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Appendix I / Utilities / Quaternions
_ _
22.10.10 Quaternion from Direction Cosines
DC_TO_Q takes a transform from frame (A) to frame (B) and returns the
normalised quaternion equivalent.
DC_TO_Q
B B
Q B
( T , Q ) ≡ ϕ ( T )
A
Equation 22.10-60
A numerically stable implementation that avoids q(0) → 0 suggested by Dr.
B.J. Hartley is as follows,
⎛
⎜
⎜
⎝
⎛
⎜ T
⎜
⎝
⎛
⎜ T
⎜
⎝
ϕ
B
A
B
A
ϕ
: =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 + tr
ϕ
B ( T )
A
,
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
22.10-10
A
( 0 ) ≥ max ( ϕ )
B
B
( 0 ) TA
( 8 ) − TA
( 6 )
,
B
2
4 ⋅ q ( 0 )
B ( 8 ) − TA
( 6 )
B
4 ⋅ q ( 1 )
A
B ( 2 ) + TA
( 4 )
B
4 ⋅ q ( 2 )
A
,
,
A
ϕ
ϕ
,
1 + 2 ⋅ T
1 + 2 ⋅ T
1 + 2 ⋅ T
T
B
A
( 1 ) ≥ max(
ϕ )
ϕ
⇒
B
A
B
A
B
A
DC
A
B
( 1 ) − tr ( T )
A
B
( 5 ) − tr ( T )
A
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
( ) ( ) ⎟ ⎟⎟⎟⎟⎟⎟
B
9 − tr T
Q
: =
B ( 3 ) − TA
( 7 )
B
4 ⋅ q ( 0 )
⇒
A
Q
B
A
: =
B
B
( 1 ) TA
( 2 ) + TA
( 4 )
,
B
2
4 ⋅ q ( 1 )
( 2 ) ≥ max(
ϕ )
T
B
A
B ( 3 ) − TA
( 7 )
B
4 ⋅ q ( 2 )
A
,
⇒
A
Q
ϕ
B
A
B
A
: =
,
,
A
T
⎞
⎠
Equation 22.10-61
T
B
A
⎞
( ) ( )
( ) ⎟ ⎟
B
4 − TA
2
B
4 ⋅ q 0
A
Equation 22.10-62
T
B
A
B ( 7 ) + TA
( 3 )
B
4 ⋅ q ( 1 )
A
⎟ ⎟
⎞
⎠
Equation 22.10-63
( ) ( ) ( )
( ) ⎟ ⎟
B
B
2 T 6 + T 8
⎞
A
A
,
B
2
4 ⋅ q 2
A
Equation 22.10-64
⎠
⎠