M.Sc. (Statistics)

M.Sc. (Statistics)

This dissertation mainly consists of three parts. The first part proposes generalized linear minimum mean-square error (GLMMSE) estimation for nonlinear point estimation. The second part proposes a recursive joint decision and estimation (RJDE) algorithm for joint decision and estimation (JDE). The third part analyzes the performance of sequential probability ratio test (SPRT) when the log-likelihood ratios (LLR) are independent but not identically distributed. The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. It searches for the best estimator in the set of all estimators that are linear in the measurement. A GLMMSE estimation framework is proposed in this disser- tation. It employs a vector-valued measurement transform function (MTF) and finds the best estimator among all estimators that are linear in MTF. Several design guidelines for the MTF based on a numerical example were provided. A RJDE algorithm based on a generalized Bayes risk is proposed in this dissertation for dynamic JDE problems. It is computationally efficient for dynamic problems where data are made available sequentially. Further, since existing performance measures for estimation or decision are effective to evaluate JDE algorithms, a joint performance measure is proposed for JDE algorithms for dynamic problems. The RJDE algorithm is demonstrated by applications to joint tracking and classification as well as joint tracking and detection in target tracking. The characteristics and performance of SPRT are characterized by two important functions—operating characteristic (OC) and average sample number (ASN). These two functions have been studied extensively under the assumption of independent and identically distributed (i.i.d.) LLR, which is too stringent for many applications. This dissertation relaxes the requirement of identical distribution. Two inductive equations governing the OC and ASN are developed. Unfortunately, they have non-unique solutions in the general case. They do have unique solutions in two special cases: (a) the LLR sequence converges in distributions and (b) the LLR sequence has periodic distributions. Further, the analysis can be readily extended to evaluate the performance of the truncated SPRT and the cumulative sum test.

Apresenta-se o desenvolvimento de modelos matemáticos e algoritmos de fusão sensorial para navegação terrestre usando uma unidade de medição inercial (UMI) de baixo desempenho e o Filtro Estendido de Kalman. Os modelos foram desenvolvidos com base nos sistemas de navegação inercial strapdown (SNIS). O termo baixo desempenho refere-se à UMIs que por si só não são capazes de efetuar o auto- alinhamento por girocompassing. A incapacidade de se navegar utilizando apenas uma UMI de baixo desempenho motiva a investigação de técnicas que permitam aumentar o grau de precisão do SNIS com a utilização de sensores adicionais. Esta tese descreve o desenvolvimento do modelo completo de uma fusão sensorial para a navegação inercial de um veículo terrestre usando uma UMI de baixo desempenho, um hodômetro e uma bússola eletrônica. Marcas topográficas (landmarks) foram instaladas ao longo da trajetória de teste para se medir o erro da estimativa de posição nesses pontos. Apresenta-se o desenvolvimento do Filtro de Kalman Adaptativo Suavizado (FKAS), que estima conjuntamente os estados e o erro dos estados estimados do sistema de fusão sensorial. Descreve-se um critério quantitativo que emprega as incertezas de posição estimadas pelo FKAS para se determinar a priori, dado os sensores disponíveis, o intervalo de tempo máximo que se pode navegar dentro de uma margem de confiabilidade desejada. Conjuntos reduzidos de landmarks são utilizados como sensores fictícios para testar o critério de confiabilidade proposto. Destacam-se ainda os modelos matemáticos aplicados à navegação terrestre, unificados neste trabalho. Os resultados obtidos mostram que, contando somente com os sensores inerciais de baixo desempenho, a navegação terrestre torna-se inviável após algumas dezenas de segundos. Usando os mesmos sensores inerciais, a fusão sensorial produziu resultados muito superiores, permitindo reconstruir trajetórias com deslocamentos da ordem de 2,7 km (ou 15 minutos) com erro final de estimativa de posição da ordem de 3 m. / This work presents the development of the mathematical models and the algorithms of a sensor fusion system for terrestrial navigation using a low-grade inertial measurement unit (IMU) and the Extended Kalman Filter. The models were developed on the basis of the strapdown inertial navigation systems (SINS). Low-grade designates an IMU that is not able to perform girocompassing self-alignment. The impossibility of navigating relying on a low performance IMU is the motivation for investigating techniques to improve the SINS accuracy with the use of additional sensors. This thesis describes the development of a comprehensive model of a sensor fusion for the inertial navigation of a ground vehicle using a low-grade IMU, an odometer and an electronic compass. Landmarks were placed along the test trajectory in order to allow the measurement of the error of the position estimation at these points. It is presented the development of the Smoothed Adaptive Kalman Filter (SAKF), which jointly estimates the states and the errors of the estimated states of the sensor fusion system. It is presented a quantitative criteria which employs the position uncertainties estimated by SAKF in order to determine - given the available sensors, the maximum time interval that one can navigate within a desired reliability. Reduced sets of landmarks are used as fictitious sensors to test the proposed reliability criterion. Also noteworthy are the mathematical models applied to terrestrial navigation that were unified in this work. The results show that, only relying on the low performance inertial sensors, the terrestrial navigation becomes impracticable after few tens of seconds. Using the same inertial sensors, the sensor fusion produced far better results, allowing the reconstruction of trajectories with displacements of about 2.7 km (or 15 minutes) with a final error of position estimation of about 3 m.

