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

Appendix I / Utilities / Quaternions
_ _
B
A
B
B
B B
( t + ∆t
) : = ( Q&
⋅ ∆t
) ⊗ Q : = ∆Q
Q
Q ⊗
A
Equation 22.10-20
Over time period (∆t) the rotating frame moves through an angle (ωAB.∆t)
about a vector defined by the direction cosines (ωA,B/ωA,B). If the angular
rate remains constant over this period the quaternion representing the
rotation,
∆Q
B
A
∆Q
B
A
: =
: =
⎛
⎜ ⎛ ∆t
cos ⎜ ⋅ ω
⎜
⎝ ⎝ 2
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ω
ω
B
A,
B
B
A,
B
⋅
⎛
⎜
⎝
B
A,
B
1
1 −
8
1
2
−
22.10-4
⎞
⎟
⎠
⋅
,
ω
ω
A
B
A,
B
B
A,
B
⋅ sin
B ( ω ⋅ ∆t
)
1
48
⋅
A,
B
⎛
⎜
⎝
+ K
A
∆t
⋅ ω
2
A
B
A,
B
⎞
⎟
⎠
⎞
⎟
⎟
⎠
Equation 22.10-21
( ) ⎟ ⎟⎟⎟⎟⎟
B 2 ⎞
ω ⋅ ∆t
+ K ⎟ ⋅ ∆t
Equation 22.10-22
The general expression for propagating a quaternion over time interval (∆t),
B ( Q )
A
k + 1
: = Q
B
A
+
N
∑
n : = 1
⎛
⎜ ⎡
0
n
⎜ ∆t
⎢
⎜ ⋅ ⎢ L
⎜
n + 1 ⎢ B
⎜ ⎢
ωA,
⎝ ⎣
A,
B
B
M
M
M
2
−
−
⎠
B ( ω )
A,
B
L
B [ ω × ]
A,
B
T
n
⎞
⎠
⎤
⎥
⎥
⎥
⎥⎦
⋅ Q
B
A
⎞
⎟
⎟
⎟
⎟
⎠
Equation 22.10-23
Q_RATE takes a quaternion representing the transformation from frame (A)
to frame (B), and the angular velocity vector of (B) with respect to (A). It
returns the quaternion time rate of change representing the rate of rotation of
(B) with respect to (A), using 1 st order of approximation.
B
A
Q_RATE
B B B
( Q , ω , Q&
)
A
A,
B
A
Equation 22.10-24
B B B
B B B
( − ω • q , q ( 0 ) ⋅ ω − ω q )
2 ⋅ Q : =
×
&
A,
B
A
A
A,
B
A,
B
A
Equation 22.10-25