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Abstract

Because there are always non-line-of-sight effects in signal propagation, researchers have proposed various algorithms to mitigate the measured error caused by non-line-of-sight. Initially inspired by flocking birds, particle swarm optimization is an evolutionary computation tool for optimizing a problem by iteratively attempting to improve a candidate solution with respect to a given measure of quality. In this article, we propose a new location algorithm that uses time-of-arrival measurements to improve the mobile station location accuracy when three base stations are available. The proposed algorithm uses the intersections of three time-of-arrival circles based on the particle swarm optimization technique to give a location estimation of the mobile station in non-line-of-sight environments. An object function is used to establish the nonlinear relationship between the intersections of the three circles and the mobile station location. The particle swarm optimization finds the optimal solution of the object function and efficiently determines the mobile station location. The simulation results show that the proposed algorithm performs better than the related algorithms in wireless positioning systems, even in severe non-line-of-sight propagation conditions.

Keywords

Mobile station base station time of arrival non-line-of-sight particle swarm optimization

Introduction

Recently, the requirement for more accurate position determination has extensively increased. Many researchers have concentrated on how to achieve higher accuracy in positioning. An increasing number of studies have discussed how to locate mobile stations (MSs) in wireless cellular communication systems. The techniques enable us to obtain the location information of MSs. An estimation algorithm that uses the received signal strength (RSS) with a known path-loss attenuation model can calculate the distance between the MS and the base station (BS).^1,2 The time-of-arrival (TOA) estimation is a widely used method that calculates the distance between the MS and the BS by measuring the propagation time of the signals and multiplying it by the velocity of light.^3,4 The angle of arrival (AOA) can be estimated using an antenna array.^5,6 The time difference of arrival (TDOA) is also widely used, which measures the time difference between the signals and eliminates the timing offset between MS and BS networks.^7,8 There are some differences between time-based and angle-based categories, and each has its own advantages and limitations. The time-based schemes generally provide much better positioning accuracy, but these techniques require at least three BSs and synchronization among the networks. The angle-based schemes only require at least two BSs without synchronization. Many applications in wireless location techniques have been well developed, including vehicle navigation, auto-billing system, and intelligent transportation systems (ITS).⁹ Emergency 911 (E-911) is an important issue, where the public safety officer must know the caller’s phone number and accurate location.

MS location with high accuracy can be obtained if there is a line-of-sight (LOS) propagation in the transmission path. In urban or suburban areas, the LOS path is not always available. Some factors affect the accuracy of the location estimation, such as additive noise, multipath propagation, multiple access interference, and non-line-of-sight (NLOS) errors. However, significant errors in time and angle measurements occur because of NLOS propagation. Because the signals reflect or diffract during transmission, the direct path, or LOS propagation, is usually unavailable. For angle-based location systems, the measured angle is degraded by scatterers and obstacles. For time-based location systems, NLOS induces errors as positive biases over the true range. Several studies have proposed methods to mitigate the NLOS effects, such as⁵ by minimizing a proposed joint TOA/AOA constrained objective function. Another method was presented in Wylie and Holtzman,¹⁰ where the true range of the estimated MS location is reconstructed by the sample statistics of range measurements over a period of time. In Venkatraman and Caffery,¹¹ a hybrid TOA/AOA technique uses three TOA measurements and one AOA measurement to estimate the MS location. In our previously proposed method, resilient back-propagation (Rprop) neural networks are used to estimate the MS location when TOA and AOA measurements are available from two BSs¹² and three BSs.¹³

Hearability is an important issue that affects the choice of location scheme in cellular communication systems.¹⁴Hearability is defined as the ability to simultaneously receive signals with sufficient power from neighboring BSs. In a cellular communication system, the MS location is estimated based on the received measurements. However, because of the cell distribution and signal loss, MS may not always be able to reliably detect the neighboring cells. For example, the hearability of the MS is notably low in rural areas because each BS always covers a large area. A longer distance makes the MS unable to clearly receive the signal from the BS. The lack of available BSs limits the location service and impairs the positioning schemes. To improve hearability, the fractional load technique is proposed to detect a large number of geographically displaced cells.¹⁵

