Page 129 - Innovations in Intelligent Machines

120 A. Pongpunwattana and R. Rysdyk
The feedback information of the states x(k) known at time t k and the
V
¯ V
¯ V
commanded inputs u(q)=[¯ (q +1), ˙z (q +1), d (q + 1)] T for all
z
q ∈{k, k +1,...,N − 1} is given. We assume that the vehicle’s guidance
system can follow its commanded trajectory. Thus, the predicted position
and velocity of the vehicle is equivalent to the commanded inputs:
V
V
˜ z (q +1) ≡ ¯z (q + 1) (3)
˜ V ¯ V
˙ z (q +1) ≡ ˙z (q +1)
for all q ∈{k, k +1,...,N − 1}. Assuming the location of each obstacle is
independent of the location of all other obstacles, the dynamic propagation
of the expected health state ξ ˜ V of the vehicle is given by

N o

O
˜ O
˜ v
˜ V
˜ V
ξ (q +1) = ξ (q) 1 − B (q +1)ξ (q)η j (4)
j
j
j=1
˜ v
Here B (q +1) is the probability that the vehicle collides or intersects with an
j
˜ O
obstacle j during the time t q <t ≤ t q+1 . The variable ξ is the expected value
j
of the health state of obstacle j,and η O is the effectiveness of the obstacle j
j
in destroying a vehicle if they make contact. The value of η O is in the range
j
E
˜ v
V ˜ V
[0, 1]. B (q + 1) is a function of ˜ , ˙z , ˜ and the environment uncertainty
z
z
j
˜ v
x
parameter vector σ . The details of how to compute B are given in Section 3.
j
The position of site j ∈{1, 2,... ,N S } at time t k is a random variable
which can be written as
E
E
z (k)= ˜z (k)+ ε x j (5)
j
j
x
E
z
where ˜ is the expected value of the position. ε is assumed to be a random
j j
variable with zero mean with a probability density function

x 2 x
ρ x x, σ x = 1/(π σ j ), x ≤ σ j (6)
j j x
0, x >σ
j
x
Here σ is a given parameter specifying the uncertainty radius of site j.The
j
x
E
area within the circle with center location ˜ and radius σ contains all pos-
z
j j
sible locations of the site.
The velocity of site j ∈{1, 2,... ,N S } known at time t k is also a random
variable expressed by
E
˜ E
˙ z (k)= ˙z (k)+ ε v j (7)
j
j
v
where ˙z ˜ E is the expected value of the velocity and ε is assumed to be a
j
j
random variable with zero mean and a probability density function

v 2 ), v ≤ σ v
1/(π σ
ρ v j v, σ j v = j j v (8)
0, v >σ j