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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