This master’s thesis describes a way to represent vehicles when tracking them through an image sequence. Vehicles are described with a state containing their position, velocity, size, etc.. The thesis highlights the properties of homographies due to their suitability for estimation of projective transformations. The idea is to approximatively represent vehicles with planes based on feature points found on the vehicles. The purpose with this approach is to estimate the displacement of a vehicle by estimating the transformation of these planes. Thus, when avehicle is observed from behind, one plane approximates features found on the back and one plane approximates features found on the side, if the side of the vehicle is visible. The projective transformations of the planes are obtained by measuring the displacement of feature points. The approach presented in this thesis builds on the prerequisites that a camera placed on a vehicle provides an image of its field of view. It does not cover how to find vehicles in an image and thus it requires that the patch which contains the vehicle is provided. Even though this thesis covers large parts of image processing functionalities, the focus is on how to represent vehicles and how to design an appropriate filter for improving estimates of vehicle displacement. Due to noisy features points, approximation of planes, and estimated homographies, the obtained measurements are likely to be noisy. This requires a filter that can handle corrupt measurements and still use those that are not. An unscented Kalman filter, UKF, is utilized in this implementation. The UKF is an approximate solution to nonlinear filtering problems and is here used to update the vehicle’s states by using measurements obtained from homographies. The choice of the unscented Kalman filter was made because of its ease of implementation and its potentially good performance. The result is not a finished implementation for tracking of vehicles, but rather a first attempt for this approach. The result is not better than the existing approach, which might depend on one or several factors such as poorly estimated homographies, unreliable feature points and bad performance of the UKF.

ITC/USA 2006 Conference Proceedings / The Forty-Second Annual International Telemetering Conference and Technical Exhibition / October 23-26, 2006 / Town and Country Resort & Convention Center, San Diego, California / OFDM is a preferred physical layer for an increasing number of telemetry and LAN applications. However, joint estimation of the multipath channel and frequency offset in OFDM remains a challenging problem. The Unscented Kalman Filter (UKF) is presented to solve the offset/channel tracking problem. The advantages of the UKF are that it is less susceptible to divergence than the EKF, and does not require computation of a Jacobian matrix. A hybrid analysis/simulation approach is developed to rapidly evaluate UKF performance in terms of symbol-error rate and channel/offset error for the 802.11a OFDM format.

The experimental determination of dynamic characteristics of elastic structures, in particularof space flight related structures typically is performed by experimental modal analysis(EMA) or output-only modal analysis (OMA). This document is focused on the OMA methodsand state-space modelling, the motivation for this approach is the possibility to monitorthe real loading of a structure in order to provide a loading history which may be used foran assessment of safe remaining life once the dynamic characteristics has been determined. Previous work has demonstrated that Extened Kalman Filter is not sufficient in thecase when the forces are unkown and the only resource available are the responses of thestructure. In this research a new method called Unscented Kalman Filter is investigated andimplemented, proving its capability to obtain a better approximation of the elastic structurebehaviour and a correction of the modal parameters.

<p>State estimation deals with estimation of the state of an object of interest by observing noisy measurements. The process to obtain the state estimations is called filtering. In this report several filters are compared to an existing one. The new filters deal with nonlinear process and measurement models in a different way than the existing filter. Instead of approximating the nonlinear transformations the probability densities are approximated by a set of points which undergo the nonlinear transformation.</p><p>The application for which the filters will be used is to estimate the position and orientation of a camera in a markerless environment, using data from an inertial measurement unit and a camera. It is found that the corresponding process and measurement models contain nonlinearities and therefore an accuracy improvement is expected with the new filters.</p><p>The new filters are variations of the so-called unscented Kalman filter. Also a discussion on the marginalized particle filter is presented. Instead of using randomly chosen samples as in the particle filter methods, the unscented Kalman filter uses deterministically chosen points. The marginalized particle filter partitions the variables of the system in a linear and a nonlinear part. Linear Kalman filters are applied to the linear variables and a particle filter to the nonlinear variables, thus reducing the computational load. Details of various implementations of the filters are given, as well as the motivation for the specific implementations.</p><p>Tests are carried out to assess the performance of the filters. This is done with both simulation data and real measurements. A comparison is made to the original extended Kalman filter. The tests are focussed on accuracy and computational load.</p><p>Results showed that the use of the new filters did not improve accuracy. This is mainly due to the fact that the nonlinearities are not so severe. Furthermore the filters had a higher computational load, which is an important aspect in the system reviewed in this report. Therefore the current filter need not to be replaced. The unscented Kalman filter is a good alternative to the EKF in case of new applications, since it can handle the system in a black-box manner in contrast to the EKF.</p>

