Navigation Functionalities for an Autonomous UAV Helicopter

4.2. TRAJECTORY GENERATOR 33
Fig. 4.3 depicts a path segment with the relative boundary conditions.
The segment is defined by the starting point coordinates P (0), the end
point coordinates P (1) and two vectors which represent the direction of
the segment tangent at the starting point P ′ (0) and at the end point P ′ (1)
so that the 3D path segment is defined by 12 parameters. The path planner
[18] calculates the 12 parameters ensuring the continuity of the path
and of the first order derivative at the segment joints.
Figure 4.3: Boundary conditions for a path segment.
Actually by imposing only these boundary conditions (continuity of the
path and continuity of the first order derivative at the segment joints) the
segment has two degrees of freedom undefined which are represented by
the magnitude of the tangent vectors P ′ (0) and P ′ (1).
The magnitude of the tangent vector affects the curvature of the path.
If the continuity of the second order derivative at the joints (e.g. the
curvature) is imposed then all the 12 parameters would be found and in
this case the path would be a cubic spline [21]. The two degrees of freedom
are chosen by the path planner in order to satisfy other conditions which
will not be mentioned here. For more details the reader is referred to [18].