Nonlinear Attitude Filtering: A Comparison Study

This paper contains a concise comparison of a number of nonlinear attitude filtering methods that have attracted attention in the robotics and aviation literature. With the help of previously published surveys and comparison studies, the vast literature on the subject is narrowed down to a small pool of competitive attitude filters. Amongst these filters is a second-order optimal minimum-energy filter recently proposed by the authors. Easily comparable discretized unit quaternion implementations of the selected filters are provided. We conduct a simulation study and compare the transient behaviour and asymptotic convergence of these filters in two scenarios with different initialization and measurement errors inspired by applications in unmanned aerial robotics and space flight. The second-order optimal minimum-energy filter is shown to have the best performance of all filters, including the industry standard multiplicative extended Kalman filter (MEKF)

Reducing Computational Cost in the Invariant Unscented Kalman Filtering For Attitude Estimation

This article proposes a new formulation to derive the invariant unscented Kalman filter (IUKF) algorithm for attitude estimation problem, where both state and sigma-point are considered as a transformation group parametrization of the filter. The detailed IUKF equations are presented in this article. The filter equations relie on the same ideas as the usual Unscented Kalman Filter (UKF), but it uses a geometrically adapted correction term based on an invariant output error. The specific interest of the proposed formulation is that only the invariant state estimation errors between the predicted state and each sigma point must be known to determine the predicted outputs errors. As we have already computed the set of invariant state errors during the prediction step, the computation cost to find the covariance matrix of the invariant state estimation in the update step is greatly reduced

The Invariant Unscented Kalman Filter

International audienceThis article proposes a novel approach for nonlinear state estimation. It combines both invariant observers theory and unscented filtering principles whitout requiring any compatibility condition such as proposed in the -IUKF algorithm. The resulting algorithm, named IUKF (Invariant Unscented Kalman Filter), relies on a geometrical-based constructive method for designing filters dedicated to nonlinear state estimation problems while preserving the physical invariances and systems symmetries. Within an invariant framework, this algorithm suggests a systematic approach to determine all the symmetry- preserving terms without requiring any linearization and highlighting remarkable invariant properties. As a result, the estimated covariance matrices of the IUKF converge to quasi-constant values due to the symmetry-preserving property provided by the invariant framework. This result enables the development of less conservative robust control strategies. The designed IUKF method has been successfully applied to some relevant practical problems such as the estimation of attitude for aerial vehicles using low-cost sensors reference systems. Typical experimental results using a Parrot quadrotor are provided in this pape

Lie Algebraic Unscented Kalman Filter for Pose Estimation

An unscented Kalman filter for matrix Lie groups is proposed where the time propagation of the state is formulated on the Lie algebra. This is done with the kinematic differential equation of the logarithm, where the inverse of the right Jacobian is used. The sigma points can then be expressed as logarithms in vector form, and time propagation of the sigma points and the computation of the mean and the covariance can be done on the Lie algebra. The resulting formulation is to a large extent based on logarithms in vector form, and is therefore closer to the UKF for systems in $\mathbb{R}^n$. This gives an elegant and well-structured formulation which provides additional insight into the problem, and which is computationally efficient. The proposed method is in particular formulated and investigated on the matrix Lie group $SE(3)$. A discussion on right and left Jacobians is included, and a novel closed form solution for the inverse of the right Jacobian on $SE(3)$ is derived, which gives a compact representation involving fewer matrix operations. The proposed method is validated in simulations