This paper develops a new constrained model predictive control for a dynamic path tracking of an off-road mobile robot with a double steering axle. The controller is based on a dynamic model that includes wheel-ground lateral slippage and terrain geometry parameters. It is formulated as an optimization problem that computes at each time-step the optimal front and rear steering angles required to perform a desired path, with respect to multiple constraints, essentially the steering joint limits and the tire adhesion area bounds (i.e., pseudo-sliding zone limits). The capabilities of such a path tracking controller are shown and discussed through numerical simulations and experiments on a real off-road mobile robot at different speeds.