In this article, we propose a general symbolic dynamic modelling framework devoted to Mobile Multibody Systems subject to hard persistent contacts. In particular, all rigid planar and spatial wheeled vehicles belong to this class of systems. To illustrate the approach we apply it to a realistic model of the three dimensional bicycle. Though being a familiar object for everybody, deriving the fully nonlinear dynamics of this system in a closed symbolic form is far from being trivial. Using a Newton-Euler algorithm coupled to a projective approach based on an explicit model of the contacts, the approach is successfully applied to the simulation of a free bicycle. It shows how the passive asymptotic stabilisation of the bicycle can be naturally ensured when it is thrown with sufficient initial velocities.