  Accurate tracking of planned trajectories in the presence of perturbations is
an important problem in control and robotics. Symmetry is a fundamental
mathematical feature of many dynamical systems and exploiting this property
offers the potential of improved tracking performance. In this paper, we
investigate the tracking problem for systems on homogeneous spaces, manifolds
which admit symmetries with transitive group actions. We show that there is
natural manner to lift any desired trajectory of such a system to a lifted
trajectory on the symmetry group. This construction allows us to define a
global tracking error and apply LQR design to obtain an approximately optimal
control in a single coordinate chart. The resulting control is then applied to
the original plant and shown to yield excellent tracking performance. We term
the resulting design methodology the Equivariant Regulator (EqR). We provide an
example system posed on a homogeneous space, derive the trajectory
linearisation in error coordinates and demonstrate the effectiveness of EqR
compared to standard approaches in simulation.
