Page 131 - Innovations in Intelligent Machines

122 A. Pongpunwattana and R. Rysdyk
˜
where C is the function used to compute the expected operation cost of the
<t ≤ t N .
vehicle traveling along its planned path Q(s p ) during time t s p
Substituting Equation 14 into Equation 15, the objective function becomes
N T N−1

˜
˜
˜ V
˜ i
F

F
J = α (q)˜x (q) B (q +1)ξ (q)η V − C(Q(s p )) (16)
i i υ
i=1 q=s p
˜
The cost function C is defined as

˜
Q
˜ V
˜ V
C(Q(s p )) = α V ξ (s p ) − ξ (N) + α (1 − F(N)) (17)
V
Q
where α is the vehicle cost weighting factor, and α is the path cost weight-
ing factor. F(N) is the ratio of the amount of fuel remaining in the vehicle’s
fuel tank at time t N to the full capacity of the tank. The weighting factors
F
V
α , α ,and α Q are parameters whose values are set based on the opera-
i
tor’s assessment of the task accomplishment, cost of vehicle loss, and fuel
consumption.
From the description of the stochastic world model provided above, we can
see that the states of the world, which include task states, vehicle states, and
states of the environment, are coupled. For example, the probability of success
in executing a task at a time during the mission depends on the probability
of survival of the vehicles at that time. This, in turn, depends on the states of
the obstacles intersecting with the vehicles along the path. Hence, to evaluate
˜
the utility function J given in Equation 16, it is necessary to run a simulation
to predict the expected values of the world states at each of the discretized
time steps.
3 Probability of Intersection
The position of each site i is assumed to be known with limited certainty.
It is described by the expected value ˜ E and a probability density func-
z
i
x
tion ρ . An intersection of a site and a vehicle is predicted probabilistically.
i
This section presents the approach proposed by Rathbun and Capozzi [20] to
approximate the probability of intersection of a vehicle and a site with limited
knowledge about the location of the site.
In general, a collision or intersection of two objects has an effect on both
objects. In our model, the effects of an intersection depend on the effective
ranges of the site and the vehicle. We define the effective range of a site as
the radius of the circular area where the site can affect the states of any vehicle
within that area. The effective range of a vehicle is defined as the radius of the
circular area where the vehicle can affect the states of any site being within
O
that area. For any pair of site i with an effective range R and a path Q υ of a
i
V
vehicle υ with an effective range R , we define the site-to-vehicle intersection
υ