Unmanned aerial vehicles (UAV's) have been the focus of significant research interest in both military and commercial areas since they have a variety of practical applications including reconnaissance, surveillance, target acquisition, search and rescue, patrolling, real-time monitoring, and mapping, to name a few. To increase the autonomy and the capability of these UAV's and thus to reduce the workload of human operators, typical autonomous UAV's are usually equipped with both a navigation system and a tracking system. The navigation system provides high-rate ownship states (typically ownship inertial position, inertial velocity, and attitude) that are directly used in the autopilot system, and the tracking system provides low-rate target tracking states (typically target relative position and velocity with respect to the ownship). Target states in the global frame can be obtained by adding the ownship states and the target tracking states. The data estimated from this combination of the navigation system and the tracking system provide key information for the design of most UAV guidance laws, control command generation, trajectory generation, and path planning. As a baseline system that estimates ownship states, an integrated navigation system is designed by using an extended Kalman filter (EKF) with sequential measurement updates. In order to effectively fuse various sources of aiding sensor information, the sequential measurement update algorithm is introduced in the design of the integrated navigation system with the objective of being implemented in low-cost autonomous UAV's. Since estimated state accuracy using a low-cost, MEMS-based IMU degrades with time, several absolute (low update rate but bounded error in time) sensors, including the GPS receiver, the magnetometer, and the altimeter, can compensate for time-degrading errors. In this work, the sequential measurement update algorithm in smaller vectors and matrices is capable of providing a convenient framework for fusing the many sources of information in the design of integrated navigation systems. In this framework, several aiding sensor measurements with different size and update rates are easily fused with basic high-rate IMU processing. In order to provide a new mechanism that estimates ownship states, a new nonlinear filtering framework, called the unscented Kalman filter (UKF) with sequential measurement updates, is developed and applied to the design of a new integrated navigation system. The UKF is known to be more accurate and convenient to use with a slightly higher computational cost. This filter provides at least second-order accuracy by approximating Gaussian distributions rather than arbitrary nonlinear functions. This is compared to the first-order accuracy of the well-known EKF based on linearization. In addition, the step of computing the often troublesome Jacobian matrices, always required in the design of an integrated navigation system using the EKF, is eliminated. Furthermore, by employing the concept of sequential measurement updates in the UKF, we can add the advantages of sequential measurement update strategy such as easy compensation of sensor latency, easy fusion of multi-sensors, and easy addition and subtraction of new sensors while maintaining those of the standard UKF such as accurate estimation and removal of Jacobian matrices. Simulation results show better performance of the UKF-based navigation system than the EKF-based system since the UKF-based system is more robust to initial accelerometer and rate gyro biases and more accurate in terms of reducing transient peaks and steady-state errors in ownship state estimation. In order to estimate target tracking states or target kinematics, a new vision-based tracking system is designed by using a UKF in the scenario of three-dimensional air-to-air tracking. The tracking system can estimate not only the target tracking states but also several target characteristics including target size and acceleration. By introducing the UKF, the new vision-based tracking system presents good estimation performance by overcoming the highly nonlinear characteristics of the problem with a relatively simplified formulation. Moreover, the computational step of messy Jacobian matrices involved in the target acceleration dynamics and angular measurements is removed. A new particle filtering framework, called an extended marginalized particle filter (EMPF), is developed and applied to the design of a new vision-based tracking system. In this work, only three position components with vision measurements are solved in particle filtering part by applying Rao-Blackwellization or marginalization approach, and the other dynamics, including the target nonlinear acceleration model, with Gaussian noise are effectively handled by using the UKF. Since vision information can be better represented by probabilistic measurements and the EMPF framework can be easily extended to handle this type of measurements, better performance in estimating target tracking states will be achieved by directly incorporating non-Gaussian, probabilistic vision information as the measurement inputs to the vision-based tracking system in the EMPF framework.

