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Chapter 3 / Inertial Navigation
_ _
B
A
DC
E
E B
E B
( 0.
5 ⋅ ( ϕ ( Q ( t − ∆ ) ) + ( Q ) ) )
T : = ϕ
ϕ
Q
3.3-7
A
R
Q
A
Equation 3.3-22
The functions converting Quaternions to Euler angles to direction cosines
are defined in §22.10 and §22.4.7 respectively. The Earth and tilt rate errors
induced by taking current rather than average values are small in
comparison with the average angular rate of frame (B) with respect to frame
(A) over a sub-frame hence,
800
ω
B
A,
B
AR
B
B
( T ( t − ∆ ) , T , )
: = ϕ
∆
The specific acceleration along the Alignment axes is,
f : = A + A − g − ω × ω ×
r, u
r,
o
o,
u
A
u
R
C,
E
A
C,
E
R
P
Equation 3.3-23
r,
u
Equation 3.3-24
Since point (o) is fixed on the Earth, and there is no angular acceleration of
frame (A) with respect to frame (C),
o,
u
Combining terms,
A : = ω × ω ×
r,
o
o,
u
C,
E
C,
E
o,
u
C,
E
P
C,
E
r,
o
A : = P&
& + 2 ⋅ ω ⋅ P&
+ ω × ω ×
B
r, u
B
A
C,
E
Equation 3.3-25
P
o,
u
A
A
( &
A E
E G
P&
u + 2 ⋅ ( TE
⋅ ωC,
E ) × P&
u − TG
g )
f : = T ⋅
⋅
800
B
r, u
B
B
( f r, u ( t − ∆R
) f r, u )
f : = 0.
5 ⋅
+
Equation 3.3-26
u
Equation 3.3-27
Equation 3.3-28
The reference incremental angle turned through over a sub-frame is then,
800
B
C
∆ E : = ∆ ⋅
R
800
f
B
r, u
Equation 3.3-29