Total least-squares formulation of Wahba´s problem

The Wahba´s problem is widely recognised as a least-square problem in three-axis attitude estimation from vector measurements. However, both the observation and the reference vectors can be corrupted by errors, so a total least-square problem considering both errors should be formulated to be rigorous. It is found that for errors with heteroscedastic and isotropic covariance structures, the total least-square problem degenerates to be the Wahba´s problem. A method for solving this problem is presented. Errors corrupting both observation and reference vectors can be equivalent to pseudo errors corrupting only observation vectors in the case of isotropic covariance structures, so the Wahba´s problem can also be formulated as a least-square problem adjusting these pseudo errors in a somewhat simpler manner.