A Survey of Code Optimization Methods for Kalman Filter Extensions
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paper_title: Recursive bayesian estimation using gaussian sums

paper_content:

The Bayesian recursion relations which describe the behavior of the a posteriori probability density function of the state of a time-discrete stochastic system conditioned on available measurement data cannot generally be solved in closed-form when the system is either non-linear or nongaussian. In this paper a density approximation involving convex combinations of gaussian density functions is introduced and proposed as a meaningful way of circumventing the difficulties encountered in evaluating these relations and in using the resulting densities to determine specific estimation policies. It is seen that as the number of terms in the gaussian sum increases without bound, the approximation converges uniformly to any density function in a large class. Further, any finite sum is itself a valid density function unlike many other approximations that have been investigated. The problem of determining the a posteriori density and minimum variance estimates for linear systems with nongaussian noise is treated using the gaussian sum approximation. This problem is considered because it can be dealt with in a relatively straightforward manner using the approximation but still contains most of the difficulties that one encounters in considering non-linear systems since the a posteriori density is nongaussian. After discussing the general problem from the point-of-view of applying gaussian sums, a numerical example is presented in which the actual statistics of the a posteriori density are compared with the values predicted by the gaussian sum and by the Kalman filter approximations.
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paper_title: UWB Positioning with Generalized Gaussian Mixture Filters

paper_content:

Low-complexity Bayesian filtering for nonlinear models is challenging. Approximative methods based on Gaussian mixtures (GM) and particle filters are able to capture multimodality, but suffer from high computational demand. In this paper, we provide an in-depth analysis of a generalized GM (GGM), which allows component weights to be negative, and requires significantly fewer components than the traditional GM for ranging models. Based on simulations and tests with real data from a network of UWB nodes, we show how the algorithm's accuracy depends on the uncertainty of the measurements. For nonlinear ranging the GGM filter outperforms the extended Kalman filter (EKF) in both positioning accuracy and consistency in environments with uncertain measurements, and requires only slightly higher computational effort when the number of measurement channels is small. In networks with highly reliable measurements, the GGM filter yields similar accuracy and better consistency than the EKF.
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paper_title: Modeling Infectious Diseases in Humans and Animals

paper_content:

By Matthew James Keelingand Pejman RohaniPrinceton, NJ: Princeton University Press,2008.408 pp., Illustrated. $65.00 (hardcover).Mathematical modeling of infectious dis-eases has progressed dramatically over thepast 3 decades and continues to ﬂourishat the nexus of mathematics, epidemiol-ogy, and infectious diseases research. Nowrecognized as a valuable tool, mathemat-ical models are being integrated into thepublic health decision-making processmore than ever before. However, despiterapid advancements in this area, a formaltraining program for mathematical mod-eling is lacking, and there are very fewbooks suitable for a broad readership. Tosupport this bridging science, a commonlanguage that is understood in all con-tributing disciplines is required.
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paper_title: A Bayesian approach to problems in stochastic estimation and control

paper_content:

In this paper, a general class of stochastic estimation and control problems is formulated from the Bayesian Decision-Theoretic viewpoint. A discussion as to how these problems can be solved step by step in principle and practice from this approach is presented. As a specific example, the closed form Wiener-Kalman solution for linear estimation in Gaussian noise is derived. The purpose of the paper is to show that the Bayesian approach provides; 1) a general unifying framework within which to pursue further researches in stochastic estimation and control problems, and 2) the necessary computations and difficulties that must be overcome for these problems. An example of a nonlinear, non-Gaussian estimation problem is also solved.
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paper_title: Estimation with Applications to Tracking and Navigation

paper_content:

From the Publisher: ::: "Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations. It explains state estimator design using a balanced combination of linear systems, probability, and statistics." "The authors provide a review of the necessary background mathematical techniques and offer an overview of the basic concepts in estimation. They then provide detailed treatments of all the major issues in estimation with a focus on applying these techniques to real systems." "Suitable for graduate engineering students and engineers working in remote sensors and tracking, Estimation with Applications to Tracking and Navigation provides expert coverage of this important area."--BOOK JACKET.
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paper_title: State space regularization in the nonstationary inverse problem for diffuse optical tomography

paper_content:

In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT.
---
paper_title: Gaussian filters for nonlinear filtering problems

paper_content:

