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

Appendix I / Utilities / Quaternions
_ _
Q &
B
A
: =
1
2
⎛ 0
⎜
⎜
⎜
⎜ 0
⋅ ⎜
⎜
⎜ 0
⎜
⎜
⎝ 0
,
,
,
,
− q
q
q
B
A
B
A
− q
B
A
B
B
( 1 ) , − q ( 2 ) , − q ( 3 )
B
B
( 0 ) , − q ( 3 ) , q ( 2 )
B
B
( 3 ) q ( 0 ) , − q ( 1 )
B
A
B
B
( 2 ) , q ( 1 ) , q ( 0 )
22.10.6 Quaternion Propagation by Euler Increments
22.10-5
A
A
A
A
A
A
A
A
⎞
⎟
⎟
⎟
⎛ 0
⎟ ⎜
⎟ ⋅ ⎜
⎟ ⎜ B
⎟ ⎝ ωA,
⎟
⎟
⎠
B
⎞
⎟
⎟
⎟
⎠
Equation 22.10-26
DQ_EULER takes the quaternion representing the orientation of frame (B)
with respect to frame (A), and returns its value after time interval (∆t) using
current and previous Euler angle increments.
DQ_EULER
B B B
( Q , ∆E
, ∆E
)
A
Equation 22.10-27
Dropping nomenclature for convenience, extending the results presented in
§22.10.5, and expanding in a Taylor series about the current quaternion
approximating to the 3 rd order,
Q
k+
1
: =
∆Q
k
⊗
Q
k
: =
Q
k
C
+ ∆t
⋅ Q
A
L
A
& −1
2
1 3
k 2 t Q&
& −
+ ⋅ ∆ ⋅ k + 6 ⋅ ∆t
⋅
Equation 22.10-28
Expressing the time derivatives in terms of the angular rate and acceleration
of frame (B) with respect to frame (A) - courtesy J.D. Stanley,
Q &
k
: =
q
ω
B
A,
B
⊗
Q
k
where
Differentiating and dropping the full notation,
Since,
q
ω
B
A,
B
: =
⎛
⎜
⎜
⎜
⎝
− 2
0
−1 B
⋅ ωA,
B
⎞
⎟
⎟
⎟
⎠
&Q
&
Equation 22.10-29
Q&
& q q
q
q q
: = ω&
⊗ Q + ω ⊗ Q&
: = ω&
⊗ Q + ω ⊗ ω ⊗ Q&
q
ω ⊗
q
ω ⊗
Q
≡
−
q q ( ω • ω ) ⋅ Q
Equation 22.10-30
Equation 22.10-31
k