  In this paper, we address the problem of uncertainty propagation through
nonlinear stochastic dynamical systems. More precisely, given a discrete-time
continuous-state probabilistic nonlinear dynamical system, we aim at finding
the sequence of the moments of the probability distributions of the system
states up to any desired order over the given planning horizon. Moments of
uncertain states can be used in estimation, planning, control, and safety
analysis of stochastic dynamical systems. Existing approaches to address moment
propagation problems provide approximate descriptions of the moments and are
mainly limited to particular set of uncertainties, e.g., Gaussian disturbances.
In this paper, to describe the moments of uncertain states, we introduce
trigonometric and also mixed-trigonometric-polynomial moments. Such moments
allow us to obtain closed deterministic dynamical systems that describe the
exact time evolution of the moments of uncertain states of an important class
of autonomous and robotic systems including underwater, ground, and aerial
vehicles, robotic arms and walking robots. Such obtained deterministic
dynamical systems can be used, in a receding horizon fashion, to propagate the
uncertainties over the planning horizon in real-time. To illustrate the
performance of the proposed method, we benchmark our method against existing
approaches including linear, unscented transformation, and sampling based
uncertainty propagation methods that are widely used in estimation, prediction,
planning, and control problems.