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests demonstrate that new filters significantly improve the extended Kalman filter with no additional cost, and the new Gaussian sum filter has a nearly optimal performance.
---
paper_title: Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks

paper_content:

Particle filters (PFs) are powerful sampling-based inference/learning algorithms for dynamic Bayesian networks (DBNs). They allow us to treat, in a principled way, any type of probability distribution, nonlinearity and non-stationarity. They have appeared in several fields under such names as "condensation", "sequential Monte Carlo" and "survival of the fittest". In this paper, we show how we can exploit the structure of the DBN to increase the efficiency of particle filtering, using a technique known as Rao-Blackwellisation. Essentially, this samples some of the variables, and marginalizes out the rest exactly, using the Kalman filter, HMM filter, junction tree algorithm, or any other finite dimensional optimal filter. We show that Rao-Blackwellised particle filters (RBPFs) lead to more accurate estimates than standard PFs. We demonstrate RBPFs on two problems, namely non-stationary online regression with radial basis function networks and robot localization and map building. We also discuss other potential application areas and provide references to some finite dimensional optimal filters.
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paper_title: Cubature Kalman Filters

paper_content:

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter. Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters.
---
paper_title: New developments in state estimation for nonlinear systems

paper_content:

Based on an interpolation formula, accurate state estimators for nonlinear systems can be derived. The estimators do not require derivative information which makes them simple to implement.
---
paper_title: State space regularization in the nonstationary inverse problem for diffuse optical tomography

paper_content:

In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT.
---
paper_title: The unscented Kalman filter for nonlinear estimation

paper_content:

This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples.
---
paper_title: Gaussian Filter based on Deterministic Sampling for High Quality Nonlinear Estimation

paper_content:

In this paper, a Gaussian filter for nonlinear Bayesian estimation is introduced that is based on a deterministic sample selection scheme. For an effective sample selection, a parametric density function representation of the sample points is employed, which allows approximating the cumulative distribution function of the prior Gaussian density. The computationally demanding parts of the optimization problem formulated for approximation are carried out off-line for obtaining an efficient filter, whose estimation quality can be altered by adjusting the number of used sample points. The improved performance of the proposed Gaussian filter compared to the well-known unscented Kalman filter is demonstrated by means of two examples.
---
paper_title: Gaussian filters for nonlinear filtering problems

paper_content:

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests demonstrate that new filters significantly improve the extended Kalman filter with no additional cost, and the new Gaussian sum filter has a nearly optimal performance.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---
paper_title: Optimization of the Simultaneous Localization and Map Building Algorithm for Real Time Implementation

paper_content:

Addresses real-time implementation of the simultaneous localization and map-building (SLAM) algorithm. It presents optimal algorithms that consider the special form of the matrices and a new compressed filler that can significantly reduce the computation requirements when working in local areas or with high frequency external sensors. It is shown that by extending the standard Kalman filter models the information gained in a local area can be maintained with a cost /spl sim/O(N/sub a//sup 2/), where N/sub a/ is the number of landmarks in the local area, and then transferred to the overall map in only one iteration at full SLAM computational cost. Additional simplifications are also presented that are very close to optimal when an appropriate map representation is used. Finally the algorithms are validated with experimental results obtained with a standard vehicle running in a completely unstructured outdoor environment.
---
paper_title: Nonlinear Kalman Filtering for Force-Controlled Robot Tasks

paper_content:

Introduction.- Literature Survey: Autonomous Compliant Motion.- Literature Survey: Bayesian Probability Theory.- Kalman Filters for Nonlinear Systems.- The Non-Minimal State Kalman Filter.- Contact Modelling.- Geometrical Parameter Estimation and CF Recognition.- Experiment: A Cube-In-Corner Assembly.- Task Planning with Active Sensing.- General Conclusions.
---
paper_title: Reduced Sigma Point Filtering for Partially Linear Models

paper_content:

A method for performing unscented Kalman filtering with a reduced number of sigma points is proposed. The procedure is applicable when either the process or measurement equations are partially linear in the sense that only a subset of the elements of the state vector undergo a nonlinear transformation. It is shown that for such models second-order accuracy in the moments required for the unscented Kalman filter recursion can be obtained using a number of sigma points determined by the number of nonlinearly transformed elements rather than the dimension of the state vector. A procedure for computing the sigma points is developed. An application of the proposed method to smoothed target state estimation from bearings measurements is presented.
---
paper_title: State space regularization in the nonstationary inverse problem for diffuse optical tomography

paper_content:

