Aerospace and Electronic Systems Magazine August 2017 - 13
Watson et al.
Figure 2.
Inertial navigation propagation schematic.
similated data in RTGx's other applications (GNSS constellations,
LEO satellites, etc.).
Attitude Update
To integrate attitude, a third-order Runge-Kutta integrator was
used to propagate the platform's body-to-ECI quaternion [43], as
shown in (3). Here, I is an identity matrix; β is composed of the raw
gyroscope measurements, as defined in (4); and the subscript t signifies the IMU time step (e.g., t − 2 is using data from two discrete
sample intervals in the past):
(
)
1 ˆ
1
1
qˆt = I +
βt + 4 βˆt −1 + βˆt − 2 + I + βˆt
12
12
4
1
1
+ βˆt βˆt −1 − βˆt − 2 qˆt − 2
12
2
ˆ ˆ
β t −1β t − 2
(3)
With the 3rd-order integrator, attitude estimates are predicted at
half the IMU data rate (i.e., 200-Hz IMU data yield 100-Hz attitude
βˆt +1− n
AUGUST 2017
−3 δθ
1
=
−3 δθ 2
−3 δθ3
(
(
(
( )
( )
(3(δθ )
0
t +1 − n
t +1 − n
( )
t +1 − n
1
+ (δθ1 )t − n
( )
+ δθ 2
t −n
+ (δθ 3 )t − n
3
2
Velocity Update
Using the updated INS attitude, the IMU specific force measurements are transformed from the body frame to the ECI frame
i
by multiplying with the body-to-ECI DCM, Cb, as shown in (5),
where the estimated body-to-ECI DCM is determined from the
updated quaternion, using the relationships found in many texts
[44]:
Δv i = Cbi Δv b
− (δθ1 )t − n
2
t +1− n
3
t +1 − n
t +1− n
3
2
t +1− n
t −n
t −n
(5)
With the specific force measurement transformed into the ECI reference frame, the velocity is calculated via Euler integration:
) (3(δθ )
0
(3(δθ )
+ (δθ ) )
− (δθ ) ) ( −3 (δθ )
t +1 − n
)
) ( −3(δθ )
) (3(δθ )
estimates). The accumulation of integration and sensor errors can
cause the integrated quaternion to lose its required orthonormality.
To prevent this from happening, a periodic normalization of the
quaternion was implemented.
1
( )
t −n
( )
t −n
− δθ 2
− δθ3
0
t +1− n
IEEE A&E SYSTEMS MAGAZINE
) (3(δθ )
) ( −3(δθ )
(3(δθ )
)
3
t +1− n
2
1
+ (δθ1 )t − n
t +1 − n
t +1 − n
0
)
+ (δθ ) )
− (δθ ) )
( )
− δθ3
2
t −n
t −n
(4)
1 t −n
13
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