Integration of GPS with Other Sensors and Network Assistance - Understanding GPS: Principles and Applications - page 520

ure conditions by the filter. Generally, gross discrepancies between the GPS
measurements (in this case, a Doppler or velocity component) and the reference
speed and heading may indicate such a failure; however, distinguishing between
slipping or skidding and a large Doppler error (as could be induced by tracking a
reflected signal or tracking beyond the limits of the lock detector) is not straightfor-
ward. Failures are generally detected by a Kalman filter through a statistical test
applied to the measurement residual, as explained in Section 9.2.

if ( D 2 res r scale r var ) bypass this Doppler measurement (9.33)

where D res is the Doppler residual for the current satellite represented in meters per
second, and r var is the Kalman filter computed residual variance [in (m/s) 2 ]. The
parameter r scale is typically set to 9, implying that the probability of a residual failing
the test (under the assumed unfailed error conditions of a Gaussian process) is
roughly 0.01. If the failure condition is the reference trajectory (as would be the case
if significant tire slipping or skidding was occurring), then several, or perhaps all,
Doppler measurement residuals should fail. This is therefore a way to distinguish
skidding and slipping from Doppler failure, since it is unlikely that several or all
Doppler measurements would fail at the same time. In this case, two approaches can
limit the errors induced in the integrated trajectory: reinitialization to a GPS position
and velocity (if that is possible, given the GPS coverage at the time of the failure), or
addition of sufficient process noise such that measurement rejections no longer
occur. The appropriate level can be determined through experiments conducted
with test data, or it may be possible (depending on the number of Doppler measure-
ments available during the failure condition, to solve for the needed amount of pro-
cess noise:

T 2

hQh =−

var

r D (9.34) res

The vector h in (9.34) represents the measurement gradient for each measure-
ment that produces a detected failure using the test of (9.33). Since (9.34) is a single
equation, each residual that produces a failure detection through (9.33) should be
included to enable a possible solution for the process noise increment Q , which

will generally have more than a single nonzero component. An overdetermined set
of equations for

Q may be ensured if we limit the increment to the horizontal
velocity components, or further limit the increment to a speed adjustment or a scale
factor adjustment to the a priori process noise levels. Once determined, the
covariance propagation can be repeated and the Doppler measurements repro-
cessed, if sufficient processor throughput exists.

ABS
Integration of the sensed wheel speeds, or distances traveled from an ABS in a vehi-
cle, is perhaps the most cost-effective augmentation of GPS, since no additional sen-
sors are required. A commonly selected state vector for the Kalman filter is given as
(9.35) in row vector form:

[ ]

T T

x = p δδδ vvv (9.35)

L Rz