State estimation deals with estimation of the state of an object of interest by observing noisy measurements. The process to obtain the state estimations is called filtering. In this report several filters are compared to an existing one. The new filters deal with nonlinear process and measurement models in a different way than the existing filter. Instead of approximating the nonlinear transformations the probability densities are approximated by a set of points which undergo the nonlinear transformation. The application for which the filters will be used is to estimate the position and orientation of a camera in a markerless environment, using data from an inertial measurement unit and a camera. It is found that the corresponding process and measurement models contain nonlinearities and therefore an accuracy improvement is expected with the new filters. The new filters are variations of the so-called unscented Kalman filter. Also a discussion on the marginalized particle filter is presented. Instead of using randomly chosen samples as in the particle filter methods, the unscented Kalman filter uses deterministically chosen points. The marginalized particle filter partitions the variables of the system in a linear and a nonlinear part. Linear Kalman filters are applied to the linear variables and a particle filter to the nonlinear variables, thus reducing the computational load. Details of various implementations of the filters are given, as well as the motivation for the specific implementations. Tests are carried out to assess the performance of the filters. This is done with both simulation data and real measurements. A comparison is made to the original extended Kalman filter. The tests are focussed on accuracy and computational load. Results showed that the use of the new filters did not improve accuracy. This is mainly due to the fact that the nonlinearities are not so severe. Furthermore the filters had a higher computational load, which is an important aspect in the system reviewed in this report. Therefore the current filter need not to be replaced. The unscented Kalman filter is a good alternative to the EKF in case of new applications, since it can handle the system in a black-box manner in contrast to the EKF.

Small satellites are becoming increasingly appealing as technology advances and shrinks in both size and cost. The development time for a small satellite is also much less compared to a large satellite. For small satellites to be successful, the navigation systems must be accurate and very often they must be autonomous. For lunar navigation, contact with a ground station is not always available and the system needs to be robust. The extended Kalman filter is a nonlinear estimator that has been used on-board spacecraft for decades. The filter requires linear approximations of the state and measurement models. In the past few years, the unscented Kalman filter has become popular and has been shown to reduce estimation errors. Additionally, the Jacobian matrices do not need to be derived in the unscented Kalman filter implementation. The intent of this research is to explore the capabilities of the extended Kalman filter and the unscented Kalman filter for use as a navigation algorithm on small satellites. The filters are applied to a satellite orbiting the Moon equipped with an inertial measurement unit, a sun sensor, a star camera, and a GPS-like sensor. The position, velocity, and attitude of the spacecraft are estimated along with sensor biases for the IMU accelerometer, IMU gyroscope, sun sensor and star camera. The estimation errors are compared for the extended Kalman filter and the unscented Kalman filter for the position, velocity and attitude. The analysis confirms that both navigation algorithms provided accurate position, velocity and attitude. The IMU gyroscope bias was observable for both filters while only the IMU accelerometer bias was observable with the extended Kalman filter. The sun sensor biases and the star camera biases were unobservable. In general, the unscented Kalman filter performed better than the extended Kalman filter in providing position, velocity, and attitude estimates but requires more computation time. / text

An important aspect for precision guidance, navigation, and control for lunar operations is environmental modeling. In particular, consider gravity field modeling. Available gravity field models for the Moon reach degree and order 165 requiring the use and storage of approximately 26,000 spherical harmonic coefficients. Although the high degree and order provide a means by which to accurately predict trajectories within the influence of the Moon's gravitational field, the size of these models makes using them computationally expensive and restricts their use in design environments with limited computer memory and storage. It is desirable to determine reduced complexity realizations of the gravitational models to lower the computational burden while retaining the structure of the original gravitational field for use in rapid design environments. The extended Kalman filter and the unscented Kalman filter are used to create reduced order models and are compared against a simple truncation based reduction method. Both variations of the Kalman filter out perform the truncation based method as a means by which to reduce the complexity of the gravitational field. The extended Kalman filter and unscented Kalman filter were able to achieve good estimates of position while reducing the number of spherical harmonic coefficients used in gravitational acceleration calculations by approximately 5,400, greatly increasing the speed of the calculations while reducing the required computer allocation. / text

O filtro de Kalman estendido tem sido a mais popular ferramenta de filtragem não linear das últimas quatro décadas. É de fácil implementação e apresenta baixo custo computacional. Nos casos nos quais as não linearidades do sistema dinâmico são significativas, porém, o filtro de Kalman estendido pode apresentar resultados insatisfatórios. Nessas situações, o filtro de Kalman unscented substitui com vantagens o filtro de Kalman estendido, pois pode apresentar melhores estimativas de estado, embora ambos os filtros exibam complexidade computacional de mesma ordem. A qualidade das estimativas de estado do filtro unscented está intimamente ligada à sintonia dos parâmetros que controlam a transformada unscented. A versão escalada dessa transformada exibe três parâmetros escalares que determinam o posicionamento dos pontos sigma e, consequentemente, afetam diretamente a qualidade das estimativas produzidas pelo filtro. Apesar da importância do filtro de Kalman unscented, a sintonia ótima desses parâmetros é um problema para o qual ainda não há solução definitiva. Não há nem mesmo recomendações heurísticas que garantam o bom funcionamento do filtro unscented na maior parte dos problemas tratáveis por meio de filtros Gaussianos. Essa carência e a importância desse filtro para a área de filtragem não linear fazem da busca por mecanismos de sintonia automática do filtro unscented área de pesquisa ativa. Assim, este trabalho propõe técnicas para sintonia automática dos parâmetros da transformada unscented escalada. Além da sintonia desses parâmetros, também é abordado o problema de sintonizar as matrizes de covariância dos ruídos de processo e de medida demandadas pelo modelo do sistema dinâmico usado pelo filtro unscented. As técnicas propostas cobrem então a sintonia automática de todos os parâmetros do filtro. / The extended Kalman filter has been the most popular nonlinear filter of the last four decades. It is easy to implement and exhibits low computational cost. When nonlinearities are significant, though, the extended Kalman filter can display poor state estimation performance. In such situations, the unscented Kalman filter can yield better state estimates, while displaying the same order of computational complexity as the extended Kalman filter. The quality of the state estimates produced by the unscented Kalman filter is directly influenced by the tuning of the scalar parameters that govern the unscented transform. The scaled version of the unscented transform features three scalar parameters that determine the positioning of the sigma points, thus directly affecting the filter state estimation performance. Despite the importance of the unscented Kalman filter, the optimal tuning of the scaled unscented transform parameters is still an open problem. This work hence discusses algorithms for the automatic tuning of the unscented transform parameters. The discussion includes the tuning of the needed noise covariance matrices, thus covering the automatic tuning of all parameters of the unscented Kalman filter.

