  The trajectory tracking problem is a fundamental control task in the study of
mechanical systems. A key construction in tracking control is the error or
difference between an actual and desired trajectory. This construction also
lies at the heart of observer design and recent advances in the study of
equivariant systems have provided a template for global error construction that
exploits the symmetry structure of a group action if such a structure exists.
Hamiltonian systems are posed on the cotangent bundle of configuration space of
a mechanical system and symmetries for the full cotangent bundle are not
commonly used in geometric control theory. In this paper, we propose a group
structure on the cotangent bundle of a Lie group and leverage this to define
momentum and configuration errors for trajectory tracking drawing on recent
work on equivariant observer design. We show that this error definition leads
to error dynamics that are themselves ``Euler-Poincare like'' and use these to
derive simple, almost global trajectory tracking control for fully-actuated
Euler-Poincare systems on a Lie group state space.
