Two frameworks based on Model Predictive Control (MPC) for obstacle avoidance with autonomous vehicles are presented. A given trajectory represents the driver intent. An MPC has to safely avoid obstacles on the road while trying to track the desired trajectory by controlling front steering angle and differential braking. We present two different approaches to this problem. The first approach solves a single nonlinear MPC problem. The second approach uses a hierarchical scheme. At the high-level, a trajectory is computed on-line, in a receding horizon fashion, based on a simplified point-mass vehicle model in order to avoid an obstacle. At the low-level an MPC controller computes the vehicle inputs in order to best follow the high level trajectory based on a nonlinear vehicle model. This article presents the design and comparison of both approaches, the method for implementing them, and successful experimental results on icy roads.