Localization using Distance Geometry : Minimal Solvers and Robust Methods for Sensor Network Self-Calibration

In this thesis, we focus on the problem of estimating receiver and sender node positions given some form of distance measurements between them. This kind of localization problem has several applications, e.g., global and indoor positioning, sensor network calibration, molecular conformations, data visualization, graph embedding, and robot kinematics. More concretely, this thesis makes contributions in three different areas.First, we present a method for simultaneously registering and merging maps. The merging problem occurs when multiple maps of an area have been constructed and need to be combined into a single representation. If there are no absolute references and the maps are in different coordinate systems, they also need to be registered. In the second part, we construct robust methods for sensor network self-calibration using both Time of Arrival (TOA) and Time Difference of Arrival (TDOA) measurements. One of the difficulties is that corrupt measurements, so-called outliers, are present and should be excluded from the model fitting. To achieve this, we use hypothesis-and-test frameworks together with minimal solvers, resulting in methods that are robust to noise, outliers, and missing data. Several new minimal solvers are introduced to accommodate a range of receiver and sender configurations in 2D and 3D space. These solvers are formulated as polynomial equation systems which are solvedusing methods from algebraic geometry.In the third part, we focus specifically on the problems of trilateration and multilateration, and we present a method that approximates the Maximum Likelihood (ML) estimator for different noise distributions. The proposed approach reduces to an eigendecomposition problem for which there are good solvers. This results in a method that is faster and more numerically stable than the state-of-the-art, while still being easy to implement. Furthermore, we present a robust trilateration method that incorporates a motion model. This enables the removal of outliers in the distance measurements at the same time as drift in the motion model is canceled

Time-based Location Techniques Using Inexpensive, Unsynchronized Clocks in Wireless Networks

The ability to measure location using time of flight in IEEE 802.11 networks is impeded by the standard clock resolution, imprecise synchronization of the 802.11 protocol, and the inaccuracy of available clocks. To achieve real-time location with accuracy goals of a few meters, we derive new consensus synchronization techniques for free-running clocks. Using consensus synchronization, we improve existing time of arrival (TOA) techniques and introduce new time difference of arrival (TDOA) techniques. With this common basis, we show how TOA is theoretically superior to TDOA. Using TOA measurements, we can locate wireless nodes that participate in the location system, and using TDOA measurements, we can locate nodes that do not participate. We demonstrate applications using off-the-shelf 802.11 hardware that can determine location to within 3m using simple, existing optimization methods. The synchronization techniques extend existing ones providing distributed synchronization for free-running clocks to cases where send times cannot be controlled and adjusted precisely, as in 802.11 networks. These location and synchronization techniques may be applied to transmitting wireless nodes using any communication protocol where cooperating nodes can produce send and receive timestamps

Toward Collinearity-Avoidable Localization for Wireless Sensor Network

In accordance with the collinearity problem during computation caused by the beacon nodes used for location estimation which are close to be in the same line or same plane, two solutions are proposed in this paper: the geometric analytical localization algorithm based on positioning units and the localization algorithm based on the multivariate analysis method. The geometric analytical localization algorithm based on positioning units analyzes the topology quality of positioning units used to estimate location and provides quantitative criteria based on that; the localization algorithm based on the multivariate analysis method uses the multivariate analysis method to filter and integrate the beacon nodes coordinate matrixes during the process of location estimation. Both methods can avoid low estimation accuracy and instability caused by multicollinearity

