Page 104 - Innovations in Intelligent Machines

94 I.K. Nikolos et al.
proposed formulation. The lower and upper boundaries of each independent
design variable are predefined by the user. Velocity boundaries depend on the
flight envelope of each UAV. For the first segment of each control polygon
seg angle 1,j upper and lower boundaries can be selected such as to define an
initial flight direction. Additionally, by selecting lower and upper boundaries
◦
for the rest of seg angle k,j variables close to 0 degrees (for example −30 to
◦
30 ), abrupt turns may be avoided.
4.3 Objective Function Formulation

The optimum flight path calculation for each UAV is formulated as a mini-
mization problem. The objective (cost) function to be minimized is formulated
as the weighted sum of five different terms
5

f = w i f i , (10)
i=1
where w i are the weights and f i are the corresponding terms described below.
Term f 1 corresponds to the single route objective of short flight paths and
is defined as the sum of the non-dimensional lengths of all N flight paths
(B-Spline curves)
N

f 1 = l j , (11)
j=1
where l j is the non-dimensional length of the j th path, given as

L j
− 1. (12)
l j =
2 2
(x target − x 0,j ) +(y target − y 0,j )
In Eq. 12 L j is the length of the j th path, x target ,y target are the coordinates
of the target point and x 0,j , y 0,j are the coordinates of the j th starting point.
In Eq. 12, for the calculation of the non-dimensional length l j , the distance
between the starting and target points is subtracted, in order to obtain zero
f 1 value for straight line paths.
Term f 2 is a penalty term, designed in order to materialize the general
constraint of collision avoidance between UAVs and the ground. All N flight
paths are checked whether or not pass through each one of the M ground
obstacles. Discrete points are taken along each B-Spline path and they are
checked whether or not they lie inside an obstacle. If this is true for a discrete
point of the path line, a constant penalty is added to term f 2 . Consequently,
term f 2 is proportional to the number of discrete points that lie inside obsta-
cles. Additionally, for each path line, a high penalty is added in case that even
one discrete point of the corresponding path lies inside an obstacle.
Term f 3 was designed in order to take into account the second coordination
objective, i.e. the target approach from different directions. For each flight