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/).

The practical deployment of wireless positioning systems requires minimizing the calibration procedures while improving the location estimation accuracy. Received Signal Strength localization techniques using propagation channel models are the simplest alternative, but they are usually designed under the assumption that the radio propagation model is to be perfectly characterized a priori. In practice, this assumption does not hold and the localization results are affected by the inaccuracies of the theoretical, roughly calibrated or just imperfect channel models used to compute location. In this paper, we propose the use of weighted multilateration techniques to gain robustness with respect to these inaccuracies, reducing the dependency of having an optimal channel model. In particular, we propose two weighted least squares techniques based on the standard hyperbolic and circular positioning algorithms that specifically consider the accuracies of the different measurements to obtain a better estimation of the position. These techniques are compared to the standard hyperbolic and circular positioning techniques through both numerical simulations and an exhaustive set of real experiments on different types of wireless networks (a wireless sensor network, a WiFi network and a Bluetooth network). The algorithms not only produce better localization results with a very limited overhead in terms of computational cost but also achieve a greater robustness to inaccuracies in channel modeling.

The capability of gathering enough data from the environment and the users enables the existence of intelligent spaces, which are able to process the collected information in order to provide useful services or information. Intelligent spaces often feature a collection of sensors and sensors networks, which collect the required information (e.g., environmental parameters such as temperature or humidity, biometric information from personal sensors,

Although several technologies can be used to estimate the position of the different objects [

In the literature, two main approaches have been proposed to solve the localization problem using RSS measurements: channel modeling based methods and fingerprint strategies. In the first one [

With respect to channel model based techniques, they are built on the fact that a channel model is a theoretical, simplified and non-perfect approach to describe the behavior of a complex propagation environment. The model-based localization approach entails a much simpler calibration phase, as it only requires the calculation of the channel model, which can be estimated from a few measurements in the deployment area or even established theoretically without the need of a previous measurement phase. The more effort is put into the calibration, the better accuracy is obtained in the localization results, as the channel model will be better adapted to the particularities of the real propagation environment. But ideally, the calibration processes should be minimized in order to make the system deployment easier and less time consuming. Therefore, a solution to these inconvenient calibration needs is the design of positioning algorithms that are robust to the inaccuracies in the channel estimation; otherwise said, strategies capable of obtaining accurate location estimates in spite of working on non-accurately calibrated channel models.

In this paper we propose and evaluate the use of two weighted least squares techniques to calculate the position of a mobile node from the estimated distances to some reference nodes. The standard RSS-based localization techniques for wireless networks do not consider the individual accuracies of the different measurements to construct a better estimator. The proposed algorithms aim at enhancing the accuracy of position estimates while reducing their sensitivity to an imperfectly modeled channel. Although weighted least squares techniques are very well-known, to our knowledge the application of these techniques to the RSS-based localization problem and, in particular, to make localization more robust to imperfect channel models, has not been presented in detail before. Our work includes an exhaustive analysis based on both simulated and empirical tests, which shows that the location results are not only more accurate, as expected for a weighting technique, but also more robust to channel estimation errors. As explained above, this fact makes these techniques very attractive from a practical point of view.

The structure of the paper is as follows. In Section 2 the related state of the art is reviewed and in Section 3 the fundamentals of channel model based localization methods are described. Sections 4 and 5 describe the proposed positioning algorithms, the weighted hyperbolic technique and the weighted circular positioning technique. Section 6 includes a performance analysis of the proposed methods with numerical simulations and Section 7 analyzes the performance of the methods through real experiments with three different wireless networks: a WiFi network, a wireless sensor network and a Bluetooth network. With this experimental validation we show that the proposed techniques reduce the localization error with respect to the standard hyperbolic and circular positioning algorithms and that they have a bigger robustness to inaccuracies in the channel estimation. We also analyze the computational load of the algorithms, an issue which may be critical when considering embedded implementations. Section 8 concludes the paper.

