Optimal Combination of Orientation Measurements Under Angle, Axis and Chord Metrics

Orientation measurements of attitudes estimate relative rotations of objects. The non-commutative algebra of rotations makes transference of techniques inspired by the usual vector-based approaches for translations non-trivial.

We treat three different metrics that may be used to compare orientations, compute the corresponding optimal averages, and relate them in a unified framework. The metrics are based on measuring differences in angular arc, axis tilt (or bivector) and rotational chord. We also compute the optimal combination of orientation estimates according to their local variances, as may be employed in a Kalman filter update step.

Our use of the geometric algebra characterization of rotations (through the bivector angle of rotors) allows us to perform computations and comparisons in a coordinate-free manner, and thus to compare and evaluate the alternative parametrizations. We briefly discuss how this subsumes and extends the traditional quaternion representation of rotations.