The improvement for searching MS locations with TOA and AOA measurement is an optimization problem. To solve the optimization problem in a large quantity of data, some recursive algorithms have been proposed to search for near-optimal solutions, such as the genetic algorithm (GA),¹⁶ simulated annealing (SA),¹⁷ and particle swarm optimization (PSO).^18–20 GA is a heuristic search that mimics the process of natural selection. SA is a generic probabilistic meta-heuristic to search for the global minimum of an object function. PSO is a population-based stochastic approach based on bird and fish flock movements that aims to optimize a problem by iteratively attempting to improve a candidate solution. The popularity of each recursive algorithm for the optimization problems is easy to implement with good performance. Thus, optimization algorithms can be easily applied to various applications and become the flexible and effective tools on design problems. For example, GA is an increasingly popular method of optimization, which is motivated by Darwin’s theories of evolution. Mutation, crossover, and selections are the primary operators for optimization to promote the evolution of a population. PSO applies the social behavior of a swarm of birds and fish to achieve a robust stochastic evolutionary computation technique. PSO has a strong ability to find global optimistic results and performs better in terms of convergence and computation time in engineering fields and computer science.²¹ A design of a multi-machine power system stabilizers using PSO is developed to search for optimal settings.²² A new competitive approach for learning agents using the PSO technique is to train neural networks to predict the desirability of states.²³ To improve performance, researchers often adjust the algorithmic parameters. A new rule for updating a particle’s velocity is used by adding constants in the component range of the PSO.²⁴ The hybrid Nelder–Mead–PSO algorithm intends to produce faster and more accurate convergence.²⁵ Macro et al.²⁶ presented the results and insights from some PSO variants and discussed the differences between the mechanisms and the update of the particle’s velocity. In their study, other factors that affect the performance are considered, such as the selection of population topology, population size, and maximum number of function evaluations.

It is well known that a TOA measurement can be represented by a circle centered at a BS. The intersections of these circles provide the MS location estimation. In this article, the MS location is estimated under the condition that an MS is only heard by three BSs. The TOA measurement from each BS forms a circle that is geometrically centered at the BS. Because of the positive biases over the true TOA value, three TOA circles overlap one another and form different intersections on the plane. The MS location should be inside the region enclosed by the overlap of three circles. A nonlinear object function is defined as the sum of square of the distance from the MS to these intersections.¹⁴ PSO is applied to minimize the object function and estimate the MS location. After the optimization problem is solved, the global minimum of the objective function is the estimated coordinates of the MS. The optimization solution can be derived by PSO, which is considered the estimated MS location. To verify the performance of the proposed PSO-based algorithm in detail, different NLOS error models are demonstrated. The simulation results show that the proposed PSO-based algorithm provides a much better location estimation than the Taylor series algorithm (TSA),^27,28 linear lines of position (LLOP) algorithm,²⁹ and range-scaling algorithm (RSA).¹⁴

In this article, we propose a method to reduce the effect of NLOS propagation in a mobile environment, which is one of the major factors that affect the positioning performance. A new localization scheme based on PSO is addressed to increase the location accuracy. With the TOA measurement, using PSO reduces the object function value of the positioning square error. The results show that the proposed method outperforms the conventional location algorithms in various NLOS models.

The remainder of this article is organized as follows. In section “Three related localization algorithms,” we introduce three related MS positioning methods: TSA, LLOP, and RSA. Section “PSO” describes the PSO algorithm. In section “Proposed location scheme,” we propose an algorithm based on PSO to estimate the MS location. Section “Simulation results” discusses the simulations on the test of the proposed algorithm and compares the results with those of other methods. Finally, section “Conclusion” draws the conclusions.

