Euler Angles
According to Euler's rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices B, C, and D, then a general rotation can be written as A=DCB.
Rotations around the z�x�z axes.
cos sin 0
sin cos 0
0 0 1
B
φ φφ φ
− =
1 0 0
0 cos sin
0 sin cos
C θ θθ θ
= −
cos sin 0
sin cos 0
0 0 1
D
ψ ψψ ψ
− =
cos( ) sin( ) sin( ) cos( )
cos( )
cos( ) cos( ) sin( )
cos(
- - -
+ -
cos( ) sin( ) cos( ) sin( ) sin( )
sin( ) cos( ) sin( ) cos( )) cos( ) sin( )
sin( ) sin( ) cos
sin( ) sin( ) cos( )
sin( ) cos( )
o (- )+ c s
CBDA
θ θ θθ θ θ
θ
φ φ φ φφ φ φ φ
φ φ
ψ ψ
θ
ψ ψ ψψ ψ ψ ψ ψ
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅
=
=( )θ
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