Aerospace and Electronic Systems Magazine September 2016 - 45

Fu, Zhang, and Yu
where mmax, Pmax, and Pi are the maximum mutation rate, the maximum probability, and the solution probability, respectively. Following the mutation process, both high and low HSI solutions are
likely to be modified, by which the diversity of the population can
be enhanced. The BBO algorithm for mutation is illustrated in Figure 17 of the Appendix.
Remark 2. After a finite number of emigrations and mutations in the population, the best solution with the minimum cost
is selected as the reference trajectory to evade multiple intruders.
Varying values of immigration rate l and emigration rate m can
improve the exploration ability in global optimization, reducing
the computation cost of avoiding multiple collisions. Since features are shared in the set of solutions, the optimal trajectory can be
guaranteed in case of trapping in local minima. Randomly selecting a mutation rate decreases the number of grid points searching
in the flight space, which is capable of solving multiple collisions
resolution in real time. The elitism process prevents elite solutions
from disappearing during immigration. As compared to other AH
algorithms in [20]-[23], BBO can provide superior performance,
especially in the case of multiple intruders.
Remark 3. As opposed to POMDP, BBO possesses superior
performance in convergence. By assigning different emigration
and immigration rate to each habitat, the information between habitats can be shared and the exploration is improved consequently.
In the process of generations, the low-HSI habitats accept the habitants living in high-HSI habitats, allowing the high convergence
rate of BBO.

where Ξ1, Ξ2, Ξ3 are three smooth mapping and F(i) is the ith derivative of F. The parameterization of the flight envelope (here the
pitch rate is considered) in function of the flat outputs F is an important unit in the path smoothing problem. The flight envelope
during a mission can be expressed by the desired trajectories. For
the UAV model given in (1) and (2), the system is flat with flat
outputs F1 = x, F2 = z, F3 = q. In this study, the reference trajectories
are designed as:
(23)
where wn is the natural frequency. Ri (i = 1, 2, 3) is the amplitude of
x, z and q, respectively.
denotes the initial position. In order to
generate the parameterization of the pitch rate in the flat output,
is derived according to the time derivative of (23):
(24)
Furthermore, the relationship between the natural frequency wn
[28] and the settling time of the reference trajectory ts can be approximated by:
(25)
Thus, (24) can be rewritten as:

(26)

THE INTEGRATION OF UAV DYNAMICS IN THE PLANNED PATH
An optimal trajectory planned by BBO may require aggressive maneuvers, such as sharp turning and suddenly climbing to a high altitude. However, each UAV has its own physical constraints, which
are indicated by the maximal Euler angular rates, turn radius, etc.
Focusing on the differential flatness algorithm, the profile of the
trajectory is tuned to ensure that the reference path from an initial position to a final position is followed by the UAV smoothly.
Therefore, constraints on flight envelope and control inputs need
to be explicitly considered in this study.
Supposing a UAV to be controlled for avoiding collisions in
a longitudinal plane, the limit of pitch rate among flight envelope
is taken into account firstly. A flatness-based nonlinear system is
defined as:
(21)

To determine the time when the maximal pitch rate is required,
it is essential to calculate the extrema of the pitch rate. First, the
time derivative of the pitch rate is derived as:
(27)
By setting t = ts/5.83, the extrema of the pitch rate are:
(28)
where the extrema denote the maximal or the minimal of the function. It is of importance to check the value of the pitch angle at the
beginning and at the end of the mission. After comparing these
three solutions, when t = ts/5.83, the corresponding solution is the
maximal one. In order to ensure that
has to satisfy:

where x Î Rn and u Î Rm represent system states and control inputs,
respectively. The system is flat only if flat outputs F Î Rm fulfill the
requirements such as:

(22)

SEPTEMBER 2016

(29)
As indicated in (21), the stability is influenced by the states
(flight envelope) and control inputs. Since the flight envelope is
limited by (29), it is necessary to impose the bounds on the maximal control inputs to avoid the UAV from performing an unrealistic maneuver.

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine September 2016