Incremental smoothing and mapping (iSAM) is presented, a novel approach to the simultaneous localization and mapping (SLAM) problem. SLAM is the problem of estimating an observer's position from local measurements only, while creating a consistent map of the environment. The problem is difficult because even very small errors in the local measurements accumulate over time and lead to large global errors. iSAM provides an exact and efficient solution to the SLAM estimation problem while also addressing data association. For the estimation problem, iSAM provides an exact solution by performing smoothing, which keeps all previous poses as part of the estimation problem, and therefore avoids linearization errors. iSAM uses methods from sparse linear algebra to provide an efficient incremental solution. In particular, iSAM deploys a direct equation solver based on QR matrix factorization of the naturally sparse smoothing information matrix. Instead of refactoring the matrix whenever new measurements arrive, only the entries of the factor matrix that actually change are calculated. iSAM is efficient even for robot trajectories with many loops as it performs periodic variable reordering to avoid unnecessary fill-in in the factor matrix. For the data association problem, I present state of the art data association techniques in the context of iSAM and present an efficient algorithm to obtain the necessary estimation uncertainties in real-time based on the factored information matrix. I systematically evaluate the components of iSAM as well as the overall algorithm using various simulated and real-world data sets.

Apresenta-se o desenvolvimento de modelos matemáticos e algoritmos de fusão sensorial para navegação terrestre usando uma unidade de medição inercial (UMI) de baixo desempenho e o Filtro Estendido de Kalman. Os modelos foram desenvolvidos com base nos sistemas de navegação inercial strapdown (SNIS). O termo baixo desempenho refere-se à UMIs que por si só não são capazes de efetuar o auto- alinhamento por girocompassing. A incapacidade de se navegar utilizando apenas uma UMI de baixo desempenho motiva a investigação de técnicas que permitam aumentar o grau de precisão do SNIS com a utilização de sensores adicionais. Esta tese descreve o desenvolvimento do modelo completo de uma fusão sensorial para a navegação inercial de um veículo terrestre usando uma UMI de baixo desempenho, um hodômetro e uma bússola eletrônica. Marcas topográficas (landmarks) foram instaladas ao longo da trajetória de teste para se medir o erro da estimativa de posição nesses pontos. Apresenta-se o desenvolvimento do Filtro de Kalman Adaptativo Suavizado (FKAS), que estima conjuntamente os estados e o erro dos estados estimados do sistema de fusão sensorial. Descreve-se um critério quantitativo que emprega as incertezas de posição estimadas pelo FKAS para se determinar a priori, dado os sensores disponíveis, o intervalo de tempo máximo que se pode navegar dentro de uma margem de confiabilidade desejada. Conjuntos reduzidos de landmarks são utilizados como sensores fictícios para testar o critério de confiabilidade proposto. Destacam-se ainda os modelos matemáticos aplicados à navegação terrestre, unificados neste trabalho. Os resultados obtidos mostram que, contando somente com os sensores inerciais de baixo desempenho, a navegação terrestre torna-se inviável após algumas dezenas de segundos. Usando os mesmos sensores inerciais, a fusão sensorial produziu resultados muito superiores, permitindo reconstruir trajetórias com deslocamentos da ordem de 2,7 km (ou 15 minutos) com erro final de estimativa de posição da ordem de 3 m. / This work presents the development of the mathematical models and the algorithms of a sensor fusion system for terrestrial navigation using a low-grade inertial measurement unit (IMU) and the Extended Kalman Filter. The models were developed on the basis of the strapdown inertial navigation systems (SINS). Low-grade designates an IMU that is not able to perform girocompassing self-alignment. The impossibility of navigating relying on a low performance IMU is the motivation for investigating techniques to improve the SINS accuracy with the use of additional sensors. This thesis describes the development of a comprehensive model of a sensor fusion for the inertial navigation of a ground vehicle using a low-grade IMU, an odometer and an electronic compass. Landmarks were placed along the test trajectory in order to allow the measurement of the error of the position estimation at these points. It is presented the development of the Smoothed Adaptive Kalman Filter (SAKF), which jointly estimates the states and the errors of the estimated states of the sensor fusion system. It is presented a quantitative criteria which employs the position uncertainties estimated by SAKF in order to determine - given the available sensors, the maximum time interval that one can navigate within a desired reliability. Reduced sets of landmarks are used as fictitious sensors to test the proposed reliability criterion. Also noteworthy are the mathematical models applied to terrestrial navigation that were unified in this work. The results show that, only relying on the low performance inertial sensors, the terrestrial navigation becomes impracticable after few tens of seconds. Using the same inertial sensors, the sensor fusion produced far better results, allowing the reconstruction of trajectories with displacements of about 2.7 km (or 15 minutes) with a final error of position estimation of about 3 m.