In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT.
---
paper_title: A rao-blackwellised unscented Kalman filter

paper_content:

The Unscented Kalman Filter oflers sign$- cant improvements in the estimation of non-linear discrete- time models in comparison to the Extended Kalman Fil- ter 1121. In this paper we use a technique introduced by Casella and Robert (2), known as Rao-Blackwellisation, to calculate the tractable integrations that are found in the Unscented Kalman Filter: We show that this leads to a re- duction in the quasi-Monte Carlo variance, and a decrease in the computational complexity by considering a common tracking problem.
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paper_title: Gaussian Filtering using state decomposition methods

paper_content:

State estimation for nonlinear systems generally requires approximations of the system or the probability densities, as the occurring prediction and filtering equations cannot be solved in closed form. For instance, Linear Regression Kalman Filters like the Unscented Kalman Filter or the considered Gaussian Filter propagate a small set of sample points through the system to approximate the posterior mean and covariance matrix. To reduce the number of sample points, special structures of the system and measurement equation can be taken into account. In this paper, two principles of system decomposition are considered and applied to the Gaussian Filter. One principle exploits that only a part of the state vector is directly observed by the measurement. The second principle separates the system equations into linear and nonlinear parts in order to merely approximate the nonlinear part of the state. The benefits of both decompositions are demonstrated on a real-world example.
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paper_title: An Unscented Transformation for Conditionally Linear Models

paper_content:

A new method of applying the unscented transformation to conditionally linear transformations of Gaussian random variables is proposed. This method exploits the structure of the model to reduce the required number of sigma points. A common application of the unscented transformation is to nonlinear filtering where it used to approximate the moments required in the Kalman filter recursion. The proposed procedure is applied to a nonlinear filtering problem which involves tracking a falling object.
---
paper_title: Optimization of the Simultaneous Localization and Map Building Algorithm for Real Time Implementation

paper_content:

Addresses real-time implementation of the simultaneous localization and map-building (SLAM) algorithm. It presents optimal algorithms that consider the special form of the matrices and a new compressed filler that can significantly reduce the computation requirements when working in local areas or with high frequency external sensors. It is shown that by extending the standard Kalman filter models the information gained in a local area can be maintained with a cost /spl sim/O(N/sub a//sup 2/), where N/sub a/ is the number of landmarks in the local area, and then transferred to the overall map in only one iteration at full SLAM computational cost. Additional simplifications are also presented that are very close to optimal when an appropriate map representation is used. Finally the algorithms are validated with experimental results obtained with a standard vehicle running in a completely unstructured outdoor environment.
---
paper_title: Nonlinear Kalman Filtering for Force-Controlled Robot Tasks

paper_content:

Introduction.- Literature Survey: Autonomous Compliant Motion.- Literature Survey: Bayesian Probability Theory.- Kalman Filters for Nonlinear Systems.- The Non-Minimal State Kalman Filter.- Contact Modelling.- Geometrical Parameter Estimation and CF Recognition.- Experiment: A Cube-In-Corner Assembly.- Task Planning with Active Sensing.- General Conclusions.
---
paper_title: Kullback-Leibler Divergence Approach to Partitioned Update Kalman Filter

paper_content:

Kalman filtering is a widely used framework for Bayesian estimation. The partitioned update Kalman filter applies a Kalman filter update in parts so that the most linear parts of measurements are applied first. In this paper, we generalize partitioned update Kalman filter, which requires the use of the second order extended Kalman filter, so that it can be used with any Kalman filter extension such as the unscented Kalman filter. To do so, we use a Kullback-Leibler divergence approach to measure the nonlinearity of the measurement, which is theoretically more sound than the nonlinearity measure used in the original partitioned update Kalman filter. Results show that the use of the proposed partitioned update filter improves the estimation accuracy.
---
paper_title: Particle filter and smoother for indoor localization

paper_content:

We present a real-time particle filter for 2D and 3D hybrid indoor positioning. It uses wireless local area network (WLAN) based position measurements, step and turn detection from a hand-held inertial sensor unit, floor plan restrictions, altitude change measurements from barometer and possibly other measurements such as occasional GNSS fixes. We also present a particle smoother, which uses future measurements to improve the position estimate for non-real-time applications. A light-weight fallback filter is run in the background for initialization, divergence monitoring and possibly re-initialization. In real-data tests the particle filter is more accurate and consistent than the methods that do not use floor plans. An example is shown on how smoothing helps to improve the filter estimate. Moreover, a floor change case is presented, in which the filter is capable of detecting the floor change and improving the 2D accuracy using the floor change information.
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paper_title: A TDOA Gaussian mixture model for improving acoustic source tracking

paper_content:

