Equivariant Observers for Second-Order Systems on Matrix Lie Groups

Abstract

This paper develops an equivariant symmetry for second order kinematic systems on matrix Lie groups and uses this symmetry for observer design. 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 derived and used to design nonlinear observers on the lifted state space using nonlinear constructive design techniques. The observer design is specialised to kinematics on groups that themselves admit a semi-direct product structure and include applications in rigid-body motion amongst others. A simulation based on an ideal hovercraft model verifies the performance of the proposed observer architecture.