Bias estimation for invariant systems on Lie groups with homogeneous outputs

In this paper, we provide a general method of state estimation for a class of invariant systems on connected matrix Lie groups where the group velocity measurement is corrupted by an unknown constant bias. The output measurements are given by a collection of actions of a single Lie group on several homogeneous output spaces, a model that applies to a wide range of practical scenarios. The proposed observer consists of a group estimator part, providing an estimate of a bounded state evolving on the Lie group, and a bias estimator part, providing an estimate of the bias in the associated Lie algebra. We employ the gradient of a suitable invariant cost function on the Lie group as an innovation term in the group estimator. We design the bias estimator such that it guarantees uniform local exponential stability of the estimation error dynamics around the zero error state. We propose a systematic methodology for the design of suitable cost functions on Lie groups by lifting invariant cost functions from the homogeneous output spaces. We show that the resulting observer is implementable based on available sensor measurements if the homogeneous output spaces are reductive. As an example, we derive an observer for rigid body attitude using vector and gyro measurements with unknown constant gyro bias.