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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 this work, an acoustic sensor network for a relative localization system is analyzed by reporting the accuracy achieved in the position estimation. The proposed system has been designed for those applications where objects are not restricted to a particular environment and thus one cannot depend on any external infrastructure to compute their positions. The objects are capable of computing spatial relations among themselves using only acoustic emissions as a ranging mechanism. The object positions are computed by a multidimensional scaling (MDS) technique and, afterwards, a least-square algorithm, based on the Levenberg-Marquardt algorithm (LMA), is applied to refine results. Regarding the position estimation, all the parameters involved in the computation of the temporary relations with the proposed ranging mechanism have been considered. The obtained results show that a fine-grained localization can be achieved considering a Gaussian distribution error in the proposed ranging mechanism. Furthermore, since acoustic sensors require a line-of-sight to properly work, the system has been tested by modeling the lost of this line-of-sight as a non-Gaussian error. A suitable position estimation has been achieved even if it is considered a bias of up to 25 of the line-of-sight measurements among a set of nodes.

The computation of relative positions among mobile computing devices [

Several positioning algorithms can be used to determine the relative position of objects, depending on different parameters: the accuracy required for the estimation, the number of observations available at each node and the computational load. In indoor spaces, multilateration techniques are often developed by solving a set of equations that consider all the measurements carried out among nodes. One of the most common algorithms in relative positioning systems is the classic or metric Multidimensional Scaling (MDS) technique [

Many relative localization systems use data collected from a ranging mechanism, based on acoustic sensors, by measuring the times-of-flight (TOF) or difference-times-of-flight (DTOF) [

In this work, the accuracy achieved in the position estimation by the proposed acoustic sensor network is analyzed. Object positions are computed by MDS algorithm, starting from the distance measurements. Finally, after considering the different possible errors and time delays involved in the computation of temporary relations, refined positions are computed by applying the Levenberg-Marquardt algorithm (LMA) to the non-linear equations which describe the temporary relations among emissions provided by the S-RTOF.

The rest of the paper is organized as follows: Section 2 shows the proposed acoustic sensor network architecture. Also, in Section 2, the used encoding scheme and S-RTOF mechanism are described, considering the different parameters involved in the measurement process. Section 3 explains the used positioning algorithms (MDS and LMA). In Section 4 Monte-Carlo simulations are done to determine the accuracy that can be obtained in the position estimation. Finally, some conclusions are discussed in Section 5.

The system architecture is depicted in _{q}

The computation of the distances _{ql}

A remarkable capability for relative positioning is the measurement in a short time of all the spatial relations among objects, by using a common temporary reference. In order to measure the spatial relations among objects in the shortest time, it is necessary to use multi-user schemes that allow to simultaneously discriminate the emissions from every user or sensor.

In most cases, Direct-Sequence Code-Division Multiple-Access (DS-CDMA) techniques are used to discriminate the node emissions, by encoding every emitter with binary sequences and transmitting it by a simple phase modulation. These encoded signals are detected in a receptor by performing the correlation with every available sequence in the proposed system. Thus, the effectiveness depends on the properties of the used codes [

In the proposed system, an encoding scheme based on Complementary Set of

As opposed to the common encoding schemes used in localization, the encoding of emissions by

According to these signal processing techniques, the emitting stage implemented at the node hardware architecture is shown in

Finally, a processing unit has been implemented at each node, which controls the emitting stage according to the results obtained at the correlator outputs. It also computes the temporary relations among the emissions carried out by every node. Moreover, this unit performs the distribution of the data collected by every node by means of a communication block, and it computes the position among the objects.

