Navigation Functionalities for an Autonomous UAV Helicopter

4.2. TRAJECTORY GENERATOR 39
radius (picture (a) of Fig. 4.4). The solution for this flight condition is given
by 4.13 where R b z = ∞, ˙u = 0 and R b y = constant. This condition is called
trimmed flight because the first derivative of the flight parameters are zero
( ˙ φ = ˙ θ = ˙r = ˙ T = ˙u = 0). For this flight condition it is straightforward to
calculate the maximum flight speed allowed. Since the maximum values of
r, φ, θ and T are limited for safety reasons, the maximum path velocity u
can be calculated from the system 4.13:
u1 = |R b yrmax|
u2 = | gθmax
|
Xu
�
u3 = |φmaxgRb y| (4.14)
u4 = (| − g(Tmax + g)(R b y) 2 |) 1
4
The minimum of these four velocities can be taken as the maximum
speed for the path:
umax = min(u1, u2, u3, u4) (4.15)
The path generated by the path planner is represented by a cubic polynomial.
The curvature radius in general is not constant for such a path,
which means that the helicopter never flies in trimmed conditions but it
flies instead in maneuvered flight conditions.
Let’s now examine a maneuver in the vertical plane (R b y = ∞, R b z =
constant). From the second equation in 4.12 and the second equation
in 4.13 we can observe that there is no constant velocity solution ( ˙u = 0)
which satisfies both. This means that when the helicopter climbs, it loses
velocity.
To make the PFCM more flexible in the sense of allowing vertical climbing
and descending at a constant speed, we have to remove the constraint
˙θ = u/R b z and w = ˙w = 0. In other words the fuselage will not be aligned
to the path during a maneuver in the vertical plane as it is shown in Fig. 4.4
(c). The helicopter instead will follow the path as it is shown in Fig. 4.4
(b).