The field measurements depend on the state, as well as a set of parameters
which are unknown:
zk = g(xk, ak) + &k.
If the set of unknown coe?±cients is constant or slow time-varying, we add the
vector a to the overall system state, with an evolution governed by ak+1 =
ak +vk. The noise covariances are given by E[?»k?»T
k ] = R1k
, E[&k&TK
] = R2k
, and
E[vkvTK
] = Q2
k. A simultaneous sampling and navigation estimation problem
is formulated using the extended Kalman Filter (EKF), using an overall state
vector Xk = (xk, ak), a measurement vector Zk = (yk, zk):
Xk+1 = F(Xk, uk) + pk, Zk = H(Xk) + nk, (7)
E[pkpT
k ] = ?µ T2Q1
k 0
0 Q2
k ?¶ = Qk, E[nknT
k ] = ?µR1
k 0
0 R2
k ?¶ = Rk. (8)
The Extended Kalman Filter Equations from nonlinear functions F and
H, could be written using the Jacobians FX(.) ?‰? aF
aX (.), HX(.) ?‰? aH
aX (.) :
Time Update:
?†X
??’k+1 = F( ?†X
k, uk), (9)
?† Z??’k+1 = H( ?†X
??’k+1), (10)
P??’k+1 = FX( ?†X
k, uk)PkFT
X( ?†X
k, uk) + Q. (11)
Measurement Update:
(Pk+1)??’1 = (P??’k+1)??’1 + HT
X(X??’k+1)R??’1HT
X(X??’k+1) (12)
or
50
matrix Q . Additional navigation measurements are then given by:
5 Adaptive Sampling using the Extended
usually nonlinear, and describes the state evolution as:
Kalman Filter
Chapter 2 Algorithms for Robotic Deployment of WSN
Pk+1 =
P??’k+1 ??’ P??’k+1HX(X??’k+1)HTX
(X??’k+1)P??’k+1
HTX
(X??’k+1)P??’k+1HX(X??’k+1) + R
, (13)
?†X
k+1 = ?†X
??’k+1 + Pk+1HT
X( ?†X
??’k+1)R??’1(Zk+1 ??’ ?† Z??’k+1).
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