In this paper, we present a robust and adaptive model predictive control (MPC) framework for uncertain nonlinear systems affected by bounded disturbances and unmodeled nonlinearities. We use Gaussian Processes (GPs) to learn the uncertain dynamics based on noisy measurements, including those collected during system operation. As a key contribution, we derive robust predictions for GP models using contraction metrics, which are incorporated in the MPC formulation. The proposed design guarantees recursive feasibility, robust constraint satisfaction and convergence to a reference state, with high probability. We provide a numerical example of a planar quadrotor subject to difficult-to-model ground effects, which highlights significant improvements achieved through the proposed robust prediction method and through online learning.

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