  This paper compares two basic approaches to solving ordinary differential
equations, which form the basis for attitude computation in strapdown inertial
navigation systems, namely, the Taylor series expansion approach that was used
in its low-order form for deriving all mainstream algorithms and the functional
iterative integration approach developed recently. They are respectively
applied to solve the kinematic equations of major attitude parameters,
including the quaternion, the Rodrigues vector and the rotation vector.
Specifically, the mainstream algorithms, which have relied on the simplified
rotation vector without exception, are considerably extended by the Taylor
series expansion approach using the exact rotation vector and recursive
calculation of high-order derivatives. The functional iterative integration
approach is respectively implemented on both the normal polynomial and the
Chebyshev polynomial. Numerical results under the classical coning motion are
reported to assess all derived attitude algorithms. It is revealed that in the
relative frequency range when the coning to sampling frequency ratio is below
0.05-0.1 (depending on the chosen polynomial truncation order), all algorithms
have the same order of accuracy if the same number of samples are used to fit
the angular velocity over the iteration interval; in the range of higher
relative frequency, the group of Quat/Rod/RotFIter algorithms (by the
functional iterative integration approach combined with the Chebyshev
polynomial) perform the best in both accuracy and robustness, thanks to the
excellent numerical stability and powerful functional representation capability
of the Chebyshev polynomial.
