22. Earth Orientation Quaternion

In Švehla 2006, it was proposed for the first time to represent Earth orientation and rotation by means of an Earth Orientation Quaternion (EOQ). Quaternions are a very practical way to represent the Earth’s orientation parameters (EOPs), because the transformation between the terrestrial and the inertial system can be performed without calculating rotation matrices. Most importantly, the use of EOPs stored in the form of a quaternion avoids the use of the latest models and standards available from the IERS Conventions, as in the case of the EOP/ERP parameters provided by IGS and IERS. In this way, information about the Earth’s rotation/orientation is straightforward and the transformation can be performed much in the same way as for satellite attitude. This idea that was originally presented in Švehla (2006), was included in the recommendations of the Workshop on Precise Orbit Determination for the future ESA Earth observation missions, held at ESA/ESTEC in 2007 (Švehla 2007c). Following this recommendation, the ESA Core Mission GOCE provides Earth Orientation Quaternions as a separate product accompanying the kinematic and reduced-dynamic orbit. The sampling rate of Earth Orientation Quaternions, as provided in the scope of the GOCE mission.

The four Euler symmetric parameters written in the form of a quaternion are a minimal set of parameters for defining non-singular mapping to the corresponding rotation matrix. Besides their symmetrical properties, modeling finite rotations using quaternions has many advantages compared to using Euler angles since any interpolation or integration can be performed on the sphere, preserving the orthonormality of the rotation transformation (Švehla 2006).

Hamilton or quaternion algebra avoids the use of a rotation matrix and any sequence of successive rotations can be represented very elegantly by the quaternion operator. This also holds for the derivatives of the successive rotations and the treatment of the kinematic equation of rotation. We show how to interpolate and extrapolate the Earth orientation quaternions preserving the orthonormality of the transformation. We introduce a transition quaternion derived from the kinematic equation of rotation.

In the field of numerical solutions of ordinary differential equations, geometric integration is defined as a numerical method that preserves the geometric properties of the exact flow of a differential equation. Therefore, when talking about integrating quaternions on the sphere and preserving orthonormality of the rotation transformation at the same time, we are actually talking about using geometric integration.