Representation of the rotation parameter estimation errors in the Helmert transformation model

Abstract

In the Helmert transformation model, the rotation is more difficult to be treated in terms of representation, estimation, and error analysis. First, two classes of representations of the rotation, i.e. the redundant class including the direction cosine matrix and the unit quaternion, and the minimum class including the rotation vector, the Gibbs vector, the modified Rodrigues parameters, and the Euler angles, are reviewed. It is concluded that in general the redundant class should be preferred as they are transcendental-function-free, singularity-free, and discontinuity-free. Second, two classes of estimation errors, i.e. the additive and the multiplicative errors, are defined and compared in detail. While the multiplicative errors are more convenient, the relationship among different representations and the relationship with their additive counterparts are also explored from first principle. It can be seen as a review paper; however, the content concerning the relationship between the additive and the multiplicative errors is believed new.