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We present a framework for the stabilization of nonholonomic systems that do not possess special properties such as flatness or exact nilpotentizability. Our approach makes use of two tools: an iterative control scheme and a nilpotent approximation of the system dynamics. The latter is used to compute an approximate steering control which, repeatedly applied to the system, guarantees asymptotic stability with exponential convergence to any desired set point, under appropriate conditions. For illustration, we apply the proposed strategy to design a stabilizing controller for the plate-ball manipulation system, a canonical example of nonflat nonholonomic mechanism. The theoretical performance and robustness of the controller are confirmed by simulations, both in the nominal case and in the presence of a perturbation on the ball radius.