Three related localization algorithms

TSA

First, we define the true distance between BS i and the MS. It can be expressed as

$r_{i} = \sqrt{{(x - X_{i})}^{2} + {(y - Y_{i})}^{2}}$ (1)

where $(x, y)$ and $(X_{i}, Y_{i})$ are the locations of the MS and BS i, respectively. If $(x_{v}, y_{v})$ is the initial estimated position, let $x = x_{v} + δ_{x}$ and $y = y_{v} + δ_{y}$ . By linearizing the TOA equations using Taylor series expansion with second-order terms, we have

$A δ ≅ z$ (2)

where $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{matrix}]$ ; $δ = [\begin{matrix} δ_{x} \\ δ_{y} \end{matrix}]$ ; $z = [\begin{matrix} r_{1} - r_{v 1} \\ r_{2} - r_{v 2} \\ r_{3} - r_{v 3} \end{matrix}]$ ; $a_{i 1} = {\frac{\partial r_{i}}{\partial x} |}_{x_{v}, y_{v}}$ ; $a_{i 2} = {\frac{\partial r_{i}}{\partial y} |}_{x_{v}, y_{v}}$ ; and $r_{vi} = \sqrt{{(x_{v} - X_{i})}^{2} + {(y_{v} - Y_{i})}^{2}}, i = 1, 2, 3$ .

The least-squares (LS) solution to the estimation problem is given by

$δ = (A^{T} A)^{- 1} A^{T} z$ (3)

The recursive process starts with an initial guess for the MS station and repeats the computations in the iteration until the solution is converged.^30,31

LLOP

This scheme uses a simpler linear equation that derives from the original three nonlinear TOA circular equations. Instead of the circular LOP, the LLOP equation passes through the intersections of two circular ones for TOA measurements. The linear equations can be found by squaring and subtracting the distances obtained in equation (1). The MS location is determined by³²

$G ϕ = h$ (4)

where $ϕ = [\begin{matrix} x \\ y \end{matrix}]$ is the MS location, $G = [\begin{matrix} X_{2} & 0 \\ X_{3} & Y_{3} \end{matrix}]$ , and $h = \frac{1}{2} [\begin{matrix} r_{1}^{2} - r_{2}^{2} + X_{2}^{2} \\ r_{1}^{2} - r_{3}^{2} + X_{3}^{2} + Y_{3}^{2} \end{matrix}]$ .

Again, the LS solution of equation (4) is given by

$ϕ = (G^{T} G)^{- 1} G^{T} h$ (5)

RSA

An algorithm was proposed in Venkatraman et al.¹⁴ for MS location estimation using TOA measurements from three BSs. The normalized scale factors are used to correct the positive NLOS errors in TOA measurements. The constrained optimization algorithm attempts to find the normalized scale factors with some constraints based on the geometry of the cell layout, where the circles depict the ranges of three BSs.

PSO

To achieve high accuracy in a wireless location system, researchers proposed various methods to obtain the MS location. Generally, it is difficult to find the optimal solution in many possible solutions. To improve the searching efficiency, many studies have proposed near-optimal searching methods based on swarm intelligence, such as GA, SA, and PSO. Based on the concept of swarm intelligence, GA and PSO have different implementations. GA¹⁶ is a stochastic search for the optimization problem based on the mechanics of natural selection from the concept of “survival of the fittest.” The search procedure of GA begins with a population of random chromosomes. Next, the GA evaluates these string structures of the population of candidate solutions (chromosomes). Chromosomes with better performance in the fitness function of design problem are selected to reproduce offspring. In the selection, fitter offsprings are produced for biological crossover and mutation. Offsprings with less fitness are removed. Using the operators, the chromosomes exchange their characteristics in crossover and mutation over a number of iterations to produce better chromosomes, and a suitable solution is defined with respect to the current optimization. Finally, the best value of the fitness function is selected as the optimization solution of GA over a number of iterations.

The authors of this article proposed a new positioning algorithm based on GA to determine the MS location in an NLOS environment.³³ Because PSO performs better than GA,^34–36 it is particularly suitable for determining the MS location in a wireless positioning system. PSO is a simpler search method because of its simple conception, easy implementation, and efficient computation. In previous studies,^18,30,37 the PSO algorithm was used in various fields to search for the optimal solution, such as music composition, medical applications, computer graphics, and biological studies. PSO methods to solve optimal problems include the nonlinear approach and multi-objective problem. Because of its simplicity and practicality, PSO has attracted great attention of researchers in various fields. The original idea of PSO is to simulate a simplified social behavior such as bird flocking or fish schooling. In PSO, each candidate solution and the entire population are called a particle and a swarm, respectively. Through evolutionary computation, each particle searches for both the local optimal solutions and the global optimal solution. The best solution is calculated from each particle’s solution.

