  Learning-based control aims to construct models of a system to use for
planning or trajectory optimization, e.g. in model-based reinforcement
learning. In order to obtain guarantees of safety in this context, uncertainty
must be accurately quantified. This uncertainty may come from errors in
learning (due to a lack of data, for example), or may be inherent to the
system. Propagating uncertainty forward in learned dynamics models is a
difficult problem. In this work we use deep learning to obtain expressive and
flexible models of how distributions of trajectories behave, which we then use
for nonlinear Model Predictive Control (MPC). We introduce a deep quantile
regression framework for control that enforces probabilistic quantile bounds
and quantifies epistemic uncertainty. Using our method we explore three
different approaches for learning tubes that contain the possible trajectories
of the system, and demonstrate how to use each of them in a Tube MPC scheme. We
prove these schemes are recursively feasible and satisfy constraints with a
desired margin of probability. We present experiments in simulation on a
nonlinear quadrotor system, demonstrating the practical efficacy of these
ideas.
