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Appendix I / Utilities / Quaternions
_ _
q
q
C
A
C
A
q
C B C B
C B
C B
( 1 ) : = q ( 0 ) ⋅ q ( 1 ) + q ( 1 ) ⋅ q ( 0 ) + q ( 3 ) ⋅ q ( 2 ) − q ( 2 ) ⋅ q ( 3 )
B
A
B
22.10-3
A
B
A
B
A
Equation 22.10-13
C B
C B
C B C B
( 2 ) : = q ( 0 ) ⋅ q ( 2 ) + q ( 2 ) ⋅ q ( 0 ) − q ( 3 ) ⋅ q ( 1 ) + q ( 1 ) ⋅ q ( 3 )
C
A
B
A
B
A
B
A
B
A
Equation 22.10-14
C B C B C B C B
( 3 ) : = q ( 0 ) ⋅ q ( 3 ) + q ( 3 ) ⋅ q ( 0 ) + q ( 2 ) ⋅ q ( 1)
− q ( 1)
⋅ q ( 2 )
22.10.4 Quaternion Normalisation
B
A
B
A
B
A
B
A
Equation 22.10-15
A quaternion of unit magnitude satisfies the normalisation condition:
Q
2
: =
1
: =
q
2
0
+ q • q
Equation 22.10-16
Propagating a quaternion over time leads to errors and this condition is no
longer satisfied. Q_NORMAL takes in quaternion (Q) and normalises it,
returning the corrected quaternion ( C Q).
From Equation 22.10-16,
Q_NORMAL
Q
2
Q
( Q ) ≡ ϕ ( Q )
NORM
2 ( 1 + ξ ) ≅ 1 + ⋅ ξ
: = 2
The corrected quaternion is related to the input by,
Therefore,
22.10.5 Quaternion Propagation
C
Q
≅
C ( 1 + ξ ) ⋅ Q
Q ≅ 2 ⋅ Q +
2 ( 1 Q )
Equation 22.10-17
Equation 22.10-18
Equation 22.10-19
The orientation of frame (B) with respect to frame (A) can be propagated
between time (t) and (t+∆t) using quaternion multiplication,