In this paper, the accurate control of mobile robots is investigated in the framework of generic edge following. It proposes a new predictive control approach, based on the minimization of the lateral error along a distance of prediction. This permits to consider a distance of convergence independently from the yaw dynamics and the longitudinal velocity, allowing to achieve harsh maneuvers, with respect to potentially kinetically unachievable paths. The proposed algorithm is generic and permits to address the control of different kinds of robots (skid-steering, car-like, etc.) in a common framework and to consider independently the speed regulation. The control proposed here allows an accurate and reactive path tracking, even if the environment is complex and narrow. The efficiency of the approach is investigated through full-scale experiments in various conditions.