Equivariant Systems Theory and Observer Design for Second Order Kinematic Systems on Matrix Lie Groups

TL;DR

This paper develops an equivariant systems theory and observer design for second order kinematic systems on matrix Lie groups, leveraging symmetry and group actions to improve observer performance, demonstrated via hovercraft simulation.

Contribution

It introduces a novel equivariant framework and observer design methodology for second order kinematic systems on matrix Lie groups, utilizing group actions and symmetry.

Findings

Observer performs well in hovercraft simulation

Group actions ensure equivariance of second order kinematics

Framework generalizes to systems on matrix Lie groups

Abstract

This paper presents the equivariant systems theory and observer design for second order kinematic systems on matrix Lie groups. The state of a second order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of the group with the inputs lying in the tangent of the tangent bundle known as the double tangent bundle. We provide a simple parameterization of both the tangent bundle state space and the input space (the fiber space of the double tangent bundle) and then introduce a semi-direct product group and group actions onto both the state and input spaces. We show that with the proposed group actions the second order kinematics are equivariant. An equivariant lift of the kinematics onto the symmetry group is defined and used to design a nonlinear observer on the lifted state space using nonlinear constructive design techniques. A simple hovercraft simulation verifies the performance of our observer.