Since all nodes are equal in their architecture and functionality, anyone of them can start the ranging process (it is called

It is assumed that node _{M}, y_{M}, z_{M}_{M}_{q}_{l}

In response to the

According to the described principle of measurement, it is possible to determine temporary relations among the emissions carried out from the different nodes, by measuring the differences between the correlation maximum values detected at every node. These values do not exactly describe the propagation time of the acoustic emissions, since there is an offset caused by the rising time of the emitted signal, the latency of signal processing algorithms, and the response time of the transducers [_{Spk}_{Mic}_{M}_{−}_{q}_{CODE} is the temporary length of the emitted encoded signal; ‖ ‖ is the Euclidean norm between the considered position vectors (_{M}_{q}_{Spk}_{Mic}_{M}_{q}

On the other hand, in the case of the pTOFs measured between slave nodes, e.g., node _{q}_{−}_{l}

In (1), the computed pTOF depends on the distance between the _{M}_{l}_{M}_{q}_{l}_{q}_{l}

According to (1) and (2), to determine the node positions, it is necessary to solve a non-linear equation system. The parameters to be estimated are described in (3), i.e., the coordinates of every node as well as the time delays associated to the signal processing algorithms in the emission and reception stages:
_{q} are the coordinate positions (_{q},y_{q},z_{q}

The diagram of the algorithms used to compute the relative positions among nodes is shown in

The MDS algorithm allows a geometric configuration of the object positions to be achieved in the smallest number of dimensions, by means of the Law of Cosines and linear algebra, when only the inter-node distances are known [_{M}_{−}_{q}_{q}_{−}_{l}

Assuming that the moving velocities of the nodes are much lower than the propagation speed of the acoustic waves, it is possible to compute the distance between the _{M}_{q}_{M}_{Spk M}_{Mic M}_{Spk q}_{Mic q}

On the other hand, with the data collected by the slave nodes, it is possible to compute their distances by (6) and the reciprocal pTOFs measured. Again, assuming that the node velocity is much lower than the propagation speed of acoustic waves, this distance is:

If an additive Gaussian noise is considered in the measurements of pTOFs, in some particular cases, the slave nodes—see (2)—could detect a negative value of _{q}_{−}_{l}

With all the computed distances, a matrix _{ij},d_{iq},d_{jl},d_{mn}

Also

The MDS provides a suitable estimation of the node distribution and positions by converting a whole matrix of distance measurements into a topology with two or three dimensions. Nevertheless, the exact object position cannot be achieved, since there are different error sources in the detection of acoustic signals (environmental effects such as room reverberation, acoustic transducer bandwidth, accuracy in hardware clock signals, etc). These errors corrupt the matrix ^{2}, the uncertainty of the observation vector

A suitable method to solve this kind of problems is the Levenberg-Marquardt Algorithm, which allows to find a local minimum by using a particular starting point. The results obtained with the MDS algorithm are a suitable starting point to estimate the coordinates by means of the LMA.

According to (14), the number of unknown coordinates as well as the time delays associated to every node should be considered. Nevertheless the estimation of all the time delays increases the complexity of the equation system to be solved so the time delays have been considered equal in all the nodes; under this assumption, the number of parameters to be estimated is:

In order to solve the equation system, the number of observations carried out in the system should be higher than

In order to compare the accuracy achieved with the architecture previously described and the used positioning algorithms, some simulations have been carried out. In this analysis, different types of errors have been modeled.

A first analysis has been performed by considering a node topology in 2_{2}), providing the coordinate reference system (see ^{2}. Furthermore, the time delays have been considered time-invariant and determined by an off-line characterization, with values _{Mic}_{Spk}

If the positioning algorithms for _{4} are computed changing the number of nodes in the topology described in _{4}. The CDF is plotted in

In the previous tests the signal processing delays have been considered time-invariant. Nevertheless, the true value of

In order to verify the behaviour of the positioning process with these considerations, the algorithms have been tested with the topology described in _{tp}_{tp}

In this case, it is observed that the MDS allows a higher precision than the LMA (see _{4} is depicted with a continuous line, only for errors in the determination of

In the previous analysis it is assumed that the node topology is fixed. In the topology described in _{7}_{7}

The use of acoustic signals as ranging mechanism requires that the signal propagates along the line-of-sight (LOS) path between nodes. If LOS is blocked by obstacles, becoming a non-line-of-sight (NLOS) link, the signal may reach the receiver due to reflections in the environment. In this case the length of the reflected path is longer than the line-of-sight, reason why the measurements with NLOS errors can considerably bias the real pTOF value and provide inaccurate location estimation. These kinds of errors are non-Gaussian and they are more difficult to solve since they strongly depend on unknown environment conditions.

