Global Positioning System Reference
In-Depth Information
squared at the end of the turn, or roughly 1.4º, 1 sigma, compared to the actual
error, which is more than double this predicted 1-sigma value. The reason for this
optimistic prediction is that the filter assumes a white noise model, such that the
error accumulation root-sum-squares from second to second. However, the actual
error is a bias, which adds each second. A way to force the filter to be more conser-
vative, and thus more realistic, is to assume a maneuver duration associated with
the heading change and scale the process noise variance by this amount. If a 3-sec-
ond average maneuver change is assumed, the resulting prediction will be 2.4º, 1
sigma, closer to the actual induced roll error.
Gyro/Odometer
Integration of a vertical gyro (to sense heading changes) with an interface to the
vehicle's odometer is one of the first GPS augmentations considered [54] and still
one of the most popular options. A commonly selected state vector for the Kalman
filter is given as (9.32) in row vector form:
[
]
x
T
=
p
T
δδδ
vvHbs
(9.32)
o
z
HH
where
p is the three-dimensional position error vector,
δ
v o is a scale factor error
associated with the odometer,
H is heading error, and
b H and s H are the gyro bias and scale factor errors, respectively. In (9.32), position
errors are represented in meters, velocity errors in meters per second, heading error
in radians, and gyro bias error in radians per second. In general, temperature error
curves can be derived and applied for both the gyro bias and scale factor errors, if
temperature information in the vicinity of the gyro is available. The state vector def-
inition in (9.32) implies a centralized filter approach—where a single filter (eight
state) is used—however, adequate performance can be obtained using a decentral-
ized approach [54]. In this system, individual, mostly single-state filters are used.
Appropriate levels of process noise are required to force the filter to track varia-
tions in average tire pressure due to changes in temperature and driving conditions,
which affect the scale factor error associated with the odometer. In addition, if the
odometer cannot accurately track very low velocities due to sensor limitations (see
Section 9.3.2.4), additional process noise can be injected into the horizontal posi-
tion error states directly (the velocity error is therefore represented as the sum of the
odometer scale factor induced error plus other, unmodeled effects that are repre-
sented as white noise). Any filter designed to operate with sensors that derive veloc-
ity information from the vehicle's wheels must deal with the anomalous sensor
performance induced by wheel skidding and slipping. As mentioned in Section
9.3.2.4, the preferred solution to tire slipping is to derive information from the
nondriven wheels; however, this may not always be possible. For tire skidding,
there may be an indication of ABS activity (if the car has an ABS), which can be
made available to the filter. This serves as an alert, and conservatism would dictate
that additional process noise should be injected to keep the filter aware of potential
error in its propagation. The amount should be derived from test experience.
For either skidding or slipping, then, the Kalman filter may have to adjust to a
potentially significant and unmodeled source of error. Since its a priori levels of pro-
cess noise do not reflect the presence of either condition, they must be treated as fail-
δ
v z is vertical velocity error,
δ
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