Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 488

assuming that lock has been lost. This can obviously become computationally
intractable very quickly, especially as most of the channels are near thresholds and
passing in and out of a lock state. Use of the quasi-linear model for the code (or other
tracking loop) detector as described here can make the design highly resistant to
missed loss-of-lock detection, as the loop gain becomes zero as that condition is
approached. Thus appropriate modeling of the code (or carrier) loop nonlinearity
can remove loss-of-lock detection as a critical design issue.

9.2.5 Reliability and Integrity
It is difficult for a GPS receiver to determine the precise point at which it loses lock.
A capability that prevents erroneous measurements from entering the filter is thus
required. With filter processing, a check can be made using the observed error in the
GPS signal compared to the predicted covariance to see if the measurement being
processed is within reasonable limits. The extrapolated covariance at the time of the
measurement is calculated in a similar manner to the formulation used in our filter,
P . Since we are dealing with a single measurement, the equa-

tion of the range and range rate can be separated from the matrix calculation and
reduced as follows:

() t

−

= P

T

+ Q () t

1 0 0

pppp
pppp
pppp

0
0
1

00 01 02 03

[ ]

T 10 11 12 13

ΦΦ P 0 += q 001 φ + q

33 23 33

20 21 22 23

pppp φ

30 31 32 33 23

0
0
1

(

( )( )

= p + φ pp + φ p

20 23 30 21 23 31

+ q

) ] )(

33

p + φ p p + φ p

22 23 32 23 23 33

φ

23

( )

=+ p φ p + φ p + φ p + q

22 23 32 23 23 23 33 33

Since p ij = p ji , let

()

2

α

ρ =+

p 2 φ p + φ p + q

22 23 23 23 33 33

One can do the same thing for the GPS velocity measurement and get

α & = pq + p

33 33 44

The error in the observation that we earlier called the innovations process we
will denote as gamma ( ), which can be calculated by subtracting the predicted

observation Hx () t $ () t

−

from the observation. The value

2

is compared to m 2 α ,

1

2

exceeds the m 2

where m is the m σ limit (typically 3 ≤ m ≤ 6). If α limit, the mea-

surement is declared bad and not processed by the filter.