On higher order integration of angular velocities using quaternions

• We study the reconstruction of rotational quaternions from known angular velocities.
• Analytical solution is found using the theory of analytic differential systems.
• Numerical methods based on Taylor series and various maps from pure to unit quaternions are presented.
• A special care is taken in deriving the higher order methods.
• Two novel computationally efficient maps from pure to unit quaternions are described.

In the present paper the integration of angular velocities is studied. Both exact and approximate results are expressed in terms of rotational quaternions. Analytical solution is found using the theory of analytic differential systems. This exact solution serves as a suitable basis for derivation of various numerical methods. Approximative approaches based on Taylor series and several maps from pure to unit quaternions are presented. A special care is taken in describing the higher order approximations. The computational performance and comparison of numerical methods is demonstrated by examples.