Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Appendix C / Axis Transforms
_ _
B
B
A
A
16-6
B A
[ I 3 − ∆E
× ] X
X : = ∆T
⋅ X ≡
⋅
B [ ∆E
× ]
A
≡
⎡ 0
⎢
⎢
⎢ B
∆ΨA
⎢
⎢
⎢
⎣ − ∆Θ
B
A
,
,
,
− ∆Ψ
0
∆Φ
B
A
B
A
A
,
,
,
∆Θ
− ∆Φ
0
Equation 16.2-2
B
A
B
A
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Equation 16.2-3
Consider a misalignment in the transformation from frame (A) to frame (B),
B ~
A
B ~
B
B
A
B B
[ ] A
~
I 3 − ∆E
B × T
T : = ∆T
⋅ T : =
⋅
B
A
B ~
B
B ~
A
B [ ] ~
B
A
~
I 3 + ∆E
B × T
T : = − ∆T
⋅ T : =
⋅
Equation 16.2-4
Equation 16.2-5
This use of the skew symmetric form using the positive sign is not advised;
reversal of the frame identifier is preferred,
B
A
B [ ] ~
B
I − ∆E
× T
T : =
⋅
16.3 Euler Angles as a Function of Direction Cosines
3
B ~
A
Equation 16.2-6
Euler angles are obtained from the transformation matrix direction cosines.
B
Θ ∈ − π 2 , π 2 ,
For example, for a YPR Euler triplet where ] [
E
B
A
: =
⎛
⎜
⎝
tan
−1
⎛ T
⎜
⎝ T
8
9
⎞
⎟
⎠
,
sin
−1
A
⎞ ⎞
⎟
⎠ ⎠
( ) ⎟ −1
−
⎜ 4
T , tan ⎟
7
⎛ T
⎜
⎝ T
1
Equation 16.3-1
The implementation of this equation must accommodate zero pitch and roll
angles when defining the arc-tangent function.