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The 3 angles z, y, x about new axes (intrinsic) in the order z-y-x are found from the rotation matrix R_AB. The angles (called Euler angles or Tait–Bryan angles) are defined by the following procedure of successive rotations:

  1. 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).

  2. 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.

  3. 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.





a 3x3 rotation matrix (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


z,y,x angles of rotation about new axes (rad)


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.


Kenneth Gade A Nonsingular Horizontal Position Representation. The Journal of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.

See also


zyx2R(rad(1), rad(-2), rad(-3))
#>            [,1]        [,2]        [,3]
#> [1,] 0.99923861 -0.01560227 -0.03575975
#> [2,] 0.01744177  0.99850932  0.05171974
#> [3,] 0.03489950 -0.05230407  0.99802120