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Author: Hanna E. Nyqvist
Publisher: Linköping University Electronic Press
ISBN: 9176856283
Category :
Languages : en
Pages : 92
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Book Description
Pose (position and orientation) tracking in room-scaled environments is an enabling technique for many applications. Today, virtual reality (vr) and augmented reality (ar) are two examples of such applications, receiving high interest both from the public and the research community. Accurate pose tracking of the vr or ar equipment, often a camera or a headset, or of different body parts is crucial to trick the human brain and make the virtual experience realistic. Pose tracking in room-scaled environments is also needed for reference tracking and metrology. This thesis focuses on an application to metrology. In this application, photometric models of a photo studio are needed to perform realistic scene reconstruction and image synthesis. Pose tracking of a dedicated sensor enables creation of these photometric models. The demands on the tracking system used in this application is high. It must be able to provide sub-centimeter and sub-degree accuracy and at same time be easy to move and install in new photo studios. The focus of this thesis is to investigate and develop methods for a pose tracking system that satisfies the requirements of the intended metrology application. The Bayesian filtering framework is suggested because of its firm theoretical foundation in informatics and because it enables straightforward fusion of measurements from several sensors. Sensor fusion is in this thesis seen as a way to exploit complementary characteristics of different sensors to increase tracking accuracy and robustness. Four different types of measurements are considered; inertialmeasurements, images from a camera, range (time-of-flight) measurements from ultra wide band (uwb) radio signals, and range and velocity measurements from echoes of transmitted acoustic signals. A simulation study and a study of the Cramér-Rao lower filtering bound (crlb) show that an inertial-camera system has the potential to reach the required tracking accuracy. It is however assumed that known fiducial markers, that can be detected and recognized in images, are deployed in the environment. The study shows that many markers are required. This makes the solution more of a stationary solution and the mobility requirement is not fulfilled. A simultaneous localization and mapping (slam) solution, where naturally occurring features are used instead of known markers, are suggested solve this problem. Evaluation using real data shows that the provided inertial-camera slam filter suffers from drift but that support from uwb range measurements eliminates this drift. The slam solution is then only dependent on knowing the position of very few stationary uwb transmitters compared to a large number of known fiducial markers. As a last step, to increase the accuracy of the slam filter, it is investigated if and how range measurements can be complemented with velocity measurement obtained as a result of the Doppler effect. Especially, focus is put on analyzing the correlation between the range and velocity measurements and the implications this correlation has for filtering. The investigation is done in a theoretical study of reflected known signals (compare with radar and sonar) where the crlb is used as an analyzing tool. The theory is validated on real data from acoustic echoes in an indoor environment.
Author: Hanna E. Nyqvist
Publisher: Linköping University Electronic Press
ISBN: 9176856283
Category :
Languages : en
Pages : 92
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Book Description
Pose (position and orientation) tracking in room-scaled environments is an enabling technique for many applications. Today, virtual reality (vr) and augmented reality (ar) are two examples of such applications, receiving high interest both from the public and the research community. Accurate pose tracking of the vr or ar equipment, often a camera or a headset, or of different body parts is crucial to trick the human brain and make the virtual experience realistic. Pose tracking in room-scaled environments is also needed for reference tracking and metrology. This thesis focuses on an application to metrology. In this application, photometric models of a photo studio are needed to perform realistic scene reconstruction and image synthesis. Pose tracking of a dedicated sensor enables creation of these photometric models. The demands on the tracking system used in this application is high. It must be able to provide sub-centimeter and sub-degree accuracy and at same time be easy to move and install in new photo studios. The focus of this thesis is to investigate and develop methods for a pose tracking system that satisfies the requirements of the intended metrology application. The Bayesian filtering framework is suggested because of its firm theoretical foundation in informatics and because it enables straightforward fusion of measurements from several sensors. Sensor fusion is in this thesis seen as a way to exploit complementary characteristics of different sensors to increase tracking accuracy and robustness. Four different types of measurements are considered; inertialmeasurements, images from a camera, range (time-of-flight) measurements from ultra wide band (uwb) radio signals, and range and velocity measurements from echoes of transmitted acoustic signals. A simulation study and a study of the Cramér-Rao lower filtering bound (crlb) show that an inertial-camera system has the potential to reach the required tracking accuracy. It is however assumed that known fiducial markers, that can be detected and recognized in images, are deployed in the environment. The study shows that many markers are required. This makes the solution more of a stationary solution and the mobility requirement is not fulfilled. A simultaneous localization and mapping (slam) solution, where naturally occurring features are used instead of known markers, are suggested solve this problem. Evaluation using real data shows that the provided inertial-camera slam filter suffers from drift but that support from uwb range measurements eliminates this drift. The slam solution is then only dependent on knowing the position of very few stationary uwb transmitters compared to a large number of known fiducial markers. As a last step, to increase the accuracy of the slam filter, it is investigated if and how range measurements can be complemented with velocity measurement obtained as a result of the Doppler effect. Especially, focus is put on analyzing the correlation between the range and velocity measurements and the implications this correlation has for filtering. The investigation is done in a theoretical study of reflected known signals (compare with radar and sonar) where the crlb is used as an analyzing tool. The theory is validated on real data from acoustic echoes in an indoor environment.
