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Data Fusion via Kalman Filter 127

When the design uses a large Q, the variance of the estimated position is
large, but the estimator rapidly adjusts the estimated state so that the estimate
does not significantly lag the actual state following the vehicle maneuver. When
the design uses a small value for Q, the variance of the estimated position is
smaller; however, the estimated state significantly lags the actual state following
the vehicle maneuver.
Figure 3.2a and b are intentionally placed side-by-side to emphasize the
fact that there is no single optimal choice for the design parameter Q. The
desirable setting of Q depends on the application and maneuvering conditions.
Some receivers allow the user to effect the receiver estimation procedure (either
the model structure or the value of Q) through the user interface. It is the
responsibility of the user to understand the settings and their tradeoffs relative
to the application. This is especially true when the state estimate is being used
as the input to a control system.
Due to the structure of the k matrix, if the GPS H matrix has a null
direction d such that Hd = 0, then position, velocity, and acceleration errors
parallel to d will not be observable from the GPS measurements. Note that
the rows of the H matrix contain the line-of-sight unit vectors between the
receiver antenna and the satellite. Therefore, to accurately and rapidly track the
platform motion during (and after) a maneuver, the receiver must be tracking at
least one satellite located in a direction such that the line-of-sight unit vector has
a significant component in the same direction as the acceleration unit vector;
otherwise, the GPS measurements will be insensitive to the acceleration. In
1
particular, if a receiver is operating in an urban canyon type of environment
and accelerates parallel to the direction in which the satellite signals are blocked
then the position, velocity, and acceleration accuracy in that direction will
deteriorate.
No amount of signal processing can help, unless additional sensors
(e.g., inertial, wheel speed, vision, precision clock) are added.
Finally, it is critical to note that estimation errors, even restricted to the
GPS measurement epochs, are correlated. They are not white discrete-time
processes. This is clearly illustrated in Figure 3.2b for small values of Q, but is
also true for larger values of Q. The fact that the position estimation errors are
not white is critical to understanding one of the drawbacks of using the GPS
position estimates to aid an INS (see Section 3.5.2.1).

3.3.3.6 Summary
The approaches discussed in the previous three sections have several aspects
that should be pointed out. First, as discussed following Equation (3.50),

1 This is a canyon created by the urban environment (e.g., a road between tall buildings) that may
block satellite signals in specific directions [37–39].

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 127 — #29