This article presents a robust and almost global feedback attitude tracking control scheme in conjunction with a robust deterministic estimator that constructs state estimates for feedback. These control and the estimation schemes use the natural and globally unique representation of rigid body attitude provided by rotation matrices. Attitude and angular velocity state estimates are constructed from discrete (possibly multi-rate) measurements and a deterministic filtering scheme. Computational implementation of this estimator-based tracking scheme is carried out with a Lie group variational integrator, which preserves the rotation matrix structure without need for reprojection. Numerical simulation results obtained using this integrator show the robust and almost global tracking properties of this scheme. We also compare the performance of this attitude tracking control scheme with a quaternion observer-based feedback attitude tracking control scheme that has appeared in recent literature. Numerical simulation results are obtained for tracking an oscillating angular velocity spin maneuver with both the attitude tracking schemes for a satellite in circular Earth orbit. These results demonstrate the advantages of our scheme in attitude tracking of continuous rotational motions.