Using Rotations to Build Aerospace Coordinate Systems - Defence ...

DSTO–TN–0640xyz Angle–Axis to EulersIf the xyz rotation order of (B1) is used, a single rotation matrix R can be converted tothree rotations using the Euler angles α, β, γ, where s can be chosen as either +1 or −1:sinα =sR 32√1 − R231, cos α = sR 33√1 − R231sinβ = −R 31 ,√cos β = s 1 − R312sinγ =sR 21√1 − R231, cos γ = sR 11√1 − R231.(B6)The value of s is immaterial in the sense that either choice will give three angles thathave the same effect as the original rotation. Care is needed to avoid oversimplifying theseequations: we must remember that (B6) does not imply that e.g. α = tan −1 (R 32 /R 33 ),since the tan −1 function only gives angles in the range −90 ◦ → +90 ◦ . Nor does it implythat β must equal − sin −1 R 31 , because of a similar restriction on the sin −1 function.These equations can be implemented in Matlab using the code:alpha = atan2(s*R(3,2), s*R(3,3));beta = atan2(-R(3,1), s*sqrt(1-R(3,1)^2));gamma = atan2(s*R(2,1), s*R(1,1));24