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Starting with a direction cosine matrix (DCM), we need to determine the three Euler angles. The connection is made by comparing the DCM elements with the combined Euler 313 sequence. It is important to note that the 12 combinations of Euler angles for a given sequence can be found from a given DCM. The DCM matrix is
![$\displaystyle DCM = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \ A_{21} & A_{22} & A_{23} \ A_{31} & A_{32} & A_{33} \end{array} \right]$](http://images.physicslibrary.org/cache/objects/82/l2h/img1.png "$\displaystyle DCM = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \ A_{21} & A_{22} & A_{23} \ A_{31} & A_{32} & A_{33} \end{array} \right]$") |
(1) |
The Euler 313 sequence is
![$\displaystyle R_3(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi... ...\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right]$](http://images.physicslibrary.org/cache/objects/82/l2h/img2.png "$\displaystyle R_3(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi... ...\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right]$") |
(2) |
If we examine the element in coloum 3 row 3, then by inspection
Solving for  yields
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(3) |
Care must now be taken when evaluating the inverse cosine. It is a multivalued function. Analytically, the convention is to choose the principle value such that
If a numerical program is used, a function acos() usually does this for us. The next step is to analyze the ratio
 . Using these values from the Euler sequence we get
Rearranging and using the tangent yields
Solving the quadrant ambiquity caused by the inverse tangent is done by examining the signs of the numerator and denominator. Denoting y as the numerator and x as the denominator, then the quadrant is chosen by:
quadrant 1
![$\left[ \begin{array}{cc} y & x \ + & + \ \end{array} \right] $](http://images.physicslibrary.org/cache/objects/82/l2h/img10.png "$\left[ \begin{array}{cc} y & x \ + & + \ \end{array} \right] $")
so if x and y are both positive, then
 . Similarily for the other quadrant possibilites
quadrant 2
![$\left[ \begin{array}{cc} y & x \ + & - \ \end{array} \right] $](http://images.physicslibrary.org/cache/objects/82/l2h/img12.png "$\left[ \begin{array}{cc} y & x \ + & - \ \end{array} \right] $")
quadrant 3
![$\left[ \begin{array}{cc} y & x \ - & - \ \end{array} \right] $](http://images.physicslibrary.org/cache/objects/82/l2h/img13.png "$\left[ \begin{array}{cc} y & x \ - & - \ \end{array} \right] $")
quadrant 4
![$\left[ \begin{array}{cc} y & x \ - & + \ \end{array} \right] $](http://images.physicslibrary.org/cache/objects/82/l2h/img14.png "$\left[ \begin{array}{cc} y & x \ - & + \ \end{array} \right] $")
Ofcourse, it is much simplier to use a calculator or numerical program that uses the atan2() function which will choose the principle value. In a similar fashion we can find the final Euler angle by looking at the ratio
 which gives the relation
Canceling terms and rearrangeing gives us
which uses the same method to resolove quadrant ambiguity as above. To summarize, we will give the formulas for the conversion from a direction cosine matrix to the Euler 313 angles in Matlab syntax. Be careful how you implement this in other numerical programs. In Matlab it goes atan2(y,x) and in Mathematica it is ArcTan[x, y].
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