1, sampling measures can be obtained for parameter-linear fields by using a
Kalman Filter to set up the recursive equations. A similar simple case appears
in [28, pp.137]. First, if there is no localization uncertainty, the state and
output equations can be written as:
Ak+1 = Ak, Zk = GkAk + vk, where
Gk = (1 ?· ?· ?· gi(Xk) ?· ?· ?· gm(Xk)), E[vkvT
k ] = R.
Because of the simple state update equation, the error covariance update
can be reduced to:
A0 ??? ( ?? A0, PA0 ), P0 = PA0 , P??’1
k+1 = P??’1
k + GT
k+1R??’1Gk+1,
?† Ak+1 = ?† Ak + Pk+1GT
k+1R??’1(Zk+1 ??’ Gk+1 ?† Ak).
The error covariance is similar to the least squares solution, and can be
directly calculated by Pk = (P??’1
o +Pk
j=1 GTj
R??’1Gj)??’1.
Because the field is linear in the parameter form, we don??™t have to use
the Taylor series expansion in the EKF, even in the presence of localization
uncertainty:
?µXk+1
Ak+1 ?¶ = ?µXk
Ak ?¶+ ?µI
0?¶Uk + ?µwk
0 ?¶ = ?µXk
Ak ?¶+ BUk + #k,
?µYk
Zk ?¶ = ?µ I 0
0 (1 g(XT
k ))?¶?µXk
Ak ?¶+ ??k,
where
E[#k#T
k ] = Q = ?µQ1 0
0 0 ?¶,E[??k??T
k ] = R = ?µR1 0
0 R2 ?¶
are the white noise covariances of the state and output. The nonlinear Kalman
filter equations become:
P??’
k+1 = Pk + Q, ?µX??’
k+1
A??’
k+1 ?¶ = ?µ ?†X
k
?† Ak ?¶+ BUk, Pk+1 =
1
1
P??’
k+1
+ HT
k+1
1
RHk+1
,
52
Chapter 2 Algorithms for Robotic Deployment of WSN
Hk = ?µ I 0
0 (1 g(XT
k ))?¶, ?µX??’
k+1
A??’
k+1 ?¶ = ?µXk
Ak ?¶+ BUk,
?µ ?†X
k+1
?† Ak+1 ?¶ = ?µX??’
k+1
A??’
k+1 ?¶+ Pk+1HT
k+1R??’1(?µYk+1
Z??’
k+1 ?¶??’ Hk+1 ?µX??’
k+1
A??’
k+1 ?¶).
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