathbf{P}}_k$ replaces ${\mathbf{P}}_{k-1}$ of equation (20). Then, the EKF repeats the process and the update states.

As indicated in equations (20) and (24), the performance of the EKF heavily depends on the accuracies of equations (21) and (25). Therefore, the EKF produces fairly accurate location estimates even with noisy TDoA measurements if ${\mathbf{Q}}_{k-1}$ and ${\mathbf{R}}_k$ are exactly known. Note that ${\mathbf{Q}}_{k-1}$ and ${\mathbf{R}}_k$ are time-varying vectors as indicated in equations (21) and (25), respectively. Usually, it is not easy to accurately estimate the time-varying vectors at each iteration. In the case of location estimation, the process noise components have the Gaussian property in equation (18). Therefore, the covariance matrix ${\mathbf{Q}}_{k-1}$ of equation (21) exhibits a stationary property: ${\mathbf{Q}}_{k-1}=\mathbf{Q}$ . This indicates that the covariance matrix of the process noise vector can easily be managed in the case of location estimation. However, the measurement noise vector ${\mathbf{V}}_k$ of equation (23) is affected by TDoA distortions under fading environments. Moreover, the fading effects rapidly change as the objects move with high speed. In other words, the covariance matrix ${\mathbf{R}}_k$ of equation (25) exhibits a nonstationary property. In principle, the EKF is applicable to even the nonstationary case. ¹³ However, the estimation of ${\mathbf{R}}_k$ is required in each iteration, which may lead to a failure of real-time location estimation.

Proposed RTLS

System model

Figure 2 illustrates the system model for the proposed RTLS. In the figure, the system model consists of transmitters, receivers, one control server, and one locating server. In Figure 2, transmitters and receivers follow the IEEE802.15.4a specification. ¹⁸ Therefore, they utilize UWB signals. In the system model under consideration, each mobile object employs one transmitter for generation of the UWB signal. Therefore, each transmitter is attached on the corresponding mobile object for location estimation. For the positioning and the identification of each object, the corresponding transmitter generates the UWB signal and delivers the identification number, respectively. As depicted in Figure 2, the receivers collect the UWB signal and the identification number from each mobile object. Using the received UWB signal, each receiver finds the arrival time-stamp of the signal. The arrival time-stamp indicates the ToA measurement, which is denoted as $T_i$ in equation (1). Therefore, each station (including the reference station) employs one receiver in order to measure the ToA value from the received UWB signal. In Figure 2, all the receivers are connected to the control server. The server collects the ToA values from the receivers. Among the received ToA values, the server selects N + 1 smallest ToA values. A smaller ToA value implies shorter path between station and mobile object. The UWB signal tends to be more reliable over a shorter path. Therefore, a smaller ToA value is more likely to be accurate. After the selection, the control server calculates N TDoA values using equation (1). In the calculation, the control server uses the smallest ToA as the reference ToA $(T_0)$ of equation (1). Then, the server computes N distance values using the N TDoA values as indicated in equation (2). Finally, the control server sends the N distance values to the locating server as shown in Figure 2. As soon as the locating server receives the N distance values, it estimates the location of the mobile object associated with the distances.

Figure 2.

The system model for the proposed RTLS.

The UWB signals usually exhibit accurate ToA estimation of less than 3 ns, which indicates less than 1 m distance uncertainty. ¹⁹ In such case, the locating server of Figure 2 can employ the conventional method (direct method or EKF). However, the UWB accuracy (less than 3 ns) can just be achieved in the case of a stationary target under a single-path additive white Gaussian noise (AWGN) channel. ¹⁹ Therefore, the conventional locating method is unavailable in the case of high-speed mobile objects under a heavily fading channel. For accurate location estimation of high-speed mobile objects under fading environments, the modified EKF method is proposed in this article. Thus, the locating server of Figure 2 employs the modified EKF for the accurate location estimation.

Modified EKF

The modified EKF is also composed of prediction and update steps like the conventional EKF. The modified version follows the same prediction step as the conventional EKF since the covariance matrix of equation (21) is stationary in the location estimation. However, the covariance matrix of equation (25) is nonstationary in fading environments. Therefore, the update step is revised in order to deal with the nonstationary situation.

