Dynamics on Lie Groups with Applications to Attitude Estimation

The problem of filtering—propagation of states through stochastic differential equations (SDEs) and association of measurement data using Bayesian inference—in a state space that 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. We present a novel proof that the so-called group-affine property of the state evolution is necessary and sufficient for linearity of the dynamics on the associated Lie algebra, in turn implying that CGs are invariant under such evolution. We then show how a putative SDE on the group can be 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, which we detail in a new approach called the tangent space filter (TSF). We provide example calculations for several Lie groups that arise in the problem of estimating the position, velocity, and orientation of a rigid body from a noisy, potentially biased inertial measurement unit (IMU). For the specific problem of attitude estimation, we further demonstrate through numerical experiments that TSF-based approaches are more accurate and robust than a widely used baseline attitude filtering technique.