  Rigid motion computation or estimation is a cornerstone in numerous fields.
Attitude computation can be achieved by integrating the angular velocity
measured by gyroscopes, the accuracy of which is crucially important for the
dead-reckoning inertial navigation. The state-of-the-art attitude algorithms
have unexceptionally relied on the simplified differential equation of the
rotation vector to obtain the attitude. This paper proposes a Functional
Iteration technique with the Rodrigues vector (named the RodFIter method) to
analytically reconstruct the attitude from gyroscope measurements. The RodFIter
method is provably exact in reconstructing the incremental attitude as long as
the angular velocity is exact. Notably, the Rodrigues vector is analytically
obtained and can be used to update the attitude over the considered time
interval. The proposed method gives birth to an ultimate attitude algorithm
scheme that can be naturally extended to the general rigid motion computation.
It is extensively evaluated under the attitude coning motion and compares
favorably in accuracy with the mainstream attitude algorithms. This work is
believed having eliminated the long-standing theoretical barrier in exact
motion integration from inertial measurements.
