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Appendix I / Utilities / Quaternions
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22.10 Quaternion Utilities
The author would like to acknowledge P.H. Jones, R.E. Daisley and J.D.
Stanley for the many interesting and rigorous discussions concerning
Quaternions, in particularly their evolution using gyroscope incremental
angle data.
22.10.1 Quaternion Conjugation
Q_CONJ takes a quaternion representing the transformation from frame (A)
to frame (B), and returns its conjugate quaternion representing the
transformation from (B) to (A).
Q_CONJ
22.10.2 YPR Euler Angles to Quaternions
A
B
22.10-1
B A ( ) , Q
A B Q
B B ( q q )
Q : = −
A
A
Equation 22.10-1
Equation 22.10-2
E_TO_Q takes a Euler triplet representing the orientation of frame (B) with
respect to frame (A) and provides the equivalent quaternion.
E_TO_Q
B B
Q B
( E , Q ) ≡ ϕ ( E )
A
Equation 22.10-3
The relationship between quaternion components and Euler angles is given
by the following half angle equations.
Q
B
A
: =
⎡ cos
⎢
⎢
⎢ sin
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Q B
A
B ( Φ 2 )
B ( Φ 2 )
0
0
A
A
Φ
A
: = Q ⊗ Q ⊗ Q
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡ cos
⎢
⎢
⎢
⎢
⊗ ⎢
⎢ sin
⎢
⎢
⎢
⎣
Θ
E
B ( Θ 2 )
0
B ( Θ 2 )
0
A
A
Ψ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
A
⎡ cos
⎢
⎢
⎢
⎢
⊗ ⎢
⎢
⎢
⎢
⎢
⎣ sin
Equation 22.10-4
B ( Ψ 2 )
0
0
A
⎤
( ) ⎥⎥⎥⎥⎥⎥⎥⎥⎥ B
Ψ 2
A
⎦
Equation 22.10-5