Invariant unscented Kalman filter with application to attitude estimation

This paper addresses the problem of nonlinear attitude estimation for mini-UAVs and proposes a novel formulation of the Unscented Kalman Filter (UKF), named Invariant UKF (IUKF), adapted to dynamical systems possessing symmetries. A first attempt for designing a symmetry-preserving observer based on an UKF-like technique has been proposed by the authors recently, leading to the development of the so-called n-IUKF estimation algorithm. The IUKF presented in this article relies on the same ideas as the usual UKF, but instead of using a linear correction term based on a linear output error, it uses a geometrically adapted correction term based on an invariant output error; similarly the gain matrix is not updated from linear error terms (in state and output), but from invariant ones. Therefore, its mathematical formulation differs from the one associated with former n-IUKF observer in not requiring the definition of a compatibility condition. From a practical point of view, the advantage of KF-like methods for designing any invariant state observer is their automatic tuning of the correction gains based on the computation of estimation error covariance matrices. Thus, this paper investigates a systematic approach to construct invariant filters derived from the unscented transform. The invariance properties, as well as the filtering performances, of the proposed IUKF are evaluated by post-processing real data gathered from experiments carried out on a quadrotor mini-UAV equipped with low-cost sensors. The results obtained validate our design.