Dynamics on Lie groups with applications to attitude estimation

Abstract

The problem of filtering - propagation of states through stochastic differential equations (SDEs) and association of measurement data using Bayesian inference - in a state space which forms a Lie group is considered. Particular emphasis is given to concentrated Gaussians (CGs) as a parametric family of probability distributions to capture the uncertainty associated with an estimated state. The so-called group-affine property of the state evolution is shown to be necessary and sufficient for linearity of the dynamics on the associated Lie algebra, in turn implying CGs are invariant under such evolution. A putative SDE on the group is then reformulated as an SDE on the associated Lie algebra. The vector space structure of the Lie algebra together with the notion of a CG enables the leveraging of techniques from conventional Gaussian-based Kalman filtering in an approach called the tangent space filter (TSF). We provide example calculations for several Lie groups that arise in the problem of estimating position, velocity, and orientation of a rigid body from a noisy, potentially biased inertial measurement unit (IMU). For the specific problem of attitude estimation, numerical experiments demonstrate that TSF-based approaches are more accurate and robust than another widely used attitude filtering technique.