Author: Kristoffer Bergman
Publisher: Linköping University Electronic Press
ISBN: 9176850579
Category :
Languages : en
Pages : 112
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Book Description
During the last decades, motion planning for autonomous systems has become an important area of research. The high interest is not the least due to the development of systems such as self-driving cars, unmanned aerial vehicles and robotic manipulators. In this thesis, the objective is not only to find feasible solutions to a motion planning problem, but solutions that also optimize some kind of performance measure. From a control perspective, the resulting problem is an instance of an optimal control problem. In this thesis, the focus is to further develop optimal control algorithms such that they be can used to obtain improved solutions to motion planning problems. This is achieved by combining ideas from automatic control, numerical optimization and robotics. First, a systematic approach for computing local solutions to motion planning problems in challenging environments is presented. The solutions are computed by combining homotopy methods and numerical optimal control techniques. The general principle is to define a homotopy that transforms, or preferably relaxes, the original problem to an easily solved problem. The approach is demonstrated in motion planning problems in 2D and 3D environments, where the presented method outperforms both a state-of-the-art numerical optimal control method based on standard initialization strategies and a state-of-the-art optimizing sampling-based planner based on random sampling. Second, a framework for automatically generating motion primitives for lattice-based motion planners is proposed. Given a family of systems, the user only needs to specify which principle types of motions that are relevant for the considered system family. Based on the selected principle motions and a selected system instance, the algorithm not only automatically optimizes the motions connecting pre-defined boundary conditions, but also simultaneously optimizes the terminal state constraints as well. In addition to handling static a priori known system parameters such as platform dimensions, the framework also allows for fast automatic re-optimization of motion primitives if the system parameters change while the system is in use. Furthermore, the proposed framework is extended to also allow for an optimization of discretization parameters, that are are used by the lattice-based motion planner to define a state-space discretization. This enables an optimized selection of these parameters for a specific system instance. Finally, a unified optimization-based path planning approach to efficiently compute locally optimal solutions to advanced path planning problems is presented. The main idea is to combine the strengths of sampling-based path planners and numerical optimal control. The lattice-based path planner is applied to the problem in a first step using a discretized search space, where system dynamics and objective function are chosen to coincide with those used in a second numerical optimal control step. This novel tight combination of a sampling-based path planner and numerical optimal control makes, in a structured way, benefit of the former method’s ability to solve combinatorial parts of the problem and the latter method’s ability to obtain locally optimal solutions not constrained to a discretized search space. The proposed approach is shown in several practically relevant path planning problems to provide improvements in terms of computation time, numerical reliability, and objective function value.
Author: Daniel Arnström
Publisher: Linköping University Electronic Press
ISBN: 9179296920
Category : Electronic books
Languages : en
Pages : 72
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Book Description
In model predictive control (MPC) an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. The primary contribution of this thesis is a method which determines which sequence of subproblems a popular class of such active-set algorithms need to solve, for every possible QP instance that might arise from a given linear MPC problem (i.e, for every possible state and reference signal). By knowing these sequences, worst-case bounds on how many iterations, floating-point operations and, ultimately, the maximum solution time, these active-set algorithms require to compute a solution can be determined, which is of importance when, e.g, linear MPC is used in safety-critical applications. After establishing this complexity certification method, its applicability is extended by showing how it can be used indirectly to certify the complexity of another, efficient, type of active-set QP algorithm which reformulates the QP as a nonnegative least-squares method. Finally, the proposed complexity certification method is extended further to situations when enhancements to the active-set algorithms are used, namely, when they are terminated early (to save computations) and when outer proximal-point iterations are performed (to improve numerical stability).
