Page 132 - Innovations in Intelligent Machines

Evolution-based Dynamic Path Planning for Autonomous Vehicles 123
v
region as the region in space Z of all points within a distance R O from the
i
i
path Q υ ,
υ O
Z = {z : z − s <R , ∀s ∈ S} (18)
i
i
where S is the set of all points along the path Q v . If the center location of
E
the site, z , is within the site-to-vehicle intersection region, the states of the
i
vehicle can be changed by the site. The vehicle-to-site intersection region is
i
defined as the region in space Z of all points within a distance R V from the
v v
path Q v ,
i V
Z = {z : z − s <R , ∀s ∈ S} (19)
v v
If the center location of the site is within the vehicle-to-site intersection region,
the states of the site can be changed by the vehicle.
Let denote ρ i the probability density function of the position of site i in
2-dimension space. The probability that site i hits vehicle v is the integral of
the density function over the site-to-vehicle intersection region:

˜ v
B = ρ i (z)dxdy (20)
i
Z v
i
The probability that vehicle v hits site i is the integral of the density function
over the vehicle-to-site intersection region:

˜ i
B = ρ i (z)dxdy (21)
v
Z i v
We do not know the solutions to the Equation 20 and 21 for a piecewise set
of path segments generated by our path planner. Instead, a computationally
efficient technique is used to approximate the probability of intersection. We
call this method, Field Integral Approximation [20]. In this method, we define
i
two probability density fields β i v and β related to the probability density
v
v
function ρ i of the site. The probability density field β is defined such that
i
the probability that site i hits vehicle v can be approximated by integrating
the probability density field along the path Q v ,

v
˜ v
B = β ds (22)
i i
Q v
i
Likewise, the probability density field β is defined the same way. The prob-
v
ability that vehicle v hits site i can then be written in the form

i
˜ i
B = β ds (23)
v v
Q v
v
i
The remaining question is how to determine the fields β and β that
i v
provide a good approximation of the exact probabilities. Work by Rathbun
and Cappozi [20] suggests an approximation method for a site having con-
stant probability density at all points located at the same distance from the