Location estimation plays a key role in many applications, and is today attracting a great interest from both industry and academia. This process is sustained by the increasing widespread use of interconnected smart devices, which have the potential to enable seamless position capabilities at significantly reduced development costs. Indeed, besides the undisputed importance of the ultimate localization accuracy, the low-complexity requirement has been recently regarded as a fundamental need in most resource-constrained operational contexts, such as body area networks and wireless sensor networks (WSNs). In this respect, the design of localization solutions able to exploit the received radio frequency signals in an opportunistic way, without requiring any additional hardware, is of great practical interest.
Several approaches have been proposed in the literature to tackle the problem of node localization by leveraging information extracted from existing terrestrial technologies [1
]. Among others, range-based solutions have been widely employed in popular terrestrial localization systems, especially in WSNs, where they are preferred to range-free techniques [7
] thanks to their reduced complexity. Range information can be obtained by exploiting different characteristics of the received signals, namely received signal strength (RSS) [9
], time (difference) of arrival (TOA/TDOA) [17
], and angle of arrival (AOA) [21
]. Hybrid approaches combining different types of measurements have also been proposed. For instance, references [26
] investigated novel localization schemes based on the combined use of RSS and AOA measurements, which are able to achieve improved estimation performance. Although interesting, these techniques require enhanced hardware and processing capabilities that are usually not available on-board low-cost sensor nodes employed in typical WSN deployments; more precisely, AOA-based methods are not appealing in such contexts owing to the increased complexity and costs associated with multiple antennas.
TOA/RSS solutions better fit the requirements of low-cost contexts, thanks to their simplicity and ready availability, and the hybrid (joint) approach is interesting for its improved accuracy [33
]. A theoretical study of the localization performance achievable using joint TOA/RSS measurements has been conducted in [33
]. By providing the exact expression of the Cramér–Rao lower bound (CRLB), authors showed that such hybrid schemes can offer significant improvements in the estimation accuracy with respect to the case where RSS-only or TOA-only measurements are used. In [34
], authors addressed the problem of target localization in presence of mixed line-of-sight (LOS)/non-line-of-sight (NLOS) scenarios. In a first step, the algorithm classifies all the links as either LOS or NLOS according to a parametric Nakagami distribution; then, the position is estimated through a weighted least squares (WLS) approach that considers the sole TOA measurements for those links identified as LOS and the RSS measurements for the NLOS paths. A novel algorithm to tackle the problem of target localization in adverse NLOS environments has been proposed in [35
]. In this work, authors tried to mitigate the detrimental effects of NLOS biases on both RSS and TOA measurements by recasting the original (non-convex) problem as an approximated generalized trust region sub-problem. The ultimate target position estimate is then obtained by applying a bisection procedure based on an iterative squared range WLS approach. In [36
], same authors addressed the more challenging scenario in which all paths are in NLOS. By assuming a partial knowledge of the NLOS biases affecting the collected measurements, the highly-dimensional localization problem can be approximated to a min-max problem in which the NLOS biases are treated as known nuisance parameters. Similarly to [35
], the final target location is estimated by resorting to a robust bisection procedure within a generalized trust region framework.
It is worth noting that almost all the aforementioned localization schemes based on hybrid TOA/RSS measurements require that a partial or even perfect knowledge of the model parameters is available. Moreover, in most cases all the nodes in the network are implicitly assumed to be perfectly synchronized among each other. Although these assumptions are typically used to reduce the high dimensionality of the cost functions involved in the position estimation process, in practical WSN deployments, nodes are not synchronized to the same clock and all the model parameters are unknown, hence the need to be adaptively estimated “on the field”. Furthermore, in most operational scenarios, such as indoor environments and urban areas, the presence of dynamic obstacles (people, vehicles, etc.) introduces significant modifications in the surrounding environments, leading to channel characteristics that are non-stationary over time. Therefore, to address the problem of target localization in real applications, a preliminary calibration phase aimed at estimating the unknown channel parameters at run-time should be performed. On the one hand, such a procedure must be as simple as possible in order to suit multiple real-time re-calibrations even in low-cost applications; on the other hand, it must be fully automatic, that is, it should not require any external assistance, including human interventions and frequent configurations—for this reason, such a feature can be also called self-calibration.
