Nonlinear Attitude Filtering: A Comparison Study

This paper contains a concise comparison of a number of nonlinear attitude filtering methods that have attracted attention in the robotics and aviation literature. With the help of previously published surveys and comparison studies, the vast literature on the subject is narrowed down to a small pool of competitive attitude filters. Amongst these filters is a second-order optimal minimum-energy filter recently proposed by the authors. Easily comparable discretized unit quaternion implementations of the selected filters are provided. We conduct a simulation study and compare the transient behaviour and asymptotic convergence of these filters in two scenarios with different initialization and measurement errors inspired by applications in unmanned aerial robotics and space flight. The second-order optimal minimum-energy filter is shown to have the best performance of all filters, including the industry standard multiplicative extended Kalman filter (MEKF)

Continuous stirred tank reactor fault detection using higher degree Cubature Kalman filter

Continuous Stirred Tank Reactor (CSTR) plays a major role in chemical industries, it ensures the process of mixing reactants according to the attended specification to produce a specific output. It is a complex process that usually represent with nonlinear model for benchmarking. Any abnormality, disturbance and unusual condition can easily interrupt the operations, especially fault. And this problem need to detect and rectify as soon as possible. A good knowledge based fault detection using available model require a good error residual between the measurement and the estimated state. Kalman filter is an example of a good estimator, and has been exploited in many researches to detect fault. In this paper, Higher degree Cubature Kalman Filter (HDCKF) is proposed as a method for fault detection by estimation the current state. Cubature Kalman filter (CKF) is an extension of the Kalman filter with the main purpose is to estimate process and measurement state with high nonlinearities. It is based on spherical radial integration to estimate current state by generating cubature points with specific value. Conventional CKF use 3rd degree spherical and 3rd degree radial, here we implement Higher Degree CKF (HDCKF) to have better accuracy as compared to conventional CKF. High accuracy is required to ensure no false alarm is detected and furthermore good computational cost will improve its detection. Finally, a numerical example of CSTR fault detection using HDCKF is presented. Implementation of HDCKF for fault detection is compared with other filter to show effective results

Adaptive and robust fractional gain based interpolatory cubature Kalman filter

In this study, we put forward the robust fractional gain based interpolatory cubature Kalman filter (FGBICKF) and the adaptive FGBICKF (AFGBICKF) for the development of the state estimators for stochastic nonlinear dynamics system. FGBICKF introduces a fractional gain to interpolatory cubature Kalman filter to increase the robustness of state estimation. AFGBICKF is developed to enhance the state estimation adaptive to stochastic nonlinear dynamics system with unknown process noise covariance through recursive estimation. The simulations on re-entry target tracking system have shown that the performance of FGBICKF is superior to that of cubature Kalman filter and interpolatory cubature Kalman filter, and standard deviation of FGBICKF is closer to posterior Cramér-Rao lower bound. Moreover, our simulations have also demonstrated that AFGBICKF remains stable even when the initial process noise covariance increase, proving its adaptiveness, robustness, and effectiveness on state estimation

The Sparse-grid based Nonlinear Filter: Theory and Applications

Filtering or estimation is of great importance to virtually all disciplines of engineering and science that need inference, learning, information fusion, and knowledge discovery of dynamical systems. The filtering problem is to recursively determine the states and/or parameters of a dynamical system from a sequence of noisy measurements made on the system. The theory and practice of optimal estimation of linear Gaussian dynamical systems have been well established and successful, but optimal estimation of nonlinear and non-Gaussian dynamical systems is much more challenging and in general requires solving partial differential equations and intractable high-dimensional integrations. Hence, Gaussian approximation filters are widely used. In this dissertation, three innovative point-based Gaussian approximation filters including sparse Gauss-Hermite quadrature filter, sparse-grid quadrature filter, and the anisotropic sparse-grid quadrature filter are proposed. The relationship between the proposed filters and conventional Gaussian approximation filters is analyzed. In particular, it is proven that the popular unscented Kalman filter and the cubature Kalman filter are subset of the proposed sparse-grid filters. The sparse-grid filters are employed in three aerospace applications including spacecraft attitude estimation, orbit determination, and relative navigation. The results show that the proposed filters can achieve better estimation accuracy than the conventional Gaussian approximation filters, such as the extended Kalman filter, the cubature Kalman filter, the unscented Kalman filter, and is computationally more efficient than the Gauss-Hermite quadrature filter

