Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 473

action by the loop and, eventually, force the loop to lose lock. Thus, the clock insta-
bility sets a floor for the aided bandwidth. One additional observation can be made
using this simple model. If the INS aiding signal is expressed as the sum of the true
range rate plus a range rate error induced by INS velocity error, it can be shown that
the tracking loop error is a function only of the INS errors. Thus, the aiding

makes the loop's performance insensitive to the actual motion (i.e., velocity and
acceleration) of the host, replacing it with the dynamics of the INS errors.
To demonstrate what we have described, we will implement a simplified GPSI
system with a single gyro and a single-channel GPS receiver whose antenna is collo-
cated with the IMU, eliminating the need for lever-arm compensation. (Lever-arm
compensation is required when the GPS receiver antenna and IMU do not share the
same origin in a 3-axis right-handed coordinate system.) The inertial components
that have been purchased for this system have an uncompensated drift uncertainty
of 10º/hour. A single channel receiver refers to a GPS receiver with the capability to
measure pseudorange and pseudorange rate from only one satellite at a time. In
addition to providing pseudorange and pseudorange rate, the receiver also forwards
the position of the current satellite, the velocity of the current satellite, the time of
the GPS measurement, and the deviations

δρ

σ & of both the pseudorange and
pseudorange rate. Ionospheric, relativistic, satellite clock, and tropospheric correc-
tions are applied to the pseudorange data within the receiver prior to its forwarding
to

σ ρ and

the navigation processor. We will denote corrected pseudorange and

pseudorange rate as ρ cor , respectively. In order to implement this system,

we must first formulate the states, the observation models, and the noise model sta-
tistics, and initialize our error covariance matrix P 0 and the estimated error state
vector

ρ cor and

$

x 0 . We ignore, for simplicity, implementation issues related to numerical sta-
bility that would often motivate more complicated forms of the filter in practice.

9.2.4.1 States
The states selected are position error ( x direction, δ x ), velocity error ( x direction,

x ), GPS receiver clock bias ( t u ), and clock drift ( &

δ & t u . This minimum set has been

selected to minimize the number of mathematical calculations that have to be per-
formed. Receiver clock bias and drift are required because they will eventually be
subtracted from the corrected pseudorange and pseudorange rate measurements,
respectively, to form the measurement-based satellite-to-user range r m and range
rate

)

r m . In this example, we have chosen position and velocity along our x -axis as
our system outputs. Therefore, the error state vector is

&

δ
δ
δ
δ

x
x
t
t

&

x =

&

Units for the states consists of kilometers for position, meters/second for veloc-
ity, meters for clock bias, and meters/second for clock drift. All data is referenced to
an ECEF reference frame.