Author: Per Boström-Rost
Publisher: Linköping University Electronic Press
ISBN: 9176850757
Category :
Languages : en
Pages : 106
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Book Description
This thesis studies a class of sensor management problems called informative path planning (IPP). Sensor management refers to the problem of optimizing control inputs for sensor systems in dynamic environments in order to achieve operational objectives. The problems are commonly formulated as stochastic optimal control problems, where to objective is to maximize the information gained from future measurements. In IPP, the control inputs affect the movement of the sensor platforms, and the goal is to compute trajectories from where the sensors can obtain measurements that maximize the estimation performance. The core challenge lies in making decisions based on the predicted utility of future measurements. In linear Gaussian settings, the estimation performance is independent of the actual measurements. This means that IPP becomes a deterministic optimal control problem, for which standard numerical optimization techniques can be applied. This is exploited in the first part of this thesis. A surveillance application is considered, where a mobile sensor is gathering information about features of interest while avoiding being tracked by an adversarial observer. The problem is formulated as an optimization problem that allows for a trade-off between informativeness and stealth. We formulate a theorem that makes it possible to reformulate a class of nonconvex optimization problems with matrix-valued variables as convex optimization problems. This theorem is then used to prove that the seemingly intractable IPP problem can be solved to global optimality using off-the-shelf optimization tools. The second part of this thesis considers tracking of a maneuvering target using a mobile sensor with limited field of view. The problem is formulated as an IPP problem, where the goal is to generate a sensor trajectory that maximizes the expected tracking performance, captured by a measure of the covariance matrix of the target state estimate. When the measurements are nonlinear functions of the target state, the tracking performance depends on the actual measurements, which depend on the target’s trajectory. Since these are unavailable in the planning stage, the problem becomes a stochastic optimal control problem. An approximation of the problem based on deterministic sampling of the distribution of the predicted target trajectory is proposed. It is demonstrated in a simulation study that the proposed method significantly increases the tracking performance compared to a conventional approach that neglects the uncertainty in the future target trajectory.
Author: Gustav Zetterqvist
Publisher: Linköping University Electronic Press
ISBN: 9180758304
Category :
Languages : en
Pages : 93
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Book Description
Direction of arrival (DOA) estimation is a well-established problem in signal processing. It involves determining the direction from which a signal reaches a sensor array, and is fundamental in applications like radar, sonar, and acoustics. Traditionally, DOA estimation relies on comparing the time of arrival of the signal across different sensors in the array. However, this approach is sensitive to the time difference of arrival (TDOA) between sensors, which can be challenging to estimate accurately. Additionally, precise synchronization among the sensors is essential, but this can be difficult to achieve in certain environments or applications. In this thesis, we explore a novel approach to DOA estimation based on the received signal power at the sensors. The method exploits the directional sensitivity of the microphones in the array, which defines how effectively each microphone captures sound from different directions. To model the directional sensitivity, we use a Fourier series (FS) model. The model is then used to estimate the DOA of a sound source across various environments, and for different types of signals. The parametric model enables Cramér-Rao lower bound (CRLB) analysis of the DOA estimation problem. Our findings demonstrate that the directional sensitivity exhibits a significant variation in accordance with the frequency content of the signal, and we exploit this to estimate the DOA for different types of sounds. The proposed method has been validated with a range of signals, including gunshots, elephant trumpets, sirens, and female screams. The results show that the developed method achieves high accuracy in estimating the DOA for the above-mentioned signals. Furthermore, the method performs similarly well in outdoor scenarios with realistic background noise levels. When compared to state-of-the-art DOA estimation techniques, our approach performs better or equally well for the investigated sounds. A key advantage of this method is that it does not require any TDOA measurement between the microphones, enabling the design of smaller, more compact devices. This opens up new possibilities for estimating DOA in environments where traditional methods