Traditionally, time difference of arrival (TDOA) based acoustic source tracking consists of two stages, more precisely, estimation of TDOAs followed by a tracking algorithm. In general, these two stages are performed separately and presume that (1) TDOAs can be estimated reliably; and (2) the errors in detection behave in a well-defined fashion. The presence of noise and reverberation, however, leads to multimodal TDOA distributions and causes larger errors in the estimates, which ultimately lowers the tracking performance. To counteract this effect, we propose an approach that enhances TDOA estimation by (1) accounting for the multimodal aspect through a Gaussian mixture model and (2) integrating knowledge that has been obtained in the tracking stage. In doing so, this approach tightly couples the two stages. Experimental results on the AV16.3 corpus show that the proposed approach significantly improves the tracking performance compared to various other tracking algorithms.
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paper_title: The huge microphone array

paper_content:

The Huge Microphone Array project began in February 1994 to design, construct, debug, and test a real-time 512-microphone array system and to develop algorithms for it. Analysis of known algorithms indicated that signal-processing performance of over 6 Gflops would be required, while the need for portability-fitting it into a small van-also set an upper limit to the power required. These trade-offs and many others have led to a unique design in both hardware and software. This two-part article presents the full design and its justifications. The authors also discuss performance for a few important algorithms relative to the use of processing capability, response latency, and difficulty of programming.
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paper_title: System for robust 3D speaker tracking using microphone array measurements

paper_content:

A system for three-dimensional passive acoustic speaker localization and tracking using a microphone array is presented and evaluated. Initial speaker position estimates are provided by a time-delay-based localization algorithm. These raw estimates are spatially smoothed by a multiple model adaptive estimator consisting of three extended Kalman filters running in parallel. The performance of the proposed system is evaluated for real data in a common office environment. The reference trajectory of the moving speaker is delivered by visually tracking a color marker on the speaker's forehead by a stereo-camera system. The proposed acoustic source tracker shows robustness and accuracy in a variety of different scenarios.
---
paper_title: Design of Sigma-Point Kalman Filter with Recursive Updated Measurement

paper_content:

In this study, the authors focus on improving measurement update of existing nonlinear Kalman approximation filter and propose a new sigma-point Kalman filter with recursive measurement update. Statistical linearization technique based on sigma transformation is utilized in the proposed filter to linearize the nonlinear measurement function, and linear measurement update is applied gradually and repeatedly based on the statistically linearized measurement equation. The total measurement update of the proposed filter is nonlinear, and the proposed filter can extract state information from nonlinear measurement better than existing nonlinear filters. Simulation results show that the proposed method has higher estimation accuracy than existing methods.
---
paper_title: Recursive Update Filtering for Nonlinear Estimation

paper_content:

Nonlinear filters are often very computationally expensive and usually not suitable for real-time applications. Real-time navigation algorithms are typically based on linear estimators, such as the extended Kalman filter (EKF) and, to a much lesser extent, the unscented Kalman filter. This work proposes a novel nonlinear estimator whose additional computational cost is comparable to (N-1) EKF updates, where N is the number of recursions, a tuning parameter. The higher N the less the filter relies on the linearization assumption. A second algorithm is proposed with a differential update, which is equivalent to the recursive update as N tends to infinity.
---
paper_title: Design of Sigma-Point Kalman Filter with Recursive Updated Measurement

paper_content:

In this study, the authors focus on improving measurement update of existing nonlinear Kalman approximation filter and propose a new sigma-point Kalman filter with recursive measurement update. Statistical linearization technique based on sigma transformation is utilized in the proposed filter to linearize the nonlinear measurement function, and linear measurement update is applied gradually and repeatedly based on the statistically linearized measurement equation. The total measurement update of the proposed filter is nonlinear, and the proposed filter can extract state information from nonlinear measurement better than existing nonlinear filters. Simulation results show that the proposed method has higher estimation accuracy than existing methods.
---
paper_title: Recursive Update Filtering for Nonlinear Estimation

paper_content:

