The variance of ?† A can now be calculated based on the variance of (x, y)
through Jacobians:
var( ?† A) = ?? a ?† A
a(x, y)!var(x, y)?? a ?† A
a(x, y)!T
,
where the Jacobian is given by
a ?† A
a(x, y)
= " aF1
axi
aF1
ayi
aF2
axi
aF2
ayi #, and
aF1
axj
=
f0
af1
axj ??’f1
af0
axj
f2
0
, aF2
axj
=
f0
af2
axj ??’f2
af0
axj
f2
0
,
af1
axj
= (Pn
i=1 yi) ?· 2xj ??’ (Pn
i=1 xiyi) ??’ (Pn
i=1 xi) ?· yj ,
af0
axj
= 2nxj ??’ 2Pn
i=1 xi, af2
axj
= ??’(Pn
i=1 yi) + nyj .
.
Similarly,
aF1
ayj
= 1
f0
af1
ayj
= 1
f0
{(Pn
i=1 x2i
) ??’ (Pn
i=1 xi) ?· xj},
aF2
yj
= 1
f0
af2
ayj
= 1
f0
{??’(Pn
i=1 xi) + nxj}.
Even though we obtained closed-form algebraic solutions for the unknown
parameter covariance, we can see that introducing localization uncertainty
to our sampling problem makes it hard to obtain closed-form solutions for a
large number of parameters, or for nonlinear fields. Using closed-form variance
solutions to determine optimal sampling locations is only feasible if no
localization uncertainty is considered.
49
Dan O. Popa and Frank L. Lewis
Due to di?±culties in estimating the combined parametric and localization
uncertainty using closed-form solutions, we employ an iterative estimation
approach to select optimal sampling points. Let xk denote the 3D position of
one of the mobile robots at sample number k. A model for the mobile node is
xk+1 = xk + Tf(xk, uk) + Twk
where T is the sampling rate, uk is the control input to the vehicle and wk is
state measurement noise, assumed to be white, with zero mean, and covariance
1
k
yk = h(xk) + ?»k.
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