In the update state of the modified EKF, $\mathbf{h}({\mathbf{X}}_k^{-})$ of equation (23) is defined as follows

$$\mathbf{h}({\mathbf{X}}_k^{-})=\left[\begin{array}{c}\sqrt{{(p_{x,k}^{-}-x_1)}^2+{(p_{y,k}^{-}-y_1)}^2}-\sqrt{{(p_{x,k}^{-}-x_0)}^2+{(p_{y,k}^{-}-y_0)}^2}\hfill \\ \multicolumn{1}{c}{\sqrt{{(p_{x,k}^{-}-x_2)}^2+{(p_{y,k}^{-}-y_2)}^2}-\sqrt{{(p_{x,k}^{-}-x_0)}^2+{(p_{y,k}^{-}-y_0)}^2}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\sqrt{{(p_{x,k}^{-}-x_N)}^2+{(p_{y,k}^{-}-y_N)}^2}-\sqrt{{(p_{x,k}^{-}-x_0)}^2+{(p_{y,k}^{-}-y_0)}^2}}\end{array}\right](29)$$

(29)

where $p_{x,k}^{-}$ and $p_{y,k}^{-}$ denote a priori x-position and y-position, respectively, of the target at the kth iteration (current iteration). From equation (17), ${\mathbf{X}}_k^{-}$ is defined as follows

$${\mathbf{X}}_k^{-}={\left[p_{x,k}^{-}{p}_{y,k}^{-}{v}_{x,k}^{-}{v}_{y,k}^{-}{a}_{x,k}^{-}{a}_{y,k}^{-}\right]}^T$$

(30)

where $v_{x,k}^{-}$ , $v_{y,k}^{-}$ , $a_{x,k}^{-}$ , and $a_{y,k}^{-}$ denote the a priori x-direction velocity, y-direction velocity, x-direction acceleration, and y-direction acceleration, respectively, of the target at the kth iteration (current iteration). Using equations (29) and (30), $\mathbf{H}$ of equation (26) is expressed as follows

$$\mathbf{H}=\left[\begin{array}{cccc}{A}_1\hfill & B_1\hfill & 0\hfill & \begin{array}{cc}\begin{array}{cc}0\hfill & 0\hfill \end{array}\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{A_2}& B_2\hfill & 0\hfill & \begin{array}{cc}\begin{array}{cc}0\hfill & 0\hfill \end{array}\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & \begin{array}{cc}\begin{array}{cc}⋮\hfill & ⋮\hfill \end{array}\hfill & ⋮\hfill \end{array}\hfill \\ \multicolumn{1}{c}{A_N}& B_N\hfill & 0\hfill & \begin{array}{cc}\begin{array}{cc}0\hfill & 0\hfill \end{array}\hfill & 0\hfill \end{array}\hfill \end{array}\right]$$

(31)

where the expressions of $A_i(i=1,2,\dots N)$ and $B_i(i=1,2,\dots N)$ are given in equations (32) and (33), respectively

$$A_i=\frac{p_{x,k}^{-}-x_i}{\sqrt{{(p_{x,k}^{-}-x_i)}^2+{(p_{y,k}^{-}-y_i)}^2}}-\frac{p_{x,k}^{-}-x_0}{\sqrt{{(p_{x,k}^{-}-x_0)}^2+{(p_{y,k}^{-}-y_0)}^2}}$$

(32)

where $i=1,2,\dots N$

$$B_i=\frac{p_{y,k}^{-}-y_i}{\sqrt{{(p_{x,k}^{-}-x_i)}^2+{(p_{y,k}^{-}-y_i)}^2}}-\frac{p_{y,k}^{-}-y_0}{\sqrt{{(p_{x,k}^{-}-x_0)}^2+{(p_{y,k}^{-}-y_0)}^2}}$$

(33)

where $i=1,2,\dots N$ .

Since it is difficult to find the nonstationary covariance matrix ${\mathbf{R}}_k$ of equation (25) in real-time mode, the $N\times N$ stationary covariance matrix $\mathbf{R}$ is used instead of ${\mathbf{R}}_k$ in the modified EKF as follows

$$\mathbf{R}=\left[\begin{array}{cccc}{\sigma }_d^2\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\sigma }_d^2\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \cdots \hfill & {\sigma }_d^2\hfill \end{array}\right]$$