The problem of range-based localization has been studied for many years, especially in the field of radar and sonar, where the range measurements are usually obtained from time of arrival (TOA) or time difference of arrival (TDOA) measurements, and more recently in cellular networks, motivated by the FCC E-911 norm. In these areas, weighting techniques have been previously used to solve TOA/TDOA-based localization problems. For example, in [

In ad hoc and sensor networks, the position of the nodes is typically computed from RSS measurements, which are then converted into distances using a channel model. This problem is in fact a range-based localization problem; however, very few works have studied the use of weighting techniques for RSS-based localization. The authors of [

However, these and most of the existing works in RSS-based channel modeling localization consider that the radio propagation model is known a priori, either because a certain model is assumed (for example, perfect free-space propagation) or because the parameters of the channel model are supposed to be estimated in the deployment environment prior to the real operation of the system. The first assumption is not realistic, as the propagation conditions in the real scenario may differ from those predicted by the theoretical model. On the other hand, taking measurements in the deployment area to estimate the channel model is not always possible (for example, in hostile environments) and, if possible, the number of measurements that can be taken is limited, thus the estimated model is usually a poor representation of the real channel. Consequently, to be applied in a real deployment a localization technique should take into consideration these concerns, either by trying to model the environment in a more realistic way or by using a positioning technique that is more robust to the model inaccuracies.

Few studies have been carried out to develop techniques in which the channel model is calculated during localization, or periodically updated, in order to reduce the localization errors that are produced by the inaccuracies of the propagation model. A common approach leverages RSS measurements between reference nodes with known positions to periodically estimate updated channel models and avoid using old models that no longer represent the channel behavior. These updated channels may also be different depending on the position of the nodes. In this way, these techniques use channel models that try to be consistent with the actual propagation characteristics. Some examples of localization algorithms using this approach can be found in [

Another approach consists in assuming that the radio propagation is characterized by a given model and estimate the value of the channel parameters together with the position of the mobile nodes. For example, in [

In this section we briefly describe a typical localization method based on channel modeling. Let us consider a wireless network composed of both mobile and fixed (or anchor) nodes. In an ad hoc network, these fixed nodes are ordinary nodes that are chosen for this purpose. In a WiFi network, these nodes could be the access points. The position of a mobile node can be calculated using the model-based approach in the following way. First, the mobile node measures the RSS received from the anchor nodes. Second, a channel model is used to estimate, from the RSS measurements, the distances between the mobile node and each anchor node. And finally, the position of the node is determined from these distances using a multilateration algorithm.

Many channel models have been proposed for outdoor and indoor environments [_{RX}^{2}) is a zero-mean gaussian random variable with standard deviation _{TX}_{0}, and has to be experimentally determined. On the other hand, the path loss exponent

In a real localization application, the radio channel has to be either predicted theoretically or, in the best case, estimated from a limited and usually small set of RSS measurements. As a result, the estimated model, characterized for example by the parameters

As said before, once the distances to different anchor nodes are estimated, a positioning algorithm must be applied in order to calculate the position of the mobile node. Two of the simplest and most common positioning algorithms that have been used for RSS-based localization are the circular positioning algorithm [_{i}_{i}_{i}

The position (

This method requires an initial location estimation.

The hyperbolic positioning algorithm converts this problem into a linear problem that can be solved with a least squares estimator, as we explain next. Using the previous notation, the square of the distance between the mobile node and anchor node

Without loss of generality, the origin of coordinates can be taken at anchor node _{1} = _{1} = 0. Thus, for

Expressing

In the case of RSS-localization, we do not know the real distances _{i}_{i}

Therefore, the position of the mobile node can be calculated as the least-squares solution of this equation, given by

Note that this hyperbolic algorithm does not directly minimize the error given by

These two typical positioning algorithms used for RSS-based localization give the same weight to the different distance estimations. But, since the RSS does not depend linearly on the distance between the nodes, the same error in the RSS measurement will produce larger errors in the distance estimation if the distance between the nodes is higher, as it can be deduced from

The linear problem in _{i}_{i}_{i}

Assuming that the channel is lognormal, it can be rapidly derived from _{i}_{i}_{d}_{i}_{d}_{d}

