Geometric Integration of Quaternions

This paper employs a geometric integration algorithm to propagate the quaternion kinematics in order to preserve the unit norm. While many studies have focused on this aspect specifically while considering other systems, this work has the additional objective of studying result accuracy. Many applications including space object tracking and asteroid cataloguing require state propagation over longer time intervals where only sparse observation data is available. It is during these intervals that the quaternion norm drifts and accuracy decreases as a result of error accumulation. Quaternion trajectories obtained using third and fourth order Crouch-Grossman Lie group methods are compared with those calculated using the classical third and fourth order Runge-Kutta algorithms using different time steps. Results show that the use of the Crouch-Grossman Lie group method better preserves the quaternion unit norm for the larger time steps considered. It is also found that the fourth order Crouch-Grossman algorithm is more accurate than its Runge-Kutta counterpart except for the smallest time step used.