In this paper, we address the problem of target localization based on the combined use of TOA and RSS measurements under the realistic case of self-calibrated parameters. This work is motivated by the analysis conducted in [43
], where it is highlighted the biased nature of the distance estimates obtained via RSS-only ranging in case of self-calibrated channel parameters. The main objective of this paper is to investigate how a dynamic estimation of the TOA and RSS model parameters will impact the ultimate position estimation performance of joint TOA/RSS range-based localization. To this aim, we extended our previous work [44
] by considering, in addition to the hybrid TOA/RSS range estimator proposed therein, a preliminary self-calibration stage and a final position estimation stage. In particular, differently from [44
] where all the parameters are assumed to be known, in this paper, we consider the more realistic case where the model parameters are estimated “on the field” according to a self-calibration procedure (described in Section 3.1
). Moreover, we explicitly address the problem of nodes synchronization by including the clock offsets in all the considered models, as opposed to [44
], where nodes are assumed to be synchronized. Our results show that using estimated parameters in place of the ideal (true) ones introduces detrimental non-linear effects that cannot be easily mitigated, and may lead to localization errors that can be significantly bigger than those obtained assuming a perfect knowledge of the environment; nonetheless, the provided algorithm is able to perform the whole localization task in a real environment and in a fully automatic (unsupervised) way.
Summarizing, the paper provides a two-fold contribution. First, we formalize the hybrid TOA/RSS localization problem in presence of synchronization errors among all the nodes in the network, and derive the corresponding joint maximum likelihood (ML) estimator of target position and nuisance parameters. As we will see, such an estimator does not admit a closed form solution; moreover, due to the large number of nuisance parameters in the final cost function, a numerical optimization is practically unfeasible. To circumvent such issues, as a second contribution we design a novel two-step algorithm with reduced complexity to solve the joint TOA/RSS range-based localization problem; more specifically, we split the original problem into two separated phases:
a first calibration phase carried out by the different nodes in the network to estimate all the unknown RSS and TOA model parameters, including the synchronization offsets;
a second localization step that leverages the combination of hybrid TOA/RSS ranging and iterative least squares (ILS) approach to estimate the unknown target position.
It is worth highlighting that the proposed approach, in each of the two separated phases involved in the localization task, exploits cooperation between nodes with known positions (anchors) and the node to be localized (blind node). More precisely, anchors need to exchange their collected TOA/RSS measurements and to communicate the time instants they start to transmit to the blind node. In this respect, we assume that all nodes in the network are equipped with low-power radio technologies (e.g., based on IEEE 802.15.4 standard, such as ZigBee, 6LoWPAN, etc.) that enable communications in a range of a few meters, while still preserving the required power efficiency.
The rest of the paper is organized as follows. In Section 2
we provide the problem formulation and discuss the resolution of the joint estimation problem using both TOA and RSS measurements. In Section 3
we derive in details the proposed two-step localization approach with self-calibration. Section 4
is devoted to the performance assessment of the proposed approach also in comparison to natural competitors. We conclude the paper in Section 5
4. Performance Assessment
In this section, we evaluate the performance of the proposed algorithm by means of Monte Carlo simulations. We consider, as a reference scenario, a WSN deployed over an area of
m; in particular, the N
anchors are distributed in the space so as to provide a good coverage in terms of connectivity, while the blind node is randomly located within the considered area and its position varies in each simulation trial. We consider for the RSS measurements the parameter
= −30 dB, while for the path-loss exponent the two cases
are evaluated. As to the synchronization offsets
, they are generated according to a uniform distribution in the interval
m represent the equivalent offsets expressed in terms of distances, respectively. In this analysis, we assume without loss of generality that both RSS and TOA measurements have the same accuracy, i.e.,
. Thus, all measurements have the same variance
, i.e., neither measurement has better “quality” than the other. This is not always the case, of course, but this choice has the advantage of reducing the parameter space for the simulations to the most interesting case: in fact, in [44
] it is shown that if TOA and RSS variances are (significantly) different, then one of the two sources of information always dominates; hence, the hybrid estimator boils down to the TOA-only or RSS-only estimator, accordingly. For the assessment we consider values of
ranging from 3 to 6, which are representative of several operational scenarios of WSNs, from outdoor open spaces to propagation environments full of obstructions (e.g., people, furniture, …).