There are two simple rules in PSO swarm. First, each particle must return to its own previous location. Second, each particle attempts to imitate its neighbors by successfully flying to their positions. The overall behavior of the swarm has a rapid concentration in promising regions of the search space. The procedure of searching for the optimal solution is described as follows. First, random particles initialize the population. Second, each particle updates its position according to its individual experiences and the successful positions of its neighbors. Mutations are the basic operators that provide the necessary variety in a swarm. As in other comparable gradient-based algorithms, PSO usually requires more object functions. PSO generally uses a large number of parallel processors and more powerful solution generations to search for the global solution. In recent years, instead of investigating how to apply mutation operators and determining the type of diversity that should be provided in the swarm in the PSO process, more researchers have investigated the design of new mutation operators.

Researchers have applied the PSO method in various applications to solve the optimization problems. However, some optimization problems are very difficult to solve because the computational complexity is very large. The current position, memory, and social knowledge of a swarm affect the particle movement. The PSO algorithm is a useful method to solve stochastic optimization problems through finding a global or a near-optimal solution. Each particle attempts to move to its own position, depending on its previous location and the successful positions of its neighbors. The objective function determines the quality of the particle position.

At the time index k, the position and velocity of particle i are $x_{i}$ and $v_{i}$ , respectively. During the iterative process, the velocity and position of particle i are updated as follows

$v_{i} (k + 1) = v_{i} (k) + ϕ_{1} [γ_{1, i} (P_{i} - x_{i} (k))] + ϕ_{2} [γ_{2, i} (G_{i} - x_{i} (k))]$ (6)

$x_{i} (k + 1) = x_{i} (k) + v_{i} (k)$ (7)

where $ϕ_{1}$ and $ϕ_{2}$ are the acceleration coefficients and $γ_{1, i}$ and $γ_{2, i}$ are the random numbers in the interval [0,1] applied to the ith particle. P stores the best position found by the ith particle, which is called the personal best. G is the best position found by swarm methods, which is known as the global best. Both acceleration coefficients are commonly set to 2.

Commonly, an inertia term is added in equation (6), as shown in equation (8).³¹ The inertia function is commonly taken as a constant or a decreasing function in k, and this approach has been shown to be effective for the algorithm. If the decreasing function is used, the effect of the past velocity decreases. Depending on the following procedures, the swarm consists of several particles that search through the feasible space to explore the optimal solution

$\begin{array}{l} v_{i} (k + 1) = w (k) v_{i} (k) + ϕ_{1} [γ_{1, i} (P_{i} - x_{i} (k))] \\ + ϕ_{2} [γ_{2, i} (G_{i} - x_{i} (k))] \end{array}$ (8)

PSO is widely used as a form of evolutionary computation because of its advantages over other optimization techniques. First, it is an algorithm without any derivative procedure, so it is easy to implement with basic mathematical operations. In addition, it is less sensitive to the limitation of the objective function, such as convexity or continuity.³² To make PSO more efficient, the particles should move toward the border of the solution space. Three solutions were proposed in Robinson and Rahmat-Samii:³⁸ the particles arrive at the domain boundary with a velocity, the particles hit a boundary of perfect reflecting surfaces and reflect back to the solution space, and the particles move from the solution space without calculating the object function. Finally, the particles return to the domain. Deepalaxmi et al. summarized some key features of PSO. First, PSO only requires a fitness function to measure the “quality” of a solution, which reduces the computational complexity.³⁷ Second, PSO is less sensitive to a good initial solution. Third, PSO is easily incorporated with other optimization tools. Fourth, PSO can escape the local minima problem. Finally, the particles return to the domain. Therefore, the PSO algorithm has become popular in recent years because it is simple to implement in various fields and robust for multi-objective problems. The PSO is easy to study and implement. Both discrete and continuous parameters and parallel computing implementation are achieved using the PSO.^39,40 Because of these advantages, PSO is used as the optimization search method in our study.

Proposed location scheme

In the TOA approach, the distance between the MS and the BS is measured by finding the propagation time between the MS and the BS. According to the geometric approach viewpoint, the TOA measurement generates a circle, which is centered at the BS. Then, the MS position is estimated using the circles produced by TOA measurements at multiple BSs. The coordinates of BS1, BS2, and BS3 are $(X_{1}, Y_{1}) = (0, 0)$ , $(X_{2}, Y_{2}) = (X_{2}, 0)$ , and $(X_{3}, Y_{3})$ , respectively. Each of the following three equations describes a circle for the TOA measurement, as shown in Figure 1

$Circle 1 : x^{2} + y^{2} = {r_{1}}^{2}$ (9)

$Circle 2 : {(x - X_{2})}^{2} + y^{2} = {r_{2}}^{2}$ (10)

$Circle 3 : {(x - X_{3})}^{2} + (y - Y_{3})^{2} = {r_{3}}^{2}$ (11)

Figure 1.

Geometric layout of the three circles.

If there is no error, the three circles will intersect at a common point, which is the true MS location. However, the NLOS error is common and can seriously degrade the location accuracy. In addition, the NLOS error always appears as a positive bias; thus, we expect that the MS location can be found at the overlap of the circles. Figure 1 also shows a scenario where the MS location should be inside the overlap of three circles. As mentioned above, these points $(U, V, W)$ are defined as feasible intersections. The feasible intersections must simultaneously satisfy the following inequalities

$x^{2} + y^{2} \leq {r_{1}}^{2}$ (12)

${(x - X_{2})}^{2} + y^{2} \leq {r_{2}}^{2}$ (13)

${(x - X_{3})}^{2} + (y - Y_{3})^{2} \leq {r_{3}}^{2}$ (14)

Because the MS location must satisfy the above conditions, the MS location can be obtained by the feasible intersections of the three circle equations. A nonlinear object function was proposed in Venkatraman et al.¹⁴ and can be defined as the sum of the square of the distance from the MS to the feasible intersections of the circle equations. The feasible intersections can be obtained by solving the three circle equations. The coordinates of $U, V, W$ are $(Ux, Uy)$ , $(Vx, Vy)$ , and $(Wx, Wy)$ , respectively, and the object function can be represented as follows

$\begin{array}{l} f (x, y) = (x - U x)^{2} + (y - U y)^{2} + (x - V x)^{2} \\ + (y - V y)^{2} + (x - W x)^{2} + (y - W y)^{2} \end{array}$ (15)

To eliminate NLOS errors, PSO is used to minimize the nonlinear object function. In the PSO algorithm, each candidate solution and the entire population are called a particle and the swarm, respectively. The algorithm guides these particles to search for globally optimal solutions. Because of the NLOS errors, the measured distance between MS and BS is always greater than the actual distance. The MS location should be inside the region enclosed by the overlap of the three circles. Therefore, the initial populations are randomly distributed in the overlap of three circles. The particles update their position according to their personal experience and their neighbors’ experiences. The optimization solution that can minimize the object function can be derived using the PSO procedures. The function is represented as the variables x and y, which are the coordinates of the estimated location. After a number of iterations, the value of the object function gradually decreases and converges to a minimum. Finally, the coordinate of the final iteration is the MS location estimation. During the iterations, the value of the object function gradually decreases. If the object function decreases, the effect of the past velocity decreases. Generally, the value will converge after a number of generations when the solution does not change after a specific number of generation.

Simulation results

Computer simulations were performed to examine the performance of the proposed location methods. Without loss of generality, the coordinates of the BSs were set to BS1: (0, 0), BS2: (1732 m, 0), and BS3: (866 m, 1500 m). Moreover, simulations were performed for cellular environments, assuming that BS 1 is the serving BS, and the MS location was randomly selected according to a uniform distribution in the region formed by the points BS1, I, J, and K, as shown in Figure 2. The result did not significantly change when we adjusted the parameter of PSO because of the good convergence of the object function. Therefore, we set the parameter as follows: population size = 20, $φ_{1} = 2$ , and $φ_{2} = 2$ . If $abs (G_{i + 1} - G_{i}) < 10^{- 4}$ , the program terminates and sets a certain length of epochs as the final epoch.

