  Attitude estimation is often a prerequisite for control of the attitude or
orientation of mechanical systems. Current attitude estimation algorithms use
coordinate representations for the group of rigid body orientations. All
coordinate representations of the group of orientations have associated
problems. While minimal coordinate representations exhibit kinematic
singularities for large rotations, non-minimal coordinates like quaternions
require satisfaction of extra constraints. A deterministic attitude estimation
problem for a rigid body with bounded measurement errors is considered here. An
attitude estimation algorithm that globally minimizes the attitide estimation
error, is obtained. Assuming that the initial attitude, the initial angular
velocity and measurement noise lie within given ellipsoidal bounds, an
uncertainty ellipsoid that bounds the attitude and the angular velocity of the
rigid body is obtained. The center of the uncertainty ellipsoid provides point
estimates, and the size of the uncertainty ellipsoid measures the accuracy of
the estimates. The point estimates, and the uncertainty ellipsoids are
propagated using a Lie group variational integrator, and its linearization,
respectively. The attitude estimation is optimal in the sense that the attitude
estimation error and the size of the uncertainty ellipsoid is minimized.
