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Appendix I / Utilities / Axis Transforms
_ _
T
B
A
DC_TO_YPR
Φ
B
A
: =
tan
B B
E B
( T , E ) ≡ ϕ ( T )
−1
22.4-7
A
A
DC
B
B
( T ( 8 ) T ( 9 ) )
A
B
−1
B
( 7 ) ≤ 0.
999999 ⇒ Θ : = sin − T ( 7 )
Θ
B
A
: =
Ψ
B
A
T
sin
B
A
−1
: =
( 7 )
>
A
0.
999999
A
⇒
A
Equation 22.4-41
Equation 22.4-42
( )
B
( − 0.
999999 ⋅ sign ( 1 , T ( 7 ) ) )
tan
−1
A
B
B
( T ( 4 ) T ( 1 ) )
22.4.10 Cartesian to Direction Cosine Transformation,
A
A
A
Equation 22.4-43
Equation 22.4-44
Equation 22.4-45
XYZ_TO_DC takes the Cartesian position of point (b) with respect to point
(a) both expressed in frame (A). The utility returns the transformation
matrix from (A) to frame (B), where (B) is orientated with respect to (A)
such that the Euler roll angle is zero using the results of §16.13 with point
(m) replaced by the generic point (b).
XYZ_TO_DC
A B
DC A
( P , T ) ≡ ϕ ( P )
22.4.11 Direction Cosines to Cartesian Transformation
b
A
X
b
Equation 22.4-46
DC_TO_XYZ takes the transformation matrix from frame (A) to frame (B)
and returns the Cartesian components of a unit vector emanating from point
(a) whose projection passes through point (b).
i
∈
DC_TO_XYZ
{ ( 1:
X ) , ( 2 : Y ) , ( 3:
Z ) }
B A
X B
( T , P , k ) ≡ ϕ ( T )
A
⇒
b
⎧
⎪
⎨
⎪
⎩
k
k
=
≠
0
0
⇒
⇒
DC
P
iA
b
P
iA
b
A
Equation 22.4-47
: =
: =
T
T
B
A
B
A
( 1,
i )
⎫
⎪
⎬
( ) ⎪ i , 1
⎭
Equation 22.4-48