tr_3dq function solves asymmetric and symmetric 3D similarity transformation based on quaternions.

The transformation parameters- 3 translations, 3 rotation angles and one scale factor- between two Cartesian coordinate systems are estimated with least-squares and total least-squares for asymmetric and symmetric cases, respectively. It works for any rotation angle, converges without any condition error (if the constellation of the reference points is proper), and does not need good starting values of the transformation parameters to initialize the iterative estimation.

Please refer to the following article for more information:

Uygur SO, Aydin C, Akyilmaz O, (2020), Evaluating Euler rotation angles from 3D Coordinate transformations based on quaternions, Journal of Spatial Science, https://doi.org/10.1080/14498596.2020.1776170