The proposed hybrid estimator is computed using
as in [44
], and it is compared against the most natural competitor algorithms based on Newton–Raphson iterations, given in [53
] and in the following labeled as NR1 and NR2. The basic difference among the NR1 and NR2 methods consists in the number of Newton–Raphson iterations performed to compute the distance estimate. More precisely, the NR1 algorithm uses a single iteration, while the NR2 algorithm employs two iterations. Even though such algorithms were originally designed for range estimation, the ultimate blind node position can be easily retrieved by resorting to the ILS procedure in Equation (39
) based on the N
distance estimates obtained during the ranging phase. For comparison purposes, we also report the performance of the (range-free) joint ML position estimator in Equation (8
), obtained via a two-dimensional grid search. We, however, remark that the latter cannot be taken as a direct competitor since it is not available in closed-form; hence, it will be considered only as a benchmark. Moreover, we also compare the performance of the hybrid estimators against the more traditional algorithms based only on RSS or TOA information, denoted by “RSS-only” and “TOA-only”, respectively. To assess the ultimate algorithms performance, we consider the mean square error (MSE) metric, computed from the position error based on 1000 trials.
We start the analysis by investigating the computational complexity of all the considered approaches. More specifically, we have recorded the runtimes of the estimators executed on the same hardware platform, for a network with
anchors. A grid of
evaluation points per dimension is considered for the joint ML estimator. A comparison between the average runtimes of the different estimators (normalized to the fastest one, that is the NR-based method,) is provided in Figure 3
As expected, the joint ML requires by far the longest runtime due to the two-dimensional search involved in the solution of Equation (8
). Remarkably, the proposed hybrid TOA/RSS approach has only slightly higher complexity compared to NR, while it is significantly less complex than the optimal joint ML.
In Figure 4
, we report the MSEs of all the considered algorithms as function of the measurement error
. The solid curves represent the ideal estimation performance obtained by assuming that the TOA/RSS model parameters
are perfectly known a priori, while the dashed curves show the corresponding versions with the unknown parameters replaced by their estimates obtained from the preliminary calibration phase presented in Section 3.1
. The obtained results clearly show that the proposed hybrid estimator outperforms the state-of-the-art competitors for all the considered range of
and, remarkably, can achieve almost the same accuracy of the joint ML benchmark, but at a significantly reduced computational cost. To better highlight the obtained results, in Figure 5
we depict the considered
anchors for a given position of the target, together with a graphical representation of the estimated circular error probable (CEP) (the CEP is usually defined as the radius of the circle, centered in the average estimated position, which is expected to include 50% of the estimated positions) resulting from the application of the proposed hybrid TOA/RSS approach for the two specific cases of
As expected, the actual performance in the more realistic case of self-calibrated algorithms are much worse compared to that obtained under ideal knowledge of the models parameters, especially for high values of
. To corroborate such results, we investigate the performance of the self-calibration procedure adopted to estimate the unknown TOA/RSS model parameters. In Figure 6
, we depict the estimation errors obtained during the calibration phase for the two values of the measurement error
, which are representative of a “low-noise” and “high-noise” scenarios, respectively.