Composite Disturbance Filtering: A Novel State Estimation Scheme for Systems With Multi-Source, Heterogeneous, and Isomeric Disturbances

State estimation has long been a fundamental problem in signal processing and control areas. The main challenge is to design filters with ability to reject or attenuate various disturbances. With the arrival of big data era, the disturbances of complicated systems are physically multi-source, mathematically heterogenous, affecting the system dynamics via isomeric (additive, multiplicative and recessive) channels, and deeply coupled with each other. In traditional filtering schemes, the multi-source heterogenous disturbances are usually simplified as a lumped one so that the "single" disturbance can be either rejected or attenuated. Since the pioneering work in 2012, a novel state estimation methodology called {\it composite disturbance filtering} (CDF) has been proposed, which deals with the multi-source, heterogenous, and isomeric disturbances based on their specific characteristics. With the CDF, enhanced anti-disturbance capability can be achieved via refined quantification, effective separation, and simultaneous rejection and attenuation of the disturbances. In this paper, an overview of the CDF scheme is provided, which includes the basic principle, general design procedure, application scenarios (e.g. alignment, localization and navigation), and future research directions. In summary, it is expected that the CDF offers an effective tool for state estimation, especially in the presence of multi-source heterogeneous disturbances

Filtragem Não Linear Adaptativa e Seguimento Radar Ótimo de Veículos Aeroespaciais

