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Appendix C / Axis Transforms
_ _
When referring to individual direction cosines the notation is consistent with
the FORTRAN array mapping structure.
T B
A
This is equivalent to,
T B
A
≡
: =
⎡ T
⎢
⎢
⎢ T
⎢
⎢
⎢
⎣ T
⎡ T
⎢
⎢
⎢ T
⎢
⎢
⎢
⎣ T
( 1 ) , T ( 4 ) , T ( 7 )
( 2 ) , T ( 5 ) , T ( 8 )
( ) ( ) ( )⎥ ⎥⎥⎥⎥⎥
3 , T 6 , T 9
( 1,
1 ) , T ( 1,
2 ) , T ( 1,
3 )
( 2 , 1 ) , T ( 2 , 2 ) , T ( 2 , 3 )
( ) ( ) ( ) ⎥ ⎥⎥⎥⎥⎥
3 , 1 , T 3 , 2 , T 3 , 3
16-5
⎤
⎦
Equation 16.1-8
⎤
⎦
Equation 16.1-9
If the full notation becomes laborious when dealing with direction cosines it
is permissible to drop the frame reference providing that the meaning of the
transform is clearly defined. Since transformations defined by direction
cosines result in orthogonal matrices,
T
A
B
16.2 Small Angle Approximations
≡
[ ] [ ] 1
B T
B −
T ≡ T
A
A
Equation 16.1-10
When the angular rotation between frames is small the transformation
matrix reduces to a skew symmetric form using small angle approximations
highlighted by the delta notation. A transformation defined by small
angular rotations decomposes into the identity matrix and an anti-symmetric
matrix defined by an Euler triplet,
∆T
B
A
: =
B [ I − ∆E
× ]
3
A
≡
⎡ 1
⎢
⎢
⎢ − ∆Ψ
⎢
⎢
⎢ B
⎣ ∆ΘA
B
A
,
,
,
∆Ψ
1
B
A
− ∆Φ
B
A
,
,
,
− ∆Θ
∆Φ
1
B
A
B
A
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Equation 16.2-1