  This paper introduces a global uncertainty propagation scheme for rigid body
dynamics, through a combination of numerical parametric uncertainty techniques,
noncommutative harmonic analysis, and geometric numerical integration. This
method is distinguished from prior approaches, as it allows one to consider
probability densities that are global, and are not supported on only a single
coordinate chart on the manifold. The use of Lie group variational integrators,
that are symplectic and stay on the Lie group, as the underlying numerical
propagator ensures that the advected probability densities respect the
geometric properties of uncertainty propagation in Hamiltonian systems, which
arise as consequence of the Gromov nonsqueezing theorem from symplectic
geometry. We also describe how the global uncertainty propagation scheme can be
applied to the problem of global attitude estimation.
