Comparison of body rotations using Euler angles and quaternions



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Many comparisons between Euler angle and quaternion representations of body rotations have been done in the past, but what are additionally investigated in this article are the handling of the generally important non-zero starting conditions, and a demonstration of the correctness of the Euler to quaternion and quaternion to Euler conversions despite giving remarkably different Euler angle and quaternion values for the same set of starting values. Two Euler configurations are also investigated and compared to demonstrate that the findings are generally valid. The first is the often-used yaw-pitch-roll (inner) and the second the less known roll-yaw-pitch (inner) configuration. Some of the test scenarios were chosen such that both the Euler angle configurations had to move through their gimbal lock positions. All transformation matrices, using two sets of Euler angles, and three sets of quaternion values, gave the same projections in an inertial axes system of a point fixed to the rotating body doing a chosen set of rotations, but there are accuracy differences. Therefore there is an optimal solution for each Euler configuration. The findings of this paper are also important in tracking loops where these types of transformations are used in the relative geometry calculations where the angles, for example, between the target and the camera sightline, must be determined accurately.