International Journal of Navigation and Observation

Volume 2015, Article ID 562680, 16 pages

http://dx.doi.org/10.1155/2015/562680

## Reduced-Complexity Algorithms for Indoor Map-Aware Localization Systems

^{1}Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Via Pietro Vivarelli 10, 41125 Modena, Italy^{2}Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via Amendola 2, 42122 Reggio Emilia, Italy

Received 26 May 2015; Revised 13 September 2015; Accepted 28 September 2015

Academic Editor: Gonzalo Seco-Granados

Copyright © 2015 Francesco Montorsi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The knowledge of environmental maps (i.e., map-awareness) can appreciably improve the accuracy of optimal methods for position estimation in indoor scenarios. This improvement, however, is achieved at the price of a significant complexity increase with respect to the case of map-unawareness, specially for large maps. This is mainly due to the fact that optimal map-aware estimation algorithms require integrating highly nonlinear functions or solving nonlinear and nonconvex constrained optimization problems. In this paper, various techniques for reducing the complexity of such estimators are developed. In particular, two novel strategies for restricting the search domain of map-aware position estimators are developed and the exploitation of state-of-the-art numerical integration and optimization methods is investigated; this leads to the development of a new family of suboptimal map-aware localization algorithms. Our numerical and experimental results evidence that the accuracy of these algorithms is very close to that offered by their optimal counterparts, despite their significantly lower computational complexity.

