September 2010

Document Type


Degree Name



Dept. of Biomedical Engineering


Oregon Health & Science University


The sigma-point Kalman filter (SPKF) is proved to be a more accurate alternative to the extended Kalman filter (EKF) in numerous nonlinear state estimation related applications. However, nonlinear smoothing algorithms based on the sigma-point Kalman filtering technique are not well established. We have derived new fixed-interval and fixed-lag smoothing algorithms using the sigma-point methodology and extended these nonlinear smoothers to a common family of algorithms, called sigma-point Kalman smoothers (SPKS). While the fixed-interval SPKS (FI-SPKS) operates on a fixed set of observations, the fixed-lag SPKS(FL-SPKS) sequentially operates on the buffered blocks of measurements as they become available. Both the FI-SPKS and FL-SPKS make use of the forward-backward (FB) and Rauch-Tung-Striebel (RTS) approaches to perform smoothing. In the FB method, a standard SPKF is used as a forward filter. The backward filter requires the use of the inverse dynamics of the forward filter. While smoothers based on the EKF simply invert the linearized dynamics, with the SPKF the forward nonlinear dynamics are never analytically linearized. Thus the backward nonlinear dynamics are not well defined. In this work, we make use of the relationship between the SPKF and weighted statistical linear regression (WSLR) to pseudo-linearize the nonlinear dynamics. The independent forward and backward estimates are then statistically combined to generate the smoothed results. The WSLR linearized dynamics are also incorporated in the RTS method to derive the backward smoothing gain which operates on the forward SPKF estimates to produce the smoothed states. We have applied the proposed SPKS to the challenging areas of probabilistic inference, such as indoor localization and multiharmonic frequency tracking, and evaluated the performance by comparing to the state-of-the-art tracking engines. Furthermore, we have successfully extended the theoretical understanding of the SPKF by analyzing its estimation-error bounds for the discrete-time nonlinear dynamical system. We have derived the mean-square error lower bound using the well-known Cramér-Rao theory. The upper error bound, which is also termed as the "stability bound", exponentially converges to the steady state if certain conditions on the state dynamics and system noises are satisfied. The theoretical derivations are experimentally verified using practical examples.




School of Medicine



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