Figure 2.

Cell layout of the relationship between the true ranges and the inter-BS distances.

Regarding the NLOS effects in the simulations, two common propagation models are used in this article: the circular disk of scatterer model (CDSM)⁴¹ and the uniformly distributed noise model.¹⁴ The CDSM model assumes that when the signal travels between MS and BSs, there are scatterers spreading around the MS, such as buildings and other obstacles. Then, the signal undergoes a single reflection at the scatterers. The measured ranges are the sum of the distances between the BS and the scatterer and between the MS and the scatterer. The accuracy of the MS location is measured in terms of root-mean-square (RMS) error between the actual MS location and the estimated MS location. Compared to conventional methods, the effect of the radius of the CDSM on the average location error of various methods is shown in Figure 3. Regardless of the selected number of upper bounds on NLOS errors, the average location errors of TSA, LLOP, and RSA are always larger than the proposed PSO-based algorithm. The proposed algorithm thus helps obtain a more accurate MS location and reduce the RMS errors.

Figure 3.

Average location error versus the radius of scatterers.

The location accuracy using the proposed PSO-based algorithm is also shown in the cumulative distribution function (CDF) curves of location errors, as shown in Figure 4. Here, the radius of the scatterers was set to be 200 m. Compared with the existing methods, the accuracy of the MS location indeed improved with the proposed algorithm. TSA and LLOP clearly predict the MS location with poor accuracy, and the proposed algorithm always achieves the best performance.

Figure 4.

CDF of the location error on the CDSM.

The NLOS propagation model is called the uniformly distributed noise model, where the TOA measurement error is assumed to be uniformly distributed over $(0, U_{i})$ , for $i = 1, 2, 3$ , where $U_{i}$ is the upper bound of the error. Figure 5 shows the effect of various methods with the upper bound of NLOS error on the average location error. The positioning accuracy, which is measured in terms of RMS error, decreases when the upper bound of NLOS error increases. For almost every value of the upper bound of NLOS errors, the proposed algorithm performs better than the TSA, LLOP, and RSA.

Figure 5.

Average location error versus the upper bound of NLOS errors.

Conclusion

This article presents a new positioning algorithm based on the PSO technique to determine the MS location in NLOS environments. PSO is an evolutionary computation that imitates the movement of organisms such as a flock of birds or a school of fish. The PSO algorithm is a population-based computational method to solve optimization problems by iteratively improving the quality of the candidate solutions. Because signal measurements are limited by hearability, only three BSs are available in this article. The estimated MS location is inside the overlapping region of three different circles. The initial particles are generated from this overlapping region. By updating the original positions to better ones, which are found by other particles, the value of the object function is reduced, and the swarm is moved toward the best solutions. After a number of iterations, the value of object function decreases. Finally, the optimal result of the PSO can be considered the estimated MS location in wireless communication systems. It is confirmed that the proposed algorithm always performs better than TSA, LLOP, and RSA in different levels of NLOS errors and different NLOS models. The simulation results show that the PSO-based algorithm reduces the NLOS errors and provides a more accurate positioning. The performance of the proposed location method is compared with those of other methods such as TSA, LLOP, and RSA. The numerical simulations also demonstrate that the proposed location system improves the performance of mitigating the NLOS ranging error.

In this article, we apply the existing technique (PSO) to a new field (wireless positioning) and form a new application. Most positioning methods proposed by other researchers originate from their creations and ideas. Although moderate novelties are achieved, they lack a strict theatrical background. Because PSO has been thoroughly investigated in other fields, we can expect a predictable outcome when it is used to increase the positioning accuracy. The simulation results are consistent for different simulating parameters.

Moreover, in the future, the Internet of Things (IOT) will be an important and critical issue. IOT has widespread applications in fields such as intelligent transportation, intelligent housing, industry monitoring, nursing and tracking services for elders, and personal e-health management. These services require a highly accurate positioning technology. Therefore, how to implement localization method in the IOT scheme is an important topic. This article provides a potential solution to the researchers in this field.

Footnotes

Academic Editor: Gang Wang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

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