Definition of the Euler Transformation Matrices

To transform motions about the Mir body axes into the inertial reference frame, define three Euler angles , , and as arguments of three rotation matrices. These matrices will be combined into one matrix to provide for a general multiaxis rotation. The Euler matrices , and rotate from the inertial unprimed frame to the primed frame use the form where is a vector in inertial space. First rotate about the inertial space axis z to get to the first primed frame.

At this point . Now we rotate about the primed frame y axis, by .

At this point  . Now rotate the double primed frame axis about the primed frame z axis (note, not x, since this is the Euler system).

Finally . Define to be the combined solid body rotation matrix and be its inverse such that