In normal on-road situations, autonomous vehicles will be expected to have smooth trajectories with relatively little demand on the vehicle dynamics to ensure passenger comfort and driving safety. However, the occurrence of unexpected events may require vehicles to perform aggressive maneuvers, near the limits of their dynamic capacities. In order to ensure the occupant's safety in these situations, the ability to plan controllable but near-limits trajectories will be of very high importance. One of the main issues in planning aggressive maneuvers lies in the high complexity of the vehicle dynamics near the handling limits, which effectively makes state-of-the-art methods such as Model Predictive Control difficult to use. This article studies a highly precise model of the vehicle body to derive a simpler, constrained second-order integrator dynamic model which remains precise even near the handling limits of the vehicle. Preliminary simulation results indicate that our model provides better accuracy without increasing computation time compared to a more classical kinematic bicycle model. The proposed model can find applications for contingency planning, which may require aggressive maneuvers, or for trajectory planning at high speed, for instance in racing applications.