We will
not consider the case when three beacons are colinear, since the possibility is
quite low. Thus we have:
P(S is resolved) ?? P(S can be reached by at least three beacons)
=
Nq
Xd=3
P(S can be reached by d beacons)
=
Nq
Xd=3??Nq
d !pd(1 ??’ p)Nq??’d, where p =
??R2
L2
!
Nq
Xd=3
??d
d!
?· e??’??, as N ! 1, where ?? = Nqp, p =
??R2
L2
= 1 ??’ e??’??(1 + ?? + ??2/2) = f(??).
Since ?? = Npq >0, f???(??) = ??2e??’??/2 > 0, which indicates that f(??) will
increase as ?? increases. Therefore the larger number of beacons (Nq), the
higher probability that a sensor gets localized using our TPSS scheme.
Next we study the impact of the inaccuracy of TDoA measurements on the
localization errors. The first result is reported in Figure 7 for a network of 400
nodes with 20% of initial beacons. For each sensor that has been resolved, the
estimated location is linked with the corresponding real position. We observe
that as the epoch increases, the position error tends to increase. This trend can
also be observed in a further study given on the impact of di?®erent epochs and
measurement errors on position errors. The result is given in Figure 8, which
shows the computation errors (averaged over 100 tests) after one or three
epochs for di?®erent network density with the same initial beacon percentage
25%.
The increase of positioning errors along with epochs shows the e?®ect of
cumulative errors. Recall that once a sensor gets localized, it will use its
computed position to help other sensors on position estimation.
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