Strapdown Attitude Computation: Functional Iterative Integration versus Taylor Series Expansion

TL;DR

This paper compares the Taylor series expansion and functional iterative integration methods for solving attitude kinematic equations in strapdown inertial navigation, highlighting their accuracy, stability, and applicability under different motion frequencies.

Contribution

It extends mainstream algorithms by incorporating exact rotation vectors and high-order derivatives, and evaluates the performance of both approaches with numerical simulations.

Findings

Chebyshev polynomial-based iterative method outperforms others at high frequencies.

All algorithms have similar accuracy at low frequencies when using the same sample size.

Chebyshev polynomial provides superior numerical stability and functional representation.

Abstract

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.