  Trajectory tracking for the kinematic unicycle has been heavily studied for
several decades. The unicycle admits a natural $\SE(2)$ symmetry, a key
structure exploited in many of the most successful nonlinear controllers in the
literature. To the author's knowledge however, all prior work has used a
body-fixed, or left-invariant, group error formulation for the study of the
tracking problem. In this paper, we consider the spatial, or right-invariant,
group error in the design of a tracking controller for the kinematic unicycle.
We provide a physical interpretation of the right-invariant error and go on to
show that the associated error dynamics are drift-free, a property that is not
true for the body-fixed error. We exploit this property to propose a simple
nonlinear control scheme for the kinematic unicycle and prove almost-global
asymptotic stability of this control scheme for a class of persistently
exciting trajectories. We also verify performance of this control scheme in
simulation for an example trajectory.