Condition monitoring of hydraulic systems is an area that has grown substantially in the last few decades. This thesis presents a scheme that automatically generates the fault symptoms by on-line processing of raw sensor data from a real test rig. The main purposes of implementing condition monitoring in hydraulic systems are to increase productivity, decrease maintenance costs and increase safety. Since such systems are widely used in industry and becoming more complex in function, reliability of the systems must be supported by an efficient monitoring and maintenance scheme. This work proposes an accurate state space model together with a novel model-based fault diagnosis methodology. The test rig has been fabricated in the Process Automation and Robotics Laboratory at UBC. First, a state space model of the system is derived. The parameters of the model are obtained through either experiments or direct measurements and manufacturer specifications. To validate the model, the simulated and measured states are compared. The results show that under normal operating conditions the simulation program and real system produce similar state trajectories. For the validated model, a condition monitoring scheme based on the Unscented Kalman Filter (UKF) is developed. In simulations, both measurement and process noises are considered. The results show that the algorithm estimates the iii system states with acceptable residual errors. Therefore, the structure is verified to be employed as the fault diagnosis scheme. Five types of faults are investigated in this thesis: loss of load, dynamic friction load, the internal leakage between the two hydraulic cylinder chambers, and the external leakage at either side of the actuator. Also, for each leakage scenario, three levels of leakage are investigated in the tests. The developed UKF-based fault monitoring scheme is tested on the practical system while different fault scenarios are singly introduced to the system. A sinusoidal reference signal is used for the actuator displacement. To diagnose the occurred fault in real time, three criteria, namely residual moving average of the errors, chamber pressures, and actuator characteristics, are considered. Based on the presented experimental results and discussions, the proposed scheme can accurately diagnose the occurred faults.

Nonlinear estimation techniques play an important role for process monitoring since some states and most of the parameters cannot be directly measured. There are many techniques available for nonlinear state and parameter estimation, i.e., extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filtering (PF) and moving horizon estimation (MHE) etc. However, many issues related to the available techniques are to be solved. This dissertation discusses three important techniques in nonlinear estimation, which are the application of unscented Kalman filters, improvement of moving horizon estimation via computation of the arrival cost and different implementations of extended Kalman filters. First the use of several estimation algorithms such as linearized Kalman filter (LKF), extended Kalman filter (EKF), unscented Kalman filter (UKF) and moving horizon estimation (MHE) are investigated for nonlinear systems with special emphasis on UKF as it is a relatively new technique. Detailed case studies show that UKF has advantages over EKF for highly nonlinear unconstrained estimation problems while MHE performs better for systems with constraints. Moving horizon estimation alleviates the computational burden of solving a full information estimation problem by considering a finite horizon of the measurement data; however, it is non-trivial to determine the arrival cost. A commonly used approach for computing the arrival cost is to use a first order Taylor series approximation of the nonlinear model and then apply an extended Kalman filter. The second contribution of this dissertation is that an approach to compute the arrival cost for moving horizon estimation based on an unscented Kalman filter is proposed. It is found that such a moving horizon estimator performs better in some cases than if one based on an extended Kalman filter. It is a promising alternative for approximating the arrival cost for MHE. Many comparative studies, often based upon simulation results, between extended Kalman filters (EKF) and other estimation methodologies such as moving horizon estimation, unscented Kalman filter, or particle filtering have been published over the last few years. However, the results returned by the extended Kalman filter are affected by the algorithm used for its implementation and some implementations of EKF may lead to inaccurate results. In order to address this point, this dissertation investigates several different algorithms for implementing extended Kalman filters. Advantages and drawbacks of different EKF implementations are discussed in detail and illustrated in some comparative simulation studies. Continuously predicting covariance matrix for EKF results in an accurate implementation. Evaluating covariance matrix at discrete times can also be applied. Good performance can be expected if covariance matrix is obtained from integrating the continuous-time equation or if the sensitivity equation is used for computing the Jacobian matrix.