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A general method for planning and orbitally stabilizing periodic motions for impulsive mechanical systems with underactuation one is proposed. For each such trajectory, we suggest a constructive procedure for defining a sufficient number of nontrivial quantities that vanish on the orbit. After that, we prove that these quantities constitute a possible set of transverse coordinates. Finally, we present analytical steps for computing linearization of dynamics of these coordinates along the motion. As a result, for each such planned periodic trajectory, a hybrid transverse linearization for dynamics of the system is computed in closed form. The derived impulsive linear system can be used for stability analysis and for design of exponentially orbitally stabilizing feedback controllers. A geometrical interpretation of the method is given in terms of a novel concept of a moving Poincare section. The technique is illustrated on a devil stick example.