Riemannian cubics in quadratic matrix Lie groups Abstract   Quadratic matrix Lie groups are subgroups of the general linear group that satisfy a quadratic matrix identity. The main purpose of this paper is to consider Riemannian cubics in quadratic matrix Lie groups with leftinvariant metrics. Results for Riemannian cubics in quadratic matrix Lie groups extend those in SO(n) since the group SO(n) with biinvariant metric is a very special case. By examining Riemannian cubics in SO(2, 1) and SO(3, 1), we find that the socalled null Lie quadratics in     so  (  p  ,  q  )     (pxc2xa0u003exc2xa00, qxc2xa0u003exc2xa00), and even more generally for any quadratic matrix Lie group, can be given in closed forms in terms of Lie quadratics in     so  (  p  )     and     so  (  q  )    . Further, we present some quantitative analyses of nonnull Lie quadratics in     so  (  p  ,  q  )    .

Invariant extended Kalman filter on matrix Lie groups Abstract   We derive symmetry preserving invariant extended Kalman filters (IEKF) on matrix Lie groups. These Kalman filters have an advantage over conventional extended Kalman filters as the error dynamics for such filters are independent of the group configuration which, in turn, provides a uniform estimate of the region of convergence. The proposed IEKF differs from existing techniques in literature on the account that it is derived using minimal tools from differential geometry that simplifies its representation and derivation to a large extent. The filter error dynamics is defined on the Lie algebra directly instead of identifying the Lie algebra with an Euclidean space or defining the error dynamics in local coordinates using exponential map, and the associated differential Riccati equations are described on the corresponding space of linear operators using tensor algebra. The proposed filter is implemented for the attitude dynamics of the rigid body, which is a benchmark problem in control, and its performance is compared against a conventional extended Kalman filter (EKF). Numerical experiments support that the IEKF is computationally less intensive and gives better performance than the EKF.