are impractical. A limitation, however, is that the method requires knowledge of the microphone’s directional sensitivity, which necessitates calibration in an anechoic chamber. Nevertheless, this calibration has proven to be robust, and only needs to be performed once to create a model applicable across different environments. Additionally, this thesis explores a different application of DOA estimation, where geophones are used to estimate the DOA to elephants. As elephants move, they generate ground vibrations, and these signals can be captured by geophones. We show that a traditional delay-and-sum beamformer can accurately estimate the DOA of elephants at distances up to 40 meters. By determining when elephants are approaching and from which direction, park rangers can take early measures to avoid conflicts between humans and elephants, which is a major problem in some parts of the world. Förmågan att höra var ett ljud kommer ifrån, något vi ofta tar för givet, kallas för riktningsuppfattning. Den gör det möjligt för oss att snabbt avgöra om någon ropar på oss och från vilket håll ljudet kommer. Denna förmåga är viktig för att kunna orientera sig i omgivningen och uppfatta hot eller andra viktiga ljud. Våra öron samarbetar genom att jämföra hur ljud når varje öra, både när det gäller ljudets intensitet och hur lång tid det tar för ljudet att nå dem. Det här kallas för interaural tids- och nivåskillnad. Vissa ljud kan dock vara svåra att uppfatta, till exempel om ljudet är kort och impulsivt, eller om det är i en stadsmiljö med mycket bakgrundsljud och reflektioner. I den här avhandlingen undersöker vi nya metoder för att uppskatta ljudets riktning. Vi använder mikrofoner för att mäta ljudet och beräknar därefter riktningen som ljudet kommer ifrån. Traditionella metoder fokuserar på tidsskillnaden mellan ljud som registreras i olika mikrofoner. Vi tar istället en annan väg och undersöker hur ljudets styrka kan användas för att avgöra riktningen, oavsett tidsskillnader mellan mikrofonerna. Vår metod bygger på att vi skapar en modell av mikrofonernas riktningskänslighet, det vill säga hur väl de uppfattar ljud från olika håll. Modellen skapas genom att mäta mikrofonens riktningskänslighet i ett ekofritt rum. Genom att först mäta detta i en kontrollerad miljö, utan ekon, kan vi sedan använda modellen för att beräkna ljudriktningen i mer varierande miljöer och för olika typer av ljud. Till exempel har vi använt ljud såsom pistolskott, elefanttrumpeter, sirener och skrik för att testa vår metod. Resultaten visar att vår metod kan beräkna riktningar med hög noggrannhet för de ovan nämnda ljuden, även i en utomhusmiljö med mer realistiska nivåer av bakgrundsljud. När vi jämfört vår metod med traditionella metoder, presterar vår lösning lika bra eller bättre för de testade ljuden. En stor fördel med vår metod är att den inte kräver att mikrofonerna är placerade på ett visst avstånd från varandra, vilket innebär att vi kan bygga mindre och mer kompakta enheter. Detta kan leda till nya typer av produkter för att identifiera ljudriktningar i olika situationer. En nackdel är dock att mikrofonernas riktningskänslighet måste kalibreras i ett ljudlabb, men denna kalibrering har visat sig vara robust och det räcker att utföra en kalibrering som kan användas i flera olika miljöer. I avhandlingen inkluderas även en annan tillämpning av riktningsskattning, nämligen att uppskatta riktningen till elefanter med hjälp av geofoner som mäter vibrationer i marken. Elefanter är stora djur som skapar tydliga vibrationer i marken när de går. Genom att mäta dessa vibrationer med geofoner kan vi uppskatta riktningen till elefanten. Vi visar att traditionella metoder kan uppskatta riktningen med hög noggrannhet på ett avstånd upp till 40 meter. Genom att avgöra när elefanter närmar sig människor och varifrån de kommer kan parkvakter vidta åtgärder för att undvika konflikter mellan människor och elefanter, vilket är ett stort problem i vissa delar av världen.
Author: Erik Hedberg
Publisher: Linköping University Electronic Press
ISBN: 9179297404
Category : Electronic books
Languages : en
Pages : 83
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Book Description
The two topics at the heart of this thesis are how to improve control of industrial manipulators and how to reason about the role of models in automatic control. On industrial manipulators, two case studies are presented. The first investigates estimation with inertial sensors, and the second compares control by feedback linearization to control based on gain-scheduling. The contributions on the second topic illustrate the close connection between control and estimation in different ways. A conceptual model of control is introduced, which can be used to emphasize the role of models as well as the human aspect of control engineering. Some observations are made regarding block-diagram reformulations that illustrate the relation between models, control and inversion. Finally, a suggestion for how the internal model principle, internal model control, disturbance observers and Youla-Kucera parametrization can be introduced in a unified way is presented.
