Page 105 - Autonomous Mobile Robots

88 Autonomous Mobile Robots

The power reflected from the second feature, P REF2 is given by

2 ˆ ])
P t GA I ϒ R 2 (1 −[ ˆ ϒ R 1 + ˆ ϒ a 1
P REF2 = 2 (2.31)
ˆ
(4π) R (R 2 − R 1 )
2 ˆ 2 ˆ
1
The power then transmitted back to the first feature from the second feature is
given by

3 ˆ
P t GA I ϒ R 2 (1 −[ ˆ ϒ R 1 + ˆ ϒ a 1 ])
P INC21 = (2.32)
4
3 ˆ 2 ˆ
ˆ
(4π) R (R 2 − R 1 )
1
The power, P INC21 then passes through feature 1 and is given by
]) (2.33)
P TRAN21 = P INC21 (1 −[ ˆ ϒ R 1 + ˆ ϒ a 1

The power returned from the second feature is then P ˆ = P TRAN21 A e /
TRAN21
ˆ 2
(4πR )
1
3 ˆ 2
P t GA I A e ϒ R 2 (1 −[ ˆ ϒ R 1 + ˆ ϒ a 1 ])
P ˆ = (2.34)
TRAN21 4
ˆ
4 ˆ 4 ˆ
(4π) R (R 2 − R 1 )
1
In general, the predicted power from the ith feature can be written as
KA I (2i−1) ˆ ϒ R i (k + 1|k)
ˆ
P TRANi1 (k + 1|k) =
(4π) 2i
i−1 2
j=0 [1 −( ˆ ϒ R j (k+1|k)+ ˆ ϒ a j (k + 1|k))]
× (2.35)
i−1 ˆ ˆ 4
j=0 (R j+1 (k + 1|k) − R j (k + 1|k))
2 ˆ
where K = P t GA e , A e = Gλ /4π, ˆ ϒ R 0 = ˆ ϒ a 0 = R 0 = 0 and, for the ith
ˆ
feature, R i is related to the augmented state by Equation (2.25).
Equation (2.25), Equation (2.26), and Equation (2.35) between them com-
prise the observation. In order to generate realistic predictions of the range bins,
knowledgeofthepowerandrangenoisedistributionsisnecessary. Thishasbeen
studied extensively in previous work, and can be found in Reference 15.
The range and power noise are experimentally obtained [15]. The noise
in range is the phase noise, which is obtained by observing the range bins

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c002” — 2006/3/31 — 17:29 — page 88 — #48