Nonlinear filters are often very computationally expensive and usually not suitable for real-time applications. Real-time navigation algorithms are typically based on linear estimators, such as the extended Kalman filter (EKF) and, to a much lesser extent, the unscented Kalman filter. This work proposes a novel nonlinear estimator whose additional computational cost is comparable to (N-1) EKF updates, where N is the number of recursions, a tuning parameter. The higher N the less the filter relies on the linearization assumption. A second algorithm is proposed with a differential update, which is equivalent to the recursive update as N tends to infinity.
---
paper_title: Cubature Kalman Filters

paper_content:

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter. Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters.
---
paper_title: Nonlinear Kalman Filtering for Force-Controlled Robot Tasks

paper_content:

Introduction.- Literature Survey: Autonomous Compliant Motion.- Literature Survey: Bayesian Probability Theory.- Kalman Filters for Nonlinear Systems.- The Non-Minimal State Kalman Filter.- Contact Modelling.- Geometrical Parameter Estimation and CF Recognition.- Experiment: A Cube-In-Corner Assembly.- Task Planning with Active Sensing.- General Conclusions.
---
paper_title: New developments in state estimation for nonlinear systems

paper_content:

Based on an interpolation formula, accurate state estimators for nonlinear systems can be derived. The estimators do not require derivative information which makes them simple to implement.
---
paper_title: The unscented Kalman filter for nonlinear estimation

paper_content:

This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples.
---
paper_title: Gaussian Filter based on Deterministic Sampling for High Quality Nonlinear Estimation

paper_content:

In this paper, a Gaussian filter for nonlinear Bayesian estimation is introduced that is based on a deterministic sample selection scheme. For an effective sample selection, a parametric density function representation of the sample points is employed, which allows approximating the cumulative distribution function of the prior Gaussian density. The computationally demanding parts of the optimization problem formulated for approximation are carried out off-line for obtaining an efficient filter, whose estimation quality can be altered by adjusting the number of used sample points. The improved performance of the proposed Gaussian filter compared to the well-known unscented Kalman filter is demonstrated by means of two examples.
---
paper_title: Gaussian filters for nonlinear filtering problems

paper_content:

We develop and analyze real-time and accurate filters for nonlinear filtering problems based on the Gaussian distributions. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We also discuss the mixed Gaussian filters in which the conditional probability density is approximated by the sum of Gaussian distributions. A new update rule of weights for Gaussian sum filters is proposed. Our numerical tests demonstrate that new filters significantly improve the extended Kalman filter with no additional cost, and the new Gaussian sum filter has a nearly optimal performance.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---

Title: A Survey of Code Optimization Methods for Kalman Filter Extensions

Section 1: Introduction
Description 1: Introduce the application areas of Bayesian state estimation and Kalman Filter Extensions (KFEs), and outline the workflow of the paper.

Section 2: Notations
Description 2: List and describe the variables and notations used throughout the paper.

Section 3: Background
Description 3: Provide the background of discrete-time Bayesian filtering and various Kalman Filter Extensions.

Section 4: Partially Linear Functions
Description 4: Discuss algorithms for treating different setups of UKF and improving computation efficiency with partially linear state and measurement models.

Section 5: Conditionally Linear Measurements
Description 5: Present algorithms for handling conditionally linear state variables using sigma-points and other optimization techniques.

Section 6: Part of State Is Unobserved and Static
Description 6: Examine situations where some state variables are unobserved and static, and describe the corresponding optimization algorithms.

Section 7: Part of State is Unobserved and Whole State Is Static
Description 7: Explore optimization methods for cases where all state variables are static and partially unobserved.

Section 8: Block Diagonal Measurement Covariance
Description 8: Describe the optimization of the Kalman gain computation using block diagonal measurement covariance.

Section 9: Applying the Matrix Inversion Lemma to Innovation Covariance
Description 9: Explain the optimization of the inverse of the innovation covariance using the matrix inversion lemma.

Section 10: Pedestrian Dead Reckoning
Description 10: Provide a case study on applying the surveyed optimization techniques to pedestrian dead reckoning systems.

Section 11: Optimization of Iterative KFEs
Description 11: Discuss the application of the optimizations to iterative KFE algorithms like Recursive Update Filter (RUF).

Section 12: Conclusions
Description 12: Summarize the contributions of the survey, highlighting the unification and general notation of various optimization methods.