Page 127 - Autonomous Mobile Robots

110 Autonomous Mobile Robots

under these assumptions were briefly discussed in the previous sections. In
many applications either the measurement model, the system dynamics, or
both are nonlinear. In these cases the KF may not be the optimal estimator.
Nonlinearestimation isa difficult problemwithouta general solution. Nonlinear
estimation methods are discussed, for example, in References 25 and 26. For
the navigation systems which are the main focus of this chapter, the EKF has
proved very useful because the linearized dynamic and measurement models are
accurate for the short periods of time between measurements. Due to its utility
in the applications of interest, the EKF is reviewed in this section.
Such navigation systems can be described by the nonlinear stochastic
differential equation

˙ x(t) = f(x, u, t) + g(x, t)w (t) (3.34)
with a measurement model of the form

y(t) = h(x, t) + v (t) (3.35)
where f is a known nonlinear function of the state x, the signal u, and time; g is

a known nonlinear function of the state and time; and w and v are continuous-

time white noise processes.
For its covariance propagation and measurement updates, the EKF will
use a linearization of Equation (3.34) and Equation (3.35). The linearization is
performed relative to a reference trajectory x (t) which is a solution of
∗
∗
∗
˙ x (t) = f(x , u, t)
between the measurement time instants. The corresponding reference measure-
ment is

∗
∗
y (t) = h(x , t)
The error state vector is defined as

δx = x − x ∗

and the measurement residual vector as

∗
δy(t) = y(t) − y (t)
The linearized dynamics of the error state vector are

δ˙ x(t) = F(t)δx(t) + G(t)w (t) + e x (t) (3.36)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c003” — 2006/3/31 — 16:42 — page 110 — #12