Super-resolved localisation in multipath environments

In the last few decades, the localisation problems have been studied extensively. There are still some open issues that remain unresolved. One of the key issues is the efficiency and preciseness of the localisation in presence of non-line-of-sight (NLoS) path. Nevertheless, the NLoS path has a high occurrence in multipath environments, but NLoS bias is viewed as a main factor to severely degrade the localisation performance. The NLoS bias would often result in extra propagation delay and angular bias. Numerous localisation methods have been proposed to deal with NLoS bias in various propagation environments, but they are tailored to some specif ic scenarios due to different prior knowledge requirements, accuracies, computational complexities, and assumptions. To super-resolve the location of mobile device (MD) without prior knowledge, we address the localisation problem by super-resolution technique due to its favourable features, such as working on continuous parameter space, reducing computational cost and good extensibility. Besides the NLoS bias, we consider an extra array directional error which implies the deviation in the orientation of the array placement. The proposed method is able to estimate the locations of MDs and self-calibrate the array directional errors simultaneously. To achieve joint localisation, we directly map MD locations and array directional error to received signals. Then the group sparsity based optimisation is proposed to exploit the geometric consistency that received paths are originating from common MDs. Note that the super-resolution framework cannot be directly applied to our localisation problems. Because the proposed objective function cannot be efficiently solved by semi-definite programming. Typical strategies focus on reducing adverse effect due to the NLoS bias by separating line-of-sight (LoS)/NLoS path or mitigating NLoS effect. The LoS path is well studied for localisation and multiple methods have been proposed in the literature. However, the number of LoS paths are typically limited and the effect of NLoS bias may not always be reduced completely. As a long-standing issue, the suitable solution of using NLoS path is still an open topic for research. Instead of dealing with NLoS bias, we present a novel localisation method that exploits both LoS and NLoS paths in the same manner. The unique feature is avoiding hard decisions on separating LoS and NLoS paths and hence relevant possible error. A grid-free sparse inverse problem is formulated for localisation which avoids error propagation between multiple stages, handles multipath in a unified way, and guarantees a global convergence. Extensive localisation experiments on different propagation environments and localisation systems are presented to illustrate the high performance of the proposed algorithm compared with theoretical analysis. In one of the case studies, single antenna access points (APs) can locate a single antenna MD even when all paths between them are NLoS, which according to the authors’ knowledge is the first time in the literature.Open Acces

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Distributed Kalman Filters over Wireless Sensor Networks: Data Fusion, Consensus, and Time-Varying Topologies

Kalman filtering is a widely used recursive algorithm for optimal state estimation of linear stochastic dynamic systems. The recent advances of wireless sensor networks (WSNs) provide the technology to monitor and control physical processes with a high degree of temporal and spatial granularity. Several important problems concerning Kalman filtering over WSNs are addressed in this dissertation. First we study data fusion Kalman filtering for discrete-time linear time-invariant (LTI) systems over WSNs, assuming the existence of a data fusion center that receives observations from distributed sensor nodes and estimates the state of the target system in the presence of data packet drops. We focus on the single sensor node case and show that the critical data arrival rate of the Bernoulli channel can be computed by solving a simple linear matrix inequality problem. Then a more general scenario is considered where multiple sensor nodes are employed. We derive the stationary Kalman filter that minimizes the average error variance under a TCP-like protocol. The stability margin is adopted to tackle the stability issue. Second we study distributed Kalman filtering for LTI systems over WSNs, where each sensor node is required to locally estimate the state in a collaborative manner with its neighbors in the presence of data packet drops. The stationary distributed Kalman filter (DKF) that minimizes the local average error variance is derived. Building on the stationary DKF, we propose Kalman consensus filter for the consensus of different local estimates. The upper bound for the consensus coefficient is computed to ensure the mean square stability of the error dynamics. Finally we focus on time-varying topology. The solution to state consensus control for discrete-time homogeneous multi-agent systems over deterministic time-varying feedback topology is provided, generalizing the existing results. Then we study distributed state estimation over WSNs with time-varying communication topology. Under the uniform observability, each sensor node can closely track the dynamic state by using only its own observation, plus information exchanged with its neighbors, and carrying out local computation

Sensor resource management with evolutionary algorithms applied to indoor positioning

