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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In order to enhance accuracy and reliability of wireless location in the mixed line-of-sight (LOS) and non-line-of-sight (NLOS) environments, a robust mobile location algorithm is presented to track the position of a mobile node (MN). An extended Kalman filter (EKF) modified in the updating phase is utilized to reduce the NLOS error in rough wireless environments, in which the NLOS bias contained in each measurement range is estimated directly by the constrained optimization method. To identify the change of channel situation between NLOS and LOS, a low complexity identification method based on innovation vectors is proposed. Numerical results illustrate that the location errors of the proposed algorithm are all significantly smaller than those of the iterated NLOS EKF algorithm and the conventional EKF algorithm in different LOS/NLOS conditions. Moreover, this location method does not require any statistical distribution knowledge of the NLOS error. In addition, complexity experiments suggest that this algorithm supports real-time applications.

In order to understand sensor data in spatial context or for proper navigation throughout a sensing region, automatic location of the sensors in wireless networks is a key enabling technology. Therefore, mobile location technologies, which are designated to estimate the position of a MN, have drawn a lot of attention for its various potential location-based services [

If the LOS propagation exists between the MN and all fixed nodes (FNs), which are known-location reference anchor nodes in a sensor network, a high location accuracy can usually be achieved using the conventional location algorithms [

In the open literature, many methods have been employed to mitigate the adverse effect. Generally, these methods can be divided into two categories: methods for static positioning systems and methods for mobile tracking systems. Reference [

Although the above algorithms mitigate the influence from the NLOS errors and improve the location accuracy in a certain extent, the more thorough method is directly to estimate and eliminate the NLOS bias in each measurement. Reference [

The remainder of this paper is organized as follows. The measurement model is described in Section 2. The NLOS bias estimation using constrained optimization method is developed in Section 3, and Section 4 formulates the modified EKF-based tracking algorithm with NLOS correction. Section 5 presents the LOS/NLOS identification method for decreasing computing time. The simulation results and performance analysis are discussed in Section 6. Finally, conclusions are drawn in Section 7.

Assume that sensors are capable of transmission and reception, and then unknown-location devices are able to make measurements to multiple reference nodes. In the existing wireless systems, the range between the MN and a known-location FN could be measured by time of arrival (TOA), round trip time, and signal strength measurement techniques. In this study, we assume the mobility estimation is based on TOA at the MN, and the distance measurement is obtained by multiplying the time measurement by the speed of light. Assume that the MN has detected the range signals from the _{k}, _{k}]^{T}_{i}, _{i}]^{T}_{i}(_{i}(_{i}(_{i}(

Here, we propose a least squares (LS) optimization-based technique to estimate the NLOS propagation delay without requiring any prior statistics information. With the linearization of the system using Taylor’s series approximation as discussed in [_{0} is the Jacobian matrix of _{0},

Note that the reference point _{0} should be chosen close enough to the true position in order for (9) to be valid. The reference point coordinate estimates may be determined using the simple estimator as follows. Because the MN cannot be located farther than _{i}(_{i}(_{0} = (^{(0)}, ^{(0)}) for the MN location can be obtained by finding the center of the feasible region. With three FNs, for example, it is calculated by averaging the coordinates of the points of intersection (^{(1)}, ^{(1)}), (^{(2)}, ^{(2)}) and (^{(3)}, ^{(3)}), ^{(0)} = (^{(1)} + ^{(2)} + ^{(3)})/3 and ^{(0)} = (^{(1)} + ^{(2)} + ^{(3)})/3. This initial position contains an NLOS bias error. However, the NLOS bias is estimated to recalculate the MN position in later steps, which is explained in the next section.

If the bias vector

However, in reality, _{0}

Then, the following constrained optimization problem is defined to estimate the NLOS bias errors:
_{v}^{T}_{i}_{ij}

It is obvious that (14) is a constrained LS problem, a type of quadratic programming (QP) problem. There are many algorithms developed to solve this type of problem [

Assume that a MN of interest moves on a two-dimensional plane, and the motion state at time instant _{k}, _{k}, _{k}, _{k}]^{T}_{k}, _{k}] corresponds to the horizontal and vertical Cartesian coordinates of the mobile position, [_{k}, _{k}] are the corresponding velocities. The mobile state with random acceleration can be modeled as:
_{xk}, _{yk}]^{T}^{T}

Because of the non-linear measurement equation, the EKF has to be used. Applying the linearization method used in EKF design, the linearized measurement equation becomes:

Similar to the Kalman filter, the operations of the EKF can be represented by two recursive steps. The prediction step includes the following operations:

Although the EKF is probably the most widely used estimation algorithm for nonlinear systems, it is derived basing on Gaussian noise condition. In the LOS environments where only measurement noise is considered, the EKF is an optimal estimator and can improve the tracking accuracy since measurement noise is usually assumed to obey Gaussian distribution [_{i}

