  Motivated by attitude control and attitude estimation problems for a rigid
body, computational methods are proposed to propagate uncertainties in the
angular velocity and the attitude. The nonlinear attitude flow is determined by
Euler-Poincar\'e equations that describe the rotational dynamics of the rigid
body acting under the influence of an attitude dependent potential and by a
reconstruction equation that describes the kinematics expressed in terms of an
orthogonal matrix representing the rigid body attitude. Uncertainties in the
angular velocity and attitude are described in terms of ellipsoidal sets that
are propagated through this highly nonlinear attitude flow. Computational
methods are proposed, one method based on a local linearization of the attitude
flow and two methods based on propagation of a small (unscented) sample
selected from the initial uncertainty ellipsoid. Each of these computational
methods is constructed using the Lie group variational integrator algorithm,
viewed as a discretization of the attitude flow dynamics. Computational results
are obtained that indicate (1) the strongly nonlinear attitude flow
characteristics and (2) the limitations of each of these methods, and indeed
any method, in providing effective global bounds on the nonlinear attitude
flow.
