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Data Fusion via Kalman Filter 119

The error covariance matrix for the estimated position vector is

 
0.377 0.040 0.133
R x =  0.040 1.492 0.376 
0.133 0.376 0.384

The variance of x is smaller than the variance of y due to the geometric alignment
of the satellites. Also, the estimates of x and y are correlated. Note that receivers
do not typically provide this covariance matrix as an output.
If only the first three measurements are used to estimate x, then the meas-
urements can be perfectly fit, but the error in the estimated position and the
error covariance matrix will increase.

3.3.3 KF for Stand-Alone GPS Solutions
This section discusses methods that have been proposed in the literature to
achieve improved performance by using the EKF to share information across
measurement epochs. Higher performance can be represented by increased
position accuracy ordecreased requirements on the number of required satellites
per epoch.
Sharing information across measurement epochs requires models for the
dynamics of the user receiver and the receiver clock error. The receiver
clock dynamic model is discussed briefly in Section 3.3.3.1. Various pos-
sible dynamic models for the receiver antenna position are discussed in
Section 3.3.3.2 to Section 3.3.3.4. Each of these dynamic models will be lin-
ear; however, since the GPS measurement model is nonlinear, the solution still
requires an EKF.
The use of the EKF algorithm of Section 3.2.3 to solve the GPS navigation
problem is illustrated as a block diagram in Figure 3.1. The dynamic motion
equations are integrated between measurement times to predict the receiver
antenna position at subsequent measurement times. The measurement predic-
tion equations use the predicted antenna position and the computed satellite
position to predict range measurement for each satellite. The residuals between
the GPS measurements and the predicted measurements drive the EKF which
outputs the error state estimates. The error state estimates are fed back to correct
the predicted states, which are used to initialize the prediction step for the next
epoch.
Section 3.3.3.2 to Section 3.3.3.4 discuss the advantages and disadvantages
of three possible receiver state estimation algorithms. The EKF algorithm and
Figure 3.1 are applicable to all three approaches. The main distinctions between
the approaches are the definitions of the state vector and the dynamic model for
the state vector.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 119 — #21