  This paper explores transverse coordinates for the purpose of orbitally
stabilizing periodic motions of nonlinear, control-affine dynamical systems. It
is shown that the dynamics of any (minimal or excessive) set of transverse
coordinates, which are defined in terms of a particular parameterization of the
motion and a strictly state-dependent projection operator recovering the
parameterizing variable, admits a (transverse) linearization along the target
motion, with explicit expressions stated. Special focus is then placed on a
generic excessive set of orthogonal coordinates, revealing a certain limitation
of the "excessive" transverse linearization for the purpose of control design.
To overcome this limitation, a linear comparison system is introduced, and
conditions are stated for when the asymptotic stability of its origin
corresponds to the asymptotic stability of the origin of linearized transverse
dynamics. This allows for the construction of feedback controllers utilizing
this comparison system which, when implemented on the dynamical system, renders
the desired motion asymptotically stable in the orbital sense.
