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Data Fusion via Kalman Filter 117

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there may still be significant error η x = x − ˆ x between the actual state and the
best estimate after incorporating the measurements. The measurement noise
covariance matrix is

 2 
σ 1 0 ··· 0
 . .
R ρ = cov(v) =  . . . .  (3.50)

0 ··· 0 σ 2
m
2
The value of σ for the ith satellite could be determined based on time-series
i
analysis of measurement data, the S/N ratio determined in the tracking loop for
that channel, or computed based on satellite elevation. The covariance of η x is

T
R x = cov(η x ) =[H R −1 H] −1 (3.51)

It is important to note that this matrix is not diagonal. Therefore, errors in the
GPS position estimates at a given epoch are correlated.
Theabovesolutionapproachcanberepeated(independently)foreachepoch
of measurements. This calculation of the position described so far, at each
epoch, results in a series of single-point solutions. At each epoch, at least four
simultaneous measurements are required and the solution is sensitive to the
current measurement noise. There is no information sharing between epochs.
Such information sharing between epochs could decrease noise sensitivity and
decrease the number of satellites required per epoch; however, information
sharing across epochs will require use of a dynamic model. Section 3.3.3 dis-
cusses advantages and disadvantages of alternative models and EKF solutions
for GPS-only solutions. Section 3.4 discusses methods for combining GPS and
IMU data.

Example 3.2 Throughout the remainder of this chapter we will extend the
2
example that begins here. The example will be analyzed in . By this we mean
that we are analyzing a 2D world, not a 2D solution in a 3D world. We restrict
the analysis to a 2D world for a few reasons (1) the analysis will conveniently
fit within the page constraints of this chapter; (2) graphical illustrations are
convenient; and (3) several important theoretical issues can be conveniently
illustrated within the 2D example. The main conclusions from the 2D example
have exact analogs in the 3D world (discussed in Example 3.6).
2
In a 2D world, p(t), v(t) ∈ and there is a single angular rotation angle
ψ(t) ∈ with ω(t) = ˙ ψ(t) ∈ . All positions and ranges will be in meters.
All angles are measured in degrees. The quantities ψ and ω are not used in this
example, but are defined here for completeness as they are used in Example 3.6.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 117 — #19