In order to verify the performance of the proposed algorithms with NLOS effects, the topology of _{1} and _{14} have a large observation error, whereas other nodes have clear LOS path. In practice, this situation can happen when the LOS path between the considered nodes is blocked by walls or other signal-scattering object. The NLOS has been modelled as a positive bias error with an uniformly random distribution, where the maximum value is a percentage of the real pTOF measured in the interval [0%, 50%]. The total bias and total variance in the position estimation have been obtained for the different NLOS error considered as is depicted in _{3} and _{6}. Nevertheless, when stronger NLOS effects appear between the considered nodes, the global node position estimation has a reduced performance with both algorithms.

The second test considers an NLOS effect by means of a solid object closer to _{3} (see the node distribution in _{1},_{2},_{6},_{13} to _{3}. In this case, the total bias and total variance in the position estimation for different NLOS errors is depicted in

This example shows that, due to the cooperative nature of the architecture, the NLOS effect in a set of pTOF measurement from different nodes implies errors in all the node positions. Nevertheless, the features of the ranging mechanism provide a set of redundant observation data, which, together with high-level algorithms such as geometry consistency or statistical analysis, can improve the estimation performance by identifying and rejecting those measurements with low accuracy.

The performance of a relative positioning algorithm based on the Multi-Dimensional Scaling (MDS) technique and the Levenberg-Marquardt algorithm (LMA) has been analyzed. In this case, the spatial relations among objects are determined by an acoustic sensor network where only acoustic emissions are used, providing a low-complex ranging mechanism. The MDS algorithm allows a geometric configuration of the nodes to be obtained with the minimum number of dimensions. Nevertheless, there are different error sources in the detection of acoustic signals that perturb the distance computation by inserting an error in the position estimation. In order to improve the achieved results, an optimization by LMA has been applied. In this way, the temporary relations among acoustic emissions carried out by the nodes have been used for the minimization. The characterization of the positioning procedure has been performed by Monte-Carlo simulations. The obtained results show that the LMA algorithm allows the precision in the MDS results to be improved, considering different node topologies and errors with Gaussian distribution in the determination of temporary relations. In this case, a fine-grained localization is achieved, even considering different error magnitude in the ranging mechanism (from

This work has been supported by the Ministerio de Educación y Ciencia from Spain (RESELAI project, ref. TIN2006-14896-C02-01), by the Ministerio de Fomento from Spain (DETECTREN project, ref. P13/08), by Comunidad de Madrid/Universidad de Alcalá (SUSGAB project, ref. CCG08-UAH/DPI-4072) and by Facultad de Ingeniería, Universidad Nacional de la Patagonia San Juan Bosco, Argentina.

General scheme of the proposed sensor network.

Detailed block diagram of node hardware architecture.

Principle of measurement using simultaneous Round-Trip-Time-of-Flight. (a) Emission of the request from the

Positioning Algorithm Scheme.

Position estimation for a distribution of nodes with a Gaussian noise in the pTOF measurements. (a) 95% uncertainty ellipses considering an error with standard deviation Σ = 100μs. (b) Total variance in the position estimation. (c) Total bias in the position estimation.

Cumulative distribution function in the position estimation of node _{4}

Position estimation with errors in _{4}

Position estimation when changing the topology of nodes.

Errors in the position estimation considering NLOS effect between nodes _{1} and _{14} (see the node distribution in

Errors in the position estimation considering NLOS effect between nodes _{1},_{2},_{6},_{13} and _{3} (according to the distribution shown in