(34)

where ${\sigma }_d^2$ denotes a constant variance. Therefore, $\mathbf{R}$ replaces ${\mathbf{R}}_k$ in the calculation of the Kalman gain of equation (24). In equation (34), the stationary property holds if each element of ${\mathbf{V}}_k$ is WGN in equation (23). However, the ${\mathbf{V}}_k$ elements include fading effects as well as WGN effects. To overcome the fading effects, the update of equation (27) is modified as follows

$${\mathbf{X}}_k={\mathbf{X}}_k^{-}+\mathbf{K}[{\hat{\mathbf{Y}}}_k-\mathbf{h}\left({\mathbf{X}}_k^{-}\right)]$$

(35)

where ${\hat{\mathbf{Y}}}_k=[{\hat{y}}_{1,k}{\hat{y}}_{2,k}\cdots {\hat{y}}_{N,k}{]}^T$ . The element $({\hat{y}}_{i,k})$ of ${\hat{\mathbf{Y}}}_k$ is determined as follows

$${\hat{y}}_{i,k}=\{\begin{array}{c}{h}_i,\mathrm{if}|c{\tau }_i-h_i|>\mathrm{thr}\hfill \\ \multicolumn{1}{c}{c{\tau }_i,\mathrm{otherwise}}\end{array}$$

(36)

where $i=1,2,\dots N$ . In equation (36), $\mathrm{thr}$ denotes a predefined threshold, which is proportional to ${\sigma }_d^2$ in equation (34). In equation (36), $c{\tau }_i$ and $h_i$ are the ith element $(i=1,2,\dots N)$ of ${\mathbf{Y}}_k$ in equation (22) and $\mathbf{h}({\mathbf{X}}_k^{-})$ in equation (29), respectively. As depicted in equation (36), if the difference between $c{\tau }_i$ and $h_i$ is larger than the threshold value $\mathrm{thr}$ , the value of ${\hat{y}}_{i,k}$ is set to $h_i$ . In this case, the process value $(h_i)$ is used as the value of ${\hat{y}}_{i,k}$ . Otherwise, the value of ${\hat{y}}_{i,k}$ is set to $c{\tau }_i$ . In other words, the value of ${\hat{y}}_{i,k}$ is determined by the ith TDoA measurement $({\tau }_i)$ . In equation (36), the difference between measured value $(c{\tau }_i)$ and process value $(h_i)$ indicates the ith noise element of ${\mathbf{V}}_k$ in equation (23). This implies that the noise element is profoundly affected by fading effects if the difference is larger than the threshold in equation (36). Therefore, the measured value $(c{\tau }_i)$ is discarded since it is heavily distorted. Since the measured value $(c{\tau }_i)$ is replaced by the process value $(h_i)$ , it does not have any effect on the calculation of the a posteriori state vector ${\mathbf{X}}_k$ as indicated in equations (35) and (36). This considerably contributes to the robustness of the modified EKF to severe fading effects.

Sometimes, a very deep fading causes a zero-TDoA case. During the zero-TDoA duration, only the process value $(h_i)$ is used as the value of ${\hat{y}}_{i,k}$ in equation (36) since any TDoA measurement is unavailable in the update process. The process value is usually contaminated by the process noise as indicated in equation (16). If the zero-TDoA duration is too long, the process noise is continuously accumulated. In this case, the difference between the measured value $(c{\tau }_i)$ and the process value $(h_i)$ may be larger than the threshold in equation (36) even if an available TDoA is accurate. This causes a divergence in the target localization. To avoid such a divergence, the method of equation (36) is extended as follows

$$\begin{array}{c}\mathrm{if}\mathrm{zero}\_\mathrm{TDoA}\_\mathrm{duration}>\max \_\mathrm{duration}\hfill \\ \mathrm{if}{\tau }_i\mathrm{is}\mathrm{available}\hfill \\ {\hat{y}}_{i,k}=c{\tau }_i\hfill \\ \mathrm{zero}\_\mathrm{TDoA}\_\mathrm{duration}=0\hfill \\ \mathrm{else}\hfill \\ {\hat{y}}_{i,k}=h_i\hfill \\ \mathrm{end}\hfill \\ \mathrm{else}\hfill \\ \mathrm{if}{\tau }_i\mathrm{is}\mathrm{available}\hfill \\ {\hat{y}}_{i,k}=\{\begin{array}{c}{h}_i,\mathrm{if}|c{\tau }_i-h_i|>\mathrm{thr}\hfill \\ \multicolumn{1}{c}{c{\tau }_i,\mathrm{otherwise}}\end{array}\hfill \\ \mathrm{else}\hfill \\ {\hat{y}}_{i,k}=h_i\hfill \\ \mathrm{end}\hfill \\ \mathrm{end}\hfill \end{array}$$

