A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB001 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB002 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB003 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB004 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB005 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB006 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB007 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB008 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conclusions <s> 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. <s> BIB009
In this survey, we collected different optimizations for KFEs that exploit the structure of the state and measurement models to reduce the computational BIB001 1 A Unscented Kalman Filter (UKF) BIB002 n I X Cubature Kalman Filter (CKF) BIB007 n α I X Smart Sampling Kalman Filter (S 2 KF)  n + α I X Gaussian Filter (GF) BIB006 αn I X Gauss-Hermite Quadrature Kalman filter (QKF) BIB003 α n I X Divided Difference (DD) BIB004 n N Second Order Polynomial Kalman Filter (SOKF2) BIB003 n 2 N Recursive Update Filter (RUF) BIB008 j A X Unscented Recursive Update Filter (URUF) BIB009 jn I X X load. Table 1 summarizes the structures of models that are exploited in different optimizations. These structures are present in various nonlinear estimation problems. Table 2 (17)- BIB005 . The optimization that exploits the partially unobserved and static state in Section 5.1 is better suited for algorithms that compute the J within the algorithm (A or N in Table 2 ). The main contribution of this survey is collecting the different code optimization methods into a single reference and using a general notation that allows to use them with various KFEs. In addition to giving a unifying source for code optimization methods, we make the following contributions: