As an example, consider navigation in a shallow region (lake or
measured as altitude by the vehicle.
4.1 Closed-form estimation for a field without localization
uncertainty
Let??™s first assume that we are estimating a field distribution that is known
to be linear in its parameters. For instance, the field can represent dissolved
oxygen in a body of water that varies linearly with the location of the sample
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coastal region) where smooth changes in depth provide a linear field model
without the need of a Kalman Filter. In this case, the sampling objective is
Fig. 3. Utility function with the position of node 2 varying between nodes 1 and 3
on the x axis (1d - top) and on both x and y axes (2d -bottom) for a link capacity
(a)
(b)
Chapter 2 Algorithms for Robotic Deployment of WSN
point in the (x, y) direction, and quadratically with the sample point depth
(z). Or it can represent a general field distribution that has been approximated
through a finite number of coe?±cients using a polynomial, Fourier or
Wavelet basis. The linear-in-parameters form allows us to compute a closed
form solution for the information measure used by the sampling algorithm.
Since there is no uncertainty in the vehicle localization, the unknown field
coe?±cient covariance can be calculated directly using a simple least-square
estimation. After n measurements taken at location xi, the field model depends
linearly on the coe?±cients aj via position-dependent functions, and we
can directly estimate the unknown coe?±cients from the least-square solution:
z1 = a0 + a1g1(x1) + ?· ?· ?· + amgm(x1)
z2 = a0 + a1g1(x2) + ?· ?· ?· + amgm(x2)
.
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