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UAV Path Planning Using Evolutionary Algorithms 93
(b) Approaching the target from different directions. All angles between
successive approaching directions should be as equal as possible, in order
to assure an almost uniform distribution of UAVs around the target during
their approach, for maximizing the probability of mission accomplishment.
The single coordination constraint is defined as keeping a minimum safety
distance between UAVs, in order to ensure:
(a) collision avoidance between UAVs, and
(b) a spatial separation between the corresponding flight corridors, which, for
some missions, increases the probability of survival.

4.2 Path Modeling Using B-Spline Curves
In this work each path is constructed using a B-Spline curve. Although the
resulting curve in the physical space should be a 2-D one, 3-D B-Spline curves
are utilized for the construction of each path. The two dimensions are used
for the production of the x, y coordinates in the physical space of motion
(horizontal plane), while the 3 rd dimension corresponds to the velocity c along
the path. For this reason, each B-Spline control point is defined by 3 numbers,
corresponding to x k,j ,y k,j ,c k,j (k =0,...,n,j =1,...,N,N being the number
of UAVs, while n + 1 is the number of control points in each B-Spline curve,
the same for all curves). In this way a smooth variation of velocity c is defined
along the path. The first (k =0)and last(k = n) control points of the control
polygon are the initial and target points of the j th UAV, which are predefined
by the user. The corresponding velocities c 0,j ,c n,j (launch and approaching
velocities) are also predefined by the user.
The control polygon of each B-Spline curve is defined by successive straight
line segments. For each segment, its length seg length k,j , and its direction
seg angle k,j are used as design variables (k =1,...,n − 1,j =1,...,N).
Design variables seg angle k,j are defined as the difference between the direc-
tion (in deg.) of the current segment and the previous one. For the first
segment of each control polygon seg angle is measured from x-axis. Addi-
1,j
tionally, the UAVs’ velocities c k,j at each control point are used as design
variables, except for the starting and target points (where they are prede-
fined).
Using seg length and seg angle the coordinates of each B-Spline con-
k,j k,j
trol point x k,j and y k,j can be easily calculated. The use of seg length and
k,j
seg angle as design variables instead of x k,j and y k,j was adopted for two
k,j
reasons. The first reason is the fact that abrupt turns of each flight path can
be easily avoided by explicitly imposing short lower and upper bounds for the
seg angle design variables. The second reason is that by using the proposed
k,j
design variables a better convergence rate was achieved compared to the case
with the B-Spline control points’ coordinates as design variables. The latter
observation is a consequence of the shortening of the search space, using the