Author: Andreas Bergström
Publisher: Linköping University Electronic Press
ISBN: 9179299172
Category :
Languages : en
Pages : 140
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Book Description
The measurements of radio signals are commonly used for localization purposes where the goal is to determine the spatial position of one or multiple objects. In realistic scenarios, any transmitted radio signal will be affected by the environment through reflections, diffraction at edges and corners etc. This causes a phenomenon known as multipath propagation, by which multiple instances of the transmitted signal having traversed different paths are heard by the receiver. These are known as Multi-Path Components (MPCs). The direct path (DP) between transmitter and receiver may also be occluded, causing what is referred to as non-Line-of-Sight (non-LOS) conditions. As a consequence of these effects, the estimated position of the object(s) may often be erroneous. This thesis focuses on how to achieve better localization accuracy by accounting for the above-mentioned multipath propagation and non-LOS effects. It is proposed how to mitigate these in the context of positioning based on estimation of the DP between transmitter and receiver. It is also proposed how to constructively utilize the additional information about the environment which they implicitly provide. This is all done in a framework wherein a given signal model and a map of the surroundings are used to build a mathematical model of the radio environment, from which the resulting MPCs are estimated. First, methods to mitigate the adverse effects of multipath propagation and non-LOS conditions for positioning based on estimation of the DP between transmitter and receiver are presented. This is initially done by using robust statistical measurement error models based on aggregated error statistics, where significant improvements are obtained without the need to provide detailed received signal information. The gains are seen to be even larger with up-to-date real-time information based on the estimated MPCs. Second, the association of the estimated MPCs with the signal paths predicted by the environmental model is addressed. This leads to a combinatorial problem which is approached with tools from multi-target tracking theory. A rich radio environment in terms of many MPCs gives better localization accuracy but causes the problem size to grow large—something which can be remedied by excluding less probable paths. Simulations indicate that in such environments, the single best association hypothesis may be a reasonable approximation which avoids the calculation of a vast number of possible hypotheses. Accounting for erroneous measurements is crucial but may have drawbacks if no such are occurring. Finally, theoretical localization performance bounds when utilizing all or a subset of the available MPCs are derived. A rich radio environment allows for good positioning accuracy using only a few transmitters/receivers, assuming that these are used in the localization process. In contrast, in a less rich environment where basically only the DP/LOS components are measurable, more transmitters/receivers and/or the combination of downlink and uplink measurements are required to achieve the same accuracy. The receiver’s capability of distinguishing between multiple MPCs arriving approximately at the same time also affects the localization accuracy.
Author: Du Ho
Publisher: Linköping University Electronic Press
ISBN: 9176851664
Category :
Languages : en
Pages : 116
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Book Description
In recent years, the quadcopter has become a popular platform both in research activities and in industrial development. Its success is due to its increased performance and capabilities, where modeling and control synthesis play essential roles. These techniques have been used for stabilizing the quadcopter in different flight conditions such as hovering and climbing. The performance of the control system depends on parameters of the quadcopter which are often unknown and need to be estimated. The common approach to determine such parameters is to rely on accurate measurements from external sources, i.e., a motion capture system. In this work, only measurements from low-cost onboard sensors are used. This approach and the fact that the measurements are collected in closed-loop present additional challenges. First, a general overview of the quadcopter is given and a detailed dynamic model is presented, taking into account intricate aerodynamic phenomena. By projecting this model onto the vertical axis, a nonlinear vertical submodel of the quadcopter is obtained. The Instrumental Variable (IV) method is used to estimate the parameters of the submodel using real data. The result shows that adding an extra term in the thrust equation is essential. In a second contribution, a sensor-to-sensor estimation problem is studied, where only measurements from an onboard Inertial Measurement Unit (IMU) are used. The roll submodel is derived by linearizing the general model of the quadcopter along its main frame. A comparison is carried out based on simulated and experimental data. It shows that the IV method provides accurate estimates of the parameters of the roll submodel whereas some other common approaches are not able to do this. In a sensor-to-sensor modeling approach, it is sometimes not obvious which signals to select as input and output. In this case, several common methods give different results when estimating the forward and inverse models. However, it is shown that the IV method will give identical results when estimating the forward and inverse models of a single-input single-output (SISO) system using finite data. Furthermore, this result is illustrated experimentally when the goal is to determine the center of gravity of a quadcopter.