#### 1. Introduction

Indoor localization systems have found widespread application in a number of different areas, both military and civilian [1]. In many cases, the maps of the environments where the users are supposed to lie (e.g., the floor plans of a given building or a spatial road network) are perfectly known and this form of a priori knowledge (dubbed* map-awareness* in the following) can be employed for improving localization accuracy. In particular, in the technical literature about localization systems map information is usually exploited to (a) improve the quality of position estimates generated by standard localization technologies; (b) set simple geometric constraints in the search space of* fingerprinting systems* (e.g., see [2–4]); (c) acquire an accurate knowledge of propagation of wireless signals on the basis of two-dimensional or three-dimensional* ray-tracing techniques* [5, 6]; (d) develop* map-based statistical models* for the measurements acquired by multiple anchors in a given indoor environment. It is important to point out the following:(i)Point (a) refers to the wide family of* map-matching algorithms* (see [7] and references therein), which includes both simple search techniques exploiting geometric and topological information provided by maps, and more refined techniques based on signal processing algorithms, such as Kalman filtering and particle filtering [8]. Distinct types of map-matching algorithms may offer a substantially different complexity-performance trade-off, but the improvement in localization accuracy is limited by the fact that map information is not exploited in the first stage of position estimation (i.e., in ranging). Note also that these algorithms have been mainly developed for transport applications (where they are employed to refine the estimated position provided by GPS or GPS integrated with* dead-reckoning* in outdoor environments). In principle, they can be also employed in indoor scenarios, provided that a proper map (e.g., a schematic representation of the possible user positions such as a Voronoi diagram) is available for the considered environment; for instance, map-matching methods based on particle filtering have been proposed in [9–13] for localization systems operating in indoor scenarios.(ii)* Fingerprinting methods* (see point (b)) make a limited use of the geometrical properties of the propagation scenario, since they mainly rely on a training database. In addition, the preliminary measurement campaign necessary to generate such a database can represent a formidable task in large buildings and/or time-varying scenarios.(iii)* Ray-tracing techniques* (see point (c)) can certainly outperform the above-mentioned techniques in terms of accuracy; however, real-time ray-tracing is often unfeasible because of its large computational complexity and the detailed knowledge it requires about various physical properties (e.g., electrical permittivity of walls and boundary conditions) of localization scenarios.(iv)Point (d) refers to the so-called* map-aware statistical localization techniques*, which have been developed for* time of arrival* (TOA),* time difference of arrival* (TDOA), and* received signal strength-* (RSS-) based indoor localization systems [14, Sec. 4.11.5], [15, 16]. Such techniques rely on the availability of* map-aware statistical signal models* and are characterized by the following relevant features: (a) they are able to compensate for the* non-line-of-sight* (NLOS)* bias* (i.e., for the extra delay or extra attenuation due to propagation through obstructions, mainly walls), which represents a major source of error in indoor localization; (b) their use requires measurement campaigns substantially less time consuming than those needed for fingerprinting techniques in the same scenarios. In addition, previous work illustrated in [15, 16] has evidenced that these models can be easily combined with optimal estimation techniques to derive novel localization algorithms.This paper represents a follow-up to [15], where new map-aware statistical models based on experimental measurements have been exploited to develop optimal* map-aware minimum mean square error* (MMSE) and* maximum a posteriori* (MAP) estimators. Our previous work has evidenced that these estimators may substantially outperform their map-unaware counterparts at the price, however, of a significant complexity increase, specially for large maps. In this paper, the problem of complexity reduction of map-aware MMSE and MAP estimators is tackled and the following novel contributions are developed:(1)Two novel algorithms for restricting the domain over which the agent position is searched for in map-aware estimation are devised. These algorithms, dubbed* distance-reduced domain* (DRD) and* probability-reduced domain* (PRD) in the following, can reduce the rate at which the complexity of map-aware estimators increases with map size without substantially affecting localization accuracy.(2)The application of specific mathematical tools (namely,* cubature rules* for numerical integration [17, 18] and* direct-search methods* [19, 20]), which can be exploited to reduce the implementation complexity of map-aware algorithms, is analysed.(3)Novel suboptimal map-aware algorithms resulting from the combination of the above-mentioned algorithms and mathematical tools are proposed and are compared, in terms of both* root median square error* (RMSE) and computational complexity (namely, overall number of* floating point operations* (FLOPs)), to* maximum likelihood* (ML) estimators (note that ML estimators are optimal for* map-unaware *localization and have been widely adopted for their limited computational complexity [21]). The following is worth pointing out:(i)Most of the performance results illustrated in this paper mainly rely on a set of data generated according to the statistical model developed in [15], which, in turn, is based on experimental measurements. This approach, which is commonly adopted in the technical literature (e.g., see [22, 23]), is motivated by the fact that experimental databases referring to localization systems have usually a limited size; this is mainly due to the time consuming tasks required by any measurement campaign in this field. However, some performance results evaluated on the basis of our experimental database are also shown.(ii)In our work only RSS-based localization algorithms are considered, even if the proposed approach can be easily extended to TOA and TDOA based systems, since all rely on similar statistical signal models [15].(iii)Our contribution focuses on the role played by* map-awareness* in the localization of still targets in static environments (consequently, the potential improvement deriving from the knowledge of mobility models is not taken into consideration).The remaining part of this paper is organized as follows. In Section 2, some models for indoor maps and measurements in map-aware RSS-based localization systems are illustrated. In Section 3, optimal and suboptimal localization algorithms based on the proposed models are developed, whereas in Section 4 various problems arising from their implementation are analysed and some solutions are proposed. In Section 5 various numerical results about the performance and complexity of the devised algorithms are illustrated. Finally, in Section 6, some conclusions are drawn.

*Notations*. The* probability density function* (pdf) of a random vector evaluated at the point is denoted ; denotes the pdf of a Gaussian random vector having mean and covariance matrix , evaluated at the point ; denotes a square matrix having the arguments on its main diagonal and zeros elsewhere; denotes the cardinality of the set ; denotes the th element of the vector and denotes its size; denotes the floor of ; denotes the Euclidean projection of over a domain , that is, the point which minimises the Euclidean distance with ; denotes the centre of the rectangle ; () is the unit vector along the () axis.

#### 2. Reference Scenario and Signal Model

In the following, we focus on a* two-dimensional* (2D) RSS-based localization system employing devices, called* anchors*, and whose positions are known, to estimate the position of an* agent* (see Figure 1). The mathematical models defined in [15] for the map, the wireless connectivity between the agent and the anchors, and the measurements acquired for position estimation are adopted; their essential features are summarised below.