Finally, substituting these values into

It should be noted that _{d}_{i}_{i}_{i}_{d}

This weighted technique involves the computation of the inverse of matrix ^{3}), where

The second technique that we consider is the weighted circular algorithm, which is based on the circular positioning technique but introduces a different weight for each measurement. The basics of the algorithm are explained next. The error in the distance estimation for anchor node

In order to estimate the position (_{1} _{2} … _{N}^{T}_{i}_{ii}_{i}_{i}

As it can be noticed, the function that we want to minimize in this case (

The position (

As the value of _{d}

This technique is expected to perform better than the standard ones and than the weighted hyperbolic technique, as it considers the accuracy of the distance measurements and it is based on the circular algorithm (which gives better localization results than the hyperbolic algorithm, as it minimizes directly the distance errors and not some non-linear function of them as in the hyperbolic algorithm). But on the other hand, it is an iterative method, so its computational load may be greater.

The computational complexity of both circular positioning algorithms (classical and weighted) is O(^{3}), we expect a smaller computational load for this algorithm when the value of

In the following sections we show the performance of the two weighted techniques in comparison to the standard hyperbolic and circular positioning techniques for RSS-based localization. This comparison is made both with simulated RSS data and experimental data from a real sensor network deployment, from a WiFi deployment and from a Bluetooth deployment.

In this section we evaluate through some simulations in Matlab the performance of the proposed positioning techniques in terms of the accuracy of the localization results. We compare these results with those obtained with the standard hyperbolic and circular positioning algorithms and with the Cramer-Rao lower bound. We pay a particular attention to the robustness of these techniques to errors in the estimated channel model, that is, on how the accuracy degrades as the estimated channel model differs from the optimum one. The more robust the technique is, the more attractive from the practical point of view it will be, since it will behave better in a real situation where the channel is estimated from a limited set of measurements.

We also study the computational load of the considered algorithms, which is an important factor for two reasons. On the one hand, using algorithms with low computational cost is usually interesting, especially in resource-constrained devices in which the processing capacity is quite limited. On the other hand, the computational load is an indicator of the energy that will be consumed in the computation of the algorithm. This is an important issue if the algorithm is executed in the nodes and not in a central processor, since in wireless devices energy is a limited resource that must be saved and, although the transmission and reception of radio signals is the most energy consuming task in wireless devices, the computation also contributes to the energy consumption.

The Cramer–Rao lower bound for the localization problem with RSS measurements is derived in [

Note that the Cramer–Rao bound indicates the minimum variance an unbiased estimator of the position can achieve, thus, it is not directly comparable with the Mean Square Error (MSE) of the proposed estimators, which are biased. However, in the following figures representing the MSE of the proposed algorithms, we also show as a reference the Cramer–Rao bound.

The simulation environment is the following. We consider a 100 m × 100 m room, with _{estim}_{estim}_{real}_{real}_{estim}_{real}_{estim}_{real}_{estim}_{real}_{estim}_{real}

We have carried out 1,000 simulations, varying randomly the positions of the mobile node throughout the room. _{real}_{real}

_{real}_{real}_{real}_{real}

_{real}_{real}

_{real}

To sum up, the obtained results indicate that the accuracy of the proposed methods is better than the accuracy obtained with the traditional algorithms, especially when the measurements of the RSS have a large standard deviation and when the beacon density is high. We have also observed that the weighted circular technique has a very good performance and provides better localization results than the weighted hyperbolic technique, but at the expense of much higher computational costs (around three times the computational cost of the weighted hyperbolic techniques for typical (small) values of

In the previous simulations we have considered that the channel model used for the RSS-distance conversion was the same that the real channel model, that is, the channel model that was used to generate the RSS measurements. But in a real situation, the channel is usually characterized from a set of measurements and the estimated channel model may not be the best, that is, another model with slightly different parameters could lead to more accurate results. Therefore, it is very interesting to evaluate the sensitivity of the proposed algorithms to errors in the estimation of the channel parameters.

