(14)
The EKF provides a recursive method for computing the state covariance,
and can be utilized as an information measure. The sampling algorithm will
choose the next locations in space to sample such that the covariance norm is
minimized. Note that in the time and measurement updates, no measurement
is needed to be taken for the covariance updates. The measurement Zk+1 is
only needed for the estimate measurement update (14).
This means that the equations (11) and (12) are covariance updates available
before the measurement is taken at time k+1. They can be used to predict
the usefulness of the measurement at time k + 1 before it is ever taken.
Also, they show the e?®ect not only of the measurement Zk+1 in reducing
the error covariance, they also show the cost of moving to the estimated
position ?†X
??’k+1 needed to take the measurementZk+1 in terms of uncertainty
injected due to the motion uncertainty (e.g., covariance increase in (11)). This
aspect may avoid long distance movements as appear in [40], where we are
jumping to far corners of the map to take the next measurements, and has
added benefits in that the EKF linear approximation becomes less accurate as
we move long distances. The nonlinearity can be o?®set by predicting the states
using the time update (9) and using the predicted states in (11). Finally, in
order to choose the next sampling location, we use the following algorithm:
EKF-Based Adaptive Sampling Algorithm (AS-3)
Assumptions: The field model is nonlinear but parametric, sampling locations
xj belong to a discrete sampling space ??– and can have uncertainty.
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