Deterministic Sampling for Nonlinear Dynamic State Estimation

The goal of this thesis is improving existing and suggesting novel filtering algorithms for nonlinear dynamic state estimation. Its main contributions are novel techniques for approximating continuous probability distributions by discrete distributions defined on the same continuous domain. It considers both, linear and periodic underlying state spaces. In the linear case, we propose a method for approximating Gaussian densities by using a global distance measure capable of handling continuous and discrete probability distributions simultaneously. It is based on the Localized Cumulative Distribution (LCD), an alternative to the cumulative distribution function that considers all possible local probability masses around a given point. The sample set obtained from minimizing the distance measure is applied to approximate integration, where the results are shown to outperform state-of-the-art approaches. In case of periodic state spaces, this thesis proposes a deterministic sampling scheme for the Bingham distribution, which is an antipodally symmetric distribution defined on an N-dimensional hypersphere. This state space is of particular interest due to its capability of representing angles and orientations. Unit vectors in 2D are interpreted as angles, whereas unit vectors in 4D are interpreted as unit quaternions and, thus, are a suitable representation of uncertain orientations. The sampling scheme is based on moment matching and can be thought of as a hyperspherical equivalent to the samples obtained in the Unscented Kalman filter (UKF). Thus, its applicability is not restricted to the Bingham case but it is more broadly applicable to other antipodally symmetric hyperspherical distributions. Again, the resulting set of deterministically computed samples is applied to approximate integration. This contributes and improves the current state-of-the-art in two ways. First, it is the first time that a determinXV istic sampling scheme for the hypersphere has been proposed. Second, consideration of large uncertainties is now possible, because the typical exploitation of local linearity of the underlying domain is avoided.

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