  In this paper, we will discuss new developments regarding the Geometric
Nonholonomic Integrator (GNI) [23, 24]. GNI is a discretization scheme adapted
to nonholonomic mechanical systems through a discrete geometric approach. This
method was designed to account for some of the special geometric structures
associated to a nonholonomic motion, like preservation of energy, preservation
of constraints or the nonholonomic momentum equation. First, we study the GNI
versions of the symplectic-Euler methods, paying special attention to their
convergence behavior. Then, we construct an extension of the GNI in the case of
affine constraints. Finally, we generalize the proposed method to nonholonomic
reduced systems, an important subclass of examples in nonholonomic dynamics. We
illustrate the behavior of the proposed method with the example of the
Chaplygin sphere, which accounts for the last two features, namely it is both a
reduced and an affine system.
