  In this paper we develop a novel, discrete-time optimal control framework for
mechanical systems with uncertain model parameters. We consider finite-horizon
problems where the performance index depends on the statistical moments of the
stochastic system. Our approach constitutes an extension of the original
Differential Dynamic Programming method and handles uncertainty through
generalized Polynomial Chaos (gPC) theory. The developed iterative scheme is
capable of controlling the probabilistic evolution of the dynamic system and
can be used in planning and control. Moreover, its scalable and fast converging
nature play a key role when dealing with complex, high-dimensional problems.
Based on Lagrangian mechanics principles, we also prove that Variational
Integrators can be designed to properly propagate and linearize the gPC
representations of stochastic, forced mechanical systems in discrete time. We
utilize this benefit to further improve the efficiency of our
trajectory-optimization methodology. Numerical simulations are included to
validate the applicability of our approach.
