  In this paper, a risk-aware motion control scheme is considered for mobile
robots to avoid randomly moving obstacles when the true probability
distribution of uncertainty is unknown. We propose a novel model predictive
control (MPC) method for limiting the risk of unsafety even when the true
distribution of the obstacles' movements deviates, within an ambiguity set,
from the empirical distribution obtained using a limited amount of sample data.
By choosing the ambiguity set as a statistical ball with its radius measured by
the Wasserstein metric, we achieve a probabilistic guarantee of the
out-of-sample risk, evaluated using new sample data generated independently of
the training data. To resolve the infinite-dimensionality issue inherent in the
distributionally robust MPC problem, we reformulate it as a finite-dimensional
nonlinear program using modern distributionally robust optimization techniques
based on the Kantorovich duality principle. To find a globally optimal solution
in the case of affine dynamics and output equations, a spatial branch-and-bound
algorithm is designed using McCormick relaxation. The performance of the
proposed method is demonstrated and analyzed through simulation studies using a
nonlinear car-like vehicle model and a linearized quadrotor model.