The unimodality of a function seems a simple concept. But in the Euclidean space R^m, m=3,4,..., it is not easy to define. We have an easy tool to find the minimum point of a unimodal function. The goal of this project is to formalize and support distinctive strategies that typically guarantee convergence. Support is given both by analytic arguments and simulation study. Application is envisioned in low-dimensional but non-trivial problems. The convergence of the proposed iterated grid search algorithm is presented along with the results of particular application studies. It has been recognized that the derivative methods, such as the Newton-type method, are not entirely satisfactory, so a variety of other tools are being considered as alternatives. Many other tools have been rejected because of apparent manipulative difficulties. But in our current research, we focus on the simple algorithm and the guaranteed convergence for unimodal function to avoid the possible chaotic behavior of the function. Furthermore, in case the loss function to be optimized is not unimodal, we suggest a weaker condition: almost (noisy) unimodality, under which the iterated grid search finds an estimated optimum point. / Ph. D.

The objective of this thesis is to design a relative navigation and guidance system for unmanned aerial vehicles (UAVs) for vision-based control applications. The vision-based navigation, guidance and control has been one of the most focused on research topics for the automation of UAVs. This is because in nature, birds and insects use vision as the exclusive sensor for object detection and navigation. In particular, this thesis studies the monocular vision-based navigation and guidance. Since 2-D vision-based measurements are nonlinear with respect to the 3-D relative states, an extended Kalman filter (EKF) is applied in the navigation system design. The EKF-based navigation system is integrated with a real-time image processing algorithm and is tested in simulations and flight tests. The first closed-loop vision-based formation flight has been achieved. In addition, vision-based 3-D terrain recovery was performed in simulations. A vision-based obstacle avoidance problem is specially addressed in this thesis. A navigation and guidance system is designed for a UAV to achieve a mission of waypoint tracking while avoiding unforeseen stationary obstacles by using vision information. A 3-D collision criterion is established by using a collision-cone approach. A minimum-effort guidance (MEG) law is applied for a guidance design, and it is shown that the control effort can be reduced by using the MEG-based guidance instead of a conventional guidance law. The system is evaluated in a 6 DoF flight simulation and also in a flight test. For monocular vision-based control problems, vision-based estimation performance highly depends on the relative motion of the vehicle with respect to the target. Therefore, this thesis aims to derive an optimal guidance law to achieve a given mission under the condition of using the EKF-based relative navigation. Stochastic optimization is formulated to minimize the expected cost including the guidance error and the control effort. A suboptimal guidance law is derived based on an idea of the one-step-ahead (OSA) optimization. Simulation results show that the suggested guidance law significantly improves the guidance performance. Furthermore, the OSA optimization is generalized as the n-step-ahead optimization for an arbitrary number of n, and their optimality and computational cost are investigated.