Once the NLOS bias is obtained in Section 3, the measurement updating

The unique change of the modified EKF is to subtract the NLOS bias in the updating equation, so increased computing time is small, but the NLOS bias estimation using the optimization method will increase the computational complexity. In the mixed LOS/NLOS environments, however, the above NLOS bias estimation is actually needless when the LOS situation appears. However, we do not know when and how often the LOS or NLOS conditions appear and disappear in the realistic scenarios, so it is important to check the LOS/NLOS condition for reducing unnecessary computing time. In the previous algorithms [

From the incoming measurement and the optimal prediction obtained in the previous step, the innovation sequence in the LOS case is defined as:

But in the NLOS case, the measurement equation becomes:

Correspondingly, the innovation vector turns into:

Assuming that the covariance matrix of the NLOS error

From (

Because variance calculation needs a time series of range measurements from each FN, the LOS/NLOS identification based on variance comparison is not suitable for real-time application. On the contrary, the square sum of the innovation vector, ^{T}

If not, the NLOS error is present, then we would expect for the square sum of the innovation vector to have a significantly larger deviation than the values of (31). This assumption is strongly supported by the field test results and the above analysis, which clearly indicate that the presence of the NLOS error increases the deviation of the measurements in a significant manner.

Based on the calculated square sum of the innovation vector, therefore, the hypothesis testing can decide whether NLOS components exist. The decision rule of the LOS/NLOS hypothesis testing for mobile location is chosen as follows:

Simulation results are provided in this section to assess the performance of the proposed algorithm in both large-scale and small-scale environments. Two conditions in [^{2}. The simulated trajectory has L = 2,000 time samples, and the sample interval

In this condition, the NLOS measurement noise is fixed for the whole trajectory, which is assumed to be a white Gaussian random variable with positive mean _{NLOS}_{NLOS}

_{s}, _{s}), the RMSE at the time instant k is defined as

It can be seen from

In order to evaluate the tracking precision obtained by the proposed algorithm, we have computed the RMSE of the three different trackers in the worst situation,

This group of simulations will show the great improvement in the accuracies on the range estimations and tracking trajectories in the LOS/NLOS transition condition. In order to do that, we take TOA measurements during 60 seconds in three intervals NLOS-LOS-NLOS.

Accordingly, it can be seen from the estimated trajectories in

The identification results in the LOS/NLOS mixed environment is provided in

Compared with the conventional EKF smoother, the additional computation in the proposed algorithm is primarily introduced by the estimation of NLOS bias using constrained LS optimization, which mainly involves the operations of matrix inverse and multiplication. The complexity of the matrix inverse operation is ^{3}), and the multiplication operation is ^{2}), where

From

To verify the proposed method and examine the performance in a short-range environment, the following simulations are carried out. The location area is a square with dimensions of 100 _{i}_{i}_{i}_{i}_{i}_{i}^{2}. The simulated trajectory has L = 200 time samples, and the sample interval Δ

On the other hand,

Because the NLOS propagation is ubiquitous in both indoor and outdoor positioning scenarios, a robust algorithm is required to mitigate the impact of the NLOS location estimation error. In this work, we have developed a suitable NLOS identification and mitigation algorithm for the improvement of location accuracy in EKF-based mobile location. Simulation results show that the proposed algorithm exceeds the FCC target and significantly outperforms the other two methods. Meantime, it does not depend on a particular distribution of the NLOS error. As compared to the conventional time-history method, the proposed identification approach has achieved the real-time capability by using only current innovation sequence. The complexity comparison suggests that the performance gain of the proposed method is at the expense of increasing the computer time. However, the difference of complexity is small and acceptable when considering the large performance gain it achieves. This method can still be considered as a candidate for most real-time applications. Further investigation will emphasize on the impact of different motion models and channel models for the proposed algorithm and the theoretic bound on the location estimation precision for a given set of measurements.

We are grateful to the anonymous reviewers for their helpful comments which have significantly improved the quality of the paper. This work was financially supported by the National High Technology Research and Development Program of China (863 Program) (2008AA01z227), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (10KJB510008).

Flow chart of the proposed algorithm with NLOS correction.

Comparison of the RMSE when the three FNs are all in NLOS conditions.

Zoom of the range estimation by three algorithms during 60 seconds.

The estimated trajectories of the MN by three algorithms from a single realization.

The identification results of the LOS/NLOS hypothesis testing.

Impact of the number of NLOS FNs on the location accuracy in the good node geometry.

Impact of the number of NLOS FNs on the location accuracy in the bad node geometry.

Performance comparisons among three algorithms under the different NLOS conditions.

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Error (m) | 67% | 95% | 67% | 95% | 67% | 95% |

3LOS, 0NLOS | 17.17 | 30.07 | 34.58 | 73.35 | 9.83 | 38.39 |

2LOS, 1NLOS | 32.76 | 63.96 | 105.8 | 221.1 | 309.2 | 375.1 |

1LOS, 2NLOS | 35.99 | 69.52 | 238.5 | 280.1 | 789.9 | 859.5 |

0LOS, 3NLOS | 37.37 | 76.58 | 301.9 | 315.9 | 828.5 | 916.3 |

Computer running time of the three methods.

Average Time(s) | 0.352 | 0.216 | 0.191 |

Standard deviation(s) | 0.011 | 0.012 | 0.016 |