In this paper, a constrained nonlinear predictive control scheme is proposed for a class of under-actuated nonholonomic systems. The scheme is based on fast generation of steering trajectories that inherently fulfill the contraints while showing a ""{\em translatability}"" property which is generally needed to derive stability results in receding-horizon schemes. The corresponding open-loop optimization problem can be solved very efficiently with a fixed and deterministic upper bound on the execution time making possible a real-time implementation on fast systems. The whole framework is shown to hold for the well known challenging problem of a snake board constrained stabilization. Illustrative simulations are proposed to assess the efficiency of the proposed solution.