IEEE - Aerospace and Electronic Systems - August 2022 - 25

Schwung and Lunze
TIME DELAY ESTIMATION
The delay estimator of the give-way object uses the hidden
Markov model (HMM) depicted in Figure 5 and the
Ricean fading model [12] to represent the statistics of
the physical wireless channel. Similarly, to the Gilbert-
Elliott model the background process of the HMM is
described by a two-state Markov model. Its states are
indicated as " good " and as " bad " and correspond to specific
channel conditions. The transition probabilities pgg,
pbb, pgb,and pbg are constant. The identification method
used for the parameters of the Markov model can be
found in [2]. The states have independent packet loss
probabilities that represent the channel quality and form
the sensor model of the HMM (lower part of Figure 5).
In the state " bad " the probability is considered to be
pb ¼ 1. In the state " good " the probability pgðdð~tiÞÞ
varies in dependence upon the distance dð~tiÞ between
the objects and is given by
pgðdð~tiÞÞ ¼
Z ffiffiffiffiffiffiffiffi
2Srs
p
1
fRðRsðdð~tiÞÞjPrðdð~tiÞÞ; KÞ dRsðdð~tiÞÞ:
(5)
Srs states the receiver sensitivity and Prðdð~tiÞÞ is the
received signal power. With the parameters
n2 ¼
K V
1 þK
and s2 ¼
V
2 ð1 þKÞ
where K is the Ricean factor, V ¼ EðR2
(6)
s Þ is the mean of
the total received signal power, and EðÞ is the expected
value, the function
fRðrjPrðdð~tiÞÞ; KÞ ¼
r
s2 exp
I0
r2 þ n2
2 s2
r n
s2
(7)
gives the probability density function of the Rice distribution
for the received signal amplitude. Here r ¼ Rsðdð~tiÞÞ
and V ¼ Prðdð~tiÞÞ applies and I0ðxÞ denotes the 0th order
modified Bessel function of the first kind.
An estimate of the transmission delay is derived by
tn;maxðdð~tiÞÞ ¼
M
Rmaxðdð~tiÞÞ
(8)
in which M denotes the number of bits that have to be
transmitted and Rmaxðdð~tiÞÞ states the maximum data rate
that is supported by the channel with probability 1
pe;max. As the data rate depends on the distance dð~tiÞ
between the objects, tn;maxðdð~tiÞÞ needs to be newly estimated
whenever jdð~tiÞ dðtÞj exceedsd. It must be noted
that tn;maxðdð~tiÞÞ is only the statistical mean of the time
delay. Hence, the current delay tk can be smaller or larger.
For an estimate of the overall time delay of the data
transmission, computation delays of the objects have to be
considered as well. An estimate tc;max of these delays is
derived by a worst-case execution time analysis of each
component of the objects.
The overall estimated time delay is given by
~tmaxðdð~tiÞÞ ¼ tc;max þ de tn;maxðdð~tiÞÞ
(9)
with some de > 1. The adjustment factor de ensures that
not the statistical mean value of the transmission delay
given by (8) is used, but a value greater than this mean
value. This adjustment reduces the probability of considering
a packet to be lost even though it is just received
after the time span ~tmaxðdð~tiÞÞ. The estimate (9) is used by
the event generator for triggering the events.
As both objects are able to fulfill the two functions
(stand-on and give-way), their control units are identical.
As the give-way object is responsible for the time delay
estimation, the delay estimator of the stand-on object is
not used.
PREDICTION UNIT
The prediction unit of the give-way object executes the
following two tasks.
It receives the data SSðtc;kÞ from the stand-on
object, which consists of its position pSðtc;kÞ, its
speed vSðtc;kÞ, and its future trajectory wSðtÞ.
It generates an ellipsoidal set PSðt; tr;k; tkÞ with
tr;k ¼ tc;k þ tk, which includes all possible future
positions of the stand-on object
PSðt; tr;k; tkÞ¼ pðtÞ2 R3 :
(
þ
Figure 5.
Two-state Markov model of the wireless channel.
AUGUST 2022
IEEE A&E SYSTEMS MAGAZINE
ðyðtÞySÞ2
r2
S;y
þ
S;z
ðxðtÞxSÞ2
r2
S;x
ðzðtÞzSÞ2
r2
1 0
)
(10)
where the time dependencies of the center and the
radii of the ellipsoid are omitted for simplicity.
The center of the ellipsoid given byxSðt; tr;k; tkÞ,
25
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