Aerospace and Electronic Systems Magazine August 2017 - 15

Watson et al.

i
FINS




−
=






2 g rˆ iT
rˆ
rese rˆi 2

Cˆ bi

03
03

(Cˆ fˆ )
i
b

Cˆ bi 


03 


03 
03 

03 

(16)

The system matrix F is used to calculate the discrete system transformation matrix (STM) Φ. The STM is calculated by taking the
power series expansion, and for this study, a third-order expansion
was used.

PRACTICAL IMPLEMENTATION DETAILS
Beyond the INS mechanization, several practical implementation
details are required for a proper design. This section overviews a
few of these considered in the presented implementation.

IMU to GNSS Lever Arm
The preceding INS mechanization integrates the position and velocity at the location of the IMU. Therefore, the INS solution needs
to be transposed to the same location as the GNSS solution (i.e.,
the phase center of the GNSS antenna) using the following:
r i ,G = r i , I + Cˆ bi Lb

(17)

where Cˆ bi is the estimated platform attitude, the known lever-arm
from the IMU to the GNSS antenna Lb represented in the platform's
north, east, down body axis, and the superscripts G and I represent
the GNSS antenna phase center and IMU location, respectively.
Similarly, the INS velocity solution must be transposed, as shown
in (18), with Ωbib as the skew-symmetric matrix of the IMU measured angular rate:
vki ,|Gk −1 = v i , I + Cˆ bi Ωbib Lb

(18)

This is done upon each GNSS measurement update and reversed
after the INS closed-loop feedback correction has been applied.

GNSS and IMU Time Alignment
Another implementation issue is the precise time-tag alignment of
the INS to the GNSS time tag. This issue arises because the IMU
measurements, although typically stamped with a GPS time tag,
are typically not scheduled to be precisely aligned with the GNSS
measurement epochs. This means that IMU data must be used to
predict the navigation states beyond the GNSS observation epoch
and then linearly interpolated back to the time of the measurements. Furthermore, when a GNSS update occurs, an incremental
error-state transformation matrix, which provides the propagation
mapping of INS error states, must be used to map to the error states
to the GNSS measurement epoch. Then, after the GNSS update,
the inverse of this transformation is used to map the predicted INS
error states back to the IMU time stamp to ensure that INS error
states remain consistent with the INS time tags.
AUGUST 2017

Figure 3.

Flight trajectories of the two GRAV-D GPS/INS data sets that were
processed for this study.

ALGORITHM PERFORMANCE EVALUATION
FLIGHT DATA
Two flights tests of the National Oceanic and Atmospheric Administration (NOAA) NGS Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project [47] that were flown over
Alaska are used to conduct this study. In total, 6.3 h of flight data
are processed (i.e., 2.8 h and 3.5 h). The flight data sets consist
of dual-frequency GPS pseudorange and carrier-phase observables
recorded at 1 Hz and an IMU's triaxial accelerometer and rate
gyroscope measurements recorded at a rate of 200 Hz. The commercial GNSS/INS system flown was a NovAtel Synchronized
Position Attitude Navigation (SPAN) system that is composed of
an OEM4-2G GPS receiver and a micro-inertial reference system
navigation-grade IMU. The surveyed lever arm between the GPS
antenna's phase center and the IMU was provided by NOAA NGS,
as well as the rotation angles relating the IMU's mounted orientation to the aircraft body axis.
Figure 3 shows the latitude and longitude flight profiles for the
two flights processed in this study. The maximum ranges from the
takeoff location for the two flights were 727 and 698 km, making
the possibility of using RTK to a single GNSS reference station
impractical.

REFERENCE SOLUTION STRATEGY
The reference PPP position solutions were postprocessed with
JPL's GIPSY-OASIS II and Final orbit and clock products. In addition, carrier-phase integer ambiguities were fixed by employing
JPL's Wide-Lane Phase Bias products with their single-receiver
integer-ambiguity-resolution algorithm [37]. Finally, the position

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