  Generalized Polynomial Chaos (gPC) theory has been widely used for
representing parametric uncertainty in a system, thanks to its ability to
propagate uncertainty evolution. In an optimal control context, gPC can be
combined with several optimization techniques to achieve a control policy that
handles effectively this type of uncertainty. Such a suitable method is
Differential Dynamic Programming (DDP), leading to an algorithm that inherits
the scalability to high-dimensional systems and fast convergence nature of the
latter. In this paper, we expand this combination aiming to acquire
probabilistic guarantees on the satisfaction of constraints. In particular, we
exploit the ability of gPC to express higher order moments of the uncertainty
distribution - without any Gaussianity assumption - and we incorporate chance
constraints that lead to expressions involving the state variance. Furthermore,
we demonstrate that by implementing our algorithm in a receding horizon
fashion, we compute control policies that effectively reduce the accumulation
of uncertainty on the trajectory. The applicability of our method is verified
through simulation results on a differential wheeled robot and a quadrotor that
perform obstacle avoidance tasks.