Considering the same simulation environment as in the previous experiment, that is, a 100 m × 100 m room, with _{estim}_{estim}_{real}_{real}_{estim}_{estim}_{estim}_{estim}_{real}_{estim}_{estim}_{real}^{X/10η} > 1, whereas if

In this section we describe the results of some experimental tests to evaluate the accuracy, processing time and robustness of the proposed positioning algorithms in real wireless network deployments. Three wireless networks have been used for these experiments: a Wifi network, a Bluetooth network and a wireless sensor network composed of MicaZ motes. These deployments were used to collect real RSS measurements, which were later used to process the proposed algorithms off-line.

The three experiments were carried out in an office area in which we deployed several reference nodes at different positions and a mobile node was used to test the algorithms. There were walls and furniture in the testing area that introduced non-line-of-sight propagation paths between the reference nodes and the mobile node.

In the first experiment, we deployed four WiFi access points (Aruba AP-65) in the testing area and we measured the RSS with a PDA (HP iPAQ hw6915) in various points following a 80 cm × 80 cm grid.

In the second experiment, we deployed three Bluetooth access points (minicomputers with a Belkin F8T013xx1 Bluetooth USB adapter) in the same office area and we measured the RSS from a PDA (HP iPAQ hw6915) that was situated in various points following the same grid as in the WiFi experiment.

Finally, in the third experiment, we deployed a MicaZ sensor network composed of twelve anchor nodes situated at fixed positions, five mobile nodes mounted on a platform that moved across the deployment area and a base node connected to a PC. The mobile nodes measured at every position the RSS from each anchor node and sent these measurements to the base, in order that the PC performed the RSS-distance conversion and the positioning algorithm. As the five nodes were situated at slightly different positions on the platform (they were separated between 15 and 30 cm), their measurements were considered as different test positions to be localized separately.

The three deployments work in the 2.4 GHz ISM band and with real propagation conditions, that is, people moving in the office, walls and furniture that act as obstacle for the radio signal and produce non-line-of-sight propagation,

These RSS measurements were then converted into distances through the lognormal channel model in _{estim}_{estim}

The received signal strengths in the WiFi experiment are quite high, since the transmission power of the WiFi access points was set to 20 dBm. However, in the other two experiments (Bluetooth and MicaZ) the environment was not so trivial, as the power levels were lower and often close to the sensitivity of the receivers.

The differences between the estimated values of

Finally, with the estimated distances, the proposed positioning algorithms were used to estimate the location of the PDA (in the WiFi and Bluetooth experiments) and the mobile nodes (in the sensor network experiment). For the weighted hyperbolic algorithm the final position is obtained using

As it can be noticed, the three experiments validate what we expected from theory and observed in the simulations: the weighted hyperbolic positioning technique has a better performance than the hyperbolic positioning technique and the weighted circular positioning technique has a better performance than the circular positioning technique. Again, we can also observe, as we did in the simulations, that the weighted circular positioning technique has the best performance and that the weighted hyperbolic positioning technique has a better performance than the circular positioning technique. This is a remarkable fact because this least squares technique also has a lower computational load. Therefore, this algorithm shows a good compromise between localization accuracy and computational cost.

In the experiments we described, we have used the model that was fitted to the experimental data for the RSS-distance conversion. But in a real localization situation, this is not always possible; the channel is usually characterized from a collection of previous measurements and would be different from the one we have used. Therefore, from the practical point of view, it has a great interest to evaluate the sensitivity of the proposed algorithms to variations in the estimated channel parameters, as we did with the simulations.

In the WiFi and in the wireless sensor network experiments it can be seen that the two proposed algorithms have better accuracy than the corresponding non-weighted algorithms for any value of the parameters of the channel model. That is, the weighted hyperbolic positioning algorithm always gives better localization results than the standard hyperbolic algorithm and the weighted circular algorithm always gives better localization results than the standard circular algorithm. Furthermore, when the channel model differs from the optimum one (the one for which the localization errors are the lowest, which corresponds in the WiFi case for example to _{estim}_{estim}_{estim}_{estim}

In this paper we have proposed the use of two weighted positioning algorithms to calculate the position of a mobile node in an ad hoc network from a set of distance estimations to the anchor nodes. We have extensively proved through numerical simulations and real experiments that these algorithms provide better localization results than the traditional hyperbolic and circular positioning algorithms, as they consider the accuracy of each distance estimation. Furthermore, we have seen that this improvement is especially important when the estimation of the channel model is not very accurate, which means that the proposed algorithms are more robust to inaccuracies in the channel estimation. This fact makes the algorithms very attractive for real localization applications in which the channel model must be estimated from a limited set of measurements, or cannot be calibrated at all.

