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86 I.K. Nikolos et al.
maximum UAV velocity magnitudes during their flights, predefined safety
distance between UAVs, near simultaneous arrival to the target and target
approach from different directions.
This work is an extension of a previous one [35], which used Differential
Evolution (DE) in order to find optimal paths of coordinated UAVs, with
the paths being modeled with straight line segments. The main drawback of
that approach was the need of a large number of segments for complicated
paths, resulting in a large number of design variables and, consequently, gen-
erations to converge. In this work the Differential Evolution (DE) algorithm is
combined with a Radial Basis Functions Network (RBFN), which serves as a
surrogate approximation, in order to reduce the number of exact evaluations
of candidate solutions. The candidate paths are modeled in the physical space
and evaluated with respect to the physical (working) space. B-Spline curves
are used for path line modeling, and complicated paths can be produced with
a small number of control variables.
The rest of the chapter is organized as follows: section 2 contains
B-Spline and Evolutionary Algorithms fundamentals; the solid terrain formu-
lation, used for experimental simulations, is also presented. An off-line path
planner for a single UAV will be briefly discussed in section 3, in order to
introduce the concept of UAV path planning using Evolutionary Algorithms.
Section 4 deals with the concept of coordinated UAV path planning using
Evolutionary Algorithms. The problem formulation is described, including
assumptions, objectives, constraints, objective function definition and path
modeling. Section 5 presents the optimization procedure using a combination
of Differential Evolution and a Radial Basis Functions Artificial Neural Net-
work, which is used as a surrogate model in order to enhance the converge
rate of Differential Evolution algorithm. Simulations results are presented in
section 6, followed by discussion and conclusions in section 7.

2 B-Spline and Evolutionary Algorithms Fundamentals

2.1 B-Spline Curves

Straight-line segments cannot represent a flying objects path line, as it is
usually the case with mobile robots, sea and undersea vessels. B-Splines are
adopted to define the UAV desired path, providing at least first order deriva-
tive continuity. B-Spline curves are well fitted in the evolutionary procedure;
they need a few variables (the coordinates of their control points) in order to
define complicated curved paths. Each control point has a very local effect on
the curve’s shape and small perturbations in its position produce changes in
the curve only in the neighborhood of the repositioned control point.
B-Spline curves are parametric curves, with their construction based on
blending functions [36, 37]. Their parametric construction provides the ability