Global Positioning System Reference
In-Depth Information
where
p is the three-dimensional position error vector, represented in meters;
δ
v L is
a scale factor error associated with the left wheel;
δ
v R is a scale factor error associ-
v z is vertical velocity error in meters per second. Of
course, it is possible to include information from more than two wheels: Inclusion
of separate scale factors for each wheel can then lead to observability problems for
the integration filter. Essentially, the average of the left and right scale factors is esti-
mated by comparison with GPS-derived speed, while the difference is determined
using GPS-derived heading.
Of course, ABS-determined speed and heading is also subject to failures induced
by slipping and skidding—but in a potentially more damaging way than for the
gyro/odometer system. Because heading is also determined from the wheels, the
potential exists for very large heading errors to develop (e.g., one wheel slipping
over ice while the other is stationary produces a heading error rate equal to the
wheel speed divided by the track). A slipping rate of 20 mph corresponds to a head-
ing error rate of almost 300º/hour! In general, heading errors are more of a concern
in the use of DR systems than speed errors, due to the potential for excessive error
growth as heading errors become large.
Another issue worthy of mention is the possible adjustment of the covariance
equations as heading errors become large. Because of the additional failure mecha-
nisms just discussed, heading errors exceeding the expected linear range (e.g., 10º)
can and will occur. In these cases, filter conservatism can be lost with a linear model.
In developing a linear model involving the sine and cosine of heading, the usual (lin-
ear) approximations are:
ated with the right wheel; and
δ
()
sin
δ
HH
=
δ
(9.36a)
()
cos
δ
H
=
1
(9.36b)
In (9.36), the heading error is represented in radians. As heading error becomes
large, the cosine function can be better approximated as 1
H 2 /2. The error vari-
ance propagation equations have become nonlinear, since expressions involving
error variances associated with the sine and cosine of heading error can no longer
can be linearized. These expressions can be approximated by including additional
terms that involve the variance of
− δ
H 2 /2. Its variance can be approximated using a
Gaussian assumption and noting that:
δ
()
4
var
δ
H
2
=
3
σ δ
(9.37)
H
Thus, the traditionally linear variance propagation equations can be replaced with
equations that approximate the nonlinear distortion of the statistics.
Gyro/ABS Performance Comparisons
A comparison of urban canyon performance for experimental gyro- and ABS-based
DR systems is performed in [29]. Both are integrated with two types of GPS receiv-
ers: wide and narrow correlator spacing. As discussed in Section 6.3, the receiver
with narrow correlator spacing is expected to reduce the effects of multipath on
each pseudorange measurement. Many sets of comparison data are generated and
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