Page 133 - Innovations in Intelligent Machines

124 A. Pongpunwattana and R. Rysdyk
site’s expected center location. In this method, the probability density field
at a distance r from the center location of the site is defined by a function
which provides the correct probability of intersection given a path of a single
revolution about the expected center location of the site at the distance r.
The two probability density fields are given by
O
O
p i (r + R ) − p i (r − R )
v
i
i
β (r)= (24)
i
2πr
and
V V
p i r + R v − p i r − R v
i
β (r)= (25)
v
2πr
where p i (r) is the probability that the center location of site i is within a
E
distance r of the expected location ˜ ,
z
i
r

x
p i (r)= 2πyρ (y)dy (26)
i
0
x
where ρ is the probability density function of the position of the site. For
i
r< 0, we define
p i (r)= −p i (|r|),r < 0 (27)
For a site with a probability density function,

x 2
E
z
x x 1/ π (σ ) , x − ˜ ≤ σ i x
i
i
ρ (x, σ )= (28)
i
i
E
0, x − ˜ >σ x
z
i i
the probability distribution function p(r) is given by

2 x 2 x
r / (σ ) ,r ≤ σ
p i (r)= i i x (29)
1, r > σ
i
For simple types of paths, it may be possible to determine a closed form solu-
tions to the Equation 22 and 23. For complex types of paths, the probability
of site i encountering vehicle v during time t q <t ≤ t q+1 can be approximated
v
as a summation of the probability density field β along the path which can
i
be written as
M−1
E
˜ v
˜ V
V
v
B (q +1) = β ( ˜ (T k ) − ˜ (T k ) ) ˙z (T k )∆T (30)
z
z
i i v i v
k=0
V
z
Here ˜ (T k ) is the expected position of the vehicle moving along on the path
v
E
z
= t q+1 · ˜ (T k ) is the expected center
i
Q v at time t T k with t T 0 = t q and t T M
˜ V
· ˙z (k) is the velocity of the vehicle at time
v
location of the site at time t T k
. ∆T is the time step of the numerical integration, and ∆T = T k+1 − T k .
t T k
M is the number of steps used to discretize the time period. The probability
of vehicle v hitting site i during time t q <t<t q+1 can be approximated by
M−1
i V E
˜ V
˜ i
z
B (q +1) = β ( ˜ (T k ) − ˜ (T k ) ) ˙z (T k )∆T (31)
z
v
v
v
i
v
k=0