Author: Robin Forsling
Publisher: Linköping University Electronic Press
ISBN: 9179297242
Category :
Languages : en
Pages : 110
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Book Description
Sensor networks consist of sensors (e.g., radar and cameras) and processing units (e.g., estimators), where in the former information extraction occurs and in the latter estimates are formed. In decentralized estimation information extracted by sensors has been pre-processed at an intermediate processing unit prior to arriving at an estimator. Pre-processing of information allows for the complexity of large systems and systems-of-systems to be significantly reduced, and also makes the sensor network robust and flexible. One of the main disadvantages of pre-processing information is that information becomes correlated. These correlations, if not handled carefully, potentially lead to underestimated uncertainties about the calculated estimates. In conservative estimation the unknown correlations are handled by ensuring that the uncertainty about an estimate is not underestimated. If this is ensured the estimate is said to be conservative. Neglecting correlations means information is double counted which in worst case implies diverging estimates with fatal consequences. While ensuring conservative estimates is the main goal, it is desirable for a conservative estimator, as for any estimator, to provide an error covariance which is as small as possible. Application areas where conservative estimation is relevant are setups where multiple agents cooperate to accomplish a common objective, e.g., target tracking, surveillance and air policing. The first part of this thesis deals with theoretical matters where the conservative linear unbiased estimation problem is formalized. This part proposes an extension of classical linear estimation theory to the conservative estimation problem. The conservative linear unbiased estimator (CLUE) is suggested as a robust and practical alternative for estimation problems where the correlations are unknown. Optimality criteria for the CLUE are provided and further investigated. It is shown that finding an optimal CLUE is more complicated than finding an optimal linear unbiased estimator in the classical version of the problem. To simplify the problem, a CLUE that is optimal under certain restrictions will also be investigated. The latter is named restricted best CLUE. An important result is a theorem that gives a closed form solution to a restricted best CLUE. Furthermore, several conservative estimation methods are described followed by an analysis of their properties. The methods are shown to be conservative and optimal under different assumptions about the underlying correlations. The second part of the thesis focuses on practical aspects of the conservative approach to decentralized estimation in configurations where the communication channel is constrained. The diagonal covariance approximation is proposed as a data reduction technique that complies with the communication constraints and if handled correctly can be shown to preserve conservative estimates. Several information selection methods are derived that can reduce the amount of data being transmitted in the communication channel. Using the information selection methods it is possible to decide what information other actors of the sensor network find useful.
Author: Kamiar Radnosrati
Publisher: Linköping University Electronic Press
ISBN: 9176852695
Category :
Languages : en
Pages : 121
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Book Description
The possibilities for positioning in cellular networks has increased over time, pushed by increased needs for location based products and services for a variety of purposes. It all started with rough position estimates based on timing measurements and sector information available in the global system for mobile communication (gsm), and today there is an increased standardization effort to provide more position relevant measurements in cellular communication systems to improve on localization accuracy and availability. A first purpose of this thesis is to survey recent efforts in the area and their potential for localization. The rest of the thesis then investigates three particular aspects, where the focus is on timing measurements. How can these be combined in the best way in long term evolution (lte), what is the potential for the new narrow-band communication links for localization, and can the timing measurement error be more accurately modeled? The first contribution concerns a narrow-band standard in lte intended for internet of things (iot) devices. This lte standard includes a special position reference signal sent synchronized by all base stations (bs) to all iot devices. Each device can then compute several pair-wise time differences that corresponds to hyperbolic functions. Using multilateration methods the intersection of a set of such hyperbolas can be computed. An extensive performance study using a professional simulation environment with realistic user models is presented, indicating that a decent position accuracy can be achieved despite the narrow bandwidth of the channel. The second contribution is a study of how downlink measurements in lte can be combined. Time of flight (tof) to the serving bs and time difference of arrival (tdoa) to the neighboring bs are used as measurements. From a geometrical perspective, the position estimation problem involves computing the intersection of a circle and hyperbolas, all with uncertain radii. We propose a fusion framework for both snapshot estimation and filtering, and evaluate with both simulated and experimental field test data. The results indicate that the position accuracy is better than 40 meters 95% of the time. A third study in the thesis analyzes the statistical distribution of timing measurement errors in lte systems. Three different machine learning methods are applied to the experimental data to fit Gaussian mixture distributions to the observed measurement errors. Since current positioning algorithms are mostly based on Gaussian distribution models, knowledge of a good model for the measurement errors can be used to improve the accuracy and robustness of the algorithms. The obtained results indicate that a single Gaussian distribution is not adequate to model the real toa measurement errors. One possible future study is to further develop standard algorithms with these models.
