An approach for introduction of transverse coordinates in a vicinity of a periodic trajectory is presented. The approach allows finding by numerical integration periodic normalized mutually-orthogonal vector-functions that form a continuously differentiable basis on moving Poincare sections for a given periodic solution of a nonlinear dynamical system. The found moving frame is used to define new local (transverse) coordinates for an associated affine nonlinear control system in a neighborhood of the trajectory, and to proceed with orbital stability analysis and/or synthesis of a stabilizing feedback control law. As a demonstrating example of the approach, the problem of orbital stabilization of a trajectory of a multibody car system is considered. The results of computer simulations of the system are presented.