(37)

As indicated in equation (37), if the zero-TDoA duration exceeds the predefined maximum duration $(\max \_\mathrm{duration})$ and any TDoA value is available, the TDoA measurement is immediately used for ${\hat{y}}_{i,k}$ . Then, the zero-TDoA duration is reset, which enables the method of equation (36) at the next iteration as depicted in equation (37). This restrains the case that the difference $(|c{\tau }_i-h_i|)$ is always larger than the threshold of equation (36) after the predefined zero-TDoA duration, which in turn avoids a tracking divergence.

Experimental evaluation

Experimental evaluation exhibits the effectiveness of the proposed RTLS. For the experiment of high-speed objects, the mobile objects move with the maximum speed of 60 km/h. Each mobile object is equipped with the transmitter, which generates UWB signals. Figure 3(a) and (b) illustrates the transmitter and the receiver, respectively, for the RTLS. As illustrated in Figure 3(b), the receiver is installed on high tower, which increases the detection probability of the LOS component. The transmitter and the receiver follow the IEEE802.15.4a standard for the UWB signals. For reliable collection of the UWB signals, lots of receivers are installed around the experimental track, which is illustrated in Figure 4.

Figure 3.

The transmitter and the receiver for the proposed RTLS: (a) transmitter and (b) receiver.

Figure 4.

The receivers around the experimental track for reliable collection of the UWB signals.

Table 1 summarizes the experimental parameters for the proposed RTLS. The channel exhibits frequency-selective fading and fast fading because of large bandwidth (of UWB signal) and high speed (of mobile objects), respectively. For accurate localization of mobile objects, the transmitter of Figure 3(a) sends 30 UWB signals within 1 s. As indicated in Table 1, the number of the available receivers is larger than 50 in Figure 4. Table 1 also reveals that the control server generates 6 TDoA measurements using the received UWB signals. In other words, N is set to 6 in Figure 2. In this experiment, about 10 mobile objects move on the track independently.

Table 1.

Experimental parameters.

Parameter	Property/value
Channel	Frequency-selective fading/fast fading
Time-length of 30 UWB signals	1 s
Number of receivers	>50
Number of TDoA measurements	6
Number of mobile objects	About 10

UWB: ultra-wideband; TDoA: time difference of arrival.

Figure 5 illustrates the various tracks for the experimental evaluation of the proposed RTLS. As shown in the figure, four types of tracks are used in this experiment. Figure 5(a) indicates the track whose distance is 1000 m. Figure 5(b) shows the track of 1200 or 1300 m. In Figure 5(c), the distance of the track is 1400 m. Figure 5(d) shows the track whose distance is 1700, 1800, 1900, or 2000 m. In Figure 5, S and G denote the starting position and the goal position (ending position), respectively.

Figure 5.

The various tracks for the experimental evaluation of the proposed RTLS: (a) 1000 m, (b) 1200 or 1300 m, (c) 1400 m, and (d) 1700, 1800, 1900, or 2000 m.

Figures 6 and 7 exhibit the experiment results. In the figures, the proposed method is compared with the direct method and the conventional EKF. As stated in the “Related works” section, the current location algorithms^{3–5,11–13} can be classified into the direct method and the conventional EKF.

Figure 6.

The comparison of the location estimates for the track of 1000 m in the cases of (a) the direct method, (b) the conventional EKF, and (c) the modified EKF.

Figure 7.

The comparison of the location estimates for the track of 1200 m in the cases of (a) the direct method, (b) the conventional EKF, and (c) the modified EKF.

Figure 6(a)–(c) shows a comparison of the location estimates for the track of 1000 m (Figure 5(a)) in the cases of the direct method, the conventional EKF, and the modified EKF, respectively. As illustrated in Figure 6(a), the direct method produces a lot of estimation errors. In the figure, the dashed boxes enclose the erroneous estimates. The errors occur due to noisy TDoA measurements. Some estimates have large error differences (more than 200 m). As stated earlier, the frequency-selective fading significantly intensifies the noisy effects due to the ISI. This leads to the large error values. The conventional EKF mitigates the noisy effects. As shown in Figure 6(b), many erroneous estimates are corrected. However, the conventional EKF still produces noticeable distortions in the localization. This is due to the fact that the conventional EKF is susceptible to the nonstationary property of the covariance matrix ${\mathbf{R}}_k$ in equation (25). In Figure 6(b), the dashed boxes highlight the distortions. Figure 6(c) reveals that the modified EKF entirely compensates the noticeable distortions. Using the proposed method of equation (36), the modified EKF becomes robust to the nonstationary property of the covariance matrix for TDoA measurements.

