Variational unscented Kalman filter on matrix Lie groups

In this paper, several estimation algorithms called the variational unscented Kalman filters (UKF-Vs) are proposed for matrix Lie groups. The proposed filters are inspired by the unscented Kalman filter in Euclidean space and they exhibit advantages over conventional methods, as the prediction step and the measurement update step are established on the Lie algebra and its dual space, which therefore avoids direct operations on highly nonlinear Lie group configuration spaces. Correspondingly, the proposed UKF-Vs exhibit significant improvements in terms of the estimation error and mean square error. This also makes it possible to construct a computationally efficient geometric integrator for the filtering dynamics. The obtained formulation is independent of the nonlinear Lie group state-space, and it sheds light on fully predicting and updating on the Lie algebra and its dual which are endowed with vector space structures. In particular, these formulations can avoid singularities or the well-known gimbal lock in the attitude estimation problem. Furthermore, the stochastic stability of the proposed filters is studied. The performances of the proposed filters are demonstrated for the satellite attitude estimation problem, which is an important benchmark from a control perspective. Numerical results show that the proposed UKF-Vs keep less computational complexity and perform significantly high accuracy compared with two unscented Kalman filters on Lie groups.