The obtained performance is in agreement with the findings in [43
] (which however addressed the specific problem of RSS-only ranging): in fact, higher values of
amplify the errors in the model parameters estimation and, in turn, introduce severe non-linear effects that are detrimental to the final positioning task. From this analysis, it clearly emerges that in practical WSN deployments—where the true values of the environmental parameters are unknown—the ultimate localization accuracy is also affected by the quality of the parameter estimation task.
To further investigate how a realistic calibration affects the localization process, in Figure 7
we report the algorithms performance as a function of the number of anchors N
, now for the case
. As it can be noticed, for a sufficient number of anchors N
, almost all the algorithms exhibit a satisfactory localization accuracy, in spite of the presence of a quite high noise level
. Interestingly, for
the dashed-curves (estimated parameters) closely attain the ideal ones (known parameters). Again, the proposed hybrid TOA/RSS position estimator offers the best performance, with an MSE below 10 m for almost all the considered span of N
. These results are reasonable since for higher values of N
the increasing amount of information collected within the network enables a much more accurate estimation of the unknown models parameters (through the self-calibration procedure proposed in Section 3.1
); conversely, as N
decreases, the algorithms performance experience a rapid degradation until achieving unacceptable results for a number of anchors
. In addition to a higher amount of information in the network (anchor-to-blind measurements scale linearly with N
, anchor-to-anchor measurements scale quadratically with N
), increasing the number of anchors N
also leads to a better coverage of the considered area, resulting in turn into more favorable geometric conditions for the estimation of the unknown blind position. To quantify the impact of a better distribution of the anchors in the space, in Figure 8
we evaluate the horizontal dilution of precision (HDOP) for each different number of anchors N
As it can be seen, increasing the density of anchors from 8 to 14 yields an accuracy improvement in terms of HDOP of about 40%.
As concerns the poor performance of the NR-based methods, it can be attributed to the fact that the hybrid TOA/RSS ranging problem is solved via a suboptimal approach based on Newton–Raphson iterations. More precisely, from the closed-form expressions of the estimators provided in [53
], it can be noticed that the Newton–Raphson procedure is initialized with a coarse range estimate obtained using the sole TOA measurement; hence, it is reasonable to expect that the performance of both NR1 and NR2 approaches heavily depend on the accuracy of such data. Such theoretical considerations are in agreement with the performance of the NR1 method, whose performance curves are superimposed to the ones provided by the TOA-only estimator, hence are not visible in the figures, meaning that in the considered scenario one iteration of the Newton–Raphson method is indeed not sufficient to find a good approximate solution to the joint TOA/RSS ML problem.
To complete the analysis, in Figure 9
we show the empirical cumulative distribution functions (ECDFs) of the algorithms localization errors for the case
and assuming a number of anchors
. As it can be noticed, a perfect knowledge of the models parameters would lead to positioning errors that are always below 7 m for all the hybrid estimators; conversely, in case of a realistic self-calibration, all the algorithms suffer evident impairments in the achieved performance, with errors that can even reach 20 m (tails of the ECDFs). Remarkably, the proposed hybrid estimator closely follows the joint ML benchmark, with a position error that is lower than 5 m in
of the cases. As a confirmation, in Figure 10
, we report the ECDFs of the localization errors for the challenging case of
As it could be expected, the obtained performance are generally worse than the ones in the previous case, especially for the RSS-only and TOA-only estimators which cannot mitigate the negative effects of an increased noise. Nevertheless, the proposed hybrid estimator still outperforms all the competitors, achieving localization performance that remains very close to the one provided by the joint ML. Notice also that for some percentile the joint ML has inferior performance; this is partially due to the fact that it is obtained by a numerical optimization of a non-linear function, which may not always terminate at the absolute maximum. Furthermore, in [44
] it is shown that the hybrid TOA/RSS range estimator achieves a smaller MSE than the joint ML estimator (this is because of the bias-variance trade-off, i.e., it has been possible to design an estimator that is slightly biased but overall has a reduced value of MSE.); hence, this may, in turn, impact on the corresponding localization accuracy assessed here.