In terms of accuracy, the circular algorithms perform better than the hyperbolic algorithms, as in the second case the cost function to be minimized is a non-linear function of the error. However, due to their non-iterative nature, the hyperbolic algorithms have the advantage of a lower computational cost (in the weighted hyperbolic algorithm this is true for small values of

For example, in terms of computational cost, the cost of the weighted hyperbolic technique is only slightly higher than the cost of the hyperbolic algorithm, and lower than the cost of the circular algorithm for the typical characteristics of real deployments. Therefore, the use of this algorithm will increase the accuracy of the localization system, at the expense of a little increase in the computational cost. Again, in a real deployment, where the channel estimation may not be very accurate, the use of this positioning algorithm may definitively improve the localization system performance. The weighted circular technique would be useful in applications that need a great accuracy in the localization and do not have energy restrictions or temporal constraints.

In further work we are planning to empirically evaluate the computational cost of these algorithms in terms of energy consumption. In particular, we have recently implemented embedded versions of the four algorithms (in MicaZ motes, under TinyOS 2.1) to evaluate with real-field experiments the energy consumption during the localization when the algorithms are executed in these resource-constrained devices. Preliminary results show, as otherwise expected, that whereas circular algorithms are more suitable for applications in which the localization is done occasionally, the weighted hyperbolic technique is appropriate and practical for real-time location tracking, as its computation time is very short and almost identical to that of the classical hyperbolic technique.

This work has been supported by the Government of Madrid under grant S-0505/TIC-0255 and by the Spanish Ministry of Science and Innovation under grant TIN2008-06742-C02-01. Paula Tarrío acknowledges the Spanish Ministry of Science and Innovation under grant FPI BES-2006-13954.

Average localization error (calculated as the square root of the MSE) as a function of the standard deviation _{real}

Average localization error (calculated as the square root of the MSE) of the four positioning methods as a function of _{real}_{real}

Average localization error (calculated as the square root of the MSE) of the four positioning methods as a function of _{real}

Average processing time of the different positioning algorithms as a function of the number of anchor nodes (_{real}

Average localization error for different positioning algorithms and for different values of the estimated parameters _{estim}_{estim}

Cumulative distribution function of the localization error in the wireless sensor network experiment using the four positioning algorithms.

Average localization error in the WiFi experiment for different values of _{estim}_{estim}

Average localization error in the wireless sensor network experiment for different values of _{estim}_{estim}

Average localization error and average processing time for the different positioning algorithms in the three experiments.

WiFi (N = 4) | ||||

| ||||

Hyperbolic | Weighted hyperbolic | Circular | Weighted circular | |

| ||||

Average localization error | 7.73 m | 3.69 m | 4.06 m | 3.03 m |

Average processing time | 0.04 ms | 0.11 ms | 2.24 ms | 4.04 ms |

| ||||

Bluetooth (N = 3) | ||||

| ||||

Hyperbolic | Weighted hyperbolic | Circular | Weighted circular | |

| ||||

Average localization error | 4.58 m | 4.58 m | 3.93 m | 2.70 m |

Average processing time | 0.04 ms | 0.10 ms | 2.44 ms | 3.02 ms |

| ||||

WSN (N = 12) | ||||

| ||||

Hyperbolic | Weighted hyperbolic | Circular | Weighted circular | |

| ||||

Average localization error | 10.59 m | 2.80 m | 4.01 m | 2.14 m |

Average processing time | 0.03 ms | 0.13 ms | 1.48 ms | 3.53 ms |