Premio Extraordinario de Doctorado de la UAH en el año académico 2016-2017Esta tesis pretende contribuir a la mejora de la gestión de recursos en sistemas de sensores aplicados a localización en interiores. Mediante esta gestión pueden abordarse dos temas, la colocación de estos sensores y su uso óptimo una vez colocados, centrándose la tesis en el primero de ellos. Durante la tesis se considera el uso de un sistema de posicionamiento en interiores basado en señales infrarrojas con medida de diferencia de fase. Estas medidas de fase son posteriormente transformadas en distancias, con lo cual nuestro problema es el de trilateración hiperbólica utilizando medidas de diferencia de distancia. Aunque se describe un modelo para el error en diferencia de distancias del enlace infrarrojo, podemos abstraernos de este y simplemente considerar que utilizamos medidas de diferencia de distancia que están normalmente distribuidas con una varianza dada por el modelo usado. De hecho, el trabajo expuesto en esta tesis podría ser usado con cualquier otro sistema del cual obtengamos un modelo de los errores de medida, ya sea empleando además trilateración esférica o angulación. La gran mayoría de trabajos que mejoran la precisión de un sistema de posicionamiento colocando sensores optimizan funciones de coste basadas en el límite inferior de Cramér-Rao, enfoque que adoptamos también en este trabajo. En el capítulo de la tesis dedicado al estado del arte hacemos un repaso de las diferentes propuestas existentes, que concluye explicando qué pretendemos aportar sobre las contribuciones existentes en la literatura científica. En resumen, podemos clasificar las propuestas actuales en tres clases. La primera de ellas trata de determinar una configuración óptima para localizar un objetivo, normalmente utilizando el determinante de la matriz de información de Fisher o la dilución de la precisión. Estos métodos pueden obtener expresiones analíticas que proporcionan una explicación sobre como intervienen las características de los sensores y su colocación en la precisión obtenida. Sin embargo, carecen de aplicabilidad en situaciones reales. El segundo tipo de propuestas emplea métodos numéricos para optimizar la colocación de sensores considerando varios objetivos o un área entera. Los métodos propuestos en esta tesis encajan dentro de esta categoría. Por último, existen métodos que utilizan técnicas de selección de sensores para obtener configuraciones óptimas. Entre las distintas propuestas encontramos varias deficiencias, como la simplificación del modelo de error de la medida para obtener expresiones fácilmente tratables, la consideración de un solo criterio de precisión de la localización, colocación de un número determinado y fijo de sensores, o su despliegue en áreas simples que no presenten problemas de oclusiones. Nuestra primera aportación trata de solucionar la consideración de un único criterio de precisión, que normalmente es el determinante o la traza de la matriz de covarianza o información de la estimación. Cada métrica obtenida de estas matrices tiene un significado práctico distinto, y la consideración de solo una de ellas puede dar lugar a soluciones que presenten deficiencias en las otras, como la obtención de elipses de error muy alargadas. Nuestra propuesta implica el uso de algoritmos evolutivos multifunción que optimicen varias de estas métricas, como el error cuadrático medio en todo el área, la isotropía de la solución, y la máxima desviación que puede aparecer. Esto nos permite tener un conjunto de soluciones dadas en un frente de Pareto, que permitirán al gestor de la red de sensores visualizar las posibles soluciones y elegir entre ellas según las necesidades. También permite obtener colocaciones que mejoren la convergencia de algunos estimadores. La segunda contribución de la tesis se ocupa de la colocación de sensores en zonas más complejas, donde existan obstáculos que provoquen oclusiones a algunos sensores. De esta manera, podemos introducir el problema de intentar cubrir la mayor cantidad de puntos del espacio con el número mínimo de sensores necesario para calcular la posición de un objetivo. Dicho número influirá en el porcentaje de área cubierto y en la precisión obtenida, además de aumentar el coste del sistema. Debido a esto, también será un objetivo a optimizar junto a la cobertura y la incertidumbre de la posición estimada. Para llevar a cabo esta optimización se propone una mejora sobre el algoritmo utilizado en la aportación anterior basada en el uso de subpoblaciones y añadiendo operadores genéticos que modifiquen el número de sensores según la cobertura y condensación en los distintos puntos de la zona a cubrir. Cada uno de los capítulos dedicado a las aportaciones descritas contiene resultados y conclusiones que confirman el buen funcionamiento de los métodos propuestos. Finalmente, la tesis concluye con una lista de propuestas que serán estudiadas en un futuro