Figure 7(a), (b), and (c) exhibits a comparison of the location estimates for the track of 1200 m (Figure 5(b)) in the cases of the direct method, the conventional EKF, and the modified EKF, respectively. The direct method also generates many estimation errors due to noisy TDoA measurements in Figure 7(a). The noise effects of Figure 7(a) are more profound than those of Figure 6(a). The largest error difference is higher than about 400 m in Figure 7(a). As illustrated in Figure 7(b), the conventional EKF also corrects many erroneous estimates. However, more distortions exist in Figure 7(b) than Figure 6(b) since the TDoA measurements include more noisy effects as indicated in Figure 7(a). Figure 7(c) also shows that the modified EKF can completely compensate the noticeable distortions. Using the proposed method of equation (36), the modified EKF overcomes the vulnerability to the nonstationary property of the covariance matrix for TDoA measurements. For the track of 1400 m (Figure 5(c)), note that the experiment results are similar to those of Figure 7.

Figure 8 shows the location estimates for the track of 1800 m (Figure 5(d)) in the zero-TDoA case. Figure 8(a) exhibits the zero-TDoA case (which is highlighted using the dashed circle) as well as lots of erroneous estimates in the case of direct method. In Figure 8(a), TDoA values are unavailable for about 100 m, which indicates the time duration of about 6 s assuming the maximum speed (60 km/h) of the mobile object. In other words, a very deep fading continues for about 6 s in the figure. Even though the conventional EKF can correct many erroneous estimates, there are still distinct distortions as illustrated in Figure 8(b). Moreover, the zero-TDoA case is not fully corrected, which is highlighted using the dashed box in Figure 8(b). The incomplete correction leads to a discontinuity in the time axis as shown in Figure 8(c). The discontinuity misleads to false information that the mobile object can move with an extremely high velocity. Figure 8(d) displays the location estimates in the case that the modified EKF does not employ the proposed method of equation (37). As shown in the figure, a tracking divergence occurs due to the long-time zero-TDoA duration (about 6 s). Figure 8(e) exhibits the location estimates in the case that the modified EKF employs the presented approach of equation (37). This figure demonstrates that the proposed technique of equation (37) can completely avoid such a tracking divergence in the case of modified EKF under zero-TDoA situations.

Figure 8.

The comparison of the location estimates for the track of 1800 m in the cases of the direct method, the conventional EKF, and the modified EKF: (a) direct method, (b) conventional EKF (x vs y), (c) conventional EKF (time vs x), (d) modified EKF: the case without the method of equation (37), and (e) modified EKF: the case with the method of equation (37).

Conclusion

This article proposes a novel RTLS for localization of high-speed mobile objects in fading environments. For the localization of mobile objects, the RTLS utilizes TDoA measurements, which are achieved using the UWB signals based on the IEEE802.15.4a. The new RTLS employs a presented EKF method, which is a modification of the conventional EKF scheme. The modified EKF considerably mitigates the frequency-selective fading and the deep fading effects, which causes severe ISI and zero-TDoA, respectively. For suppression of the frequency-selective fading, the modified EKF excludes the TDoA measurements in the update process if they are excessively deviated from the process values. For a long-time zero-TDoA case, the modified EKF conditionally utilizes the exclusion approach, which can avoid a possible tracking divergence.

Experimental evaluation shows that the proposed EKF approach (modified EKF) is superior to the direct method and the conventional EKF technique. For realistic evaluation of high-speed objects, the novel RTLS localizes the mobile objects with the maximum speed of 60 km/h in the various tracks. The experiment results show that the modified EKF completely compensates erroneous location estimates in the tracks. They also reveal that the presented EKF approach can eliminate a possibility of tracking divergence in the case of long-time zero-TDoA. The experimental evaluation confirms that the novel RTLS (which employs the modified EKF) is very suitable for localization of high-speed mobile objects under fading environments.