  In this paper, we study an affine connection approach to realizing
nonholonomic mechanical systems mediated by viscous friction forces with large
coefficients, viewed as a singular perturbation of the nonholonomic system. We
show that the associated slow manifold is represented coordinate-free as the
image of a section over the nonholonomic distribution. We propose a novel
invariance condition based on covariant derivatives and prove that this
condition is equivalent to the classical invariance condition based on time
derivatives. Accordingly, we propose a novel recursive procedure to approximate
the slow manifold based on the covariant derivatives of a formal power series
expansion of the section. Using this recurrence relation, we derive, up to
second order, approximations of the slip velocities residing in the slow
manifold, as well as the associated approximated dynamics up to first order.
Lastly, we illustrate our approach with a case study of a vertical rolling
disk.
