Gradient-Like Observers on Semidirect Products

This paper proposes a simple full state observer design for invariant state space systems where the state is evolving on a semidirect product of connected, finite-dimensional Lie groups. The design is based on a pair of cost functions defined on the Lie groups and consists of a copy of the observed system and a gradient-like innovation term. Under mild conditions the observer displays almost global exponential convergence. We illustrate the construction by an application in pose estimation.