A filtragem não-linear é um dos tópicos mais importantes e complexos em engenharia, especialmente quando aplicada a situações de tempo-real em ambientes altamente não-lineares. Este é o cenário da maioria das aplicações aeroespaciais nomeadamente, aviso de colisão, seguimento radar, vigilância, orientação, navegação e controlo de veículos aeroespaciais, sendo que o principal objetivo é a estimação dos estados de um determinado alvo (seja este uma aeronave, satélite, míssil ou outro) a partir de medições ruidosas. A maior dificuldade está em desenvolver métodos que sejam capazes de lidar não só com a não-linearidade dos modelos, mas também com as incertezas associadas aos instrumentos de medições e às perturbações existentes no meio envolvente que afetam diretamente o sistema e, na sua maioria, são difíceis de prever e computar. Uma das estratégias mais utilizadas para garantir o ajuste dinâmico e ótimo dos métodos de filtragem face a todas estas adversidades é a implementação de algoritmos adaptativos. Assim sendo, a abordagem mais utilizada para lidar com esta problemática é a filtragem de Kalman. O seu sucesso, principalmente na área de engenharia, deve-se na sua maioria ao filtro de Kalman estendido (EKF – Extended Kalman Filter). Este assenta no pressuposto de que a linearização é suficiente para representar localmente a não-linearidade do sistema e, por conseguinte, o algoritmo utiliza o modelo linearizad0 em substituição ao modelo original não-linear. A linearização é um processo relativamente fácil de compreender e aplicar, o que justifica a popularidade do filtro. Contudo, ao lidar com sistemas altamente não-lineares, o EKF tende a apresentar algumas limitações, tais como, estimativas erráticas, comportamentos instáveis e por vezes até divergentes. De forma a colmatar algumas destas limitações, esta tese apresenta um filtro de Kalman estendido melhorado e adaptativo, denominado por improved Extended Kalman Filter (iEKF), onde para além da adaptabilidade clássica das matrizes de ruído, é proposto uso da norma de Frobenius como fator de correção da estimativa da covariância a priori e é também proposto um novo ponto de linearização. Desta forma, o iEKF adapta as matrizes de transição dos modelos através do novo ponto de linearização e adapta as informações estatísticas através da matriz de covariância proposta. A principal intenção é manter a simplicidade e estrutura pelo qual o EKF é conhecido, porém melhorar o seu desempenho e precisão com conceitos simples, eficazes e adaptativos. Um outro foco desta tese é analisar o desempenho da filtragem no seguimento radar. Assim sendo, tanto o EKF como o iEKF foram implementados e analisados em quatro aplicações deste âmbito, sendo estas: a estimação de uma órbita de um satélite artificial, a estimação de uma transferência orbital (transferência de Hohmann), a estimação de uma reentrada na atmosfera, e por fim, a estimação da trajetória de uma aeronave comercial, em que objetivo é estimar a posição e velocidade do veículo. Tanto o EKF como o iEKF foram analisados e comparados com base no RMSE (Root Mean Square Error). Os resultados demonstram que o iEKF fornece estimativas superiores. O algoritmo é, em geral, mais preciso, estável e confiável, demonstrando ser uma alternativa conveniente ao clássico EKF. Em suma, esta tese propõe um novo método de filtragem não-linear adaptativo, denominado por iEKF. Os resultados indicam que este deve ser tido em consideração para a estimação de estados não-linear tanto para o seguimento radar, como para qualquer outra área que necessidade de um algoritmo de filtragem eficiente.Nonlinear filtering is an important and complex topic in engineering, especially when applied to real-time applications with a highly nonlinear environment. This scenario involves most aerospace applications, such as surveillance, guidance, navigation, attitude control, collision warning and target tracking, where the main objective consists of estimating the states of a moving target (aircraft, satellite, missile, spacecraft, etc.) based on noisy measurements. The challenge is to develop methods that are capable to cope, not only with the nonlinearities of the models but also with the instrumental inaccuracies related to the data acquisition system and the environmental perturbations that are unwanted and, in most cases, difficult to compute. One of the promising strategies to dynamically adjust and guarantee filter optimality is the computation of adaptative algorithms. A very well-known framework to deal with those problems is the Kalman filter algorithms, whose success in engineering applications is mostly due to the Extended Kalman Filter (EKF). The EKF is based on the assumption that a local linearization of the system may be a sufficient description of nonlinearities, therefore the linearized model is used instead of the original nonlinear function. Such approximations are easy to understand and apply, which explains the popularity of the filter. However, when dealing with highly nonlinear systems, the EKF estimates suffer serious problems, such as unstable and quickly divergent behaviours and/or erratic estimates. To address those limitations, this thesis proposes an improved Extended Kalman filter (iEKF) with an adaptative structure, where a new Jacobian matrix expansion point is proposed, and a Frobenius norm of the covariance matrix is suggested as a correction factor for the a priori estimates. Therefore, the iEKF does not only update the statistical information based on the proposed covariance matrix but also updates the state and measurements transitions matrices based on the new Jacobian expansion point. The core idea is to maintain the EKF structure and simplicity but improve the overall performance with simple yet effective concepts. Another objective of this thesis was to evaluate the performance of the filtering methods on radar tracking applications. Thus, the effectiveness of EKF and iEKF were analysed and compared in four radar tracking applications: an artificial satellite orbit estimation, a Hohmann orbit transfer, an atmospheric reentry estimation, and a commercial aircraft trajectory estimation, where the position and velocity of the aerospace vehicle were computed. The EKF and iEKF were compared based on the RMSE (Root Mean Square Error). Simulations results suggest that the iEKF provides a considerably higher accuracy on the overall results. The algorithm is more precise, stable, and reliable, which make it an attractive alternative to the classic EKF. In summary, this thesis proposed an improved Extended Kalman Filter with an adaptative structure. This algorithm is a promising method for nonlinear state estimation, not only for radar tracking applications but any applications that require an efficient nonlinear filter

A SLAM Algorithm Based on Adaptive Cubature Kalman Filter

We need to predict mathematical model of the system and a priori knowledge of the noise statistics when traditional simultaneous localization and mapping (SLAM) solutions are used. However, in many practical applications, prior statistics of the noise are unknown or time-varying, which will lead to large estimation errors or even cause divergence. In order to solve the above problem, an innovative cubature Kalman filter-based SLAM (CKF-SLAM) algorithm based on an adaptive cubature Kalman filter (ACKF) was established in this paper. The novel algorithm estimates the statistical parameters of the unknown system noise by introducing the Sage-Husa noise statistic estimator. Combining the advantages of the CKF-SLAM and the adaptive estimator, the new ACKF-SLAM algorithm can reduce the state estimated error significantly and improve the navigation accuracy of the SLAM system effectively. The performance of this new algorithm has been examined through numerical simulations in different scenarios. The results have shown that the position error can be effectively reduced with the new adaptive CKF-SLAM algorithm. Compared with other traditional SLAM methods, the accuracy of the nonlinear SLAM system is significantly improved. It verifies that the proposed ACKF-SLAM algorithm is valid and feasible