Globally exponentially convergent observer for systems evolving on matrix Lie groups

Abstract

The estimate of a system state, arrived at using measurements, is often used in design of state controllers in robotics. These measurements are often biased and contain noise. Many such systems usually evolve on matrix Lie groups. In this paper, we propose a globally exponentially convergent observer for systems evolving on matrix Lie groups with bounded velocity. The design of observers on the Lie group prohibits continuous globally convergent observers, which we sidestep by designing the observer in the ambient Euclidean space of the group and show exponential convergence of the observer to the state of the system. The performance of the observer is shown using an example of the rigid body rotation and translation system evolving on the special Euclidean group. We also compare the proposed observer with an observer present in the literature and show the improvements afforded by our observer.