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Appendix I / Utilities / Quaternions
_ _
DQE_DT takes the quaternion error, and the estimated angular rate of frame
(B) with respect to frame (A), and returns the time rate of change of the
quaternion error.
DQE_DT
22.10.8 Quaternion Transformation Matrix
22.10-8
B B B
( ∆Q
, ωˆ
, ∆Q&
)
A
A,
B
A
Equation 22.10-49
Q_TO_DC takes a quaternion representing the orientation of frame (B) with
respect to frame (A), and provides the transform between the two frames.
Q_TO_DC
B B
DC B
( Q , T ) ≡ ϕ ( Q )
A
Equation 22.10-50
Expressing the direction cosines in terms of quaternion components,
T
B
A
: =
A
B 2 ( ( ) ) ⎟
⎞ ⎛ B B T
B
B
q
⋅ + ⋅ ( ) ( ) [ ] ⎞
A 0 1 I 3 2 ⎜ q ⋅ q − qA
0 ⋅ q × ⎟
⎠ ⎝ A A
A ⎠
⎜
⎛ 2 ⋅ −
⎝
Equation 22.10-51
The skew symmetric quaternion operator is defined such that it is equivalent
to the vector cross product,
T
B
A
⎡ q
⎢
⎢
: = ⎢ 2 ⋅
⎢
⎢
⎢
⎣ 2 ⋅
2
0
+ q
[ q × ]
− q
− q
: =
⎡ 0
⎢
⎢
⎢ q3
⎢
⎢
⎢
⎣ − q
2 ⋅
2
,
,
,
− q
0
q
1
3
Q
,
,
,
A
q2
⎤
⎥
⎥
− q1
⎥
⎥
⎥
0 ⎥
⎦
Equation 22.10-52
( q ⋅ q + q ⋅ q ) 2 ⋅ ( q ⋅ q − q ⋅ q )
( ) ( )
( ) ( ) ⎥ ⎥⎥⎥⎥⎥
2 2 2 2
q1
⋅ q2
− q0
⋅ q3
q0
− q1
+ q2
− q3
2 ⋅ q2
⋅ q3
+ q0
⋅ q1
2 2 2 2
q ⋅ q + q ⋅ q 2 ⋅ q ⋅ q − q ⋅ q q + q − q + q
1
2
1
3
2
2
0
2
3
2
1
2
2
3
0
0
3
1
0
1
1
3
Equation 22.10-53
The main diagonal of the direction cosine matrix is sometimes expressed in
a form that takes into account the normalisation condition,
2
0
2
3
⎤
⎦