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GPS and INS have complementary properties which have motivated various
researchers to study methods to fuse the data from the two systems. The object-
ive is to attain high performance for a higher percentage of the time than either
approach could attain independently. Thischapteruses GPS and INS to illustrate
the use of the KF for data fusion. Section 3.2 reviews the KF, EKF, and various
properties and application issues. Section 3.3 reviews the various issues related
to the GPS system. Of particular interest will be various assumed dynamic
models and issues affecting state estimation accuracy. Section 3.4 provides a
brief review of INS and their error models. The main objective is to present
the model information necessary to analyze alternative methods for combining
GPS and INS information, which is done in Section 3.5.

3.2 KALMAN FILTER
Since R. E. Kalman published his idea in the early 1960s [16,17], the KF
has been the subject of extensive research. It has been applied successfully to
solve many practical problems in different fields, particularly in the area of
autonomous navigation. The KF involves two basic steps: use of the system
dynamic model to predict the evolution of the state between the times of the
measurements and use of the system measurement model and the measurements
to optimally correct the estimated state at the time of the measurements. It is
well known that the KF is recursive, computationally efficient, and optimal in
the sense of the minimum mean of the squared errors [18].
This section contains three subsections. Section 3.2.1 reviews the linear
dynamic system models that are required for the prediction and measurement
update steps of the KF. Section 3.2.2 reviews the KF algorithm, a few of its
properties, and methods to address various implementation issues. Section 3.2.3
reviews the EKF algorithm which is applicable when either the dynamic or
measurement model of the system is not linear. The EKF is needed in GPS–INS
data fusion applications since the INS dynamic model is nonlinear and the GPS
measurement model may be nonlinear.

3.2.1 Stochastic Process Models

Because the state of most physical systems evolve in continuous time,
continuous-time dynamic models are of interest. The dynamic behavior of
a linear continuous-time system driven by a random process ω(t) may be
described mathematically by a set of ordinary differential equations:

˙ x(t) = F(t)x(t) + G(t)ω(t) (3.1)

y(t) = H(t)x(t) + v(t) (3.2)

© 2006 by Taylor & Francis Group, LLC

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