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.

Keywords:

• Helmert transformation
• Rotation
• Direction cosine matrix
• Quaternion
• Rotation vector
• Gibbs vector
• Modified Rodrigues parameters
• Euler angles
• Multiplicative errors
• Additive errors

Acknowledgements

We would like to dedicate this work to our teacher Professor Yuanxi Yang on his 60th birthday. We are grateful to the two anonymous reviewers for their insightful comments which improved the manuscript significantly.

Additional information

Funding

This work is supported by the Fundamental Research Funds for the Central Universities (2015QNA31).