Uncertainty Propagation of Correlated Quaternion and Euclidean States Using the Gauss-Bingham Density
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abstract
A new probability density function, called the Gauss-Bingham density, is proposed and studied in the context of uncertainty propagation. The Gauss-Bingham density quantifies the correlation between a quaternion and Euclidean states on the cylindrical manifold on which these states naturally exist. The Gauss-Bingham density, including its canonical form, is developed. In order to approximate the temporal evolution of the uncertainty, an unscented transform for the Gauss-Bingham density is first developed. The sigma points are then transformed according to given (potentially) nonlinear system dynamics, and the maximum weighted log-likelihood parameters of the Gauss-Bingham density are recovered. Uncertainty propagation using the Gauss-Bingham density does not rely on a small angle assumption to project the uncertainty in the quaternion into a three parameter representation as does the predictor of the multiplicative extended Kalman filter, so its accuracy does not suffer when propagating large attitude uncertainties. Two simulations are presented to show the process and efficacy of uncertainty propagation using the Gauss-Bingham density and to compare it to the multiplicative extended Kalman filter.
