The rotation matrix
R_AB is created based on 3 angles
about new axes (intrinsic) in the order z-y-x.
The angles (called Euler angles or Tait–Bryan angles) are defined by the following procedure
of successive rotations:
Given two arbitrary coordinate frames A and B, consider a temporary frame T that initially coincides with A. In order to make T align with B, we first rotate T an angle z about its z-axis (common axis for both A and T).
Secondly, T is rotated an angle y about the NEW y-axis of T.
Finally, T is rotated an angle x about its NEWEST x-axis. The final orientation of T now coincides with the orientation of B.
The signs of the angles are given by the directions of the axes and the right hand rule. Note that if A is a north-east-down frame and B is a body frame, we have that z=yaw, y=pitch and x=roll.
zyx2R(z, y, x)
Angle of rotation about new z axis
Angle of rotation about new y axis
Angle of rotation about new x axis
3x3 rotation matrix R_AB (direction cosine matrix) such that the relation between a vector v decomposed in A and B is given by: v_A = R_AB * v_B
Kenneth Gade A Nonsingular Horizontal Position Representation. The Journal of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.
#> [,1] [,2] [,3] #> [1,] 0.8137977 -0.4409696 0.37852231 #> [2,] 0.4698463 0.8825641 0.01802831 #> [3,] -0.3420201 0.1631759 0.92541658