Equivariant Systems Theory and Observer Design

TL;DR

This paper develops the foundational theory for designing observers and filters for equivariant systems on homogeneous spaces, expanding the scope of symmetry-based methods in robotics and vision applications.

Contribution

It provides a comprehensive theoretical framework for equivariant systems on homogeneous spaces, enabling advanced observer design beyond Lie-group models.

Findings

Framework for observer design on homogeneous spaces

Extension of symmetry-based methods to new manifold types

Foundation for future practical algorithms

Abstract

A wide range of system models in modern robotics and avionics applications admit natural symmetries. Such systems are termed equivariant and the structure provided by the symmetry is a powerful tool in the design of observers. Significant progress has been made in the last ten years in the design of filters and observers for attitude and pose estimation, tracking of homographies, and velocity aided attitude estimation, by exploiting their inherent Lie-group state-space structure. However, little work has been done for systems on homogeneous spaces, that is systems on manifolds on which a Lie-group acts rather than systems on the Lie-group itself. Recent research in robotic vision has discovered symmetries and equivariant structure on homogeneous spaces for a host of problems including the key problems of visual odometry and visual simultaneous localisation and mapping. These discoveries motivate a deeper look at the structure of equivariant systems on homogeneous spaces. This paper provides a comprehensive development of the foundation theory required to undertake observer and filter design for such systems.