Finding Euler/Tait Bryan Angles from Rotation Matrix

1. Any Rotation Matrix representing Rotation by an Arbirary Angle around any Arbirary Axis can be factored into 3 Elementary Rotation Matrices. Each such Elementary Rotation Matrix represents a Rotation Around One of the Coordinate Axes (\(X\), \(Y\) or \(Z\)) by some Angle.
2. The Angles of Rotation corresponding to each such Elementary Rotation Matrix is named based on the Set of Axes used for Rotations.
   
   If the Axis of Rotation for the Rotation that is Applied First is Same as Axis of Rotation for the Rotation Applied Third, then the Set of Angles in the corresponding Elementary Rotation Matrices are called the Euler Angles.
   
   If the Axis of Rotation for all the 3 Elementary Rotation Matrices are Different , then the Set of Angles in the corresponding Elementary Rotation Matrices are called the Tait-Bryan Angles.
3. The following gives the All 6 Possible Combinations of Axes of Rotation for Elementary Rotation Matrices correponding to Euler Angles
   
   \(XYX\), \(XZX\), \(YZY\), \(YXY\), \(ZXZ\), \(ZYZ\)
   
   The following gives the All 6 Possible Combinations of Axes of Rotation for Elementary Rotation Matrices correponding to Tait-Bryan Angles
   
   \(XYZ\), \(ZYX\), \(XZY\), \(YZX\), \(YXZ\), \(ZXY\)
4. For each of the above given 12 Combinations of Axes of Rotation, the Set of Euler or Tait-Bryan Rotation Angles can be Calculated in 2 ways i.e either Clockwize or Counter Clockwize.
   
   And, for each Set Counter Clockwise or Clockwise Euler or Tait-Bryan Rotation Angles , there are 2 ways to apply Rotations.
   
   The First way is to apply Rotation corresponding to 1st Axis and associated Angle First, followed by 2nd Axis and associated Angle and then the 3rd Axis and associated Angle. Any Rotation Matrix Configuration obtained using this method is called a Pre-Rotation Matrix.
   
   The Second way is to apply Rotation corresponding to 3rd Axis and associated Angle First, followed by 2nd Axis and associated Angle and then the 1st Axis and associated Angle. Any Rotation Matrix Configuration obtained using this method is called a Post-Rotation Matrix.
   
   Hence, there are a total of \(48\hspace{1mm}(12 \times 2 \times 2)\) different types of Configurations Possible for Euler/Tait-Bryan Rotation Matrices.
5. For each of the 48 Configurations of Euler/Tait-Bryan Rotation Matrices, atleast 2 Sets of Euler/Tait Bryan Angles can be Calculated. Hence, a total of \(96\hspace{1mm}(48 \times 2)\) different Sets of Euler/Tait-Bryan Angles can be Calculated.
   
   However for any Combination of Axes of Rotation corresponding to Euler Angles, the Set of Angles obtained from Pre-Rotation Matrices and Post-Rotation Matrices are same.
   
   And for Combination of Axes of Rotation corresponding to Tait-Bryan Angles, the Set of Angles obtained from Pre-Rotation Matrices of \(XYZ\) configuration is same as Set of Angles obtained from Post-Rotation Matrices \(ZYX\) configuration, and vice versa (the same applies for the pairs \(XZY\), \(YZX\) and \(YXZ\), \(ZXY\)).
   
   Hence, for any given 3D Rotation Matrix, a net total of \(48\) different Sets of Euler/Tait-Bryan Angles can be Calculated.