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Data Fusion via Kalman Filter 113

The EKF iteratesSteps1to7forthe duration ofthe application. Steps1and2
perform computations related to time propagation of the state and the matrix P.
Steps 4 to 7 perform computations related to the measurement update.
In the EKF algorithm, the computation, use, resetting, and time propagation
of δˆ x k|k often causes confusion. The above algorithm is a total state implement-
ation. In an alternative error state implementation of the algorithm, Step 6 could
be removed. Without Step 6, Equation (3.39) of Step 2 would have to be imple-
mentedtotimepropagatetheerrorstateandthesimplificationtoEquation(3.42)
of Step 5 would not be possible. In this alternative implementation, it is pos-
∗
sible that, over time, δˆ x k|k could become large. In this case, x is not near the
actual state. In this case, the linearized equations may not be accurate. The
∗
EKF algorithm as presented (using Step 6) includes δˆ x k|k in x resulting in
a more accurate linearization. The total and error state implementations are
discussed in greater detail in References 20 and 27.

3.3 GPS NAVIGATION SYSTEM
The purpose of this section is to discuss the advantages and disadvantages
of various EKF approaches to state estimation using GPS measurements.
Section 3.3.1 presents background information about GPS that is necessary for
the subsequent discussions. Section 3.3.2 discusses position estimation based
on GPS measurements. The EKF approaches to solving the GPS equations are
compared in Section 3.3.3.

3.3.1 GPS Measurements
The GPS is designed to provide position, velocity, and time estimates to users
at all times, in all weather conditions, anywhere on the Earth. The existing
GPS signal for each satellite consists of a spectrum spreading code and data
bits modulated onto a carrier signal. By accurately measuring the transit time
of the code signal, the receiver can form a measurement of the pseudorange
between the satellite and the receiver antenna. This measurement is referred
to as a pseudorange as it is also affected by receiver and satellite clock errors.
By processing the data bits to determine the clock error model and ephemeris
data, the receiver can compute the satellite position and clock errors as a func-
tion of time. Tracking the satellite signal requires that the receiver acquire
either frequency or phase lock to the satellite carrier signal. Phase information
from the tracking loop has utility as an additional range measurement and the
change in the phase measurement over a known period of time (referred to
in the GPS literature as a Doppler measurement) can be used to estimate the
receiver velocity. The GPS satellites broadcast signals on two frequencies: L1

© 2006 by Taylor & Francis Group, LLC

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