Double rotor parameterisation

Assuming the axes are orthogonal for the model, the size of the set of non-redundant parameters is 15. To eliminate the redundant parameters, the geometry of the system can be used to construct a 3D eigenframe of the motion consisting of three Euler angles:

x-axis
This axis of the eigensystem can be defined as a vector parallel to the 1 rotor axis.
y-axis
This axis of the eigensystem can be defined as a vector parallel to the 2 rotor axis.
z-axis
This can be defined as a vector parallel to the line of shortest distance connecting the two rotor axes. As x and y are orthogonal by definition of the model, the line of shortest distance will be orthogonal to both x and y.

The two pivot points defining the position in space of the two rotor axes define the system. Using the above eigenframe, these can be parameterised using only four parameters:

1 pivot point
This is defined using three coordinates and is located at the intersection of the 1 rotor axis and the line of shortest distance between the axes.
2 pivot point
This is defined as the intersection of the 2 rotor axis and the line of shortest distance. Using the z-axis of the eigenframe and the 1 pivot, the 3D position can be defined as a simple displacement, pd.

The set of all parameters of the system is therefore

where Pi are the average domain position translations and rotations, Ei are the eigenframe Euler angles, pi are the coordinates of the 1 pivot point, pd is the displacement for the 2 pivot point, and σmax, i are the two torsion half-angles of the rotors.

The relax user manual (PDF), created 2020-08-26.