Equivariant Observers for Second-Order Systems on Matrix Lie Groups

This article 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 semidirect 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 specialized to kinematics on groups that themselves admit a semidirect 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.